Medical Engineering & Physics 27 (2005) 103–113

Technical note

Validation of a finite element model of the human metacarpal D.S. Barkera,∗ , D.J. Netherwayb , J. Krishnana , T.C. Hearna a

Department of Orthopaedic Surgery, Repatriation General Hospital and Flinders University of South Australia, Australia b Institute of Craniofacial Studies and the University of Adelaide, Australia Received 15 November 2002; received in revised form 18 August 2004; accepted 7 October 2004

Abstract Implant loosening and mechanical failure of components are frequently reported following metacarpophalangeal (MCP) joint replacement. Studies of the mechanical environment of the MCP implant-bone construct are rare. The objective of this study was to evaluate the predictive ability of a finite element model of the intact second human metacarpal to provide a validated baseline for further mechanical studies. A right index human metacarpal was subjected to torsion and combined axial/bending loading using strain gauge (SG) and 3D finite element (FE) analysis. Four different representations of bone material properties were considered. Regression analyses were performed comparing maximum and minimum principal surface strains taken from the SG and FE models. Regression slopes close to unity and high correlation coefficients were found when the diaphyseal cortical shell was modelled as anisotropic and cancellous bone properties were derived from quantitative computed tomography. The inclusion of anisotropy for cortical bone was strongly influential in producing high model validity whereas variation in methods of assigning stiffness to cancellous bone had only a minor influence. The validated FE model provides a tool for future investigations of current and novel MCP joint prostheses. © 2004 IPEM. Published by Elsevier Ltd. All rights reserved. Keywords: Finite element; Human; Metacarpal; Validation; Strain gauge; In vitro; Anisotropic

1. Introduction A range of implants have been designed to replace the metacarpophalangeal (MCP) joint, however, stress/strain analyses of the intact metacarpal and the metacarpal containing a prosthesis in situ are uncommon [1–5]. Gunter et al. [4] used FE analysis to study the optimal stem length for the metacarpal component of a titanium osseointegrated prosthesis. The authors considered an intact metacarpal, and three different lengths of prosthesis: short, medium and long. The authors demonstrated that the medium length prosthesis produced the smallest change in von Mises bone stresses compared to the intact situation. Gayet et al. [5] performed a FE analysis of a cemented metal metacarpal implant and compared the bone stresses with the intact metacarpal. The authors concluded that the introduction of the prosthesis caused ∗ Corresponding author. Present address: Portland Orthopaedics Ltd., 3/44 McCauley St., Matraville, NSW 2036, Australia. E-mail address: [email protected] (D.S. Barker).

stress shielding throughout the metacarpal and also introduced a non-physiological flexion moment. Although these models provided valuable insights into the mechanical behaviour of MCP prostheses, there was no validation of the models with physical specimens. The lack of pre-clinical mechanical assessment may be a factor in the poor clinical outcomes of MCP joint replacement when compared to hip and knee joint replacement [6–17]. Non-physiological load transfer and overstressing of material components are two factors associated with implant failure that can be identified through pre-clinical analysis. The risk of premature failure can then be reduced by optimising the implant design. In vitro strain gauge (SG) and finite element (FE) analyses are the mainstay of pre-clinical stress/strain assessment of an implant-bone construct. SG analysis enables an assessment of biological variation but is limited to measuring surface strain at discrete points. In contrast, FE analysis provides a map of stress/strain throughout the construct but contains several assumptions and simplifications regarding the geometrical,

1350-4533/$ – see front matter © 2004 IPEM. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.medengphy.2004.10.001

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material and boundary conditions [18]. The impact of these assumptions and in particular the values assigned to parameters used in the model should be tested through judicious use of physical validation testing and parametric variation. Several investigators have compared SG and FE models of intact bone and implant-bone constructs, either descriptively, or quantitatively through regression analysis [19–27]. In general, there was a high degree of correlation between the SG and FE data, with slopes close to unity, however, the mechanical representation of cortical bone as an isotropic material was a frequent assumption. This assumption may have been reasonable in the aforementioned studies as the majority of bones were not subjected to significant torsion. Cortical bone is anisotropic with approximately equal stiffness in the perpendicular axes of the transverse plane (transversely isotropic) with magnitude 60% of that in the longitudinal direction [28–30]. Thus, FE models neglecting cortical bone anisotropy may fail to produce accurate results when shear and hoop strains are present due to torsion or the presence of an implant [26]. The choice of an appropriate method for assigning material properties to cancellous bone is also arbitrary. The relationship between computed tomography (CT) number and stiffness of cancellous bone varies depending on location and other factors and the exponent of density has been reported as ranging between one and three [31–36]. In the absence of direct measurement, a finite element parameter containing uncertainty, such as the mechanical properties of human metacarpal bone, should be subject to variation within physiological limits. The employment of such a sensitivity analysis determines the level of importance which should be given to the parameter in question. This study has been motivated by the need for such an analysis in the context of MCP joint replacement. The aim of this study was to create a three dimensional model of the human metacarpal and validate the model by comparing principal strain output from the FE analysis with a SG model of the same metacarpal. A further aim was to assess different methods of assigning mechanical properties of the metacarpal in the absence of direct testing.

