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Three-dimensional finite element modelling of the human ACL: simulation of passive knee flexion with a stressed and stress-free ACL G. Limberta,b,c,*, M. Taylorc, J. Middletona a
Biomechanics Research Unit, University of Wales, College of Medicine, The Cardiff Medicentre, CF14 4UJ Cardiff, United Kingdom b FIRST Numerics Ltd., Cardiff, United Kingdom c Bioengineering Sciences Research Group, School of Engineering Sciences, University of Southampton, SO17 1BJ Southampton, United Kingdom Accepted 30 January 2004
Abstract In this study, a three-dimensional finite element model of the human anterior cruciate ligament (ACL) was developed and simulations of passive knee flexion were performed. The geometrical model of the ACL was built from experimental measurements performed on a cadaveric knee specimen which was also subjected to kinematics tests. These experiments were used to enforce the particular boundary conditions used in the numerical model. A previously developed transversely isotropic hyperelastic material model was implemented and the ability to pre-stress the ligament was also included. The model exhibited the key characteristics of connective soft tissues: anisotropy, nonlinear behaviour, large strains, very high compliance for compressive or bending loading along the collagen fibres and incompressibility. Simulations of passive knee flexion were performed, with and without pre-stressing the ACL. The resultant force generated by the ACL was monitored and the results compared to existing experimental data. The stress distribution within the ligament was also assessed. When the ACL was pre-stressed, there was a good correlation between the predicted and experimental resultant forces reported in the literature over the entire flexion–extension range. The stress distribution in the pre-stressed and stress-free ACL were similar, although the magnitudes in the pre-stressed ACL were higher, particularly at low flexion angles. r 2004 Elsevier Ltd. All rights reserved. Keywords: ACL; Constitutive law; Transversely isotropic hyperelasticity; Finite element; Resultant force
1. Introduction The anterior cruciate ligament (ACL) is the most commonly injured ligament of the body (Fetto and Marshall, 1980; Johnson, 1982), especially during sport (Speer et al., 1995) and motor vehicle accidents (Crowninshield and Pope, 1976) and therefore the biomechanics of the ACL is of interest. Direct measurement of the stress or strain distribution within the ACL is difficult and various techniques have been used in the past: strain gauges (Henning et al., 1985), displacements (Markolf et al., 1990; Renstrom et al., 1986), buckle transducers (Barry and Ahmed, 1986) or optical *Corresponding author. Biomechanics Research Unit, University of Wales, College of Medicine, The Cardiff Medicentre, Cardiff CF14 4UJ, United Kingdom. Tel.: +44-29-2068-2162; fax: +44-29-20682161. E-mail address:
[email protected] (G. Limbert). URL: www.firstnumerics.com. 0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2004.01.030
methods such as Roentgen stereophotogrammetry (Meijer et al., 1989; van Dijk et al., 1979). There are a number of limitations with the existing experimental methods. Experimental sensors tend to alter the natural geometry of the ligament and thus cause errors in the measurements. Also, in the majority of cases, the measurements are taken at discrete locations, rather than continuously over the entire surface of the ligament. The fibrous nature of ligaments means that deformation patterns in the ACL are not uniform and vary significantly according to the location of the measurement. Finally, various studies have tested the ACL in isolation in uniaxial tension, but such tests are not representative of the complex strain patterns that occur during flexion–extension (Amis and Dawkins, 1991). As Hirokawa and Tsuruno (2000) noted there may be no conditions where all of the fibre bundles are uniformly stretched. Various studies have attempted to measure the ACL strains in situ. Yamamoto et al. (1998)
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measured the surface strains using the photoelastic method. The limitation of this study was that they applied the photoelastic coating at 10 of flexion and this was assumed to be the zero strain state for the ACL, whereas in reality various fibre bundles were likely to have been strained in this position. Also, assessment of the surface strains was restricted to the lateral side of the ACL because of the way the experimental apparatus was built. This shortcoming was later overcome by the same authors (Hirokawa et al., 2001) in a study where they used a new experimental jig allowing a panoramic view of the ACL surface at any angle of knee flexion. A significant alternative to experimental studies is the use of computer methods. In particular, the use of finite element analysis is popular as it allows the threedimensional structure to be analysed and the stress/ strain distribution to be visualised anywhere within the model. Comparatively few full three-dimensional (3D) finite element (FE) models of the ACL have been developed (Daniel, 2001; Hirokawa and Tsuruno, 2000; Limbert, 2001; Limbert and Taylor, 