Building and using a statistical 3D motion atlas for analyzing myocardial contraction in MRI Nicolas Rougon, Caroline Petitjean and Fran¸coise Prˆeteux ARTEMIS Project Unit GET / Institut National des T´el´ecommunications 9, Rue Charles Fourier – 91011 Evry, FRANCE ABSTRACT We address the issue of modeling and quantifying myocardial contraction from 4D MR sequences, and present an unsupervised approach for building and using a statistical 3D motion atlas for the normal heart. This approach relies on a state-of-the-art variational non rigid registration (NRR) technique using generalized information measures, which allows for robust intra-subject motion estimation and inter-subject anatomical alignment. The atlas is built from a collection of jointly acquired tagged and cine MR exams in short- and long-axis views. Subject-specific non parametric motion estimates are first obtained by incremental NRR of tagged images onto the end-diastolic (ED) frame. Individual motion data are then transformed into the coordinate system of a reference subject using subject-to-reference mappings derived by NRR of cine ED images. Finally, principal component analysis of aligned motion data is performed for each cardiac phase, yielding a mean model and a set of eigenfields encoding kinematic variability. The latter define an organ-dedicated hierarchical motion basis which enables parametric motion measurement from arbitrary tagged MR exams. To this end, the atlas is transformed into subject coordinates by reference-to-subject NRR of ED cine frames. Atlas-based motion estimation is then achieved by parametric NRR of tagged images onto the ED frame, yielding a compact description of myocardial contraction during diastole. Keywords: Non rigid registration, generalized information measures, exclusive f -information, statistical cardiac motion atlas, model-based motion estimation, myocardial contractile function, cardiac MRI.

1. INTRODUCTION MRI is a key modality for dynamically imaging the heart anatomy and function. It provides a valuable investigation tool for early diagnosis and clinical/therapeutical follow-up, delivering all the necessary information for assessing a large variety of cardio-vascular pathologies, including those related to morphology and flow rate of coronary arteries, myocardial perfusion and cardiac function. In particular, the relevance of MRI for identifying and localizing acute and chronic myocardial ischemia through the detection of kinematic abnormalities has been established.1 Accordingly, abundant research efforts have been devoted to cardiac motion recovery and deformation assessment from MR image sequences.2, 3 The reference MR modality for imaging myocardial strain is tagged MRI.4, 5 Measuring myocardial deformations from tagged MR sequences relies on estimating a dense displacement field consistent with the motion of the structured tagging pattern. Classical methods comprise differential and phase-based optical flow techniques, and segmentation-based approaches combining tag extraction, sparse motion estimation along tags and dense motion interpolation over the image domain.6 However, their clinical applicability is questioned by several limitations: differential optical flow fails for large displacements; phase-based optical flow is limited to 1D tags and sensitive to artifacts; tag segmentation often requires supervision to deal with tag fading out and tag destruction along endocardial boundaries induced by blood flows within the cardiac chambers. Recently, intensity-based non rigid registration (NRR) methods7, 8 have been proposed independently by Petitjean et al.9 and Chandrashekara et al.10 as robust, segmentation-free techniques to obtain dense displacement Further author information: (Send correspondence to Nicolas Rougon) Nicolas Rougon: E-mail: [email protected], Telephone: +33 (0)1 60 76 46 44

