Heterogeneous modeling based prosthesis design with porosity and material variation

Journal of the Mechanical Behavior of Biomedical Materials 87 (2018) 124–131

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Heterogeneous modeling based prosthesis design with porosity and material variation Sudhir Kumar Singh, Puneet Tandon

T

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Mechanical Engineering Discipline, PDPM Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, India

A R T I C LE I N FO

A B S T R A C T

Keywords: Heterogeneous modeling Stress shielding Osseointegration Consolidation analysis

The work proposes the development of heterogeneous bio-implants with the aim to minimize stress shielding effect and enhance bone ingrowth. Stress shielding in the implant can be minimized by reducing the overall stiffness of the implant, which is achieved here by varying the material based on stress distribution across the prosthesis. To increase overall stability of the implant by simultaneous enhancing osseointegration and reducing stress shielding, the work proposes the design of heterogeneous prosthesis with graded porosity and material having radial, axial and mixed (simultaneous radial and axial) variations. Static analysis for material variation models and consolidation analysis for graded porosity and material variation models are performed. After comparisons of results among different models, radial variation model was observed to deliver the results.

1. Introduction Heterogeneous objects consist of different constituent materials with varying composition and/or microstructure, thus, producing gradation in their properties (Kumar et al., 1999). Heterogeneous objects can be multi-material objects with discrete material distribution; objects with sub-objects embedded and functionally graded materials (FGM) with continuous material distribution (Gupta et al., 2010). Heterogeneous objects are believed to acquire superior properties in the applications, where multiple functional requirements are simultaneously expected. With gradual material variations (Kou and Tan, 2007), different properties and advantages of various materials can be extracted and traditional limitations due to materials incompatibilities can be eliminated With special features and advantages, heterogeneous objects have great potentials in numerous applications, like design of mechanical components, electronics, power plant, aerospace technology, and in biotechnology (Ozbolat and Koc, 2011). Heterogeneous modeling could be used as a tool to optimize the material selection and its distribution, based on the criteria of loading and functional requirements. User defined material distribution with smooth variation of material using the concepts such as gradient reference approach could lead to computationally robust and efficient designs (Gupta et al., 2012). Bones can be considered as heterogeneous objects (Cheng and Lin, 2005), due to their composite structures. Bones contain hydroxyapatite and collagen as organic and inorganic components, with variation in

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strength, stiffness, porosity, and density in cortical and trabecular (spongy) bones (Pan et al., 2011). The cortical bones are stiffer and able to resist higher ultimate stresses than trabecular bones, but they are relatively more brittle in nature. The mechanical response to loading differs widely between cortical and trabecular bones (Osterhoff et al., 2016). In addition to infection, aseptic loosening between implant and bones is one of the major post-surgery concerns. Stress shielding is one of the main causes of aseptic loosening (Joshi et al., 2000). Large stiffness difference between the bones and the prosthesis is the major cause of stress shielding phenomenon. Stress shielding causes aseptic loosening, which increases micro-motion. This may cause not only pain to patients and but also lead to frequent implant replacements (Ridzwan et al., 2007). By merging a stiff material with a flexible one, it is possible to obtain a design with preferable controlled stiffness (Simoes, Marques, 2005). Efforts have been made to use fixation devices made of glass fiber reinforced composites with no risk of debris and adverse tissue reactions (Moritz et al., 2014). Moreover combination of polymer based composites with bioactive glass granules reports interesting results for strength and osseointegration (Puska et al., 2016). Porous structure could provide a desirable environment for bone ingrowth to achieve a tight fixation between the implant and the surrounding bones, and enhance the transport of fluids. Pores are necessary for bones tissue formation because they allow migration and proliferation of osteoblast and mesenchymal cells, as well as

Correspondence to: PDPM Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, Dumna Airport Road, Jabalpur 482 005 India. E-mail addresses: [email protected] (S.K. Singh), [email protected] (P. Tandon).

https://doi.org/10.1016/j.jmbbm.2018.07.029 Received 8 June 2018; Received in revised form 17 July 2018; Accepted 18 July 2018 Available online 21 July 2018 1751-6161/ © 2018 Elsevier Ltd. All rights reserved.