2001b; Pioletti, 1997; Pioletti et al., 1998a) Pioletti and co-workers (Pioletti, 1997; Pioletti et al., 1998b) developed a full thermodynamic formulation of a constitutive law, based on an incompressible isotropic hyperelastic formulation, for the ACL and derived material data from experiments (uniaxial tensile tests) and implemented it into a commercial finite element code. As shown by Limbert and Taylor (2001b), the assumption of isotropy of Pioletti’s model was a severe limitation of the constitutive model when the ACL is subjected to compression along the collagen fibres. Limbert and Taylor (2001b) implemented an isotropic hyperelastic formulation and studied the stresses in the ACL as a result of passive knee flexion. An isotropic formulation was found to generate unrealistically high compressive stresses in the ligament which conflict with experimental observations. Daniel (2001) developed and implemented a 3D orthotropic viscohyperelastic finite element model of the ACL which was an extension of the Pioletti model (Pioletti, 1997; Pioletti et al., 1998b). The obvious advantage of this model was the inclusion of the transverse isotropy and time-dependent material behaviour. However, the model was shown not to be thermodynamically admissible for the whole strain range considered. This model was tested for a simulated 5 mm sidestep motion of the knee. This offered no insight into the behaviour of the ACL when the knee is subjected to passive flexion. The only quantitative results from the finite element analyses were the von Mises stresses at the surface of the ACL for the sidestep motion considered and emphasis of the study was placed on the development of the constitutive material model. Hirokawa and Tsuruno (2000) developed a structurally based continuum phenomenological constitutive model of the ACL that encompasses the full 3D and
finite-strain regime. The ligament was assumed to be a compliant Mooney–Rivlin material reinforced by two distinct families of collagen fibres. The model was implemented in a finite element code and was tested for a passive flexion and a drawer test. Although this model was the first published attempt to develop a continuum anisotropic constitutive law for the ACL, the anisotropic directions chosen for the two families of fibres were arbitrary and did not necessarily reflect the natural orientation of the collagen fibres. This limited, somehow, the usefulness of the model. Moreover, no quantitative data of the numerical model that could be directly compared to experimental studies were reported. The major limitation of finite element-based studies is the lack of adequate validation. In the majority of finite element studies of the ACL to date only the stress distribution patterns have been reported for simulated passive knee flexion or drawer tests (Daniel, 2001; Hirokawa and Tsuruno, 2000; Pioletti, 1997; Pioletti et al., 1998a). The predicted stress and strain distributions are difficult to validate, because of the problems in experimentally measuring these parameters, as discussed earlier. However, the force generated by the ACL during various activities has been directly measured experimentally (Roberts et al., 1994; Wascher et al., 1993). If a finite element model is unable to predict the resultant force generated during flexion–extension, then the stress/strain distribution is also likely to be incorrect. The force generated by the ACL could be used as an indirect measure of the performance of a finite element model, but to date none of the numerical studies in the literature have reported this parameter. A further limitation of previous FE models of the ACL is that they assume that the ligament has a stressfree state, usually at full extension or at 10 of flexion (Daniel, 2001). Various experimental studies have suggested that the ACL does not have a stress-free state at any knee flexion angle (Durselen . et al., 1996). Using a direct measurement technique (strain gauges), Bach et al. (1997) measured in vitro the strains in the anteromedial and posterolateral bands of the ACL from 10 of hyperextension to full flexion. On a sample of ten cadaveric knee specimens, the ranges of strains at full knee extension were 3.2–5.2% for the anteromedial band of the ACL and 6–8.8% for the posterolateral band. This is in direct contradiction with the modelling assumptions of the previously cited authors (Daniel, 2001; Hirokawa and Tsuruno, 2000; Pioletti, 1997; Pioletti et al., 1998a). The aim of the present study was to verify the performance of an FE model of the ACL, implementing an existing transversely isotropic hyper elastic material formulation (Limbert, 2001) by comparing the predicted force generated by the ligament during passive flexion
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exercises with existing experimental data (Roberts et al., 1994; Wascher et al., 1993). As part of this study, the concept that the ACL has no stress-free state was also examined. The ACL was pre-stressed to varying degrees at full extension and the influence of the predicted force generated by the ligament analysed over the passive flexion–extension range. Comparisons were also made between the stress distributions generated by the stressfree and pre-stressed ligaments.