estimates in a non-supervised way. Here, the use of information-theoretic similarity criteria11, 12 allows to robustly deal with tag fading out without specifying a demagnetization model. Based on Rueckert’s work,13 the approach in10, 14, 15 delivers parametric motion estimates by maximizing normalized mutual information16 over a space of free form deformations (FFD). Our approach9, 17 relies on the Ali-Silvey class of generalized information measures for which a generic variational optimization framework, encompassing both non parametric18 and parametric19 displacement spaces, has been developed. This class, which comprises mutual information20, 21 and normalized mutual information16 as subcases, extends Shannon modeling by allowing to explicitely account for image statistics through the adaptation of the information metrics. Within this framework, the relevance of exclusive f -information9, 19 with the Iα information metrics for recovering cardiac motion has been established.17 Relying on these works, we address in this paper the issue of compactly modeling and quantifying myocardial contraction from 4D MR sequences, and present a generic approach for building and using a non parametric statistical 3D motion atlas for the normal heart. The atlas is built from a collection of jointly acquired tagged MR and cine MR exams in short- and long-axis views. Subject-specific non parametric motion estimates are first obtained by incremental NRR of tagged images onto the end-diastolic (ED) frame. Individual motion data are then transformed into the coordinate system of a reference subject using subject-to-reference mappings derived by NRR of cine ED images. Finally, principal component analysis of aligned motion data is performed for each cardiac phase to produce a statistical atlas, comprising a mean motion model and a set of eigenfields encoding kinematic variability. The latter define an organ-dedicated hierarchical motion basis which enables parametric motion measurement from arbitrary tagged MR exams. To this end, the atlas is transformed into subject coordinates by reference-to-subject NRR of ED cine frames. Atlas-based motion estimation is then achieved by parametric NRR of tagged images onto the ED frame, yielding a compact description of myocardial contraction during diastole. This paper is organized as follows. The theoretical foundations of NRR using generalized information measures are recalled in Section 2. The atlas construction scheme is detailed in Section 3. In Section 4, atlas-based motion estimation from arbitrary healthy MR exams is presented, allowing quantitative assessment of myocardial deformation as explained in Section 5. Results are presented and discussed in Section 6 before concluding.

2. NRR USING GENERALIZED INFORMATION MEASURES Given two images I1 and I2 defined over a domain D ⊂ Rn with values in an intensity interval Ω, registering I2 onto I1 consists in finding a smooth displacement field u within some function space T over Rn , so that I2u (x) := I2 (x + u(x)) is similar to I1 (x) for any x ∈ D. The field u is usually decomposed as the sum Θ u := uΘ global + ulocal of a global (usually affine) parametric component uglobal defined by a low-dimensional p parameter Θ ∈ R , and of a local component ulocal which, depending on the applicative requirements, is modeled as a non parametric or a parametric transform∗ . These two components are identified independently in a sequential fashion. A regularized approach for solving this ill-posed problem defines a solution as a minimizer u∗ of a functional J (u) := S(u) + λR(u) over T . The latter combines a similarity functional S quantifying the discrepancy between I1 and I2u , and a regularizing functional R ensuring smooth solutions. λ > 0 is a weighting coefficient. Finding u∗ is achieved iteratively by gradient descent over T . Non parametric motions yield a flow over D: ∂u = −∂u J (u(x, t)) ∂t with initial condition u(·, 0) = 0, where ∂u J is the 1st variation of J over T . For linear parametric motions, such that uΘ (x) := B(x)Θ and B(x) is a matrix encoding a local basis of T , a minimizer Θ∗ is derived from the flow over Rp : ∂Θ = −∂Θ J (uΘ(t) ) ∂t ∗

with initial condition Θ(0) = 0, where ∂Θ J is the 1st variation of J w.r.t. Θ. This leads to defining u∗ = uΘ . ∗

In the latter case, it will be denoted by uΘ local .

2.1. Exclusive f -information Robust NRR can be performed in a statistical framework by modeling I1 and I2u as random variables (RV) over u u Ω (with marginal p.d.f. pI1 and pI2 , and joint p.d.f. pI1 ,I2 ), and by defining S as a statistical similarity measure between probability distributions. Classical choices are mutual information (MI)20, 21 and normalized mutual information (NMI)16 which provide universal answers for unknown image statistics. Generalized information measures22 allow to explicitely account for data statistics by tuning information measurement to observations. An important subclass is f -information† which is defined for any convex continuous mapping f over R+ as22 :  I1 ,I u 
p 2 (i1 , i2 ) u I I2u If (I1 , I2 ) := p1 (i1 )p (i2 )f (1) di1 di2 u pI1 (i1 )pI2 (i2 ) Ω2 and includes MI as a special case associated to the Kullback metrics fKL (x) := x log x. In order to deal robustly with lacunar joint statistics occurring for partially overlapping image supports, we have introduced a derived class, called exclusive f -information and defined as9, 18 : Zf (I1 , I2u ) := If (I1 , I1 ) + If (I2u , I2u ) − 2If (I1 , I2u )

.

