Sajad Arabnejad KhanokiPhD Student

e-mail sajadarabnejadkhanokimailmcgillca

Damiano Pasini1Associate Professor

e-mail Damianopasinimcgillca

Mechanical Engineering Department

McGill University

Montreal Quebec Canada H3A 0C3

Multiscale Designand Multiobjective Optimizationof Orthopedic Hip Implantswith Functionally GradedCellular MaterialRevision surgeries of total hip arthroplasty are often caused by a deficient structuralcompatibility of the implant Two main culprits among others are bone-implant interfaceinstability and bone resorption To address these issues in this paper we propose a noveltype of implant which in contrast to current hip replacement implants made of either afully solid or a foam material consists of a lattice microstructure with nonhomogeneousdistribution of material properties A methodology based on multiscale mechanics anddesign optimization is introduced to synthesize a graded cellular implant that can mini-mize concurrently bone resorption and implant interface failure The procedure isapplied to the design of a 2D left implanted femur with optimized gradients of relativedensity To assess the manufacturability of the graded cellular microstructure a proof-of-concept is fabricated by using rapid prototyping The results from the analysis areused to compare the optimized cellular implant with a fully dense titanium implant and ahomogeneous foam implant with a relative density of 50 The bone resorption and themaximum value of interface stress of the cellular implant are found to be over 70 and50 less than the titanium implant while being 53 and 65 less than the foam implant[DOI 10111514006115]

Keywords total hip replacement cellular microstructure lattice material finite elementmethod asymptotic homogenization multiobjective optimization

1 Bone Replacement Implants

Current orthopedic prostheses are generally made of uniformdensity homogenous material such as 316L stainless steel cobaltchromium alloys titanium-based alloys and tantalum Over thelast few decades the design of orthopedic prostheses has beenimproved to achieve long-term fixation and easy osseointegrationAlthough technological advances have made current total hiparthroplasty successful over 13 of hip prostheses still require re-vision surgeries as a result of bone resorption and aseptic loosen-ing of the implant [1] Revision surgery is a much more complexprocedure than the first total hip arthroplasty (THA) due to bonedegradation around the first implant Bone degradation compro-mises bone ability to adequately secure the new implant Althoughpatient-related factors such as sickle cell anemia [2] poor bonequality [3] and high body mass index may predispose the patientto prosthetic failures mechanical rather than medical factors aremajor causes of implant failure [4] Excluding biocompatibilityrequirements further three requirements can be identified as indi-cators of implant success (i) implant stability in the short andlong term (ii) preservation of bone tissue around the implantfrom resorption and (iii) high wear and corrosion resistance ofthe articulating surfaces

The volumetric amount of wear particle and its clinical conse-quences have been reduced considerably by the development of

extremely wear-resistant polymers [56] or the design of metal-on-metal implants [78] Yet the reduction of bone-implant inter-face instability and bone resorption in the long term still remains achallenge

Current orthopedic implants are generally stiffer than the boneadjacent to the prosthesis Due to its high stiffness an implant pre-vents the applied stress from being transferred to the adjacentbone thereby resulting in bone resorption around the implantThis weakens the implant support which leads to bone fractureand implant loosening Over the last three decades alternativeimplant designs have been proposed to reduce stress shielding andminimize the associated clinical consequences [9] Recent implantdesigns have only been partially successful as the solution of oneproblem has given rise to another one For example to overcomethe mismatch between a stiff stem and the adjacent bone compos-ite and isoelastic hip stems were introduced [1011] The results ofthese studies show an undesired increase of the shear stressbetween the implant and the bone an outcome that increases therisk of interface motion [1213] These attempts help elucidate theantagonist nature of stress shielding and bone-implant interfacestability

In their seminal work Kuiper and Huiskes identified the con-flict existing between stress shielding and interface shear stress[1415] and attempted to find a trade-off design for a bi-dimen-sional hip implant They showed that one solution to this issue isan implant whose material properties vary locally throughout thestructure A non-homogeneous distribution of elastic propertieswithin the hip stem could contribute to minimizing the probabilityof interface failure while concurrently limiting the amount ofbone loss In their approach however the solution of the multiob-jective problem was simplified reformulated and solved as a sin-gle objective optimization problem As a result the whole set of

1Corresponding author Room 372 MacDonald Engineering Building Mechani-cal Engineering Department McGill University Montreal Quebec Canada H3A0C3

Contributed by the Bioengineering Division of ASME for publication in the JOUR-

NAL OF BIOMECHANICAL ENGINEERING Manuscript received August 30 2011 finalmanuscript received February 10 2012 accepted manuscript posted February 212012 published online March 21 2012 Assoc Editor Stephen Klisch

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-1Copyright VC 2012 by ASME

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trade-off designs could not have been captured Hedia et al[1617] attempted to reconcile the conflicting nature of theseobjective functions by proposing the use of three bioactive materi-als hydroxyapatite Bioglass and collagen to design a gradedcementless hip stem Although their implant design reduced boneresorption and bone-implant interface stresses the use of suchbioactive materials have limitations due to their brittleness andinsufficient strength when applied to load-bearing applications[18ndash20] In a more recent study Fraldi et al [21] applied a maxi-mum stiffness topological optimization strategy to re-design a hipprostheses with the goal of reducing stress shielding in the femurAccording to this method elements with intermediate volumefraction (between 0 and 1) are penalized to limit their presence inthe final solution For regions with intermediate relative densitycertain microstructures should be proposed to match those materi-als in terms of effective elastic properties Laser micro-drilling issuggested to create the required micro-porosity an option that canbe used only on the implant surface not throughout the implant

Other advances in total hip replacement have used a micro-structured material over a fully dense material Hip implants withporous tantalum have been proposed in knee and hip replacementsurgery [22] Tantalum foam is an excellent material due to itsbiocompatibility high volumetric porosity and modulus of elas-ticity similar to that of bone To create the tantalum foam puretantalum is chemically deposited on a carbon skeleton As a resultthe microstructure of a tantalum foam implant has an almost uni-form and random distribution of pore shape and size [23] through-out the implant These material characteristics however havebeen demonstrated to be incapable of solving the conflicting na-ture of the physiological phenomena occurring in an implant[1415] Indeed whereas the reduced stiffness of the foamdecreases bone resorption the uniform distribution of cells has theundesired effect of increasing the interface stresses

Recent advances in additive manufacturing such as Electron-Beam Melting (EBM) Selective Laser Melting (SLM) Stereoli-thography Apparatus (SLA) and other rapid prototyping techni-ques offer the possibility of novel bone-replacement implantswith a controlled cellular microstructure [24ndash28] As a result cel-lular components with tailored microstructures can be built with ahigh level of quality accuracy and reliability Besides providingan exceptional degree of control over the mechanical propertiessuch manufacturing processes are capable of building graded cel-lular structures As demonstrated by the work of Kuiper andHuiskes [1415] this feature is an asset for bone-replacementimplants since the internal skeleton of the prosthesis can bedesigned to ease osseointegration as well as to match the local me-chanical properties of the femoral bone By properly selecting to-pology size and relative density of the unit cell of the implant itis thus possible to (a) fabricate implants which can provide me-chanical properties mimicking those of the host bone (b) manu-facture three-dimensional structures with an interconnectedporosity and pore sizes suitable to bone ingrowth and vasculariza-tion and (c) customize implants for each patient by using CT scandata of the patientrsquos bone

While the aforementioned research is promising for the supportof cellular microstructure based bone-replacement implants theliterature is limited to the manufacturing aspects or the mechani-cal testing of biomedical implants with periodic cellular micro-structure [24ndash28] No study has been found which deals with themultiscale mechanics aspects of an implant with a functionallygraded cellular material as well as the multiobjective nature of thefeatures that their design requires

This paper proposes a systematic methodology for the design ofbone-replacement implants with improved structural stability Fortotal hip arthroplasty we propose to design an implant with tai-lored gradients of lattice material that can simultaneously mini-mize bone resorption and bone-implant interface stress Theprocedure that hinges on multiscale mechanics theory and multi-objective optimization is applied to the design of a bi-dimensionalfemoral hip implant with optimal graded cellular microstructure

Its biocompatibility performance is discussed with respect to thatof currently-used hip implants

We note that in this work the terms ldquocellularrdquo ldquocellrdquo and ldquowallcellrdquo refer exclusively to the artificial porous material of theimplant These terms neither refer to body cells nor have a biolog-ical or medical connotation

2 Methodology

Kuiper and Huiskes [1415] showed that the use of a gradedmaterial in an orthopedic stem can lead to a reduction of bothstress shielding and bone-implant interface stress To this endhierarchical computational procedures [29ndash32] can be imple-mented to design an optimum material distribution within theimplant These strategies might generally require a high computa-tional cost besides yielding a microstructure which is difficult tofabricate In this work we suggest to design gradients of materialproperties through a tailored lattice microstructure whose geo-metrical parameters are optimized in each region of the implant toachieve minimum bone loss and implant interface failure

The mechanical properties of a cellular structure depend on therelative density and the geometric parameters of the unit cell asdescribed for example by the expression of the Youngrsquos modulus[33]

E frac14 CEsqqs

m

(1)

where E is the effective Youngrsquos modulus of the unit cell q isthe density of the unit cell and Es and qs are the Youngrsquos modulusand density of the constitutive material respectively In additionm has a value varying from 1 to 3 as determined by the mechani-cal failure mode of the unit cell and C is a geometric constant ofthe unit cell By changing the relative density of the lattice micro-structure it is thus possible to obtain desired values of mechanicalproperties in any zone of the implant

Figure 1 summarizes the procedure proposed here to design a cel-lular implant with controlled gradients of mechanical propertiesThe method integrates a multiscale mechanics approach to deal withthe scale-dependent material structure and a multiobjective optimi-zation strategy to handle the conflicting nature of bone resorptionand implant interface failure The main steps identified by the num-bers reported in the flow chart of Fig 1 are described here

bull (1-2) A finite element model of the bone is created by proc-essing CT-scan data of a patient bone The design domain ofthe prosthesis is assumed to possess a 3D lattice microstruc-ture where the unit cell ie the building block can be ofany arbitrary topology (Fig 1) The microscopic parametersof the unit cell geometry and the macroscopic shape of theimplant are the design variables of the vector b The unit cellis assumed to be locally periodic and its field quantities suchas stress and strain to vary smoothly through the implant

bull (3) The characteristic length of the unit cell in the cellularimplant is assumed to be much smaller than the characteristiclength of the macrodimensions of the implant Hence themicrostructure can be replaced with a homogeneous mediumwhose equivalent mechanical properties in particular the ho-mogenized stiffness tensor of each unit cell are calculatedthrough asymptotic homogenization theory [34ndash38]

bull (45) The homogenized stiffness tensors are then used to con-struct the stiffness matrix which will be the input to the FiniteElement (FE) solver As a result the average strains andstresses throughout the bone and the structure of the prosthe-sis are calculated

bull (6) To obtain the microscopic stress field for each unit cellfrom the macroscopic strain the stress recovery procedureillustrated in Refs [3438ndash40] is used

bull (7) If the microscopic stress level is below a predefinedfailure criterion the macroscopic stresses and strains

031004-2 Vol 134 MARCH 2012 Transactions of the ASME

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representing the mechanical behavior of the implant are usedto evaluate bone loss (mrethbTHORN) and interface failure (FethbTHORN) Inthe formulation of the multiobjective optimization problemthe constraints are set on the average porosity of the cellularimplant ethbTHORN the mean pore size P and the minimum thick-ness of cell walls tmin In particular ethbTHORN 40 and50 lm P 800 lm are selected to ease bone ingrowth[4142] The thickness of the cell walls is selected to begreater than the minimum resolution tmin offered by a givenmanufacturing process For example tmin is 100 lm and 70lm respectively for SLM and SLA [2743]

bull (8-9) If the solutions of the optimization have not convergedthen the vector b of the design variables are updated to findthe set of non-dominated solutions of the Pareto front

bull (9) If the unit cell fails at the microscale level the cell wallswill be iteratively increased to reduce the microscopic stresses

As described above multiscale mechanics and multiobjective opti-mization are integrated aspects of the method proposed in this paperTheir descriptions are separately detailed in the following subsections

21 Multiscale Analysis and Design of a CellularImplant The deformation and failure mechanisms of a structurewith heterogeneous material can occur at both the macroscopicand microscopic scales In a full scale simulation the heterogene-ities are explicitly modeled at the microscale to guarantee high ac-curacy The computational effort however can be very lengthyand time-consuming As an alternative the microstructure can bereplaced by a homogeneous medium and the mathematical theory ofhomogenization can be used to characterize the mechanical behaviorof heterogeneous media [44] As shown in Fig 2 a body Xe with aperiodic microstructure subjected to the traction t at the tractionboundary Ct a displacement d at the displacement boundary Cd anda body force f can be replaced by a homogenized body X with theprescribed external and traction boundaries applied to Xe

without ge-ometrical details and voids of the local coordinate system

The homogenized properties and strength of a cellular structurecan be obtained by performing either analytical or numerical or ex-perimental approaches [33ndash3945ndash53] Extensive efforts have beendevoted to the derivation of the equivalent mechanical properties

Fig 1 Flow chart illustrating the design of a graded cellular hip implant minimizing boneresorption and implant interface failure

Fig 2 Homogenization concept of a cellular structure

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-3

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by structural analysis [3345ndash50] In these studies the effectivemoduli and yield strength of a cellular material are generally mod-eled by assuming that the cell walls flex like beams Despite thesimplicity of this method in calculating the overall mechanicalproperties the results are reliable only if the relative density islower than 03 [47] Furthermore the actual stress distributionwithin the unit cell walls cannot be captured As an alternativemethod the asymptotic homogenization method is used in thiswork to deal with the multiscale analysis of the cellular implantThis technique is applied to calculate the homogeneous stiffnessmatrix of the unit cell for different values of relative density as wellas to determine the microscopic stresses and strains [343839] Theasymptotic homogenization method has been widely used in topol-ogy optimization [3454ndash58] and hierarchical design of materialsand structures [29ndash32] In this work unlike hierarchical topologyoptimization a predefined unit cell topology with parametric geom-etry is considered as the microstructure of the implant then theoptimization algorithm searches for the optimum unit cell geometryof the lattice to minimize the antagonist objective functions under aset of constraints This procedure is similar to the one developed byBendsoslashe and Kikuchi [58] and is not limited to any cell topologywe selected the hollow square cell as an example to demonstratethe methodology A brief explanation about the asymptotic homog-enization technique and the main steps to calculate the effectivestiffness and microscopic stress and strain are provided below

The approach assumes that each field quantity depends on twodifferent scales one on the macroscopic level x and the other onthe microscopic level yfrac14 xe where e is a magnification factorthat enlarges the dimensions of a unit cell to make it comparablewith the dimensions of the material The field quantities such asdisplacement stress and strain are assumed to vary smoothly atthe macroscopic level being periodic at the microscale [35ndash39]The effective properties of the periodic material are determinedthrough the solution of local problems formulated on the represen-tative volume element (RVE) of the material As a result the ho-mogenized stiffness tensor (EH

ijkl) of a cellular material may bedefined as follow [39]

EHijkl frac14

1

Yj j

ethYC

EijpmMpmkldY (2)

where Yj j is the volume of the entire unit cell with void Yc is thesolid part of the cell Eijkl is the local elasticity tensor that dependson the position within the representative volume element ie Eijkl

is equal to the elasticity tensor of the material located in the cellwalls and it vanishes in the voids Mijkl is the local structure ten-sor which relates the macroscopic strains (e) to the local or micro-structural strains (e) through the relation

eij frac14 Mijklekl (3a)

Mijkl frac141

2ethdikdjl thorn dildjkTHORN ekl

ij (3b)

where dij is the Kronecker f and eklij is the microstructural strain

corresponding to the component kl of the macroscopic strain ten-sor (ekl) ekl

ij is the solution of the following equation

ethYC

Eijpme1ijethvTHORNekl

pmethuTHORNdY frac14eth

YC

Eijkle1ijethvTHORNekldY (4)

where e1ijethvTHORN is the virtual strain In general ekl can be an arbitrary

macroscopic strain tensor Considering the assumption of smalldeformation and linear material behavior ekl may be written as alinear combination of unit strains For a two-dimensional case theunit strains are defined as

e11 frac14 1 0 0frac12 T e22 frac14 0 1 0frac12 T e12 frac14 0 0 1frac12 T (5)

To calculate the effective mechanical properties of a cellular ma-terial the first task is to obtain the matrix Mijkl After discretizing

the RVE domain the unit strains are applied to each element ofthe FE model Periodicity of the strain field is ensured by impos-ing periodic boundary conditions on the RVE edges The directmethod [59] is selected here to derive periodic boundary condi-tions The microscopic strain field (ekl

ij ) inside the RVE isobtained by solving Eq (4) The results are substituted into Eq(3b) to calculate the local structure tensor Mijkl for each elementof the RVE Finally the effective stiffness tensor EH

ijkl is obtainedby calculating Eq (2) Once the local structure tensor Mijkl isobtained the microscopic strains and stresses corresponding tothe macroscopic strain can be obtained via Eq (3a) and the consti-tutive equation of the cell wall material

The steps described above are used to compute the homoge-nized stiffness tensor for each unit cell of the cellular hip implant(Fig 1) These tensors are used to construct the global stiffnessmatrix for the FE solver to obtain macroscopic stress and straindistribution within bone and implant The values are then post-processed to evaluate the objective functions of the multiobjectiveoptimization problem whose formulation is described in the fol-lowing section

22 Formulation of the Multiobjective OptimizationProblem For the design of an optimum implant we impose thesimultaneous minimization of the amount of bone loss around theprosthesis and the probability of mechanical failure at the bone-implant interface As illustrated in Fig 1 the multiobjective opti-mization problem can be formulated as

Minimize mrethbTHORNFethbTHORN

bone loss

interface failure

Subject to

ethbTHORN 4050lm P 800lm

t tmin

average porosity

meanpore size

cell wall thickness

(6)

The amount of bone loss around the stem is determined by assess-ing the amount of bone that is underloaded Bone can be consid-ered locally underloaded when its local strain energy (Ui) per unit

of bone mass (q) averaged over n loading cases (S frac14 1n

Pnifrac141

Ui

q )

is beneath the local reference value Sref which is the value of Swhen no prosthesis is present However it has been observed thatnot all the underloading leads to resorption rather a certain frac-tion of underloading (the threshold level or dead zone s) is toler-ated In fact bone resorption starts when the local value of S isbeneath the value of eth1 sTHORNSref [1460] Using this definition theresorbed bone mass fraction mr can be obtained from

mrethbTHORN frac141

M

ethV

gethSethbTHORNTHORNqdV (7)

where M and V are the original bone mass and volume respec-tively and gethSethbTHORNTHORN is a resorptive function equal to unity if thelocal value of S is beneath the local value of eth1 sTHORNSref and equalto 0 if eth1 sTHORNSref lt S In this study the value of the dead zone sis assumed to be 05 [14]

The other objective is to minimize the probability of local inter-face failure which is expressed by the following functional of theinterface stress distribution

FethbTHORN frac14 1

n

Xn

ifrac141

ethP

f ethrbi THORNdP (8)

where FethbTHORN is the global interface function index rbi is the inter-

face stress at the loading case i depending on the design variableb P is the interface area and f ethrb

i THORN is the local interface stressfunction which is defined based on the multi-axial Hoffman fail-ure criterion [61] This function is used to determine where local

031004-4 Vol 134 MARCH 2012 Transactions of the ASME

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debonding might occur along the bone-implant interface [1462]The probability of local interface failure f ethrTHORN is given by

f ethrTHORN frac14 1

StScr2

n thorn1

St 1

Sc

rn thorn

1

S2s

s2 (9)

where St and Sc are the uniaxial tensile and compressive strengthsrespectively Ss is the shear strength and rn and s are normal andshear stresses at the bone-implant interface respectively Forf ethrTHORN 1 a high probability of failure is expected whereas forf ethrTHORN 1 the risk of interface failure is low Tensile compressionand shear strengths of bone can be expressed as a function of bonedensity following the power law relations obtained by Pal et al[62]

St frac14 145q171 Sc frac14 324q185 Ss frac14 216q165 (10)

The distribution of bone density can be obtained through a CT-scan data of bone and then used in Eq (10) to find the effectivemechanical properties of the bone from which the local interfacefailure can be determined via Eq (9) Finally the interface failureindex FethbTHORN is evaluated by means of Eq (8) It should be notedthat FethbTHORN serves as a qualitative indicator only lower values ofFethbTHORN indicate reduced probability of local interface failure in theimplant

The expressions of bone resorption after implantation and me-chanical failure of bone-implant interface described in this section

are used in the finite element analysis to evaluate the objectivefunctions to be optimized The following sections shows the appli-cation of the method for the design of a two dimensional cellularimplant

3 Application of the Methodology to design a 2D

Femoral Implant with a Graded Cellular Material

31 FEM model of the Femur at the Macroscale The lefthand side of Fig 3 shows the geometry of the left femur consid-ered in this work along with the appropriate loads and boundaryconditions These data have been obtained from the work ofKuiper and Huiskes [14] The 3D geometry of the femur is simpli-fied into a 2D model where the thickness of the stem and bonevaries such that the second moment of area about the out-of-planeaxis does not differ in both models [16] Furthermore in this pa-per the implant material is designed to be an open cell lattice toease bone in growth in the implanted stem and obtain a full bondAlthough bone ingrowth does not exist in a postoperative situa-tion it can appear later if local mechanical stability is guaranteedThe minimization of interface stress reduces the possibility ofoccurrence of interface micromotion and instability [4] There-fore to decrease the computational cost required by a stabilityanalysis based on a nonlinear frictional contact model the pros-thesis and the surrounding bone are assumed fully bonded

The load case represents daily static loading during the stancephase of walking [63] The distal end of the femur is fixed toavoid rigid body motion For the material properties of the modelwe consider 20 GPa as the Youngrsquos modulus of the cortical boneand 15 GPa for the proximal bone The Poissonrsquos ratio is set to be03 [14]

32 FEM model of the Cellular Implant at theMicroscale The right hand side of Fig 3 illustrates the model ofa cementless prosthesis implanted into the human femur The griddepicts the domain of the implant to be designed with a lattice ma-terial of graded properties Figure 4 shows the unit cell geometryused for the tessellation of the whole implant The gradients ofmaterial properties are governed by the lattice relative densitywhich is a variable controlled by the cell size and wall thicknessof the hollow square In future work the performance of alterna-tive cell topologies suitable for bone tissue scaffolding will beinvestigated as well as combinations of dissimilar unit cells in thetessellation [64]

For the material property of the implant we consider Ti6Al4V[24] which is a biocompatible material commonly used in EBMIts mechanical properties are the following 900 MPa for the yieldstrength of the solid material 120 GPa for Youngrsquos modulus and03 for Poissonrsquos ratio

33 Design of the Graded Cellular Material of theImplant The procedure introduced in this paper for the design ofa graded cellular implant requires both multiscale analysis andmultiobjective optimization as described in Secs 21 and 22 andshown in Fig 1 The variables of the lattice model are the relativedensities attributed to 130 sampling points 26 rows along the

