POLITECNICO DI TORINO III Facolt`a di Ingegneria Corso di Laurea in Ingegneria delle Telecomunicazioni

Tesi di Laurea

Segmentazione delle Displasie Corticali Focali su Immagini di Risonanza Magnetica Nucleare

Relatori: Prof. Gabriella Olmo Ing. Enrico Magli Candidato: Tommaso Mansi

Luglio 2005

McGill University Montreal Neurological Institute

Automatic Segmentation of Focal Cortical Dysplasia Lesions on MR Images using Geometric Deformable Models Tommaso Mansi

The present work was carried out under the supervision of Dr. Andrea Bernasconi, MD, and Dr. Olivier Colliot, PhD, within the Montreal Neurological Institute, McGill University.

Summary Focal cortical dysplasia (FCD), a malformation of cortical development, has been increasingly recognized as a major cause of drug-refractory epilepsy. Neurosurgery, consisting in the removal of the FCD lesion, is an effective treatment for these patients. However, the outcome is closely related to the resection of the whole lesion. Their precise delineation is thus crucial for surgical planning in epilepsy. FCD are characterized on T1 MRI by a cortical thickening, a blurring of the gray/white matter interface, and a hyper-intense signal within the dysplastic tissue. Nevertheless, they are difficult to distinguish from healthy cortex because of their subtleties, and the absence of evident boundaries makes their delineation challenging. This thesis presents a method for segmenting FCD lesions on T1-weighted MRI based on 3D deformable models implemented using the level set framework. Several questions are then considered. How to segment a structure which is ill-defined, heterogeneous and unshaped? How to estimate boundaries between dysplastic and healthy tissues when standard analyses are unable to define them objectively? Which MRI information can be used to successfully segment these lesions? Since gray levels are insufficient to distinguish FCD from non-lesional cortex, the first proposed method makes use of deformable models driven by the FCD features. Lesions and healthy tissues are modeled by probability maps estimated through a supervised learning over a set of subjects suffering from FCD. Then, the deformable model evolves so as to enclose the likely lesional voxels, while avoiding those lying within healthy tissues. We propose next to enhance this first model by incorporating a boundary-based knowledge. The ill-defined lesional interfaces are estimated and an additional force, based on gradient vector flow, is subsequently applied to pull the deformable model towards the FCD boundaries. The proposed methods were tested on 18 patients suffering from FCD and their performances were qualitatively and quantitatively evaluated by comparison with manual tracings painted by experts. The validation showed that the similarity between the automated segmentations and the manual labels is excellent, the lesion coverage very good, and the false positives satisfactory. This new approach may become a useful tool for the presurgical evaluation of patients with intractable epilepsy.

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Sommario Le displasie corticali focali (FCD) sono delle malformazioni dello sviluppo cerebrale sempre pi` u riconosciute come una causa maggiore di epilessia refrattaria ai farmaci. La chirurgia, che consiste nella resezione della lesione, `e allora l’unico trattamento disponibile per i pazienti che ne soffrono. L’esito dell’operazione `e tuttavia strettamente legato all’ablazione completa dei tessuti lesionati. La loro precisa delimitazione costituisce dunque una tappa cruciale del trattamento chirurgico. Le displasie corticali sono caratterizzate sulle immagini T1-pesate di risonanza magnetica nucleare (RMN) da un ispessimento della corteccia cerebrale, da una sfumatura della giunzione tra la sostanza grigia e la sostanza bianca, e da un’iperintensit`a del tessuto lesionato. Tuttavia, le FCD rimangono difficili da distinguere dalla corteccia sana a cause della loro sottigliezza, e l’assenza di contorni evidenti ne rende la delimitazione ardua. Questa tesi presenta un metodo per la segmentazione delle displasie corticali su immagini di RMN T1, basato su modelli deformabili 3D implementati mediante il metodo level set. Alcune problematiche sono allora considerate. Come delimitare una lesione che `e sfumata, eterogenea, e senza forma precisa? Come stimare i limiti tra la lesione ed i tessuti sani quando analisi visive tradizionali sono incapaci di definirli oggettivamente? Quali informazioni di RMN si possono usare per segmentare efficientemente le FCD? Poich´e i livelli di grigio sono insufficienti per distinguere le FCD dalla corteccia cerebrale sana, un primo metodo utilizza dei modelli deformabili guidati dalle caratteristiche RMN delle lesioni. Le FCD ed i tessuti sani sono modellati da funzioni probabilistiche stimate mediante un apprendimento supervisionato. Il modello deformabile evolve quindi in modo da segmentare i voxel probabilmente lesionali, evitando quelli giacendo nelle strutture sane. In seguito, proponiamo di perfezionare questo primo modello aggiungendo un’informazione di contorno. I limiti sfumati ed impliciti delle FCD sono stimati, ed una forza, basata sul gradient vector flow, `e inserita nel modello per attirarlo verso i contorni delle lesioni. I metodi sviluppati sono stati sperimentati sulle immagini di 18 pazienti che soffrono di FCD. I risultati sono stati valutati qualitativamente e quantitativamente, III

paragonandoli a delle etichette manuali costruite da esperti. La convalidazione dimostr`o che la similarit`a tra le segmentazione automatiche e le etichette manuali `e eccellente, la copertura delle lesioni molto buona, ed i falsi positivi soddisfacenti. Questo nuovo metodo potrebbe dunque costituire un aiuto molto utile alla valutazione pre-chirurgica dei pazienti che soffrono di epilessia farmacoresistente.

IV

Acknowledgments I would first, and foremost, like to thank Andrea Bernasconi and Olivier Colliot, my supervisors at the Montreal Neurological Institute. The ideas and suggestions they have come up with, and the discussions we have had, have greatly contributed towards the development of this thesis. Their comments on the work performed and on drafts of this manuscript have been invaluable. I have learned a lot from their great scientific knowledge and, above all, I have really appreciated their availability and their great personal qualities. I would like to thank all the members of the Neuroimaging of Epilepsy Laboratory, and especially Neda Bernasconi and Samson Antel, for the invaluable work performed together which has been crucial for the successful achievement of this thesis. I would also acknowledge gratefully my supervisors at the Politecnico di Torino, Prof. Gabriella Olmo and Ing. Enrico Magli, for their priceless comments on the present manuscript and in particular on the Italian sections. Moreover, I have appreciated the correspondences we have had during the development of the thesis. At last, I would like to thank the members of the jury for their participation in the presentation of the performed work. From a more personal point of view, I would like to thank, once again, Neda Bernasconi, Andrea Bernasconi and Olivier Colliot for their warm welcome. I have really appreciated their kindness and, above all, their friendship, which have made my stay at Montreal unforgettable. Finally, I would also like to thank all my family and my girlfriend, Marta. In spite of the great distance which has separated us, they provided me priceless support, encouragement and diversion from the stresses evolved.

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Table of contents Introduction

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Segmentazione delle displasie corticali focali mediante modelli deformabili 1 Epilessia e displasie corticali focali . . . . . . . . . . . . . . . . . . . . 1.1 Breve introduzione all’epilessia . . . . . . . . . . . . . . . . . . 1.2 Le displasie corticali focali . . . . . . . . . . . . . . . . . . . . 1.3 Alcuni metodi esistenti per migliorare la diagnosi delle FCD . 2 Breve introduzione alle superfici attive ed al metodo level-set . . . . . 2.1 I modelli deformabili a variabili geometriche . . . . . . . . . . 2.2 Il metodo level-set . . . . . . . . . . . . . . . . . . . . . . . . 3 Segmentazione delle FCD mediante modelli deformabili guidati dalle caratteristiche di RMN . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Modellazione probabilistica delle caratteristiche delle FCD . . 3.2 Definizione del modello deformabile . . . . . . . . . . . . . . . 3.3 Evoluzione level-set . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Brevi indicazioni d’implementazione . . . . . . . . . . . . . . . 3.5 Sperimentazione . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Discussione dei risultati . . . . . . . . . . . . . . . . . . . . . 4 Modelli deformabili perfezionati per la segmentazione delle FCD . . . 4.1 Una descrizione probabilistica pi` u completa delle caratteristiche RMN delle FCD . . . . . . . . . . . . . . . . . . . . . . 4.2 Modellazione dei contorni delle FCD . . . . . . . . . . . . . . 4.3 Due modelli basati sulle caratteristiche RMN delle FCD e la modellazione dei loro contorni . . . . . . . . . . . . . . . . . . 4.4 Sperimentazione e risultati . . . . . . . . . . . . . . . . . . . . 4.5 Discussione dei risultati . . . . . . . . . . . . . . . . . . . . . 1 Epilepsy, focal cortical dysplasia and magnetic resonance imaging 1.1 Overview of epilepsy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Historical background . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The basics of epilepsy . . . . . . . . . . . . . . . . . . . . . . VII

5 5 5 6 7 8 8 9 11 11 12 13 13 14 16 17 17 18 19 21 21 25 25 25 28

1.2 Focal cortical dysplasia in focal epilepsy . . . 1.2.1 Focal epilepsy . . . . . . . . . . . . . . 1.2.2 Malformations of cortical development 1.2.3 Focal cortical dysplasia . . . . . . . . . 1.3 Diagnosis aid systems for FCD detection . . . 1.3.1 Enhancement of the visual detection . 1.3.2 Automated detection . . . . . . . . . . 1.4 Open issues . . . . . . . . . . . . . . . . . . .

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2 Image segmentation using geometric deformable models 2.1 Brief overview of parametric deformable models . . . . . . 2.1.1 Mathematical formulation of the curve evolution . . 2.1.2 External forces . . . . . . . . . . . . . . . . . . . . 2.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . 2.2 Theorical background of geometric deformable models . . . 2.2.1 Theory of curve evolution . . . . . . . . . . . . . . 2.2.2 The level set method . . . . . . . . . . . . . . . . . 2.2.3 From parametric to geometric deformable models . 2.3 Image segmentation using geometric active contours . . . . 2.3.1 Shape modeling with geometric deformable models 2.3.2 Region-based deformable models . . . . . . . . . . 2.3.3 Shape, geometrical and topological constraints . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Segmentation of FCD using a feature-based deformable model 3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Overall picture of the approach . . . . . . . . . . . . . . . 3.1.2 Probabilistic modeling of the MRI characteristics of FCD . 3.1.3 Geometric deformable model using MRI-feature knowledge 3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 General considerations . . . . . . . . . . . . . . . . . . . . 3.2.2 Numerical schemes . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Initialization: computation of the signed distance map . . 3.2.4 The narrow-band method . . . . . . . . . . . . . . . . . . 3.2.5 The speed function into consideration . . . . . . . . . . . . 3.2.6 Automatic stopping criterion for the level set evolution . . 3.3 Experiments and results . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Subjects and Image preparation . . . . . . . . . . . . . . . 3.3.2 Manual segmentation . . . . . . . . . . . . . . . . . . . . . 3.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII

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4 Enhancing the feature-based deformable model 4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Biological considerations . . . . . . . . . . . . . . . . . . . . 4.1.2 Design of the boundary information . . . . . . . . . . . . . . 4.1.3 Feature-based deformable models with Gradient Vector Flow 4.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Computation of the Gradient Vector Flow . . . . . . . . . . 4.2.2 Numerical schemes . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Considerations on the automatic stopping criterion . . . . . 4.3 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Subjects, image preparation and protocol . . . . . . . . . . . 4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Solving the false-positive issue . . . . . . . . . . . . . . . . . . . . . 4.4.1 Biological considerations . . . . . . . . . . . . . . . . . . . . 4.4.2 Definition of new probabilistic models of FCD features . . . 4.4.3 Feature-based deformable model with Gradient Vector Flow 4.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Conclusions A Introduction to brain anatomy 1.1 Brain organization . . . . . . 1.1.1 The cerebrum . . . . . 1.1.2 The cerebellum . . . . 1.1.3 The brain stem . . . . 1.2 Cellular description . . . . . . 1.2.1 The neurons . . . . . . 1.2.2 The glial cells . . . . .

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Introduction Nowadays epilepsy is considered as the most prevalent neurological ailment and affects about 1% of the population. Several drugs have been developed to control the epileptic seizures, but 30% of the patients still suffers from medically intractable epilepsy. Malformations of cortical development, and in particular focal cortical dysplasia (FCD), have been increasingly recognized as a major cause of drug-refractory seizures. Neurosurgery, consisting in the removal of the lesional tissue, is then the only available treatment for those patients. Their living condition can be improved and patients may become free from seizures. However, surgerical outcome is closely related to the resection of the whole lesion. The precise estimation of the FCD extent is thus crucial for the surgical treatment. Magnetic resonance imaging (MRI) has greatly contributed towards the improvement of FCD detection. In vivo analysis has allowed the identification of FCD in an increasing number of patients, sparing them invasive and time-consuming procedures. However, standard MRI evaluation still fails to identify the lesions in a large number of cases because of their subtlety and the complexity of the cortex convolution. Furthermore, the delineation of the lesional tissue is made difficult by the ill-defined boundaries characterizing it. Segmenting these lesions constitutes thus a challenging image analysis application. Recently, advanced image processing techniques have been developed to recognize FCD on T1-weighted MRI. These methods rely on various types of voxel-wise analyses and aim at emphasizing the lesions. [Bernasconi et al., 2001; Antel et al., 2002] developed computational models of the FCD characteristics to improve the visual identification of the lesions. Afterwards, these models were integrated in an automated detection procedure relying on a Bayesian classifier [Antel et al., 2003]. Another approach proposed by [Wilke et al., 2003] consists in applying voxel-based morphometry to unveil differences in cortical thickness between a given subject and a group of controls, differences that may betray the presence of an FCD lesion. However, although these techniques successfully identify the dysplastic tissues in a majority of patients, they provide very limited lesion coverage and are thus inappropriate for FCD segmentation. Therefore, to our knowledge, the question of the FCD segmentation has never been addressed. 1

∗ ∗ ∗ Our purpose was to design an automated method for segmenting FCD lesions on T1-weighted MRI. However, several challenging questions have to be considered. How to segment a structure which is by definition ill-defined, heterogeneous, with variable size and shape, and that can be located almost everywhere throughout the cortex? How to estimate boundaries between dysplastic and healthy tissues when standard analyses are unable to define them objectively? Which MRI information can be used to successfully segment these lesions? We propose in this thesis a method relying on 3D geometric deformable models implemented using the level set method. Such a choice was motivated by the useful properties of these techniques, particularly appropriate for our purpose. Contrary to the parametric active surfaces, geometric deformable models are defined using geometric variables only, such as curvature, and do not require any parameterization. The level set method can thus be applied to compute the surface evolution. As a result, geometric deformable models can handle topology changes naturally and are robust with respect to noise and initialization. Moreover, they can make use of boundary information and other advanced knowledge about the objects to segment (like region or shape knowledge) to effectively detect ill-defined or noisy edges. Since gray levels are insufficient to distinguish FCD from healthy cortex, we propose here to drive the deformable model using a probabilistic modeling derived from the FCD features. An additional force, describing the lesional boundaries, will be subsequently applied to pull the active surface towards the interfaces between dysplastic tissues and healthy structures. The method is evaluated on eighteen patients suffering from FCD. The resulting automated segmentations are then compared qualitatively and quantitatively with manual labels previously painted by experts. Furthermore, since the question of FCD segmentation has never been addressed, we will compare our solution with the FCD classifier developed by [Antel et al., 2003] though it was not designed for segmentation purposes but for FCD detection. ∗ ∗ ∗ The thesis starts with a detailed Italian summary of the performed work, followed next by the complete description of the approach, organized as follows. Chapter 1 is mainly devoted to the biological considerations. The basics of epilepsy are first presented. Then, malformations of cortical development and focal cortical dysplasia are described in details. Finally, a short outline of the existing imaging analysis techniques enhancing the FCD detection is provided. 2

The underlying principles of geometric deformable models and level set framework are explained in chapter 2. A short overview of parametric active surfaces introduces the philosophy of the method. Then, a detailed description of the geometric deformable models and curve evolution theory is given, followed by the description of the level set method. At last, we close with a non-exhaustive presentation of existing deformable models. Their advantages and drawbacks are discussed within the context of FCD segmentation. Chapter 3 provides a first model for segmenting FCD. It relies on relevant region information derived from the MRI features of the FCD lesions. The underlying principles are first explained and the main implementation points outlined. Then, the validation protocol is described and the results are discussed. Chapter 4 presents two more advanced deformable models using boundarybased attraction in addition to the region force. The final results are given and discussed. These models constitute the outcome of our work, validating the proposed approach and opening new fascinating prospects. The conclusions summarize the results and present possible directions for future work. Finally, Appendix A outlines briefly some basics of brain anatomy and explains with more details the vocabulary used all along this manuscript. ∗ ∗ ∗ The contributions of this thesis have been presented in the following papers and abstracts: • A level set driven by MR features of focal cortical dysplasia for lesion segmentation, in Proceedings of Medical Image Understanding and Analysis MIUA 2005, Bristol, UK, July 2005 (In Press). • Segmentation of focal cortical dysplasia lesions using a feature-based level set, in Proceedings of Medical Image Computing and Computer-Assisted Intervention, MICCAI 2005, Palm Springs, California, USA, October 2005 (In Press). • Automatic delineation of focal cortical dysplasia lesion on MRI, Submitted to Annual Meeting of the American Epilepsy Society AES 2005 (abstract). A journal paper is currently under preparation.

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Segmentazione delle displasie corticali focali mediante modelli deformabili Introduzione Questo capitolo riassume brevemente le diverse parti della presente tesi. La prima sezione `e dedicata alla presentazione dell’epilessia e delle displasie corticali focali. Le problematiche biologiche legate a queste lesioni saranno presentate, seguite dalla descrizione delle tecniche esistenti facilitando la loro diagnostica. Il secondo paragrafo introduce i modelli deformabili ed il metodo level-set. La terza sezione propone un primo modello per segmentare automaticamente le displasie su immagini di risonanza magnetica nucleare (RMN), basato su informazioni probabilistiche calcolate dalle caratteristiche RMN delle FCD. Infine, l’ultima parte descrive un metodo pi` u elaborato che usa, oltre all’informazione probabilistica, una modellizzazione implicita dei contorni delle lesioni. Per lo sviluppo dettagliato di questi argomenti, il lettore si riferisca ai capitoli successivi.

1 1.1

Epilessia e displasie corticali focali Breve introduzione all’epilessia

Oggigiorno, l’epilessia `e la malattia neurologica la pi` u diffusa nel mondo. In Italia, circa l’1% della popolazione soffre regolarmente di crisi epilettiche, crisi che possono nuocere gravemente alla vita delle persone malate, fisicamente ma anche socialmente. Trattamenti medici sono stati sviluppati, ma il 30% dei pazienti ancora soffre di crisi incurabili. Un attacco epilettico `e un fenomeno temporale ed improvviso dovuto ad un’attivit`a elettrica anormale di una regione pi` u o meno estesa del cervello. Quest’attivit`a pu`o essere generale, in tal caso tutto il cervello `e colpito, o parziale (detta anche focale). In questo secondo caso, la crisi nasce in una zona ben precisa della corteccia 5

cerebrale e pu`o poi propagarsi verso altre zone o, nel peggiore dei casi, colpire tutto il cervello. Lo sviluppo di tecniche avanzate, come l’elettroencefalografia (EEG) negli anni ‘30 e la risonanza magnetica nucleare (RMN) negli anni ‘80, ha considerevolmente migliorato la comprensione della malattia, favorito lo sviluppo di nuovi trattamenti, e permesso la diagnosi delle regioni cerebrali coinvolte in un numero di casi sempre maggiore. La RMN permette, in effetti, la visualizzazione in vivo del cervello, mentre l’EEG rivela le zone attivate durante gli attacchi epilettici. La neurochirurgia, che consiste nella rimozione dei tessuti colpiti, diventa cos`ı sempre pi` u comune e fattibile, premettendo di curare pazienti i cui attacchi sono incontrollabili mediante farmaci.

1.2

Le displasie corticali focali

Le displasie corticali focali (FCD dall’inglese focal cortical dysplasia) sono diagnosticate in un crescente numero di pazienti che soffrono di epilessia focale farmacoresistente. Descritte per la prima volta in [Taylor et al., 1971], le FCD sono malformazioni locali della corteccia cerebrale causate da un’anormale proliferazione di cellule nervose durante lo sviluppo del cervello. Sono caratterizzate essenzialmente da una disorganizzazione totale della struttura a strati della corteccia (figure 1.5, p37 e 1.7, p39). Inoltre, la regione colpita possiede cellule non differenziate, con forme strane, orientate casualmente e di dimensione anormalmente grande. Infine, si possono anche trovare in alcuni casi delle cellule giganti, chiamate balloon cells, con una fine membrana ed un nucleo eccentrico [Tassi et al., 2002; Palmini et al., 2004] (figura 1.6, p38). Grazie alla RMN, le FCD possono essere identificate pi` u rapidamente, e senza l’uso di metodi invasivi ad alto rischio per il paziente. In effetti, esse possiedono caratteristiche ben specifiche sulle immagini RMN T1 (figura 1.8, p40), tra cui: • un ispessimento focale dell’area corticale colpita, • una sfumatura della giunzione tra la sostanza grigia e bianca, • un’iper-intensit`a della sostanza grigia lesionale. Per`o, anche se queste caratteristiche sono spesso trovate nelle lesioni visibili, la semplice analisi visiva delle immagini RMN non permette l’identificazione delle FCD in un numero ancora troppo importante di casi, per colpa della loro sottigliezza e dell’alta complessit`a del cervello. Inoltre, quando le lesioni sono visibili, la valutazione oggettiva ed affidabile delle loro dimensioni `e resa difficile dalla sfumatura dei contorni. Al giorno d’oggi, la chirurgia `e l’unico trattamento disponibile per le persone che soffrono di epilessia focale dovuta alle FCD. L’operazione consiste nel rimuovere la lesione senza danneggiare i tessuti circondanti. La riuscita dell’intervento chirurgico 6

1 – Epilessia e displasie corticali focali

`e quindi strettamente legata alla precisione della valutazione pre-operatoria delle dimensioni della FCD. Tuttavia, quest’ultima tappa `e molto complessa per colpa della sfumatura dei loro contorni sulle immagini RMN, ed `e dunque fatta manualmente.

1.3

Alcuni metodi esistenti per migliorare la diagnosi delle FCD

Negli ultimi anni, delle tecniche di elaborazione d’immagini mediche hanno migliorato significativamente l’identificazione delle FCD. Queste tecniche, fondate su analisi voxel per voxel, permettono di dare risalto ad anomalie ben precise che potrebbero tradire la presenza di una FCD. Un primo metodo, proposto da [Bernasconi et al., 2001; Antel et al., 2002], consiste nell’elaborazione di modelli quantitativi delle tre caratteristiche principali delle FCD sulle immagini RMN. Un’immagine di thickness traduce la misura dello spessore della corteccia cerebrale, stimato mediante un metodo differenziale (figura 1.11D, p44). Valori elevati simbolizzano le zone ad alto spessore e vice-versa. Un’altra immagine, chiamata relative intensity, permette di indicare le regioni della corteccia ad alta intensit`a (figura 1.11E, p44). Infine, un’immagine che modella l’ampiezza del gradiente evidenzia la sfumatura della giunzione tra la sostanza grigia e la sostanza bianca (figura 1.11C, p44). Queste tre immagini, chiamate feature maps, hanno migliorato sensibilmente la diagnostica delle FCD, permettendone l’identificazione in una gran maggioranza di soggetti malati. Successivamente, [Antel et al., 2003] svilupparono un metodo automatico per localizzare le lesioni. Questo metodo utilizza un classificatore bayesiano basato sulle tre sopraccennate feature maps (figura 1.13, p48). Il tasso di rilevamento ottenuto era molto convincente, ma i contorni trovati dall’algoritmo non corrispondevano ai limiti evidenziati dagli esperti (circa l’80% delle lesioni non era classificato correttamente). Questo metodo `e dunque ben adattato al rilevamento delle FCD ma non alla loro segmentazione. Un altro algoritmo fu proposto da [Wilke et al., 2003]. L’idea `e di paragonare voxel per voxel l’immagine RMN di un paziente con un’immagine media d’individui sani, per identificare le zone della corteccia il cui spessore `e anormalmente massiccio. L’ispessimento focale della corteccia `e allora evidenziato e la FCD localizzata (figura 1.12, p46). Questa procedura offre anch’essa un buon tasso di rilevamento ma non delimitava con esattezza le lesioni. Al meglio della nostra conoscenza, non esiste nessuna tecnica automatica che segmenta con efficienza ed affidabilit`a le lesioni FCD. Il lavoro svolto in questa tesi sembra dunque essere una prima proposta d’algoritmo evoluto dedicato interamente alla segmentazione di queste malformazioni, e non unicamente al loro rilevamento. Lo scopo quindi `e di trovare i limiti del tessuto lesionato, di maniera automatica ed affidabile. 7

2

Breve introduzione alle superfici attive ed al metodo level-set

La segmentazione delle immagini mediche gioca un ruolo sempre pi` u importante nei vari campi di ricerca medica, e soprattutto in neurologia. In effetti, la delimitazione automatica di strutture cerebrali permette, da un lato ai ricercatori di realizzare studi quantitativi su una gran quantit`a di soggetti, e dall’altro ai medici di identificare delle lesioni fino ad oggi invisibili. Tuttavia, la segmentazione delle immagini di risonanza magnetica rimane sempre un problema difficile a causa del rumore spesso forte e della presenza di difetti di acquisizione (come per esempio gli effetti dei movimenti dei pazienti). Anche se molti algoritmi sono disponibili, pochi danno risultati soddisfacenti. Tecniche basate su istogrammi per esempio sono sufficienti per segmentare i tessuti principali (fluido cerebrospinale, sostanza grigia e sostanza bianca), ma non sono del tutto adatti alla delimitazione automatica delle FCD.

2.1

I modelli deformabili a variabili geometriche

I modelli deformabili, o contorni attivi, fanno parte degli algoritmi avanzati pi` u utilizzati nella segmentazione delle immagini mediche. Introdotti per la prima volta da [Kass et al., 1987], sono curve (2D) o superfici (3D) che si deformano, sotto l’effetto di varie forze, verso i contorni dell’oggetto da identificare. Diverse informazioni possono essere utilizzate, di tipo locale (come l’intensit`a di un voxel, il gradiente, ecc.) o globale (come una caratteristica regionale dell’oggetto, una conoscenza a priori di forma o di struttura, ecc.). Il metodo `e inoltre robusto al rumore presente nelle immagini, ed i risultati sono automaticamente regolarizzati e lisci. Introdotti da [Caselles et al., 1993] e [Malladi et al., 1995], i modelli deformabili geometrici sono superfici attive definite unicamente tramite variabili geometriche come la curvatura. Cos`ı, contrariamente ai contorni attivi parametrici, non richiedono nessuna parametrizzazione esplicita, e possono usufruire del metodo level-set per il calcolo dell’evoluzione (prossima sezione). Sia S la superficie che evolve nel tempo t con una parametrizzazione X(s,t) qualunque, N(s,t) il vettore normale e unitario interno, e κ(s,t) la curvatura (figura 2.1, p58). L’evoluzione del modello deformabile geometrico S `e modellato tramite la seguente equazione di evoluzione: ∂X = F (κ)N. ∂t

(1)

Il termine F (κ) `e chiamato funzione velocit`a (e anche forza o deformazione) e contiene le informazioni necessarie per attirare la superficie verso i contorni dell’oggetto desiderato. Esistono due tipi principali di funzione velocit`a: 8

2 – Breve introduzione alle superfici attive ed al metodo level-set

• La forza constante, F (κ) = F0 , guida il modello deformabile in una direzione fissa. Se la costante F0 `e positiva, allora il modello si restringe, mentre se F0 `e negativa, allora S si dilata. • La seconda forza, chiamata forza di curvatura F (κ) = ακ, modula la deformazione della superficie in funzione della sua curvatura nel punto X(s,t) considerato. Questa forza `e fondamentale poich´e permette il conseguimento automatico di un risultato regolarizzato. Un esempio fondamentale di funzione velocit`a costruita per la segmentazione delle immagini `e quella proposta da [Caselles et al., 1993; Malladi et al., 1995], F (κ) = g · (c + κ),

(2)

dove c `e una costante negativa, κ la curvatura, e g una funzione, il cui scopo `e di fermare il modello deformabile sui punti ad alto gradiente, definito da g(x,y) =

1 , 1 + |∇(Gσ ⊗ I(x,y))|p

(3)

con p ∈ {1,2}, e Gσ ⊗ I `e la convoluzione dell’immagine I con una gaussiana di deviazione standard σ. In zone omogenee, il gradiente `e debole e quindi g vale circa 1. II modello deformabile evolve dilatandosi grazie alla costante c. Quando il modello arriva in zone ad alto gradiente, vale a dire sui margini dell’oggetto da segmentare, allora g tende a zero, e la deformazione si ferma. La modellazione di questa funzione `e dunque cruciale per la segmentazione delle immagini, poich´e contiene tutte le informazioni necessarie per trovare i limiti dell’oggetto desiderato.

2.2

Il metodo level-set

Sviluppato da [Osher and Sethian, 1988] e utilizzato per la prima volta in elaborazione delle immagini da [Caselles et al., 1993; Malladi et al., 1995], il metodo level-set permette di risolvere in maniera efficiente l’equazione di evoluzione di qualsiasi modello deformabile a variabili geometriche. L’idea principale `e di immergere la superficie attiva S in una funzione 3D, chiamata funzione implicita e denotata come φ. S `e allora definita come l’insieme dei punti della funzione φ che verificano φ = 0. L’algoritmo fa evolvere nel tempo questa funzione, deformando di maniera implicita la superficie S. Il risultato `e allora ottenuto estraendo S alla fine dell’evoluzione selezionando i punti con valori uguali a zero. 9

Di maniera pi` u formale, la funzione implicita `e definita da φ[X(s,t)] = 0,

(4)

negativa dentro la superficie e positiva fuori (figura 3.6, p84). Di solito, si sceglie φ uguale alla funzione distanza calcolata dal modello deformabile iniziale St=0 , il valore di ogni voxel di φ essendo allora uguale alla distanza che li separa da St=0 . Il passaggio dall’equazione di evoluzione alla formulazione level-set si fa tramite l’equivalenza seguente:

⇐⇒

∂X = F (κ)N ∂t ∂φ = F (κ)|∇φ|. ∂t

(5)

La funzione implicita φ evolver`a quindi secondo quest’ultima equazione. La risoluzione dell’equazione level-set (5) si fa mediante un algoritmo iterativo. Delle approssimazioni numeriche ben specifiche sono utilizzate per garantirne la stabilit`a. Il lettore si riferisca al capitolo 2 ed al libro [Sethian, 1999a] per maggiori informazioni sul tema.

In conclusione, i modelli deformabili geometrici presentano molteplici vantaggi particolarmente utili per il nostro problema. Possono usufruire di conoscenze di basso ed alto livello per evidenziare con precisione i contorni delle lesioni FCD. Inoltre, i risultati ottenuti sono sempre regolari e chiusi, rendendo il metodo robusto al rumore. Il metodo level-set utilizzato per risolvere l’equazione di evoluzione d`a altri importanti vantaggi. La modellazione implicita permette i cambi di topologia. In effetti, la superficie attiva pu`o dividersi o unirsi senza alterare la funzione implicita che rimane valida durante tutta la procedura. Inoltre, le variabili geometriche possono essere calcolate direttamente a partire della funzione implicita, accelerando la procedura. Nondimeno, quest’algoritmo ha una complessit`a pi` u elevata rispetto ai contorni attivi a parametrizzazione esplicita, per colpa dell’addizione di una dimensione al problema. Un’implementazione ingenua `e di conseguenza non applicabile per le problematiche 3D. Fortunatamente, esistono algoritmi avanzati che risolvono questo problema, come il metodo a banda stretta o narrow-band [Adalsteinsson and Sethian, 1995]. Utilizzeremo quindi questo metodo per accelerare significativamente il programma (vedere prossima sezione). 10

3 – Segmentazione delle FCD mediante modelli deformabili guidati dalle caratteristiche di RMN

3

Segmentazione delle FCD mediante modelli deformabili guidati dalle caratteristiche di RMN

La segmentazione delle FCD `e un problema difficile. I contorni sfumati della lesione impediscono la stima precisa ed oggettiva delle loro dimensioni. Inoltre, l’intensit`a dei voxel lesionali non `e omogenea e pu`o cambiare di maniera drastica da una regione della lesione all’altra. Infine, non esistono frontiere esplicite tra la FCD e la corteccia cerebrale sana. Di conseguenza, le informazioni locali, come il gradiente o l’intensit`a dei livelli di grigio, sono insufficienti per delimitare efficientemente le lesioni. Il metodo presentato in questa sezione si basa su una descrizione probabilistica delle FCD per guidare il modello deformabile verso i contorni delle lesioni. Rimandiamo il lettore al capitolo 3 del presente documento per i dettagli completi del algoritmo, e per le figure che mostrano alcuni risultati.

3.1

Modellazione probabilistica delle caratteristiche delle FCD

L’idea sottesa dal nostro algoritmo consiste nella guida del modello deformabile mediante descrizioni probabilistiche relativi ai diversi tessuti cerebrali, descrizioni calcolate dalle feature map sviluppate da [Antel et al., 2002]. Come detto nella sezione precedente, le FCD sono evidenziate su immagini T1pesate di RMN da tre caratteristiche: un ispessimento focale della corteccia cerebrale, una sfumatura della giunzione tra la sostanza grigia e bianca, e una iper-intensit`a della sostanza grigia lesionale. Queste tre caratteristiche sono comuni alla maggioranza delle lesioni visibile e rappresentano dunque una buona descrizione delle zone lesionali. Vediamo ora come stimare i modelli probabilistici. Sia Ω il paziente la cui lesione sar`a delimitata sull’immagine di RMN. Le sue feature map sono calcolate per formare un vettore di caratteristiche f (M) = (T h(M),RI(M),Gr(M)), con M un punto nell’immagine e T h, RI e Gr la thickness map, la relative intensity map ed il gradient map rispettivamente. Consideriamo poi un insieme di soggetti che soffrono di FCD, detto insieme d’apprendimento, e calcoliamone le feature map. Le descrizioni probabilistiche relative ai diversi tessuti cerebrali del paziente Ω sono allora stimate usando un apprendimento supervisionato a partire dall’insieme di apprendimento, e tenendo conto del vettore f (M) associato a Ω. Quattro classi c di strutture cerebrali sono definite: il fluido cerebrospinale (CSF), la sostanza grigia (GM), la sostanza bianca (WM) e la FCD (L). Quattro modelli probabilistici Pc (M) sono quindi calcolati. L’insieme di apprendimento `e prima segmentato in quattro classi distinte corrispondenti alle classi L,GM,WM e CSF. La sostanza grigia, bianca ed il fluido 11

cerebrospinale sono estratti mediante un metodo ad istogramma con soglia automatica, mentre la classe L `e identificata manualmente da un esperto (figura 3.3, p80). Le probabilit`a condizionali P (f (M)|c) sono allora modellate mediante distribuzioni normali a tre parametri. Ne stimiamo i parametri tramite il metodo di massima probabilit`a (maximum likelihood ), dalle feature map, e dalle immagini a quattro classi dell’insieme di apprendimento. Infine, grazie alla regola di Bayes, otteniamo le descrizioni probabilistiche desiderate Pc (M) = P (c|f (M). La figura 3.4, p80 mostra le descrizioni probabilistiche ottenute per un paziente con FCD.