(2)

Exclusive f -information represents the information exclusively contained in I1 and I2u when observed jointly, in the sense of the information metrics f . Since Zf is minimal for maximally correlated RVs, we set: S(u) := Zf (I1 , I2u )

.

(3)

2.2. Gradient flows of exclusive f -information u

u

The first variation of S(u) is obtained under closed-form by approximating pI1 ,I2 and pI2 via global Parzen estimates over the region of overlap D of the two images21, 23 :
1 I1 ,I2u p (i1 , i2 ) ≈ K2 (I1 (x) − i1 , I2u (x) − i2 ) dx (4) |D| D
u 1 pI2 (i2 ) ≈ K1 (I2u (x) − i2 ) dx (5) |D| D where K2 (resp. K1 ) is a 2D (resp. 1D) density kernel, and |D| denotes the volume of D. The kernel widths, which have a critical impact on convergence and accuracy,24 can be automatically determined using a leave-k-out cross-validation technique applied to marginal histograms.25 For non parametric motions, one shows: u

∂u S(u) = −V I1 ,I2 (x) ∇I2u (x)

(6)

9, 19

where the velocity of the registration flow is given by :

     u|i   u   2 ∂Lu ∂L2 1 1 ∂Lu 1 3 I1 ,I2u u u I1 (x), I2 (x) + EI1 I2 (x) . (x) := − K1  (I2 (x)) V K2  K1  |D| ∂i2 ∂i2 |D| ∂i2 (7) Here, EI1 (·) denotes expectation w.r.t. the RV I1 and the symbol  refers to 2D (resp. 1D) convolution u I1 ,I2

(i1 ,i2 )  u u u u  u over Ω2 (resp. Ω). Letting ρu (i1 , i2 ) := pI1 , one has: Lu u 1 := f (ρ ) , L2 := f (ρ ) − ρ f (ρ ) , p (i1 ) pI2 (i2 )  u    u|i 1 pI2 − f  I12u and L2 1 (·) := Lu Lu u I2 3 := 2f 2 (i1 , ·), ∀i1 ∈ Ω. p

p

For linear parametric motions, the flow in the parameter space is as follows19 :
  u t ∂Θ S(uΘ ) = − V I1 ,I2 (x) (∇I2u ) B (x) dx .

(8)

D

In both cases, it can be proven that using f = fKL allows to exploit the statistics of I2u in a more extensive fashion than MI9, 18 . †

Here, we confine ourselves to integral f -information. Non-integral f -information is addressed in .9, 19

2.3. First-order regularization Non-parametric and high-order parametric motion models require explicit regularization. We use the NagelEnkelmann oriented smoothness constraint26 :
  1 Tr ∇uT TI1 ∇u dx (9) R(u) := 2 D which is a geometric stabilizer involving the Jacobian ∇u of u and the image-dependent tensor TI1 : TI1 :=

   1 β + ||∇I1 ||2 Id − ∇I1 ∇I1T ||∇I1 ||2 + 2β

(10)

where β > 0 is a contrast parameter. Contrary to the classical linear elasticity constraint which acts uniformly over the domain D, this stabilizer promotes intra-region smoothing, yielding motion estimates localized within image objects. For non-parametric motions, its first variation is: ∂u R(u) = −∇ · (TI1 ∇u)

.

(11)

For linear parametric motions, ∂Θ R(uΘ ) linearly depends on Θ and is computed analytically.

2.4. Information metrics optimization An important class of mappingsαf consists of parametric information metrics fα depending on a parameter α > 0. which leads to Iα -information and fα (x) := |x − 1|α which defines Instances comprise fα (x) := x −αx+α−1 α(α−1) χα -information‡ .22 These metrics are limiting cases of the Kullback metrics: lim fα = fKL . Considering J as a α→1

functional J (u, α) depending both on u and α, an optimal value for α can be searched as: α∗ = arg min J (u, α). Estimating α∗ by gradient descent yields a flow over R+ :

α

∂α = −∂α J (u, α) ∂t with initial condition α(0) = 1 (i.e. fα(0) = fKL ) , where the first variation of J w.r.t. α is given by:    

 I  Iu 2 ∂fα 2 ∂fα 1 1 ∂α J (u, α) = p1 (i1 ) p2 (i2 ) di1 + di2 u ∂α pI1 ∂α pI2 Ω Ω
u ∂fα u (ρ (i1 , i2 )) di1 di2 −2 pI1 (i1 )pI2 (i2 ) ∂α 2 Ω

(12)

(13)

Coupling (13) with the regularized motion flow (6,11) or (8,11) leads to an alternated minimization scheme where each variable of the pair (u, α) is iteratively updated independently, keeping the other constant and equal to its estimate at the previous iteration.

