Three-Dimensional Finite Element Analysis
of a Total Knee Prosthesis

RAJ BASU and BONEY MATHEW

Mathson Industries Inc.
1845 Thunderbird Street
Troy, MI 48084

A common problem with knee prostheses is wear failure of the polyethylene tibial plateau. The objective of this study was to compute the stresses generated in the polyethylene and to determine how the stresses varied with changes in the minimum polyethylene thickness and the joint load. High stresses generated in the polyethylene thickness and the joint load. High stresses generated in the polyethylene lead to failure by wear.

A three-dimensional finite element model of a metal-backed polyethylene tibial component with a conforming femoral component was generated. Nonlinear GAP elements were used to define the contact between the articulating femoral and tibial surfaces. Four load cases were considered. Two cases simulated level walking and two simulated stair climbing. For each activity, one load case considered the joint load as being equally shared between the two condyles, while the other case assumed that the joint load was acting only on one condyle. Six different model cases were considered each having a different minimum polyethylene thickness (tmin). Each model case was analyzed for each of the four load cases.

The maximum Von Mises stress exceeded the yield strength of polyethylene for all thicknesses for the extreme load case of stair climbing. For tmin = 8 mm, the maximum Von Mises stress was less than the yield strength for al the other load cases. The maximum shear stress was less than the failure value for all cases. The optimum minimum polyethylene thickness for the tibial component of a knee prosthesis was concluded as 8 mm. With this thickness, the polyethylene would not yield during normal level walking and during stair climbing, yielding and subsequent wear of the polyethylene would be reduced. For a minimum thickness greater than 8 mm, the undesirable effect associated with over-resection of bone become dominant.

INTRODUCTION

The knee is the largest synovial joint of the human body. The movement at the knee joint involves mainly flexion-extension, as well as medial-lateral rotation and abduction-adduction in the frontal plane. There is also limited gliding of the femoral condyles on top of the tibial plateau.

There are instances when the knee joint becomes degenerated due to disease or damaged due to injury. Joint degeneration due to osteoarthritis and rheumatoid arthritis is a common problem in the elderly. Total knee replacement is often a successful treatment.

The primary indications for total knee arthroplasty are pain, stiffness, instability and deformity at the joint. However, whether the patient requires a knee replacement or not also depends on other factors such as age and life-style, the precise nature of the under-lying disease and the clinical state of the affected knee.

A total knee prosthesis includes a tibial component and a femoral component. The femoral component consists of a bicondylar surface, made of a biocompatible metallic material, such as Co-Cr alloy, that articulates on top of the tibial plateau. The tibial component consists of an ultra-high molecular weight polyethylene (UHMWPE) bicondylar plateau fixed on top of a metallic tray with a central stem or short pegs on the underside.

It has been found that a prosthesis will primarily fail due to the failure of the tibial component. More often than not, a tibial component will fail due to polyethylene wear of the tibial plateau.

Wright, et al (1), in 1982, studied the mechanism of wear, from the analysis of retrieved implants. Whenever materials are in moving contract and transmitting large loads, material damage occurs. The resulting damage may occur in different modes that are collectively termed wear.

BACKGROUND

Kilgus, et al (2), in 1991, reported 8 revisions (4.5% of the series) in a series of 176 PCA knees, all due to tibial polyethylene wear, at an average of 60 months. Nine additional knees (5.1%) had thinning of greater than 30% of the initial polyethylene thickness. Jones, et al (3), in 1992, in their 5 year follow-up study reported that extensive wear on the medial side of the polyethylene surface led to failure in 5 (4.6%) of a series of 108 uncemented PCA knees. Engh, et al (4) in 1992, observed severe wear of the polyethylene tibial plateau in 51% of the 86 metal-backed tibial components retrieved after an average of 39.5 months in situ.

Tsao, et al (5), in 1993, reported a 6.6% failure rate due to severe polyethylene wear in a series of 487 PCA knees, all due to polyethylene wear, after a period of 4.5 years. Kilgus, et al (2), in 1991, discussed the factors that play an important role in the wear failure of the polyethylene. The factors are manufacturing processes, polyethylene thickness, material property of the polyethylene, tibio-femoral articular geometry, knee alignment, femoral component surface bearing material and modularity of the tibial inserts and trays.

Bartel, et al (6,7) in 1985 and 1986, used an elasticity solution to express the contact stresses as a function of the tibial plateau thickness and recommended a minimum polyethylene thickness of 8 mm. Finite element analysis (FEA) was used to compare the maximum contact stress and to determine the maximum principal and maximum shear stresses. Collier, et al (8) in 1991, used a finite element method to study the maximum contact and Von Mises stresses on the polyethylene plateau as a function of the polyethylene thickness and recommended a minimum polyethylene thickness of 6 mm. The criterion for failure was that yielding occurred when the Von Mises stress exceeded the uniaxial yield strength.