Journal of the Mechanical Behavior of Biomedical Materials 87 (2018) 124–131

S.K. Singh, P. Tandon

vascularization (Xie et al., 2017). With higher Young's moduli compared to bones, dense biomaterials pose the problem of stress shielding. On the other hand, biomaterials with low Young's moduli and porous behavior exhibit the potential of bone ingrowth, which would depend on parameters such as porosity, pore interconnectivity, and pore size. Unfortunately, highly porous biomaterials have poor mechanical properties. Thus, for optimum performance biomaterials with desirable biological and mechanical properties, graded porosity, pore size, and/ or composition had been developed (Miao and Sun, 2009). From the literature, it was observed that simultaneous analysis of graded materials and porosity variation was not given due attention. Little attention had been paid to the optimization of material biological stimuli, as bones area complex structure with variable mechanical and biological properties. Further, consolidation analysis of bio-implants was hardly carried out. The objective of the present work is to carry out an appropriate computational analysis procedure to determine the prototype stiffness variation in order to reduce the stress shielding effect. The work unfolds the design of heterogeneous prosthesis with the aim to increase the stability of the implant, by simultaneous varying the implant material and graded porosity in different patterns. Fig. 1. Simplified prosthesis prototype model used in this study.

2. Methodology

shielding (Oshkour et al., 2014). The stress shielding effect (SSE) is calculated by the following formula

The work presented here consists of two modules. The first module, titled ‘prosthesis design models’ discusses the importance of different patterns of material variation in prosthesis design and its effect in reducing the stress shielding effect. The second module titled ‘porous prosthesis design models’ discusses the poroelastic behavior of boneimplant behavior and the role of porosity in bone ingrowth.

SSE =

Hip prosthesis mainly fails due to the design of the implant. Stress shielding is related to the large difference between Young's modulus of the implant material and the bone. This elasticity mismatch causes an increase in induced bones resorption, which diminishes the density of adjacent bones and consequently, generates a higher risk of fracture (Torres et al., 2014). In intact bones, the load of the body is transferred through the bones. However, when the implant is inserted into the bones, they behave like a composite bar. Both implant and the bones share the load of the body. Due to the higher stiffness of the implant, the load borne by the implant is more as compared to that of the bones. This leads to a reduction in load transferred to the bones after implantation in comparison to that before implantation and is called stress shielding effect. The simplified model of the prosthesis which is used in this study are represented with the help of Fig. 1. If two materials are bonded (e.g., bones and implant)and under application of force, then based on strain equality (in both the bars) and Hooke's law, the load shared on each part of a structure can be obtained using the following equation:

Ai Ei F Ai Ei + Ab Eb

Fb =

Ab Eb F Ai Ei + Ab Eb

Average volume equivalent stress Σ(Vol. of each element) * (Eqvi. stress on each element) = Σ(Vol. of each element)

where Mi Mbending

Fi Fnormal

Mi Ii Ei = Mbending Ii Ei + Ib Eb

(4)

However, the loading on the prosthesis is complex which involves torsion, bending and other loadings. Thus, by varying the biomaterials according to the distribution of the load on the implant, one can reduce the stress shielding effect by reducing the overall stiffness of the implant. In this work, the heterogeneous modeling of the implant is done by simplifying design and varying the materials in different patterns, i.e., in radial, axial and mixed (simultaneous radial and axial) directions. Modeling is carried out with the help of open source software, OpenSCAD. Fig. 2 represents the simplified models of three heterogeneous prosthesis. In these figures, the red and blue colors denote the most and the least stiff materials respectively. For radial variation, stiffer material is used in the circumferential boundary region, and stiffness decreases in radial inward direction of the prosthesis. This material distribution is attributed to the fact that outer circumferential regions are heavily loaded in the presence of bending and torsional loadings. For axial variation, proximal end is modeled with stiffer material and stiffness decreases towards the distal end as most significant periprosthetic bone loss occurs in proximal zone and least in distal zone (Tavakkoli Avval et al., 2015). Oshkur et al. has also adopted the similar material variation pattern (Oshkour et al., 2014). In mixed variation model, materials are varied by combining radial and axial variation patterns. In this study material distribution in heterogeneous prosthesis has been governed by the distance based potential function f(X) which depend upon the expression, domain and expression (Gupta and