3.2

Definizione del modello deformabile

Una volta stimate le descrizioni probabilistiche di ogni tessuto cerebrale, possiamo utilizzarle per guidare la superficie attiva verso i contorni della lesione. L’idea `e di applicare un procedimento simile al metodo di massimo a posteriori. Un voxel dell’immagine `e detto appartenere alla classe c se la probabilit`a legata a questa classe `e la pi` u elevata. Partendo da questo principio, il modello deformabile evolver`a per includere i voxel appartenenti alla lesione, mentre i voxel appartenenti agli altri tessuti saranno esclusi. A tale fine, ci ispiriamo alla competizione di regioni, introdotta da [Zhu and Yuille, 1996], e definiamo il seguente modello deformabile per la segmentazione delle FCD ⎧   ∂u ⎪ ⎪ (u) − R (u) N(u) + κ(u)N(u), = α R N L L ⎨ ∂t (6) RL (u) = PL (u), ⎪ ⎪   ⎩ RN L (u) = max Pc (u), c ∈ {CSF,GM,WM} dove N(u) `e il vettore normale unitario, interno alla superficie, κ(u) la curvatura, α ed  due costanti positive, e RL (u) e RN L (u) le funzioni di appartenenze ad i   tessuti lesionati e sani rispettivamente. Il termine α RN L (u) − RL (u) N(u) sar`a chiamato nel seguito forza di regione. I calcoli che permettono di arrivare a questa formulazione sono descritti nel capitolo 3.1.3. Vediamo come si comporta questo modello. Se RN L (u) > RL (u), allora la probabilit`a del voxel u di appartenere ad una classe non lesionale `e pi` u elevata di quella relativa alla FCD. u appartiene dunque ad un tessuto sano. RN L (u) − RL (u) `e positivo e la superficie si dirige nella direzione di N(u), cio`e si restringe, ed esclude questo punto. Inversamente, se RN L (u) < RL (u), allora la probabilit`a di essere nella FCD `e la pi` u elevata. u appartiene dunque alla lesione. RN L (u) − RL (u) `e allora negativo e la superficie si dirige nella direzione di opposta a N(u), cio`e si dilata, ed esclude questo punto. Infine, il termine κ(u)N(u) corrisponde ad una deformazione di curvatura, e fornisce risultati regolarizzati e lisci. 12

3 – Segmentazione delle FCD mediante modelli deformabili guidati dalle caratteristiche di RMN

3.3

Evoluzione level-set

La formulazione level-set dell’equazione (6) si ottiene facilmente mediante l’equivalenza (5). In effetti, se poniamo   F (κ) = α RN L (u) − RL (u) + κ(u), allora otteniamo la seguente equazione level-set   ∂φ (M) = α RN L (M) − RL (M) |∇φ(M)| + κ(M)|∇φ(M)|, ∂t

(7)

con φ la funzione implicita, definita al tempo zero come la funzione distanza calcolata a partire della superficie attiva iniziale. Chiameremo questo modello FBDM (Feature-Based Deformable Model ). Le approssimazioni numeriche del FBDM sono riportate nel capitolo 3.2.2

3.4

Brevi indicazioni d’implementazione

Metodo della banda stretta L’equazione level-set (7) `e stata implementata usando il metodo a banda stretta (narrow band method ). Quest’algoritmo consiste nel calcolo dell’evoluzione unicamente all’interno di una banda stretta, definita intorno alla superficie attiva, e calcolata all’inizio del procedimento. Quest’astuzia evita di calcolare inutilmente il valore di φ nei punti lontani dalla superficie. Dobbiamo tuttavia verificare durante l’evoluzione che la banda stretta sia sempre valida, in altre parole che il modello deformabile sia sempre dentro della banda. In effetti, la superficie attiva si muover`a durante la procedura per trovare i contorni della lesione, e pu`o dunque uscirne. Per risolvere questo problema, si verifica ad ogni iterazione se la superficie non si avvicina troppo ad un bordo della banda. Se ci`o avviene, quest’ultima `e ricalcolata a partire della nuova posizione della superficie attiva. Quest’algoritmo ci ha permesso di convalidare il nostro metodo di maniera quantitativa su un importante numero di pazienti. Il lettore si riferisca al capitolo 3.2.4 per ottenere maggior informazioni. Re-inizializzazione della funzione implicita L’equazione di evoluzione del modello deformabile `e definita unicamente sulla superficie. Quando si passa alla formulazione level-set, il problema diventa tridimensionale e tutti i punti della funzione implicita evolvono secondo quest’equazione. Tuttavia, la funzione velocit`a ha senso solo sulla superficie e non altrove. Di conseguenza, se si implementa l’equazione ingenuamente, si pu`o ottenere un’instabilit`a numerica 13

dovuta al movimento degli altri level-set, cio`e degli altri insiemi di punti con valore qualunque. Diverse soluzioni sono state sviluppate per risolvere questo problema. Siccome lavoriamo su immagini 3D di grande dimensione, abbiamo deciso di utilizzare un metodo rapido ed efficiente. L’algoritmo consiste nella re-inizializzazione regolare della funzione implicita φ per evitarne la degenerazione. Inoltre, un metodo sub-voxelico `e stato applicato per evitare movimenti non desiderati della superficie durante questa fase. Ulteriori dettagli sono riportati nella sezione 3.2.5. Determinazione della fine dell’evoluzione L’ultimo punto importante dell’implementazione `e la definizione di un criterio efficiente per fermare l’algoritmo una volta compiuta l’evoluzione. La competizione di regioni definisce una misura di energia E legata alla superficie attiva S. Quest’energia `e minima, in teoria nulla, quando S si trova sui contorni della lesione, e positiva quando S si trova altrove. Possiamo allora fermare l’algoritmo quando E `e minima. Per valutare l’andamento di E, si considerano i cinque ultimi valori. La loro deviazione standard deve inferiore ad una certa soglia (cio`e l’energia deve essere la pi` u costante possibile), e l’ultimo valore di E deve essere compreso tra la pi` u alta e la pi` u bassa delle altre quattro energie (figura 3.14, p99). Infine, questo criterio deve essere verificato cinque volte consecutive per poter fermare l’algoritmo e dare il risultato finale (capitolo 3.2.6)

3.5

Sperimentazione

Pazienti selezionati e preparazione delle immagini Abbiamo selezionato 24 soggetti con FCD visibili sulle immagini RMN T1. Il comitato di etica dell’Istituto Neurologico di Montreal approv`o lo studio, ed i pazienti consentirono l’uso delle loro RMN. Tutte le immagini sono state corrette dai difetti d’intensit`a e di movimento. Il cranio `e poi stato rimosso dalle immagini, ed un’uniformazione delle dimensioni ed orientazioni `e stata effettuata mettendo le immagini in uno spazio standard. Inizializzazione del modello deformabile Il modello deformabile fu inizializzato utilizzando il classificatore automatico proposto da [Antel et al., 2003]. Sui 24 soggetti selezionati, le lesioni di 18 dei pazienti sono state correttamente identificate. Infine, un esperto verific`o i risultati per essere sicuri che l’inizializzazione fosse effettuata di maniera soddisfacente. Per quanto riguarda i sei soggetti non identificati, abbiamo provato ad inizializzarli manualmente ma l’algoritmo non ha dato risultati rilevanti. Le caratteristiche di queste lesioni 14

3 – Segmentazione delle FCD mediante modelli deformabili guidati dalle caratteristiche di RMN

sembravano in fatti troppo sottili e insufficientemente discriminanti. La valutazione del metodo `e dunque stata effettuata sui 18 pazienti identificati. Segmentazione manuale Le lesioni dei 18 pazienti selezionati sono state segmentate manualmente da due esperti mediante un programma di disegno 3D. Denoteremo queste etichette come M1 e M2 . Poi, gli esperti si sono riuniti e, con la collaborazione di due neurologi, si sono messi d’accordo sulle zone ambigue definite come lesionali da uno di loro ma non dall’altro. Le etichette ottenute, chiamate “di consenso”, furono utilizzate come riferimento durante la valutazione quantitativa della segmentazione automatica. Risultati Per valutare l’efficienza del nostro programma, abbiamo utilizzato tre misure diverse. La prima, chiamata similarit`a `e definita da S =2×

|A ∩ B| . |A| + |B|

(8)

S stima la somiglianza tra due etichette A e B, etichette che possono essere manuali (M1 o M2 ), o automatiche (che denoteremo come A). Abbiamo poi calcolato la percentuale di copertura ottenuta con il nostro algoritmo C = 100 ×

|A ∩ M| . |M|

(9)

Questa misura valuta l’area della lesione correttamente segmentata dal modello deformabile. Infine, una misura di falsi positivi valuta la percentuale di tessuti sani classificati come lesionali |A \ M| . (10) Fp = 100 × |A| La similarit`a media tra i due insiemi d’etichette M1 e M2 era di 0.62. Questo valore corrisponde ad un buon risultato, specialmente nel caso delle FCD dove la sfumatura dei limiti impedisce una delimitazione esatta dei contorni. Abbiamo poi paragonato i risultati della segmentazione automatica con le etichette di consenso. I 18 pazienti sono stati segmentati, usando un procedimento leave-one out. Durante la segmentazione della lesione di un certo paziente, lo toglievamo dall’insieme d’apprendimento per stimarne le modellizzazioni probabilistiche. L’insieme di apprendimento era allora composto dai 17 altri individui. La similarit` a media ottenuta era di 0.65, la percentuale di copertura uguale a 57% ed il tasso di falsi positivi di 16%. Abbiamo anche calcolato queste misure rispetto alle etichette 15

M1 e M2 , i risultati sono presentati nella tabella 3.2, pagina 104. Infine, per valutare il miglioramento fornito dal nostro algoritmo, abbiamo calcolato queste misure per i risultati del classificatore automatico. La similarit`a media era allora di 0.26, la copertura di 16% ed il tasso di falsi positivi di 0.26%. Le figure 3.17, p105, a 3.24, p110, mostrano alcuni esempi di segmentazioni automatiche ottenute.

3.6

Discussione dei risultati

La similarit`a tra le segmentazioni manuali `e buona ma riflette la difficolt`a di definire di maniera precisa ed oggettiva i contorni delle FCD. Questo valore giustifica la costruzione di etichette di consenso che, grazie alle discussioni tra gli esperti, sono meno soggette a variabilit`a, e dunque pi` u affidabili. I risultati provveduti dal nostro algoritmo sono buoni. La similarit` a, in rapporto alle etichette di consenso, `e buona. I falsi positivi sono ottimi e la copertura soddisfacente. Un calcolo fatto nella discussione del capitolo 3 ci indica in effetti che un tasso di falsi positivi circa uguale a 20% corrisponde ad uno sbaglio equivalente alla dilatazione dell’etichetta di un voxel. Una tale variazione `e dunque insignificante nel nostro caso, ulteriormente confermato dall’analisi visiva dei risultati. La copertura ottenuta `e eccellente se paragonata a quella fornita dal classificatore automatico (57% contro 16%). Pero, un tale valore non `e sufficiente per le applicazioni di segmentazione. Infatti, le feature map sembrano insufficienti ed incapaci di estrarre la totalit`a della lesione per colpa dell’eterogeneit`a delle caratteristiche di RMN. Questo problema costituisce il principale difetto del metodo. La prossima sezione ne presenter`a una soluzione. Le similarit`a in rapporto alle etichette manuale M1 e M2 , e l’analisi visuale dei risultati, hanno dimostrato che questo metodo permette di svelare zone lesionali che potrebbero facilmente sfuggire all’attenzione dell’esperto. La figura 3.16, p104, d`a un esempio di tale situazione. Questa propriet`a `e molto interessante poich´e permette l’uso del metodo in ambiente clinico, facilitando la diagnostica e la valutazione delle dimensioni delle FCD. L’influenza dei parametri sui risultati `e stata studiata di maniera quantitativa. E’ stato dimostrato che la segmentazione automatica non dipende dalle etichette utilizzate durante la fase di stima dei modelli probabilistici. Cos`ı, non sono necessarie etichette molto precise, e qualunque persona preparata alla delimitazione manuale delle lesioni pu`o crearle. Il criterio di fine evoluzione funziona bene e non ferma prematuramente l’algoritmo. I parametri del modello sono stati studiati. Quelli utilizzati sono soddisfacenti ma possono essere cambiati senza alterare i risultati, purch´e la deformazione di curvatura resti sufficientemente debole rispetto alla forza costante al fine di evitare che la superficie attiva restringesse le piccole lesioni. Rimandiamo il lettore al capitolo 3 per tutti i dettagli degli esperimenti effettuati. 16

4 – Modelli deformabili perfezionati per la segmentazione delle FCD

Il metodo proposto fornisce risultati buoni e promettenti. Essendo automatizzato, fornisce risultati oggettivi e permette di rivelare sottili zone lesionali. Nonostante la difficolt`a a delimitare i limiti delle FCD, la similarit`a media `e buona ed i falsi positivi deboli. Tuttavia, il tasso medio di copertura deve essere migliorato per utilizzare quest’algoritmo come metodo di segmentazione automatica. La prossima sezione fornir`a una soluzione a questo problema.

4

Modelli deformabili perfezionati per la segmentazione delle FCD

Presentiamo in questa sezione la versione finale dei due modelli deformabili proposti per segmentare efficientemente le FCD. Il capitolo 4 della presente tesi descrive le tappe successive che ci hanno permesso di arrivare a queste soluzioni. In poche parole, abbiamo applicato le seguenti modellizzazioni con le descrizioni probabilistiche dei tessuti cerebrali presentate nella sezione precedente. I risultati ottenuti avevano per`o troppi falsi positivi e dunque non erano soddisfacenti per applicazioni mediche. Abbiamo allora dovuto introdurre un modello probabilistico pi` u completo per risolvere questo problema. Rimandiamo il lettore al capitolo 4 per la presentazione del metodo, e descriviamo qui unicamente le principali idee dei modelli deformabili finali.

4.1

Una descrizione probabilistica pi` u completa delle caratteristiche RMN delle FCD

Considerazioni biologiche La sfumatura della giunzione tra la sostanza grigia e bianca `e una delle principali caratteristiche RMN delle FCD. Tuttavia, la sua unica presenza non basta per distinguere le lesioni dalla corteccia sana. In effetti, alcune parti della corteccia cerebrale (pensiamo alla zona centrale) hanno un contrasto debole ed una leggera sfumatura dei contorni. La presenza nei dintorni delle lesioni di strutture sane con sfumatura dei limiti `e dunque possibile. Questi tessuti possono allora indurre in errore il modello deformabile che le segmenter`a, creando dei falsi positivi. Di conseguenza, dobbiamo modellare le giunzioni sane tra sostanza grigia e bianca, e tra sostanza grigia e fluido cerebrospinale, al fine di evitare questi problemi. Definizione di nuove descrizioni probabilistiche Il metodo utilizzato per modellare i tessuti cerebrali `e identico a quello descritto nel capitolo precedente, tranne l’addizione di due classi supplementari come proposto da 17

[Antel et al., 2003]. Queste classi corrispondono, in effetti, ai due tipi di giunzione: fluido cerebrospinale / sostanza grigia (CSF/GM) e sostanza grigia / sostanza bianca (GM/WM). Le immagini RMN dei pazienti dell’insieme di apprendimento sono quindi segmentate in sei classi c, le quattro precedenti (CSF, GM, WM, L) e le due di transizione (CSF/GM e GM/WM). La lesione `e sempre identificata manualmente da un esperto, e i tre tessuti principali (FCD, GM e WM) determinati da un metodo ad istogramma con soglia automatica. Le classi di transizione sono definite a partire di questa segmentazione intermedia come segue. Per ogni voxel M dell’immagine si considera una finestra 3 × 3 × 3 centrata in M. Se almeno il 30% dei voxel attigui a M fanno parte della classe CSF, e altri 30% fanno parte della classe GM, allora M `e classificato come transizione CSF/GM. Si ottiene la classe GM/WM in maniera similare (figura 4.10, p135). Una volta calcolate le immagini a sei classi, possiamo effettuare l’apprendimento supervisionato per stimare i parametri delle probabilit`a condizionale P
(f (M)|c) (f (M) `e il vettore di feature associato al paziente da segmentare). La regola di Bayes ci d`a allora la probabilit`a desiderata Pc
(M) = P
(c|f (M)) (capitolo 4.4.2). La figura 4.11, p135, mostra le probabilit` a relative alle sei classi.

4.2

Modellazione dei contorni delle FCD

Una forza supplementare `e richiesta per aumentare la percentuale di copertura ottenuta col modello deformabile precedente. La forza proposta si basa su una modellizzazione implicita dei contorni della lesione, per guidare la superficie attiva pi` u efficientemente. Considerazioni biologiche Prima di spiegare i principi della nuova forza, torniamo brevemente alle considerazioni biologiche. Com’`e stato detto nella prima sezione di questo capitolo, le FCD provocano una disorganizzazione totale della struttura a strati della corteccia cerebrale [Tassi et al., 2002; Palmini et al., 2004]. Di conseguenza, la lesione si estende su tutta la sezione della corteccia, vale a dire dalla giunzione tra fluido cerebrospinale e sostanza grigia, fino a quella tra sostanza grigia e bianca. Inoltre, `e comunemente ipotizzato il fatto che le cellule malformate e non differenziate presente nelle FCD seguono durante lo sviluppo cerebrale la direzione di migrazione delle cellule sane. [Rakic, 1995] dimostr`o che la migrazione si effettua in direzione radiale, dalla matrice germinale situata attorno ai ventricoli verso la corteccia. Quest’ipotesi ci permette allora di pensare che le frontiere tra la corteccia sana e la lesione siano ortogonali alle giunzioni CSF/GM e GM/WM. La nuova forza guider`a quindi il modello deformabile verso i limiti CSF/GM e GM/WM, ortogonalmente, per includere tutta la corteccia malata (figura 4.3, p122). 18

4 – Modelli deformabili perfezionati per la segmentazione delle FCD

Modellazione dei contorni delle FCD Non possiamo utilizzare metodi diretti per estrarre i contorni delle lesioni per colpa della sfumatura che la caratterizza. Tuttavia, un metodo implicito ci consente di approssimarli di maniera soddisfacente. Se consideriamo la segmentazione a tre classi ottenuta col metodo ad istogramma (CSF, GM e WM), e se calcoliamo la proporzione delle lesioni classificate come CSF, GM o WM, troviamo che circa l’88% dei voxel lesionali cade nella sostanza grigia (tabella 4.1, p119). Possiamo dunque attirare la superficie attiva verso i limiti rivelati dalla segmentazione ad istogramma. In effetti, anche se non corrispondono necessariamente alla realt`a, se ne avvicinano sufficientemente per essere considerati come una buona approssimazione. Dobbiamo ora creare una forza che guidi il modello deformabile verso questi limiti impliciti ma di maniera ortogonale alle giunzioni, come proposto dalla seconda ipotesi biologica. A tal fine, proponiamo di utilizzare il gradient vector flow (GVF) sviluppato da [Xu and Prince, 1998a]. Il GVF `e un campo di vettori orientati verso i contorni dell’oggetto da segmentare, calcolato mediante un’equazione di diffusione (equazione 4.1, p120) che propaga l’informazione perpendicolarmente ai contorni dell’oggetto. Una forza utilizzando il GVF pu`o allora guidare ortogonalmente il modello deformabile verso i contorni della lesione (figura 4.3, p122). Questa forza, che denoteremo come FGV F , `e quindi particolarmente adatta al nostro problema

FGV F (u) N(u) = v ˆ(u) · N(u) N(u), (11) con v ˆ(u) il GVF normalizzato (la norma del campo di vettore vale 1 in ogni punto dell’immagine) e N(u) il vettore normale unitario interno. Rimandiamo il lettore al capitolo 4.1.2 per i dettagli della modellizzazione. Le figure 4.2 e 4.4, p121, mostrano le diverse tappe della costruzione della forza GVF.

4.3

Due modelli basati sulle caratteristiche RMN delle FCD e la modellazione dei loro contorni

Due modelli deformabili, basati sulla forza di regione e la forza GVF, sono stati proposti. Tuttavia, le due deformazioni non hanno lo stesso obiettivo. Il GVF ha come scopo l’attrazione del modello deformabile verso i limiti della lesione. E’ basato unicamente su supposizioni biologiche, e non su caratteristiche esplicite estratte dall’immagine. Di conseguenza, questa forza non deve influenzare l’informazione di regione. Sar`a dunque attivata unicamente quando il modello deformabile si trover`a fuori delle zone probabilmente lesionali (cio`e zone la cui probabilit`a massima `e quella relativa alla lesione). 19

Definizione di una nuova forza di regione Vediamo, prima di presentare i due modelli proposti per segmentare le FCD, le modificazioni introdotte nella forza di regione. Infatti, abbiamo cambiato unicamente il termine relativo ai tessuti sani dove sono state aggiunte le probabilit`a delle classi di transizione. Otteniamo quindi le seguenti funzioni d’appartenenza: 

RL
(u) = PL
(u), 

RN L (u) = max Pc (u), c ∈ {CSF,GM,WM,CSF/GM,GM/WM}

(12)



mentre la forza di regione rimane α RN L (u) − RL (u) N(u). Modello deformabile esteso Il primo modello deformabile consiste nell’aggiungere direttamente nel FBDM la forza GVF utilizzando un “interruttore”. Quando la superficie attiva si trova in
zone probabilmente lesionali (cio`e RL
(u) > RN L (u)), allora si tiene conto unicamente della forza di regione ed il modello deformabile si dilata. Al contrario, se la superficie
attiva si trova in zone sane (cio`e RL
(u) < RN L (u)), allora si disattiva la forza di regione e si attiva il GVF, il modello deformabile si diriger`a verso le giunzioni tra i vari tessuti cerebrali. Il modello pu`o essere scritto matematicamente come segue 

 ∂u = α1 1 − H RN L (u) − RL (u) RN L (u) − RL (u) N(u) ∂t  

+ β1 H RN L (u) − RL (u) v ˆ(u) · N(u) N(u) + 1 κ(u)N(u),

(13)

con H la funzione di Heaviside, e α1 , β1 e 1 delle costanti positive. Noteremo questo modello eFBDM (extended Feature-Based Deformable Model ). Modello deformabile in due tappe Il secondo modello propone di dividere l’algoritmo in due tappe. La prima parte consiste nella segmentazione delle FCD con il FBDM, modificato per tener conto delle classi di transizione. Poi, si parte dal risultato ottenuto e si applica un secondo modello, chiamato BBDM (Boundary-Based Deformable Model). In zone probabilmente lesionali si utilizza unicamente la forza di regione, la superficie attiva si dilata. Per`o, in zone probabilmente sane, si utilizzano le due forze simultaneamente, la forza GVF per guidare il modello deformabile verso le giunzioni CSF/GM e GM/WM, e la forza di regione, pi` u debole del GVF, per controllare la dilatazione 20

4 – Modelli deformabili perfezionati per la segmentazione delle FCD

dovuta a quest’ultima. Il BBDM si scrive allora

∂u = α2 RN L (u) − RL (u) N(u) ∂t  

ˆ(u) · N(u) N(u) + β2 H RN L (u) − RL (u) v + 2 κ(u)N(u).

(14)

con H la funzione di Heaviside, e α2 , β2 e 2 delle costanti positive.

4.4

Sperimentazione e risultati

Abbiamo provato i due modelli sui 18 pazienti selezionati e paragonato le segmentazioni automatiche con le etichette di consenso, tramite le stesse misure quantitative (similarit`a, percentuale di copertura e tasso di falsi positivi). I dettagli del protocollo di test sono dati nel capitolo 4.5. La seguente tabella riporta i diversi risultati. Table 1: Risultati per i modelli eFBDM e BBDM rispetto alle etichette di consenso. Sono riportati come media±SD (min a max).

S C Fp

eFBDM BBDM 0.73 ± 0.08 (0.6 a 0.86) 0.73 ± 0.08 (0.6 a 0.86) 80.3% ± 14.1% (49.7% a 96.2%) 72.1% ± 15.7% (44.3% a 94.3%) 28.5% ± 16.6% (0.7% a 56.1%) 20.1% ± 14.8% (0.4% a 47.7%)

Inoltre, una verificazione visiva `e stata fatta con l’aiuto di due neurologi (N. Bernasconi e A. Bernasconi). La percentuale di copertura era ottima e la similarit`a eccellente. Sui 18 pazienti, 7 sono stati giudicati eccellenti (copertura ottima, nessun falso positivo), 9 molto buoni, e due leggermente meno buoni di quelli ottenuti con il FBDM per colpa dei falsi positivi un po’ grandi. Nondimeno, i falsi positivi erano ogni tanto non trascurabili, specialmente con l’eFBDM dove la forza GVF non era controllata. Le figure 4.13, p139, a 4.21, p145, mostrano alcuni esempi di risultati ottenuti.

4.5

Discussione dei risultati

I risultati ottenuti sono eccellenti sopratutto quando si considera la difficolt`a di segmentare le FCD. La similarit`a media `e molto significativa, la percentuale di copertura `e buona per applicazioni di segmentazione, ed i falsi positivi ragionevoli. L’analisi visiva dei risultati ha confermato le misure quantitative. Il GVF funziona bene, attira il modello deformabile verso le giunzioni CSF/GM e GM/WM, in direzione ortogonale, evitando cos`ı la dilatazione laterale della superficie verso la 21

corteccia sana. Inoltre, la segmentazione automatica ha permesso in alcuni casi di rivelare zone lesionali non identificate da un esperto, ma confermate dalle etichette di consenso o dopo l’analisi visiva finale. Tuttavia, in alcuni casi, delle parti dei tessuti sani adiacenti sono state considerate come lesioni. Questo problema `e spiegabile facilmente dalla difficolt`a di estrarre con precisioni i piccoli sulci o gyri (appendice A). In effetti, un sulcus mal identificato pu`o provocare l’“unione” di un gyrus sano con la lesione. Il GVF guida allora il modello deformabile verso la giunzione sbagliata. Similarmente, se la banda di sostanza bianca situata all’interno di un piccolo gyrus `e mal evidenziata, allora il GVF pu`o ignorarla e attirare la superficie aldil`a dei limiti della FCD. Nondimeno, questi difetti sono sempre molto evidenti, e l’esperto non `e indotto in errore. La sezione 4.5 d`a pi` u dettagli su questo problema.

Conclusione In questa tesi, abbiamo proposto e valutato un metodo nuovo ed originale per la segmentazione automatica su immagini di RMN T1 delle displasie corticali focali. La delimitazione precisa di queste lesioni `e fondamentale per la loro diagnostica ma soprattutto per l’esito dell’operazione chirurgica. Tuttavia, la sfumatura dei contorni e l’assenza di limiti espliciti tra la FCD e la corteccia cerebrale sana rendono la valutazione delle loro dimensioni molto ardua. Le sottili aree lesionali possono facilmente sfuggire all’analisi visiva, mentre le procedure tradizionali di segmentazione automatica sono penalizzate. Per risolvere questi problemi, abbiamo utilizzato modelli deformabili basati sulle caratteristiche RMN delle FCD, e su una modellazione implicita dei loro contorni. Inoltre, per valutare l’efficienza dell’algoritmo in maniera quantitativa ed affidabile, abbiamo costruito, con la collaborazione di quattro esperti, delle etichette di consenso. Il primo modello proposto usa unicamente una descrizione probabilistica delle caratteristiche RMN delle FCD. I risultati erano molto promettenti, con una buona similarit`a, dei falsi positivi trascurabili ed una copertura buona ma ancora troppo debole per applicazioni di segmentazione. Abbiamo allora costruito una forza basata su una stima implicita dei contorni delle lesioni. Questa forza `e derivata dal gradient vector flow e attira il modello deformabile verso le giunzioni della corteccia. Due modelli sono stati proposti, il primo procede in un’unica tappa ma usa un interruttore per usare l’una o l’altra forza, mentre l’altra si svolge in due tappe, con la forza di regione utilizzata per controllare l’espansione GVF. I risultati ottenuti con i due modelli sono simili. Il primo, eFBDM, fornisce una maggior copertura ma con leggermente pi` u falsi positivi, mentre l’altro, il BBDM, 22

4 – Modelli deformabili perfezionati per la segmentazione delle FCD

copre meno la lesione ma con meno falsi positivi. In ogni caso, la similarit`a ottenuta `e eccellente, la copertura buonissima ed i falsi positivi ragionevoli. I modelli proposti sono robusti a tutti i parametri. Non dipendono dalle etichette manuali relative ai pazienti dell’insieme d’apprendimento (usate durante la fase di stima delle descrizioni probabilistiche). Qualunque persona preparata per disegnare questo tipo di lesione pu`o quindi costruire le etichette. Inoltre, i parametri dei modelli (αi , i e βi ) non richiedono nessuna regolazione particolare. Bisogna solo scegliere i sufficientemente piccolo rispetto ad αi e βi per evitare che il modello restringesse le piccole FCD. Abbiamo dimostrato in questa tesi l’efficienza dei modelli deformabili geometrici nella segmentazione delle FCD. I risultati ottenuti ci incoraggiano a considerare questa tecnica come un utile aiuto alla diagnosi visiva ma anche alla valutazione prechirurgica, grazie all’eccellente copertura e similarit`a fornite dal metodo. A nostra conoscenza, tale programma non esiste e questo lavoro, compiuto nell’ambito della tesi di laurea, costituisce dunque un primo algoritmo automatico procurando risultati rilevanti. Vogliamo ricordare al lettore che questo capitolo `e un riassunto molto breve del lavoro svolto. I capitoli successivi descrivono i dettagli completi dei modelli proposti, la loro implementazione, ed i risultati ottenuti.

23

24

Chapter 1 Epilepsy, focal cortical dysplasia and magnetic resonance imaging 1.1

Overview of epilepsy

1.1.1

Historical background

Epilepsy through the Ages While epilepsy is nowadays commonly accepted as a neurological affection, it was widely misunderstood during the past centuries. The first descriptions date back the earliest writings. Indeed, the upsetting characteristics of the seizures attracted much attention and a lot of discussions and controversies are linked with this disease. Some Mesopotamian documents of the fifth millennium B.C already mentioned them. Epilepsy was called at that time the “Falling Disease” or the ”Sacred Disease” and was closely linked with demons and evil spirits because of its strange outward signs. Epileptic persons were thus often treated by priests through divine invocations. The Bible itself mentions these practices by showing how Jesus Christ casts out a devil from a young man with epilepsy. “Teacher, I brought you my son, who is possessed by a spirit that has robbed him of speech. Whenever it seizes him, it throws him to the ground. He foams at the mouth, gnashes his teeth, and becomes rigid. I asked your disciples to drive the spirit out, but they could not.” — Gospel According to Mark (9:14-29) In 400 B.C., Hippocrates wrote the first scientific book about epilepsy. He was then the first philosopher and physician to characterize it as a brain disorder and not as a mere magical manifestation. 25

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

“It is thus with regard to the disease called Sacred: it appears to me to be nowise more divine nor more sacred than other diseases, but has a natural cause like other affections. . . ” — Hippocrates But in spite of everything, epilepsy has been associated for centuries with superstition, and people who suffered from it were also victim of the ignorance of the others. In 1494 for example, two Dominican friars wrote the Malleus Maleficarum, a handbook on witch-hunting. It was explained that witches could be identified by their epileptic seizures. The consequences were dramatic, a wave of persecution arose and more than 200,000 women were killed unfairly. Later, in the early 1900s, the disease was still considered as madness and people suffering from it were cared for in asylum. And while they were not considered to be mad, they were seldom allowed to get married or become parents, indeed sometimes sterilized like in some US states. The beginning of epilepsy surgery Epilepsy had to wait the end of the nineteenth century to be recognized as a neurological disease. At that time, neurology was only an emerging field and its main objective was the localization of cortical functions to specific brain regions. In 1873, a physician of the National Hospital for Neurology and Neuro-surgery of London, John Hughlings Jackson, described for the first time epileptic seizures as “occasional sudden excessive, rapid and local discharges of gray matter”, that is as local abnormal activity of the brain [Jackson, 1873]. This intuition, which was then confirmed by Ferrier’s experiments in primates [Ferrier, 1874], had important consequences on the understanding of epilepsy. Brain functions began to be localized and a close link appeared between clinical characteristics of the seizures and the localization of the site of origin. Besides, thanks to the observation of these symptoms, William Macewen, a neurosurgeon of Glasgow, localized and removed in 1879 a tumor from the brain of one of his patients with epilepsy. The operation was then a success; the patient survived and became seizure free. Following this experiment, other such attempts were led but often with bad outcomes because of the poor medical means at that time. However, the principles of epilepsy surgery were established and the removal of the seizure focus began to be a viable way to treat the disease. It became then quickly evident that a good localization of these foci was critical to get the best possible outcome. The modern era During the twentieth century, epilepsy treatment and brain understanding improved as new techniques arose. The first major breakthrough is certainly the electroencephalography (EEG). Developed by [Berger, 1929], it immediately earned the interest 26

1.1 – Overview of epilepsy

of researchers. EEG allowed the tracking and the measurement of the electrical brain activity by a totally non-invasive method; the emitted waves were picked up by electrodes placed to the scalp. In that way, abnormal activations of the brain were easily detected and the localization of epileptiform foci could be achieved by a totally objective method. Moreover, a new type of epilepsy, the temporal lobe epilepsy, was brought to light. Indeed, the absence of explicit symptoms made this form of epilepsy very difficult to diagnose and the affected area of the brain was almost impossible to find before the coming of EEG. But in an article published in 1951, [Bailey and Gibbs, 1951] reported promising results of temporal lobe resections on 25 patients whose abnormal regions were localized only by EEG analysis, without any clinical observations. These results were so encouraging that EEG became very quickly the primary method for seizure focus localization. In the same way, epilepsy surgery became a viable way to treat epilepsy, curing successfully an increasing number of patients with pharmacologically intractable epilepsy.

Figure 1.1: EEG of a 46-year-old patient. Each line corresponds with a specific electrode. Horizontal axis represents time, vertical axis voltage.

Magnetic resonance imaging (MRI) is the second key technology nowadays widely applied to human studies. First used for experimental purposes in the late 1970s, it began to be available for clinical diagnostic since the mid 1980s. It became then quickly evident that this novel technique would improve greatly the understanding of epilepsy and enhance the seizure focus localization. It allowed indeed the analysis of precise structures of the brain, managing the detection of small tumors, cortical malformations and even hippocampal sclerosis. The electric localization was then associated in many cases with a structural abnormality. However, the structural anomalies were sometimes not consistent with the electric focus. The question was then which region a neurosurgeon had to remove in order to treat the patient, the abnormal tissues or the seizure focus detected by the EEG. Still today the answer remains unclear but it seems that the removal of the structural anomaly is critical for a good post surgical outcome. 27

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

Nowadays, MRI is still in progress and the understanding of brain functions follows close behind it. Recently for example, functional MRI (fMRI) confirmed the vascular changes that occur during seizures, as intuited by Wilder Penfield years before. Other new MR techniques, like diffusion tensor imaging (DTI), seem to be promising as well by revealing relevant connections between two given cortical regions. The understanding of the influence of an abnormal region on its neighborhood for example may greatly benefit from such technologies. Moreover, the increase of computer power allowed the use of advanced image processing on MR images, introducing in this manner new diagnosis-aid systems for neurosurgery but also permitting quantitative analysis of brain structures. One thing is for sure, MRI and image processing will be crucial in the development of brain understanding and epilepsy treatment. The last century was not only marked by the coming of EEG and MRI, but also by great scientists who have considerably contributed towards the knowledge of brain function. Wilder Graves Penfield, a Canadian researcher and neurosurgeon, is certainly one of them. Born in the US, he moved to Montreal in 1928 and became quickly the first neurosurgeon of the city until 1960, when he retired. While he came at the beginning to teach at McGill University, he became in 1934 the first Director of the Montreal Neurological Institute and the associated Montreal Neurological Hospital, which he founded with the help of the Rockefeller Foundation. His works greatly improved the localization of cortical structures and the understanding of brain mechanisms. He discovered several connections between cortical areas and organs, and he was the first to bring to light the dramatic vascular changes occuring during seizures. At that time those changes were not visible, even on EEG, and the researchers had to wait the coming of advanced imaging techniques such as fMRI to confirm his intuition. Besides, they still base their studies on his works to interpret the variations of intensity on fMRI or SPECT images (single-photon emission computed tomography). Penfield died in 1976, leaving behind him a great legacy. Although he lost his life before the coming of these imaging techniques, he clearly foresaw that they would reveal many new aspects of brain functions and of epilepsy. And they do not seem to prove him wrong.

1.1.2

The basics of epilepsy

Epidemiology of epilepsy Epilepsy is the most common neurological affection after migraine. A patient is said to have epilepsy if he has suffered from two epileptic seizures at least. In that way, approximately 1% of the Canadian population is really affected [Wiebe et al., 1999], in comparison with the 5% who may have only one single epileptic seizure in the course of their life. This rate, called prevalence, is quite constant in all over the world despite the different statistical protocols used by the various countries. In 28

1.1 – Overview of epilepsy

Figure 1.2: Wilder Graves Penfield (1891-1976).