3. CONSTRUCTING A CARDIAC MOTION ATLAS

  Given a collection Iτi (1 ≤ i ≤ M, 1 ≤ τ ≤ Ni ) of 3D tagged   MR exams of M healthy subjects, comprising respectively Ni frames, subject-specific motion estimates uiτ (1 ≤ i ≤ M ) can be derived by non parametric i NRR of frames Iτi onto the ED image IN using the regularized exclusive f -information flow (6,11). Registration i i is achieved incrementally by using uτ −1 as initial condition for estimating uiτ . A statistical 4D non parametric motion atlas  is then built by performing, for each cardiac phase τ , principal component analysis (PCA) of the fields uiτ (1 ≤ i ≤ M ). Accounting for cardiac rhythm and anatomy variability requires their prior spatiotemporal alignment on a common reference coordinate system. ‡

Another example, related to non-integral f -informations, is the Renyi metrics.9, 22

Temporal alignment The inter-frame time interval is constant during MR acquisition. Since cardiac rhythm is subject-dependent, the length of MR image sequences is therefore variable. This implies a temporal alignment of subject-specific motion data sequences on a reference length N , which has been set to 17 frames. Using an ECG-gated acquisition protocol with constant time delay after the R-wave, alignment is simply achieved using linear interpolation. Spatial alignment Following temporal registration, spatial alignment of individual motion data on a common coordinate system is achieved from jointly acquired 3D cine MR data, using the generic scheme proposed by Rao et al.15 The reference coordinate system is prescribed by the ED cine frame of a reference subject. Individual motion data are transformed into this system via subject-to-reference mappings derived by sequentially performing affine and localnon  parametric NRR between cine ED images. The resulting collection of aligned ˜ iτ (1 ≤ i ≤ M, 1 ≤ τ ≤ N ). motion data is denoted by u The key difference between Rao et al. work15 and our approach relies in motion modeling. In the former, a parametric NRR method maximizing NMI over a FFD space10 is enforced to produce a parametric motion atlas. By contrast, we use exclusive f -information-based NRR over a non parametric transform space to generate a non parametric motion atlas. This approach offers the advantage of preserving the statistical diversity of the motion content of MR data in the learning stage, which is a crucial point for further modeling kinematic variability. Using parametric motion estimates yields indeed biaised statistical models, where statistical modes reflect both the data and the motion representation.  i ˜ τ (1 ≤ i ≤ M ) for each cardiac phase τ Statistical modeling Performing PCA§ of aligned motion data u (1 ≤ τ ≤ N ) finally produces a 4D statistical atlas, comprising a mean motion model: ¯ τ := u

M 1  i ˜ u M i=1 τ

(14)

and a set (vτj ) (1 ≤ j ≤ Mτ ) of Mτ < M eigenfields encoding kinematic variability. The latter are obtained ˇ iτ := u ˜ iτ − u ¯τ by singular value decomposition (SVD) of the covariance matrix Cτ of centered motion data u i i ˇ τ,k := u ˇ τ (xk ) (k ∈ Γ) and |Γ| be the image (1 ≤ i ≤ M ). Given a lexicographic indexing Γ of image pixels, let u ˇ i : = [(ˇ size. Denoting by U uiτ,k )T ]Tk∈Γ the vector of centered motion data for subject i, ordered according to Γ, τ Cτ is the |Γ| × |Γ| matrix defined as: Cτ :=