(1)

where the subscript i denote the implant and b denote the bone, F is the force applied to the integrated bone and implant model, E is Young's modulus of elasticity and A is the cross-sectional area. Further,

Fi Ai Ei = ; Fnormal Ai Ei + Ab Eb

(3)

pre − THA is the average volume von Mises stress in the intact where σVMS THA femur, i.e., before Total Hip Arthroplasty (THA) and σVMS is the average volume von Mises stress in the implant, i.e., after the THA. As von Mises stress is non-negative quantity, negative stress difference value indicates increase of stress in post-THA situation, i.e., stress shielding. A negative value of the stress shielding effect indicates an increase in stress concentration, which is of great concern (Fraldi et al., 2010). Average volume equivalent stress (von Mises stress) is given by

2.1. Prosthesis design models

Fi =

pre − THA THA (σvms ) − (σvms ) pre − THA (σvms )

(2)

the normal load is shared by implant under axial load and

is the transverse load shared by implant under bending loads.

From Eqs. (1) and (2), it is noticed that the load borne by the implant and the bones in combined loading depends upon the respective stiffness. Excessive stiffness of the implant will lead to more load being shared by the prosthesis and will consequently lead to increase in stress 125

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Table 1 Biomaterial with their approximate properties.

Proximal end

Bio-materials

Young's Modulus (GPa)

Tensile Strength (MPa)

Poisson's Ratio

Density (kg/m3)

Ti-6Al-4V (Ti64) Ti-12Mo-6Zr-2Fe (TMZF) Ti-35 Nb-5Ta-7Zr0.4 O (TNZTO)

110 75–85

900 1030

0.32 0.32

4500 5000

66

1010

0.34

5600

Table symbols Ti: Titanium, Al: Aluminum, V: Vanadium, Mo: Molybdenum, Zr: Zirconium, Fe: Iron, Nb: Niobium, Ta: Tantalum, O: Oxygen.

displacement of the prosthesis relative to the bones with respect to time is an important clinical parameter to predict the long-term behavior of prosthesis. Early accentuated migration could lead to premature fracture of the implant (Simoes, Marques, 2005). Introducing porosity into the implant seems to be an effective way to reduce Young's modulus of the implant, which leads to a reduction in stress shielding effect and improves the stability of the implant by promoting bone ingrowth. Porous implant promotes bones ingrowth and acts like an organization of vascular canals that can ensure the supply of blood and nutrients for viability of bones. However, increasing porosity drastically reduces the strength of the implant (Niinomi and Nakai, 2011). Thus, the objective is not only to design bio-inspired implants that promote bone ingrowth and reduces stress shielding, but also to optimize the mechanical and biological properties of the implant. In this work, this is taken care by designing heterogeneous bio-implants having a simultaneous variation of graded materials and porosity. Porous surface of the implant and interconnected network of internal pores provide the implant natural fixation in the body due to soft tissue and bones ingrowth (Imwinkelried, 2007). Performance of porous surface relies on several topological features, including size and shape of pores, pores orientation, porosity and randomness with which pores are distributed (Muñoz et al., 2018). Attempts has been made to distinct differences in osteointegration based on the size of the particles of the implants (Kulkova et al., 2014). A devoted efforts has been given to create osseoconductive surfaces with hydroxyapatite and bioactive glass granules exposed to excimer laser (Kulkova et al., 2017). Ideally for implant with a graded porosity, largest porosity should be on the periosteal surface (circumferential surface) so that bones ingrowth can be enhanced and as one moves towards the center, the porosity decreases to provide strength to the implant (Miao and Sun, 2009). Boneimplant-contact (BIC) ratio is considered one of the primary measures to osseointegration, which measures the degree how bone ingrows and can be determined by the proportion of elemental areas with Young's modulus higher than the threshold of the mature bone to the total element volume of connecting tissues (Chen et al., 2013). While designing a porous prosthesis, one needs to consider the poroelastic theory for the functional optimization of the prosthesis. Poroelastic theory has broad potential to make one understand the fracture healing, bone remodeling, implant osseointegration and cartilage degeneration (Stokes et al., 2010). Both solid matrix and pore network are assumed to be continuous. In addition to varying the material properties by the governing mathematical equations mentioned in the prosthesis design model section, by using the additional parameter for porosity one could obtain the heterogeneous models with simultaneous material and porosity. Porosity in the rth element can be stated as:

Distal end (a) Radial

(b) Axial

(c) Mixed

Fig. 2. Heterogeneous prosthesis models with different materials distribution patterns.

Tandon, 2017). To model the simplified prosthesis, cylindrical coordinate system has been used and so X depends on ρ (radial distance), θ (azimuth) and z (axial coordinate). Volume fraction of rth material (Vr) at a point can be stated as (Gupta and Tandon, 2015):

Vr = f(X )(Mer − Msr ) + Msr for 0 ≤ f(X ) ≤ 1

(5)

where, Msr and Mer are material composition of rth material at the start and end positions in the defined material distribution. In this study instead of materials, materials properties such as Young's modulus of elasticity, Poisson's ratio and density were varied to achieve the material heterogeneity. For radial distribution prosthesis model

f(X ) = f(ρ , θ , z ) =

(ρ−ρi ) z (ρo −ρi )

(6)

where ρ b ≤ρo ≤ ρt , 0 < ρ ≤ρo , 0 ≤ ρi <ρo and 0 ≤ z ≤zo and ρi and ρo are the inner and outer radius, ρ b is the bottom outer radius at z = 0, ρt is the top outer radius at z = zo and zo is the axial length of truncated cone. For axial variation prosthesis model, the model has been modeled into two parts namely the truncated cone and toroidal.

f(X) = f(ρ , θ , z ) = zρ

(7)

for 0 ≤ z ≤z o and 0 ≤ ρ ≤ ρo for truncated cone and

f(X ) = f(ρ , θ , z ) = θρ

(8)

π ≤θ≤ 3

for 0 and 0≤ρ ≤ ρo for toroidal section. For mixed variation prosthesis model, combinations of the Eqs. 6–8 were used. The common feature of biomaterials is that as they are used in intimate contact with the living body, so it is essential that the implanted materials do not cause any harmful effect. To serve safely and for a longer time, without any side effect, a metallic implant must possess the important properties, like, it should be non-toxic, inert, strong resistant to corrosion, non-magnetic, biocompatible, etc. For this study, biomaterials that are considered are recently developed but widely used. They are listed in Table 1 with their approximate properties (Geetha et al., 2009; Gilbert et al., 2009).

Pr = f (X)(Per −Psr )+Psr for 0 ≤ f(X ) ≤ 1

(9)

where Psr and Per are the porosity of rth material at the start and end positions. The distance based potential function f(x) remain the same for radial distribution as mentioned in the prosthesis design model

2.2. Porous prosthesis design models Besides stress shielding, the incidence of migration, i.e., permanent 126

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(Chung and Mansour, 2014). Thus, from Eq. (10) and Eq. (11), the constitutive response for the porous solid is given as

section and for axial it includes only the truncated cone parts. 2.3. Consolidation analysis