Fig 3 2D Finite element models of the femur (left) and theprosthesis implanted into the femur (right)

Fig 4 2D hollow square unit cell for given values of relative density

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-5

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prosthetic length and 5 columns along the radial direction asshown in the right side of Fig 3 The number of sampling pointshas been chosen to be 130 to limit the computational time requiredfor the analysis and optimization while providing a reasonable re-solution for the relative density distribution For a more refined den-sity distribution the number of sample points can be increasedTheir values have been constrained in the range 01 q 1 [47]to prevent elastic buckling from occurring in the unit cell prior toyielding The values of the relative density between the samplingpoints are obtained through linear interpolation Although not con-sidered in the current research the shape of the implant could beincluded in the vector b as a design variable (Fig 1)

To calculate the stress and strain regime of the implant thesteps explained in Sec 21 are followed Initial values of relativedensity are assigned to each element of the FE model created inANSYS (Canonsburg Pennsylvania USA) The stiffness matrixof each element is calculated through a computational code2DHOMOG developed in-house which obtains the homogenizedstiffness matrix of the square unit cell as a function of relativedensity Figure 5 shows the results which are first obtained at dis-crete points of relative density To obtain the continuous functionsfor the properties we calculate through the least squares methodthe expressions (Table 1) for two ranges of relative density ieq lt 03 and q gt 03 These functions enable to assign the valuesof the stiffness for a given relative density to each of the samplepoints modeled in the FE solver We note that the expressions ofYoungrsquos moduli and Poissonrsquos ratios in the x1 and x2 directions donot change since the cell thickness is uniform

Once the stress and strain regimes of the cellular material havebeen calculated the non-dominated sorting genetic (NSGA-II)

algorithm [65] is employed to solve the multiobjective optimiza-tion problem described in Sec 22 The strain energy within thebone and the stress distribution at the bone-implant interface isthen calculated and used in Eqs (7) and (8) to evaluate the objec-tive functions The initial population is then sorted based on thenondomination front criterion A population of solutions calledparents are selected from the current population based on theirrank and crowding distance Then genetic operators are appliedto the population of parents to create a population of off-springsFinally the next population is produced by taking the best solu-tions from the combined population of parents and off-springsThe optimization continues until the user-defined number of func-tion evaluations reaches 25000 [65] The computational costrequired to run the optimization process in a single 24 GHz Intelprocessor with 4 GB RAM was about 300000 CPU s about threeand a halfdays We plan to use parallel computing with a PC clus-ter in future work this will considerably reduce the computationaltime since each function evaluation can be performedindependently

For each point in the objective function space the stress recov-ery procedure is applied to verify whether the stresses are admissi-ble To apply this procedure the average macroscopic straininside each unit cell is found The position of each unit cell withinthe implant is obtained after imposing a proper cell tessellationwhich in this work has been set to be uniform The size of the unitcell is selected as small as possible to capture the relative distribu-tion contour with higher resolution For a relative density of 01the square cell sizes for the cell wall thickness of either 70 or 100lm have been selected respectively to be 136 and 18 mm Oncethe position of the unit cells has been obtained 3 3 Gauss pointsare assigned to each cell The values of relative density and mac-roscopic strain at these points are obtained from the relative den-sity distribution and macroscopic strain field For Gauss pointslocated outside the implant border the values are linearly extrapo-lated from those located at the neighboring points inside theimplant domain Using a Gaussian quadrature integration [66] theaverage relative density and macroscopic strain of each cell arecalculated The local stress distribution and the yield safety factorof each cell are obtained through the von Mises stress criterionThe procedure is applied to all unit cells of the selected optimaldesign located on the Pareto frontier and the minimum local safetyfactor of a cell is specified as design safety factor

4 Results and Discussion

The advantage of multiobjective optimization with a posterioriarticulation of preference is that a set of optimum solutions areavailable without requiring the designer to choose in advance anyweighting factors to the objective functions Once the whole set ofPareto solutions has been determined the designer has the free-dom to select the desired solution based on the importance ofbone mass preservation relative to the amount of interface stressFigure 6 shows all the optimum solutions ie the relative densitydistribution for a hip stem implant with graded cellular materialThe x axis represents the amount of bone resorption for theimplanted hip on the y axis is the interface failure index Amongthe optimal solutions we examine three representative relativedensity distributions the extreme points A and C of the Paretofrontier for which one objective function has importance factor 0and the other 100 and a solution B characterized by a 50weight factor For each solution Fig 6 gives the following per-formance metrics bone resorption (mr) interface failure index(FethbTHORN) maximum interface failure (f ethrTHORNmax) average porosity ofeach stem ( ) and design safety factor (SF) after implementingthe stress recovery procedure The maximum interface failuref ethrTHORNmax is included since FethbTHORN which quantifies only the overalleffect of the implant stiffness on the interface stresses is not suffi-cient to provide information on the probability of failure

As seen from the performance metrics in Fig 6 the porosity ofsolutions A B and C is greater than 40 which is satisfactory

Fig 5 Effective Youngrsquos modulus of 2D square lattice versusrelative density Solution points obtained through homogeniza-tion theory are fitted with the least squares method

Table 1 Effective Mechanical Properties of the Square UnitCell as a Function of Relative Density

q lt 03 03 lt q lt 1

E1

Esfrac14 E2

Es058ethqTHORN1046

127ethqTHORN2 052ethqTHORN thorn 023

G

Es0093ethqTHORN307

126ethqTHORN3 133ethqTHORN2 thorn 051ethqTHORN 0065

21

sfrac14 12

s07ethqTHORN105

068ethqTHORN2 thorn 028ethqTHORN thorn 006

031004-6 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

for bone ingrowth [41] By comparing implants C and A weobserve that a raise of the implant porosity from point C to Aresults in an implant stiffness decrease which on one hand low-ers bone loss and on the other hand enhances the risk of interfacefailure When solution B is compared to C a reduction of 8 ofbone resorption is noted with a slight increase of the peak value ofthe interface failure On the other hand by contrasting solution Bto A we note a significant increase (60) of the peak value ofinterface failure which is below the Hoffman failure strength anda minor reduction (2) of the amount of bone resorption Themain benefit of solution A is the maximum porosity of the micro-structure that can promote bone ingrowth Solution B on the otherhand might be the preferred solution with respect to low boneresorption and interface failure These observations emerge onlyby inspection of the objective functions selected in this work weremark here that other parameters should be taken into accountfor the selection of the best implant These include patientrsquos bone

characteristics the range of activity age and desired level ofbone mass preservation after implantation

For prescribed geometric loading and constraint conditions wenow compare the metrics of resorbed bone mass (mr) and distribu-tion of interface stress (f ethrbTHORN) of the optimal solution B with thoseof (i) a currently-used fully dense titanium stem and (ii) a cellularimplant with a uniformly distributed relative density of 50 Fig-ures 7 and 8 illustrate the results of the comparison

For the solid titanium stem the amount of bone resorption cal-culated through Eq (7) is 67 and the interface failure indexFethbTHORN obtained from Eq (8) is 133 Using the distribution of f ethrTHORNgenerated around the titanium stem (Fig 8(a)) we observe thatthe maximum value of interface failure (051) occurs at the distalend of the implant As expected this implant is much stiffer thanthe surrounding bone thereby resulting in a higher amount ofbone resorption For the numerical validation the interface shearstress of the titanium implant at the proximal region is also

Fig 6 Trade-off distributions of relative density for the optimized cellular implant

Fig 7 Distribution of bone resorption around (a) fully dense titanium implant (b) cellularimplant with uniform relative density of 50 (c) graded cellular implant (solution B in Fig 6)

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-7

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

compared with the one obtained by Kowalczyk for a 3D model[4] The mean and the maximum values of interface shear stressfor the 3D titanium implant in the work by Kowalczyk [4] are057 and 28 MPa respectively These values are respectively031 and 215 MPa for the titanium implant in this paper The con-tribution to the higher level of shear stress in the 3D model ofKowalczyk is the distribution of shear force on a smaller area InKowalczykrsquos study [4] the implant and bone are bonded only atthe proximal region while in our work the whole bone-implantinterface is bonded which results in a decrease of the mean andthe maximum values of interface shear stress

The cellular implant with uniform relative density of 50 isapproximately three times more flexible than the titanium stemThis implant can qualitatively simulate the behavior of an implantmade out of tantalum foam For this stem the amount of boneresorption and the interface failure index are about 34 and 287respectively and the interface failure is maximum (071) at theedge of proximal region Compared to the solid titanium implantthe amount of bone resorption decreases by 50 whereas themaximum interface failure increases about 40 This shows that adecrease of the implant stiffness with uniform porosity distribu-tion aiming at reducing bone resorption has the undesirable effectof increasing the risk of interface failure at the proximal regionThis outcome confirms the findings of the previous work byKuiper and Huiskes [14]

Figure 7(c) and 8(c) show the results for the graded cellularimplant B Its bone resorption and interface failure index are 16and 115 respectively The peak value of the local interface fail-ure is 025 Compared to the titanium stem both the amount ofbone resorption and the peak of interface failure decrease of 76and 50 respectively With respect to the uniformly-distributedcellular implant the decrease of bone resorption and interface fail-ure peak is of 53 and 65 respectively A graded cellularimplant with optimized relative density distribution is thus capa-ble of reducing concurrently both the conflicting objective func-tions In particular bone resorption reduces as a result of thecellular material which makes the implant more compliant theinterface stress on the other hand is minimized by the optimizedgradients of cellular material

A proof-of-concept implant was built to verify the manufactur-ability of the optimum grade lattice material Figure 9 shows thepolypropylene prototype of solution B which was manufacturedwith a 3D printer Objet Connex500 [67] A uniform tessellationand a square unit cell of 18 mm size were assumed to draw themodel The cell geometry was calculated from the average rela-tive density obtained from the method described in this paper AnSTL file of the graded cellular implant solution B was finallyused for rapid prototyping

The limitations of the method proposed in this paper are herediscussed First the method tackles the design of an implant stati-cally loaded in the stance phase of walking Further research isrequired to extend it to variable loading and fatigue life design

Second the accuracy of the asymptotic homogenization needsto be investigated at the vicinity of the implant borders Althoughmacroscopic mean stress fields on the boundaries are satisfied inthe global solutions [4068] microscopic stresses do not satisfythe Y-periodicity assumption These boundary effects howevermay be captured by applying a boundary layer corrector [6869] aspatially decaying stress localization function [70] or an adaptivemultiscale methodology [71] To investigate the accuracy of thesetechniques it is required to construct a detailed finite elementmodel and compare the actual stressstrain distribution inside thecell walls with the microscopic stressesstrains estimated from thehomogenization method This task will be performed in future

Third to extend the proposed procedure to three dimensions a3D cell topology with high porosity interconnected pore struc-ture large surface area with suitable textures and good

Fig 8 Distribution of local interface failure f ethrTHORN around (a) fully dense titanium implant(b) cellular implant with uniform relative density of 50 (c) graded cellular implant (solu-tion B in Fig 6)

Fig 9 Polypropylene proof-of-concept of the optimal gradedcellular implant (solution B)

031004-8 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

mechanical properties should be selected [7273] The asymptotichomogenization will be applied to characterize tissue scaffoldingcell topologies for a wide range of relative density Their effectivemoduli yield and ultimate surfaces for multiaxial loading condi-tions will be obtained to create a standard library from which toselect suitable cell geometries for a 3D cellular implant design

Fourth bone has been here considered as consisting of a cancel-lous and a cortical part with isotropic mechanical properties Fur-ther study is necessary to improve the constitutive material modelof bone and to consider bone as a graded cellular material

Finally in the optimization procedure a simple formulation hasbeen used to quantitatively measure both bone resorption andimplant stability In a future study detailed simulations of implantstability [74ndash77] and bone remodeling process [6077ndash80] can beperformed to assess both the short and long term performance ofthe implant

5 Concluding Remarks

This work has presented a methodology integrating multiscaleanalysis and design optimization to design a novel hip implantmade of graded cellular material The method can contribute tothe development of a new generation of orthopedic implants witha graded cellular microstructure that will reduce the clinical con-sequences of current implants

In the first part of the paper the homogenization method hasbeen reviewed and then used to capture the mechanics of theimplant at the micro and macro scale In the second part multiob-jective optimization has been applied to find optimum gradients ofmaterial distribution that minimize concurrently bone resorptionand bone-implant interface stresses The results have shown thatthe optimized cellular implant exhibits a reduction of 76 ofbone resorption and 50 of interface stress with respect to a fullydense titanium implant

References[1] Kurtz S Ong K Lau E Mowat F and Halpern M 2007 ldquoProjections of

Primary and Revision Hip and Knee Arthroplasty in the United States from2005 to 2030rdquo J Bone Jt Surg Am Vol 89(4) pp 780ndash785

[2] Vichinsky E P Neumayr L D Haberkern C Earles A N Eckman JKoshy M and Black D M 1999 ldquoThe Perioperative Complication Rate ofOrthopedic Surgery in Sickle Cell Disease Report of the National Sickle CellSurgery Study Grouprdquo Am J Hematol 62(3) pp 129ndash138

[3] Kobayashi S Saito N Horiuchi H Iorio R and Takaoka K 2000 ldquoPoorBone Quality or Hip Structure as Risk Factors Affecting Survival of Total-HipArthroplastyrdquo The Lancet 355(9214) pp 1499ndash1504

[4] Kowalczyk P 2001 ldquoDesign Optimization of Cementless Femoral Hip Pros-theses Using Finite Element Analysisrdquo J Biomech Eng 123(5) pp 396ndash402

[5] Moen T C Ghate R Salaz N Ghodasra J and Stulberg S D 2011 ldquoAMonoblock Porous Tantalum Acetabular Cup Has No Osteolysis on Ct at 10Yearsrdquo Clin Orthop Relat Res 469(2) pp 382ndash386

[6] Kurtz S Gawel H and Patel J 2011 ldquoHistory and Systematic Review ofWear and Osteolysis Outcomes for First-Generation Highly Crosslinked Poly-ethylenerdquo Clin Orthop Relat Res 469(8) pp 2262ndash2277

[7] Grubl A Marker M Brodner W Giurea A Heinze G Meisinger VZehetgruber H and Kotz R 2007 ldquoLong Term Follow up of Metal on MetalTotal Hip Replacementrdquo J Orthop Res 25(7) pp 841ndash848

[8] Neumann D R P Thaler C Hitzl W Huber M Hofstadter T and DornU 2010 ldquoLong-Term Results of a Contemporary Metal-on-Metal Total HipArthroplasty A 10-Year Follow-up Studyrdquo J Arthroplasty 25(5) pp700ndash708

[9] Glassman A Bobyn J and Tanzer M 2006 ldquoNew Femoral Designs DoThey Influence Stress Shieldingrdquo Clin Orthop Relat Res 453(12) pp64ndash74

[10] Adam F Hammer D S Pfautsch S and Westermann K 2002 ldquoEarly Fail-ure of a Press-Fit Carbon Fiber Hip Prosthesis with a Smooth Surfacerdquo JAntroplasty 17(2) pp 217ndash223

[11] Trebse R Milosev I Kovac S Mikek M and Pisot V 2005 ldquoPoorResults from the Isoelastic Total Hip Replacementrdquo Acta Orthop 76(2) pp169ndash176

[12] Harvey E Bobyn J Tanzer M Stackpool G Krygier J and Hacking S1999 ldquoEffect of Flexibility of the Femoral Stem on Bone-Remodeling and Fix-ation of the Stem in a Canine Total Hip Arthroplasty Model without CementrdquoJ Bone Jt Surg 81(1) pp 93ndash107

[13] Huiskes R Weinans H and Rietbergen B 1992 ldquoThe Relationship betweenStress Shielding and Bone Resorption around Total Hip Stems and the Effectsof Flexible Materialsrdquo Clin Orthop Relat Res 274(1) pp 124ndash134

[14] Kuiper J and Huiskes R 1992 ldquoNumerical Optimization of Hip-ProstheticStem Materialrdquo Recent Advances in Computer Methods in Biomechanics andBiomedical Engineering J Middleton G N Pande and K R Williams edsBooks and Journals International Ltd Swansea pp 76ndash84

[15] Kuiper J H and Huiskes R 1997 ldquoMathematical Optimization of ElasticProperties Application to Cementless Hip Stem Designrdquo J Biomech Eng119(2) pp 166ndash174

[16] Hedia H Shabara M El-Midany T and Fouda N 2006 ldquoImproved Designof Cementless Hip Stems Using Two-Dimensional Functionally Graded Materi-alsrdquo J Biomed Mater Res Part B Appl Biomater 79(1) pp 42ndash49

[17] Hedia H S and Mahmoud N-A 2004 ldquoDesign Optimization of Function-ally Graded Dental Implantrdquo Biomed Mater Eng 14(2) pp 133ndash143

[18] Watari F Yokoyama A Saso F Uo M and Kawasaki T 1997ldquoFabrication and Properties of Functionally Graded Dental Implantrdquo Compo-sites Part B 28(1-2) pp 5ndash11

[19] Katti K S 2004 ldquoBiomaterials in Total Joint Replacementrdquo Colloids Surf B39(3) pp 133ndash142

[20] Thompson I and Hench L 1998 ldquoMechanical Properties of BioactiveGlasses Glass-Ceramics and Compositesrdquo Proc Inst Mech Eng Part H JEng Med 212(2) pp 127

[21] Fraldi M Esposito L Perrella G Cutolo A and Cowin S 2010ldquoTopological Optimization in Hip Prosthesis Designrdquo Biomech ModelMechanobiol 9(4) pp 389ndash402

[22] Bobyn J D Poggie R Krygier J Lewallen D Hanssen A Lewis RUnger A Orsquokeefe T Christie M and Nasser S 2004 ldquoClinical Validationof a Structural Porous Tantalum Biomaterial for Adult Reconstructionrdquo J BoneJt Surg 86(Supplement 2) pp 123ndash129

[23] Bobyn J Stackpool G Hacking S Tanzer M and Krygier J 1999ldquoCharacteristics of Bone Ingrowth and Interface Mechanics of a New PorousTantalum Biomaterialrdquo J Bone Jt Surg Br Vol 81(5) pp 907ndash914

[24] Parthasarathy J Starly B Raman S and Christensen A 2010 ldquoMechanicalEvaluation of Porous Titanium (Ti6al4v) Structures with Electron Beam Melt-ing (Ebm)rdquo J Mech Behav Biomed Mater 3(3) pp 249ndash259

[25] Heinl P Muller L Korner C Singer R F and Muller F A 2008ldquoCellular Ti-6al-4v Structures with Interconnected Macro Porosity for BoneImplants Fabricated by Selective Electron Beam Meltingrdquo Acta Biomater4(5) pp 1536ndash1544

[26] Stamp R Fox P Orsquoneill W Jones E and Sutcliffe C 2009 ldquoThe Devel-opment of a Scanning Strategy for the Manufacture of Porous Biomaterials bySelective Laser Meltingrdquo J Mater Sci Mater Med 20(9) pp 1839ndash1848

[27] Yang S Leong K Du Z and Chua C 2002 ldquoThe Design of Scaffolds forUse in Tissue Engineering Part Ii Rapid Prototyping Techniquesrdquo TissueEng 8(1) pp 1ndash11

[28] Murr L Gaytan S Medina F Lopez H Martinez E Machado B Hernan-dez D Martinez L Lopez M and Wicker R 2010 ldquoNext-Generation Biomed-ical Implants Using Additive Manufacturing of Complex Cellular and FunctionalMesh Arraysrdquo Philos Trans R Soc London Ser A 368(1917) pp 1999ndash2032

[29] Coelho P Fernandes P Guedes J and Rodrigues H 2008 ldquoA HierarchicalModel for Concurrent Material and Topology Optimisation of Three-Dimensional Structuresrdquo Struct Multidiscip Optim 35(2) pp 107ndash115

[30] Rodrigues H Guedes J and Bendsoe M 2002 ldquoHierarchical Optimizationof Material and Structurerdquo Struct Multidiscip Optim 24(1) pp 1ndash10

[31] Goncalves Coelho P Rui Fernandes P and Carrico Rodrigues H 2011ldquoMultiscale Modeling of Bone Tissue with Surface and Permeability ControlrdquoJ Biomech 44(2) pp 321ndash329

[32] Coelho P G Cardoso J B Fernandes P R and Rodrigues H C 2011ldquoParallel Computing Techniques Applied to the Simultaneous Design of Struc-ture and Materialrdquo Adv Eng Software 42(5) pp 219ndash227

[33] Gibson L J and Ashby M F 1999 Cellular Solids Structure and Proper-ties Cambridge University Press Cambridge UK

[34] Guedes J and Kikuchi N 1990 ldquoPreprocessing and Postprocessing for Mate-rials Based on the Homogenization Method with Adaptive Finite ElementMethodsrdquo Comput Methods Appl Mech Eng 83(2) pp 143ndash198

[35] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization I-Homogenization Theory for Media with PeriodicStructurerdquo Comput Struct 69(6) pp 707ndash717

[36] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Opimization I I-Analytical and Numerical Solution of HomogenizationEquationsrdquo Comput Struct 69(6) pp 719ndash738

[37] Fang Z Starly B and Sun W 2005 ldquoComputer-Aided Characterization forEffective Mechanical Properties of Porous Tissue Scaffoldsrdquo Comput-AidedDes 37(1) pp 65ndash72

[38] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics 6th ed Butterworth-Heinemann Burlington

[39] Hollister S and Kikuchi N 1992 ldquoA Comparison of Homogenization andStandard Mechanics Analyses for Periodic Porous Compositesrdquo ComputMech 10(2) pp 73ndash95

[40] Pellegrino C Galvanetto U and Schrefler B 1999 ldquoNumerical Homogeni-zation of Periodic Composite Materials with Non Linear MaterialComponentsrdquo Int J Numer Methods Eng 46(10) pp 1609ndash1637

[41] Bragdon C Jasty M Greene M Rubash H and Harris W 2004ldquoBiologic Fixation of Total Hip Implants Insights Gained from a Series of Ca-nine Studiesrdquo J Bone Jt Surg 86(Supplement 2) pp 105ndash117

[42] Harrysson O L A Cansizoglu O Marcellin-Little D J Cormier D R andWest Ii H A 2008 ldquoDirect Metal Fabrication of Titanium Implants with Tai-lored Materials and Mechanical Properties Using Electron Beam Melting Tech-nologyrdquo Mater Sci Eng C 28(3) pp 366ndash373

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-9

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

trade-off designs could not have been captured Hedia et al[1617] attempted to reconcile the conflicting nature of theseobjective functions by proposing the use of three bioactive materi-als hydroxyapatite Bioglass and collagen to design a gradedcementless hip stem Although their implant design reduced boneresorption and bone-implant interface stresses the use of suchbioactive materials have limitations due to their brittleness andinsufficient strength when applied to load-bearing applications[18ndash20] In a more recent study Fraldi et al [21] applied a maxi-mum stiffness topological optimization strategy to re-design a hipprostheses with the goal of reducing stress shielding in the femurAccording to this method elements with intermediate volumefraction (between 0 and 1) are penalized to limit their presence inthe final solution For regions with intermediate relative densitycertain microstructures should be proposed to match those materi-als in terms of effective elastic properties Laser micro-drilling issuggested to create the required micro-porosity an option that canbe used only on the implant surface not throughout the implant