M 1  ˇi ˇi T ˇ τU ˇT U (U ) = U τ M i=1 τ τ

ˇ τ := [U ˇ1 ...U ˇ M ] is the |Γ| × M matrix with column vectors U ˇ i . Computing Cτ is intractable in where U τ τ τ practice because of its high dimension. However, its SVD can be efficiently extracted using the dimensionality reduction technique proposed by Turk and Pentland.27 Letting (wτj ) (resp. (λjτ )) (1 ≤ j ≤ M ) denote the ˇ τ , one indeed shows that27 : ˇTU eigenvectors (resp. eigenvalues) of the M × M matrix U τ ˇ τ wτj vτj = U

(15)

and that (λjτ ) (1 ≤ j ≤ M ) are the singular values of Cτ . The complexity of SVD computation is therefore reduced to a M × M eigenvalue problem with M  |Γ|. Ordering the eigenvalues in decreasing order (λkτ ≥ λlτ , 1 ≤ k ≤ l ≤ M ), the SVD is finally truncated so as to retain the Mτ most significant eigenfields according to a total variance criterion:    |λkτ |2   1≤k≤M     Mτ := arg min ≤ ϑ (16)  M  |λjτ |2 1≤j≤M §

An alternative solution is independent component analysis (ICA).

where ϑ is a predefined fraction of the variance στ :=



|λjτ |2 of centered motion data at phase τ .

1≤j≤M

4. ATLAS-BASED MYOCARDIAL MOTION ESTIMATION The eigenfields (vτj ) define an organ-dedicated, time-adaptive, hierarchical motion basis. This allows to define a linear parametric representation uΘτ for myocardial displacements at any phase τ (1 ≤ τ ≤ N ): ¯ τ (x) + Vτ (x)Θτ uΘτ (x) := u

(17)

M

where Vτ (x) := [vτ1 (x) . . . vτ τ (x)] is the Mτ × d matrix with column vectors vτj (x) (x ∈ D) and Θτ ∈ RMτ . By construction, this representation is tailored to myocardial kinematics. This contrasts with classical parametric modeling approaches which enforce generic analytical function bases (e.g. FFD, wavelet splines, Fourier series) uniformly at any phase and any slice level. Based on (17), parametric motion estimation from arbitrary tagged MR exams can then be achieved using the ¯ τ and regularized, exclusive f -information-based parametric NRR flow (8,11) with initial condition u(·, 0) = u Θτ (0) = 0. Since the motion atlas is defined w.r.t. the coordinate system of the reference subject, a preliminary transformation into subject coordinates is necessary. The corresponding reference-to-subject mapping, which is applied to the mean motion and eigenfields, is derived by sequentially performing affine and local non parametric NRR between ED cine MR images.

5. QUANTIFYING MYOCARDIAL DEFORMATIONS Myocardium has a complex architecture: fibers in the mid wall are circumferential whereas subendocardial fibers are longitudinally directed. This results in inhomogeneous and complex contraction patterns, correlated to fiber structure. In particular, the left ventricle (LV) deformations comprise radial thickening, circumferential shortening, torsion, and longitudinal shortening. Local deformation information is derived either from displacement fields themselves (0th -order attributes), or from their spatial variations encoded by the strain tensor E(u) (1st -order attributes): E(u) :=

 1 ∇u + ∇uT + ∇uT ∇u 2

(18)

which is computed numerically using central finite differences. • 0th -order attributes allow for assessing directional displacements. For short-axis (SA) views, radial displacement (resp. rotation/torsion along the long axis (LA) of the heart) is defined as the component (dT u∗τ ) d of u∗τ along the unit vector field d normal (resp. tangent) to myocardial boundaries. For LA views, radial (resp. longitudinal) motion is similarly measured by considering the field normal to myocardial boudaries (resp. parallel to LA). • Relevant 1st -order attributes, derived from E(u), comprise radial and circumferential strains and shears, and eigenvectors/values which define extremal deformation directions/magnitudes. Averaging these various attributes over each region of a layered regional model of the myocardium yields segmental descriptions of the deformation. Due to the lack of consensus regarding RV segmentation, which is made difficult by its relative thinness and complex geometry, only LV segmental measurements are reported in the sequel. They refer to a 3 SA-levels, 16-segments LV model compliant with the American Heart Association (AHA) recommendations28 shown on Figure 1.