2. Materials and methods 2.1. Generation of the finite element model of the ACL The 3D geometry of the insertion sites of a human ACL was obtained from an experiment performed on a nonpathological intact cadaveric right knee specimen from a male donor of unknown age. Using a direct measurement technique (Martelli et al., 2000), the 3D motions of the tibia relative to the femur (rigidly fixed in an horizontal position) were recorded during kinematic acquisitions made by a trained orthopaedic surgeon. After completion of repeated acquisitions of the knee kinematics, the knee was dismantled and cleared from its surrounding soft tissue structures (flesh, muscle, capsular structures) in order to allow easy access to the ACL and its insertion sites. Using an electrogoniometer (FARO Arm Model B06-01/02; FARO Technologies, Lake Mary, FL, US), discrete points defining the contours of the ACL at its femoral and tibial insertions, as well as points defining the fibre orientations, were acquired by the surgeon. A file containing the point numbers and their 3D coordinates was then imported into the FE pre and post-processor Patran v8.0 (s The MacNeal Schwendler Corporation, Los Angeles, CA, USA). Prior to this, the coordinates of the points defining the tibial insertion area were transformed (Martelli et al., 2000) in order to obtain their position when the knee is at full extension. For each set of points (one for the tibial insertion, one for the femoral insertion), a surface interpolation was performed in the pre-processor software. Given that the full 3D shape of the ACL was not available and that a ‘‘reasonable’’ ACL shape does not affect significantly the results of the FE analysis (Pioletti, 1997), the ligament was reconstructed by connecting the two insertion surfaces. The solid volume reconstructed was that of the ACL when the knee is at full extension (Figs. 2(a), 2(f), 3(a), 3(f)). The solid volume representing the ACL was meshed with 8-noded hexahedral elements using Patran v8.0. Special care was taken in order to optimise the performance of the mesh for the large displacement and large strain analysis. The mesh consisted of 3297 elements and 3784 nodes.
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2.2. Constitutive law and mechanical properties Ligaments are dense connective tissues consisting of mainly parallel-fibre collagenous tissues embedded in a highly compliant solid matrix of proteoglycans (Fung, 1981). The preferred orientation of the collagen fibres induces the transversely isotropic symmetry of the ligament. Due to their natural composite structure, ligaments can be described by a continuum theory of fibre reinforced composites at finite strains (Spencer, 1992; Limbert and Taylor, 2002). The existence of a strain energy function C which depends on I1, I2 and I3, the first three principal invariants of the right Cauchy– Green deformation tensor C is postulated and so is a fourth invariant, ln0 ; corresponding to the stretch in the fibre direction characterised by the unit vector n0. This unit vector can be defined globally or pointwise and ln0 is defined by Eq. (1): ðln0 Þ2 ¼ n0 ðC n0 Þ ¼ I4 ;
ð1Þ
where I4 is an invariant firstly introduced by Ericksen and Rivlin (1954). It is assumed that the strain energy function characterising the mechanical behaviour of the ACL C could be split into the sum of a strain energy function representing the mechanical response of the ground substance Cm and a strain energy function encompassing the anisotropic behaviour introduced by the collagen fibres f ðlÞ (Spencer, 1992): Cm ðI1 ; I2 ; I3 ; lÞ ¼ Cm þ f ðlÞ;
ð2Þ
where Cm is chosen as being a neo-Hookean incompressible isotropic hyperelastic potential which is a simple extension of the classical linear isotropic elasticity for the finite strain regime. The neo-Hookean model has been shown to represent well the elastic behaviour of the ground substance of connective tissues (Weiss et al., 1996; Limbert, 2001). For an incompressible material, Cm also exhibits the property of convexity which assures stability of the material. Cm ðI1 ; I2 ; I3 Þ ¼ C1 ðI1 3Þ þ gðI3 Þ;
ð3Þ
where C1 is a material parameter and g(I3) is a simple penalty function used to enforce the kinematic condition I3=1, corresponding to total incompressibility (Limbert and Taylor, 2001a). In order to reproduce the characteristic stiffening features of the tissue constituting the ACL and its mechanical behaviour during large deformation, a transversely isotropic hyperelastic potential with an exponential law (Eq. (4b)) was used for representing the mechanical contribution of the collagen fibres. The inability of the collagen fibres to sustain compressive load along their axis was taken into