Other advances in total hip replacement have used a micro-structured material over a fully dense material Hip implants withporous tantalum have been proposed in knee and hip replacementsurgery [22] Tantalum foam is an excellent material due to itsbiocompatibility high volumetric porosity and modulus of elas-ticity similar to that of bone To create the tantalum foam puretantalum is chemically deposited on a carbon skeleton As a resultthe microstructure of a tantalum foam implant has an almost uni-form and random distribution of pore shape and size [23] through-out the implant These material characteristics however havebeen demonstrated to be incapable of solving the conflicting na-ture of the physiological phenomena occurring in an implant[1415] Indeed whereas the reduced stiffness of the foamdecreases bone resorption the uniform distribution of cells has theundesired effect of increasing the interface stresses

Recent advances in additive manufacturing such as Electron-Beam Melting (EBM) Selective Laser Melting (SLM) Stereoli-thography Apparatus (SLA) and other rapid prototyping techni-ques offer the possibility of novel bone-replacement implantswith a controlled cellular microstructure [24ndash28] As a result cel-lular components with tailored microstructures can be built with ahigh level of quality accuracy and reliability Besides providingan exceptional degree of control over the mechanical propertiessuch manufacturing processes are capable of building graded cel-lular structures As demonstrated by the work of Kuiper andHuiskes [1415] this feature is an asset for bone-replacementimplants since the internal skeleton of the prosthesis can bedesigned to ease osseointegration as well as to match the local me-chanical properties of the femoral bone By properly selecting to-pology size and relative density of the unit cell of the implant itis thus possible to (a) fabricate implants which can provide me-chanical properties mimicking those of the host bone (b) manu-facture three-dimensional structures with an interconnectedporosity and pore sizes suitable to bone ingrowth and vasculariza-tion and (c) customize implants for each patient by using CT scandata of the patientrsquos bone

While the aforementioned research is promising for the supportof cellular microstructure based bone-replacement implants theliterature is limited to the manufacturing aspects or the mechani-cal testing of biomedical implants with periodic cellular micro-structure [24ndash28] No study has been found which deals with themultiscale mechanics aspects of an implant with a functionallygraded cellular material as well as the multiobjective nature of thefeatures that their design requires

This paper proposes a systematic methodology for the design ofbone-replacement implants with improved structural stability Fortotal hip arthroplasty we propose to design an implant with tai-lored gradients of lattice material that can simultaneously mini-mize bone resorption and bone-implant interface stress Theprocedure that hinges on multiscale mechanics theory and multi-objective optimization is applied to the design of a bi-dimensionalfemoral hip implant with optimal graded cellular microstructure

Its biocompatibility performance is discussed with respect to thatof currently-used hip implants

We note that in this work the terms ldquocellularrdquo ldquocellrdquo and ldquowallcellrdquo refer exclusively to the artificial porous material of theimplant These terms neither refer to body cells nor have a biolog-ical or medical connotation

2 Methodology

Kuiper and Huiskes [1415] showed that the use of a gradedmaterial in an orthopedic stem can lead to a reduction of bothstress shielding and bone-implant interface stress To this endhierarchical computational procedures [29ndash32] can be imple-mented to design an optimum material distribution within theimplant These strategies might generally require a high computa-tional cost besides yielding a microstructure which is difficult tofabricate In this work we suggest to design gradients of materialproperties through a tailored lattice microstructure whose geo-metrical parameters are optimized in each region of the implant toachieve minimum bone loss and implant interface failure

The mechanical properties of a cellular structure depend on therelative density and the geometric parameters of the unit cell asdescribed for example by the expression of the Youngrsquos modulus[33]

E frac14 CEsqqs

m

(1)

where E is the effective Youngrsquos modulus of the unit cell q isthe density of the unit cell and Es and qs are the Youngrsquos modulusand density of the constitutive material respectively In additionm has a value varying from 1 to 3 as determined by the mechani-cal failure mode of the unit cell and C is a geometric constant ofthe unit cell By changing the relative density of the lattice micro-structure it is thus possible to obtain desired values of mechanicalproperties in any zone of the implant

Figure 1 summarizes the procedure proposed here to design a cel-lular implant with controlled gradients of mechanical propertiesThe method integrates a multiscale mechanics approach to deal withthe scale-dependent material structure and a multiobjective optimi-zation strategy to handle the conflicting nature of bone resorptionand implant interface failure The main steps identified by the num-bers reported in the flow chart of Fig 1 are described here

bull (1-2) A finite element model of the bone is created by proc-essing CT-scan data of a patient bone The design domain ofthe prosthesis is assumed to possess a 3D lattice microstruc-ture where the unit cell ie the building block can be ofany arbitrary topology (Fig 1) The microscopic parametersof the unit cell geometry and the macroscopic shape of theimplant are the design variables of the vector b The unit cellis assumed to be locally periodic and its field quantities suchas stress and strain to vary smoothly through the implant

bull (3) The characteristic length of the unit cell in the cellularimplant is assumed to be much smaller than the characteristiclength of the macrodimensions of the implant Hence themicrostructure can be replaced with a homogeneous mediumwhose equivalent mechanical properties in particular the ho-mogenized stiffness tensor of each unit cell are calculatedthrough asymptotic homogenization theory [34ndash38]

bull (45) The homogenized stiffness tensors are then used to con-struct the stiffness matrix which will be the input to the FiniteElement (FE) solver As a result the average strains andstresses throughout the bone and the structure of the prosthe-sis are calculated

bull (6) To obtain the microscopic stress field for each unit cellfrom the macroscopic strain the stress recovery procedureillustrated in Refs [3438ndash40] is used

bull (7) If the microscopic stress level is below a predefinedfailure criterion the macroscopic stresses and strains

031004-2 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

representing the mechanical behavior of the implant are usedto evaluate bone loss (mrethbTHORN) and interface failure (FethbTHORN) Inthe formulation of the multiobjective optimization problemthe constraints are set on the average porosity of the cellularimplant ethbTHORN the mean pore size P and the minimum thick-ness of cell walls tmin In particular ethbTHORN 40 and50 lm P 800 lm are selected to ease bone ingrowth[4142] The thickness of the cell walls is selected to begreater than the minimum resolution tmin offered by a givenmanufacturing process For example tmin is 100 lm and 70lm respectively for SLM and SLA [2743]

bull (8-9) If the solutions of the optimization have not convergedthen the vector b of the design variables are updated to findthe set of non-dominated solutions of the Pareto front

bull (9) If the unit cell fails at the microscale level the cell wallswill be iteratively increased to reduce the microscopic stresses

As described above multiscale mechanics and multiobjective opti-mization are integrated aspects of the method proposed in this paperTheir descriptions are separately detailed in the following subsections

21 Multiscale Analysis and Design of a CellularImplant The deformation and failure mechanisms of a structurewith heterogeneous material can occur at both the macroscopicand microscopic scales In a full scale simulation the heterogene-ities are explicitly modeled at the microscale to guarantee high ac-curacy The computational effort however can be very lengthyand time-consuming As an alternative the microstructure can bereplaced by a homogeneous medium and the mathematical theory ofhomogenization can be used to characterize the mechanical behaviorof heterogeneous media [44] As shown in Fig 2 a body Xe with aperiodic microstructure subjected to the traction t at the tractionboundary Ct a displacement d at the displacement boundary Cd anda body force f can be replaced by a homogenized body X with theprescribed external and traction boundaries applied to Xe

without ge-ometrical details and voids of the local coordinate system

The homogenized properties and strength of a cellular structurecan be obtained by performing either analytical or numerical or ex-perimental approaches [33ndash3945ndash53] Extensive efforts have beendevoted to the derivation of the equivalent mechanical properties

Fig 1 Flow chart illustrating the design of a graded cellular hip implant minimizing boneresorption and implant interface failure

Fig 2 Homogenization concept of a cellular structure

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-3

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

by structural analysis [3345ndash50] In these studies the effectivemoduli and yield strength of a cellular material are generally mod-eled by assuming that the cell walls flex like beams Despite thesimplicity of this method in calculating the overall mechanicalproperties the results are reliable only if the relative density islower than 03 [47] Furthermore the actual stress distributionwithin the unit cell walls cannot be captured As an alternativemethod the asymptotic homogenization method is used in thiswork to deal with the multiscale analysis of the cellular implantThis technique is applied to calculate the homogeneous stiffnessmatrix of the unit cell for different values of relative density as wellas to determine the microscopic stresses and strains [343839] Theasymptotic homogenization method has been widely used in topol-ogy optimization [3454ndash58] and hierarchical design of materialsand structures [29ndash32] In this work unlike hierarchical topologyoptimization a predefined unit cell topology with parametric geom-etry is considered as the microstructure of the implant then theoptimization algorithm searches for the optimum unit cell geometryof the lattice to minimize the antagonist objective functions under aset of constraints This procedure is similar to the one developed byBendsoslashe and Kikuchi [58] and is not limited to any cell topologywe selected the hollow square cell as an example to demonstratethe methodology A brief explanation about the asymptotic homog-enization technique and the main steps to calculate the effectivestiffness and microscopic stress and strain are provided below

The approach assumes that each field quantity depends on twodifferent scales one on the macroscopic level x and the other onthe microscopic level yfrac14 xe where e is a magnification factorthat enlarges the dimensions of a unit cell to make it comparablewith the dimensions of the material The field quantities such asdisplacement stress and strain are assumed to vary smoothly atthe macroscopic level being periodic at the microscale [35ndash39]The effective properties of the periodic material are determinedthrough the solution of local problems formulated on the represen-tative volume element (RVE) of the material As a result the ho-mogenized stiffness tensor (EH

ijkl) of a cellular material may bedefined as follow [39]

EHijkl frac14

1

Yj j

ethYC

EijpmMpmkldY (2)

where Yj j is the volume of the entire unit cell with void Yc is thesolid part of the cell Eijkl is the local elasticity tensor that dependson the position within the representative volume element ie Eijkl

is equal to the elasticity tensor of the material located in the cellwalls and it vanishes in the voids Mijkl is the local structure ten-sor which relates the macroscopic strains (e) to the local or micro-structural strains (e) through the relation

eij frac14 Mijklekl (3a)

Mijkl frac141

2ethdikdjl thorn dildjkTHORN ekl

ij (3b)

where dij is the Kronecker f and eklij is the microstructural strain

corresponding to the component kl of the macroscopic strain ten-sor (ekl) ekl

ij is the solution of the following equation

ethYC

Eijpme1ijethvTHORNekl

pmethuTHORNdY frac14eth

YC

Eijkle1ijethvTHORNekldY (4)

where e1ijethvTHORN is the virtual strain In general ekl can be an arbitrary

macroscopic strain tensor Considering the assumption of smalldeformation and linear material behavior ekl may be written as alinear combination of unit strains For a two-dimensional case theunit strains are defined as

e11 frac14 1 0 0frac12 T e22 frac14 0 1 0frac12 T e12 frac14 0 0 1frac12 T (5)

To calculate the effective mechanical properties of a cellular ma-terial the first task is to obtain the matrix Mijkl After discretizing

the RVE domain the unit strains are applied to each element ofthe FE model Periodicity of the strain field is ensured by impos-ing periodic boundary conditions on the RVE edges The directmethod [59] is selected here to derive periodic boundary condi-tions The microscopic strain field (ekl

ij ) inside the RVE isobtained by solving Eq (4) The results are substituted into Eq(3b) to calculate the local structure tensor Mijkl for each elementof the RVE Finally the effective stiffness tensor EH

ijkl is obtainedby calculating Eq (2) Once the local structure tensor Mijkl isobtained the microscopic strains and stresses corresponding tothe macroscopic strain can be obtained via Eq (3a) and the consti-tutive equation of the cell wall material

The steps described above are used to compute the homoge-nized stiffness tensor for each unit cell of the cellular hip implant(Fig 1) These tensors are used to construct the global stiffnessmatrix for the FE solver to obtain macroscopic stress and straindistribution within bone and implant The values are then post-processed to evaluate the objective functions of the multiobjectiveoptimization problem whose formulation is described in the fol-lowing section

22 Formulation of the Multiobjective OptimizationProblem For the design of an optimum implant we impose thesimultaneous minimization of the amount of bone loss around theprosthesis and the probability of mechanical failure at the bone-implant interface As illustrated in Fig 1 the multiobjective opti-mization problem can be formulated as

Minimize mrethbTHORNFethbTHORN

bone loss

interface failure

Subject to

ethbTHORN 4050lm P 800lm

t tmin

average porosity

meanpore size

cell wall thickness

(6)

The amount of bone loss around the stem is determined by assess-ing the amount of bone that is underloaded Bone can be consid-ered locally underloaded when its local strain energy (Ui) per unit

of bone mass (q) averaged over n loading cases (S frac14 1n

Pnifrac141

Ui

q )

is beneath the local reference value Sref which is the value of Swhen no prosthesis is present However it has been observed thatnot all the underloading leads to resorption rather a certain frac-tion of underloading (the threshold level or dead zone s) is toler-ated In fact bone resorption starts when the local value of S isbeneath the value of eth1 sTHORNSref [1460] Using this definition theresorbed bone mass fraction mr can be obtained from

mrethbTHORN frac141

M

ethV

gethSethbTHORNTHORNqdV (7)

where M and V are the original bone mass and volume respec-tively and gethSethbTHORNTHORN is a resorptive function equal to unity if thelocal value of S is beneath the local value of eth1 sTHORNSref and equalto 0 if eth1 sTHORNSref lt S In this study the value of the dead zone sis assumed to be 05 [14]

The other objective is to minimize the probability of local inter-face failure which is expressed by the following functional of theinterface stress distribution

FethbTHORN frac14 1

n

Xn

ifrac141

ethP

f ethrbi THORNdP (8)

where FethbTHORN is the global interface function index rbi is the inter-

face stress at the loading case i depending on the design variableb P is the interface area and f ethrb

i THORN is the local interface stressfunction which is defined based on the multi-axial Hoffman fail-ure criterion [61] This function is used to determine where local

031004-4 Vol 134 MARCH 2012 Transactions of the ASME

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debonding might occur along the bone-implant interface [1462]The probability of local interface failure f ethrTHORN is given by

f ethrTHORN frac14 1

StScr2

n thorn1

St 1

Sc

rn thorn

1

S2s

s2 (9)

where St and Sc are the uniaxial tensile and compressive strengthsrespectively Ss is the shear strength and rn and s are normal andshear stresses at the bone-implant interface respectively Forf ethrTHORN 1 a high probability of failure is expected whereas forf ethrTHORN 1 the risk of interface failure is low Tensile compressionand shear strengths of bone can be expressed as a function of bonedensity following the power law relations obtained by Pal et al[62]

St frac14 145q171 Sc frac14 324q185 Ss frac14 216q165 (10)

The distribution of bone density can be obtained through a CT-scan data of bone and then used in Eq (10) to find the effectivemechanical properties of the bone from which the local interfacefailure can be determined via Eq (9) Finally the interface failureindex FethbTHORN is evaluated by means of Eq (8) It should be notedthat FethbTHORN serves as a qualitative indicator only lower values ofFethbTHORN indicate reduced probability of local interface failure in theimplant

The expressions of bone resorption after implantation and me-chanical failure of bone-implant interface described in this section

are used in the finite element analysis to evaluate the objectivefunctions to be optimized The following sections shows the appli-cation of the method for the design of a two dimensional cellularimplant

3 Application of the Methodology to design a 2D

Femoral Implant with a Graded Cellular Material

31 FEM model of the Femur at the Macroscale The lefthand side of Fig 3 shows the geometry of the left femur consid-ered in this work along with the appropriate loads and boundaryconditions These data have been obtained from the work ofKuiper and Huiskes [14] The 3D geometry of the femur is simpli-fied into a 2D model where the thickness of the stem and bonevaries such that the second moment of area about the out-of-planeaxis does not differ in both models [16] Furthermore in this pa-per the implant material is designed to be an open cell lattice toease bone in growth in the implanted stem and obtain a full bondAlthough bone ingrowth does not exist in a postoperative situa-tion it can appear later if local mechanical stability is guaranteedThe minimization of interface stress reduces the possibility ofoccurrence of interface micromotion and instability [4] There-fore to decrease the computational cost required by a stabilityanalysis based on a nonlinear frictional contact model the pros-thesis and the surrounding bone are assumed fully bonded

The load case represents daily static loading during the stancephase of walking [63] The distal end of the femur is fixed toavoid rigid body motion For the material properties of the modelwe consider 20 GPa as the Youngrsquos modulus of the cortical boneand 15 GPa for the proximal bone The Poissonrsquos ratio is set to be03 [14]

32 FEM model of the Cellular Implant at theMicroscale The right hand side of Fig 3 illustrates the model ofa cementless prosthesis implanted into the human femur The griddepicts the domain of the implant to be designed with a lattice ma-terial of graded properties Figure 4 shows the unit cell geometryused for the tessellation of the whole implant The gradients ofmaterial properties are governed by the lattice relative densitywhich is a variable controlled by the cell size and wall thicknessof the hollow square In future work the performance of alterna-tive cell topologies suitable for bone tissue scaffolding will beinvestigated as well as combinations of dissimilar unit cells in thetessellation [64]

For the material property of the implant we consider Ti6Al4V[24] which is a biocompatible material commonly used in EBMIts mechanical properties are the following 900 MPa for the yieldstrength of the solid material 120 GPa for Youngrsquos modulus and03 for Poissonrsquos ratio

33 Design of the Graded Cellular Material of theImplant The procedure introduced in this paper for the design ofa graded cellular implant requires both multiscale analysis andmultiobjective optimization as described in Secs 21 and 22 andshown in Fig 1 The variables of the lattice model are the relativedensities attributed to 130 sampling points 26 rows along the

Fig 3 2D Finite element models of the femur (left) and theprosthesis implanted into the femur (right)

Fig 4 2D hollow square unit cell for given values of relative density

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-5

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prosthetic length and 5 columns along the radial direction asshown in the right side of Fig 3 The number of sampling pointshas been chosen to be 130 to limit the computational time requiredfor the analysis and optimization while providing a reasonable re-solution for the relative density distribution For a more refined den-sity distribution the number of sample points can be increasedTheir values have been constrained in the range 01 q 1 [47]to prevent elastic buckling from occurring in the unit cell prior toyielding The values of the relative density between the samplingpoints are obtained through linear interpolation Although not con-sidered in the current research the shape of the implant could beincluded in the vector b as a design variable (Fig 1)

To calculate the stress and strain regime of the implant thesteps explained in Sec 21 are followed Initial values of relativedensity are assigned to each element of the FE model created inANSYS (Canonsburg Pennsylvania USA) The stiffness matrixof each element is calculated through a computational code2DHOMOG developed in-house which obtains the homogenizedstiffness matrix of the square unit cell as a function of relativedensity Figure 5 shows the results which are first obtained at dis-crete points of relative density To obtain the continuous functionsfor the properties we calculate through the least squares methodthe expressions (Table 1) for two ranges of relative density ieq lt 03 and q gt 03 These functions enable to assign the valuesof the stiffness for a given relative density to each of the samplepoints modeled in the FE solver We note that the expressions ofYoungrsquos moduli and Poissonrsquos ratios in the x1 and x2 directions donot change since the cell thickness is uniform

Once the stress and strain regimes of the cellular material havebeen calculated the non-dominated sorting genetic (NSGA-II)

algorithm [65] is employed to solve the multiobjective optimiza-tion problem described in Sec 22 The strain energy within thebone and the stress distribution at the bone-implant interface isthen calculated and used in Eqs (7) and (8) to evaluate the objec-tive functions The initial population is then sorted based on thenondomination front criterion A population of solutions calledparents are selected from the current population based on theirrank and crowding distance Then genetic operators are appliedto the population of parents to create a population of off-springsFinally the next population is produced by taking the best solu-tions from the combined population of parents and off-springsThe optimization continues until the user-defined number of func-tion evaluations reaches 25000 [65] The computational costrequired to run the optimization process in a single 24 GHz Intelprocessor with 4 GB RAM was about 300000 CPU s about threeand a halfdays We plan to use parallel computing with a PC clus-ter in future work this will considerably reduce the computationaltime since each function evaluation can be performedindependently

For each point in the objective function space the stress recov-ery procedure is applied to verify whether the stresses are admissi-ble To apply this procedure the average macroscopic straininside each unit cell is found The position of each unit cell withinthe implant is obtained after imposing a proper cell tessellationwhich in this work has been set to be uniform The size of the unitcell is selected as small as possible to capture the relative distribu-tion contour with higher resolution For a relative density of 01the square cell sizes for the cell wall thickness of either 70 or 100lm have been selected respectively to be 136 and 18 mm Oncethe position of the unit cells has been obtained 3 3 Gauss pointsare assigned to each cell The values of relative density and mac-roscopic strain at these points are obtained from the relative den-sity distribution and macroscopic strain field For Gauss pointslocated outside the implant border the values are linearly extrapo-lated from those located at the neighboring points inside theimplant domain Using a Gaussian quadrature integration [66] theaverage relative density and macroscopic strain of each cell arecalculated The local stress distribution and the yield safety factorof each cell are obtained through the von Mises stress criterionThe procedure is applied to all unit cells of the selected optimaldesign located on the Pareto frontier and the minimum local safetyfactor of a cell is specified as design safety factor

4 Results and Discussion

The advantage of multiobjective optimization with a posterioriarticulation of preference is that a set of optimum solutions areavailable without requiring the designer to choose in advance anyweighting factors to the objective functions Once the whole set ofPareto solutions has been determined the designer has the free-dom to select the desired solution based on the importance ofbone mass preservation relative to the amount of interface stressFigure 6 shows all the optimum solutions ie the relative densitydistribution for a hip stem implant with graded cellular materialThe x axis represents the amount of bone resorption for theimplanted hip on the y axis is the interface failure index Amongthe optimal solutions we examine three representative relativedensity distributions the extreme points A and C of the Paretofrontier for which one objective function has importance factor 0and the other 100 and a solution B characterized by a 50weight factor For each solution Fig 6 gives the following per-formance metrics bone resorption (mr) interface failure index(FethbTHORN) maximum interface failure (f ethrTHORNmax) average porosity ofeach stem ( ) and design safety factor (SF) after implementingthe stress recovery procedure The maximum interface failuref ethrTHORNmax is included since FethbTHORN which quantifies only the overalleffect of the implant stiffness on the interface stresses is not suffi-cient to provide information on the probability of failure

As seen from the performance metrics in Fig 6 the porosity ofsolutions A B and C is greater than 40 which is satisfactory

Fig 5 Effective Youngrsquos modulus of 2D square lattice versusrelative density Solution points obtained through homogeniza-tion theory are fitted with the least squares method