1 7 2

8

13 14

3

9

17 15

12

6

16 11

5

10 4

Figure 1. The bull-eye representation of the 3-levels, 16-segments LV model recommanded by the AHA. Segments #1-6, #7-12 and #13-16 refer to the basal, mid and apical levels, respectively. Segment #17, associated with the apex, is optional and has not been used.

6. RESULTS AND DISCUSSION Experiments have been performed on a cardiac MR database of healthy volunteers. Tagged MR data were acquired during multiple breath holds on a 1.5T GE scanner using the SPAMM technique (285mm field of view, 256x128 acquisition matrix, 1.48x1.48mm in-plane resolution, 45ms time interval, 10mm tag spacing). For each patient, 12 base-to-apex SA and 8 LA sequences were captured. Cine MR sequences were acquired at the same slice levels in identical conditions. For atlas construction purpose, an homogeneous population of M = 11 subjects (8 men/3 women, mean age±SD = 30±4 years) was extracted from the database. Subject-specific motion estimation has been achieved by applying exclusive Iα -information-based NRR to 3D data sets generated by integrating SA and LA images. A multiresolution implementation based on Gaussian pyramids with 3 resolution levels (including full resolution) has been used. The hyperparameters of the method have been determined empirically and kept constant for all images. Empirical p.d.f. estimates have been derived from 256 bins normalized histograms, smoothed by Gaussian Parzen kernels with widths σ1 = σ2 = 3. Trilinear interpolation has been retained for subpixel computations. The Nagel-Enkelmann constrast parameter and the regularization parameter have been set to β = 10000 and λ = 20, respectively. Integration was performed via a simple Euler explicit gradient descent scheme with fixed time step ∆t = 0.015. To speed up computations, the optimal value α = 1.22 derived for the reference subject was used for processing the other exams. We have validated this strategy by checking that the values of α delivered by the metrics optimization scheme (13) were always very close to α . A typical motion estimation result ¶ is shown on Figure 2. The statistical motion atlas has then been built from subject-specific motion data as described in Section 3. Truncating the PCA decomposition to ϑ = 95% of the total variance στ has resulted into retaining Mτ = 7 λj components at any cardiac phase τ . Monitoring the ratio σττ along the cardiac cycle reveals that the contribution of each eigenfield to the total variance remains remarkably stationary. This is allustrated on Figure 3 (left) for the three largest eigenvalues (j = 1, 2, 3) which have been found to concentrate 53.1%, 18.9% et 9.4% of στ , respectively, in average during the cardiac cycle, yielding a total of 81.4% of στ . On Figure 3 (right), the mean  λjτ  ratio N τ =1 στ (1 ≤ j ≤ Mτ ) during the cardiac cycle is shown as a function of the statistical mode j for the apical, mid and basal levels. This indicates that motion variability exhibits a larger dispersion at the apex, whereas mid and basal levels behave similarly. ¯ τ of the atlas at the basal, mid and apical levels are shown on SA slices through the mean component u Figure 4. From the mean motion model, deformation attributes can be computed, yielding reference quantitative measurements at the local (Figure 5-6) and segmental (Figure 7) scales. In the sequel, we confine ourselves to 0th order attributes. SA slices through the resulting mean contraction and torsion maps are shown on Figure 5 and Figure 6, respectively. Functional observations on patient-specific data appear to be consistently incorporated in the atlas, namely an overall contraction over the whole myocardium which is enhanced on the inferior wall, and a wringing torsion behaviour comprising clockwise basal rotation and counterclockwise apical rotation. LA slices through the mean contraction and torsion maps (not shown) indicate that longitudinal contraction mostly ¶

Myocardium segmentation, required for visualization purpose, was derived from cine MR data using a classical watershed morphological segmentation scheme.