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account (Eq. (4a)): 8 qf > Z < ¼0 if lp1; qf dl such that ql f ðlÞ ¼ > ql : qf ¼ C2 ½eC3 ðl1Þ 1 if l > 1; ql l
ð4a; bÞ where l is the square of I4 and C2, C3 are material parameters. The Cauchy stress tensor, usually referred as the engineering stress tensor, is obtained by simple combined derivation of C with respect to C and pushforward operation: qC qf ðlÞ T r¼F 2 n#n þ p1; F ¼ 2 C1 b þ l qC ql ð5Þ where F is the deformation gradient tensor, 1 represents the second-order unit tensor, b is the left Cauchy–Green deformation tensor, p is the hydrostatic pressure appearing as a kinematic reaction to the incompressibility constraint, ‘‘#’’ denotes the tensor product and n is the Eulerian counterpart of n0 through the deformation as given by the relation F n0 ¼ ln:
ð6Þ
The constitutive model was implemented into the commercial FE code PAM-CRASHt (PAM Systems International, Rungis, France). The material parameters of the constitutive law presented above were identified with the tensile stress– strain curve obtained by Pioletti (1997) for human ACLs by means of a simple least-squares method (Bates and Watts, 1998) producing a set of material parameters C1, C2 and C3. The coefficient C1 governs the isotropic mechanical response of the ground substance and was assumed to be 1 MPa a priori (Ault and Hoffman, 1992). The mechanical contribution from the collagen fibres (coefficients C2 and C3) were obtained through the optimisation procedure. C3 is a factor that scales the exponential stress while C4 controls the rate of uncrimping of the collagen fibres (Weiss et al., 1996). Eventually, the following set of material coefficients were derived: C1=1 MPa, C2=0.4247 MPa and C3=22.2548 MPa.
In fact, this amounts to generating stretch and, therefore, stress without displacing the nodes of the ligament that is already in its initial pre-stressed configuration. At the insertion sites where the meshes corresponding, respectively, to bone and ligaments connect, the nodes representing the insertion surfaces of the ligament are rigidly fixed to the mesh of the bony structures. A simple way to implement the capacity to apply initial stretch to a FE model consists in applying a multiplicative decomposition of the deformation gradient based on three different states (Weiss et al., 1995). We assume the existence of a configuration w0 at which the ligament is stress-free, a state wI at which the ligament is pre-stressed but in the reference configuration (point of departure of the finite element analysis) and the current state, or deformed state, wc : The total deformation gradient, Fc0, from the stress-free to the current configuration is expressed as follows: Fc0 ¼ FcI FI0 where FcI and FI0, are, respectively, the deformation gradients from the initial configuration to the current configuration and from the stress-free configuration to the initial configuration. By assuming a special form of FI0, it is possible to calculate the total deformation at any time of the FE analysis. It is noteworthy to emphasise that FI0 is defined as an input material parameter and is constant throughout the whole computation. If we assume that an incompressible ligament has a initial pre-stretch lI in direction Z (its fibre direction) and that the two other principal directions of the ligament lie in a plane perpendicular to direction Z, then the deformation gradient, F# I0 ; in the material (or local) reference frame, takes the following form: 3 2 1 ffiffiffiffi p 0 0 7 6 l 7 6 I 7 6 7: 1 ð7Þ F# I0 ¼ 6 7 60 ffiffiffiffi p 0 7 6 l I 5 4 0
0
lI
If R represents the rotation matrix from the local coordinate frame to the global coordinate frame, FI0 is calculated by FI0 ¼ RF# I0 RT :
ð8Þ
2.3. Application of the initial stress field within the ACL at full extension
2.4. Boundary conditions
The application of a uniform initial stretch was performed using a particular numerical technique which operates a special treatment of the deformation gradient (Weiss et al., 1995). In a finite element context, application of an initial stretch to a mesh corresponding to a ligament in its already pre-stretched state is a challenging task.
Passive flexion–extension kinematics tests were performed with the knee in the neutral position (no internal or external rotation) for flexion angles of 0 , 10 , 30 , 45 , 90 , 110 and 125 that corresponds to full flexion of the knee. Although in the physical kinematic tests, the femur was rigidly embedded while the tibia was free to move in the flexion plane, these conditions were reversed
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in the finite element analyses. The successive discrete positions of the femoral insertion of the ligament were used as displacement boundary conditions and the nodes of the tibial insertion area were considered as rigidly fixed.