Table 1 Effective Mechanical Properties of the Square UnitCell as a Function of Relative Density

q lt 03 03 lt q lt 1

E1

Esfrac14 E2

Es058ethqTHORN1046

127ethqTHORN2 052ethqTHORN thorn 023

G

Es0093ethqTHORN307

126ethqTHORN3 133ethqTHORN2 thorn 051ethqTHORN 0065

21

sfrac14 12

s07ethqTHORN105

068ethqTHORN2 thorn 028ethqTHORN thorn 006

031004-6 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

for bone ingrowth [41] By comparing implants C and A weobserve that a raise of the implant porosity from point C to Aresults in an implant stiffness decrease which on one hand low-ers bone loss and on the other hand enhances the risk of interfacefailure When solution B is compared to C a reduction of 8 ofbone resorption is noted with a slight increase of the peak value ofthe interface failure On the other hand by contrasting solution Bto A we note a significant increase (60) of the peak value ofinterface failure which is below the Hoffman failure strength anda minor reduction (2) of the amount of bone resorption Themain benefit of solution A is the maximum porosity of the micro-structure that can promote bone ingrowth Solution B on the otherhand might be the preferred solution with respect to low boneresorption and interface failure These observations emerge onlyby inspection of the objective functions selected in this work weremark here that other parameters should be taken into accountfor the selection of the best implant These include patientrsquos bone

characteristics the range of activity age and desired level ofbone mass preservation after implantation

For prescribed geometric loading and constraint conditions wenow compare the metrics of resorbed bone mass (mr) and distribu-tion of interface stress (f ethrbTHORN) of the optimal solution B with thoseof (i) a currently-used fully dense titanium stem and (ii) a cellularimplant with a uniformly distributed relative density of 50 Fig-ures 7 and 8 illustrate the results of the comparison

For the solid titanium stem the amount of bone resorption cal-culated through Eq (7) is 67 and the interface failure indexFethbTHORN obtained from Eq (8) is 133 Using the distribution of f ethrTHORNgenerated around the titanium stem (Fig 8(a)) we observe thatthe maximum value of interface failure (051) occurs at the distalend of the implant As expected this implant is much stiffer thanthe surrounding bone thereby resulting in a higher amount ofbone resorption For the numerical validation the interface shearstress of the titanium implant at the proximal region is also

Fig 6 Trade-off distributions of relative density for the optimized cellular implant

Fig 7 Distribution of bone resorption around (a) fully dense titanium implant (b) cellularimplant with uniform relative density of 50 (c) graded cellular implant (solution B in Fig 6)

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-7

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

compared with the one obtained by Kowalczyk for a 3D model[4] The mean and the maximum values of interface shear stressfor the 3D titanium implant in the work by Kowalczyk [4] are057 and 28 MPa respectively These values are respectively031 and 215 MPa for the titanium implant in this paper The con-tribution to the higher level of shear stress in the 3D model ofKowalczyk is the distribution of shear force on a smaller area InKowalczykrsquos study [4] the implant and bone are bonded only atthe proximal region while in our work the whole bone-implantinterface is bonded which results in a decrease of the mean andthe maximum values of interface shear stress

The cellular implant with uniform relative density of 50 isapproximately three times more flexible than the titanium stemThis implant can qualitatively simulate the behavior of an implantmade out of tantalum foam For this stem the amount of boneresorption and the interface failure index are about 34 and 287respectively and the interface failure is maximum (071) at theedge of proximal region Compared to the solid titanium implantthe amount of bone resorption decreases by 50 whereas themaximum interface failure increases about 40 This shows that adecrease of the implant stiffness with uniform porosity distribu-tion aiming at reducing bone resorption has the undesirable effectof increasing the risk of interface failure at the proximal regionThis outcome confirms the findings of the previous work byKuiper and Huiskes [14]

Figure 7(c) and 8(c) show the results for the graded cellularimplant B Its bone resorption and interface failure index are 16and 115 respectively The peak value of the local interface fail-ure is 025 Compared to the titanium stem both the amount ofbone resorption and the peak of interface failure decrease of 76and 50 respectively With respect to the uniformly-distributedcellular implant the decrease of bone resorption and interface fail-ure peak is of 53 and 65 respectively A graded cellularimplant with optimized relative density distribution is thus capa-ble of reducing concurrently both the conflicting objective func-tions In particular bone resorption reduces as a result of thecellular material which makes the implant more compliant theinterface stress on the other hand is minimized by the optimizedgradients of cellular material

A proof-of-concept implant was built to verify the manufactur-ability of the optimum grade lattice material Figure 9 shows thepolypropylene prototype of solution B which was manufacturedwith a 3D printer Objet Connex500 [67] A uniform tessellationand a square unit cell of 18 mm size were assumed to draw themodel The cell geometry was calculated from the average rela-tive density obtained from the method described in this paper AnSTL file of the graded cellular implant solution B was finallyused for rapid prototyping

The limitations of the method proposed in this paper are herediscussed First the method tackles the design of an implant stati-cally loaded in the stance phase of walking Further research isrequired to extend it to variable loading and fatigue life design

Second the accuracy of the asymptotic homogenization needsto be investigated at the vicinity of the implant borders Althoughmacroscopic mean stress fields on the boundaries are satisfied inthe global solutions [4068] microscopic stresses do not satisfythe Y-periodicity assumption These boundary effects howevermay be captured by applying a boundary layer corrector [6869] aspatially decaying stress localization function [70] or an adaptivemultiscale methodology [71] To investigate the accuracy of thesetechniques it is required to construct a detailed finite elementmodel and compare the actual stressstrain distribution inside thecell walls with the microscopic stressesstrains estimated from thehomogenization method This task will be performed in future

Third to extend the proposed procedure to three dimensions a3D cell topology with high porosity interconnected pore struc-ture large surface area with suitable textures and good

Fig 8 Distribution of local interface failure f ethrTHORN around (a) fully dense titanium implant(b) cellular implant with uniform relative density of 50 (c) graded cellular implant (solu-tion B in Fig 6)

Fig 9 Polypropylene proof-of-concept of the optimal gradedcellular implant (solution B)

031004-8 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

mechanical properties should be selected [7273] The asymptotichomogenization will be applied to characterize tissue scaffoldingcell topologies for a wide range of relative density Their effectivemoduli yield and ultimate surfaces for multiaxial loading condi-tions will be obtained to create a standard library from which toselect suitable cell geometries for a 3D cellular implant design

Fourth bone has been here considered as consisting of a cancel-lous and a cortical part with isotropic mechanical properties Fur-ther study is necessary to improve the constitutive material modelof bone and to consider bone as a graded cellular material

Finally in the optimization procedure a simple formulation hasbeen used to quantitatively measure both bone resorption andimplant stability In a future study detailed simulations of implantstability [74ndash77] and bone remodeling process [6077ndash80] can beperformed to assess both the short and long term performance ofthe implant

5 Concluding Remarks

This work has presented a methodology integrating multiscaleanalysis and design optimization to design a novel hip implantmade of graded cellular material The method can contribute tothe development of a new generation of orthopedic implants witha graded cellular microstructure that will reduce the clinical con-sequences of current implants

In the first part of the paper the homogenization method hasbeen reviewed and then used to capture the mechanics of theimplant at the micro and macro scale In the second part multiob-jective optimization has been applied to find optimum gradients ofmaterial distribution that minimize concurrently bone resorptionand bone-implant interface stresses The results have shown thatthe optimized cellular implant exhibits a reduction of 76 ofbone resorption and 50 of interface stress with respect to a fullydense titanium implant

References[1] Kurtz S Ong K Lau E Mowat F and Halpern M 2007 ldquoProjections of

Primary and Revision Hip and Knee Arthroplasty in the United States from2005 to 2030rdquo J Bone Jt Surg Am Vol 89(4) pp 780ndash785

[2] Vichinsky E P Neumayr L D Haberkern C Earles A N Eckman JKoshy M and Black D M 1999 ldquoThe Perioperative Complication Rate ofOrthopedic Surgery in Sickle Cell Disease Report of the National Sickle CellSurgery Study Grouprdquo Am J Hematol 62(3) pp 129ndash138

[3] Kobayashi S Saito N Horiuchi H Iorio R and Takaoka K 2000 ldquoPoorBone Quality or Hip Structure as Risk Factors Affecting Survival of Total-HipArthroplastyrdquo The Lancet 355(9214) pp 1499ndash1504

[4] Kowalczyk P 2001 ldquoDesign Optimization of Cementless Femoral Hip Pros-theses Using Finite Element Analysisrdquo J Biomech Eng 123(5) pp 396ndash402

[5] Moen T C Ghate R Salaz N Ghodasra J and Stulberg S D 2011 ldquoAMonoblock Porous Tantalum Acetabular Cup Has No Osteolysis on Ct at 10Yearsrdquo Clin Orthop Relat Res 469(2) pp 382ndash386

[6] Kurtz S Gawel H and Patel J 2011 ldquoHistory and Systematic Review ofWear and Osteolysis Outcomes for First-Generation Highly Crosslinked Poly-ethylenerdquo Clin Orthop Relat Res 469(8) pp 2262ndash2277

[7] Grubl A Marker M Brodner W Giurea A Heinze G Meisinger VZehetgruber H and Kotz R 2007 ldquoLong Term Follow up of Metal on MetalTotal Hip Replacementrdquo J Orthop Res 25(7) pp 841ndash848

[8] Neumann D R P Thaler C Hitzl W Huber M Hofstadter T and DornU 2010 ldquoLong-Term Results of a Contemporary Metal-on-Metal Total HipArthroplasty A 10-Year Follow-up Studyrdquo J Arthroplasty 25(5) pp700ndash708

[9] Glassman A Bobyn J and Tanzer M 2006 ldquoNew Femoral Designs DoThey Influence Stress Shieldingrdquo Clin Orthop Relat Res 453(12) pp64ndash74

[10] Adam F Hammer D S Pfautsch S and Westermann K 2002 ldquoEarly Fail-ure of a Press-Fit Carbon Fiber Hip Prosthesis with a Smooth Surfacerdquo JAntroplasty 17(2) pp 217ndash223

[11] Trebse R Milosev I Kovac S Mikek M and Pisot V 2005 ldquoPoorResults from the Isoelastic Total Hip Replacementrdquo Acta Orthop 76(2) pp169ndash176

[12] Harvey E Bobyn J Tanzer M Stackpool G Krygier J and Hacking S1999 ldquoEffect of Flexibility of the Femoral Stem on Bone-Remodeling and Fix-ation of the Stem in a Canine Total Hip Arthroplasty Model without CementrdquoJ Bone Jt Surg 81(1) pp 93ndash107

[13] Huiskes R Weinans H and Rietbergen B 1992 ldquoThe Relationship betweenStress Shielding and Bone Resorption around Total Hip Stems and the Effectsof Flexible Materialsrdquo Clin Orthop Relat Res 274(1) pp 124ndash134

[14] Kuiper J and Huiskes R 1992 ldquoNumerical Optimization of Hip-ProstheticStem Materialrdquo Recent Advances in Computer Methods in Biomechanics andBiomedical Engineering J Middleton G N Pande and K R Williams edsBooks and Journals International Ltd Swansea pp 76ndash84

[15] Kuiper J H and Huiskes R 1997 ldquoMathematical Optimization of ElasticProperties Application to Cementless Hip Stem Designrdquo J Biomech Eng119(2) pp 166ndash174

[16] Hedia H Shabara M El-Midany T and Fouda N 2006 ldquoImproved Designof Cementless Hip Stems Using Two-Dimensional Functionally Graded Materi-alsrdquo J Biomed Mater Res Part B Appl Biomater 79(1) pp 42ndash49

[17] Hedia H S and Mahmoud N-A 2004 ldquoDesign Optimization of Function-ally Graded Dental Implantrdquo Biomed Mater Eng 14(2) pp 133ndash143

[18] Watari F Yokoyama A Saso F Uo M and Kawasaki T 1997ldquoFabrication and Properties of Functionally Graded Dental Implantrdquo Compo-sites Part B 28(1-2) pp 5ndash11

[19] Katti K S 2004 ldquoBiomaterials in Total Joint Replacementrdquo Colloids Surf B39(3) pp 133ndash142

[20] Thompson I and Hench L 1998 ldquoMechanical Properties of BioactiveGlasses Glass-Ceramics and Compositesrdquo Proc Inst Mech Eng Part H JEng Med 212(2) pp 127

[21] Fraldi M Esposito L Perrella G Cutolo A and Cowin S 2010ldquoTopological Optimization in Hip Prosthesis Designrdquo Biomech ModelMechanobiol 9(4) pp 389ndash402

[22] Bobyn J D Poggie R Krygier J Lewallen D Hanssen A Lewis RUnger A Orsquokeefe T Christie M and Nasser S 2004 ldquoClinical Validationof a Structural Porous Tantalum Biomaterial for Adult Reconstructionrdquo J BoneJt Surg 86(Supplement 2) pp 123ndash129

[23] Bobyn J Stackpool G Hacking S Tanzer M and Krygier J 1999ldquoCharacteristics of Bone Ingrowth and Interface Mechanics of a New PorousTantalum Biomaterialrdquo J Bone Jt Surg Br Vol 81(5) pp 907ndash914

[24] Parthasarathy J Starly B Raman S and Christensen A 2010 ldquoMechanicalEvaluation of Porous Titanium (Ti6al4v) Structures with Electron Beam Melt-ing (Ebm)rdquo J Mech Behav Biomed Mater 3(3) pp 249ndash259

[25] Heinl P Muller L Korner C Singer R F and Muller F A 2008ldquoCellular Ti-6al-4v Structures with Interconnected Macro Porosity for BoneImplants Fabricated by Selective Electron Beam Meltingrdquo Acta Biomater4(5) pp 1536ndash1544

[26] Stamp R Fox P Orsquoneill W Jones E and Sutcliffe C 2009 ldquoThe Devel-opment of a Scanning Strategy for the Manufacture of Porous Biomaterials bySelective Laser Meltingrdquo J Mater Sci Mater Med 20(9) pp 1839ndash1848

[27] Yang S Leong K Du Z and Chua C 2002 ldquoThe Design of Scaffolds forUse in Tissue Engineering Part Ii Rapid Prototyping Techniquesrdquo TissueEng 8(1) pp 1ndash11

[28] Murr L Gaytan S Medina F Lopez H Martinez E Machado B Hernan-dez D Martinez L Lopez M and Wicker R 2010 ldquoNext-Generation Biomed-ical Implants Using Additive Manufacturing of Complex Cellular and FunctionalMesh Arraysrdquo Philos Trans R Soc London Ser A 368(1917) pp 1999ndash2032

[29] Coelho P Fernandes P Guedes J and Rodrigues H 2008 ldquoA HierarchicalModel for Concurrent Material and Topology Optimisation of Three-Dimensional Structuresrdquo Struct Multidiscip Optim 35(2) pp 107ndash115

[30] Rodrigues H Guedes J and Bendsoe M 2002 ldquoHierarchical Optimizationof Material and Structurerdquo Struct Multidiscip Optim 24(1) pp 1ndash10

[31] Goncalves Coelho P Rui Fernandes P and Carrico Rodrigues H 2011ldquoMultiscale Modeling of Bone Tissue with Surface and Permeability ControlrdquoJ Biomech 44(2) pp 321ndash329

[32] Coelho P G Cardoso J B Fernandes P R and Rodrigues H C 2011ldquoParallel Computing Techniques Applied to the Simultaneous Design of Struc-ture and Materialrdquo Adv Eng Software 42(5) pp 219ndash227

[33] Gibson L J and Ashby M F 1999 Cellular Solids Structure and Proper-ties Cambridge University Press Cambridge UK

[34] Guedes J and Kikuchi N 1990 ldquoPreprocessing and Postprocessing for Mate-rials Based on the Homogenization Method with Adaptive Finite ElementMethodsrdquo Comput Methods Appl Mech Eng 83(2) pp 143ndash198

[35] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization I-Homogenization Theory for Media with PeriodicStructurerdquo Comput Struct 69(6) pp 707ndash717

[36] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Opimization I I-Analytical and Numerical Solution of HomogenizationEquationsrdquo Comput Struct 69(6) pp 719ndash738

[37] Fang Z Starly B and Sun W 2005 ldquoComputer-Aided Characterization forEffective Mechanical Properties of Porous Tissue Scaffoldsrdquo Comput-AidedDes 37(1) pp 65ndash72

[38] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics 6th ed Butterworth-Heinemann Burlington

[39] Hollister S and Kikuchi N 1992 ldquoA Comparison of Homogenization andStandard Mechanics Analyses for Periodic Porous Compositesrdquo ComputMech 10(2) pp 73ndash95

[40] Pellegrino C Galvanetto U and Schrefler B 1999 ldquoNumerical Homogeni-zation of Periodic Composite Materials with Non Linear MaterialComponentsrdquo Int J Numer Methods Eng 46(10) pp 1609ndash1637

[41] Bragdon C Jasty M Greene M Rubash H and Harris W 2004ldquoBiologic Fixation of Total Hip Implants Insights Gained from a Series of Ca-nine Studiesrdquo J Bone Jt Surg 86(Supplement 2) pp 105ndash117

[42] Harrysson O L A Cansizoglu O Marcellin-Little D J Cormier D R andWest Ii H A 2008 ldquoDirect Metal Fabrication of Titanium Implants with Tai-lored Materials and Mechanical Properties Using Electron Beam Melting Tech-nologyrdquo Mater Sci Eng C 28(3) pp 366ndash373

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-9

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

representing the mechanical behavior of the implant are usedto evaluate bone loss (mrethbTHORN) and interface failure (FethbTHORN) Inthe formulation of the multiobjective optimization problemthe constraints are set on the average porosity of the cellularimplant ethbTHORN the mean pore size P and the minimum thick-ness of cell walls tmin In particular ethbTHORN 40 and50 lm P 800 lm are selected to ease bone ingrowth[4142] The thickness of the cell walls is selected to begreater than the minimum resolution tmin offered by a givenmanufacturing process For example tmin is 100 lm and 70lm respectively for SLM and SLA [2743]

bull (8-9) If the solutions of the optimization have not convergedthen the vector b of the design variables are updated to findthe set of non-dominated solutions of the Pareto front

bull (9) If the unit cell fails at the microscale level the cell wallswill be iteratively increased to reduce the microscopic stresses

As described above multiscale mechanics and multiobjective opti-mization are integrated aspects of the method proposed in this paperTheir descriptions are separately detailed in the following subsections

21 Multiscale Analysis and Design of a CellularImplant The deformation and failure mechanisms of a structurewith heterogeneous material can occur at both the macroscopicand microscopic scales In a full scale simulation the heterogene-ities are explicitly modeled at the microscale to guarantee high ac-curacy The computational effort however can be very lengthyand time-consuming As an alternative the microstructure can bereplaced by a homogeneous medium and the mathematical theory ofhomogenization can be used to characterize the mechanical behaviorof heterogeneous media [44] As shown in Fig 2 a body Xe with aperiodic microstructure subjected to the traction t at the tractionboundary Ct a displacement d at the displacement boundary Cd anda body force f can be replaced by a homogenized body X with theprescribed external and traction boundaries applied to Xe

without ge-ometrical details and voids of the local coordinate system

The homogenized properties and strength of a cellular structurecan be obtained by performing either analytical or numerical or ex-perimental approaches [33ndash3945ndash53] Extensive efforts have beendevoted to the derivation of the equivalent mechanical properties

Fig 1 Flow chart illustrating the design of a graded cellular hip implant minimizing boneresorption and implant interface failure

Fig 2 Homogenization concept of a cellular structure

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-3

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

by structural analysis [3345ndash50] In these studies the effectivemoduli and yield strength of a cellular material are generally mod-eled by assuming that the cell walls flex like beams Despite thesimplicity of this method in calculating the overall mechanicalproperties the results are reliable only if the relative density islower than 03 [47] Furthermore the actual stress distributionwithin the unit cell walls cannot be captured As an alternativemethod the asymptotic homogenization method is used in thiswork to deal with the multiscale analysis of the cellular implantThis technique is applied to calculate the homogeneous stiffnessmatrix of the unit cell for different values of relative density as wellas to determine the microscopic stresses and strains [343839] Theasymptotic homogenization method has been widely used in topol-ogy optimization [3454ndash58] and hierarchical design of materialsand structures [29ndash32] In this work unlike hierarchical topologyoptimization a predefined unit cell topology with parametric geom-etry is considered as the microstructure of the implant then theoptimization algorithm searches for the optimum unit cell geometryof the lattice to minimize the antagonist objective functions under aset of constraints This procedure is similar to the one developed byBendsoslashe and Kikuchi [58] and is not limited to any cell topologywe selected the hollow square cell as an example to demonstratethe methodology A brief explanation about the asymptotic homog-enization technique and the main steps to calculate the effectivestiffness and microscopic stress and strain are provided below

The approach assumes that each field quantity depends on twodifferent scales one on the macroscopic level x and the other onthe microscopic level yfrac14 xe where e is a magnification factorthat enlarges the dimensions of a unit cell to make it comparablewith the dimensions of the material The field quantities such asdisplacement stress and strain are assumed to vary smoothly atthe macroscopic level being periodic at the microscale [35ndash39]The effective properties of the periodic material are determinedthrough the solution of local problems formulated on the represen-tative volume element (RVE) of the material As a result the ho-mogenized stiffness tensor (EH

ijkl) of a cellular material may bedefined as follow [39]

EHijkl frac14

1

Yj j

ethYC

EijpmMpmkldY (2)

where Yj j is the volume of the entire unit cell with void Yc is thesolid part of the cell Eijkl is the local elasticity tensor that dependson the position within the representative volume element ie Eijkl

is equal to the elasticity tensor of the material located in the cellwalls and it vanishes in the voids Mijkl is the local structure ten-sor which relates the macroscopic strains (e) to the local or micro-structural strains (e) through the relation

eij frac14 Mijklekl (3a)

Mijkl frac141

2ethdikdjl thorn dildjkTHORN ekl

ij (3b)

where dij is the Kronecker f and eklij is the microstructural strain

corresponding to the component kl of the macroscopic strain ten-sor (ekl) ekl

ij is the solution of the following equation

ethYC

Eijpme1ijethvTHORNekl

pmethuTHORNdY frac14eth

YC

Eijkle1ijethvTHORNekldY (4)

where e1ijethvTHORN is the virtual strain In general ekl can be an arbitrary

macroscopic strain tensor Considering the assumption of smalldeformation and linear material behavior ekl may be written as alinear combination of unit strains For a two-dimensional case theunit strains are defined as

e11 frac14 1 0 0frac12 T e22 frac14 0 1 0frac12 T e12 frac14 0 0 1frac12 T (5)

To calculate the effective mechanical properties of a cellular ma-terial the first task is to obtain the matrix Mijkl After discretizing

the RVE domain the unit strains are applied to each element ofthe FE model Periodicity of the strain field is ensured by impos-ing periodic boundary conditions on the RVE edges The directmethod [59] is selected here to derive periodic boundary condi-tions The microscopic strain field (ekl

ij ) inside the RVE isobtained by solving Eq (4) The results are substituted into Eq(3b) to calculate the local structure tensor Mijkl for each elementof the RVE Finally the effective stiffness tensor EH

ijkl is obtainedby calculating Eq (2) Once the local structure tensor Mijkl isobtained the microscopic strains and stresses corresponding tothe macroscopic strain can be obtained via Eq (3a) and the consti-tutive equation of the cell wall material

