Topology optimisation of hip prosthesis to reduce stress shielding

S. Shuib1, M. I. Z. Ridzwan1, A. Y. Hassan1 & M. N. M. Ibrahim2 1School of Mechanical Engineering, Engineering Campus, Universiti Sains Malaysia, Malaysia 2School of Chemical Sciences, Universiti Sains Malaysia, Malaysia

Abstract

This work presents the application of a topology optimisation method to the hip implant design. A three-dimensional implanted femur is modelled and defined as a design domain. The implant, modelled as type 1, is optimised while other materials i.e. cement (type 2) and bone (type 3), are not being optimised. The domain is subjected to a load case, which corresponds to the loads applied when walking. Loads are employed at the proximal end of the implant and the abductor muscle. Loads from other muscles are not considered. The goal of the study is to minimise the energy of implant compliance subjected to several sets of volume reduction. Reductions are set to be 30%, 40%, 50%, 60%, and 70% from the initial volume (Vo). The result of each set is cut into several sections about x-y plane in z-direction in order to observe the topology inside the stem. It was found that implants with 30% Vo, 40% Vo, and 70% Vo had developed open boundaries whereas 50% Vo and 60% Vo had closed boundary and produced possible shape. Therefore, these designs (50% Vo and 60% Vo) are chosen and refined. Both are analysed using the same boundary conditions as before they were optimised. Results of stresses along medial and lateral line are plotted and compared. Keywords: topology optimization, hip prosthesis, stress shielding, FE analysis.

1 Introduction

Over 800,000 artificial hip joints were implanted worldwide annually suggesting that it is a well-accepted and successful treatment [1]. The surgical procedure involves removing parts of the hip joint that have been damaged and replacing

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the joint with an artificial implant. Normally, femur carries its external load by itself. When an implant is introduced into the canal, it shares the load with the bone. As a result, the bone is subjected to reduce stress, and hence stress shielded. This phenomenon is called stress shielding. It is natural for the bone to release calcium when it is not needed anymore, and hence reduced the bone mass. Atrophy and resorption lead to a loosening of the prosthesis and failure of the implant. Nevertheless, this reaction does not occur spontaneously; instead, it is a gradual process, which may take a few years [2]. Bone receives more loads if stem can be eliminated from the implant. Consequently, a stemless implant was designed. It was fixed with several screws and cables to support the head [3, 4]. However, it was difficult to position it correctly during operation and may possible loss of initial stability [5]. In another work, hollow geometry has been introduced by increasing stem inner diameter to reduce stress shielding [6]. However, it has increased maximum stem stress dramatically when bending was applied. Stress shielding can also be decreased if stem is made from non-stiff material such as polymeric [7]. But, flexible implant may produce higher stresses along the interface [8]. This paper tries to look at the potential of optimising the stem topology in order to reduce the same problem. The idea of topology optimisation is to get the best distribution of material within a fixed domain as we applied the boundary conditions. This method is previously used in many engineering applications such as material design [9], compliant mechanisms [10], bone remodelling [11], and components in car [12].

Figure 1: (a) Design domain. (b) Meshed domain and applied boundary conditions.

Fh Fa

Implant (type 1)

Cement (type 2)

Cortical bone (type 3)

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2 Material and methods

2.1 Design domain

A study is performed using ANSYS 7.1. Figure 1(a) shows a model of implanted femur. The design was divided into three types, which corresponded to three different materials. Implant is identified as type 1 and would be optimised. Other two types would not be optimised. The model uses twenty node hexahedral finite elements (ANSYS type SOLID95). The varied element length is given to each material. Implant should be meshed smoother or otherwise, it would result in a coarser surface in the final topology. However, smooth mesh consumes a lot of computing time. There are 12201 number of elements with 7287 elements for the implant, 1498 (cement) and 3416 (bone). The distal end of the bone is rigidly fixed and loads are applied at the proximal end of the stem (Fh) and the abductor muscle (Fa). Figure 1(b) displays model of meshed implanted femur with applied boundary conditions. Applied load cases were similarly described as in Fernandes and Rodrigues [11]. However, for optimisation purposes, only one load case with maximum resultant value is considered. It is the initial load case that corresponded to walking movement as shown in Table 1. Properties of materials used are depicted in Table 2.