Isotropic model without initial stretch (Limbert et al., 2001a) No initial stretch within the ACL Initial stretch within the ACL (135 N) Wascher et al. (1993) MIN Wascher et al. (1993) MAX
150
2.5. Finite element analyses
3. Results 3.1. Resultant force within the ACL during passive knee flexion For the sake of comparison, in addition to the results in resultant force obtained from the two FE analyses, three other sets of data were reported and combined in to a single graph (Fig. 1). Limbert and Taylor (2001b) performed a FE analysis of the ACL using geometry and boundary conditions identical to those used in the current study. However, the constitutive law and mechanical properties were those used by Pioletti (1997). The second and third sets of data represent the upper and lower bounds of the envelop of the resultant force curves obtained experimentally by Wascher et al. (1993). As shown in Fig. 1, the qualitative and quantitative results for the resultant force in Limbert and Taylor (2001b) do not follow the trend recorded experimentally by Wascher et al. (1993). In this isotropic FE model, very high compressive and flexural stresses were generated at the posterior side of the insertion zone of the ACL into the tibia, which accounted for the high
125
Resultant force (N)
Using the boundary conditions described in Section 2.4 and the mechanical properties described in Section 2.2, two series of analyses were performed. A first FE analysis was carried out with a stress-free ACL at full knee extension. A second FE analysis, using exactly the same material properties and boundary conditions with the exception of an additional initial stress field within the ACL (as described in Section 2.3). The value of the uniform initial stretch (l1=1.043) was chosen such that the initial resultant force was equal to 135 N (upper value obtained experimentally by Wascher et al. (1993). An initial stretch of 1.043 (about 4.4% strain) fits in the range of strain values obtained experimentally by Bach et al. (1997) even though the ranges they considered concerned only separate fibre bundles and not a complete ACL. In this study, the range of strains at full knee extension was 3.2–5.2% for the anteromedial region of the ACL and 6–8.8% for the posterolateral region of the ACL. In all the analyses the total resultant force acting on the ACL and the first principal stress, i.e. the tensile stress in the direction of the collagen fibres, were reported through the passive flexion range.
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100 75 50 25 0 10
30
45
60
90
110
125
Angle of knee flexion (degrees)
Fig. 1. Graph comparing the resultant force within the ACL for three FE analyses and one experimental study performed by Wascher et al. (1993) on 18 cadaveric specimens. The resultant force (N) is given as a function of the angle of knee flexion.
nonphysiological values predicted for the total force in the ligament. This highlighted the severe limitation of a phenomenological isotropic model when subject to bending and compressive loading. The transversely isotropic hyperelastic model of the ACL used in this study clearly exhibits a much better mechanical response when the knee is subjected to a passive flexion as the resultant force curves (Fig. 1). Indeed, for the initially stress-free ACL, after around 15 of flexion, the predicted resultant force is close to that of the upper bound of the experimental data of Wascher et al. (1993), with the predicted force being slightly greater than that of the experimental curve. However, between 0 and 20 of flexion, the model is unable to capture the high resultant force magnitudes observed experimentally. The initially pre-stressed ACL follows a similar trend to the stress-free ACL, with the exception that it predicts similar resultant force values to the experimental data between 0 and 20 of flexion. 3.2. Stress distribution within ACL Generally, high stress values were found at full flexion in both the stressed and stress-free ACL models, essentially due to the large sagittal plane rotation of the femoral insertion area. If the ACL is loaded in crossfibre directions, strains are higher at the insertion sites than in the midsubstance. This was also observed experimentally by Yamamoto et al. (1998) in a study using a photoelasticity methodology to track strain at the surface of the ACL. As aimed in the formulation of the constitutive model, buckling of the ligament occurs as soon as
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close to the femoral insertion is also observed on both the FE analyses. 3.2.1. The stress-free ACL In the first 30 of flexion the maximum principal stresses are located in the midanterolateral portion of the ACL reaching maximum values of 1.4 and 1.77 Mpa, respectively, at 10 and 30 of flexion. As flexion progresses, the anteromedial part of the ACL becomes the most stressed part with the maximum stress region migrating from the lateral side towards the medial side of the ACL and the posterior part of the ACL starts to slacken as can be seen by the appearance of compressive stresses near the tibial insertion site (Figs 3(a–e)). In these conditions the collagen fibres do not provide any mechanical contribution. At 45 of flexion the maximal principal stresses are not only located in the anterior midportion of the ACL, but also on the anterior aspect of the ACL at the femoral insertion site. This is due to the significant rotation of the femoral insertion area of the ligament as the flexion of the knee develops. At this particular location, the ACL is loaded in directions that cross the natural orientation of the collagen fibres generating significant shear stresses. Beyond 45 of flexion the anterior part of the femoral insertion site of the ACL undergoes the maximal first principal stresses until full flexion when the maximal values of 20 MPa are recorded.