Whiteside, et al (9), in 1985, observed that resection of excessive bone from the distal femur raises the join line and alters the kniematics of knee flexion. Volz, et al (10), in 1986, observed that resection of excessive bone from the proximal tibia will leave bone with decreased strength to resist compression, thus risking that the polyethylene thickness of the widely used tibial components do not use the recommended thickness of 6 - 8 mm.

OBJECTIVE

The objective of this study was to investigate the stresses in the polyethylene tibial plateau that leads to wear. A three dimensional finite element model of a metal-backed polyethylene tibial component with a conforming femoral component was developed. The parameters that were varied were the polyethylene thickness and the applied loading conditions. The failure criteria was as per the Maximum Shear Stress Theory and the Maximum Distortion Energy Theory of failure.

The hypothesis of the study was that the maximum Von Mises stress, the maximum contact stress, the maximum principal stress and the maximum shear stress, in the polyethylene tibial plateau decreases with an increase in the minimum polyethylene thickness and the increases with an increase in the applied joint load.

MATERIALS AND METHODS

The model geometry was based on a commercial Kinemax® Total Knee manufactured by Howmedica, Inc. The 3-D finite element model was generated using the FE package. COSMOS/M. Because of the symmetry, only half of the components were modeled to save computer time and memory space (Fig 1).

The femoral component, the tibial plateau and the metal tray were modeled using 8 noded, 3-D SOLID elements. Similar element type was used to model the soft rubber-like cushioning elements. The list of material properties is show in Table 1. All the materials were assumed to be homogeneous, isotropic and linearly elastic.

The contact between the femoral and tibial components was modeled using one-noded GAP elements. The advantages of using one-noded GAP elements are that the exact location of the point of contact and the direction of the contact forces need not be known a prior and the nodal points on the contacting surfaces need not match each other.

Previous investigations by other authors have used a total resultant load of three times the body weight (3BS) at the knee joint to consider leverl walking. MOrrison (12), in 1970, estimated the compressive force at the knee as 4.8 BW for stair climbing. Two cases simulated level walking and two simulated stair climging. The joint loads considered were 3 BS and 5 BW respectively. For each activiy, one load case considered the joint load as being equally shared between the two condyles, 2 C; while the other case assumed that the joint load was acting only on one condyle, 1 C. The magnitudes of the joint load for the four cases were 1250 N (3 BW, 2 C), 2084 N (5 BW, 2 C), 2500 N (3 BW 1C) and 4167 N (5 BW, 1 C). Six different model cases were considered, each having a different minimum polyethylene thickness tmin). The values for tmin were 2, 3, 6, 8, 10 and 11 mm.

Each model case was analyzed for each of the four load cases. The analysis method used was nonlinear static analysis. A force control technique was used in the analysis where the loads were applied in incremental steps. The Newton-Raphson (NR) iterative scheme was adopted which had a high convergence rate.

RESULTS AND DISCUSSION

The Von Mises stress plots for the model case, tmin = 2 mm are shown in Figs. 2 and 3. The variation of the maximum Von Mises stresses with tmin is shown in Fig. 4. It was observed that when tmin was below 4 mm, the stresses increased rapidly with decreasing

FIG 1 MATERIAL PROPERTIES
 

Table 1. material Properties.

FIG 2 VON MISES STRESS PLOT
(Model Case: tmin = 2 mm, Load Case: 3 BW, 1, C)

.
FIG 3 VON MISES STRESS PLOT
(Model Case: tmin = 2 mm, Load Case: 5 BW, 1, C)

thickness. A transition occurred for thicknesses between 4 - 8 mm, with the stresses decreasing slowly as tmin increased from 6 mm to 8 mm. For tmin = 8 mm and greater, the drop in stresses was minimal. The other stress curves (maximum normal, maximum principal and maximum shear stress curves) were observed to have similar patterns.
Table 2 shows the maximum stresses for the model case, tmin = 8 mm. The Von Mises stress plots for the model case, tmin = 8 mm are shown in Figs 5 and 6. The maximum Von Mises, contact and principal stresses were at the top central node. The maximum shear stress was observed 1 mm to the left of the top central node.

CONCLUSIONS

The nature of the variation in the stresses with the change in the minimum polyethylene thickness and the joint load was similar to those in previous studies (6,8).