Italy for instance, about 500,000 persons are affected by epilepsy among a population of 58,000,000 inhabitants. The prevalence depends on the age. It tends to increase from childhood to adolescence, it is quite constant next and finally increases again, slightly, after age 70. The number of new detected cases follows as for it an opposite trend. It is high during childhood and decreases until adolescence. The causes of these new cases are often unknown but they may sometimes involve some genetic disorders. During adulthood, the rate stays stable and after middle age it increases again, as tumors, strokes or other degenerative diseases such as Alzheimer’s disease arise [Kucniecky and Jackson, 1995b]. In short, 30 to 50 new cases amongst 100,000 persons are detected every year. It should be mentioned that other factors such as acquired disorders, head trauma or other infections can also provoke epilepsy, but in most cases, and more especially among young patients with generalized epilepsy, they are not well identified, indeed unknown. Among the newly diagnosed patients, about 70% of them can be successfully treated with, or even without, medication. On the other hand, there are approximately 30% of them who suffer from pharmacologically intractable epilepsy [Wiebe et al., 1999], i.e. there is no treatment for their seizures (about 10%) or the current drugs are partially or completely inefficient. Surgery can then constitute a viable solution: 64% of patients who had an operation have become seizure-free or, at worst, have presented relevant clinical improvement [Wiebe et al., 1999]. The issue of intractable epilepsy is central because the epileptic fits can easily have major consequences in both patient’s health condition and social life. On the one hand, recurrent seizures can damage the brain [Sutula et al., 2003] and progressively bring the loss of cognitive ability. But accidents, indirectly provoked by seizures, can also be harmful to the patients. The loss of consciousness that 29

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

can occur during a fit, and the following fall, may injury or kill them, if the head for example is severely hit. Deaths by suffocation or drowning are also frequent, and the risk of sudden death seems to be twice as much common among people with epilepsy. This risk indeed can reach 1 death in 300 per year for people who suffer from recurrent seizures and take high doses of drugs. But unfortunately, their origins are still unknown; they can be provoked by either direct factors, such as heart-beat failures, or indirect consequences of a fit. On the other hand, epilepsy can become a “social burden”. People suffering from it are more or less dependant on their circle for the strength and the recurrence of their seizures. Moreover, the disease can prevent them from having a regular occupation or other social activities such as having a driver’s license. Epileptic seizures Epileptic seizures are transient events caused by a sudden surge of electrical activity in the brain, disturbing its normal function. In fact, they are not a disease per se but the clinical manifestations of some more profound cortical disorders. Their very various symptoms (from totally silent seizures to heavily disabling attacks) closely depend on the cortical region that is affected by the abnormal activity and their observations, in particular at their beginning, is still the first step of the diagnosis procedures. Usually, a seizure develops in three steps. First, it may warn the patient of its imminent coming. These warnings, also called auras, are not mandatory and when they exist, they may present various symptoms such as visual loss, strange feelings or modification of the perception. Headache, nausea or even panic feelings may also point out the upcoming attack. Next, there might be no seizure (simple partial seizures) but in most cases they are followed by other symptoms. The aura symptoms may continue for instance or they can change into complex partial seizures or convulsions. Finally, at the end of the fit, the brain gets progressively back to its normal activity and the patient regains his consciousness. Until now, seizures are commonly classified in two main categories, defined by the way they begin: the primary generalized seizures and the partial seizures. Primary generalized seizures are induced by a widespread electrical discharge involving at the same time both sides of the brain. Tonic-clonic seizures, also called grand mal, are the most striking example yet of this category of epileptic fits. At its beginning, the patient loses consciousness and falls down the floor. His face turns a bit blue, his muscles stiffen and bloody saliva may come from the mouth as the tongue or cheek might be bitten. After this tonic phase, the limbs begin to jerk rapidly and rhythmically, bending and relaxing. This phase, called clonic phase, lasts few minutes and then stops progressively. The patient regains progressively consciousness and a sensation of confusion, agitation or depression overcomes him at his wakening. 30

1.1 – Overview of epilepsy

Partial seizures are generated by a focal electrical discharge involving a limited area of the brain. They are classified in three main groups according to the degree of “consciousness” kept during their development. • Simple partial seizures: those seizures can be very different from one person to another but the patient still remains alert and remembers what happens. Their symptoms are very varied, from uncontrolled muscle movements (jerking of a finger for example) to sudden psychic changes (a sudden fear, feelings of deja vu, etc.), according to the related cortical area. Usually they can be controlled by drugs and many patients become seizure-free without any medication. • Complex partial seizures: they usually start in a small area of the brain but they quickly involve other regions. The alertness is affected and people with these seizures seldom remember what happens. Some aura can point out the upcoming attack but quickly the patient loses awareness and does unconsciously odd actions (strange movements, repeating words, crying). They can be treated pharmacologically but in some cases epilepsy surgery might be necessary. • Secondarily generalized seizures: those seizures begin with a partial fit but they quickly spread throughout the whole brain and become generalized (a tonicclonic seizure for example). This evolution can be so fast that the primary partial seizure may be indiscernible. About 30% of people with partial epilepsy suffer from secondarily generalized seizures. Epidemiology studies show that partial seizures, which are the clinical manifestations of focal epilepsy, are the most prevalent form of epileptic fits. These studies reveal also that 30% of people who suffer from them cannot be cured by medications. However, since focal epilepsy originates from a local area of the brain and it is often associated with structural lesion [Engel, 2001], the surgical removal of those abnormal tissues can be considered as a viable way to treat them. Nowadays, when abnormal tissues are found, 64% of the patients who underwent surgery achieve evident clinical improvements, indeed a total recovery, whereas only 10% of them achieve such results if they are treated simply with medication [Engel, 2001]. Mechanisms of epileptic seizures Epileptic fits are known to be caused by a sudden uncontrolled activity of some cortical region. However, it is still not clear what exactly triggers them and how they spread from the epileptogenetic focus to other cortical areas. Furthermore, we still do not know to which extent the seizures themselves can damage the brain. But one thing is for sure, no single mechanism explains all the epileptiform activities. On the contrary, the different forms of epilepsy are more likely related to different factors. 31

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

The principles of the epileptic seizures can be considered at different levels, from a molecular aspect to an analysis of cell-to-cell interactions. The molecular point of view brings to light the exchanges that occur within the synapses. The failure of some neuro-transmitters makes the neurons more sensitive to their environment, increasing thus their excitability. For example, the insufficiency of GABA transmitters whose function is to inhibit the spread of the neural impulses seems to lead spontaneous epileptic activities. This phenomenon is particularly evident in some cases of temporal lobe epilepsy in which the seizures originate from the hippocampus, a subcortical structure where GABA transmitters are widely used. Glutamate, which has an excitatory purpose, is another example of neurotransmitter that can be involved in epileptic fits. The comprehension of those mechanisms has permitted to design drugs that enhance the action of the faulty molecules and thus prevent the seizures. About 70% of the patients with epilepsy can partially or completely regain the control of their epileptic fits thanks to medication. Neuronal circuitry can be an agent provocateur of epilepsy too. Besides, abnormal neuronal connections are often found in patients with epilepsy. These abnormalities are often caused by other factors than epilepsy itself, such as malformations of cortical development or tumors, but the seizures can also be responsible of the damage of neuronal connections, favoring future fits. The sprouting of axonal branches, occurring as a result of seizures in hippocampus, is the most striking example yet of such reorganization forms. This sprouting indeed replaces the death cells, but the newly connected neurons appear to be easier to excite. It is thus very difficult to decide whether an abnormal tissue is the initial cause of the seizures or a consequence. Moreover, in some cases, the abnormal tissues do not coincide with the region where the seizures effectively originates, as specified by the EEG analysis. In parallel, some sensitive regions may not be foci but rather seizure triggers. Studies have shown that abnormal activities modify the molecular environment of the neurons, leading to an increase of the excitability of the region. But the whole mechanism of the seizure spread is far from being completely discovered. New techniques, such as fMRI or diffusion tensor imaging which allows the scientists to study in vivo the neuronal connections between two different cortical areas, will be certainly crucial to the understanding of all these mechanisms.

1.2 1.2.1

Focal cortical dysplasia in focal epilepsy Focal epilepsy

Two main forms of focal epilepsy can be distinguished: temporal lobe epilepsy (TLE), which is related to the sclerosis of some cortical structures of the mesial temporal lobe, and extra-temporal lobe epilepsy (ETE), which is linked as for it to malformations of cortical development (MCD). TLE is often characterized by an 32

1.2 – Focal cortical dysplasia in focal epilepsy

hippocampal atrophy but recent studies have shown that other mesial structures may also play an important role in its development. Indeed, on the one hand, one third of the patients who had one or both of their hippocampi surgically removed still suffer from residual seizures and on the other, 20% of the persons with temporal lobe epilepsy have non-atrophied hippocampus [Jackson et al., 1994; Bernasconi et al., 2000]. MR imaging and advanced image analysis techniques might then be able to extract more subtle information from the acquisitions than hippocampal volume alone, providing thus a complete view of the structural changes resulting from the disease. As TLE goes beyond the scope of this document, we refer the reader to the literature for further information [Kucniecky and Jackson, 1995c].

1.2.2

Malformations of cortical development

Malformations of cortical development and their role in intractable focal epilepsy Contrary to TLE, ETE is not related to a specific cortical area but may rather involve any regions of the brain. In fact, they are often provoked by extra-temporal lesions and more especially by malformations of cortical development. MCD are frequently involved in pharmacologically intractable epilepsy and are found in about 25% of adults [Edwards et al., 2000] and 50% of children [Paolicchi et al., 2000] referred for epilepsy surgery. Although MRI has greatly improved the presurgical assessment of MCD by allowing the neurologists to see in vivo the patient’s brain, it is still very difficult to detect exactly their extent. Moreover, MRI still fails to identify any lesion in approximately 40%-50% of patients with ETE (cryptogenic or “MRInegative” cases) [Lorenzo et al., 1995; Bernasconi, 2004]. In that case, the patients must undergo invasive investigations, such as EEG with intracranially implanted electrodes, but those diagnostic procedures are time-consuming and, above all, entail as many risks (like hemorrhages or strokes) as the epilepsy surgery itself. As epilepsy provoked by MCD involves a local region of the brain and as it is usually intractable with medication, surgery is up to now the only known way to treat it. However, the outcome is poorer than the one reached for TLE: only 40% of patients with MCD achieve seizure freedom [Sisodiya, 2000], compared to the 75% of patients with TLE [Engel, 1996]. Two main reasons may explain that. The first one is the impact of the quantity of removed tissue upon the outcomes. Some works [Edwards et al., 2000; Tassi et al., 2002] show that freedom from seizures after surgery operation is closely linked to the resection of the whole lesion. The issue so is the localization of the lesions in order to define precisely their limits and thus to remove them entirely without damaging the healthy neighboring cortical tissues. The second point is that anatomical abnormalities can extend beyond the visible lesion [Sisodiya, 2000] and thus its simple removal may not be sufficient to completely cure the patient. 33

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

Biological description of malformation of cortical development Malformations of cortical development are caused by abnormalities occurred during the cortical development. These anomalies can be the result of a wide variety of factors, like for example genetic mutations, in utero injuries or peri- or postnatal insults [Palmini et al., 1994], factors that can happen at any moment during the cortical development, affect any regions of the brain and cause lesions with variable shape and size. Normal cortical development is divided in three main steps after the apparition during the 7th gestational week of the germinal matrix. First, young undifferentiated neurons, the neuroblasts, proliferate within the germinal zone and undergo several divisions until they reach their final state. Then, they migrate to their final destination and when they have attained it, they differentiate, organize themselves in discrete layers and establish synaptic connections. MCD are classified in various types according to the timing and the severity of the injuries during the cortical development. Among them, three categories prevail and are closely linked to epilepsy: heterotopia, polymicrogyria and focal cortical dysplasia. Heterotopia result from an abnormal migration of the neuroblasts and are characterized by collections of normal cells in abnormal locations. In fact, all anomalies of neuronal migration can be considered as heteretopia since they are all characterized by abnormal islands of normal cells, islands that can be nodular or diffused. In the first case, they can be found either near the ventricles (periventricular heterotopia) or within the cortex (focal subcortical heterotopia) whereas in the second case, they can form a layer of gray matter beneath the cortical ribbon (“double cortex” heterotopia) [Barkovich and Kuziecky, 2000]. Clinically, patients with heterotopia suffer almost always from epileptic seizures but can also be affected by mild to severe motor disturbances or mental developmental delays, according to the location and the size of the lesion [Barkovich and Kuziecky, 2000]. On MRI, they appear as groups of tissue isointense with gray matter, on all imaging sequences and all modalities (T1, T2, FLAIR, etc.) (figures 1.3A and 1.3B). Furthermore, they do not alter intensity with contrast agents, which allows the neurologists to differentiate them from tumors. Double cortex are characterized as for them by a subcortical band of gray matter (figures 1.3C and 1.3D). In that case, the clinical repercussion on the patient are closely related to the thickness of the heterotopic band. The thicker the heterotopic layer is, the more anomalous is the cortex and thus the more serious the syndromes are [Barkovich and Kuziecky, 2000]. Polymicrogyria are caused by an abnormal cortical organization and are regularly diagnosed in the early years of life. They are characterized by multiple small gyri separated by shallow sulci, by an abnormal cortical lamination and by the fusion of the cortical molecular layers (figure 1.4). They may affect either a small cortical area or a wide region and are usually diagnosed near the central sulcus [Kuzniecky et al., 1994]. However, they can be found in other lobes too [Guerrini et al., 2000]. 34

1.2 – Focal cortical dysplasia in focal epilepsy

Figure 1.3: Heterotopia. Coronal (A) and sagittal (B) T1 images showing a periventricular heterotopia characterized by a localized set of voxels iso-intense with the gray matter. Coronal (C) and axial (D) images that show a “double cortex” heterotopia with an evident abnormal sub-cortical layer of gray matter.

The clinical syndromes depend on the size and the localization of the lesion. While they often provoke epileptic seizures, they are also involved in developmental delays or motor dysfunctions. 65% of the patients who suffer from epilepsy due to polymicrogyria are unfortunately pharmacologically intractable but their quality of life can be improved with surgical treatment, especially callosotomy (the corpus callosum is either partially or completely cut). Finally, focal cortical dysplasia (FCD) is a malformation caused by an abnormal cortical organization and aberrant neuroglial proliferations. They are discussed in details in next section. In addition to these three categories, other forms of MCD exist but they are less often involved in epilepsy. We refer the reader to [Kucniecky and Jackson, 1995a] for a complete presentation of these lesions.

35

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

To summarize, MCD are an important cause of pharmacologically intractable epilepsy. They usually arise in the early years of life and can provoke, in addition to epileptic fits, severe mental or motor disturbances. Their detection through MRI is crucial since it avoids the use of time-consuming invasive procedures and allows the neurologists to diagnose the lesions almost immediately. As a result, patients can undergo surgery and have their quality of life improved earlier. However, their visual detection is still difficult though advanced image processing seems to be promising in improving their localization.

Figure 1.4: Polymicrogyria. Axial T1-weighted image that illustrates the multiple small gyri characterizing the malformation.

1.2.3

Focal cortical dysplasia

Description of focal cortical dysplasia Let us now come back to focal cortical dysplasia. As mentioned in the previous section, FCD are malformations mainly caused by an abnormal cortical organization and a proliferation of aberrant cells. Their pathogenesis are identical to those of MCD, that is genetic mutations, in utero injuries or perinatal insults [Palmini et al., 1994]. They were identified for the first time in 1971 by [Taylor et al., 1971] in specimens resected from the brain of 10 patients with intractable partial epilepsy. At that time, they were described as focal developmental anomalies of the cortex with no evident visible and palpable characteristics, but described histologically by the presence of a focal cortical chaos. Among those patients, the lobectomy of the affected cortical region greatly improved the quality of life of nine of them and 5 became fit-free, suggesting then that the neuro-surgery could be a promising treatment. From then on, FCD was increasingly recognized as a major cause of drugrefractory focal epilepsy and, nowadays, epidemiology studies show that about 30% 36

1.2 – Focal cortical dysplasia in focal epilepsy

of patients with identified MCD suffer from FCD [Barkovich et al., 2001; Sisodiya, 2000]. However, it is still very difficult to assess their extent, indeed to detect them, because of their subtleties. Furthermore, their aspect can be totally different from one lesion to another. They include a wide range of cortical anomalies, sharing however two common patterns: an abnormal cortical lamination and the presence of aberrant and undifferentiated cells. It is the degree of these malformations and the presence of other abnormalities that makes them very difficult to discern and classify. Although they are almost indiscernible visually or by palpation, they are well defined histologically. Before presenting those characteristics, let us briefly look at the histology of a healthy cortex in order to better understand the malformations related to FCD. The cortex is made up of gray matter which consists of nerve and glial cells (appendix A). However, the density of these cells is different throughout the cortical tissue. Thanks to some stains, such as the Nissl one that can reveal the neuron’s body, it has been showed that the cortical ribbon is organized into three to six distinct layers, according to the brain area. The neocortex for example, which covers the main part of the brain and where FCD are usually found, is made up of six layers numbered from the outer part to the white matter (see figure 1.5). The layer I (molecular ) contains very few neurons and is mainly made up of neuropil, the brain tissue lying between the cell bodies. Layers II (outter granular ) and III (outter pyramidal) hold primarily connections between two cortical nerve cells (corticocortical connections). Layers IV (inner granular ) consists of stellate neurons with ramifying axons and layers V (inner pyramidal) and VI (fusiform) are made up of pyramidal neurons whose axons are destined to go out the cortical ribbon. This organization allows each layer to carry out specific tasks, according to the brain area. As an illustration, the signals sent by the thalamus are first received by the stellate nerve cells of the fourth layer of sensory regions, and then transmitted to the other layers.

Figure 1.5: Cortical lamination of the neocortex: schematic drawing (left) and histological section (right).

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1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

The most striking feature of FCD is the presence of a cortical dislamination within the lesion. This cortical chaos is always visible among the different types of FCD though its intensity can vary significantly from one lesion to another because of the various anomalies that could have occurred during the cortical organization. In addition to that, strangely oriented healthy neurons can be found in wrong places and odd unshaped cells can be scattered through the cortex. Among them, four main categories can be distinguished [Palmini et al., 2004; Tassi et al., 2002]. • The undifferentiated cells, also called immature neurons, are oval cells with a big immature nucleus lying inside a body a little bigger than it (figure 1.6a). • The dysmorphic neurons are characterized as for them by their abnormal size, shape and orientation. Their dendrites do not work as they would have to do, which can increase the epileptogenic sensibility of the area (figure 1.6b). • The giant neurons are neurons of increased size but with normal shape, often pyramidal (figure 1.6c). • Finally, the balloon cells are abnormal and often big cells with thin and illdefined membrane, a pale cytoplasm and an eccentric nucleus. They have an unknown lineage (they can show either neuronal or glial cell features) and are particuliarly present at the gray-white matter junctions (figure 1.6d).

Figure 1.6: Thionin-stained histological sections of: A- immature neurones, B- giant pyramidal neurone, C- dysmorphic neurones and D- balloon cell [Tassi et al., 2002].

Two main categories of FCD can be differentiated according to the degree of the cortical dislamination and the presence of dysmorphic neurons: FCD and Taylortype FCD [Palmini et al., 2004]. FCD, type I This category of FCD is mainly characterized by the absence of dysmorphic neurons or balloon cells. They have only isolated architectural abnormalities, that is a focal cortical dislamination associated (type IB) or not (type IA) with giant or immature neurons. Patients suffering from them can have epilepsy or not. But as they are often so subtle that they cannot be seen on MRI, it is still difficult to assess their actual 38

1.2 – Focal cortical dysplasia in focal epilepsy

involvement in the disease. Nevertheless, some patients with pharmacologically intractable epilepsy have diagnosed type I FCD. Taylor-type FCD, type II Contrary to FCD, Taylor-type FCD (TFCD), whose name is a tribute to Taylor who first identified them [Taylor et al., 1971], can hold dysmorphic neurons. They are subdivided into two categories according to the absence (type IIA) or the presence (type IIB) of balloon cells. The lesional cortex is thicker, the dislamination serious and the gray/white matter junction blurred because of the presence of abnormal cells between the nerve fibers. Throughout the cortex, dysmorphic cells and giant neurons are found too. All these abnormal entities bring an imbalance between the inhibitory and excitatory neurotransmissions. The inhibitory one decreases within the lesion meanwhile the excitatory increases, resulting thus in an amplification of the electrical sensitivity of the affected region. Drugs try to minimize this imbalance but TFCD are increasingly recognized as a major cause of pharmacologically intractable partial epilepsy. The neuro-surgery is a promising way to treat patients suffering from them but the outcomes are not always satisfactory. Indeed, even if the entire lesion is removed, some patients continue suffering from seizures. Other classifications can be found in the literature. [Tassi et al., 2002] for example differentiates three main forms of FCD, the architectural dysplasia (equivalent to FCD-IA), cytoarchitectural dysplasia (FCD-IB) and Taylor-type cortical dysplasia (TFCD). However, the great majority of the current classifications share these differentiations and only their name and organization change. In the following, Palmini’s terminology will be used, conforming to the customs of our laboratory.

Figure 1.7: Histological section of a normal neocortex, type I FCD (with a mild cortical dislamination), type IIA TFCD (significant dislamination and presence of dysmorphic neurons) and type IIB TFCD (severe dislamination and presence of dysmorphic neurons and balloon cells) [Palmini et al., 2004].

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1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

Magnetic resonance imaging Thanks to MR-imaging, FCD and TFCD can be diagnosed earlier and without invasive procedures. Their very typical characteristics can provoke significant variations on MR images according to the severity of the lesion, but unfortunately they stay undetectable in most cases despite the coming of high-resolution imaging. In some studies [Tassi et al., 2002], cryptogenic patients can represent approximately 40% of people with FCD. When an FCD is diagnosed, it is characterized by one or more of the following features: (I) focal thickening of the cortex [Chan et al., 1998; Bastos et al., 1999], due to an abnormal accumulation of neurons, (II) blurring of the gray/white matter junction [Bronen et al., 1997], due to an abnormal accumulation of neurons and the presence of abnormal cells, (III) hyper-intensity of the gray matter within the lesion and decrease of the subcortical white matter intensity on T1-weighted images [Barkovich and Kuzniecky, 1996], probably caused by a demyelination of the nerve fibers, (IV) hyper-intensity of the cortex and subcortical white matter on T2-weigthed, proton-density (PD) or fluid-attenuated inversion recovery (FLAIR) images [Bronen et al., 1997], also provoked by a demyelination of the nerve fibers, (V) Hyper-intense white matter tapering towards the ventricles caused by the extension of cortical tissue (transmantle dysplasia) [Barkovich et al., 1997].

Figure 1.8: Taylor-type FCD. A- T1 coronal image that shows the increase of the cortical thickness, the blurring of the gray-white matter junction, the hypo-intensity of the underlying white matter and the hyper-intensity of the gray-matter. B- T2 coronal image that shows the hyper-intensity within the lesion and the tapering towards the ventricles. C- FLAIR coronal image showing the same features as the T2 image.

Taylor FCD are revealed by all or almost all of these features. However, their contribution can considerably vary from one lesion to another, and within the lesion itself [Tassi et al., 2002; Colombo et al., 2003; Palmini et al., 2004]. Contrary to 40

1.3 – Diagnosis aid systems for FCD detection

these studies, all based on the analysis of experts, [Colliot et al., 2004] used image processing techniques on T1-weighted MRI to assess these variations, avoiding in that way the subjectivity inevitably introduced by a human analysis. Among their 39 patients with diagnosed FCD, they found that 84% of them had an increase of the cortical thickness, 84% an hyper-intensity of the gray-matter, and 94% a blurring of the gray-white matter junction. In addition to the differences encountered throughout the lesions, these MR features are also inhomogeneous within a single FCD. Indeed, for a given characteristic, some voxels lying inside a lesion can have normal values whereas others show evident abnormal intensities. All these variations are surely an important reason why FCD and TFCD are so difficult to detect and delineate. In the next section we shall see how new techniques based on image processing can improve their detection.

1.3

Diagnosis aid systems for FCD detection

With the increase of computer power, new image processing techniques helping the detection of FCD are developed. The first methods improved their visual detection and evaluation by using for instance computed volumetric data, more intuitive MRI visualization or texture information. But quickly some attempts at automatically detecting them were made, trying to avoid in this way the subjectivity naturally introduced by a human analysis. This section begins with a description of the main methods used to enhance the visual localization of the FCD, followed next by the presentation of two automatic detection approaches.

1.3.1

Enhancement of the visual detection

Qualitative and quantitative analyses of T1-weighted MR images The first image processing applications developed to help the localization of MCD date back the mid 1990s. In 1995, [Sisodiya et al., 1995] demonstrated that such techniques were promising in FCD detection and might greatly enhance their understanding. In that article, the authors made use of different methods to analyze the T1-weighted MRI of 55 patients and 30 controls. 45 patients had extra temporal lobe epilepsy, among whom 15 had visible lesions. They computed first the 3D rendering of the cortex in order to better assess the gyral abnormalities that might be found on its surface. Indeed, as a neurologist usually looks at the 2D sections of an MRI, he looses the 3D information. The purpose was then to demonstrate that 3D analyses could improve the detection rate. The results confirmed this intuition. While no abnormalities were found on the cortical surface of the healthy controls, three gyral abnormalities were detected in the 15 patients with visible FCD and 14 (47%) in the others, nine of whom were corroborated by EEG analyses. 41

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

Some volumetric data were computed next from the brain images. The authors divided the MRI into blocks and calculated for each block the volume distribution of gray matter and subcortical white matter. The evaluation of those values allowed them to identify non visible extra-lesional abnormalities of the volume distribution in 24 of the 29 patients with visible lesions. These results confirmed thus that image processing not only could help the localization of the lesions but also might be able to reveal non visible extra-lesional abnormalities. From then on, image processing played an increasing role in FCD studies. In 1999, [Bastos et al., 1999] published an MRI visualization tool more intuitive than the multiplanar reformatting commonly used (axial, sagittal, coronal). The main idea was already introduced by Sisodiya et al., 3D MRI evaluation can give more information than traditional 2D analyses. However, instead of computing 3D renderings of the cortex, the authors reformatted in a different way the images by slicing them curvilinearly. Therefore, the natural geometry of the brain was kept and the artifacts introduced by the rectilinear visualization avoided (the cortex can look thicker or thinner if it is not parallel to the slice, figure 1.9).

Figure 1.9: Cortical thickening artifact caused by the rectilinear slice [Bastos et al., 1999].

To compute the curvilinear reformatting from the MRI, an expert first manually delineated the contours of the hemispheres on coronal planes. He defined in that way a surface which was next used to extrapolate the curvilinear slices at different depths, slices almost always perpendicular to the gyri. Once computed, they were displayed in either 2D or 3D images (figure 1.10). This method allowed the authors to detect lesions in their five cryptogenic cases (no lesion were visible on neither 2D nor 3D multiplanar MRI but confirmed by EEG) by pointing out gyral abnormalities and more evident FCD MR features. Nowadays, this technique is often used by the neurologists when no lesion is visible on multiplanar imaging, helping thus the FCD diagnosis. 42

1.3 – Diagnosis aid systems for FCD detection

Figure 1.10: Curvilinear reformatting. A- Manual delineation on coronal planes. B- Generative surface. C- Example of curvilinear slice perpendicular to the gyri. D- 2D Representation of the surface. E- 3D rendering [Bastos et al., 1999].

Computational models of the biological characteristics of FCD So far, image processing has been used for displaying the MRI in different ways or to compute volumetric data for quantitative evaluations. In [Bernasconi et al., 2001] and [Antel et al., 2002], a new approach was developed. The main idea was to extract interesting features from T1-weighted MR images in order to emphasize imaging abnormalities that can reveal an FCD. In this way, their MRI characteristics, that are the increase of the cortical thickness, the blurring of the gray / white matter junction and the hyper-intensity of the lesional gray matter, were modeled by simple image processing operators. In [Bernasconi et al., 2001], a feature map was computed for each MRI characteristic. First, the MR images were segmented into three classes: cerebro-spinal fluid (CSF), gray matter (GM) and white matter (WM). Then, the cortical thickening was evaluated through a run-length method, encoding with high values regions with increased thickness. Next, the blurring of the GM/WM transition was modeled by the absolute gradient of gray level intensities. Consequently, the blurring of the GM/WM junction was emphasized by a decrease of the gradient intensity. As for the hyper-intensity of the lesional gray matter, it was modeled by a relative intensity map. Let Bg be the gray level intensity at the boundary between GM and WM (obtained through an automatic histogram-based method). The relative intensity of a voxel g lying within the GM was then equal to 100 ∗ (1 − |Bg − g|/Bg ). In this way, the more g was intense, that is close to Bg , the higher the value of 43

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

its relative intensity index was. Finally, to help the analysis of those three different feature maps, a composite image was computed according to the expression (GM thickness ∗ Relative intensity)/Gradient map. This formula was justified by the fact that FCD is characterized by high values of cortical thickness and relative intensity, and small values of gradient amplitude (figure 1.11).

Figure 1.11: Example of ratio map. A- Axial T1 image, the FCD is indicated by the arrow. B-

Ratio map: the region with high values emphasizes the dysplastic lesion. C- Gradient map: the FCD is characterized by a decrease of the gradient. D- Cortical thickness map where high values point out the increase of the cortical thickness within the FCD. E- Relative intensity modeling the hyper-intensity of the lesional gray-matter.

To evaluate the method, the MRI and the ratio maps of 16 patients and 20 controls were presented independently to two trained experts. A lesion was considered as detected if it was localized by both experts simultaneously. Eight patients had visible FCD on MRI and no lesion was found in controls. By using the ratio maps, 14 FCD were detected (87.5%) and one false positive cluster was identified in one control, confirming thus that the ratio maps could considerably increase the FCD detection rate. However, this first paper suffered from a non-trivial problem: the poor contrast of the composite images. This had led to the detection of a false positive in one control, 44

1.3 – Diagnosis aid systems for FCD detection

an healthy region was pointed out as lesional. This limitation was mainly caused by the cortical thickness model which did not evaluate the gray matter thickness with enough accuracy. The run-length approach might have created artifacts, especially when the cortical ribbon was aligned along a particular search direction. [Antel et al., 2002] solved this issue by using a more advanced algorithm. The authors estimated the cortical thickness by modeling the cortex as an electrostatic field limited by the CSF/GM and GM/WM junctions. Then, Laplace’s equation was used to compute the thickness at each voxel lying within the cortical ribbon [Jones et al., 2000]. The computation of these feature maps will be explained with more details later in this thesis. This approach avoided the shortcoming of the previous paper and significantly improved the contrast of the ratio maps. The detection rate increased while the false positives decreased, confirming thus that the technique was promising. Nowadays, the composite maps constitute a tool used routinely by our group for the clinical evaluation of FCD lesions in new cases.

1.3.2

Automated detection

Recently, image analysis techniques have been developed to automatically detect FCD on T1-weighted MR images, relying on different voxel-wise analyses. Some methods apply the voxel-based morphometry [Ashburner and Friston, 2000] to FCD detection, while others make use of the computational models previously described to automatically classify the various brain tissues. They will be shortly presented in this section. We refer the reader to the mentioned papers for further information. Voxel-based morphometry applied to FCD detection Voxel-based morphometry (VBM) is a general approach whose aim is to compare voxel by voxel the concentration of gray matter (or of another tissue altogether) between separate groups of subjects. In that way, this technique can detect local volumetric abnormalities of the studied gray matter structure. The basic idea consists in normalizing every subject of separate groups into a stereotaxic space and to segment the resulting images into two classes corresponding to the gray matter and the white matter. The spatial normalization step is crucial since it corrects the differences in shape and orientation between the various brains and allows thus their direct comparison. A statistical analysis is then performed between the groups to reveal volumetric abnormalities of the brain tissues. Though this approach was especially dedicated to group studies, [Kassubek et al., 2002] first, and [Wilke et al., 2003] next, applied it to automatically detect FCD by comparing patients with sets of normal subjects. In [Kassubek et al., 2002], seven patients were compared to 30 healthy controls. Each of the 37 T1-weighted MR images was normalized into a stereotaxic space. Next, the gray matter was automatically segmented and smoothed by using a Gaussian kernel. Each voxels of 45

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

the resulting image, called “gray matter density” map, represented then the average concentration of neighboring gray matter. At last, an average density map and standard deviation map were computed from the 30 healthy controls. In order to detect the lesions, the density map of each patient was subtracted to the average density map. The resulting image was then compared to the standard deviation map: the values lower than one standard deviation were set to zero while the others were squared. Therefore, the abnormalities of the gray matter volume distribution were emphasized by high values. Local variations of the cortical thickness were revealed and regions likely to be lesional pointed out (figure 1.12).

Figure 1.12: Voxel-based morphometry. A- Brain MRI in stereotaxic space. B- Density map.

C- Difference map. D- Difference map with the most significant abnormalities emphasized [Kassubek et al., 2002].

The detection rate reached by this method was promising. Among the seven patients, six were correctly detected. The seventh had local maxima within the lesion but the global maximum of its density map was not consistent with the visual analysis. This method was further evaluated in [Wilke et al., 2003], from the same laboratory, where the VBM procedure was slightly enhanced and all the parameters of the algorithm were iteratively weighed up in order to get the best possible results. This study confirmed the good detection rate of the VBM (16 of the 20 patients were successfully detected) and proposed a general procedure to detect other volumetric abnormalities of brain structures. However, this method has two main drawbacks: the poor coverage of the dysplastic tissue and the high number of false positives. On the one hand, only a small part of the lesion can be detected since it emphasizes just the abnormal increases of the gray matter volume. On the other, the presence of false positives could be confusing when detecting FCD. In fact, while VBM correctly detects FCD lesions, other local maxima may be found too, making the choice of the good cluster difficult. A solution can be to increase the threshold used for filtering the results, in order to get the most significant maxima only. But while some local maxima are lower than the ones pointing out the lesional tissues, others can be of the same order, indeed 46

1.3 – Diagnosis aid systems for FCD detection

higher. It is thus impossible to discard them. Even so, this method is nowadays of a great help to the neurologists and increasingly used to detect FCD lesion and other cortical abnormalities, such as hippocampal atrophy. Automated detection of FCD using MRI feature maps While VBM relies only on the volumetric variations of the cortical gray matter, [Antel et al., 2003] proposed a novel technique to automatically detect FCD on T1weighted images. The suggested method is based on the feature maps previously developed in [Bernasconi et al., 2001; Antel et al., 2002] and on second order texture maps. These maps are then used by two different Bayesian classifiers to detect FCD on 18 patients (11 with visible lesion) and 20 healthy controls. The MRIs were first pre-processed. The brains were spatially normalized into a stereotaxic space, the non-uniformity of the intensity caused by the MR acquisition was corrected and the brain extracted. The images were then segmented into seven classes: background, cerebrospinal fluid (CSF), gray matter (GM), white matter (WM), CSF/GM transition, GM/WM transition and FCD lesion. The GM, WM and CSF classes were computed through an automated histogram-based algorithm. The transition classes were obtained from this preliminary segmentation by estimating, voxel by voxel, the partial volumes of each tissue lying inside a window of fixed size centered on each voxel. Finally, the lesion class was manually painted by a trained expert. (figure 1.13B) After these pre-processing steps, the three computational models proposed by [Antel et al., 2002] (gradient, thickness and relative intensity maps, figure 1.11CDE) were calculated. Next, three second order texture maps (contrast, angular momentum and difference entropy, figure 1.13DEF) were computed by using co-occurrence matrices of fixed size. The purpose of those texture maps was to extract non visible MRI characteristics that could reveal the histological chaos caused by FCD. The classification of a patient was then composed of two steps. First, the subject was classified by the primary classifier trained on the computational models, resulting in a seven-class segmented image. Next, the voxels classified as lesional by the first classifier were reclassified by the second one to distinguish the lesion from the gray matter according to the texture information. The final output was then a cluster map where each cluster pointed out a region likely to be lesional. Finally, a simple post-processing step was performed to remove the noisy small clusters (figure 1.13C). This algorithm was able to detect 15 of 18 patients (83%) in comparison with the 11 subjects with visible lesions. No lesion was found in controls. Nevertheless, the lesion coverage was still poor (16%) because of the ill-defined borders of FCD and the significant heterogeneity of the MRI features within the lesions themselves. This method is thus not suitable for FCD segmentation though it constitutes a viable way to detect them owing to its substantial detection rate. 47

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

Figure 1.13: Example of texture maps and automated detection computed using the FCD classifier [Antel et al., 2003]. Upper panels: A- T1-weighted MRI, the arrow indicates the FCD; B- 7-class segmented map; C- automated detection computed using the FCD classifiers. Lower panels, the texture maps: D- angular second momentum; E- contrast; F- difference entropy.