Fig. 2. Anterior view of the maximum principal tensile stresses developed in the ACL during the various stages of the passive knee flexion: full knee extension, 10 , 45 , 90 of flexion and full flexion. Results are presented as an anterior view of the ligament. The plot in the left (a–d) and right (e–h) columns are, respectively, for the stressfree and pre-stressed ACL at full extension. The white zones correspond to the location where the principal stresses are compressive.
compression or bending is developed with respect to the fibre orientation in the ACL (Fig. 2). After 90 of flexion, as reported experimentally (Girgis et al., 1975), the typical necking/buckling of the ACL in the region
3.2.2. The pre-stressed ACL At full extension of the knee, the first principal stresses in the pre-stressed ACL are uniformly distributed along the natural fibre orientations of the taut ligament. A 135 N resultant force generates a uniform stress field of 1.44 MPa (Figs. 2(f) and 3(f)). The existence of a residual stress at full knee extension delays the appearance of compressive stresses on the posterior aspects of the ACL as clearly exhibited when one compares Figs. 3(b) and (g), 3(c) and (h), and 3(d) and (i). At 10 , 30 , 45 and 60 of flexion the maximal principal stresses are 2.46, 3.15, 4.42 and 5.65 MPa whereas they are, respectively, 1.40, 1.77, 2.58 and 3.27 MPa in the case of a stress-free ACL at full knee extension. This shows that, although there is slackening of posterior fibres (as can be seen by a decrease of tensile stress in this area), the first principal tensile stresses in the ACL are higher in the case of a pre-stressed ligament, thus confirming that the anterior part of the ACL is taut during the flexion. The initial stress field does not seem to affect significantly the stress distribution within the ACL but significant variations in magnitude are found during the flexion cycle: at 30 of flexion the stresses in the pre-stressed ACL are 78% higher than for the stress-free case whereas the relative difference is only 0.9% at 110 of knee flexion.
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Fig. 3. Contour plot of the maximum principal tensile stresses developed in the ACL during the various stages of the passive knee flexion: full knee extension, 10 , 45 , 90 of flexion and full flexion. Results are presented as a posterior view of the ligament. The plot in the left (a–d) and right (e–h) columns are, respectively, for the stressfree and pre-stressed ACL at full extension. The white zones correspond to the location where the principal stresses are compressive.
4. Discussion Verification of FE models of ligament structures is difficult to perform, as there are few parameters that can be easily measured experimentally. The majority of studies that have simulated the behaviour of the ACL have reported the stress distribution within the ligament (Daniel, 2001; Hirokawa and Tsuruno, 2000; Pioletti,
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1997; Pioletti et al., 1998a). However, as discussed earlier, this is extremely difficult to measure in situ. Various studies have measured the resultant force generated by the ACL during passive flexion. The aim of this study was to assess the performance of a 3D finite element model of the ACL, which implemented a previously developed transversely isotropic hyperelastic material model, by comparing the predicted resultant force generated by the ligament with those reported in the literature. The predicted resultant force generated by the stressfree and stressed ACL were in good qualitative and quantitative agreement with the values reported in the literature, particularly at flexion angles greater than 20 . At lower flexion angles, the stress-free model of the ACL was unable to capture the forces that were observed experimentally. Only by pre-stressing the ACL was the model able to capture the resultant force behaviour observed experimentally in the first 20 of flexion. This observation, supports the theory that the ACL has no stress-free state at any flexion angle. It is also clear that the incompressible transversely isotropic hyperelastic constitutive law proposed for the ACL provides an excellent answer to the shortcomings of the existing 3D FE isotropic models of the ACL found in literature (Limbert and Taylor, 2001b; Pioletti, 1997). As can be seen in Fig. 1, assuming that the ACL is isotropic leads to the generation of nonphysiological forces through out the flexion–extension range. This suggests that if the resultant force is inaccurate, then so will the predicted stress/strain distribution. Having verified the performance of the model, at least in terms of the resultant force generated by the ACL, the stress distribution within the ligament was then examined, for the stressed and stress-free ACL. At low flexion angles, although the general distribution of stresses are similar, the stressed ACL obviously generates higher stresses than the stress-free model. As the flexion angle increases, the difference in the peak stress magnitude