The steps described above are used to compute the homoge-nized stiffness tensor for each unit cell of the cellular hip implant(Fig 1) These tensors are used to construct the global stiffnessmatrix for the FE solver to obtain macroscopic stress and straindistribution within bone and implant The values are then post-processed to evaluate the objective functions of the multiobjectiveoptimization problem whose formulation is described in the fol-lowing section

22 Formulation of the Multiobjective OptimizationProblem For the design of an optimum implant we impose thesimultaneous minimization of the amount of bone loss around theprosthesis and the probability of mechanical failure at the bone-implant interface As illustrated in Fig 1 the multiobjective opti-mization problem can be formulated as

Minimize mrethbTHORNFethbTHORN

bone loss

interface failure

Subject to

ethbTHORN 4050lm P 800lm

t tmin

average porosity

meanpore size

cell wall thickness

(6)

The amount of bone loss around the stem is determined by assess-ing the amount of bone that is underloaded Bone can be consid-ered locally underloaded when its local strain energy (Ui) per unit

of bone mass (q) averaged over n loading cases (S frac14 1n

Pnifrac141

Ui

q )

is beneath the local reference value Sref which is the value of Swhen no prosthesis is present However it has been observed thatnot all the underloading leads to resorption rather a certain frac-tion of underloading (the threshold level or dead zone s) is toler-ated In fact bone resorption starts when the local value of S isbeneath the value of eth1 sTHORNSref [1460] Using this definition theresorbed bone mass fraction mr can be obtained from

mrethbTHORN frac141

M

ethV

gethSethbTHORNTHORNqdV (7)

where M and V are the original bone mass and volume respec-tively and gethSethbTHORNTHORN is a resorptive function equal to unity if thelocal value of S is beneath the local value of eth1 sTHORNSref and equalto 0 if eth1 sTHORNSref lt S In this study the value of the dead zone sis assumed to be 05 [14]

The other objective is to minimize the probability of local inter-face failure which is expressed by the following functional of theinterface stress distribution

FethbTHORN frac14 1

n

Xn

ifrac141

ethP

f ethrbi THORNdP (8)

where FethbTHORN is the global interface function index rbi is the inter-

face stress at the loading case i depending on the design variableb P is the interface area and f ethrb

i THORN is the local interface stressfunction which is defined based on the multi-axial Hoffman fail-ure criterion [61] This function is used to determine where local

031004-4 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

debonding might occur along the bone-implant interface [1462]The probability of local interface failure f ethrTHORN is given by

f ethrTHORN frac14 1

StScr2

n thorn1

St 1

Sc

rn thorn

1

S2s

s2 (9)

where St and Sc are the uniaxial tensile and compressive strengthsrespectively Ss is the shear strength and rn and s are normal andshear stresses at the bone-implant interface respectively Forf ethrTHORN 1 a high probability of failure is expected whereas forf ethrTHORN 1 the risk of interface failure is low Tensile compressionand shear strengths of bone can be expressed as a function of bonedensity following the power law relations obtained by Pal et al[62]

St frac14 145q171 Sc frac14 324q185 Ss frac14 216q165 (10)

The distribution of bone density can be obtained through a CT-scan data of bone and then used in Eq (10) to find the effectivemechanical properties of the bone from which the local interfacefailure can be determined via Eq (9) Finally the interface failureindex FethbTHORN is evaluated by means of Eq (8) It should be notedthat FethbTHORN serves as a qualitative indicator only lower values ofFethbTHORN indicate reduced probability of local interface failure in theimplant

The expressions of bone resorption after implantation and me-chanical failure of bone-implant interface described in this section

are used in the finite element analysis to evaluate the objectivefunctions to be optimized The following sections shows the appli-cation of the method for the design of a two dimensional cellularimplant

3 Application of the Methodology to design a 2D

Femoral Implant with a Graded Cellular Material

31 FEM model of the Femur at the Macroscale The lefthand side of Fig 3 shows the geometry of the left femur consid-ered in this work along with the appropriate loads and boundaryconditions These data have been obtained from the work ofKuiper and Huiskes [14] The 3D geometry of the femur is simpli-fied into a 2D model where the thickness of the stem and bonevaries such that the second moment of area about the out-of-planeaxis does not differ in both models [16] Furthermore in this pa-per the implant material is designed to be an open cell lattice toease bone in growth in the implanted stem and obtain a full bondAlthough bone ingrowth does not exist in a postoperative situa-tion it can appear later if local mechanical stability is guaranteedThe minimization of interface stress reduces the possibility ofoccurrence of interface micromotion and instability [4] There-fore to decrease the computational cost required by a stabilityanalysis based on a nonlinear frictional contact model the pros-thesis and the surrounding bone are assumed fully bonded

The load case represents daily static loading during the stancephase of walking [63] The distal end of the femur is fixed toavoid rigid body motion For the material properties of the modelwe consider 20 GPa as the Youngrsquos modulus of the cortical boneand 15 GPa for the proximal bone The Poissonrsquos ratio is set to be03 [14]

32 FEM model of the Cellular Implant at theMicroscale The right hand side of Fig 3 illustrates the model ofa cementless prosthesis implanted into the human femur The griddepicts the domain of the implant to be designed with a lattice ma-terial of graded properties Figure 4 shows the unit cell geometryused for the tessellation of the whole implant The gradients ofmaterial properties are governed by the lattice relative densitywhich is a variable controlled by the cell size and wall thicknessof the hollow square In future work the performance of alterna-tive cell topologies suitable for bone tissue scaffolding will beinvestigated as well as combinations of dissimilar unit cells in thetessellation [64]

For the material property of the implant we consider Ti6Al4V[24] which is a biocompatible material commonly used in EBMIts mechanical properties are the following 900 MPa for the yieldstrength of the solid material 120 GPa for Youngrsquos modulus and03 for Poissonrsquos ratio

33 Design of the Graded Cellular Material of theImplant The procedure introduced in this paper for the design ofa graded cellular implant requires both multiscale analysis andmultiobjective optimization as described in Secs 21 and 22 andshown in Fig 1 The variables of the lattice model are the relativedensities attributed to 130 sampling points 26 rows along the

Fig 3 2D Finite element models of the femur (left) and theprosthesis implanted into the femur (right)

Fig 4 2D hollow square unit cell for given values of relative density

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-5

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prosthetic length and 5 columns along the radial direction asshown in the right side of Fig 3 The number of sampling pointshas been chosen to be 130 to limit the computational time requiredfor the analysis and optimization while providing a reasonable re-solution for the relative density distribution For a more refined den-sity distribution the number of sample points can be increasedTheir values have been constrained in the range 01 q 1 [47]to prevent elastic buckling from occurring in the unit cell prior toyielding The values of the relative density between the samplingpoints are obtained through linear interpolation Although not con-sidered in the current research the shape of the implant could beincluded in the vector b as a design variable (Fig 1)

To calculate the stress and strain regime of the implant thesteps explained in Sec 21 are followed Initial values of relativedensity are assigned to each element of the FE model created inANSYS (Canonsburg Pennsylvania USA) The stiffness matrixof each element is calculated through a computational code2DHOMOG developed in-house which obtains the homogenizedstiffness matrix of the square unit cell as a function of relativedensity Figure 5 shows the results which are first obtained at dis-crete points of relative density To obtain the continuous functionsfor the properties we calculate through the least squares methodthe expressions (Table 1) for two ranges of relative density ieq lt 03 and q gt 03 These functions enable to assign the valuesof the stiffness for a given relative density to each of the samplepoints modeled in the FE solver We note that the expressions ofYoungrsquos moduli and Poissonrsquos ratios in the x1 and x2 directions donot change since the cell thickness is uniform

Once the stress and strain regimes of the cellular material havebeen calculated the non-dominated sorting genetic (NSGA-II)

algorithm [65] is employed to solve the multiobjective optimiza-tion problem described in Sec 22 The strain energy within thebone and the stress distribution at the bone-implant interface isthen calculated and used in Eqs (7) and (8) to evaluate the objec-tive functions The initial population is then sorted based on thenondomination front criterion A population of solutions calledparents are selected from the current population based on theirrank and crowding distance Then genetic operators are appliedto the population of parents to create a population of off-springsFinally the next population is produced by taking the best solu-tions from the combined population of parents and off-springsThe optimization continues until the user-defined number of func-tion evaluations reaches 25000 [65] The computational costrequired to run the optimization process in a single 24 GHz Intelprocessor with 4 GB RAM was about 300000 CPU s about threeand a halfdays We plan to use parallel computing with a PC clus-ter in future work this will considerably reduce the computationaltime since each function evaluation can be performedindependently

For each point in the objective function space the stress recov-ery procedure is applied to verify whether the stresses are admissi-ble To apply this procedure the average macroscopic straininside each unit cell is found The position of each unit cell withinthe implant is obtained after imposing a proper cell tessellationwhich in this work has been set to be uniform The size of the unitcell is selected as small as possible to capture the relative distribu-tion contour with higher resolution For a relative density of 01the square cell sizes for the cell wall thickness of either 70 or 100lm have been selected respectively to be 136 and 18 mm Oncethe position of the unit cells has been obtained 3 3 Gauss pointsare assigned to each cell The values of relative density and mac-roscopic strain at these points are obtained from the relative den-sity distribution and macroscopic strain field For Gauss pointslocated outside the implant border the values are linearly extrapo-lated from those located at the neighboring points inside theimplant domain Using a Gaussian quadrature integration [66] theaverage relative density and macroscopic strain of each cell arecalculated The local stress distribution and the yield safety factorof each cell are obtained through the von Mises stress criterionThe procedure is applied to all unit cells of the selected optimaldesign located on the Pareto frontier and the minimum local safetyfactor of a cell is specified as design safety factor

4 Results and Discussion

The advantage of multiobjective optimization with a posterioriarticulation of preference is that a set of optimum solutions areavailable without requiring the designer to choose in advance anyweighting factors to the objective functions Once the whole set ofPareto solutions has been determined the designer has the free-dom to select the desired solution based on the importance ofbone mass preservation relative to the amount of interface stressFigure 6 shows all the optimum solutions ie the relative densitydistribution for a hip stem implant with graded cellular materialThe x axis represents the amount of bone resorption for theimplanted hip on the y axis is the interface failure index Amongthe optimal solutions we examine three representative relativedensity distributions the extreme points A and C of the Paretofrontier for which one objective function has importance factor 0and the other 100 and a solution B characterized by a 50weight factor For each solution Fig 6 gives the following per-formance metrics bone resorption (mr) interface failure index(FethbTHORN) maximum interface failure (f ethrTHORNmax) average porosity ofeach stem ( ) and design safety factor (SF) after implementingthe stress recovery procedure The maximum interface failuref ethrTHORNmax is included since FethbTHORN which quantifies only the overalleffect of the implant stiffness on the interface stresses is not suffi-cient to provide information on the probability of failure

As seen from the performance metrics in Fig 6 the porosity ofsolutions A B and C is greater than 40 which is satisfactory

Fig 5 Effective Youngrsquos modulus of 2D square lattice versusrelative density Solution points obtained through homogeniza-tion theory are fitted with the least squares method

Table 1 Effective Mechanical Properties of the Square UnitCell as a Function of Relative Density

q lt 03 03 lt q lt 1

E1

Esfrac14 E2

Es058ethqTHORN1046

127ethqTHORN2 052ethqTHORN thorn 023

G

Es0093ethqTHORN307

126ethqTHORN3 133ethqTHORN2 thorn 051ethqTHORN 0065

21

sfrac14 12

s07ethqTHORN105

068ethqTHORN2 thorn 028ethqTHORN thorn 006

031004-6 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

for bone ingrowth [41] By comparing implants C and A weobserve that a raise of the implant porosity from point C to Aresults in an implant stiffness decrease which on one hand low-ers bone loss and on the other hand enhances the risk of interfacefailure When solution B is compared to C a reduction of 8 ofbone resorption is noted with a slight increase of the peak value ofthe interface failure On the other hand by contrasting solution Bto A we note a significant increase (60) of the peak value ofinterface failure which is below the Hoffman failure strength anda minor reduction (2) of the amount of bone resorption Themain benefit of solution A is the maximum porosity of the micro-structure that can promote bone ingrowth Solution B on the otherhand might be the preferred solution with respect to low boneresorption and interface failure These observations emerge onlyby inspection of the objective functions selected in this work weremark here that other parameters should be taken into accountfor the selection of the best implant These include patientrsquos bone

characteristics the range of activity age and desired level ofbone mass preservation after implantation

For prescribed geometric loading and constraint conditions wenow compare the metrics of resorbed bone mass (mr) and distribu-tion of interface stress (f ethrbTHORN) of the optimal solution B with thoseof (i) a currently-used fully dense titanium stem and (ii) a cellularimplant with a uniformly distributed relative density of 50 Fig-ures 7 and 8 illustrate the results of the comparison

For the solid titanium stem the amount of bone resorption cal-culated through Eq (7) is 67 and the interface failure indexFethbTHORN obtained from Eq (8) is 133 Using the distribution of f ethrTHORNgenerated around the titanium stem (Fig 8(a)) we observe thatthe maximum value of interface failure (051) occurs at the distalend of the implant As expected this implant is much stiffer thanthe surrounding bone thereby resulting in a higher amount ofbone resorption For the numerical validation the interface shearstress of the titanium implant at the proximal region is also

Fig 6 Trade-off distributions of relative density for the optimized cellular implant

Fig 7 Distribution of bone resorption around (a) fully dense titanium implant (b) cellularimplant with uniform relative density of 50 (c) graded cellular implant (solution B in Fig 6)

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-7

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

compared with the one obtained by Kowalczyk for a 3D model[4] The mean and the maximum values of interface shear stressfor the 3D titanium implant in the work by Kowalczyk [4] are057 and 28 MPa respectively These values are respectively031 and 215 MPa for the titanium implant in this paper The con-tribution to the higher level of shear stress in the 3D model ofKowalczyk is the distribution of shear force on a smaller area InKowalczykrsquos study [4] the implant and bone are bonded only atthe proximal region while in our work the whole bone-implantinterface is bonded which results in a decrease of the mean andthe maximum values of interface shear stress

The cellular implant with uniform relative density of 50 isapproximately three times more flexible than the titanium stemThis implant can qualitatively simulate the behavior of an implantmade out of tantalum foam For this stem the amount of boneresorption and the interface failure index are about 34 and 287respectively and the interface failure is maximum (071) at theedge of proximal region Compared to the solid titanium implantthe amount of bone resorption decreases by 50 whereas themaximum interface failure increases about 40 This shows that adecrease of the implant stiffness with uniform porosity distribu-tion aiming at reducing bone resorption has the undesirable effectof increasing the risk of interface failure at the proximal regionThis outcome confirms the findings of the previous work byKuiper and Huiskes [14]

Figure 7(c) and 8(c) show the results for the graded cellularimplant B Its bone resorption and interface failure index are 16and 115 respectively The peak value of the local interface fail-ure is 025 Compared to the titanium stem both the amount ofbone resorption and the peak of interface failure decrease of 76and 50 respectively With respect to the uniformly-distributedcellular implant the decrease of bone resorption and interface fail-ure peak is of 53 and 65 respectively A graded cellularimplant with optimized relative density distribution is thus capa-ble of reducing concurrently both the conflicting objective func-tions In particular bone resorption reduces as a result of thecellular material which makes the implant more compliant theinterface stress on the other hand is minimized by the optimizedgradients of cellular material

A proof-of-concept implant was built to verify the manufactur-ability of the optimum grade lattice material Figure 9 shows thepolypropylene prototype of solution B which was manufacturedwith a 3D printer Objet Connex500 [67] A uniform tessellationand a square unit cell of 18 mm size were assumed to draw themodel The cell geometry was calculated from the average rela-tive density obtained from the method described in this paper AnSTL file of the graded cellular implant solution B was finallyused for rapid prototyping

The limitations of the method proposed in this paper are herediscussed First the method tackles the design of an implant stati-cally loaded in the stance phase of walking Further research isrequired to extend it to variable loading and fatigue life design

Second the accuracy of the asymptotic homogenization needsto be investigated at the vicinity of the implant borders Althoughmacroscopic mean stress fields on the boundaries are satisfied inthe global solutions [4068] microscopic stresses do not satisfythe Y-periodicity assumption These boundary effects howevermay be captured by applying a boundary layer corrector [6869] aspatially decaying stress localization function [70] or an adaptivemultiscale methodology [71] To investigate the accuracy of thesetechniques it is required to construct a detailed finite elementmodel and compare the actual stressstrain distribution inside thecell walls with the microscopic stressesstrains estimated from thehomogenization method This task will be performed in future

Third to extend the proposed procedure to three dimensions a3D cell topology with high porosity interconnected pore struc-ture large surface area with suitable textures and good

Fig 8 Distribution of local interface failure f ethrTHORN around (a) fully dense titanium implant(b) cellular implant with uniform relative density of 50 (c) graded cellular implant (solu-tion B in Fig 6)

Fig 9 Polypropylene proof-of-concept of the optimal gradedcellular implant (solution B)

031004-8 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

mechanical properties should be selected [7273] The asymptotichomogenization will be applied to characterize tissue scaffoldingcell topologies for a wide range of relative density Their effectivemoduli yield and ultimate surfaces for multiaxial loading condi-tions will be obtained to create a standard library from which toselect suitable cell geometries for a 3D cellular implant design

Fourth bone has been here considered as consisting of a cancel-lous and a cortical part with isotropic mechanical properties Fur-ther study is necessary to improve the constitutive material modelof bone and to consider bone as a graded cellular material

Finally in the optimization procedure a simple formulation hasbeen used to quantitatively measure both bone resorption andimplant stability In a future study detailed simulations of implantstability [74ndash77] and bone remodeling process [6077ndash80] can beperformed to assess both the short and long term performance ofthe implant

5 Concluding Remarks

This work has presented a methodology integrating multiscaleanalysis and design optimization to design a novel hip implantmade of graded cellular material The method can contribute tothe development of a new generation of orthopedic implants witha graded cellular microstructure that will reduce the clinical con-sequences of current implants

In the first part of the paper the homogenization method hasbeen reviewed and then used to capture the mechanics of theimplant at the micro and macro scale In the second part multiob-jective optimization has been applied to find optimum gradients ofmaterial distribution that minimize concurrently bone resorptionand bone-implant interface stresses The results have shown thatthe optimized cellular implant exhibits a reduction of 76 ofbone resorption and 50 of interface stress with respect to a fullydense titanium implant

References[1] Kurtz S Ong K Lau E Mowat F and Halpern M 2007 ldquoProjections of

Primary and Revision Hip and Knee Arthroplasty in the United States from2005 to 2030rdquo J Bone Jt Surg Am Vol 89(4) pp 780ndash785

[2] Vichinsky E P Neumayr L D Haberkern C Earles A N Eckman JKoshy M and Black D M 1999 ldquoThe Perioperative Complication Rate ofOrthopedic Surgery in Sickle Cell Disease Report of the National Sickle CellSurgery Study Grouprdquo Am J Hematol 62(3) pp 129ndash138

[3] Kobayashi S Saito N Horiuchi H Iorio R and Takaoka K 2000 ldquoPoorBone Quality or Hip Structure as Risk Factors Affecting Survival of Total-HipArthroplastyrdquo The Lancet 355(9214) pp 1499ndash1504

[4] Kowalczyk P 2001 ldquoDesign Optimization of Cementless Femoral Hip Pros-theses Using Finite Element Analysisrdquo J Biomech Eng 123(5) pp 396ndash402

[5] Moen T C Ghate R Salaz N Ghodasra J and Stulberg S D 2011 ldquoAMonoblock Porous Tantalum Acetabular Cup Has No Osteolysis on Ct at 10Yearsrdquo Clin Orthop Relat Res 469(2) pp 382ndash386

[6] Kurtz S Gawel H and Patel J 2011 ldquoHistory and Systematic Review ofWear and Osteolysis Outcomes for First-Generation Highly Crosslinked Poly-ethylenerdquo Clin Orthop Relat Res 469(8) pp 2262ndash2277

[7] Grubl A Marker M Brodner W Giurea A Heinze G Meisinger VZehetgruber H and Kotz R 2007 ldquoLong Term Follow up of Metal on MetalTotal Hip Replacementrdquo J Orthop Res 25(7) pp 841ndash848

[8] Neumann D R P Thaler C Hitzl W Huber M Hofstadter T and DornU 2010 ldquoLong-Term Results of a Contemporary Metal-on-Metal Total HipArthroplasty A 10-Year Follow-up Studyrdquo J Arthroplasty 25(5) pp700ndash708

[9] Glassman A Bobyn J and Tanzer M 2006 ldquoNew Femoral Designs DoThey Influence Stress Shieldingrdquo Clin Orthop Relat Res 453(12) pp64ndash74

[10] Adam F Hammer D S Pfautsch S and Westermann K 2002 ldquoEarly Fail-ure of a Press-Fit Carbon Fiber Hip Prosthesis with a Smooth Surfacerdquo JAntroplasty 17(2) pp 217ndash223

[11] Trebse R Milosev I Kovac S Mikek M and Pisot V 2005 ldquoPoorResults from the Isoelastic Total Hip Replacementrdquo Acta Orthop 76(2) pp169ndash176

[12] Harvey E Bobyn J Tanzer M Stackpool G Krygier J and Hacking S1999 ldquoEffect of Flexibility of the Femoral Stem on Bone-Remodeling and Fix-ation of the Stem in a Canine Total Hip Arthroplasty Model without CementrdquoJ Bone Jt Surg 81(1) pp 93ndash107

[13] Huiskes R Weinans H and Rietbergen B 1992 ldquoThe Relationship betweenStress Shielding and Bone Resorption around Total Hip Stems and the Effectsof Flexible Materialsrdquo Clin Orthop Relat Res 274(1) pp 124ndash134

[14] Kuiper J and Huiskes R 1992 ldquoNumerical Optimization of Hip-ProstheticStem Materialrdquo Recent Advances in Computer Methods in Biomechanics andBiomedical Engineering J Middleton G N Pande and K R Williams edsBooks and Journals International Ltd Swansea pp 76ndash84

[15] Kuiper J H and Huiskes R 1997 ldquoMathematical Optimization of ElasticProperties Application to Cementless Hip Stem Designrdquo J Biomech Eng119(2) pp 166ndash174

[16] Hedia H Shabara M El-Midany T and Fouda N 2006 ldquoImproved Designof Cementless Hip Stems Using Two-Dimensional Functionally Graded Materi-alsrdquo J Biomed Mater Res Part B Appl Biomater 79(1) pp 42ndash49

[17] Hedia H S and Mahmoud N-A 2004 ldquoDesign Optimization of Function-ally Graded Dental Implantrdquo Biomed Mater Eng 14(2) pp 133ndash143