1.4

Open issues

Focal cortical dysplasia, a particular form of malformations of cortical development, is increasingly recognized as a major cause of drug-refractory focal epilepsy. Nowadays, surgery is the only way to treat the patients by removing the lesional tissue. However, the outcome closely depends on the extent of the resected area. Indeed, the entire lesion must be removed without damaging any neighboring healthy tissue. The correct delineation of the lesions is thus a crucial issue for the surgical planning. Unfortunately, the visual evaluation of the actual dimensions of the FCD is very difficult because of their subtle characteristics. Although clinical observations and EEG analyses help their diagnosis and localization, they are not suitable for the estimation of their extent. MRI, as for it, greatly improves their detection and is nowadays more and more used in their in vivo evaluation. On T1-weighted MR images, FCD are characterized by an increase of the cortical thickness, a blurred 48

1.4 – Open issues

gray/white matter junction and a hyper-intensity of the lesional gray matter. However, these features are often so heterogeneous and subtle that the lesions are seldom visible to the naked eye. Furthermore, when they are observable, the objective evaluation of their extent is challenging because of their ill-defined and blurred boundaries. Some advanced image processing techniques attempt to decrease the inescapable variability throughout the human observations by introducing objective and more evident criteria. They manage to increase considerably the detection rate but in spite of everything, the assessment of the exact extent of FCD and their segmentation stay an open issue. Such a tool can be very useful for surgical planning and quantitative studies by automatically delineating the lesions or, at least, by guiding the experts in the evaluation of their actual boundaries. To our knowledge, the work described in this thesis is the first attempt at designing, implementing and validating advanced segmentation technique that may reach these purposes.

49

1 – Epilepsy, focal cortical dysplasia and magnetic resonance imaging

50

Chapter 2 Image segmentation using geometric deformable models Introduction Image segmentation is increasingly applied in medical imaging studies, especially in brain imaging. While it allows the researchers to quantitatively study the anatomy of brain structures in healthy and pathological cases, it begins to be routinely used in clinical applications such as diagnosis, treatment planning and computer-aided surgery. However, segmenting MR images, and medical images in general, is still a challenging task because of the inescapable variations in image quality and in object shape, dimension and orientation. On the one hand, the signal-noise ratio and the contrast can vary tremendously between two separate MRI machines, even if same acquisition protocols are applied. On the other hand, a same brain structure can have different characteristics according to the subjects, especially when pathological cases are considered. Added to that, medical images are often noisy and corrupted by artifacts, making thus the automated segmentation complex. A multitude of segmentation techniques can be found in literature but few are really suitable for MR images because of their poor signal noise ratio and the presence of artifacts. Advanced histogram-based methods and automated classifications are commonly used. However, though they can segment efficiently the main brain tissues for instance, they are not accurate enough to segment ill-defined structures or lesional tissues. Because of their relevant properties, deformable models, also called snakes, active contours or surfaces, etc., belongs to the advanced image processing tools which are increasingly applied in medical imaging. Appeared in the work by Kass, Witkin, and Terzopoulos [Kass et al., 1987], they are curves or surfaces that evolve within the image space, according to different forces. These forces drive the contours towards the boundaries of the desired object and control their smoothness. The results are 51

2 – Image segmentation using geometric deformable models

thus automatically regularized and, by constructing forces based on advanced knowledge, objects with ill-defined boundaries can be successfully segmented. Moreover, they can achieve sub-voxel accuracy since they are implemented on the continuum. There are two main types of deformable models: parametric deformable models [Kass et al., 1987] and geometric deformable models [Caselles et al., 1993; Malladi et al., 1995]. Parametric deformable models, also called snakes, use a parametric representation of the contour to deform it. In this way, the evolution forces act on the model directly. The convergence is then very fast and the method is suitable for real-time applications. Nevertheless, because of the parameterization, the curves cannot split or merge naturally. Geometric deformable models rely on the curve evolution theory and on the level set method [Sethian, 1999a]. In this approach, the contours are defined as the zero-level set of a higher-dimensional scalar function. The forces act on this new function and the contour is extracted at the end of the evolution. Therefore, no direct parameterization is needed and topology changes are naturally handled. We chose to make use of the 3D geometric deformable models to segment focal cortical dysplasia (FCD). As it was previously described, FCD are characterized on MRI by an unshaped increase of the cortical thickness, a blurred junction between gray and white matter, and a heterogeneous increase of the gray level intensity within the dysplastic tissue (chapter 1.2.3). Such characteristics avoid thus the use of traditional voxel-wise approaches to successfully segment them, and a region dependent technique that automatically regularizes the result is needed. 3D geometric deformable models appear to be appropriate since they can make a closed surface evolve smoothly and use regional information to detect object boundaries. This chapter is organized as follows. In section 1, parametric deformable models are briefly introduced in order to set out the concepts of active contours and evolution forces. Then, section 2 provides the theorical background of geometric deformable models. The theory of curve evolution and the level set method are described first, followed next by a comparison between geometric and parametric deformable models. Finally, the section 3 is devoted to the outline of some applications of the level set method in image processing.

2.1

Brief overview of parametric deformable models

In this section, the underlying principles of the parametric deformable models are shortly presented and discussed in the perspective of our purposes. For the sake of clarity, only the two-dimensional formulas are set out here as the three-dimensional formulations can be very complicated. We refer the reader to [Kass et al., 1987; Cohen, 1991; Xu et al., 2000] for further information. 52

2.1 – Brief overview of parametric deformable models

2.1.1

Mathematical formulation of the curve evolution

In the original work by [Kass et al., 1987], parametric deformable models evolve in order to minimize an energy defined along them. This energy derives from the image and is usually made up of two different terms: the internal energy and the potential energy. The internal energy makes the contours smooth while the potential energy drives them towards minimum values, localized by construction on the image edges. In that way, minimizing the energy along the deformable models amounts to minimize both internal and potential energy: the resulting contours are the smoothest curves coinciding with the boundaries of the desired objects. Let X(s) = (X(s),Y (s)), s ∈ [0,1], be the parametric deformable model, s being the parameter. The energy minimizing formulation can be written as E(X) = Eint (X) + Epot (X) The internal energy Eint is defined as    2 2    ∂X 2  1 1  + β(s)  ∂ X  ds α(s)  Eint (X) =  ∂s2  2 0 ∂s 

(2.1)

(2.2)

The first derivative is a tension term and prevents the curve from stretching. The second derivative corresponds to a rigidity term; it discourages the deformable model from bending. α(s) and β(s) are weighting parameters which are often constant. The potential energy Epot is computed by integrating a potential energy function P (x,y) along the model X(s) 1 P (X(s))ds (2.3) Epot (X) = 0

By definition, the potential energy function P (x,y) must be minimum on the desired object boundaries. It is often computed from the gradient of the image. It has been shown that the curve minimizing E(X) is a solution to the following Euler-Lagrange equation : Fint (X) + Fpot (X) = 0 where Fint can be understood as an internal force and is defined as follows     ∂ ∂2X ∂X ∂2 Fint (X) = α(s) − 2 β(s) 2 ∂s ∂s ∂s ∂s

(2.4)

(2.5)

and Fpot as an external force deriving from a potential energy Fpot (X) = −∇P (X) 53

(2.6)

2 – Image segmentation using geometric deformable models

To solve equation 2.4, the active contour is made dynamic, derived with respect to the time t and set equal to the right side of this equation, that is to say γ

∂X = Fint (X) + Fpot (X), ∂t

(2.7)

where γ is a constant (its purpose is to make the units of both sides consistent). When equation 2.7 reaches the equilibrium, the left side is equal to zero and thus a solution to the equation 2.4 is achieved. In that way, to segment a desired object, a deformable model is initialized in the image space and evolves according to the dynamic equation 2.7. It is worth noting that parametric deformable models can be expressed straight through equation 2.7, without using the energy expressions. This dynamic formulation keeps the internal force unchanged but allows the use of external forces which do not necessary derive from potential energies.

2.1.2

External forces

The external force plays a crucial role in the contour evolution. It drives the deformable model towards the target object and therefore the resulting segmentation closely depends on its design. This chapter shortly describes the main external forces that can be encountered in the literature, forces that can be applied to both active contours (2D) and deformable surfaces (3D). Gradient force Introduced for the first time by [Kass et al., 1987], the gradient force is an external force which derives from a potential energy function (equation 2.6) defined as P (x,y) = −∇[Gσ (x,y) ⊗ f (x,y)]2

(2.8)

Gσ (x,y) is a two-dimensional Gaussian function with standard deviation σ, ∇ is a gradient operator and ⊗ is the 2D image convolution operator. Such a potential energy is thus minimized on the desired object boundaries, characterized by high gradient values. The parameter σ controls the extent of the attracting range of the gradient force. Indeed, the bigger σ is, the larger the attraction range is. However, large values of σ involve a substantial blurring of the object edges, which can significantly shift them. The segmentation is thus less accurate. Usually, σ is small and the contour is initialized near the desired object boundaries. 54

2.1 – Brief overview of parametric deformable models

Pressure force In order to increase the attraction range without shifting the object edges, [Cohen, 1991] added to the gradient force a pressure force that drives the deformable models in a direction normal to itself. This pressure force, also called “balloon”, is defined as follows Fp (X) = wp N(X) (2.9) where N(X) is the inward unit normal of the curve at point X. When wp is positive, the contour shrinks whereas when it is negative, it grows. As a result, the deformable model can evolve even if the gradient force is too weak to guide it and does not require an initialization close to the object boundaries anymore. Nevertheless, the choice of wp may constitute a substantial drawback. If wp is too high, the model can go through the desired object boundaries, whereas if it is too low, the active contour may stop on noisy edges. Gradient Vector Flow Another approach to solve the issue of the poor attraction range was introduced by [Xu and Prince, 1998b]. Contrary to Cohen, who added a pressure constraint to the gradient force, they chose to diffuse the gradient information in regions distant from the object boundaries. In [Xu and Prince, 1998a], a generalization of this new force, called Gradient Vector Flow (GVF), was described. It allows the convergence towards narrow boundary concavities and towards the middle of thick borders. Let f be an edge map computed from the original image through some gradient operator. GVF is then defined as the equilibrium solution to the following equation:  ∂v = g(∇f )∇2v − h(∇f )(v − ∇v) ∂t (2.10) v(x,y,0) = ∇f (x,y) where ∇2 is the Laplacian operator. Let us note that this equation can easily be extended to higher dimensional cases. Usually, the functions g(r) and h(r) are defined to be  r2 g(r) = e− κ (2.11) h(r) = 1 − g(r) where κ is a scalar, or

 g(r) = µ h(r) = r 2

(2.12)

where µ is a positive scalar. This last case corresponds to the first gradient vector flow proposed in [Xu and Prince, 1998b]. GVF has been increasingly used in medical imaging as a result of its substantial attraction range and its accurate boundary detection. 55

2 – Image segmentation using geometric deformable models

Interactive force As they are fast running, parametric deformable models can be used in real-time applications. For that purpose, interactive external forces have been developed, allowing the users to interact with the active contours and to guide it towards the desired boundaries. [Kass et al., 1987] proposed two separate interactive forces which attract or drive away the curves from user-defined points. The first one, called spring force, pulls the model towards the user-defined point p. It is defined as Fs = ws (p − X)

(2.13)

where X is the closest point on the model to p. The second one, called volcano force, drives a point X on the model away from p. This force is computed in a neighborhood N(p) only, through the following equation:  Fv (X) =

2.1.3

p−S wv p−S X ∈ N(p) 3 0 X∈ / N(p)

(2.14)

Discussion

Since their development in the mid 1980s, parametric deformable models have been increasingly used in medical imaging. The resulting contours or surfaces are always closed and regularized, they can achieve sub-voxel accuracy and, above all, the segmentation can reach a significant precision, providing that the underlying forces are wisely designed. Moreover, this algorithm is very fast, allowing thus real-time interactive applications. However, some drawbacks stop us from applying them in FCD segmentation. First, parametric deformable models are very sensitive to the initialization. They must be initialized near the desired object boundaries. When advanced external forces are used, a fine-tuning of the parameters is often required to prevent the model from detecting undesired edges or, on the contrary, from going through the desired boundaries. Next, the initial active contour and the target must have similar shapes in order to avoid complex and time-consuming re-parameterization procedures. Finally, they do not handle topological changes naturally. As FCD lesions have various sizes and no typical shape, a segmentation technique robust against initialization is required. Furthermore, the management of topology changes might be useful if an initialization from several primary surfaces is considered. Such initial surfaces indeed might be given by automated procedures and then merged during the evolution of the deformable models. Parametric deformable models are thus not really appropriate to our purpose. Geometric deformable models solve these drawbacks while keeping the advantages. 56

2.2 – Theorical background of geometric deformable models

2.2

Theorical background of geometric deformable models

Geometric deformable models were developed simultaneously by [Caselles et al., 1993] and [Malladi et al., 1995]. They rely on the curve evolution theory and on the level set approach [Osher and Sethian, 1988; Sethian, 1999a]. Their fundamental advantage is their implicit formulation. Indeed, as the models evolve using geometric measures only, they are independent of the parameterization. In that way, they can be represented implicitly as a level set of a higher-dimensional function. Like in parametric deformable models, geometric active contours or surfaces are under the influence of external forces which drive them towards the desired object boundaries. However, because of the implicit formulation, those forces act on the embedding function rather than on the model itself. Therefore, the evolution algorithm makes this function evolve while the model is deformed indirectly, splitting and merging automatically. This section is organized as follows. First, the basics of curve evolution theory are presented. Next, the level set method is introduced and the mathematical expressions related to this approach are defined. Finally, geometric deformable models are compared with parametric active contours. Let us note that all the mathematical expressions are presented here in 2D. Nevertheless, they can be extended to 3D easily. At last, the following presentation relies on the book [Sethian, 1999a] and on [Xu et al., 2000]. We refer the reader to those papers for further information.

2.2.1

Theory of curve evolution

The curve evolution theory studies the deformation of curves using geometric measures only (curvature, unit normal, etc.). Let γ(t) be a curve evolving in time t. Let X(s,t), s ∈ [0,S] a parameterization of γ(t), N(s,t) the inward unit normal and κ(s,t) the parameterization of the curvature (figure 2.1). The evolution of γ(t) along its normal direction is given by ∂X = F (κ)N, ∂t

(2.15)

where F (κ), called speed function, modulates the strength of the curve evolution. This expression can implement any curve motion. Let us consider a curve evolution along an arbitrary direction M(s,t). If we write the vector M as the sum of normal and tangential component M = λ1 N + λ2 T, then it can be shown that the tangential component T affects the curve’s parameterization only. As a result, the front shape and geometry stay unchanged (under tangential forces, a circle stays a circle for example) and the motion can still be expressed by equation 2.15. The most common curve deformations are certainly the curvature deformation and the constant deformation. The curvature deformation makes the model evolve 57

2 – Image segmentation using geometric deformable models

Figure 2.1: Deformable model. Parameterization and notation. according to its curvature. The motion equation can be written as ∂X = ακN, ∂t

(2.16)

where α is a positive scalar. It has been shown that under this deformation, γ stays C ∞ : it tends to smooth and no singularity appears during evolution. The curvature plays thus the same role as the internal force does in parametric deformable models. Constant deformation as for it is defined as ∂X = F0 N, ∂t

(2.17)

where F0 is a constant. If F0 > 0, then the curve shrinks, whereas if F0 < 0, it grows. Constant deformation can be understood as balloon force in parametric deformable models. However, while the curvature deformation tends to regularize the curve (figure 2.2a), the constant deformation can create singularities such as corners (figure 2.2b). The resolution of equation 2.17, or equation 2.15 in general, requires then careful attention. In those cases, weak solutions under an entropy condition are often used to solve those equations. A weak solution of a differential equation is a solution which satisfies the integral form of this equation. On the other hand, the entropy condition ensures that no information is created during the deformation. To understand this concept, let us consider the front γ as an interface separating two regions. The curve propagation is then equivalent to wave propagation through Huygen’s principle. The solution of the equation always exists, even beyond the formation of singularities, and is calculated through the “first arrival” principle: it corresponds to the envelope of the imaginary tiny wave fronts emanated from each point on γ (figure 2.2c). Then, if the front is considered as a source for propagating flame, each point burnt by γ stays burnt once and for all, and the evolution cannot go backward: no information 58

2.2 – Theorical background of geometric deformable models

(a)

(b)

(c)

Figure 2.2: Curve evolution. (a) Curvature deformation. The front stays smooth. (b) Constant deformation. A corner appears during the evolution. (c) Curve propagation through Huygen’s principle [Sethian, 1999a].

can be created during the evolution. This hypothesis plays a crucial role in designing the numerical schemes used for solving the motion equations. A comprehensive treatment of entropy condition and weak solutions can be found in the book by [Sethian, 1999a], chapters 2 and 3. The speed function plays a crucial role in image segmentation using geometric deformable models. It allows the active contour to evolve and to stop on the boundaries of the object to segment, such as the external force in parametric deformable models. Moreover, it usually includes a curvature deformation which will make the results smooth. The design of the speed function is thus critical for image segmentation. Section 2.3 outlines some examples of speed functions used in image processing.

2.2.2

The level set method

Introduced by [Osher and Sethian, 1988; Sethian, 1989] and applied to image processing by [Caselles et al., 1993; Malladi et al., 1995], level set method allows the efficient computation of curve deformations. The basic idea is to embed the active contour in a higher-dimensional function. In that way, the one-dimensional geometric deformable model γ is defined as the zero level set of a two-dimensional scalar function, called implicit function. Then, the curve motion is obtained by deforming the implicit function instead of moving the curve itself. Although a dimension is added to the problem, which increases the computational complexity, this formulation has the useful property that the implicit function stays valid during the evolution, independently of the topology of the embedded curve. The front can thus split and merge automatically (figure 2.3). The derivation from the evolution equation 2.15 to the level set expression can be performed as follows. Let X(s,t) be the position vector of γ and φ(x,y,t) the 59

2 – Image segmentation using geometric deformable models

(a)

(b)

(c)

(d)

Figure 2.3: Geometric deformable model and level set method. (a) The initial front. (b) The implicit function with the front highlighted in black. (c-d) The front gradually splits into two separate curves. The implicit function stays single-valued [Xu et al., 2000].

implicit function. X(s,t) is the zero level set of φ(x,y,t), that is φ[X(s,t)] = 0.

(2.18)

By differentiating this equation with respect to time t and by applying the chain rule, we obtain ∂φ ∂X + ∇φ · = 0, (2.19) ∂t ∂t where ∇φ is the gradient of φ. If φ is defined to be negative inside the zero level set and positive outside, then the inward unit normal to the curve can be written as ∇φ |∇φ|

(2.20)

φxx φ2y − 2φx φy φxy + φyy φ2x ∇φ = . |∇φ| (φ2x + φ2y )3/2

(2.21)

N=− and the curvature as κ=∇·

Equation 2.15 can then be rewritten as follows ∂X = F (κ)N ∂t ∂X = F (κ)(−∇φ · N) −∇φ · ∂t Using equations 2.19 and 2.20, we obtain    ∂φ ∇φ = F (κ) −∇φ · − ∂t |∇φ| 60

2.2 – Theorical background of geometric deformable models

that is

∂φ = F (κ)|∇φ|. (2.22) ∂t Equation 2.22 is the level set formulation of the evolution equation 2.15. The implicit function, and thus the active contour, will be deformed according to this equation.

Next, an initial implicit function φ(x,y,t = 0) must be computed such that the initial active contour γ corresponds to its zero level set. Usually, φ(x,y,t = 0) is chosen to be equal to the signed distance function D(x,y) computed from γ. Several approaches calculating D(x,y) exist, like for example the Chamfer distance transform. Nevertheless, they are for the most part time-consuming. An elegant solution to this issue was developed by Sethian and Malladi who proposed an algorithm based on the Fast Marching Method to construct the signed distance function in O(N log N), where N is the number of pixels [Sethian, 1999a]. In conclusion, this method has several advantages over parametric deformable models. • Level set formulation can be easily extended to higher dimensions. An active surface for example can be defined as the zero level set of an implicit volume. • The topology changes are naturally handled. The embedded curve can split and merge without altering the implicit function, which remains single-valued during the evolution. • Entropy-satisfying weak solutions can be found in order to continue propagating the solution beyond the formation of singularities. Numerical resolution can be accurately performed through numerical schemes derived from the hyperbolic conservation laws [Sethian, 1999a]. • Geometric measures (unit normal, curvature, curve length, area, etc.) can be computed straight from the implicit function. It is worth noting that the addition of a dimension increases inevitably the computational complexity. As a result, this method is not naturally suitable for real-time applications. Nevertheless, some advanced algorithms, such as the narrow band method, improve significantly its computational efficiency, allowing thus its application to 3D medical imaging.

2.2.3

From parametric to geometric deformable models

In the founding paper written by [Caselles et al., 1995], a relationship between parametric and geometric active contours was brought to light through an energy minimizing principle. Indeed, the authors showed that geometric deformable models (with speed functions deriving from potential forces) are equivalent to parametric deformable models without rigidity term in the internal force. 61

2 – Image segmentation using geometric deformable models

Starting from those results, [Xu et al., 2000] got a stronger relationship between both methods by using a dynamic force formulation of the parametric deformable model. The speed functions did not necessarily derive from potential forces anymore. Let us write the parametric snake as follows ∂X ∂2X (2.23) = α 2 + Fp (X) + Fext (X). ∂t ∂s The first term of the right side of the equation corresponds to the tension term, Fp = wp N is Cohen’s pressure force and Fext represents the other external forces. The pressure force has been separated from the other external forces because it can create singularities during evolution and thus requires entropy-satisfying numerical schemes. As it has been said previously, the tangential motion affects the curve parameterization only. Thus, equation 2.23 can be rewritten by considering only the normal components of the external forces γ

∂X ∂2X (2.24) = α 2 + Fp (X) + Fext (X) · N, ∂t ∂s where N is the inward unit normal. Moreover, if κ is the curvature, we have the following relationship between the tension term and the curvature γ

∂2X = κN. ∂s2 Therefore, by using this correspondence, equation 2.23 can be rewritten as ∂X = (κ + Fp
+ F
ext · N)N, ∂t with  = α/γ, Fp
= wp /γ, and F
ext = Fext /γ.

(2.25)

(2.26)

Finally, if we let F (κ) = κ + Fp
+ F
ext · N and replace N according to equation 2.20, then we get the level set formulation equivalent to equation 2.23, that is ∂φ = F (κ)|∇φ| = (κ + Fp )|∇φ| − Fext · ∇φ. ∂t

(2.27)

It has been shown so that parametric active contours without rigidity term are equivalent to geometric deformable models. Therefore, in most applications, both techniques can reach the same results. However, while parametric deformable models are fast-running and allow interactive real-time applications, geometric deformable models handle topology changes automatically. Moreover, the latter is less sensitive against initialization. At last, advanced algorithms improve the efficiency of the level set method, solving thus the computational complexity issue. All those reasons have led us to use geometric deformable models to segment focal cortical dysplasia. By way of introduction to our model, some applications in image processing are briefly outlined in next section. 62

2.3 – Image segmentation using geometric active contours

2.3 2.3.1

Image segmentation using geometric active contours Shape modeling with geometric deformable models

The first image-processing application of the geometric deformable models and level set method was developed by [Caselles et al., 1993] and [Malladi et al., 1995] independently. It was based on the following motion equation ∂φ = g(c + κ)|∇φ|, ∂t

(2.28)

where φ is the implicit function (negative inside the contour, positive outside), c a scalar, κ the curvature and g a stopping function defined as follows g(x,y) =

1 . 1 + |∇(Gσ ⊗ I(x,y))|p

(2.29)

p is equal to 1 or 2, and Gσ ⊗ I is the 2D convolution of the image I with a Gaussian kernel whose variance is equal to σ. The term c represents the constant deformation. The curve shrinks if c is positive or grows if c is negative. k is the curvature deformation which smoothes the front. The purpose of g is to stop the deformable model on the object boundaries. Indeed, it has values close to zero in regions with high gradient values (corresponding to the image edges), whereas it is strictly positive in regions with poor gradient. Of course, g can be any other function, provided that it is positive, strictly decreasing and verifies g(r) → 0 as r → ∞. Moreover, other edge detectors can be used so long as they give high values on the object boundaries and low values elsewhere. An alternative stopping function can be for example g(x,y) = e−|∇(Gσ ⊗I(x,y))| . This model is particularly adapted to images with good contrast, since the gradient is an efficient edge detector in those cases. It is thus an effective measure for stopping the deformable model. Nevertheless, the active contour can leak out if the object boundaries are too weak or have gaps. Moreover, it cannot go backward because of the constant deformation. In order to solve these issues, [Caselles et al., 1995] considered the curve evolution as an energy minimizing problem. The purpose was then to find the shortest curve according to a metric computed from the image information. Owing to this strategy, he demonstrated a strong link between parametric deformable models without rigidity term and his formulation, called geodesic active contour, which is defined as follows ∂φ = g(c + κ)|∇φ| + ∇g · ∇φ. (2.30) ∂t The first term of the right side of this equation corresponds to the previous model (equation 2.28). The second term as for it attracts the model towards the object 63

2 – Image segmentation using geometric deformable models

edges, even if it passes through them. The contour follows the vector ∇g which is always directed towards the object boundaries (figure 2.4). Furthermore, this new attraction force makes the constant deformation c dispensable since it drives the model towards the object boundaries. The active contour does not require a constant propagation anymore. In fact, this attraction force is equivalent to the gradient force in parametric deformable models. Nevertheless, it has only a small attraction range (defined by the blurring term Gσ ) and does not stop the curve evolution when there are gaps on the boundaries. Figures 2.5 and 2.6 show two examples of geodesic active contours.

Figure 2.4: Illustration of the attraction force in 1D. (a) Original edge I. (b) Smoothed edge ˆI = Gσ ⊗ I. (c) Stopping function g. The arrows indicate the attraction range created by the term ∇g [Caselles et al., 1995]. Since its publication, this model has been successfully applied to various image segmentation problems. A 3D extension has been developed [Malladi et al., 1996; Caselles et al., 1997], allowing its use in medical imaging. The purpose is then to find minimal surfaces. However, contrary to the 2D case, several 3D curvatures can be computed (mean curvature, Gaussian curvature, etc.). Mean curvature is usually chosen but models using Gaussian curvature can be found in literature (Gaussian curvature is more accurate and avoids the formation of singularities on non-convex surfaces, contrary to mean curvature)[Yezzi et al., 1997]. This method has been adapted to vector-valued images too [Sapiro, 1997]. Called color snake, this extension relies on the motion equation developed by Caselles et al. (equation 2.30) but makes use of vector-valued operators to compute the stopping function. In that way, color images or multimodal images can be segmented through this approach. 64

2.3 – Image segmentation using geometric active contours

In conclusion, geodesic active contours rely on boundary information only. The model grows or expands, according to the sign of the constant deformation c, and stops on the desired object edges. 3D and multi-valued images can also be segmented using this technique. Nevertheless, they are not suitable for segmenting objects with ill-defined edges. Indeed, as the active contour (or surface) evolves in a constant direction, it can leak out the object through the weak boundaries, without being able to go backward. An additional force is added in [Caselles et al., 1995] but the attraction range closely depends on the image smoothing. A compromise must be found so, between the blurring and the shifting of the edges it introduces. Furthermore, this model is not able to fill the possible gaps on the object boundaries. At last, the initial active contour must be either completely inside or completely outside of the target object because of the constant deformation.

Figure 2.5: 3D segmentation of two tori. This example illustrates the ability of the level set method to handle topology changes [Caselles et al., 1997].

Figure 2.6: Segmentation of cyst form ultrasound breast image using level set method [Yezzi et al., 1997].

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2.3.2

Region-based deformable models

Recently, new speed functions have been developed to solve the issue of the weak boundaries. The basic idea is to introduce other image information in order to stop the active contour on the desired object edges. A common approach is to use, in addition to (or instead of) boundary information, some region-based knowledge. This knowledge can be based on the gray level intensities [Chan and Vese, 2001; Yezzi et al., 1999] or obtained through probabilities modeling discriminating features of the target object [Paragios and Deriche, 2002a;b]. This section outlines briefly two examples illustrating the two approaches. Segmenting contours without explicit edges In [Chan and Vese, 2001], a new method is developed in order to detect objects without explicit edges. It relies on an energy formulation which tends to segment the image into two homogeneous regions. Let u0 be an image and C an active contour lying within it. C divides u0 into two regions: inside and outside the curve, denoted respectively as inside(C) and outside(C). Let c1 and c2 be the average gray level inside and respectively outside the contour, and let us define the following “fitting” term: |u0 (x,y) − c1 |2 dx dy F1 (C) + F2 (C) = insice(C) (2.31) 2 + |u0 (x,y) − c2 | dx dy. outside(C)

Minimizing this fitting term is equivalent to driving the contour so that both regions inside and outside are as homogeneous as possible. By way of illustration, let us consider the following example. An image u0 is made up of a single object whose gray level intensity is constant and equal to ui0 . The gray level intensity of the background is also constant and equal to uo0 different from ui0 (figure 2.7). In that case, the contour C0 which minimizes the fitting term coincides exactly with the object boundaries (the fitting term is then null). Indeed, if the contour C is completely inside the object, then F2 (C) > 0 and F1 (C) = 0. Vice versa, if the contour is outside, then F1 (C) > 0 and F2 (C) = 0. If the curve straddles the object boundaries, then both F1 (C) and F2 (C) are strictly positive. Finally, if C is on the object boundaries, then the fitting term is null: the inside is homogeneous (F1 (C) = 0), as well as the outside (F2 (C) = 0). The authors proposed to make their active contour evolve by minimizing this fitting term. They added to their model two regularization terms based on the curve length and on the area of the enclosed region, and derived from this motion equation a level set formulation. It is worth mentioning that this approach is equivalent to the 66

2.3 – Image segmentation using geometric active contours

F1 (C) = 0, F2 (C) > 0 F1 (C) > 0, F2 (C) = 0 F1 (C) > 0, F2 (C) > 0 F1 (C) = 0, F2 (C) = 0 F itting > 0 F itting > 0 F itting > 0 F itting = 0

Figure 2.7: Values of the fitting term in all possible cases. F1 + F2 is minimized only when the contour is on the object edges [Chan and Vese, 2001].

Figure 2.8: Segmentation of simulated objects with blurred boundaries [Chan and Vese, 2001]. minimization partition problem developed by Mumford and Shah, with two classes and one interface. Figure 2.8 shows an example of segmentation using this approach. This method presents several advantages. It can detect contours without edges and it is robust against noise. It does not require any smoothing of the image, preserving in that way the edges. Moreover, because of its energy minimizing formulation, it can be initialized by contours placed anywhere within the image. Indeed, as the model is not driven by a constant deformation, it can change its propagation direction according to its position with respect to the object boundaries. Finally, it can detect interior contours automatically. This method has been extended to vector-valued images by [Chan et al., 2000]. However, it suffers from some limitations. As it uses only the gray level information to measure the homogeneity within both regions, it cannot detect objects with implicit boundaries. Nevertheless, other image information, such as texture information or curvature, can be used. Another issue is the computation at each time step of the average gray levels, meaning that the algorithm has to cover the entire image to calculate them. One solution may be to enclose the contour into an embedding box. But even so, this method may be time-consuming in 3D applications. 67

2 – Image segmentation using geometric deformable models

Geodesic active contour with region competition Another approach often used to solve the weak boundary issue is to drive the deformable contour by using probabilistic models of the object to segment. The basic idea here is to compute membership functions which estimate if a given voxel belongs to a region or another. This region-based evolution, also called region competition, was first developed by [Zhu and Yuille, 1996] and applied to image segmentation through other numerical algorithms. [Paragios and Deriche, 2002b] proposed a novel geometric deformable model based on region competition. This new model, called geodesic active regions, relies on a region-based module, which drives the active contour according to the region information, and on a boundary module, which makes the model stop on the desired object boundaries. Let us consider an active contour lying within a given image. It divides the image into two regions: inside and outside. First, the authors evaluate the boundary probability (that is the probability of being on the desired object edges) and the two region probabilities (that is the probability of being inside and outside the target object). Those membership functions are computed from the image itself through various techniques, like for example the supervised texture analysis used in that paper. Then, they make use of these probabilities to deform the active contour, starting from an energy minimization formulation. On the one hand, the boundary module, which relies on the boundary probability, minimizes the curve length. This module is equivalent to the geodesic active contour, apart from the fact that the stopping function g depends on a probabilistic information instead of some gradient knowledge. In that way, the geodesic active regions will be stopped on high values of the boundary probability. On the other hand, the region-based module relies on a region competition principle. The basic idea is to minimize the difference between two fitting terms relying on the region probabilities. By definition, they are minimum only when both regions are correctly segmented, that is to say when the deformable model coincides with the interface separating them. Next, a curve evolution model is derived from this energy formulation. The active contour evolves then according to the following motion equation:   ∂u pB (I(u)) =α log N(u) ∂t pA (I(u)) (2.32)   + (1 − α) g(pC (I(u)))κ(u) − ∇g(pC (I(u))) · N(u) N(u), where I is the image, u a point on the curve, N(u) the inward unit normal of the model at point u, κ(u) the curvature, α a positive scalar, pA and pB the region-based probabilities for region A (inside the target object) and region B (outside the target object), and pC the boundary probability. 68

2.3 – Image segmentation using geometric active contours

The first term of equation 2.32 corresponds to the region-based module. It aims at shrinking or expanding the curve according to the region probabilities associated with the current point u. If pB (I(u)) > pA (I(u)), then the point should belong to the background region and thus the deformable model shrinks in order to exclude it (log(pB (I(u))/pA (I(u))) > 0). On the contrary, if pB (I(u)) < pA (I(u)), then the point should belong to the target object and thus the deformable expands in order to include it (log(pB (I(u))/pA (I(u))) < 0). The second term of equation 2.32, as for it, is the boundary module. g is a positive strictly decreasing function which verifies g(r) → 0 as r → ∞. This term is identical to the geodesic active contour model, apart from the fact that it relies on the boundary probability pC . Finally, the model is generalized in order to segment images into several classes, and applied to supervised texture segmentation. An example of evolution using this approach is given by figure 2.9. In conclusion, this model solves the issue of the weak boundaries by adding region information to the geodesic active contour. Moreover, it allows the use of advanced image information for driving the deformable model as the probability functions can be estimated from any image data, such as gray levels, textures, curvature, etc. Finally, the position of the initial contour does not affect the results because of the absence of constant deformation in the model.

Figure 2.9: Supervised texture segmentation of a real image using geodesic active regions [Paragios and Deriche, 2002b].

2.3.3

Shape, geometrical and topological constraints in geometric deformable models

In addition to the previous models, new methods have been developed for segmenting objects with known shape [Leventon et al., 2000; Yang and Duncan, 2004]. In those cases, shape prior knowledge may be incorporated into the deformable models to improve their robustness with respect to noise. Besides, objects with ill-defined boundaries but with known shape, like the hippocampus for instance, can also benefit from those techniques. 69

2 – Image segmentation using geometric deformable models

Figure 2.10: Segmentation of corpus callosum. The black contour is the deformable model and the white one represents the estimated shape. The dotted curve corresponds to an evolution without shape prior knowledge [Leventon et al., 2000].