∇⋅(σij′ − αpδij ) + Fi = 0

For poroelastic analysis, the focus was on the parts of prosthesis which are inside the bone and hence the shape of the prosthesis was simplified similar to truncated cone. For saturated pore fluid flow, poroelastic governing equation sare based on the stress equilibrium equation and the conservation of fluid mass, as per Darcy's law (Cheng and Lin, 2005). The pore fluid diffusion structural capability is based on the extended Biot's consolidation theory. The literature reports (Otani et al., 1993) that in femurs the circumferential strains are generally lesser than the longitudinal strains before and after implantation. For the purpose of the analysis of the three dimensional (3D) poroelastic models, in this work, designs were simplified, and deformation was allowed only in the longitudinal direction of the bones as this leads to significant reduction of the computational efforts without much loss in accuracy. The materials considered in this work are assumed to be linear, isotropic and temperature independent. Although, in actual case, the property of bones varies, however, this worksassumesthe properties of bones similar to the one used by Gilbert et al. for Biot model (Gilbert et al., 2009). Since the implant is surrounded by spongy bone and to enhance the bones ingrowth environment, permeability is assumed to be same as that of spongy bones, i.e., 1 * 10–11(m4/N-s). Further, fluid density is considered as 950 kg/m3, fluid bulk modulus as 950 GPa and Poisson's ratio as 0.31 (Gilbert et al., 2009). Maximum porosity was confined to 25% (Saravana Kumar and George, 2017) and furtherincrease in the porosity led to poor mechanical properties. The Young's Modulus and density are assumed to be same as that of the materials mentioned in Table 1. Relation between matrix deformation and fluid flow can be described with the help of Biot's consolidation theory (Smit et al., 2002). It considers gradual drainage of fluid from the solid pores. In consolidation theory, the flow is considered to be single-phase fluid and assumed to be fully saturated. As per Biot and Marxhani (Biot, 1956; Merxhani, 2016), local static stress equilibrium is given as:

∇ . σij = −Fi

The flow of the fluid through the porous media is governed by the Darcy's law which relates mass flux qi to the gradient of pore pressure p in porous media (Chung and Mansour, 2014; Biot, 1941)

qi = −

∂ζ + ∇ . qi = 0 ∂t

α

The objective of this study was to determine the optimal stiffness variation in the prototype prosthesis in order to reduce the stress shielding effect. The interface effect between bone and implant was neglected and results are obtained by considering the analysis prosthesis. No torsional force due the anterior loading component of the hip joint force was taken into account. Here, static analysis of the model is carried out by fixing the distal end of the prosthesis and a load of 3000 N (Fstatic) at an angle of 20°is applied on the surface of the implant. An abductor muscle load of 1250 N (Fabductor) is applied at an angle of 20°, which represents the equivalent weight of 70 kg (Oshkour et al., 2014). Contact force was represented as a traction load distributed over a few elements. The traction load was specified in terms of force per area. Force per area is called traction intensity (traction pressure) (Campoli et al., 2012).Fig. 3 represents the boundary conditions for the heterogeneous prosthesis models. For analysis of these heterogeneous models with material variations, ANSYS 16.0 was used as a tool to solve the Finite Element (FE) models. Scripts were developed in ANSYS Parametric Design Language (APDL) to extract the mechanical properties of the elements. Prosthesis was represented by SOLID 187, three-dimensional (3D) ten nodes tetrahedral structural solid elements. For the analysis of poroelastic models with heterogeneity in porosity and materials, consolidation analysis was performed by assuming the compressibility of solid phase to be negligible compared to that of the drained bulk material, which gives α = 1 and 1 = φ . Further, while

(12)

km

ζ=

p km + αε ve km

(13)

φ α−φ 1 = + Km Kf Ks

kf

keeping bulk modulus of the fluid to be constant and varying the porosity, one can vary the Biot modulus (km). In ANSYS APDL TB, PM command activates the data table for coupled pore fluid diffusion and structural model of porous media. Hence, by varying Biot modulus and permeability, a porous structure for consolidation analysis could be obtained. For consolidation analysis, a special coupled element, CPT 217 3D, 10 noded-coupled pore pressure solid element has been chosen. The element has quadratic displacement and linear pore pressure behavior with four corner nodes and six mid nodes. The simplified model of the prosthesis is fixed at the bottom and traction pressure was applied at the top face. Bottom and top face are made impervious and circumferential face is made pervious with initial pressure as zero.