Table 1: Applied load cases.

Table 2: Properties used in the FE model.

2.2 Topology optimisation method

The theory of topology optimisation seeks to minimize or maximize the objective function (f) subject to constraints defined. Design variables (ρi) are

Loads Fx (N) Fy (N) Fz (N)

Fa -768 -726 1210 1

Fh 224 972 -2246

Fa 166 382 957 2

Fh 136 -630 -1692

Fa 383 669 547 3

Fh 457 -769 -1707

Parts Elastic modulus, E (GPa) Poisson’s ratio, ν

Implant (Titanium) 115 0.3

Cement 2 0.3 Cortical bone 20 0.3

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densities that are assigned to each finite element (i) in the topology problem. The density for each element varies from 0 to 1; where ρi = 0 represents material to be removed; and ρi = 1 represents material that should be kept. We want to minimize the energy of implant compliance for a given load case subject to its volume reduction. Minimizing the compliance is equivalent to maximizing stiffness. The optimisation problem is explained as follows: Minimize the energy of the implant compliance (Uc).

Figure 2: Topology main dialogs.

In this case, six components of load are applied as shown in Table 1. Therefore, f would be stated as,

∑=

=6

1

621

i

iciccc ,UW),....,U,Uf(U Wi ≥ 0 (1)

subject to:

Percentage of volume reductionTotal load applied inxyz directions

Multiple load cases

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0 < ρi ≤ 1 (i=1, 2, 3,….N) (2) V ≤ Vo – V* (3)

where: Wi = weight for load case with energy Uc V = computed volume Vo = original volume V* = amount of material to be removed

The reductions of volume were set to be 30%, 40%, 50%, 60% and 70% from the initial implant volume (Vo). Figure 2 shows topology main dialogs in Ansys 7.1. Objectives and constraints of optimisation are defined in the boxes below.

3 Results and discussions

3.1 Topology results and design interpretation

Table 3 exhibits the implant topology for each material constraint case about x-y plane in given z-direction. The topologies represent the value of density between 0.89-1. Although every solution is topologically different, tendencies are very similar. The main difference can be seen at the proximal end of the implant, especially between 1.85 mm to 10.90 mm. Solutions of 70% Vo, 40% Vo and 30% Vo had developed an open boundary whereas 50% Vo and 60% Vo had developed closed boundary and produced acceptable shape. Therefore, these designs are chosen for shape refinement.

3.2 Shape refinement

Topology optimisation results as shown in Figure 3(a) can be difficult to interpret since they contain zigzag border. The figure presented five different topologies obtained from one of the optimum implants. CAD software is used to refine the model. Figure 3(b) shows each layer of cross sections after being refined. Model in 3-dimension is displayed in Figure 3(c).

3.3 Comparison results

In order to see the performance of new models, stresses along medial and lateral side in intact and implanted femur are plotted and compared with optimum models. Figure 4 shows the stresses in the femur, which occurred along medial and lateral side. By comparing both graphs, we can see that maximum stress occurrences are evident at the distal end whereas minimum stress occurrences are apparent near the proximal end. Stresses produced in both optimum models are very close and are demonstrated almost like a similar line. In medial side,

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both implants increase the stress after one fifth of the femur. In lateral side, tensile stress is very low in the beginning, but it starts to increase after one third of the femur and is maintained until the end. This was probably because of the wide cross sectional area around the greater trochanter. Maximum stress occurs in the middle of the femur at the level of implant tip which means that load transfer has increased in the femur with the optimised implants compared to before optimisation.

Table 3: Implant topologies about x-y plane in z-direction with different implant volumes.

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Figure 3: Topology optimisation result for 60% Vo. (a) Before and (b) after shape refinement. (c) Optimised implant in 3 dimensions. Only half of its size was modelled to show it’s inside topology.

Table 4 shows maximum stresses and percentage of load distribution occurrences in femur along medial and lateral sides. The percentages are obtained by using a simple formula:

-% 100%

A Ao fArea

Ao= × (4)

where: Ao = Integration of intact femur Af = Integration of other graphs From table 4, we could observe that both optimum implants have shown an increment in load distribution to the femur in comparison with the conventional implant. Although the differences between optimum and conventional implants are not too far, it has been proven in Figure 4 that, optimised implants have tried to bring the stress as close as in intact femur especially along the length of implants.