[18] Watari F Yokoyama A Saso F Uo M and Kawasaki T 1997ldquoFabrication and Properties of Functionally Graded Dental Implantrdquo Compo-sites Part B 28(1-2) pp 5ndash11

[19] Katti K S 2004 ldquoBiomaterials in Total Joint Replacementrdquo Colloids Surf B39(3) pp 133ndash142

[20] Thompson I and Hench L 1998 ldquoMechanical Properties of BioactiveGlasses Glass-Ceramics and Compositesrdquo Proc Inst Mech Eng Part H JEng Med 212(2) pp 127

[21] Fraldi M Esposito L Perrella G Cutolo A and Cowin S 2010ldquoTopological Optimization in Hip Prosthesis Designrdquo Biomech ModelMechanobiol 9(4) pp 389ndash402

[22] Bobyn J D Poggie R Krygier J Lewallen D Hanssen A Lewis RUnger A Orsquokeefe T Christie M and Nasser S 2004 ldquoClinical Validationof a Structural Porous Tantalum Biomaterial for Adult Reconstructionrdquo J BoneJt Surg 86(Supplement 2) pp 123ndash129

[23] Bobyn J Stackpool G Hacking S Tanzer M and Krygier J 1999ldquoCharacteristics of Bone Ingrowth and Interface Mechanics of a New PorousTantalum Biomaterialrdquo J Bone Jt Surg Br Vol 81(5) pp 907ndash914

[24] Parthasarathy J Starly B Raman S and Christensen A 2010 ldquoMechanicalEvaluation of Porous Titanium (Ti6al4v) Structures with Electron Beam Melt-ing (Ebm)rdquo J Mech Behav Biomed Mater 3(3) pp 249ndash259

[25] Heinl P Muller L Korner C Singer R F and Muller F A 2008ldquoCellular Ti-6al-4v Structures with Interconnected Macro Porosity for BoneImplants Fabricated by Selective Electron Beam Meltingrdquo Acta Biomater4(5) pp 1536ndash1544

[26] Stamp R Fox P Orsquoneill W Jones E and Sutcliffe C 2009 ldquoThe Devel-opment of a Scanning Strategy for the Manufacture of Porous Biomaterials bySelective Laser Meltingrdquo J Mater Sci Mater Med 20(9) pp 1839ndash1848

[27] Yang S Leong K Du Z and Chua C 2002 ldquoThe Design of Scaffolds forUse in Tissue Engineering Part Ii Rapid Prototyping Techniquesrdquo TissueEng 8(1) pp 1ndash11

[28] Murr L Gaytan S Medina F Lopez H Martinez E Machado B Hernan-dez D Martinez L Lopez M and Wicker R 2010 ldquoNext-Generation Biomed-ical Implants Using Additive Manufacturing of Complex Cellular and FunctionalMesh Arraysrdquo Philos Trans R Soc London Ser A 368(1917) pp 1999ndash2032

[29] Coelho P Fernandes P Guedes J and Rodrigues H 2008 ldquoA HierarchicalModel for Concurrent Material and Topology Optimisation of Three-Dimensional Structuresrdquo Struct Multidiscip Optim 35(2) pp 107ndash115

[30] Rodrigues H Guedes J and Bendsoe M 2002 ldquoHierarchical Optimizationof Material and Structurerdquo Struct Multidiscip Optim 24(1) pp 1ndash10

[31] Goncalves Coelho P Rui Fernandes P and Carrico Rodrigues H 2011ldquoMultiscale Modeling of Bone Tissue with Surface and Permeability ControlrdquoJ Biomech 44(2) pp 321ndash329

[32] Coelho P G Cardoso J B Fernandes P R and Rodrigues H C 2011ldquoParallel Computing Techniques Applied to the Simultaneous Design of Struc-ture and Materialrdquo Adv Eng Software 42(5) pp 219ndash227

[33] Gibson L J and Ashby M F 1999 Cellular Solids Structure and Proper-ties Cambridge University Press Cambridge UK

[34] Guedes J and Kikuchi N 1990 ldquoPreprocessing and Postprocessing for Mate-rials Based on the Homogenization Method with Adaptive Finite ElementMethodsrdquo Comput Methods Appl Mech Eng 83(2) pp 143ndash198

[35] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization I-Homogenization Theory for Media with PeriodicStructurerdquo Comput Struct 69(6) pp 707ndash717

[36] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Opimization I I-Analytical and Numerical Solution of HomogenizationEquationsrdquo Comput Struct 69(6) pp 719ndash738

[37] Fang Z Starly B and Sun W 2005 ldquoComputer-Aided Characterization forEffective Mechanical Properties of Porous Tissue Scaffoldsrdquo Comput-AidedDes 37(1) pp 65ndash72

[38] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics 6th ed Butterworth-Heinemann Burlington

[39] Hollister S and Kikuchi N 1992 ldquoA Comparison of Homogenization andStandard Mechanics Analyses for Periodic Porous Compositesrdquo ComputMech 10(2) pp 73ndash95

[40] Pellegrino C Galvanetto U and Schrefler B 1999 ldquoNumerical Homogeni-zation of Periodic Composite Materials with Non Linear MaterialComponentsrdquo Int J Numer Methods Eng 46(10) pp 1609ndash1637

[41] Bragdon C Jasty M Greene M Rubash H and Harris W 2004ldquoBiologic Fixation of Total Hip Implants Insights Gained from a Series of Ca-nine Studiesrdquo J Bone Jt Surg 86(Supplement 2) pp 105ndash117

[42] Harrysson O L A Cansizoglu O Marcellin-Little D J Cormier D R andWest Ii H A 2008 ldquoDirect Metal Fabrication of Titanium Implants with Tai-lored Materials and Mechanical Properties Using Electron Beam Melting Tech-nologyrdquo Mater Sci Eng C 28(3) pp 366ndash373

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-9

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

by structural analysis [3345ndash50] In these studies the effectivemoduli and yield strength of a cellular material are generally mod-eled by assuming that the cell walls flex like beams Despite thesimplicity of this method in calculating the overall mechanicalproperties the results are reliable only if the relative density islower than 03 [47] Furthermore the actual stress distributionwithin the unit cell walls cannot be captured As an alternativemethod the asymptotic homogenization method is used in thiswork to deal with the multiscale analysis of the cellular implantThis technique is applied to calculate the homogeneous stiffnessmatrix of the unit cell for different values of relative density as wellas to determine the microscopic stresses and strains [343839] Theasymptotic homogenization method has been widely used in topol-ogy optimization [3454ndash58] and hierarchical design of materialsand structures [29ndash32] In this work unlike hierarchical topologyoptimization a predefined unit cell topology with parametric geom-etry is considered as the microstructure of the implant then theoptimization algorithm searches for the optimum unit cell geometryof the lattice to minimize the antagonist objective functions under aset of constraints This procedure is similar to the one developed byBendsoslashe and Kikuchi [58] and is not limited to any cell topologywe selected the hollow square cell as an example to demonstratethe methodology A brief explanation about the asymptotic homog-enization technique and the main steps to calculate the effectivestiffness and microscopic stress and strain are provided below

The approach assumes that each field quantity depends on twodifferent scales one on the macroscopic level x and the other onthe microscopic level yfrac14 xe where e is a magnification factorthat enlarges the dimensions of a unit cell to make it comparablewith the dimensions of the material The field quantities such asdisplacement stress and strain are assumed to vary smoothly atthe macroscopic level being periodic at the microscale [35ndash39]The effective properties of the periodic material are determinedthrough the solution of local problems formulated on the represen-tative volume element (RVE) of the material As a result the ho-mogenized stiffness tensor (EH

ijkl) of a cellular material may bedefined as follow [39]

EHijkl frac14

1

Yj j

ethYC

EijpmMpmkldY (2)

where Yj j is the volume of the entire unit cell with void Yc is thesolid part of the cell Eijkl is the local elasticity tensor that dependson the position within the representative volume element ie Eijkl

is equal to the elasticity tensor of the material located in the cellwalls and it vanishes in the voids Mijkl is the local structure ten-sor which relates the macroscopic strains (e) to the local or micro-structural strains (e) through the relation

eij frac14 Mijklekl (3a)

Mijkl frac141

2ethdikdjl thorn dildjkTHORN ekl

ij (3b)

where dij is the Kronecker f and eklij is the microstructural strain

corresponding to the component kl of the macroscopic strain ten-sor (ekl) ekl

ij is the solution of the following equation

ethYC

Eijpme1ijethvTHORNekl

pmethuTHORNdY frac14eth

YC

Eijkle1ijethvTHORNekldY (4)

where e1ijethvTHORN is the virtual strain In general ekl can be an arbitrary

macroscopic strain tensor Considering the assumption of smalldeformation and linear material behavior ekl may be written as alinear combination of unit strains For a two-dimensional case theunit strains are defined as

e11 frac14 1 0 0frac12 T e22 frac14 0 1 0frac12 T e12 frac14 0 0 1frac12 T (5)

To calculate the effective mechanical properties of a cellular ma-terial the first task is to obtain the matrix Mijkl After discretizing

the RVE domain the unit strains are applied to each element ofthe FE model Periodicity of the strain field is ensured by impos-ing periodic boundary conditions on the RVE edges The directmethod [59] is selected here to derive periodic boundary condi-tions The microscopic strain field (ekl

ij ) inside the RVE isobtained by solving Eq (4) The results are substituted into Eq(3b) to calculate the local structure tensor Mijkl for each elementof the RVE Finally the effective stiffness tensor EH

ijkl is obtainedby calculating Eq (2) Once the local structure tensor Mijkl isobtained the microscopic strains and stresses corresponding tothe macroscopic strain can be obtained via Eq (3a) and the consti-tutive equation of the cell wall material

The steps described above are used to compute the homoge-nized stiffness tensor for each unit cell of the cellular hip implant(Fig 1) These tensors are used to construct the global stiffnessmatrix for the FE solver to obtain macroscopic stress and straindistribution within bone and implant The values are then post-processed to evaluate the objective functions of the multiobjectiveoptimization problem whose formulation is described in the fol-lowing section

22 Formulation of the Multiobjective OptimizationProblem For the design of an optimum implant we impose thesimultaneous minimization of the amount of bone loss around theprosthesis and the probability of mechanical failure at the bone-implant interface As illustrated in Fig 1 the multiobjective opti-mization problem can be formulated as

Minimize mrethbTHORNFethbTHORN

bone loss

interface failure

Subject to

ethbTHORN 4050lm P 800lm

t tmin

average porosity

meanpore size

cell wall thickness

(6)

The amount of bone loss around the stem is determined by assess-ing the amount of bone that is underloaded Bone can be consid-ered locally underloaded when its local strain energy (Ui) per unit

of bone mass (q) averaged over n loading cases (S frac14 1n

Pnifrac141

Ui

q )

is beneath the local reference value Sref which is the value of Swhen no prosthesis is present However it has been observed thatnot all the underloading leads to resorption rather a certain frac-tion of underloading (the threshold level or dead zone s) is toler-ated In fact bone resorption starts when the local value of S isbeneath the value of eth1 sTHORNSref [1460] Using this definition theresorbed bone mass fraction mr can be obtained from

mrethbTHORN frac141

M

ethV

gethSethbTHORNTHORNqdV (7)

where M and V are the original bone mass and volume respec-tively and gethSethbTHORNTHORN is a resorptive function equal to unity if thelocal value of S is beneath the local value of eth1 sTHORNSref and equalto 0 if eth1 sTHORNSref lt S In this study the value of the dead zone sis assumed to be 05 [14]

The other objective is to minimize the probability of local inter-face failure which is expressed by the following functional of theinterface stress distribution

FethbTHORN frac14 1

n

Xn

ifrac141

ethP

f ethrbi THORNdP (8)

where FethbTHORN is the global interface function index rbi is the inter-

face stress at the loading case i depending on the design variableb P is the interface area and f ethrb

i THORN is the local interface stressfunction which is defined based on the multi-axial Hoffman fail-ure criterion [61] This function is used to determine where local

031004-4 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

debonding might occur along the bone-implant interface [1462]The probability of local interface failure f ethrTHORN is given by

f ethrTHORN frac14 1

StScr2

n thorn1

St 1

Sc

rn thorn

1

S2s

s2 (9)

where St and Sc are the uniaxial tensile and compressive strengthsrespectively Ss is the shear strength and rn and s are normal andshear stresses at the bone-implant interface respectively Forf ethrTHORN 1 a high probability of failure is expected whereas forf ethrTHORN 1 the risk of interface failure is low Tensile compressionand shear strengths of bone can be expressed as a function of bonedensity following the power law relations obtained by Pal et al[62]

St frac14 145q171 Sc frac14 324q185 Ss frac14 216q165 (10)

The distribution of bone density can be obtained through a CT-scan data of bone and then used in Eq (10) to find the effectivemechanical properties of the bone from which the local interfacefailure can be determined via Eq (9) Finally the interface failureindex FethbTHORN is evaluated by means of Eq (8) It should be notedthat FethbTHORN serves as a qualitative indicator only lower values ofFethbTHORN indicate reduced probability of local interface failure in theimplant

The expressions of bone resorption after implantation and me-chanical failure of bone-implant interface described in this section

are used in the finite element analysis to evaluate the objectivefunctions to be optimized The following sections shows the appli-cation of the method for the design of a two dimensional cellularimplant

3 Application of the Methodology to design a 2D

Femoral Implant with a Graded Cellular Material

31 FEM model of the Femur at the Macroscale The lefthand side of Fig 3 shows the geometry of the left femur consid-ered in this work along with the appropriate loads and boundaryconditions These data have been obtained from the work ofKuiper and Huiskes [14] The 3D geometry of the femur is simpli-fied into a 2D model where the thickness of the stem and bonevaries such that the second moment of area about the out-of-planeaxis does not differ in both models [16] Furthermore in this pa-per the implant material is designed to be an open cell lattice toease bone in growth in the implanted stem and obtain a full bondAlthough bone ingrowth does not exist in a postoperative situa-tion it can appear later if local mechanical stability is guaranteedThe minimization of interface stress reduces the possibility ofoccurrence of interface micromotion and instability [4] There-fore to decrease the computational cost required by a stabilityanalysis based on a nonlinear frictional contact model the pros-thesis and the surrounding bone are assumed fully bonded

The load case represents daily static loading during the stancephase of walking [63] The distal end of the femur is fixed toavoid rigid body motion For the material properties of the modelwe consider 20 GPa as the Youngrsquos modulus of the cortical boneand 15 GPa for the proximal bone The Poissonrsquos ratio is set to be03 [14]

32 FEM model of the Cellular Implant at theMicroscale The right hand side of Fig 3 illustrates the model ofa cementless prosthesis implanted into the human femur The griddepicts the domain of the implant to be designed with a lattice ma-terial of graded properties Figure 4 shows the unit cell geometryused for the tessellation of the whole implant The gradients ofmaterial properties are governed by the lattice relative densitywhich is a variable controlled by the cell size and wall thicknessof the hollow square In future work the performance of alterna-tive cell topologies suitable for bone tissue scaffolding will beinvestigated as well as combinations of dissimilar unit cells in thetessellation [64]

For the material property of the implant we consider Ti6Al4V[24] which is a biocompatible material commonly used in EBMIts mechanical properties are the following 900 MPa for the yieldstrength of the solid material 120 GPa for Youngrsquos modulus and03 for Poissonrsquos ratio

33 Design of the Graded Cellular Material of theImplant The procedure introduced in this paper for the design ofa graded cellular implant requires both multiscale analysis andmultiobjective optimization as described in Secs 21 and 22 andshown in Fig 1 The variables of the lattice model are the relativedensities attributed to 130 sampling points 26 rows along the

Fig 3 2D Finite element models of the femur (left) and theprosthesis implanted into the femur (right)

Fig 4 2D hollow square unit cell for given values of relative density

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-5

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

prosthetic length and 5 columns along the radial direction asshown in the right side of Fig 3 The number of sampling pointshas been chosen to be 130 to limit the computational time requiredfor the analysis and optimization while providing a reasonable re-solution for the relative density distribution For a more refined den-sity distribution the number of sample points can be increasedTheir values have been constrained in the range 01 q 1 [47]to prevent elastic buckling from occurring in the unit cell prior toyielding The values of the relative density between the samplingpoints are obtained through linear interpolation Although not con-sidered in the current research the shape of the implant could beincluded in the vector b as a design variable (Fig 1)

To calculate the stress and strain regime of the implant thesteps explained in Sec 21 are followed Initial values of relativedensity are assigned to each element of the FE model created inANSYS (Canonsburg Pennsylvania USA) The stiffness matrixof each element is calculated through a computational code2DHOMOG developed in-house which obtains the homogenizedstiffness matrix of the square unit cell as a function of relativedensity Figure 5 shows the results which are first obtained at dis-crete points of relative density To obtain the continuous functionsfor the properties we calculate through the least squares methodthe expressions (Table 1) for two ranges of relative density ieq lt 03 and q gt 03 These functions enable to assign the valuesof the stiffness for a given relative density to each of the samplepoints modeled in the FE solver We note that the expressions ofYoungrsquos moduli and Poissonrsquos ratios in the x1 and x2 directions donot change since the cell thickness is uniform

Once the stress and strain regimes of the cellular material havebeen calculated the non-dominated sorting genetic (NSGA-II)

algorithm [65] is employed to solve the multiobjective optimiza-tion problem described in Sec 22 The strain energy within thebone and the stress distribution at the bone-implant interface isthen calculated and used in Eqs (7) and (8) to evaluate the objec-tive functions The initial population is then sorted based on thenondomination front criterion A population of solutions calledparents are selected from the current population based on theirrank and crowding distance Then genetic operators are appliedto the population of parents to create a population of off-springsFinally the next population is produced by taking the best solu-tions from the combined population of parents and off-springsThe optimization continues until the user-defined number of func-tion evaluations reaches 25000 [65] The computational costrequired to run the optimization process in a single 24 GHz Intelprocessor with 4 GB RAM was about 300000 CPU s about threeand a halfdays We plan to use parallel computing with a PC clus-ter in future work this will considerably reduce the computationaltime since each function evaluation can be performedindependently

For each point in the objective function space the stress recov-ery procedure is applied to verify whether the stresses are admissi-ble To apply this procedure the average macroscopic straininside each unit cell is found The position of each unit cell withinthe implant is obtained after imposing a proper cell tessellationwhich in this work has been set to be uniform The size of the unitcell is selected as small as possible to capture the relative distribu-tion contour with higher resolution For a relative density of 01the square cell sizes for the cell wall thickness of either 70 or 100lm have been selected respectively to be 136 and 18 mm Oncethe position of the unit cells has been obtained 3 3 Gauss pointsare assigned to each cell The values of relative density and mac-roscopic strain at these points are obtained from the relative den-sity distribution and macroscopic strain field For Gauss pointslocated outside the implant border the values are linearly extrapo-lated from those located at the neighboring points inside theimplant domain Using a Gaussian quadrature integration [66] theaverage relative density and macroscopic strain of each cell arecalculated The local stress distribution and the yield safety factorof each cell are obtained through the von Mises stress criterionThe procedure is applied to all unit cells of the selected optimaldesign located on the Pareto frontier and the minimum local safetyfactor of a cell is specified as design safety factor

4 Results and Discussion

The advantage of multiobjective optimization with a posterioriarticulation of preference is that a set of optimum solutions areavailable without requiring the designer to choose in advance anyweighting factors to the objective functions Once the whole set ofPareto solutions has been determined the designer has the free-dom to select the desired solution based on the importance ofbone mass preservation relative to the amount of interface stressFigure 6 shows all the optimum solutions ie the relative densitydistribution for a hip stem implant with graded cellular materialThe x axis represents the amount of bone resorption for theimplanted hip on the y axis is the interface failure index Amongthe optimal solutions we examine three representative relativedensity distributions the extreme points A and C of the Paretofrontier for which one objective function has importance factor 0and the other 100 and a solution B characterized by a 50weight factor For each solution Fig 6 gives the following per-formance metrics bone resorption (mr) interface failure index(FethbTHORN) maximum interface failure (f ethrTHORNmax) average porosity ofeach stem ( ) and design safety factor (SF) after implementingthe stress recovery procedure The maximum interface failuref ethrTHORNmax is included since FethbTHORN which quantifies only the overalleffect of the implant stiffness on the interface stresses is not suffi-cient to provide information on the probability of failure

As seen from the performance metrics in Fig 6 the porosity ofsolutions A B and C is greater than 40 which is satisfactory

Fig 5 Effective Youngrsquos modulus of 2D square lattice versusrelative density Solution points obtained through homogeniza-tion theory are fitted with the least squares method

Table 1 Effective Mechanical Properties of the Square UnitCell as a Function of Relative Density

q lt 03 03 lt q lt 1

E1

Esfrac14 E2

Es058ethqTHORN1046

127ethqTHORN2 052ethqTHORN thorn 023

G

Es0093ethqTHORN307

126ethqTHORN3 133ethqTHORN2 thorn 051ethqTHORN 0065

21

sfrac14 12

s07ethqTHORN105

068ethqTHORN2 thorn 028ethqTHORN thorn 006

031004-6 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

for bone ingrowth [41] By comparing implants C and A weobserve that a raise of the implant porosity from point C to Aresults in an implant stiffness decrease which on one hand low-ers bone loss and on the other hand enhances the risk of interfacefailure When solution B is compared to C a reduction of 8 ofbone resorption is noted with a slight increase of the peak value ofthe interface failure On the other hand by contrasting solution Bto A we note a significant increase (60) of the peak value ofinterface failure which is below the Hoffman failure strength anda minor reduction (2) of the amount of bone resorption Themain benefit of solution A is the maximum porosity of the micro-structure that can promote bone ingrowth Solution B on the otherhand might be the preferred solution with respect to low boneresorption and interface failure These observations emerge onlyby inspection of the objective functions selected in this work weremark here that other parameters should be taken into accountfor the selection of the best implant These include patientrsquos bone

characteristics the range of activity age and desired level ofbone mass preservation after implantation

For prescribed geometric loading and constraint conditions wenow compare the metrics of resorbed bone mass (mr) and distribu-tion of interface stress (f ethrbTHORN) of the optimal solution B with thoseof (i) a currently-used fully dense titanium stem and (ii) a cellularimplant with a uniformly distributed relative density of 50 Fig-ures 7 and 8 illustrate the results of the comparison

For the solid titanium stem the amount of bone resorption cal-culated through Eq (7) is 67 and the interface failure indexFethbTHORN obtained from Eq (8) is 133 Using the distribution of f ethrTHORNgenerated around the titanium stem (Fig 8(a)) we observe thatthe maximum value of interface failure (051) occurs at the distalend of the implant As expected this implant is much stiffer thanthe surrounding bone thereby resulting in a higher amount ofbone resorption For the numerical validation the interface shearstress of the titanium implant at the proximal region is also

Fig 6 Trade-off distributions of relative density for the optimized cellular implant

Fig 7 Distribution of bone resorption around (a) fully dense titanium implant (b) cellularimplant with uniform relative density of 50 (c) graded cellular implant (solution B in Fig 6)