2. Materials and methods Following ethical approval from our institution, the right index metacarpal of an 85-year-old male was extracted at autopsy. The metacarpal was inspected physically and radiographically and was free from obvious pathologies or anomalies. The metacarpal was stored fresh at −30 ◦ C until testing began. 2.1. FE model The metacarpal was scanned using quantitative computed tomography (QCT; GE CTi scanner, GE Medical Systems, Milwaukee, WI, USA) at 0.5 mm intervals with slice thicknesses of 1.0 mm. A calibration phantom containing known

concentrations of CaCO3 (B-MAS phantom, Hitachi Medical Corporation, Tokyo, Japan) was concurrently scanned. The sectional scans were taken perpendicular to a line passing through the center of the metacarpal head and the basal notch for articulation with the trapezoid. The scanner settings were: tube voltage, 120 kV; exposure dose, 100 mA s; field of view, 512 mm × 0.1875 mm/512 mm × 0.1875 mm. The digital axial slices were imported into a software programme (Persona software, Institute of Craniofacial Studies, Adelaide, Australia) that defined the contours of the bony surface, based on linearly interpolated thresholding of the Hounsfield units. Hounsfield numbers greater than 175 were deemed to be bone. For each section, cubic splines were defined from the contour data for creation of the FE mesh (LUSAS/MYSTRO, Finite Element Analysis Ltd., Surrey, UK). Surfaces and volumes were subsequently created in the preprocessing package of the FE software to represent the macroscopic geometry. The volumes were meshed with hexahedral or pentahedral elements available in the LUSAS element library. These elements did not contain mid-side nodes but incorporated a feature that substantially reduced parasitic shear, a problem that causes uniform strain element meshes to be overly stiff in bending. The mesh density was determined through convergence tests of principal strains with a total of 4320 elements and 3988 nodes found to produce strains which were unaffected by further mesh density increases. No element had an aspect ratio greater than five. The finite element mesh is shown in Fig. 1. Two separate load cases representative of in vivo pinch were considered: torsion and combined axial/bending loading [37]. For combined axial/bending loading, a physiologically directed load of 80 N was applied at 30◦ of extension and 5◦ of radial deviation to the longitudinal axis, passing through the center of the metacarpal head. For torsion, a force couple resulting in a 0.8 N m moment was applied in the transverse plane containing the center of the metacarpal head. The mesh was restrained by conditions of zero displacement at all nodes located at 70% of the metacarpal length (70%L) proximal from the most distal point of the metacarpal. These loads were sub-maximal to prevent specimen failure in the strain gauge analysis. For the assignment of material properties, the FE mesh was divided into diaphyseal and metaphyseal bone regions. The metaphyseal region contained both cancellous and metaphyseal cortical bone and was defined as the distal 25% of the metacarpal. The remainder of the mesh contained diaphyseal cortical bone. Four methods were examined to assign material properties to elements within the FE mesh. 2.1.1. Material method 1 (isotropic-Dalstra) Elements within the diaphyseal region were assigned homogenous isotropic material properties of modulus, E = 17,000 MPa and Poisson’s ratio, ν = 0.3. Elements within the metaphyseal region were assigned material properties depending on the raw CT density (ρCT ) in Hounsfield units at the centroid of the element, determined through tri-linear in-

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Fig. 1. The finite element mesh of the intact human metacarpal bone.

terpolation. Linear regression was used to determine a relationship between ρCT and calcium equivalent density (ρCa.eq ) in units of g/cm3 (Eq. (1)). This was achieved by sampling the raw CT density within a 65 mm2 region of interest for each of the five phantom tubes containing known concentrations of calcium carbonate (range, 32.31–233.66 mg/cm3 ). Extrapolation was assumed to be valid outside the range of the phantom. ρCa.eq