The results validated the hypothesis. When tmin was below 4 mm, the stresses increased rapidly with decreasing thickness. A transition occurred from 4 - 8 mm. For tmin = 8 mm and greater, the drop in stresses was minimal. The maximum Von Mises stress always exceeded yield strength of polyethylene for the extreme load case. For the tmin = 8 mm, max. Von Mises stress was less than yield strength for all other load cases. Optimum minimum thickness of polyethylene should be 8 mm. Good compromise with undesirable problems of over-resection of bone.

FIG 4. Maximum Von Mises
Stress vs. Minimum Polyethylene Thickness
 

Table 2. Maximum Stresses Generated for Different Load Cases for the Model Case, tmin = 8 mm.

Fig. 5. Von Mises Stress Plot
(Model Case: tmin = 8 mm, Load
Case: 3 BW, 1 C).

Fig 6. Von Mises Stress Plot
(Model Case: tmin = 8 mm, Load Case: 5 BS 1 C).

ACKNOWLEDGMENTS

The authors would like to thank Dr. Mary C. Verstraete, Dr. Michael J. Askew, Dr. Glen O. Njus and Ms. Marnie M. Saunders at The University of Akron, Arkon, OH; Mr. Nicholas A. Plaxton at Mt. Sinai Medical Center, Cleveland, OH; and Ms. Maria Lotoczky and Ms. Natayla Zorina at Mathson Industries, Inc., Troy, MI for their valuable suggestions and assistance on this project.

REFERENCES

1. T. M. Wright, R.W. Hood and A. H.l Burstein, “Analysis of Material Failures,” Orthop, Clin. North Am., Vol. 13, 1982, pp 33 - 44.

2. D. J. Kilgus, J. R. Moreland, G.A. Finerman, T. T. Funahashi and J. S. Tipton, “Catastrophic Wear of Tibial Polyethylene Inserts,” Clinical Orthopaedics and Related Research, Vol. 273, 1991, pp. 223 - 231.

3. S. M. Jones, I. M. Pinder, C. G. Moran and A. J. Malcolm, “Polyethylene Wear in Uncemented Knee Replacements,” Journal of Bone and Joint Surgery - British Volume, Vol. 74, 1992, pp. 18 - 22.

4. C. A. Engh, K. A. Dwyer and C. K. Hanes, “Polyethylene Wear of Metal-backed Tibial Components in Total and Unicompartmental Knee Prosthesis,” Journal of Bone and Joint Surgery - British Volume, Vol. 74-B, 1992, pp. 9 - 17.

5. A. Tsao, L. Mintz, C. R. McRae, S. D. Stulberg and T. Wright, “Failure of the Porous-Coated Anatomic Prosthesis in Total Knee Arthroplasty due to Severe Polyethylene Wear,” Journal of Bone and Joint Surgery, Vol. 75, 1993, pp. 19 - 26.

6. D. L. Bartel, A. H. Burstein, M. D. Toda and D. L. Edwards, “The Effect of Conformity and Plastic Thickness on Contact Stresses in Metal-Backed Plastic Implants.” Journal of Biomechanical Engineering, Vol. 107, 1985, pp. 193 - 199.

7. D. L. Bartel, V. L. Bickness, M. S> Ithaca and T. M. Wright, “The Effect of Conformity, Thickness and Material on Stresses in Ultra-High Molecular Weight Components for Total Joint Replacement,” Journal of BOne and Joint Surgery, Vol. 68A, 1986, p. 1041.

8. J. P. Collier, M. B. Mayor, J. L. McNamara. V. A. Suprenant and R. E. Jensen. “Analysis of The Failure of 122 Polyethylene Inserts From Uncemented Tibial Knee Components,” Clinical Orthopaedics and Related Research, Vol. 273, 1991, pp. 232 - 242.

9. L. A. Whiteside and R. Summers. “The Effect of the LEvel of Distal Femoral Resection on Ligament Balance in Total Knee Replacement.” The Knee: Papers of the First scientific Meeting of the Knee Society, Baltimore, University Park Press, 1985, pp. 59 - 73.

10. R. G. Volz, S. G. Kantor, C. Howe and M. McMurtry, “Factors Affecting Tibial Component Stability: A comparative Study,” Total Arthroplasty of the Knee: Proceedings of the Knee Society, 1985 - 1986, Rockville, Maryland, Aspen Publishers, 1987, pp. 109 - 120.

11. K. J. Chillag and E. Barth, “An Analysis of Polyethylene Thickness in Modular Total Knee Components,” Clinical Orthopaedics and Related Research, Vol. 273, 1991, pp. 261 - 263.

12. J. B. Morrison, “The Mechanics of the Knee Joint in Relation to Normal Walking,” Journal of biomechanics, Vol. 3, 1970, pp. 51 - 61.