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-7

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

compared with the one obtained by Kowalczyk for a 3D model[4] The mean and the maximum values of interface shear stressfor the 3D titanium implant in the work by Kowalczyk [4] are057 and 28 MPa respectively These values are respectively031 and 215 MPa for the titanium implant in this paper The con-tribution to the higher level of shear stress in the 3D model ofKowalczyk is the distribution of shear force on a smaller area InKowalczykrsquos study [4] the implant and bone are bonded only atthe proximal region while in our work the whole bone-implantinterface is bonded which results in a decrease of the mean andthe maximum values of interface shear stress

The cellular implant with uniform relative density of 50 isapproximately three times more flexible than the titanium stemThis implant can qualitatively simulate the behavior of an implantmade out of tantalum foam For this stem the amount of boneresorption and the interface failure index are about 34 and 287respectively and the interface failure is maximum (071) at theedge of proximal region Compared to the solid titanium implantthe amount of bone resorption decreases by 50 whereas themaximum interface failure increases about 40 This shows that adecrease of the implant stiffness with uniform porosity distribu-tion aiming at reducing bone resorption has the undesirable effectof increasing the risk of interface failure at the proximal regionThis outcome confirms the findings of the previous work byKuiper and Huiskes [14]

Figure 7(c) and 8(c) show the results for the graded cellularimplant B Its bone resorption and interface failure index are 16and 115 respectively The peak value of the local interface fail-ure is 025 Compared to the titanium stem both the amount ofbone resorption and the peak of interface failure decrease of 76and 50 respectively With respect to the uniformly-distributedcellular implant the decrease of bone resorption and interface fail-ure peak is of 53 and 65 respectively A graded cellularimplant with optimized relative density distribution is thus capa-ble of reducing concurrently both the conflicting objective func-tions In particular bone resorption reduces as a result of thecellular material which makes the implant more compliant theinterface stress on the other hand is minimized by the optimizedgradients of cellular material

A proof-of-concept implant was built to verify the manufactur-ability of the optimum grade lattice material Figure 9 shows thepolypropylene prototype of solution B which was manufacturedwith a 3D printer Objet Connex500 [67] A uniform tessellationand a square unit cell of 18 mm size were assumed to draw themodel The cell geometry was calculated from the average rela-tive density obtained from the method described in this paper AnSTL file of the graded cellular implant solution B was finallyused for rapid prototyping

The limitations of the method proposed in this paper are herediscussed First the method tackles the design of an implant stati-cally loaded in the stance phase of walking Further research isrequired to extend it to variable loading and fatigue life design

Second the accuracy of the asymptotic homogenization needsto be investigated at the vicinity of the implant borders Althoughmacroscopic mean stress fields on the boundaries are satisfied inthe global solutions [4068] microscopic stresses do not satisfythe Y-periodicity assumption These boundary effects howevermay be captured by applying a boundary layer corrector [6869] aspatially decaying stress localization function [70] or an adaptivemultiscale methodology [71] To investigate the accuracy of thesetechniques it is required to construct a detailed finite elementmodel and compare the actual stressstrain distribution inside thecell walls with the microscopic stressesstrains estimated from thehomogenization method This task will be performed in future

Third to extend the proposed procedure to three dimensions a3D cell topology with high porosity interconnected pore struc-ture large surface area with suitable textures and good

Fig 8 Distribution of local interface failure f ethrTHORN around (a) fully dense titanium implant(b) cellular implant with uniform relative density of 50 (c) graded cellular implant (solu-tion B in Fig 6)

Fig 9 Polypropylene proof-of-concept of the optimal gradedcellular implant (solution B)

031004-8 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

mechanical properties should be selected [7273] The asymptotichomogenization will be applied to characterize tissue scaffoldingcell topologies for a wide range of relative density Their effectivemoduli yield and ultimate surfaces for multiaxial loading condi-tions will be obtained to create a standard library from which toselect suitable cell geometries for a 3D cellular implant design

Fourth bone has been here considered as consisting of a cancel-lous and a cortical part with isotropic mechanical properties Fur-ther study is necessary to improve the constitutive material modelof bone and to consider bone as a graded cellular material

Finally in the optimization procedure a simple formulation hasbeen used to quantitatively measure both bone resorption andimplant stability In a future study detailed simulations of implantstability [74ndash77] and bone remodeling process [6077ndash80] can beperformed to assess both the short and long term performance ofthe implant

5 Concluding Remarks

This work has presented a methodology integrating multiscaleanalysis and design optimization to design a novel hip implantmade of graded cellular material The method can contribute tothe development of a new generation of orthopedic implants witha graded cellular microstructure that will reduce the clinical con-sequences of current implants

In the first part of the paper the homogenization method hasbeen reviewed and then used to capture the mechanics of theimplant at the micro and macro scale In the second part multiob-jective optimization has been applied to find optimum gradients ofmaterial distribution that minimize concurrently bone resorptionand bone-implant interface stresses The results have shown thatthe optimized cellular implant exhibits a reduction of 76 ofbone resorption and 50 of interface stress with respect to a fullydense titanium implant

References[1] Kurtz S Ong K Lau E Mowat F and Halpern M 2007 ldquoProjections of

Primary and Revision Hip and Knee Arthroplasty in the United States from2005 to 2030rdquo J Bone Jt Surg Am Vol 89(4) pp 780ndash785

[2] Vichinsky E P Neumayr L D Haberkern C Earles A N Eckman JKoshy M and Black D M 1999 ldquoThe Perioperative Complication Rate ofOrthopedic Surgery in Sickle Cell Disease Report of the National Sickle CellSurgery Study Grouprdquo Am J Hematol 62(3) pp 129ndash138

[3] Kobayashi S Saito N Horiuchi H Iorio R and Takaoka K 2000 ldquoPoorBone Quality or Hip Structure as Risk Factors Affecting Survival of Total-HipArthroplastyrdquo The Lancet 355(9214) pp 1499ndash1504

[4] Kowalczyk P 2001 ldquoDesign Optimization of Cementless Femoral Hip Pros-theses Using Finite Element Analysisrdquo J Biomech Eng 123(5) pp 396ndash402

[5] Moen T C Ghate R Salaz N Ghodasra J and Stulberg S D 2011 ldquoAMonoblock Porous Tantalum Acetabular Cup Has No Osteolysis on Ct at 10Yearsrdquo Clin Orthop Relat Res 469(2) pp 382ndash386

[6] Kurtz S Gawel H and Patel J 2011 ldquoHistory and Systematic Review ofWear and Osteolysis Outcomes for First-Generation Highly Crosslinked Poly-ethylenerdquo Clin Orthop Relat Res 469(8) pp 2262ndash2277

[7] Grubl A Marker M Brodner W Giurea A Heinze G Meisinger VZehetgruber H and Kotz R 2007 ldquoLong Term Follow up of Metal on MetalTotal Hip Replacementrdquo J Orthop Res 25(7) pp 841ndash848

[8] Neumann D R P Thaler C Hitzl W Huber M Hofstadter T and DornU 2010 ldquoLong-Term Results of a Contemporary Metal-on-Metal Total HipArthroplasty A 10-Year Follow-up Studyrdquo J Arthroplasty 25(5) pp700ndash708

[9] Glassman A Bobyn J and Tanzer M 2006 ldquoNew Femoral Designs DoThey Influence Stress Shieldingrdquo Clin Orthop Relat Res 453(12) pp64ndash74

[10] Adam F Hammer D S Pfautsch S and Westermann K 2002 ldquoEarly Fail-ure of a Press-Fit Carbon Fiber Hip Prosthesis with a Smooth Surfacerdquo JAntroplasty 17(2) pp 217ndash223

[11] Trebse R Milosev I Kovac S Mikek M and Pisot V 2005 ldquoPoorResults from the Isoelastic Total Hip Replacementrdquo Acta Orthop 76(2) pp169ndash176

[12] Harvey E Bobyn J Tanzer M Stackpool G Krygier J and Hacking S1999 ldquoEffect of Flexibility of the Femoral Stem on Bone-Remodeling and Fix-ation of the Stem in a Canine Total Hip Arthroplasty Model without CementrdquoJ Bone Jt Surg 81(1) pp 93ndash107

[13] Huiskes R Weinans H and Rietbergen B 1992 ldquoThe Relationship betweenStress Shielding and Bone Resorption around Total Hip Stems and the Effectsof Flexible Materialsrdquo Clin Orthop Relat Res 274(1) pp 124ndash134

[14] Kuiper J and Huiskes R 1992 ldquoNumerical Optimization of Hip-ProstheticStem Materialrdquo Recent Advances in Computer Methods in Biomechanics andBiomedical Engineering J Middleton G N Pande and K R Williams edsBooks and Journals International Ltd Swansea pp 76ndash84

[15] Kuiper J H and Huiskes R 1997 ldquoMathematical Optimization of ElasticProperties Application to Cementless Hip Stem Designrdquo J Biomech Eng119(2) pp 166ndash174

[16] Hedia H Shabara M El-Midany T and Fouda N 2006 ldquoImproved Designof Cementless Hip Stems Using Two-Dimensional Functionally Graded Materi-alsrdquo J Biomed Mater Res Part B Appl Biomater 79(1) pp 42ndash49

[17] Hedia H S and Mahmoud N-A 2004 ldquoDesign Optimization of Function-ally Graded Dental Implantrdquo Biomed Mater Eng 14(2) pp 133ndash143

[18] Watari F Yokoyama A Saso F Uo M and Kawasaki T 1997ldquoFabrication and Properties of Functionally Graded Dental Implantrdquo Compo-sites Part B 28(1-2) pp 5ndash11

[19] Katti K S 2004 ldquoBiomaterials in Total Joint Replacementrdquo Colloids Surf B39(3) pp 133ndash142

[20] Thompson I and Hench L 1998 ldquoMechanical Properties of BioactiveGlasses Glass-Ceramics and Compositesrdquo Proc Inst Mech Eng Part H JEng Med 212(2) pp 127

[21] Fraldi M Esposito L Perrella G Cutolo A and Cowin S 2010ldquoTopological Optimization in Hip Prosthesis Designrdquo Biomech ModelMechanobiol 9(4) pp 389ndash402

[22] Bobyn J D Poggie R Krygier J Lewallen D Hanssen A Lewis RUnger A Orsquokeefe T Christie M and Nasser S 2004 ldquoClinical Validationof a Structural Porous Tantalum Biomaterial for Adult Reconstructionrdquo J BoneJt Surg 86(Supplement 2) pp 123ndash129

[23] Bobyn J Stackpool G Hacking S Tanzer M and Krygier J 1999ldquoCharacteristics of Bone Ingrowth and Interface Mechanics of a New PorousTantalum Biomaterialrdquo J Bone Jt Surg Br Vol 81(5) pp 907ndash914

[24] Parthasarathy J Starly B Raman S and Christensen A 2010 ldquoMechanicalEvaluation of Porous Titanium (Ti6al4v) Structures with Electron Beam Melt-ing (Ebm)rdquo J Mech Behav Biomed Mater 3(3) pp 249ndash259

[25] Heinl P Muller L Korner C Singer R F and Muller F A 2008ldquoCellular Ti-6al-4v Structures with Interconnected Macro Porosity for BoneImplants Fabricated by Selective Electron Beam Meltingrdquo Acta Biomater4(5) pp 1536ndash1544

[26] Stamp R Fox P Orsquoneill W Jones E and Sutcliffe C 2009 ldquoThe Devel-opment of a Scanning Strategy for the Manufacture of Porous Biomaterials bySelective Laser Meltingrdquo J Mater Sci Mater Med 20(9) pp 1839ndash1848

[27] Yang S Leong K Du Z and Chua C 2002 ldquoThe Design of Scaffolds forUse in Tissue Engineering Part Ii Rapid Prototyping Techniquesrdquo TissueEng 8(1) pp 1ndash11

[28] Murr L Gaytan S Medina F Lopez H Martinez E Machado B Hernan-dez D Martinez L Lopez M and Wicker R 2010 ldquoNext-Generation Biomed-ical Implants Using Additive Manufacturing of Complex Cellular and FunctionalMesh Arraysrdquo Philos Trans R Soc London Ser A 368(1917) pp 1999ndash2032

[29] Coelho P Fernandes P Guedes J and Rodrigues H 2008 ldquoA HierarchicalModel for Concurrent Material and Topology Optimisation of Three-Dimensional Structuresrdquo Struct Multidiscip Optim 35(2) pp 107ndash115

[30] Rodrigues H Guedes J and Bendsoe M 2002 ldquoHierarchical Optimizationof Material and Structurerdquo Struct Multidiscip Optim 24(1) pp 1ndash10

[31] Goncalves Coelho P Rui Fernandes P and Carrico Rodrigues H 2011ldquoMultiscale Modeling of Bone Tissue with Surface and Permeability ControlrdquoJ Biomech 44(2) pp 321ndash329

[32] Coelho P G Cardoso J B Fernandes P R and Rodrigues H C 2011ldquoParallel Computing Techniques Applied to the Simultaneous Design of Struc-ture and Materialrdquo Adv Eng Software 42(5) pp 219ndash227

[33] Gibson L J and Ashby M F 1999 Cellular Solids Structure and Proper-ties Cambridge University Press Cambridge UK

[34] Guedes J and Kikuchi N 1990 ldquoPreprocessing and Postprocessing for Mate-rials Based on the Homogenization Method with Adaptive Finite ElementMethodsrdquo Comput Methods Appl Mech Eng 83(2) pp 143ndash198

[35] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization I-Homogenization Theory for Media with PeriodicStructurerdquo Comput Struct 69(6) pp 707ndash717

[36] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Opimization I I-Analytical and Numerical Solution of HomogenizationEquationsrdquo Comput Struct 69(6) pp 719ndash738

[37] Fang Z Starly B and Sun W 2005 ldquoComputer-Aided Characterization forEffective Mechanical Properties of Porous Tissue Scaffoldsrdquo Comput-AidedDes 37(1) pp 65ndash72

[38] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics 6th ed Butterworth-Heinemann Burlington

[39] Hollister S and Kikuchi N 1992 ldquoA Comparison of Homogenization andStandard Mechanics Analyses for Periodic Porous Compositesrdquo ComputMech 10(2) pp 73ndash95

[40] Pellegrino C Galvanetto U and Schrefler B 1999 ldquoNumerical Homogeni-zation of Periodic Composite Materials with Non Linear MaterialComponentsrdquo Int J Numer Methods Eng 46(10) pp 1609ndash1637

[41] Bragdon C Jasty M Greene M Rubash H and Harris W 2004ldquoBiologic Fixation of Total Hip Implants Insights Gained from a Series of Ca-nine Studiesrdquo J Bone Jt Surg 86(Supplement 2) pp 105ndash117

[42] Harrysson O L A Cansizoglu O Marcellin-Little D J Cormier D R andWest Ii H A 2008 ldquoDirect Metal Fabrication of Titanium Implants with Tai-lored Materials and Mechanical Properties Using Electron Beam Melting Tech-nologyrdquo Mater Sci Eng C 28(3) pp 366ndash373

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-9

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

debonding might occur along the bone-implant interface [1462]The probability of local interface failure f ethrTHORN is given by

f ethrTHORN frac14 1

StScr2

n thorn1

St 1

Sc

rn thorn

1

S2s

s2 (9)

where St and Sc are the uniaxial tensile and compressive strengthsrespectively Ss is the shear strength and rn and s are normal andshear stresses at the bone-implant interface respectively Forf ethrTHORN 1 a high probability of failure is expected whereas forf ethrTHORN 1 the risk of interface failure is low Tensile compressionand shear strengths of bone can be expressed as a function of bonedensity following the power law relations obtained by Pal et al[62]

St frac14 145q171 Sc frac14 324q185 Ss frac14 216q165 (10)

The distribution of bone density can be obtained through a CT-scan data of bone and then used in Eq (10) to find the effectivemechanical properties of the bone from which the local interfacefailure can be determined via Eq (9) Finally the interface failureindex FethbTHORN is evaluated by means of Eq (8) It should be notedthat FethbTHORN serves as a qualitative indicator only lower values ofFethbTHORN indicate reduced probability of local interface failure in theimplant

The expressions of bone resorption after implantation and me-chanical failure of bone-implant interface described in this section

are used in the finite element analysis to evaluate the objectivefunctions to be optimized The following sections shows the appli-cation of the method for the design of a two dimensional cellularimplant

3 Application of the Methodology to design a 2D

Femoral Implant with a Graded Cellular Material

31 FEM model of the Femur at the Macroscale The lefthand side of Fig 3 shows the geometry of the left femur consid-ered in this work along with the appropriate loads and boundaryconditions These data have been obtained from the work ofKuiper and Huiskes [14] The 3D geometry of the femur is simpli-fied into a 2D model where the thickness of the stem and bonevaries such that the second moment of area about the out-of-planeaxis does not differ in both models [16] Furthermore in this pa-per the implant material is designed to be an open cell lattice toease bone in growth in the implanted stem and obtain a full bondAlthough bone ingrowth does not exist in a postoperative situa-tion it can appear later if local mechanical stability is guaranteedThe minimization of interface stress reduces the possibility ofoccurrence of interface micromotion and instability [4] There-fore to decrease the computational cost required by a stabilityanalysis based on a nonlinear frictional contact model the pros-thesis and the surrounding bone are assumed fully bonded

The load case represents daily static loading during the stancephase of walking [63] The distal end of the femur is fixed toavoid rigid body motion For the material properties of the modelwe consider 20 GPa as the Youngrsquos modulus of the cortical boneand 15 GPa for the proximal bone The Poissonrsquos ratio is set to be03 [14]

32 FEM model of the Cellular Implant at theMicroscale The right hand side of Fig 3 illustrates the model ofa cementless prosthesis implanted into the human femur The griddepicts the domain of the implant to be designed with a lattice ma-terial of graded properties Figure 4 shows the unit cell geometryused for the tessellation of the whole implant The gradients ofmaterial properties are governed by the lattice relative densitywhich is a variable controlled by the cell size and wall thicknessof the hollow square In future work the performance of alterna-tive cell topologies suitable for bone tissue scaffolding will beinvestigated as well as combinations of dissimilar unit cells in thetessellation [64]

For the material property of the implant we consider Ti6Al4V[24] which is a biocompatible material commonly used in EBMIts mechanical properties are the following 900 MPa for the yieldstrength of the solid material 120 GPa for Youngrsquos modulus and03 for Poissonrsquos ratio

33 Design of the Graded Cellular Material of theImplant The procedure introduced in this paper for the design ofa graded cellular implant requires both multiscale analysis andmultiobjective optimization as described in Secs 21 and 22 andshown in Fig 1 The variables of the lattice model are the relativedensities attributed to 130 sampling points 26 rows along the

Fig 3 2D Finite element models of the femur (left) and theprosthesis implanted into the femur (right)

Fig 4 2D hollow square unit cell for given values of relative density

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-5

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

prosthetic length and 5 columns along the radial direction asshown in the right side of Fig 3 The number of sampling pointshas been chosen to be 130 to limit the computational time requiredfor the analysis and optimization while providing a reasonable re-solution for the relative density distribution For a more refined den-sity distribution the number of sample points can be increasedTheir values have been constrained in the range 01 q 1 [47]to prevent elastic buckling from occurring in the unit cell prior toyielding The values of the relative density between the samplingpoints are obtained through linear interpolation Although not con-sidered in the current research the shape of the implant could beincluded in the vector b as a design variable (Fig 1)

To calculate the stress and strain regime of the implant thesteps explained in Sec 21 are followed Initial values of relativedensity are assigned to each element of the FE model created inANSYS (Canonsburg Pennsylvania USA) The stiffness matrixof each element is calculated through a computational code2DHOMOG developed in-house which obtains the homogenizedstiffness matrix of the square unit cell as a function of relativedensity Figure 5 shows the results which are first obtained at dis-crete points of relative density To obtain the continuous functionsfor the properties we calculate through the least squares methodthe expressions (Table 1) for two ranges of relative density ieq lt 03 and q gt 03 These functions enable to assign the valuesof the stiffness for a given relative density to each of the samplepoints modeled in the FE solver We note that the expressions ofYoungrsquos moduli and Poissonrsquos ratios in the x1 and x2 directions donot change since the cell thickness is uniform

Once the stress and strain regimes of the cellular material havebeen calculated the non-dominated sorting genetic (NSGA-II)

algorithm [65] is employed to solve the multiobjective optimiza-tion problem described in Sec 22 The strain energy within thebone and the stress distribution at the bone-implant interface isthen calculated and used in Eqs (7) and (8) to evaluate the objec-tive functions The initial population is then sorted based on thenondomination front criterion A population of solutions calledparents are selected from the current population based on theirrank and crowding distance Then genetic operators are appliedto the population of parents to create a population of off-springsFinally the next population is produced by taking the best solu-tions from the combined population of parents and off-springsThe optimization continues until the user-defined number of func-tion evaluations reaches 25000 [65] The computational costrequired to run the optimization process in a single 24 GHz Intelprocessor with 4 GB RAM was about 300000 CPU s about threeand a halfdays We plan to use parallel computing with a PC clus-ter in future work this will considerably reduce the computationaltime since each function evaluation can be performedindependently

For each point in the objective function space the stress recov-ery procedure is applied to verify whether the stresses are admissi-ble To apply this procedure the average macroscopic straininside each unit cell is found The position of each unit cell withinthe implant is obtained after imposing a proper cell tessellationwhich in this work has been set to be uniform The size of the unitcell is selected as small as possible to capture the relative distribu-tion contour with higher resolution For a relative density of 01the square cell sizes for the cell wall thickness of either 70 or 100lm have been selected respectively to be 136 and 18 mm Oncethe position of the unit cells has been obtained 3 3 Gauss pointsare assigned to each cell The values of relative density and mac-roscopic strain at these points are obtained from the relative den-sity distribution and macroscopic strain field For Gauss pointslocated outside the implant border the values are linearly extrapo-lated from those located at the neighboring points inside theimplant domain Using a Gaussian quadrature integration [66] theaverage relative density and macroscopic strain of each cell arecalculated The local stress distribution and the yield safety factorof each cell are obtained through the von Mises stress criterionThe procedure is applied to all unit cells of the selected optimaldesign located on the Pareto frontier and the minimum local safetyfactor of a cell is specified as design safety factor

4 Results and Discussion

The advantage of multiobjective optimization with a posterioriarticulation of preference is that a set of optimum solutions areavailable without requiring the designer to choose in advance anyweighting factors to the objective functions Once the whole set ofPareto solutions has been determined the designer has the free-dom to select the desired solution based on the importance ofbone mass preservation relative to the amount of interface stressFigure 6 shows all the optimum solutions ie the relative densitydistribution for a hip stem implant with graded cellular materialThe x axis represents the amount of bone resorption for theimplanted hip on the y axis is the interface failure index Amongthe optimal solutions we examine three representative relativedensity distributions the extreme points A and C of the Paretofrontier for which one objective function has importance factor 0and the other 100 and a solution B characterized by a 50weight factor For each solution Fig 6 gives the following per-formance metrics bone resorption (mr) interface failure index(FethbTHORN) maximum interface failure (f ethrTHORNmax) average porosity ofeach stem ( ) and design safety factor (SF) after implementingthe stress recovery procedure The maximum interface failuref ethrTHORNmax is included since FethbTHORN which quantifies only the overalleffect of the implant stiffness on the interface stresses is not suffi-cient to provide information on the probability of failure