0.919ρCT − 4.69 = 1000

0 (1)

Calcium equivalent densities were converted into apparent densities and element moduli following the results of Dalstra et al. [38] who investigated pelvic trabecular properties. ρapp = 1.60ρCa.eq

(2)

where ρapp is the apparent density, g/cm3 and 2.46 EDalstra = 2017.3ρapp

to all elements in the diaphyseal cortical region [39]:   19.5 11.4 12.5 0 0 0  11.4 20.1 12.5 0 0 0       12.5 12.5 30.9 0 0 0    Eij =  0 0 5.72 0 0   0   0 0 0 0 5.17 0 

(3)

where E is the element stiffness in MPa. The use of Eq. (3) led to a maximum stiffness value of 27,200 MPa (EDalsta.max ) for the maximum calcium equivalent density of 1.8 g/cm3 . To limit bone stiffness to physiological levels, EDalsta.max was set to 19,200 MPa. 2.1.2. Material method 2 (anisotropic-homogenous) An anisotropic elastic stiffness matrix, Eij (expressed in GPa), developed to model tibial cortical bone, was assigned

0

0

0

0

(4)

4.05

Metaphyseal elements were assigned homogenous isotropic material properties as used in a previous model of a human metacarpal (E = 300 MPa, ν = 0.3) [3]. 2.1.3. Material method 3 (anisotropic-Dalstra) Diaphyseal elements were assigned anisotropic material properties as defined in Section 2.1.2. Metaphyseal elements were assigned individual stiffnesses determined from Section 2.1.1. 2.1.4. Material method 4 (anisotropic-Lotz) Diaphyseal elements were assigned anisotropic material properties as defined in Section 2.1.2. Metaphyseal elements were assigned moduli following the results of Lotz et al. [35] who investigated proximal femoral trabecular bone (Eq. (5)). Calcium equivalent densities (mg/cm3 ) obtained in our study were converted to potassium equivalent densities (ρK.eq ) in mg/cm3 through the use of Eq. (6) so that Eq. (5) could be employed [40]. 1.2 ELotz = 0.5ρK.eq

(5)

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Fig. 2. Schematic diagram of gauge placement showing the 30% and 40% of L circumferential lines. The ulnar-volar face is obscured, hence, only four gauges are shown.

ρK.eq = 0.794ρCa.eq − 9.76

(6)

Six locations, corresponding to the gauge positions in the SG study (see below) were identified within the FE mesh from bony landmarks. Maximum and minimum principal strains were recorded by averaging the strain measured at four nodes located at each of these six locations. 2.2. SG model Six, electrical resistance, 1 mm, rosette strain gauges (KYOWA KFG-1-120-D17-11L1M2S, Kyowa Electronic Instruments Co. Ltd., Tokyo, Japan) were bonded on the metacarpal following a standard protocol [41]. Two circumferential lines were marked at 30% of length (30%L), and 40% of length (40%L) from the most distal point of the metacarpal head using a custom designed jig that enabled a standard positioning protocol (Fig. 2). Strain gauges must be placed on flat surfaces and the 30%L line represented the most practical distal location at which the rosettes could be placed and occurred approximately 10 mm proximal to the tubercles. The 40%L line was chosen to approximate the level at which the tip of a MCP prosthesis would lie. Rosettes were placed on the radial-volar (R), ulnar-volar (U) and dorsal (D) surfaces at both 30%L and 40%L (Fig. 2). The proximal metacarpal was restrained at 70%L proximal to the distal end of the metacarpal using a transfixing pin and potting cup, with radial set-screws and dental cement to augment the fixation. A custom loading pot designed to allow for torsion and combined axial/bending loading (Figs. 3 and 4)

Fig. 3. Combined axial/bending loading arrangement for strain gauged metacarpal: (A) loading pin; (B) loading cap; (C) distal potting cup; (D) strain gauged metacarpal; (E) proximal potting cup; (F) lockable support base.