I2

I1

∗

I2u

u∗

(a)

(b)

Figure 2. Information-theoretic NRR of (a) SA and (b) LA tagged MR data using exclusive f -information. The Iα information metrics has been used (α∗ = 1.22). A synthetic grid warped by u∗ has been overlaid to enhance tag motion recovery.

takes place at the apex whereas radial contraction is localized on the LV free walls. In any case, quantifications prove to be in accordance with well-established results regarding heart anatomy.29 Finally, atlas-based parametric motion estimation has been tested on arbitrary tagged MR exams outside the training dataset, and the results have been systematically compared to non parametric NRR with the same hyperparameter settings using a classical L2 error norm. At the apical level, the mean square error (MSE) for motion amplitude was found to be 0.15 ± 0.22 pixels, varying between 0 and 1.53 pixels. At the mid level, the MSE was 0.23 ± 0.34 pixels, varying between 0 and 3.59 pixels. Finally, at the basal level, the MSE was 0.40 ± 0.59 pixels, ranging from 0 to 3.28 pixels. The conclusion is that, compared to a non parametric NRR whose complexity equals the image size, atlas-based parametric NRR allows for deriving motion estimates with a reduced complexity (7 parameters per cardiac phase) and without notable loss of accuracy, providing the additional computational cost induced by reference-to-subject alignement.

7. CONCLUSION AND PERSPECTIVES We have developed a fully automatic method for modeling and compactly quantifying myocardial function from MR non-pathological data using generalized information-theoretic NRR. The resulting non parametric statistical 3D motion atlas consistently captures functional observations made on healthy patient-specific data. Applied to parametric motion measurement, it delivers accurate measurements which allow to account for regional inhomogeneities of velocity distribution with only 7 parameters per cardiac phase. Future work includes improving the statistical relevance of the atlas by extending the reference dataset, and investigating its use for computer-assisted diagnosis of designated pathologies, including myocardial ischemia, dilated/hypertrophic cardiomyopathies and sarcoidosis.

ACKNOWLEDGMENTS This work has been carried out in the framework of a research partnership between GET/INT-ARTEMIS and the Service de Radiologie Centrale of Groupe Hospitalier Piti´e-Salpˆetri`ere, Paris, France. The authors thank Prof. Ph. Cluzel for providing the MRI data, and for fruitful discussions and medical expertise on radiological and clinical aspects.