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0 50 100 150 200 250

0

10

20

30

40

50

60

70

Von

Mise

s stre

ss (M

Pa)

0

10

20

30

40

50

600 50 100 150 200 250 300

Von

Mise

s stre

ss (M

Pa)

Figure 4: Von Mises stresses of the femur along medial (top) and lateral

(bottom) side.

Table 4: Load distribution in femur.

Models Percentage of decrease in load

distribution at medial side (%). Ao = 11738.26 mm2

Percentage of decrease in load distribution at lateral side (%).

Ao = 10631.90 mm2

Intact femur N/A N/A Implant before optimised

24.75 23.72

60% Vo implant 20.94 20.54

50% Vo implant 20.72 20.95

Intact femur

Implant before optimised 50% Vo implant

60% Vo implant

Distance from proximal end (mm)

Intact femur

Implant before optimised

50% Vo implant

60% Vo implant

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4 Conclusion

Topology optimisation method was applied to get the best material needed for an implant in order to minimize stress-shielding problem. Stem topology was optimised to achieve minimum compliance for a given load case subjected to several sets of volume reduction. From ANSYS simulation, implant with 50% Vo and 60% Vo were chosen as the best and refined. Comparison between stress distribution in intact and implanted femur with optimal design was conducted. Results of application problem showed the advantage of the method.

Acknowledgement

This work is supported by a short term USM under grant No. 6035114.

References

[1] Li, C., Granger, C., Schutte, H.D., Biggers, S.B., Kennedy, J.M., & Latour, R.A. Failure analysis of composite femoral components for hip arthroplasty, Journal of Rehabilitation Research and Development, 40(2), pp. 131–146, 2003.

[2] Bagge, M.A. model of bone adaptation as an optimization process, Journal of Biomechanics, 33(11), pp. 1349-1357, 2000.

[3] Munting, E., & Verhelpen, M. Fixation and effect on bone strain pattern of a stemless hip prosthesis, Journal of Biomechanics, 28(8), pp. 949–961, 1995.

[4] Joshi, M.G., Advani, S.G., Miller, F., & Santare, M.H. Analysis of a femoral hip prosthesis designed to reduce stress shielding, Journal of Biomechanics, 33, pp. 1655-1662, 2000.

[5] Rietbergen, B.V., & Huiskes, R. Load transfer and stress shielding of the hydroxyapatite-ABG hip: A study of stem length and proximal fixation, The Journal of Arthroplasty, 16(8), pp. 55-63, 2001.

[6] Gross, S., & Abel, E.W. A finite element analysis of hollow stemmed hip prostheses as a means of reducing stress shielding of the femur, Journal of Biomechanics, 34, pp. 995–1003, 2001.

[7] Sumner, D.R., Turner, T.M., Igloria, R., Urban, R.M., & Galante, J.O. Functional adaptation and ingrowth of bone vary as a function of hip implant stiffness, Journal of Biomechanics, 31(10), pp. 909-917, 1998.

[8] Huiskes, H.W.J., Weinans, H., & Rietbergen, B.V. The relationship between stress shielding and bone resorption around total hip stems and the effects of flexible materials, Clinical Orthopaedics, 274, pp. 124-134, 1992.

[9] Lienemann, J., Greiner, A., Korvink, J.G., & Sigmund, O. Optimization of integrated magnetic field sensors’, Proc. of the 4th Int. Conf. on Modeling and Simulation of Microsystems, 2001.

[10] Sigmund, O. On the design of compliant mechanisms using topology optimization, Mechanics of Structures and Machines, 25(4), pp. 493-524, 1997.

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[11] Fernandes, P.R., & Rodrigues, H.C. A material optimisation model for bone remodelling around cementless hip stems, ECCM’99 – European Conference on Computational Mechanics, Munich, Germany, 1999.

[12] Shin, J.K., Lee, K.H., Song, S.I., & Park, G.J. Automotive door design with the ULSAB concept using structural optimization, Structural and Multidisciplinary Optimization, 23, pp. 320-327, 2002.

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