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analysed from the fifth cycle, when steady state hysteresis was noted. Slow loading frequencies were chosen to reflect the quasi-static nature of in vivo loading of the metacarpal. The particular loading frequencies used were dependent on the loading apparatus and chosen to provide a smooth distribution of data points throughout the loading cycle. Preliminary testing indicated that faster frequencies tended to cause clumping of data points at the peaks and troughs of the loading cycles, especially in torsion. Although bone exhibits viscoelastic behaviour, strain rate induced artefact would not be expected until much faster load frequencies. Principal strains from the FE model (εFE ) for different material models were compared with the SG model strains (εSG ) using linear regression. The SG model strain was chosen to be the dependent variable. Material assignment methods were ranked according to the closeness of the correlation coefficient and slope to unity, with greater divergence from unity corresponding to a lower ranking.

3. Results

Fig. 4. Torsion arrangement for strain gauged metacarpal: (G) torsion loading bar; (H) distal potting cup; (I) strain gauged metacarpal; (J) proximal potting cup.

was fixed to the distal half of the intact metacarpal head and was further constrained with set-screws, augmented with dental cement. Following bonding of the rosettes with cyanoacrylate, gauge leads were directly attached to strain gauge amplifiers (SCXI-1121 amplifier, National Instruments, Austin, USA). Strain signals were passed through an A/D converter and recorded on a personal computer. The metacarpal was loaded in an Instron model 8511 servohydraulic materials testing machine (Instron Pty Ltd., High Wycombe, UK) under conditions of both torsion and combined axial/bending and loading. For combined axial/bending loading the proximal potting cup was secured such that the metacarpal was aligned in 30◦ of flexion and 5◦ ulnar deviation to match the FE model. A vertical cyclic load from 20 N to 100 N was applied at a frequency of 0.1 Hz. For torsion, a transfixing pin was placed through a drill hole in the metacarpal head and through both sides of the metacarpal head potting cup. The proximal potting cup was adjusted such that the longitudinal axis of the metacarpal was parallel to the Instron axis. A universal joint and xy table were used to apply torsion through the metacarpal head, which acted to supinate the metacarpal. A cyclic load from 0.2 N m to 1.0 N m was applied at a frequency of 0.05 Hz. For both loading modes, a non-destructive cyclic loading scheme was used with data

For combined axial/bending loading, at the dorsal gauges, tensile principal strains were dominant and directed longitudinally for both the SG and FE models. At the radial-volar and ulnar-volar gauges, principal compressive strains were dominant and directed between 10◦ and 40◦ volar to the longitudinal axis and tensile stresses were directed between 10◦ and 40◦ to the transverse plane. For torsion, the tensile and compressive principal strains occurred at approximately 45◦ to the longitudinal axis. The SG data indicated the possible presence of an outlier (εSG = −884 microstrain) at the R30 gauge for combined axial/bending loading (Fig. 6), however, there was no reason to suspect that this value was artefactual and it was included in the regression analyses. The strains predicted by the FE models showed strong agreement with the experimentally measured strains of the same specimen (Figs. 5–8). The regression analyses indicated that the y-intercept was not significantly different (P > 0.05) from zero for torsion and combined axial/bending loading for all material assignment methods (Table 1). The slope of the regression line was significantly different (P < 0.05) from zero for torsion and combined axial/bending loading for all material assignment methods (Table 1). The slope was not significantly different (P > 0.05) from one for all cases except for the isotropic material assignment model (Section 2.1.1) subject to torsion (Table 1). For anisotropic models, torsion produced correlation coefficients and gradients closer to one, compared to combined axial/bending loading (Table 1). Similarly, the y-intercepts were considerably closer to zero for torsion compared to combined axial/bending loading. For both loading modes, material assignment methods anisotropic-Dalstra and anisotropicLotz, which included a CT density based approach for cancellous bone assignment and an anisotropic cortical bone model, produced correlation coefficients and gradients closer

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Fig. 5. Maximum principal strains for each gauge location for axial/bending loading.

Fig. 6. Minimum principal strains for each gauge location for axial/bending loading.

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Fig. 7. Maximum principal strains for each gauge location for torsion loading.

Fig. 8. Minimum principal strains for each gauge location for torsion loading.

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Table 1 Output of linear regression analyses comparing the four FE models with the SG data obtained from the same bone Material assignment method

R

R2

Slope

P-value, slope = 0

P-value, slope = 1

y-Intercept (microstrain)

P-value, intercept = 0

Combined axial/bending Isotropic-Dalstra Anisotropic-Dalstra Anisotropic-homogenous Anisotropic-Lotz

0.774 0.841 0.814 0.839

0.599 0.707 0.663 0.704

1.06 1.13 1.10 1.10

0.003