where Kd is drained bulk modulus of the material and Ks is the bulk modulus of solid matrix. The pore pressure is given as (Detournay and Cheng, 1994):

where storativity, St =

(19)

2.4. Boundary conditions

are the components of total and effective stress, p is where σij and pore pressure, α is Biot's coefficient and δij is Kronecker's delta. For an ideal isotropic poroelastic material, Biot's coefficient (Smit et al., 2002) is given as:,

p = Km (ζ −αε ve )

∂εev 1 ∂p + ∇. q = 0 + ∂t km ∂t

Eqs. (16) and (19) are the governing consolidation equations. Eq. (16) describes the mechanical equilibrium to the fluid pressure across the medium and Eq. (19) is a diffusion equation for fluid pressure, which is coupled with time rate of change in the volumetric strain.

(11)

Kd Ks

(18)

Combining Eqs. (15) and (18) yields

σij’

α=1−

(17)

where k′ is intrinsic permeability in m and μ is the coefficient of viscosity in (N-s/m2). Conservation of mass of the fluid in porous media gives

(10)

i

k′ ⎛ ⎞ ⎜∇p − ρg ⎟ μ⎝ ⎠ 2

where σij is the total Cauchy stress and Fi is the body force per unit volume. In porous media, when total stress is applied to a control volume then it gets partly distributed to the solid skeleton and partly to the pore fluid (Merxhani, 2016). Thus, the Cauchy stress in Eq. (10) would be modified as

σij = σij’–αpδij

(16)

(14) (15)

where Km is Biot modulus, ζ is variation of pore fluid content per unit reference volume, εev is the elastic volumetric strain, φ is the porosity, and Kf and Ks are bulk modulus of fluid and solid respectively 127

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Fig. 4. Result of stress distribution in radial model.

Fixed end

Fig. 3. Represents the boundary conditions for heterogeneous prosthesis models.

Initially, the porous media were assumed to be unconsolidated. The poroelastic model was solved using ANSYS 16.0. Horizontal degree of freedom of all the elements is fixed. Static analysis is performed with help of Newton-Raphson model with an end time of 10 s, in order to investigate the consolidation process (Mukherjee and Gupta, 2016). 3. Results and discussions The present section consists of two modules. The first module discusses the results of finite element analysis carried out on the models with heterogeneity in material variation, along the radial, axial and mixed directions. The second module discusses the results obtained from the consolidation analysis for the three models with simultaneous heterogeneity in material and porosity.

Fig. 5. Result of stress distribution in axial model.

3.1. Results of heterogeneity in material variation With objective to determine the minimum stiffness by varying the material in different patterns in order to reduce the stress shielding this section compares the results of different heterogeneous prosthesis models. Through heterogeneous modeling, material properties such as stiffness, density, and Poisson's ratio were varied. The average intact stress was taken to be 13 MPa (Bagheri et al., 2014). Stress shielding effect for each model was obtained using Eq. (3). Results of stress distributions in various models are represented in Figs. 4–6 under the boundary conditions mentioned in Section 2.4. From the above results, it is noticed that stress distribution patterns show similar behavior in all the three models. This could be from the fact that all three prostheses were under same loading constraints. There is symmetry in stress distribution across the prosthesis. Besides, similar behavior could be observed in previous works (Senalp et al., 2007). Fig. 7 represents the average volume equivalent stress obtained from these three prototype prosthetic models based on Eq. (4). Fig. 8 represents the stress shielding effect for each prosthesis prototype model. From Fig. 8, it could be observed that the radial variation prosthesis model has the least stress shielding effect and mixed model has the highest for the three models under consideration. It was

Fig. 6. Result of stress distribution in mixed model.

