Finite Element Modeling Of Knee Joint

Proceedings of IMECE’03 2003 ASME International Mechanical Engineering Congress & Exposition Washington, D.C., November 16-21, 2003 IMECE2003-43153 1.9 ms, TR: 7.1 ms, 1 NEX, Flip Angle: 40°, Scanning Time: 2 min) gradient recalled echo (GRE) sequences in the unloaded condition to provide visualization of the knee joint and reference for tracking the bone motion. Then, loads equivalent to 64% of body weight (340 N) were applied to the knee joint. Image sets were acquired using the fast GRE sequence immediately after the load was applied and again after 18 minutes of loading. For each image volume, 64 sagittal slices were collected with thickness of 1.5mm and in-plane resolution of 0.66mm¥0.66mm (field of view: 170mm; matrix: 256¥256 pixels). Motion Tracking To impose the kinematics of the bones during axial loading, the tibia and femur were first segmented on the MR image volume acquired using the fast GRE sequence in the initial unloaded position. The centroid and principle axes of each bone segmentation were then calculated, then motion tracking algorithms (Tamez-Pena 1999) were used to calculate the translation of the centroid and the rotation of the principle axes of each bone segmentation from the initial unloaded position to two subsequent loaded positions. This algorithm has been demonstrated to have an accuracy of 0.39mm in translation and 0.38° in rotation (Lerner 2003). FE Modeling A three-dimensional finite element model of tibio-meniscofemoral joint contact was created for the unloaded condition using pre processor HyperMesh 5.0 (Altair Engineering, Inc.). The model was analyzed using finite element software ABAQUS 6.3 (Hibbitt, Karlsson & Sorensen, Inc.).

FINITE ELEMENT MODELING OF KNEE JOINT CONTACT PRESSURES AND COMPARISON TO MAGNETIC RESONANCE IMAGING OF THE LOADED KNEE Jiang Yao, Art D. Salo, Monica Barbu-McInnis, and Amy L. Lerner Department of Biomedical Engineering University of Rochester Rochester, NY, 14627 INTRODUCTION A finite element model of the knee joint could be helpful in providing insight on mechanisms of injury, effects of treatment, and the role of mechanical factors in degenerative conditions. However, preparation of such a model involves many geometric simplifications and input of material properties, some of which are poorly understood. Therefore, a method to compare model predictions to actual behaviors under controlled conditions could provide confidence in the model before exploration of other loading scenarios. Our laboratory has developed a method to apply axial loads to the in vivo human knee during magnetic resonance imaging, resembling weightbearing conditions. Image processing algorithms may then be used to assess the three-dimensional kinematics of the tibia and femur during loading. A three-dimensional model of the tibio-menisco-femoral contact has been generated and the image-based kinematic boundary conditions were applied to investigate the distribution of stresses and strains in the articular cartilage and menisci throughout the loading period. In this study, our goal is to investigate the contact patterns during long term loading of up to twenty minutes in the healthy knee. Specifically, we assess the use of both elastic and poroelastic material properties in the cartilage, and compare model predictions to known loading conditions and images of tissue deformations.

Fig 2. Finite element model of the knee joint. For efficient analysis, model was trimmed to include either medial or lateral compartment.

METHODS Loading Device The femur and tibia bone surfaces are assumed rigid, with opposing articular cartilage modeled as isotropic elastic, isotropic poroelastic, or transversely isotropic poroelastic material with the material stiffer in the plane parallel to the cartilage surface than in the perpendicular direction [Table 1]. Table 1. Three types of material properties assigned to cartilage Isotropic elastic E=12MPa, n=0.45 Ref: (Jilani 1997) Isotropic HA=0.60MPa, n=0.07, K=1.14*10-15m4/NS, f m=0.20 poroelastic Ref: (Athanasiou 1991) Transversely E1= E2= 5.8MPa, E3= 0.46Mpa, v12= v23 =0.0, G13= isotropic 0.37Mpa, K=1.14*10-15m4/NS, fm=0.20 poroelastic Ref: (Athanasiou 1991; Cohen 1993)

Fig 1. In vivo axial loading device for the knee joint.