REFERENCES 1. M.J. Lipton, J. Bogaert, L.M. Boxt, R.C. Reba, “Imaging of ischemic heart disease,” European Radiology 12, pp. 1061–1080, 2002. 2. A.F. Frangi, W.J. Niessen, M.A. Viergever, “Three-dimensional modeling for functional analysis of cardiac images: A review,” IEEE Transactions in Medical Imaging 20(1), pp. 26–35, 2001. 3. A. Frangi, D. Rueckert, J. Duncan, “Three-dimensional cardiovascular image analysis,” IEEE Transactions on Medical Imaging 21(9), pp. 1005–1010, 2002. 4. E.A. Zerhouni, D.M. Parish, W.J. Rogers, A. Yang, E.P. Shapiro, “Human heart: tagging with MR imaging - A Method for noninvasive assessment of myocardial motion,” Radiology 169(1), pp. 59–63, 1988. 5. L. Axel, L. Dougherty, “Heart wall motion: improved method of spatial modulation of magnetization for MR imaging,” Radiology 171(2), pp. 749–750, 1989. 6. A.A. Amini, J.L. Prince, Measurement of cardiac deformations from MRI: Physical and mathematical models, Kluwer Academic Publishers, Dordrecht, 2001. 7. J.V. Hajnal, D.L. Hill, D.J. Hawkes (Eds), Medical Image Registration, CRC Press, Boca Raton, 2001. 8. T. M¨ akel¨ a, P. Clarysse, O. Sipil¨ a, N. Pauna, Q.C. Pham, T. Katila, I.E. Magnin, “A review of cardiac image registration methods,” IEEE Transactions on Medical Imaging 21(9), pp. 1011–1021, 2002. 9. C. Petitjean, N. Rougon, F. Prˆeteux, “Non rigid image registration using generalized information measures,” Tech. Rep. 02001, INT, Evry, France, March 2002. 10. R. Chandrashekara, R.H. Mohiaddin, D. Rueckert, “Analysis of myocardial motion in tagged MR images using nonrigid image registration,” in Proceedings SPIE Conference on Medical Imaging 2002: Image Processing, San Diego, CA, 4684, pp. 1168–1179, 2002. 11. F. Maes, D. Vandermeulen, P. Suetens, “Medical image registration using mutual information,” Proceedings of the IEEE 91(10), pp. 1699–1722, 2003. 12. J.P.W. Pluim, J.B.A. Maintz, M.A. Viergever, “Mutual-information-based registration of medical images: a survey,” IEEE Transactions on Medical Imaging 22(7), pp. 986–1004, 2003. 13. D. Rueckert, L.I. Sonoda, C. Hayes, D.L.G. Hill, M.O. Leach, D.J. Hawkes, “Nonrigid registration using free-form deformations: application to breast MR images,” IEEE Transactions on Medical Imaging 18(8), pp. 712–721, 1999. 14. D. Perperidis, A. Rao, M. Lorenzo-Vald´es, R. Mohiaddin, D. Rueckert, “Spatio-temporal alignment of 4D cardiac MR images,” in Proceedings 2nd International Workshop on Functional Imaging and Modeling of the Heart (FIMH’2003), Lyon, France - LNCS 2674, pp. 203–214, Springer-Verlag, Berlin, June 2003. 15. A. Rao, G.I. Sanchez-Ortiz, R. Chandrashekara, M. Lorenzo-Vald´es, R. Mohiaddin, D. Rueckert, “Construction of a cardiac motion atlas from MR using non-rigid registration,” in Proceedings 2nd International Workshop on Functional Imaging and Modeling of the Heart (FIMH’2003), Lyon, France - LNCS 2674, pp. 141–150, Springer-Verlag, Berlin, June 2003. 16. C. Studholme, D. Hill, D. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognition 32(1), pp. 71–86, 1999. 17. C. Petitjean, Recalage non rigide d’images par approches variationnelles statistiques. Application a ` l’analyse et ` a la mod´elisation de la fonction myocardique en IRM. PhD thesis, Universit´e Paris V - Ren´e Descartes, France, September 2003. 18. C. Petitjean, N. Rougon, F. Prˆeteux, Ph. Cluzel, Ph. Grenier, “Measuring myocardial deformations in tagged MR image sequences using informational non-rigid registration,” in Proceedings 2nd International Workshop on Functional Imaging and Modeling of the Heart (FIMH’2003), Lyon, France - LNCS 2674, pp. 162–172, Springer-Verlag, Berlin, June 2003. 19. N. Rougon, C. Petitjean, F. Prˆeteux, “Variational non rigid image registration using exclusive f-information,” in Proceedings International Conference on Image Processing (ICIP’2003), Barcelona, Spain, pp. 703–706, September 2003. 20. F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, P. Suetens, “Multimodality image registration by maximization of mutual iormation,” IEEE Transactions on Medical Imaging 16(2), pp. 187–198, 1997. 21. P. Viola, W.M. Wells III, “Alignment by maximization of mutual information,” International Journal of Computer Vision 24(2), pp. 137–154, 1997.

22. I.Vajda, Theory of Statistical Evidence and Information, Kluwer Academic Publishers, Dordrecht, 1989. 23. G. Hermosillo, C. Chefd’Hotel, O. Faugeras, “Variational methods for multimodal image matching,” International Journal of Computer Vision 50(3), pp. 329–343, 2002. 24. E. D’Agostino, F. Maes, D. Vandermeulen, P. Suetens, “A viscous fluid model for multimodal non-rigid image registration using mutual information.,” Medical Image Analysis 7, pp. 565–575, 2003. 25. G. Hermosillo, Variational methods for multimodal image matching. PhD thesis, Universit´e de Nice-SophiaAntipolis, May 2002. 26. H.H. Nagel, W. Enkelmann, “An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences,” IEEE Transactions on Pattern Analysis and Machine Intelligence 8(5), pp. 1565–593, 1986. 27. M. Turk, A. Pentland, “Eigenfaces for recognition,” Journal of Cognitive Neuroscience 3, pp. 71–86, 1991. 28. M.D. Cerqueira, N.J. Weissman, C. Dilsizian, A.K. Jacobs, S. Kaul et al., “Standardized myocardial segmentation and nomenclature for tomographic imaging of the heart,” Circulation 105, pp. 539–542, 2002. 29. C. Petitjean, N. Rougon, P. Cluzel, “Assessment of myocardial function: A review of quantification methods and results using tagged MRI,” Tech. Rep. 03009, INT, Evry, France, August 2003.

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