In [Leventon et al., 2000], the deformable model is driven by two different forces: a gradient force, which makes the curve stop on the object boundaries, and a shape prior model, which deforms the curve so that it matches an expected shape. In fact, the contour evolves towards a maximum a posteriori shape which is computed from the image information and a shape prior knowledge modeled by a Gaussian distribution. The parameters of this distribution are estimated over a set of training patterns. For that purpose, the training curves are considered as zero-level sets and the related implicit functions are computed. The estimated mean is then equal to the mean of the implicit functions and the variance is calculated using principal component analysis (PCA). [Yang and Duncan, 2004] use a similar approach to segment sub-cortical structures. However, instead of computing a prior knowledge based on shape only, they make their level set evolve according to joint shape-intensity prior models. The deformable model is thus more robust and can detect more subtle objects. The joint shape-intensity prior model is estimated by using the same procedure as the one proposed by [Leventon et al., 2000]. These algorithms have been successfully applied in medical imaging, and especially in brain imaging. As an illustration, brain structures such as the hippocampus, corpus callosum, have been successfully segmented (figure 2.10). One of their advantages is their robustness against noise. Moreover, they prevent the deformable models from leaking out the target object. However, they may not be suitable for the segmentation of distorted objects, like pathological sub-cortical structures for instance. But even so, those techniques exemplify the ability of geometric deformable models to make use of advanced knowledge for segmentation purposes. Another original approach has been proposed by [Zeng et al., 1999] for segmenting the cortex. The basic idea here is to make two coupled curves evolve in order to 70

2.3 – Image segmentation using geometric active contours

find the two cortical boundaries simultaneously. The first deformable model detects the white matter (WM) / gray matter (GM) junction; meanwhile the second one is stopped at the gray matter / cerebrospinal fluid (CSF) boundary. As the cortical thickness is quite constant (about three millimeters) throughout the cortex, they proposed to make use of a distance constraint between both curves. The length separating them is thus bounded by two values modeling the variation in cortical thickness. As a result, when the first deformable model find the boundary between GM and WM, it prevents the second curve from going too far or too close, and vice versa. Moreover, the distance between both curves is instantly computed owing to the level set method. In fact, as the implicit function of each curve is a signed distance map, the value of the first implicit function on a point of the second contour is equal to the distance between both models, and vice versa. Other geometrical constraints may be wisely used to segment other type of objects. In addition to shape and geometrical constraints, some researchers proposed to make geometric deformable models evolve under topological constraints [Han et al., 2003]. Indeed, it can be wise to preserve the topology of the contour when the target object has a known shape. In [Han et al., 2003], the single point concept taken from digital topology is applied. The algorithm verifies at each evolution step that no topological change has occurred and prevent the active contour from merging and splitting, preserving so the topology. This approach can be very interesting when the topology of the desired object is known. It benefits of the advantages of the level set method over the parametric deformable models (no parameterization is needed, non-intersecting contours are provided, etc.) while preserving the topology. Such a method can thus be efficient for segmenting very close structures simultaneously without grouping them. Figure 2.11 shows an example of topology preserving deformable model.

Figure 2.11: Segmentation of carpal bones using geometric deformable models. Upper panels: without topological constraint. Lower panels: with topological constraint [Han et al., 2003].

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2.4

Conclusion

Since their arrival, deformable models have been increasingly used in image processing. They can be applied in automatic detection, tracking of moving objects and, above all, in image segmentation. The basic idea is to make a curve (or surface) evolve towards the boundaries of the desired object. In parametric deformable models, the active contours rely on a parametric formulation and are driven by external forces (which guide them towards the edges of the target) and internal forces (which ensure the smoothness of the result). In geometric deformable models, the curves evolve using geometric measures only, in a direction normal to themselves. The motion is controlled by speed functions, which drive the contours towards the desired boundaries and ensure simultaneously their regularity. The two methods give smooth and closed results. They are more robust with respect to noise than other segmentation procedures and, as they are implemented in the continuum, they can achieve sub-voxel accuracy. However, geometric deformable models have several advantages over parametric active contours. Contrary to snakes, they are represented implicitly as a level set of some higher-dimensional function. In that way, no explicit parameterization is required anymore, topology changes are handled automatically and the results are less sensitive to the initialization. Also, advanced algorithms have been developed in order to improve the computational efficiency of the level set method. As it has been detailed in chapter 1, focal cortical dysplasias (FCD) are characterized on T1-weighted MRI by an unshaped cortical thickening, a blurred gray/white matter junction and a heterogeneous hyper-intensity within the dysplastic lesion. Moreover, these features can be very various throughout the lesions, preventing us from using voxel-wise approaches for segmenting FCD. In fact, algorithms which can make use of regional information and regularize the results should be more suitable. Then, as they satisfy such requirement, geometric deformable models may reasonably be applied to our purpose. However, an appropriate speed function must be designed in order to get as good results as possible. Among the various models that can be found in literature, few are really appropriate. Geodesic active contours for example rely on boundary information only. They give significant results on images with good contrast but, unfortunately, they cannot be used for segmenting objects with ill-defined boundaries, such as FCD lesions. More advanced knowledge, such as region information, is then required to successfully segment these lesions. Nevertheless, they must be discriminating enough in order to extract them accurately. For this reason, the model proposed by [Chan and Vese, 2001] cannot be applied here though it handles the detection of objects without explicit edges. Indeed, FCD cannot be segmented using gray level information only. The intensities within the lesions and the cortex are not so different 72

2.4 – Conclusion

and the hyper-intensity which characterizes the dysplastic tissues is visible on their weak boundaries mainly. As a result, while FCD are recognizable from cerebrospinal fluid and, to a lesser extent, from the white matter, it is impossible to differentiate them from healthy cortex by considering gray levels only. In fact, measurements of the cortical thickness and of the blurring of the gray/white matter junction are strongly required to separate FCD from healthy tissues. Therefore, such deformable models may go through the boundaries between FCD and healthy cortex, though they might detect the weak boundary between the lesion and the white matter. Furthermore, as FCD can be localized anywhere within the cortex and can have various sizes and shapes, geometrical, topological and shape constraints are inappropriate for segmenting them. Even so, region competition methods associated with wise membership functions may be promising. Based on a probabilistic description of the objects to segment, they may be able to detect boundaries even if they are weak or implicit. The issue becomes then to design probabilities which best describe the lesions. We chose then to base our method on that methodology. The following chapters describe the underlying principles of our approach, its implementation and the obtained results.

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74

Chapter 3 Segmentation of focal cortical dysplasia using a feature-based deformable model Introduction Though advanced image processing techniques have significantly increased the detection rate of focal cortical dysplasia lesions (FCD), the accurate segmentation of these developmental malformations is still an open problem (chapter 1.3.2). In this chapter, we propose a method for segmenting FCD which relies on active surfaces driven by probabilistic models of their MRI features. FCD are characterized on T1 MRI by an increase of the cortical thickness, a blurring of the gray/white matter junction and a hyper-intensity of the lesional gray matter (chapter 1.2.3). These features are often subtle but computational models have greatly improved their detection [Bernasconi et al., 2001; Antel et al., 2003]. The proposed approach makes use of these models to derive a probabilistic description of the lesional tissues. FCD are then delineated by active surfaces based on these probabilities. The curve evolution is implemented using the level set framework. No parameterization is thus needed, topology-changes are naturally handled and the segmentation is more robust with respect to initialization. An advanced algorithm is applied to improve the computational efficiency of the level set method, allowing the segmentation of 3D images and quantitative analyses of the results. This chapter is organized as follows. Section 1 describes the method. The computation of the probabilistic models is explained and the geometric deformable model defined. Section 2 is devoted to the implementation of the algorithm. The description of the experiments and the obtained results are presented in section 3. Finally, we close by discussing several points of the method in section 4. 75

3 – Segmentation of FCD using a feature-based deformable model

3.1 3.1.1

Methods Overall picture of the approach

The principle underlying our approach consists in making a geometric deformable model evolve according to probability maps derived from T1-weighted MRI. Indeed, advanced knowledge is required to successfully differentiate lesional tissue from healthy cortex because of their poor difference in term of gray level. For that purpose, we chose to make use of the computational models introduced by [Bernasconi et al., 2001; Antel et al., 2002], also called feature maps (chapter 1.3.1), to derive probabilistic descriptions related to each brain tissue. The method is made up of two separate steps: the estimation of the probability maps and the FCD segmentation. The estimation stage is performed using a supervised learning over a training set of pathological cases. Every voxel of the MRI to segment is associated with probabilities describing its membership to each tissue class (cerebrospinal fluid (CSF), gray matter (GM), white matter (WM) and FCD (L)), probabilities computed from the feature maps of the considered patient with respect to the training set. The complete procedure is described in the following section. The second step relies on a deformable model driven by the previously computed probability maps and initialized through the FCD classifier [Antel et al., 2003]. Flowchart 3.1 illustrates the separate stages of the method.

3.1.2

Probabilistic modeling of the MRI characteristics of FCD

We describe in this section the estimation of the probability maps which will drive the active surface. The computational models of FCD features are shortly presented. The probabilistic modeling is then described. It relies on a supervised learning over the computational models of a training set of subjects with FCD. At last, the design of the probability maps related to the patient to segment, denoted as Ω, is outlined. They will be derived from the probabilistic modeling and the feature maps associated with the patient Ω. Computational models of the MRI features of FCD The computation of the feature maps from the T1-weighted MRI constitutes the first stage of our method. For that purpose, we apply the algorithms developed by [Antel et al., 2002] and previously described in chapter 1.3.1. We shortly take them up here. As a preliminary step, each MR images is segmented into three classes (CSF, GM, WM) by applying a histogram-based method with automated threshold. The blurring of the GM/WM junction is modeled by a gradient map, denoted as Gr(M) where M is a voxel in the image space. The MRI is first smoothed with an 76

3.1 – Methods

Figure 3.1: Flowchart summarizing the different stages of the proposed approach. Probabilistic models of the FCD features are first computed. Then, the deformable model starts from the results given by the FCD classifier and evolves driven by probability maps.

isotropic 3D Gaussian kernel whose full width at half maximum (FWHM) is equal to the average cortical thickness (three millimeters). A gradient operator is then applied over the whole image. In that way, the ill-defined GM/WM interface is pointed out by a decrease of the gradient magnitude (see figure 3.2B). The hyper-intensity is emphasized by a relative intensity operator denoted as RI. Let Bg be the gray level at the boundary between GM and WM (determined by the histogram-based method previously used for segmenting the MRI) and g the gray level of a voxel M. The relative intensity operator is then defined as RI(M) = 1 −

|g − Bg | Bg

(3.1)

The hyper-intensity of the dysplastic tissue is thus modeled by an increased intensity in the RI map (see figure 3.2C). Indeed, if a voxel lying within the gray matter is hyper-intense, then its gray level intensity is close to Bg and RI increases. Finally, a thickness map T h(M) modeling the abnormal cortical thickening is computed by using the technique developed by [Jones et al., 2000]. The underlying 77

3 – Segmentation of FCD using a feature-based deformable model

Figure 3.2: Computational models of FCD features. A- T1-weighted MRI. The arrow points out the FCD lesion. B- Gradient map. FCD is recognizable by the decrease of the gradient amplitude. C- Relative intensity. FCD is revealed by an increase of the relative intensity. D- Thickness map. High values betray the lesional tissue.

principle is to consider the cortex as an electrostatic field Ψ bounded by the inner and the outer cortical surfaces. These interfaces are set to random but non-equal constants and the Laplace’s equation, defined as follows, is solved. ∂2Ψ ∂2Ψ ∂2Ψ + + =0 ∂x2 ∂y 2 ∂z 2

(3.2)

A series of iso-potential surfaces lying between both cortical boundaries are then obtained and a unit vector field is computed over the cortical ribbon. In this way, the value of the cortical thickness T h(M) at a voxel M is equal to the length of the streamline, defined by the vector field, which passes through it and connects two points of the inner and the outer interfaces. It is worth mentioning that the cortical thickness is defined within the gray matter only, the thickness value of voxels lying within WM and CSF being null (see figure 3.2D). At last, a vector-valued feature map f (M) = (T h(M),RI(M),Gr(M)) is defined from those three feature images at each voxel M in the image space. Probabilistic modeling of FCD features Once the computational models calculated, we have to estimate the probability maps for the patient Ω. These functions indicate the probability that a given voxel belongs to some tissue class according to its feature value f (M). Four brain tissue classes, denoted as c, are considered: cerebrospinal fluid (CSF), gray matter (GM), white matter (WM) and lesional (L). Therefore, four probability maps Pc (M) are computed for each tissue according to the following equation Pc (M) = P (c|f (M)), 78

(3.3)

3.1 – Methods

where c is a tissue class and M a point in the image space. P (c|f (M)), called posterior probability, models the probability of belonging to the class c knowing the feature vector f (M). Bayes rule is applied in order to transform the unknown posterior probability into an expression that can be easily estimated. P (c|f (M)) =

P (f (M)|c)P (c) P (f (M))

(3.4)

P (f (M)|c) is the conditional probability of class c. It is modeled by a trivariate normal distribution N3 (mc ,Σc ) whose parameters are estimated by using the maximum likelihood framework and a training set of subjects affected by the disease. P (f (M)|c) =N3 (mc ,Σc )  t   −1 1 1 = e− 2 f (x)−mc Σc f (x)−mc (2π)3 det(Σc )

(3.5)

First, a four-class map is computed for each patient of the training dataset. A histogram-based approach with automated threshold segments the MRI into CSF, GM and WM, and an expert draws manually the fourth class corresponding to the lesion (see figure 3.3B). Then, the parameters of N3 (mc ,Σc ) are estimated from the feature values f (M) of each patient, where the voxel M belongs to the class c. As a result, mc = (mc,Gr ,mc,RI ,mc,T h ) corresponds to the vector of the mean values of Gr, RI and T h of class c, and Σc is the covariance matrix of class c. The term P (c) of equation 3.4 corresponds to the prior probability, that is the probability of having a voxel belonging to the class c among all the possible brains. However, this information is difficult to assess by nature. As the size of FCD lesion is variable, we assume equal prior probabilities for the different classes: they are set equiprobable, P (c) = p, ∀c. Finally, P (f (M)) can be written as   P (f (M)) = P (f (M)|c)P (c) = p P (f (M)|c) (3.6) c

c

The probability maps Pc (M) of patient Ω are calculated therefore according to the following expression P (f (M)|c) Pc (M) = P (c|f (M)) =  k P (f (M)|k)

(3.7)

Figures 3.4 illustrate the different probability maps of a patient suffering from FCD.

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Figure 3.3: 4-class map of a patient of the training set. A- T1-weighted MRI where the arrow indicates the FCD lesion. B- 4-class map. Blue: cerebrospinal fluid. Green: gray matter. Yellow: white matter. Red: manual tracing of the FCD lesion.

Figure 3.4: Probabilistic modeling of FCD features. A- T1-weighted MRI where the arrow indicates the FCD lesion. B-C-D-E- Probability maps of the L (FCD), CSF, GM and WM classes (values close to 0 in black, values close to 1 in white).

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3.1.3

Geometric deformable model using MRI-feature knowledge

Feature-based deformable model In this section, we design the feature-based deformable model used for segmenting FCD. As there is no obvious boundary information, only regional knowledge should allow a successful segmentation. Region competition, proposed first by [Zhu and Yuille, 1996], seems thus to be appropriate to our purpose. The basic idea of this method is to make regions of the image space evolve by moving the interfaces between them. The motion is controlled by functions modeling the membership of a voxel to one or another region. For example, if M is a voxel lying between two regions A and B, then the interface between the two regions moves to include M into A if its membership value associated with the region A is higher than the one related to the region B. This approach was generalized afterwards by [Paragios and Deriche, 2002b] in order to segment texture objects with geometric deformable models (chapter 2.3.2). Let us consider a 3D problem with n classes and define a set of membership functions as follows  Rc : R × R → R (3.8) c ∈ {1, . . . ,n} where R is some region of the image I. These functions model each region and are maximized by the voxels correctly segmented. Then, the region competition framework consists in minimizing the following energy E(∂R) = −α

n   c=1

Rc

 Rc (I(x,y)) dx dy dz

+

n   c=1

 dσc ,

(3.9)

∂Rc

where Rc is the region associated with the class c, ∂Rc its boundaries and ∂R the set of all the interfaces. α and  are positive weighting scalars. The first term of this equation corresponds to the region competition and is minimized when all the voxels are correctly segmented. As for the second sum, called regularization term, it aims at minimizing the area of the interface, making the automated segmentation smooth. The solution to this minimization problem is computed through a gradient descent method. Stokes theorem applied to equation 3.9, gives the following EulerLagrange equations [Caselles et al., 1997; Zhu and Yuille, 1996] ∀c ∈ {1, . . . ,n},   ∂uc = α R0c (uc ) − Rc (uc ) Nc (uc ) + κc (Rc )Nc (uc ) ∂t 81

(3.10)

3 – Segmentation of FCD using a feature-based deformable model

where R0c is the membership function of the background region R0c related to Rc , uc a voxel lying between the regions Rc and R0c , Nc (uc ) the inward unit normal of the interface ∂Rc at point uc and κc (uc ) its mean curvature. We get thus n motion equations, each one of them related to one separate region. Those equations have two different terms: the first one corresponds to the region force while the other controls the regularity of the deformable model. The region force can be understood as follows: if R0c (uc ) > Rc (uc ), then the voxel uc should belong to the background and, as R0c (uc ) − Rc (uc ) > 0, the model shrinks and excludes it. Vice versa, if R0c (uc ) < Rc (uc ), then the voxel uc should belong to the region Rc and, as R0c (uc ) − Rc (uc ) < 0, the model grows and includes it. Let us now apply this formulation to our problem. The MR image is made up of four different classes c ∈ {CSF,GM,WM,L}. However, we are interested in segmenting the lesion alone and not the other classes. Therefore, only the front related to the FCD region RL evolves. The motion equation which concerns us is then   ∂uL (3.11) = α RN L (uL) − RL (uL ) NL(uL ) + κL (uL)NL (uL ), ∂t where uL is a voxel lying between the lesional region RL and the background RN L (which consists of the non lesional tissues), RL (uL) and RN L (uL) the membership functions associated with RL and RN L respectively, NL(uL ) the inward unit normal and κL (uL ) the curvature of the deformable model at point uL . To define the membership functions RL and RN L , we make use of the maximum a posteriori rule. Under this assumption, the voxel M belongs to the class c maximizing the posterior probability Pc (c|f (M)), computed according to equation 3.7. Hence, the membership function related to the lesion class can be formulated as: RL (uL) = P (L|f (uL)) = PL (uL ).

(3.12)

This formulation verifies the membership function condition. Indeed, RL (uL ) is maximized when P (L|f (uL)) reaches its maximum, that is to say when the point uL belongs to the FCD class. The same argument is applied to define a membership function for the background region RN L . A voxel uL belongs to the background if it maximizes the posterior probability of one of the non-lesional regions, that is to say CSF, GM or WM. In this way, uL is classified as a healthy if argmax c ∈ {CSF,GM,WM,L}

⇐⇒

argmax c ∈ {CSF,GM,WM,L}

P (c|f (uc)) ∈ {CSF,GM,WM}

(3.13a)

Pc (uc ) ∈ {CSF,GM,WM}

(3.13b)

As a result, we define the membership function related to RN L as   RN L (uL ) = max P (c|f (uL)), c ∈ {CSF,GM,WM}   = max Pc (uL ), c ∈ {CSF,GM,WM} 82

(3.14a) (3.14b)

3.1 – Methods

This function is indeed maximized when the point uL belongs to one of the nonlesional class. Finally, by writing u = uL, we get the evolution equation of the feature-based deformable model ⎧   ∂u ⎪ ⎪ = α RN L (u) − RL (u) N(u) + κ(u)N(u), ⎨ ∂t (3.15) RL (u) = PL (u), ⎪ ⎪   ⎩ RN L (u) = max Pc (u), c ∈ {CSF,GM,WM} where N(u) is the inward unit normal of the interface at point u, κ(u) the curvature, and α and  two positive weighting constants. From now on, the active surface will be denoted as S instead of ∂RL .

Figure 3.5: Membership functions. A- T1-weighted MRI of a patient with FCD. The arrow points out the lesion. B- Membership function related to the FCD lesion RL (values close to 0 in dark, values close to 1 in white). C- Membership function related to the healthy tissues RN L (values close to 0 in dark, values close to 1 in white). D- Region competition RN L − RL (values close to -1 in dark, values close to 1 in white).

Level set evolution The motion equation 3.15 is implemented using the level set method [Osher and Sethian, 1988; Sethian, 1999a]. We apply here the calculation presented in chapter 2.2.2 in order to derive the level set formulation of the feature-based deformable model. The basic idea is to define the active surface S as the zero level set of a 3D function φ(x,y,z), called implicit function. φ(S(t),t) = 0.

(3.16)

φ(x,y,z,t = 0) is usually a signed distance map computed from the initial deformable model S. The voxel values are equal to their distance to the surface S. 83

3 – Segmentation of FCD using a feature-based deformable model

Figure 3.6: Geometric deformable model. Sign of the implicit function: negative inside the surface, positive outside. By convention, the voxels lying inside S are negative while those lying outside are positive (figure 3.6). In that way, the inward unit normal can be rewritten as follows N=−

∇φ |∇φ|

(3.17)

Owing to the calculi done in section 2.2.2, a geometric deformable model defined as ∂u = F (κ(u))N(u) ∂t

(3.18)

can be reformulated as

∂φ (u) = F (κ(u))|∇φ(u)| (3.19) ∂t Moreover, equation 3.19 is defined everywhere within the image space. The level set formulation of equation 3.18 is thus ∂φ (M) = F (κ(M))|∇φ(M)|, ∂t where M is some point in the image space. We can now apply this formula to our motion equation. If we set   F (κ(M)) = α RN L (M) − RL (M) + κ(M),

(3.20)

(3.21)

then we get the following level set formulation of the feature-based deformable model   ∂φ (M) = α RN L (M) − RL (M) |∇φ(M)| + κ(M)|∇φ(M)|. (3.22) ∂t The surface evolution will be then performed according to this equation. The numerical schemes and the algorithm related to the level set method are explained in the next section. 84

3.2 – Implementation

3.2 3.2.1

Implementation General considerations

The proposed approach relies on several modules (section 3.1.1) running in a Linux environment. Software and libraries developed within the Montreal Neurological Institute are used to manage the MR images. Indeed, a file format especially tailored to this type of images (and also for other forms of medical data) has been designed in order to provide a homogeneous environment to the researcher of the MNI. This file format, called minc, is associated with several tools which allow the MRI visualization, some basic processing through command lines, and the development of advanced C/C++ algorithms (C libraries providing minc I/O procedures and image management functions). Therefore, all our programs are based on the minc file format and tools (www.bic.mni.mcgill.ca/software). Some modules are bash scripts using minc tools (the computational model procedures) while others are C programs (the FCD classifier for example). The level set procedure was developed in a Linux environment by using C++ and minc libraries. The GIS file format was also handled in order to display the results using Anatomist, the visualization tool provided by Brainvisa package (http://brainvisa.info). The project is made up of several classes. Broadly speaking, one class embeds the I/O and image management methods as a front-end to the minc functions, and four classes are related to the three-dimensional implementation of the level set framework (the basic algorithm, the narrow band algorithm, the motion equations and a class holding various useful methods). Finally, shell scripts were written in order to put all the separate modules together. The following sections explain in detail the critical points of the implementation. First, the numerical schemes needed to approximate the level set formulation of the motion equation are defined, followed by a brief description of the algorithm computing the signed distance map (the initial implicit function). Next, the narrow band method is introduced. This technique improves the computational efficiency of the level set method. Then, the implementation of the feature-based speed function and the related issues are detailed. Finally, we conclude by explaining the stopping criteria used by our algorithm to automatically terminate the surface evolution.

3.2.2

Numerical schemes

Accurate numerical schemes are critical for implementing the level set method. Solutions to the motion equation must be found even beyond the formation of singularities such as corners. Besides, these singularities must be propagated without smoothing or deteriorating them. [Osher and Sethian, 1988; Sethian, 1999a] used the link between Hamilton-Jacobi equations and hyperbolic conservation laws to design numerical schemes which are able to find entropy-satisfying weak solutions 85

3 – Segmentation of FCD using a feature-based deformable model

to the level set evolution (chapter 2.2.1). Indeed, the level set formulations are particular cases of the Hamilton-Jacobi equations and such schemes can be applied here. In this section, the numerical approximations are presented, followed next by a discussion about their stability and the domain boundary conditions. Numerical implementation of the level set equation Let us consider a surface S evolving under two motions: a constant expansion with speed −F0 (F0 > 0), and a curvature-driven deformation κ. ∂u = −F0 N + κN, ∂t

(3.23)

where u is a point lying within the surface S and N the inward unit normal. According to chapter 2.2.2, the level set formulation of this equation is ∂φ = −F0 |φ| + κ|φ| ∂t

(3.24)

How to approximate this equation? There are three first order numerical approximations of the partial derivatives: • Forward finite difference: ∂Γ/∂v =

Γ (v+k)−Γ (v) k

• Backward finite difference: ∂Γ/∂v = • Central finite difference: ∂Γ/∂v =

= D +v Γ

Γ (v)−Γ (v−k) k

Γ (v+k)−Γ (v−k) 2k

= D −v Γ = D 0v Γ

We have to use the appropriate numerical scheme for each term of equation 3.24. The time derivative is approximated by a forward scheme, that is φ(t + ∆t) − φ(t) ∂φ = , ∂t ∆t

(3.25)

where ∆t is the time step. Let us now consider the spatial derivatives. As it has been said in chapter 2.2.1, the first term of the right side of equation 3.24 may create singularities. Therefore, it must be approximated by schemes satisfying the entropy condition. The basic idea here is to design a numerical approximation which can find entropy-satisfying weak solutions, that is solutions to the integral form of equation 3.24 (without the curvature motion). Such a scheme is obtained through hyperbolic conservation laws and Riemann solvers. Moreover, adequate approximations must be selected according to the sign of F0 . Indeed, the spatial differences must be computed from values upwind of the direction of the surface propagation. As an illustration, if a onedimensional curve propagates from left to right, then the spatial derivative related to this curve must be computed according to a backward scheme which propagate 86

3.2 – Implementation

the information from left (v − k) to right (v) too. As for the second term of equation 3.24, it behaves like a non-linear heat equation. It can be thus approximated by central finite differences. A complete review of the underlying principles of these numerical schemes can be found in [Sethian, 1999b]. As a result, the first order numerical approximation of equation 3.24 is ⎡

⎤ − max(F0ijk ,0)∇+ + min(F0ijk ,0)∇− n ⎣   0x 2  ⎦ φn+1 ijk = φijk + ∆t 0y 2 n 0z 2 1/2 ) + (Dijk ) (Dijk ) + (Dijk + Kijk

(3.26)

n where Kijk is the central approximation of the curvature κ at point (i,j,k) (see + bellow), ∇ and ∇− the entropy-satisfying approximations defined as follows −x +x ,0)2 + min(Dijk ,0)2 + ∇+ = [ max(Dijk −y +y max(Dijk ,0)2 + min(Dijk ,0)2 + −z +z ,0)2 + min(Dijk ,0)2 ]1/2 max(Dijk

(3.27)

+x −x ∇− = [ max(Dijk ,0)2 + min(Dijk ,0)2 + +y −y max(Dijk ,0)2 + min(Dijk ,0)2 + +z −z ,0)2 + min(Dijk ,0)2 ]1/2 max(Dijk

(3.28)

0µ and Dijk a shorthand notation for D 0µ φnijk , with µ ∈ {x,y,z}. The min and max in equation 4.21 are used to select the appropriate upwind scheme, according to the sign of F0ijk .

Finally, the numerical approximation of the motion equation 3.22 is obtained by replacing F0ijk into equation 4.21 by   (3.29) F0ijk = −α RN L (i,j,k) − RL (i,j,k) One of the advantages of the level set method is that geometrical variables, such as curvature, can be directly calculated from the implicit function. Several curvatures exist for 2D surfaces, such as mean curvature, Gaussian curvature, etc. However, since the Euler-Lagrange of the regularization term of equation 3.9 leads to the mean curvature, denoted as κm , we chose to make use of this expression in our model. We get then the following expression of κm , expressed in term of the implicit function φ:  (φyy + φzz ) φ2x + (φxx + φzz ) φ2y + (φxx + φyy ) φ2z κM = ∇ ·

∇φ = |∇φ|

− 2φx φy φxy − 2φx φz φxz − 2φy φz φyz  3/2 φ2x + φ2y + φ2z 87

(3.30)

3 – Segmentation of FCD using a feature-based deformable model

This expression is approximated by using central finite differences for each spatial derivative. At last, other types of geometric variables can be computed from the implicit function, such as the area of the active surface (voxels where φ is null) or the enclosed volume (voxels where φ is negative). Stability and domain boundary conditions Two issues arise when dealing with numerical schemes. The first one pertains to the explicit numerical scheme itself, and more especially to the first order approximation of the constant deformation −F0 |φ|. Indeed, the constant propagation is equivalent to an advective term and must thus verify the Courant-Friedrichs-Lewy (CFL) condition in order to ensure its numerical stability. The CFL condition states that the active surface must not cross more than one voxel at each time step, that is max(|F0 |)∆t ≤ ∆x, (3.31) I

over where |F0 | is computed  the entire image I (see section 3.2.5). In our case, we  have F0 = −α RN L − RL with RN L ,RL ∈ [0,1], and ∆x = 1, hence α≤

1 . ∆t

(3.32)

We must thus choose α and ∆t verifying this inequality to guarantee the numerical stability of the algorithm. The other issue is related to the finite dimensions of the images. Indeed, the spatial derivatives must be calculated accordingly in order to ensure their continuity at the domain boundaries. Therefore, the differential equation 3.22 must be solved under some Neumann conditions. Among them, mirror condition is usually chosen because of its straightforward implementation. It corresponds to the symmetrical extension of the image. The partial derivatives are then computed over this new extended domain while the deformable model stays within the original image space.

3.2.3

Initialization: computation of the signed distance map

As an initial implicit function, we chose to make use of the signed distance to the surface S. However, an efficient algorithm is required to compute it owing to the 3D nature of the problem. A naive approach might be to assign to each voxel M the distance value MP, where P is the closest point on the surface S to M. Nevertheless, this method is time-consuming and cannot be applied to 3D imaging. Among the advanced approaches that can be found in literature, Rosenfeld algorithm is a good compromise between computational complexity and result accuracy. The basic idea is to propagate the global distance transform by scanning twice the entire 88

3.2 – Implementation

image and applying to each voxel a local distance mask, such as Euclidian or Chamfer masks [Rosenfeld and Pfaltz, 1966; Borgefors, 1996]. A 3 × 3 × 3 local Chamfer mask defines the distance between the voxel at its center and its neighborhood. Therefore, in isotropic images, three distance values (a,b,c) can be distinguished, each one of them related to the three different types of neighbors (area, edge and point neighbors). Various coefficients were analyzed in [Borgefors, 1996] but the ones giving √ the √ most accurate results are surprisingly (0.92644,1.34062,1.65849) and not (1, 2, 3). The algorithm is made up of two main stages, an initialization stage and a propagation stage, which is itself split into two steps. First, the distance map is initialized. Null values are assigned to voxels lying within the initial deformable model and “infinite” intensities to the others. Then, the algorithm propagates in the image space, forwardly first and backwardly next, the distances to the surface by using only one half of the directions of the mask during each scan. The Chamfer algorithm described in [Borgefors, 1996] is // Initialization For z = 0 to Dz − 1 do For y = 0 to Dy − 1 do For x = 0 to Dx − 1 do If I(x,y,z) ∈ S

Then I(x,y,z) = 0 else I(x,y,z) = +∞

// First step For z = 1 to Dz − 1 do For y = 1 to Dy − 1 do For x = 1 to Dx − 1 do

I(x,y,z) = min {I(i,j,k) + dn }

// Second step For z = Dz − 2 to 0 do For z = Dy − 2 to 0 do For z = Dx − 2 to 0 do

I(x,y,z) = min {I(i,j,k) + dn }

(i,j,k)∈N

(i,j,k)∈N

where I is the image containing the initial deformable model S, Dx ,Dy ,Dz are the image dimensions, N the set of already visited neighbors, and dn the suited local distance (0.92644, 1.34062 or 1.65849). However, this method computes the unsigned distance only. We need then to scan the image once again and assign negative values to the voxels enclosed into the active surface S. This algorithm is linear in the number of voxels and constitutes thus a reasonable approach to compute the initial implicit function. Figure 3.7 shows the various steps of the method as well as the final signed distance map of a simulated initial surface. 89

3 – Segmentation of FCD using a feature-based deformable model

Figure 3.7: Example of signed distance map computed from a simulated 3D image. A- One slice

of the simulated surface. B- Result after the first step. The distance values are propagated in one direction only. C- Unsigned distance map. D- Signed distance map with negative values inside the surface.

3.2.4

The narrow-band method

A simple implementation of the level set method, called full-grid algorithm, consists in updating iteratively all the values of the implicit function according to the numerical scheme 4.21. Since the entire image is scanned at each iteration, this approach can be very time-consuming and cannot be applied to 3D images such as MRI. Indeed, the dimensions of an MRI are usually equal to 256 × 256 × 256 voxels. The algorithm must thus update more than 16 million points at each time step, which makes it inadequate for our purpose. For that reason, we made use of another technique to make the front evolve. The narrow band method was developed by [Chopp, 1993] for computing minimal surfaces. However, it was studied extensively by [Adalsteinsson and Sethian, 1995] and applied to image segmentation for the first time by [Malladi et al., 1995]. The basic idea is to define a narrow band around the deformable model and update the implicit function only within this band (figure 3.8A). The computational complexity is therefore greatly decreased and 3D applications become conceivable. The narrow band consists of the points with distance to the deformable model less than maxDist (figure 3.8B). To build it, a distance map is computed from the active surface (section 3.2.3) and the appropriate points are selected. Those points are then stored into a 2D array whose cells (xi ,yj ) contain the narrow band segments {[z1 ,z2 ], . . . ,[zn−1 ,zn ]} of the image slice (xi ,yj ) . For example, if xi = 2, yj = 5 and the cell (2,5) has the segments {[20,43],[54,72]}, then the points (2,5,20) . . . (2,5,43) and (2,5,54) . . . (2,5,72) belong to the tube. Furthermore, as the deformable model seldom takes up the whole image, it is embedded into a bounding box. This reduces the size of the 2D array and allows us to compute the distance map only within this new domain. We get in that way more computational speed-up. Once the tube built, the algorithm updates the voxels lying within the narrow band while the others are kept unchanged (their sign however corresponds to their position with respect to the deformable model, that is negative inside and positive 90

3.2 – Implementation

Figure 3.8: Narrow band method. A- Drawing of a deformable model and its narrow band. B- Details of the narrow band. The deformable model (bold line) must not go through the minefield otherwise the evolution is stopped and the tube rebuilt.

outside). Nevertheless, we must detect when the active surface goes too close to the tube boundaries. It cannot go outside as the outer voxels are not updated. Moreover, instability can emerge if the deformable model is near the edges. Two different approaches can be found in literature. The first one, proposed by [Malladi et al., 1995], consists in rebuilding regularly the narrow band. However, the number of steps between two re-initializations is fixed empirically and can change from one application to another. The second technique, the one we chose, relies on the definition of minefields near the edges [Adalsteinsson and Sethian, 1995]. They are defined during the construction of the narrow band and are made up of voxels with distance to the deformable model lower than maxDist and bigger than howClose (figure 3.8B). In addition to that, they are characterized by the constant sign of their voxels (negative in the inner minefield, positive in the outer one), which allows the instantaneous detection of the probable presence of the front into one minefield. Indeed, as voxels change sign when the deformable model crosses them, we only have to check at each time step if the sign of any of the mined voxels has been modified. In that case, the evolution is stopped and the tube re-initialized. At last, the narrow band method can still handle topology changes naturally owing to this re-initialization strategy. Therefore, the narrow band algorithm can be written as: 1. Compute the bounding box around the deformable model, the signed distance function on a narrow band and the minefields. 2. Evolve the implicit function φ on the narrow band only. 3. If the active surface goes through one minefield, stop the evolution and go to step 1. 4. Otherwise go to step 2. An important issue of this algorithm is the choice of an appropriate width maxDist. On the one hand, if maxDist is too high, the tube is never reinitialized 91

3 – Segmentation of FCD using a feature-based deformable model

and the computational complexity is equivalent to the one of the full-grid algorithm. On the other hand, if maxDist is too small, then few voxels have to be updated but the narrow band must be rebuilt almost every time step and we loose the benefits of the technique. A compromise must thus be found. We chose to use tubes built with maxDist equal to seven voxels and howClose equal to three voxels, which, according to our experiments, constitutes an appropriate compromise between re-initialization costs and update costs. This algorithm was evaluated on several 2D and 3D images using a Linux PC with two processors AthlonXP 1.8 GHz, 1 GB RAM. Two main examples are reported here. The first one is a 2D slice of a T2-weighted MRI (174 × 201 pixels). We made use of the geodesic active contour framework ([Caselles et al., 1997], section 2.3.1) to segment the ventricles. The deformable model was initialized manually (figure 3.9A, black contours), the constant deformation was c = 10, and 10000 iterations were done to ensure complete segmentation. The full-grid algorithm took 343.1s to make the deformable model evolve, whereas the narrow band method took 85.2s (about 4 times faster). The results given by both algorithms were identical and the initial contours were automatically merged during the evolution. The narrow band segmentation is shown in figure 3.9B.