A device was used to apply axial load to the knee joint within a GE Signa clinical MRI scanner (Fig 1). A 26 yr. old female subject with healthy knees was positioned supine with the knee flexed to approximately 10°. To minimize motion during scanning, the thigh was strapped securely to a wedge shaped support restricting medial/lateral motion and rotation of the femur. The axial load of 340N was applied through weights and pulleys to an ankle-foot orthotic attached to a freely sliding track. MR imaging The knee was initially imaged with both routine (TE: 17 ms, TR: 45 ms, 1 NEX, Flip Angle: 30°, Scanning Time: 16 min) and fast (TE:

In all analyses, the menisci were assigned transversely isotropic elastic material properties to represent the circumferential fiber arrangement (E1= 140MPa, E2= E3= 20Mpa, v12= 0.2, v23 =0.3, G12= 50MPa) (Whipple 1984; Fithian 1989; Skaggs 1994; Tissakht 1995). Each meniscal attachment was defined as non-compressive linear spring elements with tensile modulus as 111MPa. The average cross section area for each attachment was 50mm2 composed of 36 springs with average initial length of 3mm. This resulted in a stiffness of

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1850N/mm for each attachment (Haut Donahue 2002). The model did not include collateral, cruciate or transverse ligaments, or joint capsule. The motions of tibia and femur were imposed at their bone centroids. To reduce calculation time, the full knee model was divided into medial and lateral compartments. Contacts were modeled between femur, tibia cartilage surfaces and meniscal surfaces. Between contact surfaces no penetration was allowed and small sliding was assumed. Fluid was not allowed to flow out of the cartilage boundaries. Validation of the FE model Because kinematic boundary conditions were applied as quantified from motion tracking, it was possible to compare the reaction forces calculated in the model to the known applied loads, assuming that the entire axial load is carried by the defined contact surfaces.

We found a shift of peak contact pressure from lateral to the medial side during axial loading. Qualitatively, the model predicted the meniscal motion and deformation comparable to the MR image during axial loading (Fig 3).

Fig 3. One sagittal slice through the lateral contact region after about 10 minutes of axial loading. (left) MRI (right) hydrostatic pressure by model with transversely isotropic poroelastic cartilage.

DISCUSSION Modeling cartilage as isotropic elastic material can not capture the creep behavior of articular cartilage during axial loading up to 20 minutes. However, using the current literature values for isotropic poroelastic material properties of cartilage, our model predicts much lower reaction force than the force applied. Increasing the modulus parallel to cartilage surface increased but still underestimated the reaction force. Low reaction forces in idealized joint contact models were also reported by Wu et al, in a study we used to validate our modeling methods (Wu, 1998). The magnitude of the reaction force is highly dependent on surface curvature, material properties and boundary conditions, all of which may need further investigation. The accuracy of our model may be limited by the MR resolution as well as motion tracking method. Other limitations include fluid boundary conditions, and modeling menisci as elastic. However, the model has provided an interesting method to investigate material properties of the soft tissues in the knee. The contact pressures between the menisci and cartilage surfaces were lower than expected, which may suggest that the material properties of meniscal attachments are inappropriate. In addition, our model does not include the transverse ligament or joint capsule. Further parametric studies and more quantitative comparisons to MR images may provide a better understanding of the role of the meniscus in this and other types of loading.

RESULTS When cartilage was modeled as isotropic elastic material, the predicted model reaction forces were 220.5N and 621.3N after 1 and 19 minutes of axial loading, however the applied reaction was 340N during the whole period, which suggested that the cartilage was much stiffer at the beginning of axial loading and became softer during loading. Suprisingly, when cartilage was modeled as poroelastic material, using values reported in recent literature, the predicted reaction force was much lower than that of the elastic model. By adding transverse isotropy to the solid matrix, there was a slight increase of the predicted reaction force, but still much lower than the elastic model prediction (Table 2). Table 2. Total reaction force with three types of cartilage material properties Total Reaction Isotropic elastic Isotropic TransverselyForce (N) poroelastic isotropic poroelastic after 1min 220.53 3.22 11.74 after 19min 621.28 5.39 19.26

ACKNOWLEDGEMENTS Financial support was provided by VirtualScopics LLC. The authors are grateful for technical support provided by Bill W. Badger, Tina Benner. REFERENCES Athanasiou, K. A., et al, 1991, J. Orthop. Res, 9:330-340. Cohen, B. et al, 1993, Trans Ortho Research Soc., Chicago, IL., 185. Fithian, D. C. et al, 1989, Trans Ortho Research Society 35: 205. Haut Donahue, T. L., et al, 2002, J Biomech Eng 124: 273-280. Jilani, A., et al, 1997, The Knee 4: 203-213. Lerner, A. L. et al, 2003, J. Biomech. Eng. 125: 246-253. Skaggs, D. L., et al, 1994, J. Orthopaedic Research 12: 176-185. Tamez-Pena et al, 1999, SPIE Medical Imaging '99, Physiology and fusion from multidimensional medical images. Tissakht, M. et al, 1995, J. Biomech 28: 411-422. Whipple, R. et al, 1984, Advances in Bioengineering, American Society of Mechanical Engineering, New York. Wu, J.Z. et al, 1998, J. Biomech 31: 165-169.

Fig 2. (a,b) Hydrostatic pressure in the elastic cartilage model and (c,d) pore pressure in the transversely isotropic poroelastic cartilage model after 1min (a,c) and 19 min (b,d) of axial loading

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