128

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Average Volumetric Equivalent Stress (MPa)

420 416 415 410 405 400 395

397

395 390 385 380 Radial Variation

Axial Variation

Mixed Variation Fig. 9. Consolidation analysis of radial prosthesis model.

Fig. 7. Average equivalent stress in heterogeneous prosthesis prototype models.

-28.5 Radial Variartion

Axial Variation

Mixed Variation

Stress shielding effect

-29

-29.5

-29.38 -29.53

-30

-30.5

-31 -31 -31.5

Fig. 10. Consolidation analysis of axial prosthesis model.

Fig. 8. Stress shielding effect in heterogeneous prosthesis prototype models.

mentioned in the prosthesis design model section that larger the value THA , more would be the stress of average volume von Mises stress i.e. σVMS shielding effect, and greater the model would be prone to aseptic loosening, which means more the chances of aseptic failure. 3.2. Results of consolidation analysis This section discusses the results of consolidation analysis in which material and porosity are varied simultaneously. The objective of this analysis is to simultaneously reduce the stress shielding effect and enhance the bone ingrowth, by making the model porous. Since the variation of porosity and pore size are same in pattern and also the grain size remains constant in all the three models, it was assumed that BIC remains constant in all the three prosthesis models. As literature do not categorically report parameters to evaluate the performance of consolidation analysis models, therefore, this work considers maximum deformation of each model as output performance parameter for all the three poroelastic models. Further the deformation obtained could be related to the stress through generalized Hooke's law and so these stresses would be useful to obtain the stress shieling effect in reach model.Figs. 9–11 represent the results of consolidation analysis on the three models. From the above results, one can observe similar stress distribution in

Fig. 11. Consolidation analysis of mixed prosthesis models.

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Fig. 12. Maximum deformation in consolidation analysis for different models.

all the three models. In static analysis, the magnitude of deformation is inversely related to the stiffness of the model. So, the model with the largest deformation would be the least stiff and hence would be the most suitable for reduction in the stress shielding effect. Fig. 12 represents the maximum deformation in each model. From the Fig. 12, it was noticed that the maximum deformation was in the range of 20μm to 50μm which has been the desirable limit for bone ingrowth (Oshkour et al., 2014). Also, mixed model has the least deformation and radial model has the highest deformation. So, the radial model has the lowest stiffness, which means this model will be most effective to reduce the stress shielding effect and as it is porous, it will help to enhance bone ingrowth. This would further increase the overall stability of the implant.

4. Conclusion In the first task, heterogeneous prosthesis design with material variations in radial, axial and mixed (simultaneous radial and axial) models were completed. Stress shielding effect, which has been one of the prominent factors in aseptic loosening between the bones and implant, was taken as a selection parameter for implant design. By comparing the three heterogeneous prosthesis design models from the perspective of stress shielding effect along with average volume equivalent stress as an output parameter (obtained from finite element analysis), radial variation prosthesis was observed to be the most suitable model among the three models. In the second task, heterogeneous prosthesis design with material and porosity variation in radial, axial and mixed had been done. These models were designed to reduce the stress shielding and at the same time enhance bone ingrowth. This was achieved by graded porosity, which also helped to maintain the required mechanical strength. Consolidation analysis based on the principles of soil mechanics had been used to analyze these models. It is observed that radial variation prosthesis has the maximum deformation and mixed model has the least deformation. This observation led to the conclusion that radial prosthesis is the best-suited model for both stress shielding and bone ingrowth, as stiffness and deformation are inversely related in static analysis. The above two analyses highlighted that among the three variations, radially varying heterogeneous prosthesis model is the most effective from both stress shielding effect and bone ingrowth enhancement point of view. The results obtained might not be very precise due to lack of availability of true experimental data, but these results would help in further studies. Selecting implant based on the stress shielding effect is not sufficient and the decision may be further improved by taking additional parameters such as fatigue loading, frictional contact between the bones and prosthesis and by comparing it with 130

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