Figure 3.9: Segmentation of brain ventricles using geodesic active contour and the narrow band method (2D slice of a T2-weigthed MRI). A- Initial deformable model (black contours). B- Segmentation result (black contours). The four initial circles merged to segment the two ventricles.

The second example is a simulated 3D image (figure 3.10A, transparent object) whose dimensions are 150 × 150 × 150 voxels. The geodesic active surface framework was used to segment the 3D object, initialized by a small cube at the image center (figure 3.10A, gray object) and with a constant deformation c = 10. The fullgrid algorithm took 5112s to compute 1000 iterations, whereas the narrow band method took only 448s (11 times faster). Similarly to the previous example, the two approaches gave same results. The narrow band segmentation is shown in figure 3.10B. The significant speed-up gotten here is mainly due to the small starting front. Indeed, the initial iterations are computed very quickly because of the narrow band method. This behavior will be crucial for segmenting FCD since they are often of small size. 92

3.2 – Implementation

At last, let us mention that equivalent speed-ups were obtained when segmenting other test images, while both full-grid and narrow band results stayed always identical. Therefore, owing to its computational efficiency, the narrow band method will be exclusively used in the following sections.

Figure 3.10: Segmentation of a simulated 3D object (shown in transparent) using geodesic active surface. A- Initial deformable model (dark gray object). B- Segmentation result (dark gray object).

3.2.5

The speed function into consideration

The issue of the speed function So far, we have considered the speed function F defined on all the image space. However, it has been designed using a region-competition approach and thus it has meaning only on voxels lying between both lesional and non-lesional regions. Let us recall the level set formulation of the feature-based deformable model (equation 3.22): ∂φ (M) = αF (M)|∇φ(M)| + κ(M)|∇φ(M)|, (3.33) ∂t where F (M) = RN L (M) − RL (M) and M is some voxel in the image space. By picking the values of the speed function directly in the image space, all the level sets will move towards the FCD boundaries (they are all driven by the same force) and will bunch up around the deformable model. Therefore, as the curvature and the inward normal are computed from the implicit function, their approximation becomes less accurate and numerical instability may emerge. This can be seen on figures 3.13CD, page 98, where a simulated 3D image representing two interlaced rings (with dimensions equal to 70 × 50 × 50) was segmented by using the geodesic active contour framework [Caselles et al., 1997]. The constant deformation was set to c = 10 (equation 2.30) and 2000 iterations were computed in 474s using the narrow band method (Linux PC, bi-processor AthlonXP 1.8GHz, 1 GB RAM). An “extension velocity” is thus required, guiding the non-null level sets by appropriate speed functions in order to maintain the signed distance map during the evolution. 93

3 – Segmentation of FCD using a feature-based deformable model

Several approaches solving this issue can be found in literature. [Gomes and Faugeras, 2000] proposed to replace the level set equation by a new formulation maintaining automatically the signed distance function. Broadly speaking, they slightly modify the level set equation by incorporating a new constraint preserving the distance property. The new equation is then implemented using the entropysatisfying numerical approximation used in the level set method [Sethian, 1999a]. Another technique consists in re-initializing the implicit function every fixed time steps in order to restore the distance function. An idea may be to extract the zero-level set from the implicit function and then compute the new signed distance map. However, the extraction cannot be performed accurately since the evolution is calculated in the continuum. As an illustration, let us consider two neighboring voxels M1 and M2 with φ values equal to 0.1 and −0.2 respectively. In that way, the deformable model crosses the segment [M1 ,M2 ] without going through M1 or M2 . However, because of the sampling, one of those voxels will belong to the extracted active surface: the front is thus moved. As a result, if the re-initialization is often performed, then those errors are accumulated and the deformable model may be stopped from going towards the desired object boundaries. [Sussman and Fatemi, 1999] proposed an elegant way to re-initialize the implicit function without extracting the deformable model. They solve a differential equation with respect to φ, starting from the current distorted implicit function. However, this approach is computationally expensive since the algorithm must wait the equilibrium solution to the re-distancing equation. Finally, [Adalsteinsson and Sethian, 1999] proposed to extend the speed function to the entire image space using a Fast Marching Method, avoiding thus the extraction of the active surface and the re-initialization of the implicit function. This technique allows the fast construction of extension velocities, in O(n log n) where n is the number of voxels. An algorithm for constructing extension velocities As a first attempt, we chose to compute extension velocities from the speed function F to preserve the signed distance map. For that purpose, we applied the method introduced by [Malladi et al., 1995]. The underlying principle is to solve the following equation instead of equation 3.22: ∂φ (M) = αFext (M)|∇φ(M)| + κ(M)|∇φ(M)|, ∂t

(3.34)

where Fext (M) is the extension velocity of the speed function F (M) (equation 3.33) and defined by Fext (M) = F (P ), M being some voxel in the image space and P the closest point on the active surface to M (figure 3.11). Hence, when F (P ) is equal to zero, meaning that the deformable model stops at this voxel, Fext (M) is also null and the level set crossing M ceases moving at 94

3.2 – Implementation

Figure 3.11: Interpolation of the speed function to maintain the signed distance function. that point. It will not collide with the active surface. However, this approach must be implemented carefully since it can be computationally expensive to look for the closest point P on the deformable model. A naive approach might be to scan the narrow band, calculate for all the voxels their distance to M and, at the end, select the closest one on the zero-level set. But this approach is time-consuming, especially when dealing with 3D images. A simple calculus demonstrates that the complexity of a single iteration is O(n2 ), where n is the number of points in the tube, in comparison with the complexity O(n) of the evolution without speed function extension. We must therefore apply a more clever method. If the distance function is preserved, or the narrow band rebuilt at every time steps, then the value of the implicit function at some voxel M corresponds to the distance between this voxel and the active surface. As a result, the closest point on the deformable model to M lies within the spherical surface centered on M and whose diameter is equal to the distance value φ(M) [Yui et al., 2002]. Thus, by looking for the point P within that spherical surface only, we greatly decrease the computational complexity of the approach (it is indeed equal to O(δ 2 n), where δ is the width of the narrow band), making it applicable to MRI segmentation. We tested the method on simulated 3D images using geodesic active surfaces. We noticed then an increase of the computational complexity but the signed distance function was preserved. Figure 3.13E, page 98, shows the segmentation of the two rings previously introduced (figure 3.13A, page 98) through this approach. The constant deformation was still c = 10 and the algorithm computed the 2000 iterations in 2105s, which is four times as long as the level set method without speed function extension (475s). Nevertheless, this example highlights a significant drawback of the algorithm: the smoothing property of the method is lost and the resulting contours are very noisy and irregular (figure 3.13F, page 98). In fact, such a construction creates discontinuous extension velocities since the distance function is not differentiable. In some cases indeed, several voxels on the deformable model have the same distance to the considered point, like for example the point M
in figure 3.11, and there is no objective criterion by which one can chose one of those voxels 95

3 – Segmentation of FCD using a feature-based deformable model

instead of another. To solve this issue, [Malladi et al., 1995] proposed to re-initialize the implicit function every fixed number of iterations. But by doing this, we get no substantial advantage from extending the speed function (whose purpose was precisely to avoid re-initialization), whereas the computational complexity increases significantly. We chose thus to give up this approach and re-initialize every fixed time steps the narrow band by applying a subtle technique to prevent the zero-level set from moving. The adopted approach: sub-voxel re-initialization of the implicit function We chose to re-initialize the implicit function every fixed time steps in order to maintain the distance property. However, the new signed distance map must be computed accurately to prevent undesired motions of the embedded deformable model. The proposed method relies on the technique developed by [Krissian and Westin, 2004]. The basic idea is to make use of the Chamfer algorithm, initialized through a sub-voxel approach, to compute the distance map within the narrow band (section 3.2.3). Let us consider two voxels V1 and V2 juxtaposed to the deformable model, so that φ(V1 ) > 0 and φ(V2 ) < 0 (figure 3.12). The active surface crosses thus the segment [V1 ,V2 ] on a point P which can be estimated using a tri-linear interpolation (3.35) P = α1 V1 + α2 V2 where φ(V2 ) and α2 = 1 − α1 φ(V2 ) − φ(V1 )

α1 =

(3.36)

Then, a simple and accurate approximation of the distances to the zero-level set, denoted as d1 and d2 , corresponds to the projection of the vector V1 V2 in the gradient direction ∇φ. Therefore, the new implicit function φ! is at points V1 and V2 ! 1 ) = d1 = β12 φ(V1 ) φ(V ! 2 ) = d2 = β12 φ(V2 ) φ(V where β12 =

1 ∇φ V1 V2 · , φ(V2 ) − φ(V1 ) ∇φ

(3.37a) (3.37b)

(3.38)

the gradient being computed through a linear combination to increase the accuracy of the initialization   ∇φ  ∇φ  ∇φ = α1 + α2 (3.39) ∇φ ∇φ V1 ∇φ V2 At last, if the voxel V1 has several neighbors Vi so that φ(Vi ) < 0, then the new ! 1 ) is equal to min{β1i φ(V1 )}. intensity φ(V i

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Figure 3.12: Sub-voxel distance between juxtaposed points and the deformable model. Therefore, the algorithm computes first the distances between the deformable model and all the neighboring voxels (those voxels have intensity equal to zero or one or more direct neighbors with opposite sign). Then, a Chamfer transform is applied to propagate the distance values in the narrow band. This method preserves thus the actual distances near the deformable model and does not move it during the re-initialization procedure. Furthermore, the use of Chamfer algorithm does not increase significantly the computational complexity of the level set method. Besides, as it occurs at fixed iterations only, it has little influence upon the global complexity. Figure 3.13G shows the segmentation result of the two interlaced rings (figure 3.13A), using geodesic active surface (c = 10), the narrow band method and the re-initialization procedure. 2000 iterations were calculated and the implicit function was re-initialized every 20 iterations. The program took 527s to compute the result (in comparison with the 475s without re-initialization) and the signed distance map is preserved.

3.2.6

Automatic stopping criterion for the level set evolution

The last issue to deal with is the stopping of the level set evolution. Indeed, nothing ensures that convergence is achieved after the computation of the number of iterations given manually to the algorithm. A solution might be to force the algorithm to calculate a large number of iterations but, in that case, the user may wait for a long time before getting the result. We then chose to implement an automatic stopping criterion which may be able to detect the end of convergence and terminate the algorithm. However, the design of such stopping criterion involves finding a compromise between strong detectors, which may stop the evolution prematurely, and lax detectors, which may never terminate the evolution.

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Figure 3.13: Effect of the speed function extension on a simulated 3D image. Geodesic active surface was applied with constant deformation c = 10 and 2000 iterations. The red contours are the resulting segmentations. A- 3D object to segment. B- The real signed distance map computed in the narrow band only. C- Segmentation result obtained without using extension velocities. D- Implicit function related to C. All the level sets bunched up around the deformable model. E- Segmentation result obtained using extension velocities. The contour is noisy and irregular. F- Implicit function related to E, the signed distance map is preserved. G- Segmentation result obtained using sub-voxel re-initialization (every 20 iterations). The contour stays smooth. H- Implicit function related to G, the signed distance map is preserved.

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Figure 3.14: Automatic stopping criteria. The evolution is stopped when the energy is minimized and almost constant. The standard deviation of the 5 last energies must be lower than a given threshold and the last energy must be bounded by Emin and Emax . These two criteria have to be verified five times in a row. As it has been described in section 3.1.3, the feature-based deformable model relies on an energy minimizing problem. The proposed stopping criterion computes thus the energy related to the active surface S and stops the evolution when the minimum seems to be reached. In fact, the region-competition “energy” E, defined below, is computed at each fixed time step r through a sub-voxel approach.  E= |E(P )|, (3.40) P ∈S

where E(P ) is the energy at point P on the deformable model. In that way, if P lies between the voxels V1 and V2 (figure 3.12), then E(P ) is equal to E(P ) = α1 E(V1 ) + α2 E(V2 ),

(3.41)

E(Vi ) = RN L (Vi ) − RL (Vi ), i ∈ {1,2}

(3.42)

with and α1 and α2 defined by equation 3.36. The level set algorithm is stopped if two criteria are verified during five consecutive steps. The first criterion is validated when the standard deviation of the five last energy values is lower than a given threshold. It evaluates in that way the energy variability. Indeed, E does not change anymore when the evolution achieves convergence. The second criterion checks if E is quite constant and does not decrease or increase slightly. The last energy value must thus be bounded by the highest and the lowest energies related to the four previous energies (figure 3.14). We validated this stopping criterion on several real MRI before applying it to our problem. The results are given in section 3.3.3. 99

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3.3 3.3.1

Experiments and results Subjects and Image preparation

Subjects 24 patients with FCD lesions visible on MRI were selected among the 38 available subjects (the others patients were cryptogenic). The mean age was 24 ± 8 years, 13 were male and 11 female. The Ethics Board of the Montreal Neurological Institute (MNI) approved the study, and written informed consent was obtained from all participants. The follow-up of these patients showed no increase in size of the lesional tissues, which prevents from mistaking them for brain tumors. 16 subjects underwent surgery due to the drug-refractory epilepsy. The resected tissues were then analyzed and the FCD histologically proven. Among them, 13 were type IIB FCD ([Palmini et al., 2004], chapter 1.2.3), that is Taylor-type FCD with balloon cells, two were Taylor-type FCD without balloon cells (type IIA) and one was characterized by a cortical dyslamination with giant and immature cells (type IB).

MRI acquisition and image preparation 3D MR images were acquired on a 1.5 T scanner (Philips Medical System, Best, The Netherlands) using a T1-fast field echo sequence (TR = 18, TE = 10, 1 acquisition average pulse sequence, flip angle = 30◦ , matrix size = 256 × 256, FOV = 256, thickness = 1 mm) with an isotropic voxel size of 1 mm3 . As part of our clinical MR protocol, T2-weighted images (TR = 2100 ms, TE 20, 78 ms, gap 0.3, thickness 3.0-5.0 mm) were also obtained. Based on these two types of MR images, FCD lesions were recognized prior to surgery in all the selected patients. The images were free of visible motion artifacts and ghost artifacts. They underwent automated correction for intensity non-uniformity and intensity standardization [Sled et al., 1998] in order to produce consistent relative intensities of the gray matter, white matter and cerebrospinal fluid. Then, they were automatically registered into stereotaxic space to adjust for differences in total brain volume and brain orientation [Collins et al., 1994]. Finally, the skull was removed using a brain extraction tool (BET) [Smith, 2000]. Once the images formatted, they were segmented into three classes (gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF)) using a histogram-based method with automatic threshold, followed next by the calculation of the computational models (section 3.1.2). 100

3.3 – Experiments and results

Level set initialization The FCD classifier developed by [Antel et al., 2003] was used to initialize the featurebased deformable model. The 24 selected subjects were classified and the lesions of 18 of them were successfully identified (75%). The obtained cluster maps were then analyzed by an expert to ensure the reliability of the automated segmentation. In that way, the probable clusters misclassifying healthy tissues as lesional were manually taken out so as to keep only those pointing out the FCD lesions. We tried to segment the six undetected subjects by initializing manually the deformable model but the automated segmentation failed. The active surface disappeared because of the weak regional force and the curvature deformation. We then analyzed the results of the FCD classifier and the probability maps related to those subjects to try to understand the reasons of those failures. The FCD of four patients were not segmented because their features were not sufficiently discriminating. The other two cases were classified by the two main stages of the automated classifier but the post-processing procedure removed the good clusters because of their small size (section 1.3.2). Indeed, one case had actually a small lesion (1.3 cm3 of the total brain volume in comparison with the average size of FCD equal to 7 cm3 ), whereas the other had more than 66% of the lesional tissue classified as white matter according to the GM/WM classification, which led to invalid feature maps and probability maps. As a result, we chose to evaluate the feature-based deformable model on the 18 detected patients only.

3.3.2

Manual segmentation

The 18 lesions were delineated on T1-weighted MRI by two trained raters, V. Naessens and D. Klironomos. They made use of a software allowing the 3D visualization of MRI and the painting in each of the three spatial dimensions. The two sets of labels are denoted as M1 and M2 respectively. Once the lesions delineated, the two experts designed consensus labels with the collaboration of two neurologists, N. Bernasconi and A. Bernasconi, in order to define a more reliable reference. They discussed about their paintings and came to an agreement about the various ambiguous regions, those identified as lesional by one expert and not by the other. The proposed automated segmentation will be compared with those consensus paintings, denoted as Mc (figure 3.15). An inter-rater agreement was calculated in order to assess the variability between the paintings of both experts M1 and M2 . For that purpose, we made use of a similarity index S derived from a reliability measure known as kappa statistic [Zijdenbos et al., 1994]. |A ∩ B| S =2× , (3.43) |A| + |B| A and B denote two different labels, and the operator |.| computes the number 101

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of voxels enclosed into a given label. This expression can assess the differences in position and size between two different labels. If the labels A and B are identical, then |A ∩ B| = |A| = |B|, leading to S = 1. If A and B have no overlap, then |A ∩ B| is equal to 0 and thus S = 0. Finally, if A and B overlap completely, with |A| = 2|B|, then S is equal to 2/3. Therefore, the higher S is, the more similar the labels are. In a general point of view, a similarity index higher than 0.7 can be considered as an excellent agreement [Bartko, 1991]. However, its interpretation is not universal and depends on the application. As an illustration, a similarity index equal to 0.7 is said good when dealing with the manual segmentation of multiple sclerosis [Zijdenbos et al., 1994], whereas similarities lower than 0.9 may be considered as unsatisfactory when considering the segmentation of well-defined structures, like brain ventricles for instance. At last, we evaluated the intra-rater similarity. To this end, D. Klironomos delineated the lesions again, two weeks after the first set of labels. In order to avoid any bias in the results, he was then not allowed to see his first paintings.

Figure 3.15: Consensus between manual labels. The two raters “overlooked” different parts of the lesion but after discussion, they agreed to include them into the consensus label. A- T1weighted MRI, the arrow indicates the FCD lesion. B- Label M1 . C- Label M2 . D- Consensus label Mc .

3.3.3

Results

Validation measurements Three indices were used to evaluate quantitatively the results of the feature-based deformable model. A similarity index S, defined by equation 3.43, assessed the similarity between the manual labels and the automatic segmentation. A coverage index C was defined by equation 3.44 in order to quantify the lesional voxels successfully segmented, according to the consensus labels. C = 100 ×

|A ∩ M| , |M|

102

(3.44)

3.3 – Experiments and results

where M corresponds to the manual label and A to the automated segmentation. Finally, a false positive index Fp counted the number of voxels segmented as lesional but lying outside the manual label. Fp = 100 ×

|A \ M| . |A|

(3.45)

Protocol The feature-based deformable model was evaluated on a Linux PC with bi-processor AthlonXP 1.8GHz, 1 GB RAM. The feature priors were computed from the labels M1 (section 3.1.2) using a leaveone out approach. The probability maps of the patient to segment were estimated from a training set consisting of all the selected subjects but him. This method allowed thus a consistent estimation of the various probabilities without introducing any bias in the results. The deformable model evolved using the narrow band method and the sub-voxel re-initialization. The tube thickness was equal to seven voxels and the time step ∆t = 0.05. The maximum number of iterations was set to 10000, the implicit function was re-initialized every 100 iterations and the stopping function criteria evaluated every 100 time steps. Finally, the parameters α and  of the feature-based deformable model (equation 3.15) were chosen without particular care. In fact, our only concern was to assign to  a small value with respect to α in order to prevent the active surface from shrinking the small lesions. In that way, all our segmentations were performed with α = 0.8 and  = 0.2. Manual segmentation For the 18 manual labels, the mean inter-rater similarity index was 0.62 ± 0.19. This value constitutes a good agreement, especially in the case of FCD lesions. The two raters disagreed about one patient (S = 0.22), whereas the highest similarity was S = 0.84. The intra-rater similarity index was equal to 0.78 ± 0.1 (range from 0.6 to 0.92), which corresponds to an excellent agreement. Evaluation of the feature-based deformable model The feature-based deformable model was evaluated by computing for each patient the similarity indices, the lesion coverage and the false positives with respect to the consensus labels Mc . The results obtained through the FCD classifier were also compared to the labels Mc with a view to evaluating the improvement provided by our approach. The results are summarized in table 3.1. At last, we evaluated these indices with respect to the manual labels M1 and M2 in order to assess the influence of the inter-rater similarity upon the results (table 3.2). 103

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Figure 3.16 presents the segmentation obtained in a patient with FCD, compared to the manual and consensus labels. Figures 3.17 to 3.24 shows other examples of level set segmentation in two subjects. Table 3.1: Results for the feature-based deformable model and the FCD classifier [Antel et al., 2003] with respect to the consensus labels Mc . They are reported as mean±SD (min to max). S C Fp

Level set 0.65 ± 0.13 (0.40 to 0.82) 0.26 ± 57.1% ± 18.1% (24.7% to 81.9%) 15.7% ± 15.8% ± 12.5% (0% to 41.4%) 0.26% ±

FCD Classifier 0.14 (0.05 to 0.46) 9.7% (2.7% to 30.2%) 0.75% (0% to 3.1%)

Table 3.2: Results for the feature-based deformable model with respect to the two manual tracings M1 and M2 . They are reported as mean±SD (min to max).

S C Fp

M1 M2 0.63 ± 0.12 (0.43 to 0.79) 0.63 ± 0.12 (0.43 to 0.77) 62% ± 21.8% (22.8% to 95.6%) 64.2% ± 18.2% (30.8% to 89.5%) 24% ± 22.2% (0% to 80.1%) 33% ± 16.1% (2.8% to 68.5%) Inter-rater similarity (M1 vs. M2 ): 0.62 ± 19 (0.22 to 0.84)

Figure 3.16: How the deformable model may unveil overlooked lesional regions, identified as such

only after agreement. A- T1-weighted MRI. The arrow indicates the FCD lesion. B- Automated segmentation. C- Consensus label. D- Manual labels (green M1 , purple M2 ). S = 0.80, C = 73.1% and Fp = 11.6%.

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Figure 3.17: Axial slices illustrating the automated segmentation of a patient with FCD. Left: feature-based deformable model (red) and its initialization (yellow). Right: The consensus labels Mc (blue). S = 0.80, C = 73.1% and Fp = 11.6%.

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Figure 3.18: Coronal slices illustrating the automated segmentation of a patient with FCD. Left: feature-based deformable model (red) and its initialization (yellow). Right: The consensus labels Mc (blue).

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Figure 3.19: Sagittal slices illustrating the automated segmentation of a patient with FCD. Left: feature-based deformable model (red) and its initialization (yellow). Right: The consensus labels Mc (blue).

Figure 3.20: 3D rendering of the brain. The automated segmentation of the FCD lesion is pointed out in red.

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Figure 3.21: Axial slices illustrating the automated segmentation of a patient with FCD. Left: feature-based deformable model (red) and its initialization (yellow). Right: The consensus labels Mc (blue). S = 0.69, C = 56.4% and Fp = 10.7%.

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Figure 3.22: Coronal slices illustrating the automated segmentation of a patient with FCD. Left: feature-based deformable model (red) and its initialization (yellow). Right: The consensus labels Mc (blue).

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Figure 3.23: Sagittal slices illustrating the automated segmentation of a patient with FCD. Left: feature-based deformable model (red) and its initialization (yellow). Right: The consensus labels Mc (blue).

Figure 3.24: 3D rendering of the brain. The automated segmentation of the FCD lesion is pointed out in red.

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Robustness with respect to the learning stage As it has been presented in section 3.1.2, manual paintings are required to estimate the probability maps used to drive the feature-based deformable model. We evaluated thus the robustness of the proposed approach with respect to the paintings by estimating two additional sets of probability maps from the labels M2 and Mc . Then, the FCD lesions were automatically delineated by using the new probability maps and the similarity indices estimated with respect to the consensus Mc . The resulting similarities were almost equal, 0.65 ± 0.12 (range = 0.39 to 0.80) for M2 and 0.65 ± 0.13 (range = 0.38 to 0.80) for Mc , in comparison with 0.65 ± 0.13 (range = 0.40 to 0.82) for M1 . Evaluation of the automatic stopping criterion To evaluate the automatic stopping criterion, we segmented all the patients by forcing the program to compute a large number of iterations (10000). The average similarity index S was then compared with the one obtained using the automatic stopping. Furthermore, we assessed the provided time-saving by calculating the elapsed time during each segmentation procedure and the number of iterations actually computed. The results are reported in table 3.3. Table 3.3: Influence of the automatic stopping criterion (ASC) on the feature-based deformable model. The similarity indices with respect to Mc , the elapsed time and the number of computed iterations are reported here as mean±SD.

With ASC Without ASC

S Elapsed time 0.65 ± 0.12 217s ± 425s 0.65 ± 0.12 784s ± 875s

Number of iterations 2144 ± 880 10000

Study of the level set parameters The robustness of the level set segmentation with respect to the parameters α and  was also assessed. All the patients were segmented using various pairs of parameters (α,) verifying  = 1 − α, with α varying from 0.9 to 0.5. Values lower than 0.5 were not interesting since the prevailing motion would have been the curvature deformation. All the other parameters were kept unchanged. The average similarity indices, the lesion coverage and the false positives were computed. When the level set was unable to segment a patient, that is to say when the curvature deformation was too strong and shrank the active surface, the similarity index and the lesion coverage were assigned zero, while the false positive index was equal to 100% (1 111

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patient was affected for (0.65,0.35), 3 for (0.6,0.4) and 9 for (0.5,0.5)). Figure 3.25 illustrates their variations with respect to the parameters. At last, we evaluated the time-saving obtained when using the narrow-band method, in comparison with the full-grid algorithm which updates the implicit function over the entire image space. We randomly chose a patient and applied the two algorithms using the same parameters. The automatic stopping criterion was activated and evaluated every 100 time steps, the implicit function was re-initialized every 100 iterations and the maximum number of iterations was set equal to 5000. The two algorithms stopped after 1400 iterations. The full-grid algorithm took 3h28 to perform the evolution whereas the narrow-band took only 48s. Visually, the two results were identical. Quantitatively, their similarity indices with respect to Mc were equal to 0.66 for both methods, and the similarity between the two automated segmentations was equal to 1.

Figure 3.25: Influence of the level set parameters α and  upon the results. The indices were computed with respect to the consensus labels.

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3.4 – Discussion

3.4

Discussion

In this chapter, we segmented focal cortical dysplasia lesions using a geometric deformable model relying on their MRI-visible characteristics. Probability maps corresponding to the lesion and the healthy tissues were estimated by using a supervised learning over a training set of subject with FCD. A region-competition framework was then applied to drive the level set according to these probabilities. Manual segmentation No gold standard was available because of the nature of FCD lesions. The blurred gray / white matter junction, the heterogeneous hyper-intensity of the lesional voxels and the lack of explicit boundaries between FCD and healthy cortex make this category of cortical malformation difficult to delineate accurately, even manually. Furthermore, subjectivity must be taken into account when dealing with manual paintings, especially at the blurred gray/white matter interface where the assessment of the dividing line between lesional and healthy tissues is challenging. We first studied the inter-rater similarity in order to quantify the variability between two manual labels. To this end, two trained experts delineated independently the lesions of the eighteen selected subjects. The inter-rater similarity was then equal to 0.62, which, though it constitutes a substantial agreement, provides a good indication of the variability that may exist between two different raters. Moreover, the intra-rater variability was also studied. An expert, D. Klironomos, painted twice all the lesions, two weeks apart. The similarity index between the two set of labels was equal to 0.78, which is an excellent agreement but confirms the difficulty to assess objectively the actual extent of FCD. In order to minimize such variability, consensus labels were designed. The two raters agreed, with the collaboration of two additional trained experts (N. Bernasconi and A. Bernasconi), about the ambiguous regions. They decided all together to include or exclude regions from the paintings, creating in that way consensus labels summarizing the opinions of the various experts. The influence of the subjectivity is thus decreased and the reliability of the manual labels improved. Validation of the automated segmentation The feature-based deformable model was compared to the consensus paintings. The obtained similarity was S = 0.65, which corresponds to a substantial agreement. The false positive index was poor, Fp = 15.8%, especially when keeping in mind the difficulty of FCD delineation. Let us perform a simple calculation to better understand this value. Let A be an imaginary spherical label whose size is equal to the average size of the eighteen labels, and B another spherical label overlapping A, whose radius is one voxel longer than A’s. Then, the false positive index of B 113

3 – Segmentation of FCD using a feature-based deformable model

with respect to A is around 17%. This means that slight differences between the automated segmentation and the reference label may lead to false positive values of the order of 20%. Since such variations are inescapable, particularly at the illdefined gray/white matter interface, false positive indices lower than 20% will be considered as excellent. It is worth mentioning that visual assessment of the results confirmed this observation. The lesions whose automated segmentation presented false positives lower than 20% revealed no significant misclassified healthy regions. The obtained lesion coverage C was equal to 57.1%. As there is, to our knowledge, no published work dealing with FCD segmentation, we compared these results with those obtained using the automated FCD classifier developed by our group [Antel et al., 2003]. Besides, this allowed us to evaluate at the same time the added value of the proposed approach since the automated classifier was used to initialize the level set procedure. The achieved improvement of the lesion coverage was significant (from C = 15.7% for the classifier to C = 57.1% for the feature-based deformable model). However, such a value is poor for segmentation purposes. In fact, the computational models of the FCD features seem to be unable to extract the entire cortical section of the lesion because of their heterogeneity within the lesional tissue [Colliot et al., 2004]. This issue constitutes the main drawback of the proposed approach. Next chapter will be devoted to its solution.

The computation of the similarity indices with respect to the manual labels M1 (S = 0.63) and M2 (S = 0.63) and the visual assessment of the automated delineation revealed that this method might unveil overlooked lesional regions. The differences between the similarities computed from the manual paintings (M1 and M2 ) and the consensus show that a significant portion of the variations between the automated segmentation and the manual delineation may be due to the inter-rater variability. The level set may indeed segment lesional regions overlooked by one of the experts, creating in that way false positives and decreasing the similarity index. Figure 3.16 illustrates this phenomenon. While the similarity indices with respect to the two set of labels are almost equal (S = 0.77), the coverage and the false positives are quite different (CM1 = 70.8% and CM2 = 84.9%, FpM1 = 15.6% and FpM2 = 29.1%). But when these indices were computed with respect to the consensus labels, we got S = 0.80, C = 73.1% and Fp = 11.6%, which corresponds to an excellent result. The level set achieved thus a good segmentation of the FCD lesion, validated only thanks to the consensus. This confirms the difficulty to delineate accurately this type of lesions and the need of consensus between raters to efficiently validate the proposed approach. The level set is totally objective and may help the user to find lesional regions overlooked beforehand. However, the coverage must be improved in order to use it as automated segmentation procedure. Next chapter will deal with this issue. 114

3.4 – Discussion

Robustness with respect to the learning stage The robustness of the feature-based deformable model with respect to its parameters was also assessed. First, we evaluated the influence of the manual paintings upon the estimation of the probability maps. For that purpose, the lesions of the selected subjects were segmented twice by using two additional sets of probability maps calculated from the labels M2 and Mc . The other level set parameters were kept unchanged. The obtained similarity indices were identical and the visual assessment revealed no difference. Therefore, any labels can be used to segment the lesions, the results will not be affected. Such robustness is highly desirable since it provides a major flexibility of the method.

Validation of the stopping criterion The automatic stopping criterion was validated by disabling it and segmenting all the patients. The resulting similarity indices were identical, showing that the criterion does not prematurely stop the evolution (table 3.3). Furthermore, an important speed-up was achieved (about 3.5 times faster) since the algorithm terminated before reaching the maximum number of iteration, often chosen high in order to ensure complete convergence of the deformable model.

Evaluation of the level set parameters The influence of the level set parameters α and  upon the segmentations was also assessed. The curves 3.25 shows that the similarity index did not change significantly for values (α,) bounded by (0.82,0.18) and (0.7,0.3). We can notice however that the lesion coverage and the false positive index decreased very slightly with α. An explanation to that might be that while α diminishes,  increases and the balance between the region competition and the curvature deformation is shifted in favor of the curvature. The expansion of the level set is thus slowed down; the active surface is stopped earlier. Nevertheless, these differences are minimal and these parameters can be chosen arbitrarily, provided they are bounded by (0.82,0.18) and (0.7,0.3). For α larger than 0.85, the region competition is too strong, leading to a poor regularization. On the other hand, if  is higher than 0.3, then α is too weak and the curvature deformation prevails. The level set may shrink small lesional regions, misclassifying them as healthy structures. Finally, we segmented a patient using the full-grid algorithm in order to quantify the speed-up obtained when using the narrow band. It took 3h28 to perform the segmentation while the narrow band method took only 48s. The time-saving was thus significant (260 times faster), as expected. 115

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3.5

Conclusion

We presented here a novel approach for segmenting focal cortical dysplasia based on geometric deformable model and region competition frame-work. The active surface was driven by probabilistic modeling of the various brain tissues which were derived from computational models of the MRI features of FCD. The results were promising and the added-value with respect to other existing techniques was significant. The similarity between the automated segmentation and the manual paintings was substantial, as well as for the false positives. The lesion coverage was good but might be improved in order to delineate the entire extent of the dysplastic tissues. Indeed, the probabilistic description of the FCD lesions was penalized by the inescapable heterogeneities characterizing them. Some attempts were performed to enhance the probabilistic modeling but all of them achieved no increase of the lesion coverage. The addition to the voxelbased knowledge of more advanced information seemed thus to be crucial for solving this issue. The following chapter will develop an extension of the feature-based deformable model which will take advantage of boundary information to improve the FCD segmentation.

116

Chapter 4 Enhancing the feature-based deformable model Introduction So far, focal cortical dysplasias have been segmented using geometric deformable models driven by probability maps describing their MRI-features. The results were promising, which encourages the use of such a method for delineating this category of malformation. However, the achieved lesion coverage was not satisfactory enough for segmentation purposes though the mean similarity index between the results and the consensus labels was significant. The level set was unable to delineate the entire extent of the dysplastic tissue. This chapter is therefore devoted to the presentation of an approach solving this issue. Section 1 describes the underlying principles of the proposed solution. An additional external force is defined according to biological assumptions and incorporated into the feature-based deformable model. This new data term, based on the gradient vector flow developed by [Xu and Prince, 1998b], provides indeed complementary information to the active surface, driving it beyond the extent revealed by the probability maps. Section 2 outlines the implementation of the gradient vector flow as well as the numerical schemes required to accurately approximate the evolution equations. Section 3 presents some preliminary results, validating the method but revealing a significant increase of the false positives. Section 4 introduces then a slight modification of the model to solve this drawback. The final results are finally discussed in section 5.

4.1

Methods

The feature-based deformable model developed in the previous chapter relied exclusively on probabilistic descriptions of the various brain tissues. However, this 117

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method was unable to delineate the whole extent of the lesion because of the inescapable heterogeneity of the FCD characteristics. An additional force based on complementary MRI information is thus required to drive the deformable model farther than the region competition alone would have done. To this end, we propose here to make use of a boundary force that will drive the active surface towards the cortical interfaces while controlling the spreading into neighboring healthy tissues.