becomes less significant between the stress-free and pre-stressed ACL. Direct comparison with similar published studies is difficult, due to differences in ACL geometry/constitutive formulation and the parameters examined. The general mechanical behaviour of the ACL model reported here is very similar to that of Hirokawa and Tsuruno (2000) and Limbert and Taylor (2001b): necking/buckling of the ACL appearing in the region near the femoral insertion, high stress values observed in the regions close to the femoral insertion. Since flexion–extension motions can produce different lengths and hence tension patterns in the ACL, it was found experimentally that the anteromedial band was taut throughout the entire flexion–extension cycle (Brantigan and Voshell, 1941; Fick, 1904), or at least to lengthen as the knee is extended (Arms et al., 1984;
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Beynnon et al., 1989). Various studies have shown that the posterior band was slack in flexion and became taut only as the knee was brought to full extension (Brantigan and Voshell, 1941; Fick, 1904; Girgis et al., 1975). The models followed a similar trend as in this study, in that the posterolateral part of the ACL slackens during flexion whilst the anterolateral part tightens in agreement with the aforementioned experimental studies. The methodology adopted to apply a pre-stretch to the FE mesh is based on the assumption that the initial stretch is uniform (i.e. identical for each element of the mesh) within the ACL. In reality, at full knee extension, the strain and hence the stress distribution is unlikely to be uniform but heterogeneous according to the fibre bundle considered and the location within the bundle (Amis and Dawkins, 1991). In the FE model, the direction of the initial stretch in each element is given by the local element geometry and a state of homogeneous deformation is assumed for the pre-stressed state. A local strain at a particular location of the ACL does not influence necessarily the strain pattern at another location. So, the resultant force within the ACL at full extension found experimentally (Roberts et al., 1994; Wascher et al., 1993) may be the result of a pre-stretched state existing only in few fibre bundles of the ACL. This aspect was not accounted for in the present model. The 3D geometry of the areas of the ACL in to the bone is subject to various errors: first, when the points are acquired on the cadaveric knee specimen by using an electrogoniometer. This introduces an error (the precision of the apparatus is in the order of 0.1 mm) which may be amplified by the action of the human operator. A second error is introduced during the geometrical reconstruction when performing an interpolation in order to define a surface from a cloud of points. These errors are difficult to quantify and condition the validity of the numerical model of the ACL. The kinematic tests were performed on a single cadaveric knee specimen whose mechanical properties for the ACL are probably different from the ones measured by Pioletti (1997) and used by Limbert and Taylor (2001b). However, as it was found (Limbert, 2001) that their influence on the resultant force within the ACL and on the magnitude and distribution of stress and strain were negligible, this gives us confidence on the present FE results and legitimates the choice made for the mechanical properties. The developed transversely isotropic model of the ACL was found suitable to represent the key mechanical features of this ligament: arbitrary kinematics (finite deformations and displacements), transversely isotropic symmetry, nonlinear and nearly incompressible behaviour, very low stiffness in compression and bending along the (collagen) fibre directions.
The present finite element models not only reproduces qualitative mechanical behaviour of the ACL (buckling under very small compressive or flexural stresses, zone of maximal stress, etc.) but also outputs quantitative data (resultant force) comparable to those obtained from physical cadaveric specimen testing. This numerical study using a continuum mechanics approach is the first, to our knowledge, in which total resultant force within the ACL is reported for a passive knee flexion. Inclusion of a pre-stress improved the prediction of resultant force, particularly at low flexion angles (20 or less) and thus appears to be a relevant parameter to be considered when simulating the mechanical behaviour of the ACL. This tends to support the hypothesis that the ACL has no stress-free position. The finite element model is therefore a promising tool that will allow to perform further numerical studies, thus providing new insights into the mechanics of the ACL.
Acknowledgements The authors would like to thank Dr. Sandra Martelli, Dr. Vera Pinskerova and Prof. Michael A.R. Freeman for providing the experimental data. We would also like to acknowledge the University of Southampton and the company ESI-Group for their financial and technical support.
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