4.1.1

Biological considerations

Focal cortical dysplasia is a malformation of cortical development due to abnormal neuroglial proliferation. It is characterized by a cortical dyslamination and by the presence of non-differentiated and dysmorphic cells, with sometimes abnormal cells (section 1.2.3). Histological studies showed that all the cortical layers, from the outer part of the cortex to the junction with the underlying white matter, are affected by the disease [Tassi et al., 2002; Palmini et al., 2004]. Moreover, it is usually hypothesized that the abnormal cells created during the proliferation process follow the direction of the migration. This phenomenon is notably represented by the transmantle abnormalities sometimes associated with FCD. Therefore, owing to the works of [Rakic, 1995] stating that the neuroglial cells migrate along the radial direction during brain development, these abnormalities may be disseminated within a cone-shaped cortical section (figure 4.1). These observations allow us to make two assumptions useful for our segmentation purpose Hyp. 1 FCD lesions extend over the entire cortical section. Hyp. 2 The boundaries between dysplastic tissue and healthy cortex might be normal to the gray matter interfaces (cerebrospinal fluid / gray matter and gray matter / white matter interfaces). The principle that may be used to improve the lesion coverage would be thus to drive the feature-based deformable model towards the cortical interfaces in a direction normal to the cortex. In that way, all the various layers may be enclosed into the automatic segmentation.

4.1.2

Design of the boundary information

We model here the above-mentioned biological observations in order to improve the lesion coverage. The basic idea is to derive from the MRI a force that can successfully drive the deformable model towards the cortical boundaries enclosing the lesion. 118

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Figure 4.1: Migration of the neuroblasts along the radial direction. The FCD lesion affects the entire section of the cortical ribbon while the borders between healthy and lesional gray matter may be perpendicular to the cortical interfaces.

Modeling of the FCD boundaries The determination of the FCD boundaries, and in particular the gray-white matter junction, is challenging because of the blurring characterizing them. Speed functions based on gradient operators are not appropriate here and another approach is required. Because of this issue, we propose to make use of an implicit estimation of the FCD boundaries. Let us consider the segmentation results given by the histogrambased method used when segmenting the MR images into three classes (gray matter GM, white matter WM and cerebrospinal fluid CSF). Then, let us calculate the proportion of the lesions classified as CSF, GM or WM. Table 4.1 summarizes the results. We can notice that the vast majority of the lesional voxels are automatically identified as belonging to gray matter (about 88%). As a result, the FCD interfaces revealed by this simple segmentation method can be considered as an adequate estimation of the actual boundaries. Though they do not correspond necessarily with the biological boundaries, they are not so far from them and can be reasonably considered as such. Accordingly, the deformable model will be driven by some force pulling it towards these interfaces in addition to the MRI-feature knowledge. Table 4.1: Mean proportion of the lesional volume lying within the various brain tissues according to the consensus labels. The values are in percentage of the average lesion volume (about 16.8 cm3 for the 18 selected subjects). They are reported as mean±SD.

% of voxels in CSF 1.76% ± 1.46% % of Voxels in GM 87.92% ± 7.87% % of Voxels in WM 9.79% ± 7.77%

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Gradient vector flow Several approaches can be used for driving the active surface towards the cortical interfaces revealed by the three-class maps. Nevertheless, a force with high attraction range is required and the edges must not be shifted by any blurring to get as much accuracy as possible. Speed functions based on gradient operators alone are therefore not appropriate. [Xu and Prince, 1998b] developed an external force for the parametric deformable model relying on the diffusion of the gradient information over regions far from the desired edges. This force can be easily adapted to geometric active surface [Xu et al., 1999]. Two stages are required to design the boundary information. The first one consists in computing an edge map f describing the desired interfaces. To this end, the GM class is first extracted from the three-class map of the patient to segment and slightly smoothed by a 3D Gaussian kernel in order to remove the noisy voxels (FWHM=2 voxels). It is worth mentioning that this smoothing is low enough to avoid any alteration of the boundaries (figure 4.2C). The edge map f is then obtained by applying 3×3×3 Prewitt operators. f represents thus the interfaces between gray and white matter as well as the junction between cerebrospinal fluid and gray matter (figure 4.2D). The second step relies on the iterative resolution of the following equation to diffuse the gradient vectors ⎧ ⎨ ∂v = g(∇f )∇2v − h(∇f )(v − ∇f ) (4.1) ∂t ⎩ v(x,y,z,0) = ∇f (x,y,z) where ∇2 is the Laplacian operator, and g(r) and h(r) are defined as r

g(r) = e− K h(r) = 1 − g(r)

(4.2a) (4.2b)

The equilibrium solution of equation 4.1 is called gradient vector flow (GVF) [Xu and Prince, 1998a]. At each point M(x,y,z) in the image space corresponds a vector v pointing at the closest cortical boundary, along the normal direction (figure 4.4A). The first term of the right side of equation 4.1 is a smoothing term and tends to produce a smoothly varying vector field. As for the second term, it forces the vector field v to be close to ∇f . The functions g(r) and h(r) weight the contribution of each term with respect to the position of the considered point. Near the edges, ∇f  is high and h prevails. The vector field closely conforms to the edge map f . In homogeneous regions, ∇f  is low and g prevails, v varies smoothly. Finally, the positive scalar K determines the weight of each member of the diffusion equation. If K is high, then g is close to 1 and the resulting GVF is very smooth. On the other hand, if K is low, then g is close to 0 and the resulting GVF conforms to the gradient. 120

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However, the norm of GVF vectors far from the cortical boundaries can be too low to successfully drive the deformable model (the motion may be so subtle that the automatic stopping criterion does not detect it and stops the evolution). In order to solve this shortcoming, the GVF is normalized in a new gradient vector v flow v ˆ = v (figure 4.4B). At last, a speed function (or force) FGV F is derived from the normalized vector field as follows

FGV F (u) N(u) = v ˆ(u) · N(u) N(u), (4.3) where u is a point on the active surface and N(u) the inward unit normal at point u. Thanks to this force, the deformable model S will follow the direction given by the GVF and stopped at the desired boundaries. This speed function satisfies the assumptions made from the biological considerations. When the normal to the active surface S is collinear to the gradient vector flow, the force FGV F is strong and S evolves along the GVF direction. Such a situation arise when parts of S are parallel to the closest cortical boundary. Those parts are then attracted by the interface, which corresponds exactly to our purpose. On the other hand, when N(u) and v ˆ(u) are perpendicular, that is to say when the active surface is tangent to the vector field, FGV F is almost zero and stops the evolution, preventing the deformable model from going through the healthy cortex. Figure 4.3 illustrates these two situations.

Figure 4.2: Computation of the boundary information from a patient with FCD. A- Axial slice

of a T1 MRI. The arrow indicates the FCD lesion. B- The three-class map (blue: CSF, green: GM, yellow: WM). C- GM map slightly blurred. D- The edge map f representing the cortical interfaces.

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Figure 4.3: Gradient vector flow with respect to the cortical interfaces. The deformable model will be attracted by the CSF/GM and GM/WM boundaries but will not expand laterally. At the end of the evolution, the active surface will enclose all the cortical layers.

A

B

Figure 4.4: Computation of the boundary information from a patient with FCD. A- GVF v computed from the edge map. B- Details of the normalized GVF v ˆ.

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4.1.3

Feature-based deformable models with Gradient Vector Flow

We have now to design a surface evolution equation which takes into account the two types of deformation, the region competition based on the MRI-features of FCD and the gradient vector flow which drives the deformable model towards the cortical boundaries. However these two forces do not have the same purpose in FCD segmentation. As the GVF force relies on assumptions alone and not on explicit image data characterizing FCD, and as it is used for improving the lesion coverage, it is not as important as the region competition. Therefore, it should not influence the region information. When a voxel is likely to belong to the lesional class, the deformable model must include it even if the GVF force points at the opposite direction. Accordingly, gradient vector flow will be used only on healthy regions. We present here two models, distinguishable by the way the region force is used with respect to the gradient vector flow. Extended feature-based deformable model

Figure 4.5: Flowchart summarizing the different stages of the extended feature-based deformable model.

The first approach consists in driving the deformable model S using the appropriate force according to its position in the image space. In the following, a lesional region corresponds to a set of contiguous voxels whose posterior probability P (c|f (M)) is maximized when c = L (f (u) being the feature vector (T h(u),RI(u),Gr(u)) at point u). The deformable model is first initialized with the results given by the FCD classifier [Antel et al., 2003] (figure 4.5). During the surface evolution, if a point u on the active surface lies within a lesional region, then the deformable model will be driven by the region force and u will be included into the segmentation result. On the other hand, if u does not lie within a lesional region, then the deformable model 123

4 – Enhancing the feature-based deformable model

will be driven by the gradient vector flow and pulled towards the cortical boundaries. The two forces are thus switched and behave independently. Therefore, the results are at least equivalent to the feature-based deformable model, the gradient vector flow cannot shrink this segmentation. Such an approach is motivated by the poor false positives obtained with the feature-based active surface, suggesting that the resulting automated segmentation do not require further shrinkage. The evolution equation related to this model is thus  

∂u = α1 1 − H RN L (u) − RL (u) RN L (u) − RL (u) N(u) ∂t  

ˆ(u) · N(u) N(u) + β1 H RN L (u) − RL (u) v + 1 κ(u)N(u),

(4.4)

where u is a point on the active surface, N(u) the inward unit normal, κ(u) the curvature, v ˆ(u) the normalized gradient vector flow, RL (u) and RN L (u) the membership functions given by equation 3.15, and H the Heaviside function  H(r) = 0 if r ≤ 0 (4.5) H(r) = 1 if r > 0 When RN L (u) < RL (u), meaning that u should belong to the lesion class, then RN L (u) − RL (u) is negative and only the region-competition force is activated. In that case, this model is equivalent to the feature-based deformable model. Vice versa, if RN L (u) > RL (u), meaning that u is likely healthy, only the gradient vector flow is activated. The active surface goes towards the cortical boundaries without spreading laterally over the cortical ribbon. This model will be denoted as extended feature-based deformable model (eFBDM). This method is almost equivalent to a two-step deformable model. As the procedure is initialized with the FCD classifier, the initial active surface is always located within a likely lesional tissue. At the beginning of the evolution it is thus driven by the region force alone. When it reaches the boundaries revealed by the membership functions, the region force is disabled and the gradient vector flow is activated, driving the contour farther towards the cortical boundaries. However, our model prevents the GVF from shrinking lesional regions, whereas a two-step approach might not be able to avoid it. Boundary-based deformable model The second approach consists in splitting the segmentation procedure in two separate stages. First, the FCD lesion is segmented using the feature-based deformable model initialized with the FCD classifier (see chapter 3). Then, a second deformable model is applied to improve the lesion coverage, starting from the results of the first stage. 124

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Figure 4.6: Flowchart summarizing the different stages of the boundary-based deformable model. The last deformable model, which will be called boundary-based deformable model (BBDM), makes use of two forces to improve the automated delineation of the FCD lesions: the GVF force, disabled when the active surface lies within regions likely to be lesional, and the region force, always activated but less strong than the GVF deformation (figure 4.6). The surface evolution equation of the BBDM is then

∂u = α2 RN L (u) − RL (u) N(u) ∂t  

+ β2 H RN L (u) − RL (u) v ˆ(u) · N(u) N(u) + 2 κ(u)N(u).

(4.6)

The philosophy of this approach is to use the MRI-feature knowledge to control the GVF deformation, preventing in that way the active surface from going too far and creating false positives. The level set method will automatically find a compromise between the region competition, which tends to shrink the surface in order to avoid non-lesional voxels, and the gradient vector flow, whose aim is to drive the contour towards the cortical boundaries. Let us mention that [Paragios et al., 2001] proposed a model allowing the active surface to be driven by GVF even in directions almost perpendicular to the vector flow. Though this approach may be appropriate for numerous image processing applications, it is not suitable for our purpose since we precisely aim at the opposite effect, that is to say at preventing the deformable model from spreading laterally into the cortical ribbon. 125

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Level set evolution The level set formulation of the two deformable models is computed following the calculi described in chapters 2 and 3. We shortly take them up here in order to derive the level set expression of the GVF force. Let φ(M,t) be the implicit function at point M and time t. φ(M,t = 0) is the signed distance map computed from the initial active surface, verifying φ(M,t = 0) < 0 when M is inside the surface and φ(M,t = 0) > 0 when M is outside. Then, as stated in section 2.2.2, we have ∇φ , |∇φ|

(4.7)

∂u = F (κ)N ∂t

(4.8)

N=− and

becomes

∂φ = F (κ)|∇φ|. ∂t We have now to re-formulate the GVF force FGV F . We can write

(4.9)

∂u = FGV F N ∂t = (ˆ v · N) N ∂u −∇φ · = (ˆ v · N) (−∇φ · N) ∂t By using the chain rule (equation 2.19) and equation 4.7, we obtain       ∇φ ∇φ ∂φ = v ˆ· − −∇φ · − , ∂t |∇φ| |∇φ| which gives ∂φ = −ˆ v · ∇φ (4.10) ∂t Therefore, by putting equations 4.9 and 4.10 together, we get the level set formulations of the two models: • extended feature-based deformable model  

∂φ (M) = α1 1 − H RN L (M) − RL (M) RN L (M) − RL (M) |∇φ(M)| ∂t  

ˆ(M) · ∇φ(M) (4.11) − β1 H RN L (M) − RL (M) v + 1 κ(M)|∇φ(M)| 126

4.2 – Implementation

• boundary-based deformable model

∂φ (M) = α2 RN L (M) − RL (M) |∇φ(M)| ∂t  

ˆ(M) · ∇φ(M) − β2 H RN L (M) − RL (M) v + 2 κ(M)|∇φ(M)|

4.2 4.2.1

(4.12)

Implementation Computation of the Gradient Vector Flow

Since all the available MR images were isotropic, we describe here the isotropic numerical schemes used to calculate the gradient vector flow [Xu and Prince, 1998a]. An edge map f is first computed from the three-class image by using some gradient operators (3D Prewitt operators in our case). Next, equation 4.1 is solved in order to diffuse the gradient information far from the boundaries. Let us write this equation with respect to the various spatial coordinates. If v = (u,v,w), then " "   2 2 2 2 2 2 2 ut = g fx + fy + fz ∇ u − h fx + fy + fz (u − fx ) (4.13a) " "   fx2 + fy2 + fz2 ∇2 v − h fx2 + fy2 + fz2 (v − fy ) (4.13b) vt = g " "   fx2 + fy2 + fz2 ∇2 w − h fx2 + fy2 + fz2 (w − fz ) (4.13c) wt = g In order to improve the computational efficiency, equations 4.13 can be written as u t = a ∇2 u − b u + c 1 vt = a ∇2 v − b v + c2 w t = a ∇2 w − b w + c 3 where

(4.14a) (4.14b) (4.14c)

 "  " 2 2 2 2 2 2 fx + fy + fz , b = h fx + fy + fz a=g c1 = bfx , c2 = bfy , c3 = bfz

These coefficients do not depend on v = (u,v,w) and can thus be calculated at the beginning of the algorithm. Equations 4.14 are approximated by forward finite difference for the time derivatives and central differences for the space derivatives, that is n+1 n Γi,j,k − Γi,j,k

∆t  + Γ + Γ + Γ Γ 1 i+1,j,k i−1,j,k i,j+1,k i,j−1,k ∇2 Γ = 2 θ + Γi,j,k+1 + Γi,j,k−1 − 6Γi,j,k

Γt =

127

(4.15) (4.16)

4 – Enhancing the feature-based deformable model

where Γ ∈ {u,v,w} is a dummy variable, i,j,k are the finite coordinates related to x,y,z respectively, ∆t the time step and θ the isotropic spatial step. Then, we substitute these expressions into equations 4.14, obtaining in that way the following iterative solution to GVF n n n n un+1 i,j,k = (1 − bi,j,k ∆t)ui,j,k + ri,j,k (ui+1,j,k + ui−1,j,k + ui,j+1,k

+ uni,j−1,k + uni,j,k+1 + uni,j,k−1 − 6uni,j,k ) + c1i,j,k ∆t n+1 n n n n vi,j,k = (1 − bi,j,k ∆t)vi,j,k + ri,j,k (vi+1,j,k + vi−1,j,k + vi,j+1,k n n n n + vi,j−1,k + vi,j,k+1 + vi,j,k−1 − 6vi,j,k ) + c2i,j,k ∆t n+1 n n n n wi,j,k = (1 − bi,j,k ∆t)wi,j,k + ri,j,k (wi+1,j,k + wi−1,j,k + wi,j+1,k n n n n + wi,j−1,k + wi,j,k+1 + wi,j,k−1 − 6wi,j,k ) + c3i,j,k ∆t

(4.17a) (4.17b) (4.17c)

where

∆t . (4.18) θ2 These equations converge towards the desired gradient vector flow. Moreover, they are stable if b, c1 , c2 and c3 are bounded and the Courant-Friedrichs-Lewy condition verified, that is r ≤ 1/6. Besides, since θ is fixed, the CFL condition is ri,j,k = a

∆t ≤

θ2 , 6gmax

(4.19)

where gmax is the maximum value of g over the entire image space.

4.2.2

Numerical schemes

We made use of the same schemes as those described in section 3.2.2 to compute the level set evolution. The numerical approximations of the curvature deformation m κ(M)|φ(M)| and the constant deformation αm Fm (M)|φ(M)| stay thus unchanged (Fm (M) being a shorthand notation for the region competition force according to the considered model, extended feature-based deformable model m = 1 and boundarybased deformable model m = 2). We have to approximate the  gradient vector flow βm U(M) · ∇φ(M), where ˆ(M) (only the weighting scalar βm changes from U(M) = H RN L (M) − RL (M) v one model to the other). This force corresponds to a pure passive advection term and must be approximated by using upwind schemes. The appropriate first-order finite difference is selected according to the sign of each component of U(M) so as to propagate the information in the apposite direction (section 3.2.2, [Sethian, 1999a]). As a result, if we rewrite the level set formulations of the two proposed models (equations 4.11 and 4.12) as follows

∂φ (M) = αm Fm (M)|∇φ(M)| − βm U(M) · ∇φ(M) + m κ(M)|∇φ(M)|, (4.20) ∂t 128

4.3 – Preliminary results

where m ∈ {1,2}, M is a voxel at (i,j,k), αm , βm and m are positive weighting scalars, and U(M) = (xU (M),yU (M),zU (M)), then the related first order numerical approximation of equation 4.20 is

⎤ ⎡ − αm max(−Fm0ijk ,0)∇+ + min(−Fm0ijk ,0)∇− ⎧ ⎫ ⎢ −x +x ⎥ n n ⎢ ⎥ ⎪ max(x ,0)D + min(x ,0)D Uijk Uijk ⎪ ⎪ ijk ijk ⎪ ⎢ ⎨ ⎬⎥ ⎢ −y +y ⎥ n n n ⎢ − βm + max(yUijk ,0)Dijk + min(yUijk ,0)Dijk ⎥ (4.21) φn+1 ijk = φijk + ∆t ⎢ ⎪ ⎪⎥ ⎪ ⎢ ⎩ + max(z n ,0)D −z + min(z n ,0)D +z ⎪ ⎭⎥ ⎢ Uijk Uijk ijk ijk ⎥ ⎣ ⎦     1/2 0y 2 n 0x 2 0z 2 + m Kijk ) + (Dijk ) + (Dijk ) (Dijk −x n where Kijk is the central approximation of the curvature κ at point (i,j,k), Dijk and similar expressions are the various first-order finite differences of φ at point (i,j,k), and ∇+ and ∇− the entropy-satisfying approximations (see section 3.2.2).

4.2.3

Considerations on the automatic stopping criterion

We made use of the stopping criterion developed in section 3.2.6 to automatically terminate the surface evolution. Though it has been designed using an energy minimizing approach, it can be applied here without modification. Indeed, when the active surface reaches its final state, it does not move anymore. Therefore, the integral along the snake of RN L (u) − RL (u), denoted as E, stays constant. Let us point out however that this expression, which corresponded to the energy of the feature-based deformable model, cannot be considered as such here because of the presence of the GVF deformation. As a result, since the automatic stopping criterion assesses the variation of E, it can still stop the surface evolution even if the final state is not minimal. The criterion becomes then the steadiness of E, that is to say the stopping of any motion of the active surface.

4.3 4.3.1

Preliminary results Subjects, image preparation and protocol

We tested the two proposed models on the 18 selected patients as described in section 3.3. We made use of the same MR images and manual paintings. Once corrected and spatially normalized, the images were segmented into four classes (CSF, GM, WM, L) and the computational models were calculated for every patient. The probability maps were estimated with respect to the labels M1 by using the leave-one out approach in order to get consistent estimations without introducing any bias in the results. 129

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The gradient vector flow was computed by calculating 200 iterations. The parameter K was set equal to 0.05. The level sets were executed on a Linux PC with bi-processor AthlonXP 1.8GHz, 1GB RAM. The narrow-band method, the sub-voxel re-initialization and the automatic stopping criteria were used. The time step ∆t was set equal to 0.05, the maximum number of iterations was 10000, the re-initialization procedure occurred every 100 iterations and the stopping criterion was evaluated every 100 time steps. The extended feature-based deformable model was initialized with the results given by the automated Bayesian classifier [Antel et al., 2003] and the parameters α1 , β1 and 1 were chosen equal to 0.8, 0.8 and 0.2 respectively. The boundary-based deformable model as for it was initialized with the automated segmentations given by the feature-based deformable model (α = 0.8,  = 0.2). The model parameters were α2 = 0.2, β2 = 0.8 and 2 = 0.1. The quantitative assessment of the obtained results was performed using the |A∩M | previously introduced indices, that is to say the similarity index S = 2 |A|+|M , the | | | coverage index C = 100 |A∩M , and the false positive index Fp = 100 |A\M . |M | |A|

4.3.2

Results

We segmented every lesion related to the 18 selected patients with the extended feature-based deformable model and the boundary-based deformable model. The similarity indices, the lesion coverage and the false positives were calculated with respect to the consensus labels Mc . The results are summarized in table 4.2. Figure 4.7 shows the normalized gradient vector flow with the results related to the extended feature-based deformable model and the boundary-based deformable model. Figure 4.8 presents the automated segmentation of a patient with FCD by using the feature-based deformable model, the extended feature-based deformable model and the boundary-based deformable model.

Table 4.2: Results for the extended feature-based deformable model (eFBDM) and the boundarybased deformable model (BBDM) with respect to the consensus labels Mc . They are reported as mean±SD (min to max).

S C Fp

eFBDM BBDM 0.65 ± 0.15 (0.31 to 0.83) 0.73 ± 0.08 (0.6 to 0.87) 86.2% ± 12.8% (53.3% to 97.7%) 77.6% ± 13.7% (45.7% to 93.4%) 42.9% ± 22% (3% to 81.6%) 26% ± 16.6% (0.39% to 52.1%)

130

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A

B

Figure 4.7: Results of FCD segmentation. A- GVF with the result of the eFBDM (in black). B- GVF with the result of the BBDM (in black).

Figure 4.8: Results of FCD segmentation. A- Axial slice of a T1-weighted MRI. The FCD lesion

is pointed out by the arrow. B- Consensus label. C- Result of the FBDM (S = 0.69, C = 56%, Fp = 11%). D- Result of the eFBDM (S = 0.74, C = 95%, Fp = 40%). E- Result of the BBDM (S = 0.87, C = 88%, Fp = 14%).

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4.3.3

Discussion

In this section, we proposed two geometric deformable models to improve the lesion coverage, the extended feature-based deformable model (eFBDM) and the boundarybased deformable model (BBDM). They both rely on MRI-feature knowledge and boundary information given by gradient vector flow (GVF). The GVF is calculated from three-class maps which approximate the cortical interfaces. The level set is thus driven towards the CSF/GM and GM/WM junctions in order to enclose all the dysplastic cortical layers. Furthermore, the lateral cortico-cortical motion of the deformable model is prevented by GVF since this force is always normal to the boundaries. The two models greatly improved the lesion coverage (86.2% for the eFBDM, 77.6% for the BBDM, in comparison with 57.1% for the feature-based deformable model (FBDM)). Visual analyses validated the added-value provided by GVF. The deformable model was attracted by the lesional interfaces and stopped laterally, as illustrated by figure 4.7. In the same way the similarity indices increased more or less significantly. The BBDM achieved an excellent similarity index (0.73). On the other hand, though the eFBDM provided a higher improvement in terms of lesion coverage, the mean similarity index was almost equal to the FBDM similarity (0.65). This result is explained by the important rise of false positives affecting this model The BBDM provides less lesion coverage but less false positives than the eFBDM does. This highlights the important role played by the region force in containing the GVF motion. The BBDM is initialized with the automated segmentation provided by the FBDM. The delineation is then extended by the GVF force but controlled by region information. The deformable model is forced to shrink when it lies within likely healthy tissues, with variable strength according to the probabilistic descriptions. In that way, a compromise between the two forces is automatically found by the level set program, preventing the active surface from going too far. In the eFBDM, the GVF motion is uncontrolled and the segmentation relies on GVF assumptions only. A source of false positives was revealed by visual assessment of the results obtained with the three proposed models (FBDM, eFBDM and BBDM). The FBDM results showed that some healthy ill-defined gray/white matter interfaces were described as lesional by the probabilistic models and were included into the automated delineation. These misclassified regions were then blindly extended by the GVF force. Figure 4.9 illustrates this phenomenon. The conjunction of these two issues leads inescapably to the misclassification of neighboring healthy regions. Solving this problem would greatly improve the reliability of the automated segmentation. In next section, we propose to incorporate into the region force additional knowledge about healthy cortical interfaces so as to contain the boundary attraction and to reduce the false positive areas. 132

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Figure 4.9: False positive area due to wrong probabilistic descriptions. A- Patient with FCD. The arrow indicates the lesion. B- Consensus label. C- Result for the FBDM. Ill-defined but healthy neighboring tissues are identified as lesional by the probabilistic model, leading to their misclassification (S = 0.76, C = 75%, Fp = 22%). D- Result for the eFBDM. The false positive area dramatically increased (S = 0.48, C = 98%, Fp = 68%). E- Result for the BBDM. The false positive area is extended but controlled by the region force (S = 0.68, C = 92%, Fp = 46%).

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4.4

Solving the false-positive issue

The geometric deformable models presented in the previous section greatly improved the lesion coverage. However, the false positive indices increased tremendously, which is not desirable in medical applications. This section describes a more comprehensive region information in order to decrease the false positives while preserving as much as possible the achieved lesion coverage.

4.4.1

Biological considerations

Let us briefly come back to a biological perspective. As it has been shown in [Colliot et al., 2004], the blurring of the gray/white matter junction is found in a large majority of patients with FCD. However, this feature alone is not discriminating enough to distinguish the lesions from healthy tissues. As an illustration, the cortical boundaries throughout the central region of the brain often have poor contrast, which makes them ill-defined and blurred. The presence of healthy tissues with ill-defined boundaries close to the lesion may lead automated segmentation procedures to over-estimate the extent of the lesional structures, and thus create false positives. An approach modeling the cortical junctions is therefore crucial to avoid such misclassifications.

4.4.2

Definition of new probabilistic models of FCD features

According to the above-mentioned biological considerations, the segmentation procedure must be able to distinguish healthy cortical interfaces from FCD lesions. To this end, we make use of probabilistic descriptions of these interfaces, as proposed by [Antel et al., 2003], in addition to the probability maps related to the four brain tissues (CSF, GM, WM, FCD). Six classes are then considered: cerebrospinal fluid (CSF), gray matter (GM), white matter (WM), FCD lesion (L), and the two transition classes (CSF/GM) and (GM/WM), resulting in the computation of six probability maps Pc
(M) for the patient to segment P
(f (M)|c) , Pc
(M) = P
(c|f (M)) = 
k P (f (M)|k)

(4.22)

where f (M) is the feature vector (T h(M),RI(M),Gr(M)) at point M. The conditional probability P
(f (M)|c) is modeled by a trivariate normal distribution N3 (mc ,Σc ) whose parameters are estimated using a maximum likelihood frame-work over the computational models of a training set of subjects. Six-class maps are used for that purpose. A histogram-based method with automatic threshold is applied to segment the MR images into CSF, GM and WM. Then, a trained rater manually delineates the lesion class L. At last, the transition classes are extracted according to the procedure developed by [Antel et al., 2003]. A 3 × 3 × 3 134

4.4 – Solving the false-positive issue

neighborhood is defined around each voxel in the image space and a given point is identified as belonging to the CSF/GM class if at least 30% of its neighbors lie within the CSF and at least 30% of the remaining voxels are GM. The same approach is used to determine the GM/WM class. Figure 4.10B illustrates a six-class map of a patient with FCD and figure 4.11 shows examples of probability maps.

Figure 4.10: 6-class map of a patient with FCD. A- T1-weighted MRI, the arrow indicates the FCD lesion. B- 6-class map. Blue: CSF. White: CSF/GM transition. Green: GM. Purple: GM/WM transition. Yellow: WM. Red: manual tracing of the FCD lesion

Figure 4.11: Probabilistic modeling of FCD features with transition classes. A- T1-weighted MRI

where the arrow indicates the FCD lesion. B-C-D-E-F-G- Probability maps of the CSG/GM, GM/WM, CSF, GM, WM and L (FCD) classes (values close to 0 in black, values close to 1 in white).

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4 – Enhancing the feature-based deformable model

4.4.3

Feature-based deformable model with Gradient Vector Flow

The evolution equations related to the two proposed models are kept unchanged (equations 4.4 and 4.6). In fact, only the region-competition force is modified in order to take into account the transition classes. Modification of the region force We apply here the same principles as those developed in section 3.1.3. The region force relies on functions estimating the membership of a given voxel to the FCD class or to a healthy region. These membership functions derive from the aforementioned probabilistic modeling of FCD features and are based on the computational models f (u) = (T h(u),RI(u),Gr(u)) of the lesion to segment. They are defined by RL
(u) = PL
(u) = P
(L|f (u)),

(4.23)



RN L (u) = max Pc (f (u)), c ∈ Λ \ L   = max P
(c|f (u)), c ∈ Λ \ L

(4.24a)

and

(4.24b)

where Λ is the set of all the tissue classes Λ = {CSF,GM,WM,L,CSF/GM,GM/WM }. The sole difference with the previous models is therefore the presence of the tran
sition classes in the background membership function RN L . Figure 4.12 shows the
membership functions related to the lesional class RL and the background region
RN L.

Figure 4.12: Membership functions. A- T1-weighted MRI of a patient with FCD. The arrow
points out the lesion. B- Membership function related to the FCD lesion RL (values close to 0
in dark, values close to 1 in white). C- Membership function related to the healthy tissues RN L

(values close to 0 in dark, values close to 1 in white). D- Region competition RN L − RL (values close to -1 in dark, values close to 1 in white).

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4.4 – Solving the false-positive issue

Extended feature-based deformable model The evolution equation of the extended feature-based deformable model is slightly modified so as to take into account the new region force. We get thus the following expressions: 



∂u

(u) − R (u) R (u) − R (u) N(u) = α1 1 − H RN L L NL L ∂t 


ˆ(u) · N(u) N(u) + β1 H RN L (u) − RL
(u) v + 1 κ(u)N(u), 



∂φ

(M) = α1 1 − H RN (M) − R (M) R (M) − R (M) |∇φ(M)| L L NL L ∂t 


ˆ(M) · ∇φ(M) − β1 H RN L (M) − RL
(M) v + 1 κ(M)|∇φ(M)|.

(4.25)

(4.26)

Boundary-based deformable model The two steps of this approach are modified to take into account the healthy cortical interfaces. The feature-based deformable model, which is first applied to get an intermediate delineation, becomes  
∂u
= α RN L (u) − RL (u) N(u) + κ(u)N(u), ∂t 
 ∂φ
(M) = α RN L (M) − RL (M) |∇φ(M)| + κ(M)|∇φ(M)|. ∂t

(4.27) (4.28)

For the sake of clarity, this new model will be denoted in the following as six-class feature-based deformable model. Then, the boundary-based deformable model starts from the resulting segmentation and is driven towards the cortical boundaries. The evolution equation and the related level set formulation of the BBDM are then:


∂u
= α2 RN L (u) − RL (u) N(u) ∂t 



+ β2 H RN ˆ(u) · N(u) N(u) L (u) − RL (u) v + 2 κ(u)N(u),


∂φ
(M) = α2 RN L (M) − RL (M) |∇φ(M)| ∂t 



(M) − R (M) v ˆ (M) · ∇φ(M) − β2 H RN L L + 2 κ(M)|∇φ(M)|. 137

(4.29)

(4.30)

4 – Enhancing the feature-based deformable model

4.5 4.5.1

Results and discussion Results

The evaluation protocol used to validate the proposed models was kept unchanged (section 4.3). The similarity indices, the lesion coverage and the false positives were computed for each patient with respect to the consensus labels Mc . The average indices are presented in table 4.3. In addition to the quantitative validation, visual evaluation was performed with the collaboration of two experts (N. Bernasconi and A. Bernasconi). The size of the false positives and false negatives (parts of the FCD lesion not segmented by the deformable model) were assessed qualitatively as being small, medium or big. Furthermore, the nature of these misclassified regions was also evaluated. The three possible criteria were then confirmed, questionable (the experts could not decide if the level set was wrong or not) or unjustifiable (the level set was right, the label should have been corrected). For the sake of conciseness, we report here only the general observations. Globally, the results were considered as very good, with a considerable improvement in similarity and lesion coverage with respect to the FBDM and the FCD classifier. Seven results were considered as excellent (no confirmed false positive, excellent lesion coverage), whereas two results were slightly worse than those obtained with the FBDM. There were few false negative areas, the lesion coverage being globally satisfactory. However, false positives were sometimes important, especially for the eFBDM were the GVF force was not controlled. Figure 4.13 shows the normalized gradient vector flow and the results for the two proposed models. Figure 4.14 presents the automated segmentation of a patient with FCD by using the six-class feature-based deformable model, the extended featurebased deformable model and the boundary-based deformable model. Axial, coronal and sagittal slices corresponding to this example are given in figures 4.15, 4.16 and 4.17. Finally, figures 4.19, 4.20 and 4.21 show the automated segmentation of another subject with FCD. Table 4.3: Results for the extended feature-based deformable model (eFBDM) and the boundarybased deformable model (BBDM) applied with the new region knowledge, with respect to the consensus labels Mc . They are reported as mean±SD (min to max).

S C Fp

eFBDM BBDM 0.73 ± 0.08 (0.6 to 0.86) 0.73 ± 0.08 (0.6 to 0.86) 80.3% ± 14.1% (49.7% to 96.2%) 72.1% ± 15.7% (44.3% to 94.3%) 28.5% ± 16.6% (0.7% to 56.1%) 20.1% ± 14.8% (0.4% to 47.7%)

138

4.5 – Results and discussion

A

B

Figure 4.13: Results of FCD segmentation (in comparison with figure 4.7). A- GVF with the result of the eFBDM (in black). B- GVF with the result of the BBDM (in black).

Figure 4.14: Results of FCD segmentation (in comparison with figure 4.8). A- Axial slice of a T1 MRI. The FCD lesion is pointed out by the arrow. B- Consensus label. C- Result of the six-class FBDM (S = 0.64, C = 49%, Fp = 7%). D- Result of the eFBDM (S = 0.86, C = 95%, Fp = 22%). E- Result of the BBDM (S = 0.85, C = 82%, Fp = 11%).

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4 – Enhancing the feature-based deformable model

Figure 4.15: Axial slices illustrating the automated segmentation of a patient with FCD. Left: extended feature-based deformable model (black), boundary-based deformable model (red) and initialization (yellow). Right: consensus labels Mc (blue). S = 0.86, C = 95% and Fp = 22% for the eFBDM; S = 0.85, C = 82% and Fp = 11% for the BBDM.

140

4.5 – Results and discussion

Figure 4.16: Coronal slices illustrating the automated segmentation of a patient with FCD. Left: extended feature-based deformable model (black), boundary-based deformable model (red) and initialization (yellow). Right: consensus labels Mc (blue).

141

4 – Enhancing the feature-based deformable model

Figure 4.17: Sagittal slices illustrating the automated segmentation of a patient with FCD. Left: extended feature-based deformable model (black), boundary-based deformable model (red) and initialization (yellow). Right: consensus labels Mc (blue).

Figure 4.18: 3D rendering of the brain. The automated segmentation of the FCD lesion is pointed out in red. Left panel: extended feature-based deformable model. Right panel: boundary-based deformable model. 142

4.5 – Results and discussion

Figure 4.19: Axial slices illustrating the automated segmentation of a patient with FCD. Left: extended feature-based deformable model (black), boundary-based deformable model (red) and initialization (yellow). Right: consensus labels Mc (blue). S = 0.83, C = 84% and Fp = 18% for the eFBDM; S = 0.83, C = 80% and Fp = 13% for the BBDM.

143

4 – Enhancing the feature-based deformable model

Figure 4.20: Coronal slices illustrating the automated segmentation of a patient with FCD. Left: extended feature-based deformable model (black), boundary-based deformable model (red) and initialization (yellow). Right: consensus labels Mc (blue).

144

4.5 – Results and discussion

Figure 4.21: Sagittal slices illustrating the automated segmentation of a patient with FCD. Left: extended feature-based deformable model (black), boundary-based deformable model (red) and initialization (yellow). Right: consensus labels Mc (blue).

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4 – Enhancing the feature-based deformable model

4.5.2

Discussion

This section put forward a solution to the false positive issue affecting the models described in the first part of this chapter. Probabilistic descriptions of the CSF/GM and GM/WM transitions were estimated and incorporated into the region force in order to prevent the deformable model from segmenting healthy tissues. Gradient vector flow was then applied to further extend the active surface towards the cortical interfaces. The results were very good. The mean similarities related to the two models were excellent (S = 0.73), especially when keeping in mind the difficulty of FCD segmentation. The lesion coverage was very satisfactory (C = 80% for the eFBDM and C = 72% for the BBDM), considerably improved with respect to the FBDM (C = 57%) and the FCD classifier (C = 16%). At last, the false positives decreased significantly in comparison with the first models presented in this chapter, reaching acceptable values for medical applications (from Fp = 43% to Fp = 28% for the eFBDM, from Fp = 26% to Fp = 20% for the BBDM). Visual analyses confirmed the quantitative results and validated our models. The GVF drove effectively the active surface towards the CSF/GM and GM/WM interfaces while containing the “lateral” spreading through the cortex (figure 4.13). The probabilistic models of the cortical interfaces improved the reliability of the segmentation since they prevented the active surface from misclassifying healthy tissues (figure 4.22). On the whole the automated delineations were very good and seven were excellent, validating the approach. Furthermore, the automated delineation was able to unveil regions not considered as lesional by the manual paintings M1 and M2 but recognized as such during the design of the consensus labels Mc or after visual inspection of the level set results. However, some healthy regions were still segmented by the deformable models, and in particular by the eFBDM. In some cases indeed, the GVF force seemed to drive the active surface too far from the real lesional boundaries, passing through weak sulci and indistinct white matter. Two main issues are raised by this observation, both relying on the design of GVF. First, shallow sulci were imperfectly extracted and gyri neighboring the lesion could have been merged with the dysplastic tissue. As a result, the GVF did not “see” the boundaries between FCD and neighboring healthy gyri, driving hence the active surface towards the cortical boundaries of the juxtaposed gyri. Next, indistinct and hypo-intense white matter separating the FCD lesion from neighboring cortex was sometimes misclassified as gray matter, creating a link between healthy and lesional tissues. GVF was then unable to distinguish such boundaries and drove blindly the level set too far. Nevertheless, the generated false positives were often of small or medium size with respect to the lesion extent and easily identifiable. They did not penalize the overall performance of the automated segmentation. 146

4.6 – Conclusion

Figure 4.22: Effect of the new region force (in comparison with figure 4.9). A- Patient with FCD. The arrow indicates the lesion. B- Consensus label. C- Result for the six-class FBDM. The neighboring tissues are not included into the segmentation anymore (S = 0.76, C = 64%, Fp = 7%). D- Result for the eFBDM. The false positive area decreased significantly. (S = 0.66, C = 92%, Fp = 48%). E- Result for the BBDM. (S = 0.77, C = 89%, Fp = 32%).

4.6

Conclusion

We demonstrated here the effectiveness of geometric deformable models based on MRI-feature knowledge and boundary information in FCD segmentation. Two different approaches were proposed. The first one, called extended feature-based deformable model (eFBDM), used the MRI-feature knowledge and the boundary information independently according to the position of the active surface. The second approach consisted in two separated steps. The lesions were first segmented using the feature-based deformable model (FBDM) presented in chapter 3. The automated segmentation was then further improved by a geometric deformable model evolving under the influence of both MRI-characteristics knowledge and boundary information simultaneously, called boundary-based deformable model (BBDM). The obtained results were very good, especially when keeping in mind the difficulty of FCD delineation. As for the previous chapter, the automated segmentations were compared with consensus labels in order to get measurements as reliable as possible. The mean similarity indices with respect to the eighteen selected subjects were excellent, the lesion coverage significant and the false positives acceptable, though the latter might be improved for fully-automated segmentation purposes or 147

4 – Enhancing the feature-based deformable model

quantitative studies. The proposed method was not designed to take the place of manual delineation but rather to be a complementary tool. It may facilitate the manual painting of the FCD lesions, reducing the inescapable user subjectivity and, more importantly, unveiling lesional regions that could be overlooked. The visual diagnosis may thus greatly benefit from such a tool as well as surgical planning in epilepsy.

148

Conclusions In this manuscript we proposed and evaluated a novel method for automatically segmenting focal cortical dysplasia (FCD) on T1-weighted MRI. The precise delineation of these lesions is crucial for clinical diagnoses and surgical planning, but their MRI features make this task challenging. The blurred gray-white matter interface and the absence of evident boundaries between dysplastic tissues and healthy cortex may lead to misdiagnoses; lesional regions may be overlooked or over-estimated. Manual segmentation is thus highly prone to user subjectivity. In the same way, automated detection and segmentation procedures are penalized by the indistinct boundaries and the variability of the gray level intensities within the FCD lesions.

To figure out these difficulties and successfully segment the FCD lesions, we made use of geometric deformable models relying on their biological features. The regularization performed by this approach, its ability to take into account advanced knowledge for segmenting objects and the natural management of topology changes are indeed particularly appropriate for our purpose. Furthermore, the labels used to evaluate our method were designed from two sets of manual paintings in order to get a standard as objective as possible. Two trained raters delineated independently the FCD lesions and then agreed with the collaboration of two neurologists about the ambiguous areas. This approach was further justified by the similarity between the two sets of paintings which, though it was significant, highlighted the possible differences in judgment of the two raters.

We proposed first to drive the active surface using a region-based force. Membership functions related to the FCD lesion and the healthy tissues were estimated according to a maximum likelihood approach. Then, the deformable model evolved so as to enclose the voxels likely to belong to the FCD while avoiding those likely to be healthy. The achieved similarities between the automated segmentations and the consensus labels were good and the false positive areas insignificant. The lesion coverage was sound and constituted a significant improvement with respect to known existing techniques. 149

However this first model was still unable to delineate the entire extent of the FCD lesions. To solve this drawback, we proposed to make use of a boundary-based force to further extend the deformable model. Gradient vector flow was applied to drive the active surface towards the cortical interfaces while preventing it from leaking throughout the healthy cortex. Two strategies were then implemented. The first one consisted in switching the GVF and the region force according to the brain tissue underlying the active surface. The region force was used within likely lesional areas whereas GVF was activated outside those regions. In the second strategy the two forces were applied simultaneously but the region force aimed at controlling the spreading caused by GVF. These two schemes gave analogous results. The similarity indices were excellent, especially when keeping in mind the difficulty of segmenting FCD. The lesion coverage was greatly improved, reaching relevant values for segmentation purposes, and the false positives were reasonable. The first approach achieved better lesion coverage than the second one but provided slightly more false positives. Nevertheless, visual inspection of the results allowed us to conclude that the size of the false positive areas were satisfactory. The proposed models were robust with respect to all their parameters. The training dataset had no influence upon the results. This property is highly desirable since no major expertise is required for drawing the labels needed to estimate the probabilities related to the various brain tissues. Furthermore, no fine-tuning of the parameters was necessary and the same parameters were used to segment all the processed subjects. Our only concern was to set the regularization term weak enough with respect to the other forces in order to prevent the deformable model from shrinking small lesions. Finally, the algorithm was fast thanks to the use of the narrow-band method and the sub-voxel re-initialization, which allowed us to perform quantitative evaluations. We have demonstrated in this manuscript the effectiveness of geometric deformable models based on feature knowledge and boundary attraction for FCD segmentation. The obtained results encourage us to consider this technique as a useful tool for visual diagnosis. The deformable model constitutes indeed a totally objective criteria and may unveil subtle lesional areas that could have been overlooked by the expert. Moreover, since the lesion coverage is excellent and the false positives acceptable and easily identifiable, this method may be part of a pre-surgical protocol defining the extent of the tissue to resect. Such a tool is not available to our knowledge and our approach constitutes thus a first automated procedure achieving significant results. ∗ ∗ ∗ 150

Our work opens up several prospects for medical applications and bio-engineering research. At first, the proposed method may be applied to other patients with more subtle lesions or even MRI-invisible FCD, selected according to some criteria or randomly, in order to give to the neurologists an indication of its performances in a wide range of cases. A second prospect may consist in reducing the false positive areas without altering the achieved lesion coverage. This question is challenging but its resolution may further improve the reliability of the method. Two techniques seem to be promising. First, advanced structural knowledge modeling the spatial position of the various sulci may be incorporated into the deformable model. This information may prevent the active surface from jumping from the lesional gyrus to neighboring healthy gyri. The difficulty here is the extraction of the sulci and their modeling; all of them have to be recognized automatically, from the deepest to the shallowest. In a more long term, the automated gray-white matter segmentation may be improved and adapted to FCD cases. A procedure which segments the cortical ribbon while taking into account the lesional hypo-intense white matter might decrease the false positives and increase the lesion coverage. An equilibrium must be found between the accurate segmentation of thin white matter areas enclosed into small gyri and the classification as gray matter of the lesional white matter. The improvement of this procedure can then increase the performances of our automated segmentation procedure but also significantly enhance other existing tools such as the FCD classifier [Antel et al., 2003] and the VBM method [Wilke et al., 2003]. The addition of texture information into the deformable model may also constitute an original improvement of our work. Precise texture analyses might emphasize the cortical dyslamination provoked by the FCD lesions. Wavelet framework and multi-scale analyses seem to be particularly appropriate for this purpose. Then, an approach similar to the one introduced by [Paragios and Deriche, 2002b] (or another method altogether) may be applied to drive the active surface towards the interfaces between dyslaminated cortex and healthy tissues. Finally, this method may be transposed to other types of malformations of cortical development or other brain lesions. Computational models describing those lesions may be designed and used to derive a probabilistic modeling of the various tissues. A deformable model relying on these probabilistic descriptions and on boundary information may then automatically delineate the lesional structures. Polymicrogyria constitutes a good candidate since they share some T1-MRI characteristics with FCD such as the cortical thickening.

151

152

Appendix A Introduction to brain anatomy This appendix outlines briefly some basics of brain anatomy. The overall structure of the brain is first described as well as its organization, followed by the presentation of the cellular components and the neuronal communication. This appendix is also devoted to the explanation of the vocabulary used all along the manuscript.

1.1

Brain organization

The brain is the central organ of the nervous system. It is commonly accepted nowadays that it is the seat of the human consciousness, intelligence and emotions. An adult brain usually weights 1.5 kg in air and its average volume is equal to 1,600 cm3 . It is suspended in the cerebrospinal fluid (CSF), which isolates it from the outside and protects it from shocks. It is made up of three different components: the cerebrum, the cerebellum and the brain stem.

1.1.1

The cerebrum

The cerebrum is the largest part of the brain. It consists of two types of tissue: the gray matter and the white matter. The gray matter forms the cerebral cortex, the outer layer of the brain, and other structures such as hippocampus. It is mainly made up of neuron bodies and supportive cells, the glial cells, which feed the neurons, protect them against extraneous invaders and improve the electrical conduction along the nerve fibers. The cortex, which concentrates the biggest part of gray matter, looks like a folded surface with outfoldings, the gyri (singular: gyrus) and fissures, the sulci (singular: sulcus). Its average thickness is about three millimeters and its surface can reach 1,400 cm2 . The white matter lies beneath it and is mainly composed of nerve fibers that connect the different areas of the brain to the spinal cords, muscles, and etc.

153

A – Introduction to brain anatomy

The cerebrum is divided into two hemispheres. They are quite symmetric and each of them controls the opposite side of the body. For example, the right hand is controlled by the left hemisphere. However, it is not an utter division since they are connected by the corpus callosum, a bundle of fibers which relays the nerve impulses from one side to the other. Therefore, when a seizure starts from one hemisphere, it can spread through these fibers and then provoke, for instance, a secondary generalized seizure. Each hemisphere is divided into four lobes, well delimited by deep sulci (figure A.1).

Figure A.1: Drawing of the cerebrum that shows the four lobes: frontal, parietal, occipital and temporal.

• The frontal lobe is the seat of the thought (frontal part of the lobe) and of the movement control (posterior part). The muscles are also governed by it and a small region lying within it enables us to speak. • The parietal lobe, located behind the frontal lobe and separated from it by the central sulcus, contains the sensory areas. Touch and hearing are integrated within this lobe and pain and temperature are sensed thanks to parietal areas. It also enables us to manipulate objects through association regions which integrate information and feelings from the outside. • The occipital lobe is found above the parietal lobe, cut off from it by the parieto-occipital sulcus. It is the center of vision; the images get by the eyes reach it first where they are then processed. That is why a patient may have visual hallucinations when a seizure is fired from the occipital lobe. Besides, lesions in this lobe may bring strong visual impairments, indeed visual loss. • The temporal lobe lies at the side of the brain, under the Sylvian sulcus. It has several functions. It processes the incoming auditory signals for example and is involved in high-level language functions, such as speech, verbal memory and comprehension. Other more or less complex functions are also performed by temporal areas 154

1.1 – Brain organization

Well-protected by these lobes and hidden inside of them, the limbic system groups together various subcortical structures involved in emotions and memory. Those structures are present in both hemispheres and lie around the thalamus, a large mass of gray matter that transmits the incoming sensory impulses to the cortex. Two of them, the amygdala and the hippocampus, are often involved in epilepsy. They are situated within the inner part of the temporal lobe and are often associated with the parahippocampal region, making up in that way the mesial temporal lobe. The amygdala are small almond-shaped structures which are involved in learning and in feeling of various emotions (fear, pleasure, etc.). Nearby, the hippocampi play a key role in memory and especially in the episodic one (the explicit memory of events, linked to emotions and vision). They relay the information to the cortex and they recover them during the recall. The damage of these structures leads therefore to the irreparable lost of memory or the incapability to form new ones. They are also involved in spatial orientation and localization. The key roles of the hippocampi in memory and in temporal lobe epilepsy make them a favorite research field. Hippocampal atrophy is the most common pattern of this type of epilepsy and seizures are often generated within it. But some recent studies have also brought to light the role of the neighboring structures in the genesis of the seizures [Yilmazer-Hanke et al., 2000; Bernasconi et al., 2003]. Moreover, it is still not clear whether the epileptic outbursts have also a retroactive action and in that way may damage them.

Figure A.2: The limbic system, the cerebellum and the brain stem. (From www.epilepsy.com) At the base of the brain and still lying inside the limbic system, the hypothalamus controls the autonomic nervous system and the hormones. It enables us to feel hunger, thirst and controls the sexual arousal and the sleep cycles. It is closely linked to the temporal lobe, which may explain why patients with temporal lobe epilepsy may feel sudden changes in their emotions and hormonal cycles (excitement, hunger, depression, anxiety, etc.). 155

A – Introduction to brain anatomy

1.1.2

The cerebellum

The cerebellum, a round and heavily-folded organ, is located above the brain stem and below the occipital and temporal lobes. Its main functions are the estimation and the guidance of the movements according to visual information. It can also control the necessary timing to coordinate them, making them fluid and precise. It sends then the commands to the cortex which will carry them out.

1.1.3

The brain stem

The brain stem is connected to the cerebrum through the hypothalamus. It joins the brain to the spinal cord and can control some autonomic movements such as heartbeat and breathing.

1.2

Cellular description

Let us now see how the brain works at a cellular level. Two main categories of cells form the brain: the neurons and the glial cells.

1.2.1

The neurons

The neuron is the core of the nervous system. It receives, creates and spreads the neuronal impulses through its membrane, connecting the brain to organs, muscles or sensatory receptors. 100 billion nerve cells are held by the brain and each of them is connected to hundreds of other cells. It is this huge inter-connectivity that has allowed us to develop advanced cognitive skills such as intelligence, emotions, etc. A neuron can be divided into three different parts: the dendrites, the body and the axon (figure A.3). The dendrites can be understood as the input gateways of the cell. The incoming nerve impulses, also called receptor potentials, run through them towards the body. Two different types of receptor potentials can be distinguished according to their functions: the excitatory ones, whose purpose is to excite the neuron in order to create a new impulse and then propagate the neuronal information, and the inhibitory signals, whose aim is the exact opposite. The length of the dendrites is very small and the propagation through them is signal-loss. Nevertheless, the decrease of the potential intensity is very important because it allows the modulation of the signals according to the dendrite in addition to their types. The incoming impulses reach next the body and a summation is processed. If the result is lower than the neuron’s excitatory threshold, then no signal is transmitted. But if this value exceeds the threshold, then a new nerve impulse, called action potential, is generated and transmitted by the third and last part of the neuron, the axon. This element is by far the longest component of the nerve cell, reaching lengths of the order of one meter. Contrary to the dendrites, it propagates the potentials without 156

1.2 – Cellular description

signal loss because of the use of ionic channels instead of pure electric conduction. Moreover, in most axons, the transmission velocity is increased by the presence of a discontinuous sheath made of myelin. The structure of this sheath is very typical and ingenious since the ion exchanges that permit the neuronal transmission are only performed within small uncovered sections, spaced from 0.2 to 2 millimeters apart (nodes of Ranvier). The action potentials “jump” then from one node to another, traveling thus faster than they would otherwise. As an illustration, the impulse propagation speed is equal to 0.5 meters per second in a non-myelinated axon whereas it can reach 120 meters per second in a myelinated one.

Figure A.3: Schematic drawing of a neuron When an action potential comes to the tip of an axon, it must be transmitted to the following cell. Usually, the axon is connected to a dendrite but connections such as axon-body, axon-axon or even dendrite-dendrite are found too. There are two different ways to connect two neurons: they can be physically joined and then the signal is transmitted directly, or they can be separated by a small gap, the synapse. In that case, the potential goes from one cell to the other by involving specific molecules, the neurotransmitters. Although the first method is the fastest one, the second approach is by far the most flexible. Besides, brain neurons often use this type of connection. At the arrival of a potential, neurotransmitters, which were ready to use, are released into the synapse. Then, they bind to receptors on the membrane of the post-synaptic neuron and they activate them. This opens an ionic channel and specific ions cross the post-synaptic membrane. An electric potential is then created according to the type of ion that entered and is successively transmitted through the receiving dendrite. At last, the neurotransmitters leave the receptors and go back to the pre-synaptic neuron where they can be either destroyed or recycled. There are more than 60 different types of neurotransmitters, such as dopamine, glutamate or GABA. However, each one of them behaves differently: some encourage the creation of a receptor potential (excitatory neurotransmitters) whereas others penalize it (inhibitory neurotransmitters). Anyway, it is the presence or the absence 157

A – Introduction to brain anatomy

of one of them that creates an excitatory or inhibitory potential, with more or less intensity according to the amount of released molecules. Furthermore, these mechanisms are not frozen once and for all but they continuously change, in the amount of neurotransmitters, in the affinity between two neurons through their synapses, etc. Those never-ending changes are the key of our learning skill.

1.2.2

The glial cells

The second category of cortical cells, the glial cells, does not play a direct role in the neuronal conduction but help the neurons to function properly. Their presence is essential for an efficient neuronal communication. Besides, several diseases such as multiple sclerosis are closely linked to their failures. They can be classified into three main categories. The astrocytes are the most prevalent and important ones. They provide neurons with nutrients and are able to regulate their external environment by removing redundant and unused ions, they recycle the neurotransmitters released into the synapses, etc. The microglia cells are as for them mobile cells which carry out the defensive tasks. They constitute the defensive system of the brain and protect it from extraneous bodies. At last, the oligodendrocytes provides the myelin sheath by wrapping their myelinated membrane around the axons.

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Bibliography Adalsteinsson, D. and Sethian, J. A. (1995). A fast level set method for propagating interfaces. Journal of Computational Physics, 118:269–277. Adalsteinsson, D. and Sethian, J. A. (1999). The fast constructionof extension velocities in level set methods. Journal of Computational Physics, 148:2–22. Antel, S. B., Bernasconi, A., Bernasconi, N., Collins, D. L., Kearney, R. E., Shinghal, R., and Arnold, D. L. (2002). Computational models of MRI characteristics of focal cortical dysplasia improve lesion detection. NeuroImage, 17(4):1755–60. Antel, S. B., Collins, D. L., Bernasconi, N., Andermann, F., Shinghal, R., Kearney, R. E., Arnold, D. L., and Bernasconi, A. (2003). Automated detection of focal cortical dysplasia lesions using computational models of their MRI characteristics and texture analysis. NeuroImage, 19(4):1748–1759. Ashburner, J. and Friston, K. (2000). Voxel-based morphometry–the methods. 11(6 Pt 1):805–821. Bailey, P. and Gibbs, F. A. (1951). The surgical treatment of psychomotor epilepsy. JAMA, 145:365–370. Barkovich, A. and Kuzniecky, R. (1996). Neuroimaging of focal malformations of cortical development. 13(6):481–494. Barkovich, A., Kuzniecky, R., Bollen, A., and Grant, P. (1997). Focal transmantle dysplasia: a specific malformation of cortical development. 49(4):1148–1152. Barkovich, A. J. and Kuziecky, R. I. (2000). Gray matter heterotopia. 55(11):1603– 1608. Barkovich, A. J., Kuzniecky, R. I., Jackson, G. D., Guerrini, R., and Dobyns, W. B. (2001). Classification system for malformations of cortical development: update 2001. Neurology, 57(12):2168–78. Bartko, J. J. (1991). Measurement and reliability: statistical thinking considerations. Schizophrenia Bullet., 17(3):483–489. 159

BIBLIOGRAPHY

Bastos, A. C., Comeau, R. M., Andermann, F., Melanson, D., Cendes, F., Dubeau, F., Fontaine, S., Tampieri, D., and Olivier, A. (1999). Diagnosis of subtle focal dysplastic lesions: curvilinear reformatting from three-dimensional magnetic resonance imaging. Ann Neurol, 46(1):88–94. Berger, H. (1929). Uber das elektrenkephalogram des menschen. Arch Psychiatri Nervenkr, 87:527–570. Bernasconi, A. (2004). Quantitative MR imaging of the neocortex. Neuroimaging Clin N Am, 14:425–436. Bernasconi, A., Antel, S. B., Collins, D. L., Bernasconi, N., Olivier, A., Dubeau, F., Pike, G. B., Andermann, F., and Arnold, D. L. (2001). Texture analysis and morphological processing of magnetic resonance imaging assist detection of focal cortical dysplasia in extra-temporal partial epilepsy. Ann Neurol, 49(6):770–5. Bernasconi, A., Bernasconi, N., Caramanos, Z., Reutens, D. C., Andermann, F., Dubeau, F., Tampieri, D., Pike, B. G., and Arnold, D. L. (2000). T2 relaxometry can lateralize mesial temporal lobe epilepsy in patients with normal mri. 12(6):739–746. Bernasconi, N., Andermann, F., Arnold, D. L., and Bernasconi, A. (2003). Entorhinal cortex mri assessment in temporal, extratemporal, and idiopathic generalized epilepsy. 44(8):1070–4. Borgefors, G. (1996). On digital distance transforms in three dimensions. Computer Vision and Image Understanding, 64(3):368–376. Bronen, R. A., Vives, K. P., Kim, J. H., Fulbright, R. K., Spencer, S. S., and Spencer, D. D. (1997). Focal cortical dysplasia of Taylor, balloon cell subtype: MR differentiation from low-grade tumors. AJNR Am J Neuroradiol, 18(6):1141– 1151. Caselles, V., Catte, F., Coll, T., and Dibos, F. (1993). A geometric model for active contours. Numerische Mathematik, 66:1–31. Caselles, V., Kimmel, R., and Sapiro, G. (1995). Geodesic active contours. Proc. 5th Int’l Conf. Comp. Vis., pages 694–699. Caselles, V., Kimmel, R., Sapiro, G., and Sbert, C. (1997). Minimal surfaces based object segmentation. IEEE Transactions on pattern analysis and machine intelligence, 19(4):394–398. Chan, S., Chin, S., Nordli, D., Goodman, R., DeLaPaz, R., and Pedley, T. (1998). Prospective magnetic resonance imaging identification of focal cortical dysplasia, including the non-balloon cell subtype. 44:749–757. 160

BIBLIOGRAPHY

Chan, T. F., Sandberg, B. Y., and Vese, L. A. (2000). Active contours without edges for vector-valued images. Journal of Visual Communication and Image Representation, 11:130–141. Chan, T. F. and Vese, L. A. (2001). Active contours without edges. IEEE Transactions on Image Processing, 10(2):266–277. Chopp, D. L. (1993). Computing minimal surfaces via level set curvature flow. Journal of Computational Physics, 106:77–91. Cohen, L. D. (1991). On active contour models and balloons. cvgip:iu, 53(2):211– 218. Collins, D. L., Neelin, P., Peters, T. M., and Evans, A. C. (1994). Automatic 3d intersubject registration of mr volumetric data in standardized Talairach space. J. Comput. Assist. Tomogr., 18(192-205). Colliot, O., Antel, S. B., Bernasconi, N., and Bernasconi, A. (2004). Quantitative MRI profiling of focal cortical dysplasia. In Abstracts of the 58th Annual Meeting of the American Epilepsy Society AES 2004, Epilepsia, volume 45, pages 286–287, New-Orleans, USA. Colombo, N., Tassi, L., Galli, C., Citterio, A., Lo Russo, G., Scialfa, G., and Spreafico, R. (2003). Focal cortical dysplasias: MR imaging, histopathologic, and clinical correlations in surgically treated patients with epilepsy. AJNR Am J Neuroradiol, 24:724–733. Edwards, J. C., Wyllie, E., Ruggeri, P. M., Bingaman, W., L¨ uders, H., Kotagal, P., Dinner, D. S., Morris, H. H., Prayson, R. A., and Comair, Y. G. (2000). Seizure outcome after surgery for epilepsy due to malformation of cortical development. Neurology, 55:1110–1114. Engel, J. (2001). A proposed diagnostic scheme for people with epileptic seizures and with epilepsy: report of the ILAE task force on classification and terminology. 42(6):796–803. Engel, J., J. (1996). Surgery for seizures. 334(10):647–652. Ferrier, D. (1874). On the localisation of the functions of the brain. BMJ, 2:766–767. Gomes, J. and Faugeras, O. (2000). Reconciling distance functions and level sets. Journal of Visual Communication and Image Representation, 11:209–223. Guerrini, R., Barkovich, A. J., Sztriha, L., and Dobyns, W. B. (2000). Bilateral frontal polymicrogyria: a newly recognized brain malformation syndrome. 54(4):909–913. 161

BIBLIOGRAPHY

Han, X., Xu, C., and Prince, J. L. (2003). A topology preserving level set method for geometric deformable model. IEEE Transactions on pattern analysis and machine intelligence, 25(6):755–768. Jackson, G. D., Kuzniecky, R. I., and Cascino, G. D. (1994). Hippocampal sclerosis without detectable hippocampal atrophy. Neurology, 44:42–46. Jackson, J. H. (1873). Observations on the anatomical, physiological, pathological investigations of epilepsies. West Riding Lunatic Asylum Rep, 3:315. Jones, S. E., Buchbinder, B. R., and Aharon, I. (2000). Three-dimensional mapping of cortical thickness using laplace’s equation. Human Brain Mapping, 11(1):12–32. Kass, M., Witkin, A., and Terzopoulos, D. (1987). Snakes : Active contour models. ijcv, 1(4):321–331. Kassubek, J., Huppertz, H. J., Spreer, J., and Schulze-Bonhage, A. (2002). Detection and localization of focal cortical dysplasia by voxel-based 3-D MRI analysis. Epilepsia, 43(6):596–602. Krissian, K. and Westin, C.-F. (2004). Fast sub-voxel reinitialization of the distance map for level set methods. Technical report, Laboratory of mathematics in imaging, Harvard medical school. Kucniecky, R. I. and Jackson, G. D. (1995a). Disorders of neuronal migration and organization. In Magnetic Resonance in Epilepsy, pages 235–255. Raven Press, New York. Kucniecky, R. I. and Jackson, G. D. (1995b). Introduction to epilepsy. In Magnetic Resonance in Epilepsy, pages 1–14. Raven Press, New York. Kucniecky, R. I. and Jackson, G. D. (1995c). Temporal lobe epilepsy. In Magnetic Resonance in Epilepsy, pages 107–182. Raven Press, New York. Kuzniecky, R. I., Andermann, F., and Guerrini, R. (1994). The epileptic spectrum in the congenital bilateral perisylvian syndrome. cbps multicenter collaborative study. 44(3 Pt 1):379–385. Leventon, M., Grimson, E., and Faugeras, O. (2000). Statistical shape influence in geodesic active contours. In Proc. of Computer Vision and Pattern Recognition CVPR 2000. Lorenzo, N. Y., Parisi, J. E., Cascino, G. D., Jack, C. R., Marsh, W. R., and Hirschorn, K. A. (1995). Intractable frontal lobe epilepsy: pathological and MRI features. 20(2):171–178. 162

BIBLIOGRAPHY

Malladi, R., Kimmel, R., Adalsteinsson, D., Sapiro, G., Caselles, V., and Sethian, J. A. (1996). A geometric approach to segmentation and analysis of 3D medical images. Proc. MMBIA, pages 244–252. Malladi, R., Sethian, J. A., and Vemuri, B. C. (1995). Shape modeling with front propagation: A level set approach. IEEE Transactions on pattern analysis and machine intelligence, 17(2):158–174. Osher, S. J. and Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed : algorithms based on Hamilton-Jacobi formulation. Journal of Computational Physics, 79:12–49. Palmini, A., Andermann, E., and Andermann, F. (1994). Prenatal events and genetic factors in epileptic patients with neuronal migration disorders. Epilepsia, 35(5):965–973. Palmini, A., Najm, I., Avanzini, G., Babb, T., Guerrini, R., Foldvary-Schaefer, N., Jackson, G., L¨ uders, O. H., Prayson, R., R, S., and Vinters, H. V. (2004). Terminology and classification of the cortical dysplasias. Neurology, 62:S2–S8. Paolicchi, J. M., Jayakar, P., Dean, P., Yaylali, I., Morrison, G., Prats, A., Resnik, T., Alvarez, L., and Duchowny, M. (2000). Predictors of outcome in pediatric epilepsy surgery. 54(3):642–647. Paragios, N. and Deriche, R. (2002a). Geodesic active regions: A new paradigm to deal with frame partition problems in computer vision. Journal of Visual Communication and Image Representation. Paragios, N. and Deriche, R. (2002b). Geodesic active regions and level set methods for supervised texture segmentation. International Journal of Computer Vision, 46(3):223–247. Paragios, N., Mellina-Cottardo, O., and Ramesh, V. (2001). Gradient vector flow fast geodesic active contours. In 8th IEEE International Conference on Computer Vision, 2001., volume 1, pages 67–73. Rakic, P. (1995). Radial versus tangential migration of neuronal clones in the developing cerebral cortex. Proc Natl Acad Sci USA, 92(25):11323–11327. Rosenfeld, A. and Pfaltz, J. (1966). Sequential operations in digital picture processing. Journal of the Association for Computing Machinery, 13(4):471–494. Sapiro, G. (1997). 68(2):247–253.

Color snakes.

Computer vision and image understanding,

163

BIBLIOGRAPHY

Sethian, J. A. (1989). A review of recent numerical algorithms for hypersurfaces moving with curvature dependant speed. J. Differential Geometry, 31:131–161. Sethian, J. A. (1999a). Level-set methods and fast marching methods. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge University Press, 2nd edition. Sethian, J. A. (1999b). Level-set methods and fast marching methods. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, chapter 5-6. Cambridge University Press, 2nd edition. Sisodiya, S. M. (2000). Surgery for malformations of cortical development causing epilepsy. 123:1075–1091. Sisodiya, S. M., Free, S. L., Fish, D. R., and Shorvon, S. D. (1995). Increasing the yield from volumetric MRI in patients with epilepsy. Magnetic Resonance Imaging, 13(8):1147–1152. Sled, J. G., Zijdenbos, A. P., and Evans, A. C. (1998). A nonparametric method for automatic correction of intensity nonuniformity in mri data. IEEE Transact. Med. Imaging, 17:87–97. Smith, S. (2000). Robust automated brain extraction. In Sixth Int. Conf. Funct. Mapp. Hum. Brain, page 625. Sussman, M. and Fatemi, E. (1999). An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow. SIAM J. Sci. Comput., 20(4):1165–1191. Sutula, T. P., Hagen, J., and Pitkanen, A. (2003). Do epileptic seizures damage the brain ? Curr Opin Neurol., 16:189–195. Tassi, L., Colombo, N., Garbelli, R., Francione, S., G, L. R., Mai, R., Cardinale, F., Cossu, M., Ferrario, A., Galli, C., Bramerio, M., Citterio, A., and Spreafico, R. (2002). Focal cortical dysplasia: neuropathological subtypes, eeg, neuroimaging and surgical outcome. 125(Pt 8):1719–1732. Taylor, D., Falconer, M., Bruton, C., and Corsellis, J. (1971). Focal dysplasia of the cerebral cortex in epilepsy. J Neurol Neurosurg Psychiatry, 34:369–387. Wiebe, S., Bellhouse, D. R., C, F., and M, E. (1999). Burden of epilepsy: the Ontario health survey. Can. J. Neurolo. Sci., 26(4):263–270. Wilke, M., Kassubek, J., Ziyeh, S., Schulze-Bonhage, A., and Huppertz, H. J. (2003). Automated detection of gray matter malformations using optimized voxel-based morphometry: a systematic approach. NeuroImage, 20(1):330–343. 164

BIBLIOGRAPHY

Xu, C., Pham, D., and Prince, J. (2000). Medical image segmentation using deformable models. In Fitzpatrick, J. and Sonka, M., editors, Handbook of Medical Imaging, volume 2, pages 129–174. SPIE Press. Xu, C. and Prince, J. (1998a). Generalized gradient vector flow external forces for active contours. Signal Processing - An International Journal, 71(2):131–139. Xu, C. and Prince, J. (1998b). Snakes, shapes and gradient vector flow. tip, 7(3):359– 369. Xu, C., Yezzi, A., and Prince, J. L. (1999). On the relationship between parametric and geometric active contours. Technical Report 99-14, JHU/ECE. Yang, J. and Duncan, J. S. (2004). 3D image segmentation of deformable objects with joint shape-intensity prior models using level sets. Medical Image Analysis, 8:285–294. Yezzi, A., Kichenassamy, S., Kumar, A., Olver, P., and Tannenbaum, A. (1997). A geometric snake model for segmentation of medical imagery. IEEE Transactions on medical imaging, 16(2):199–209. Yezzi, A., Tsai, A., and Willsky, A. (1999). A statistical approach to snakes for bimodal and trimodal imagery. IEE Int’l Conf. Comp. Vision, page 898. Yilmazer-Hanke, D. M., Wolf, H. K., Schramm, J., Elger, C. E., Wiestler, O. D., and Blumcke, I. (2000). Subregional pathology of the amygdala complex and entorhinal region in surgical specimens from patients with pharmacoresistant temporal lobe epilepsy. 59(10):907–920. Yui, S., Hara, K., Zha, H., and Hasegawa, T. (2002). A fast narrow band method and its application in topology-adaptive 3-D modeling. In 16th International Conference on Pattern Recognition, volume 4, pages 122–125. Zeng, X., Staib, L. H., Schultz, R. T., and Duncan, J. S. (1999). Segmentation and measurement of the cortex from 3D MR images using coupled surfaces propagation. IEEE Transactions on medical imaging, 18(10):100–111. Zhu, S. C. and Yuille, A. (1996). Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Transaction on pattern analysis and machine intelligence, 18(9):884–900. Zijdenbos, A. P., Dawant, B. M., Margolin, R. A., and Palmer, A. C. (1994). Morphometric analysis of white matter lesions in MR Images: method and validation. IEEE Transactions of Medical Imaging, 13(4):716–724.

165

BIBLIOGRAPHY

166