As seen from the performance metrics in Fig 6 the porosity ofsolutions A B and C is greater than 40 which is satisfactory

Fig 5 Effective Youngrsquos modulus of 2D square lattice versusrelative density Solution points obtained through homogeniza-tion theory are fitted with the least squares method

Table 1 Effective Mechanical Properties of the Square UnitCell as a Function of Relative Density

q lt 03 03 lt q lt 1

E1

Esfrac14 E2

Es058ethqTHORN1046

127ethqTHORN2 052ethqTHORN thorn 023

G

Es0093ethqTHORN307

126ethqTHORN3 133ethqTHORN2 thorn 051ethqTHORN 0065

21

sfrac14 12

s07ethqTHORN105

068ethqTHORN2 thorn 028ethqTHORN thorn 006

031004-6 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

for bone ingrowth [41] By comparing implants C and A weobserve that a raise of the implant porosity from point C to Aresults in an implant stiffness decrease which on one hand low-ers bone loss and on the other hand enhances the risk of interfacefailure When solution B is compared to C a reduction of 8 ofbone resorption is noted with a slight increase of the peak value ofthe interface failure On the other hand by contrasting solution Bto A we note a significant increase (60) of the peak value ofinterface failure which is below the Hoffman failure strength anda minor reduction (2) of the amount of bone resorption Themain benefit of solution A is the maximum porosity of the micro-structure that can promote bone ingrowth Solution B on the otherhand might be the preferred solution with respect to low boneresorption and interface failure These observations emerge onlyby inspection of the objective functions selected in this work weremark here that other parameters should be taken into accountfor the selection of the best implant These include patientrsquos bone

characteristics the range of activity age and desired level ofbone mass preservation after implantation

For prescribed geometric loading and constraint conditions wenow compare the metrics of resorbed bone mass (mr) and distribu-tion of interface stress (f ethrbTHORN) of the optimal solution B with thoseof (i) a currently-used fully dense titanium stem and (ii) a cellularimplant with a uniformly distributed relative density of 50 Fig-ures 7 and 8 illustrate the results of the comparison

For the solid titanium stem the amount of bone resorption cal-culated through Eq (7) is 67 and the interface failure indexFethbTHORN obtained from Eq (8) is 133 Using the distribution of f ethrTHORNgenerated around the titanium stem (Fig 8(a)) we observe thatthe maximum value of interface failure (051) occurs at the distalend of the implant As expected this implant is much stiffer thanthe surrounding bone thereby resulting in a higher amount ofbone resorption For the numerical validation the interface shearstress of the titanium implant at the proximal region is also

Fig 6 Trade-off distributions of relative density for the optimized cellular implant

Fig 7 Distribution of bone resorption around (a) fully dense titanium implant (b) cellularimplant with uniform relative density of 50 (c) graded cellular implant (solution B in Fig 6)

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-7

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

compared with the one obtained by Kowalczyk for a 3D model[4] The mean and the maximum values of interface shear stressfor the 3D titanium implant in the work by Kowalczyk [4] are057 and 28 MPa respectively These values are respectively031 and 215 MPa for the titanium implant in this paper The con-tribution to the higher level of shear stress in the 3D model ofKowalczyk is the distribution of shear force on a smaller area InKowalczykrsquos study [4] the implant and bone are bonded only atthe proximal region while in our work the whole bone-implantinterface is bonded which results in a decrease of the mean andthe maximum values of interface shear stress

The cellular implant with uniform relative density of 50 isapproximately three times more flexible than the titanium stemThis implant can qualitatively simulate the behavior of an implantmade out of tantalum foam For this stem the amount of boneresorption and the interface failure index are about 34 and 287respectively and the interface failure is maximum (071) at theedge of proximal region Compared to the solid titanium implantthe amount of bone resorption decreases by 50 whereas themaximum interface failure increases about 40 This shows that adecrease of the implant stiffness with uniform porosity distribu-tion aiming at reducing bone resorption has the undesirable effectof increasing the risk of interface failure at the proximal regionThis outcome confirms the findings of the previous work byKuiper and Huiskes [14]

Figure 7(c) and 8(c) show the results for the graded cellularimplant B Its bone resorption and interface failure index are 16and 115 respectively The peak value of the local interface fail-ure is 025 Compared to the titanium stem both the amount ofbone resorption and the peak of interface failure decrease of 76and 50 respectively With respect to the uniformly-distributedcellular implant the decrease of bone resorption and interface fail-ure peak is of 53 and 65 respectively A graded cellularimplant with optimized relative density distribution is thus capa-ble of reducing concurrently both the conflicting objective func-tions In particular bone resorption reduces as a result of thecellular material which makes the implant more compliant theinterface stress on the other hand is minimized by the optimizedgradients of cellular material

A proof-of-concept implant was built to verify the manufactur-ability of the optimum grade lattice material Figure 9 shows thepolypropylene prototype of solution B which was manufacturedwith a 3D printer Objet Connex500 [67] A uniform tessellationand a square unit cell of 18 mm size were assumed to draw themodel The cell geometry was calculated from the average rela-tive density obtained from the method described in this paper AnSTL file of the graded cellular implant solution B was finallyused for rapid prototyping

The limitations of the method proposed in this paper are herediscussed First the method tackles the design of an implant stati-cally loaded in the stance phase of walking Further research isrequired to extend it to variable loading and fatigue life design

Second the accuracy of the asymptotic homogenization needsto be investigated at the vicinity of the implant borders Althoughmacroscopic mean stress fields on the boundaries are satisfied inthe global solutions [4068] microscopic stresses do not satisfythe Y-periodicity assumption These boundary effects howevermay be captured by applying a boundary layer corrector [6869] aspatially decaying stress localization function [70] or an adaptivemultiscale methodology [71] To investigate the accuracy of thesetechniques it is required to construct a detailed finite elementmodel and compare the actual stressstrain distribution inside thecell walls with the microscopic stressesstrains estimated from thehomogenization method This task will be performed in future

Third to extend the proposed procedure to three dimensions a3D cell topology with high porosity interconnected pore struc-ture large surface area with suitable textures and good

Fig 8 Distribution of local interface failure f ethrTHORN around (a) fully dense titanium implant(b) cellular implant with uniform relative density of 50 (c) graded cellular implant (solu-tion B in Fig 6)

Fig 9 Polypropylene proof-of-concept of the optimal gradedcellular implant (solution B)

031004-8 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

mechanical properties should be selected [7273] The asymptotichomogenization will be applied to characterize tissue scaffoldingcell topologies for a wide range of relative density Their effectivemoduli yield and ultimate surfaces for multiaxial loading condi-tions will be obtained to create a standard library from which toselect suitable cell geometries for a 3D cellular implant design

Fourth bone has been here considered as consisting of a cancel-lous and a cortical part with isotropic mechanical properties Fur-ther study is necessary to improve the constitutive material modelof bone and to consider bone as a graded cellular material

Finally in the optimization procedure a simple formulation hasbeen used to quantitatively measure both bone resorption andimplant stability In a future study detailed simulations of implantstability [74ndash77] and bone remodeling process [6077ndash80] can beperformed to assess both the short and long term performance ofthe implant

5 Concluding Remarks

This work has presented a methodology integrating multiscaleanalysis and design optimization to design a novel hip implantmade of graded cellular material The method can contribute tothe development of a new generation of orthopedic implants witha graded cellular microstructure that will reduce the clinical con-sequences of current implants

In the first part of the paper the homogenization method hasbeen reviewed and then used to capture the mechanics of theimplant at the micro and macro scale In the second part multiob-jective optimization has been applied to find optimum gradients ofmaterial distribution that minimize concurrently bone resorptionand bone-implant interface stresses The results have shown thatthe optimized cellular implant exhibits a reduction of 76 ofbone resorption and 50 of interface stress with respect to a fullydense titanium implant

References[1] Kurtz S Ong K Lau E Mowat F and Halpern M 2007 ldquoProjections of

Primary and Revision Hip and Knee Arthroplasty in the United States from2005 to 2030rdquo J Bone Jt Surg Am Vol 89(4) pp 780ndash785

[2] Vichinsky E P Neumayr L D Haberkern C Earles A N Eckman JKoshy M and Black D M 1999 ldquoThe Perioperative Complication Rate ofOrthopedic Surgery in Sickle Cell Disease Report of the National Sickle CellSurgery Study Grouprdquo Am J Hematol 62(3) pp 129ndash138

[3] Kobayashi S Saito N Horiuchi H Iorio R and Takaoka K 2000 ldquoPoorBone Quality or Hip Structure as Risk Factors Affecting Survival of Total-HipArthroplastyrdquo The Lancet 355(9214) pp 1499ndash1504

[4] Kowalczyk P 2001 ldquoDesign Optimization of Cementless Femoral Hip Pros-theses Using Finite Element Analysisrdquo J Biomech Eng 123(5) pp 396ndash402

[5] Moen T C Ghate R Salaz N Ghodasra J and Stulberg S D 2011 ldquoAMonoblock Porous Tantalum Acetabular Cup Has No Osteolysis on Ct at 10Yearsrdquo Clin Orthop Relat Res 469(2) pp 382ndash386

[6] Kurtz S Gawel H and Patel J 2011 ldquoHistory and Systematic Review ofWear and Osteolysis Outcomes for First-Generation Highly Crosslinked Poly-ethylenerdquo Clin Orthop Relat Res 469(8) pp 2262ndash2277

[7] Grubl A Marker M Brodner W Giurea A Heinze G Meisinger VZehetgruber H and Kotz R 2007 ldquoLong Term Follow up of Metal on MetalTotal Hip Replacementrdquo J Orthop Res 25(7) pp 841ndash848

[8] Neumann D R P Thaler C Hitzl W Huber M Hofstadter T and DornU 2010 ldquoLong-Term Results of a Contemporary Metal-on-Metal Total HipArthroplasty A 10-Year Follow-up Studyrdquo J Arthroplasty 25(5) pp700ndash708

[9] Glassman A Bobyn J and Tanzer M 2006 ldquoNew Femoral Designs DoThey Influence Stress Shieldingrdquo Clin Orthop Relat Res 453(12) pp64ndash74

[10] Adam F Hammer D S Pfautsch S and Westermann K 2002 ldquoEarly Fail-ure of a Press-Fit Carbon Fiber Hip Prosthesis with a Smooth Surfacerdquo JAntroplasty 17(2) pp 217ndash223

[11] Trebse R Milosev I Kovac S Mikek M and Pisot V 2005 ldquoPoorResults from the Isoelastic Total Hip Replacementrdquo Acta Orthop 76(2) pp169ndash176

[12] Harvey E Bobyn J Tanzer M Stackpool G Krygier J and Hacking S1999 ldquoEffect of Flexibility of the Femoral Stem on Bone-Remodeling and Fix-ation of the Stem in a Canine Total Hip Arthroplasty Model without CementrdquoJ Bone Jt Surg 81(1) pp 93ndash107

[13] Huiskes R Weinans H and Rietbergen B 1992 ldquoThe Relationship betweenStress Shielding and Bone Resorption around Total Hip Stems and the Effectsof Flexible Materialsrdquo Clin Orthop Relat Res 274(1) pp 124ndash134

[14] Kuiper J and Huiskes R 1992 ldquoNumerical Optimization of Hip-ProstheticStem Materialrdquo Recent Advances in Computer Methods in Biomechanics andBiomedical Engineering J Middleton G N Pande and K R Williams edsBooks and Journals International Ltd Swansea pp 76ndash84

[15] Kuiper J H and Huiskes R 1997 ldquoMathematical Optimization of ElasticProperties Application to Cementless Hip Stem Designrdquo J Biomech Eng119(2) pp 166ndash174

[16] Hedia H Shabara M El-Midany T and Fouda N 2006 ldquoImproved Designof Cementless Hip Stems Using Two-Dimensional Functionally Graded Materi-alsrdquo J Biomed Mater Res Part B Appl Biomater 79(1) pp 42ndash49

[17] Hedia H S and Mahmoud N-A 2004 ldquoDesign Optimization of Function-ally Graded Dental Implantrdquo Biomed Mater Eng 14(2) pp 133ndash143

[18] Watari F Yokoyama A Saso F Uo M and Kawasaki T 1997ldquoFabrication and Properties of Functionally Graded Dental Implantrdquo Compo-sites Part B 28(1-2) pp 5ndash11

[19] Katti K S 2004 ldquoBiomaterials in Total Joint Replacementrdquo Colloids Surf B39(3) pp 133ndash142

[20] Thompson I and Hench L 1998 ldquoMechanical Properties of BioactiveGlasses Glass-Ceramics and Compositesrdquo Proc Inst Mech Eng Part H JEng Med 212(2) pp 127

[21] Fraldi M Esposito L Perrella G Cutolo A and Cowin S 2010ldquoTopological Optimization in Hip Prosthesis Designrdquo Biomech ModelMechanobiol 9(4) pp 389ndash402

[22] Bobyn J D Poggie R Krygier J Lewallen D Hanssen A Lewis RUnger A Orsquokeefe T Christie M and Nasser S 2004 ldquoClinical Validationof a Structural Porous Tantalum Biomaterial for Adult Reconstructionrdquo J BoneJt Surg 86(Supplement 2) pp 123ndash129

[23] Bobyn J Stackpool G Hacking S Tanzer M and Krygier J 1999ldquoCharacteristics of Bone Ingrowth and Interface Mechanics of a New PorousTantalum Biomaterialrdquo J Bone Jt Surg Br Vol 81(5) pp 907ndash914

[24] Parthasarathy J Starly B Raman S and Christensen A 2010 ldquoMechanicalEvaluation of Porous Titanium (Ti6al4v) Structures with Electron Beam Melt-ing (Ebm)rdquo J Mech Behav Biomed Mater 3(3) pp 249ndash259

[25] Heinl P Muller L Korner C Singer R F and Muller F A 2008ldquoCellular Ti-6al-4v Structures with Interconnected Macro Porosity for BoneImplants Fabricated by Selective Electron Beam Meltingrdquo Acta Biomater4(5) pp 1536ndash1544

[26] Stamp R Fox P Orsquoneill W Jones E and Sutcliffe C 2009 ldquoThe Devel-opment of a Scanning Strategy for the Manufacture of Porous Biomaterials bySelective Laser Meltingrdquo J Mater Sci Mater Med 20(9) pp 1839ndash1848

[27] Yang S Leong K Du Z and Chua C 2002 ldquoThe Design of Scaffolds forUse in Tissue Engineering Part Ii Rapid Prototyping Techniquesrdquo TissueEng 8(1) pp 1ndash11

[28] Murr L Gaytan S Medina F Lopez H Martinez E Machado B Hernan-dez D Martinez L Lopez M and Wicker R 2010 ldquoNext-Generation Biomed-ical Implants Using Additive Manufacturing of Complex Cellular and FunctionalMesh Arraysrdquo Philos Trans R Soc London Ser A 368(1917) pp 1999ndash2032

[29] Coelho P Fernandes P Guedes J and Rodrigues H 2008 ldquoA HierarchicalModel for Concurrent Material and Topology Optimisation of Three-Dimensional Structuresrdquo Struct Multidiscip Optim 35(2) pp 107ndash115

[30] Rodrigues H Guedes J and Bendsoe M 2002 ldquoHierarchical Optimizationof Material and Structurerdquo Struct Multidiscip Optim 24(1) pp 1ndash10

[31] Goncalves Coelho P Rui Fernandes P and Carrico Rodrigues H 2011ldquoMultiscale Modeling of Bone Tissue with Surface and Permeability ControlrdquoJ Biomech 44(2) pp 321ndash329

[32] Coelho P G Cardoso J B Fernandes P R and Rodrigues H C 2011ldquoParallel Computing Techniques Applied to the Simultaneous Design of Struc-ture and Materialrdquo Adv Eng Software 42(5) pp 219ndash227

[33] Gibson L J and Ashby M F 1999 Cellular Solids Structure and Proper-ties Cambridge University Press Cambridge UK

[34] Guedes J and Kikuchi N 1990 ldquoPreprocessing and Postprocessing for Mate-rials Based on the Homogenization Method with Adaptive Finite ElementMethodsrdquo Comput Methods Appl Mech Eng 83(2) pp 143ndash198

[35] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization I-Homogenization Theory for Media with PeriodicStructurerdquo Comput Struct 69(6) pp 707ndash717

[36] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Opimization I I-Analytical and Numerical Solution of HomogenizationEquationsrdquo Comput Struct 69(6) pp 719ndash738

[37] Fang Z Starly B and Sun W 2005 ldquoComputer-Aided Characterization forEffective Mechanical Properties of Porous Tissue Scaffoldsrdquo Comput-AidedDes 37(1) pp 65ndash72

[38] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics 6th ed Butterworth-Heinemann Burlington

[39] Hollister S and Kikuchi N 1992 ldquoA Comparison of Homogenization andStandard Mechanics Analyses for Periodic Porous Compositesrdquo ComputMech 10(2) pp 73ndash95

[40] Pellegrino C Galvanetto U and Schrefler B 1999 ldquoNumerical Homogeni-zation of Periodic Composite Materials with Non Linear MaterialComponentsrdquo Int J Numer Methods Eng 46(10) pp 1609ndash1637

[41] Bragdon C Jasty M Greene M Rubash H and Harris W 2004ldquoBiologic Fixation of Total Hip Implants Insights Gained from a Series of Ca-nine Studiesrdquo J Bone Jt Surg 86(Supplement 2) pp 105ndash117

[42] Harrysson O L A Cansizoglu O Marcellin-Little D J Cormier D R andWest Ii H A 2008 ldquoDirect Metal Fabrication of Titanium Implants with Tai-lored Materials and Mechanical Properties Using Electron Beam Melting Tech-nologyrdquo Mater Sci Eng C 28(3) pp 366ndash373

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-9

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

prosthetic length and 5 columns along the radial direction asshown in the right side of Fig 3 The number of sampling pointshas been chosen to be 130 to limit the computational time requiredfor the analysis and optimization while providing a reasonable re-solution for the relative density distribution For a more refined den-sity distribution the number of sample points can be increasedTheir values have been constrained in the range 01 q 1 [47]to prevent elastic buckling from occurring in the unit cell prior toyielding The values of the relative density between the samplingpoints are obtained through linear interpolation Although not con-sidered in the current research the shape of the implant could beincluded in the vector b as a design variable (Fig 1)

To calculate the stress and strain regime of the implant thesteps explained in Sec 21 are followed Initial values of relativedensity are assigned to each element of the FE model created inANSYS (Canonsburg Pennsylvania USA) The stiffness matrixof each element is calculated through a computational code2DHOMOG developed in-house which obtains the homogenizedstiffness matrix of the square unit cell as a function of relativedensity Figure 5 shows the results which are first obtained at dis-crete points of relative density To obtain the continuous functionsfor the properties we calculate through the least squares methodthe expressions (Table 1) for two ranges of relative density ieq lt 03 and q gt 03 These functions enable to assign the valuesof the stiffness for a given relative density to each of the samplepoints modeled in the FE solver We note that the expressions ofYoungrsquos moduli and Poissonrsquos ratios in the x1 and x2 directions donot change since the cell thickness is uniform

Once the stress and strain regimes of the cellular material havebeen calculated the non-dominated sorting genetic (NSGA-II)

algorithm [65] is employed to solve the multiobjective optimiza-tion problem described in Sec 22 The strain energy within thebone and the stress distribution at the bone-implant interface isthen calculated and used in Eqs (7) and (8) to evaluate the objec-tive functions The initial population is then sorted based on thenondomination front criterion A population of solutions calledparents are selected from the current population based on theirrank and crowding distance Then genetic operators are appliedto the population of parents to create a population of off-springsFinally the next population is produced by taking the best solu-tions from the combined population of parents and off-springsThe optimization continues until the user-defined number of func-tion evaluations reaches 25000 [65] The computational costrequired to run the optimization process in a single 24 GHz Intelprocessor with 4 GB RAM was about 300000 CPU s about threeand a halfdays We plan to use parallel computing with a PC clus-ter in future work this will considerably reduce the computationaltime since each function evaluation can be performedindependently

For each point in the objective function space the stress recov-ery procedure is applied to verify whether the stresses are admissi-ble To apply this procedure the average macroscopic straininside each unit cell is found The position of each unit cell withinthe implant is obtained after imposing a proper cell tessellationwhich in this work has been set to be uniform The size of the unitcell is selected as small as possible to capture the relative distribu-tion contour with higher resolution For a relative density of 01the square cell sizes for the cell wall thickness of either 70 or 100lm have been selected respectively to be 136 and 18 mm Oncethe position of the unit cells has been obtained 3 3 Gauss pointsare assigned to each cell The values of relative density and mac-roscopic strain at these points are obtained from the relative den-sity distribution and macroscopic strain field For Gauss pointslocated outside the implant border the values are linearly extrapo-lated from those located at the neighboring points inside theimplant domain Using a Gaussian quadrature integration [66] theaverage relative density and macroscopic strain of each cell arecalculated The local stress distribution and the yield safety factorof each cell are obtained through the von Mises stress criterionThe procedure is applied to all unit cells of the selected optimaldesign located on the Pareto frontier and the minimum local safetyfactor of a cell is specified as design safety factor

4 Results and Discussion

The advantage of multiobjective optimization with a posterioriarticulation of preference is that a set of optimum solutions areavailable without requiring the designer to choose in advance anyweighting factors to the objective functions Once the whole set ofPareto solutions has been determined the designer has the free-dom to select the desired solution based on the importance ofbone mass preservation relative to the amount of interface stressFigure 6 shows all the optimum solutions ie the relative densitydistribution for a hip stem implant with graded cellular materialThe x axis represents the amount of bone resorption for theimplanted hip on the y axis is the interface failure index Amongthe optimal solutions we examine three representative relativedensity distributions the extreme points A and C of the Paretofrontier for which one objective function has importance factor 0and the other 100 and a solution B characterized by a 50weight factor For each solution Fig 6 gives the following per-formance metrics bone resorption (mr) interface failure index(FethbTHORN) maximum interface failure (f ethrTHORNmax) average porosity ofeach stem ( ) and design safety factor (SF) after implementingthe stress recovery procedure The maximum interface failuref ethrTHORNmax is included since FethbTHORN which quantifies only the overalleffect of the implant stiffness on the interface stresses is not suffi-cient to provide information on the probability of failure

As seen from the performance metrics in Fig 6 the porosity ofsolutions A B and C is greater than 40 which is satisfactory

Fig 5 Effective Youngrsquos modulus of 2D square lattice versusrelative density Solution points obtained through homogeniza-tion theory are fitted with the least squares method

Table 1 Effective Mechanical Properties of the Square UnitCell as a Function of Relative Density

q lt 03 03 lt q lt 1

E1

Esfrac14 E2

Es058ethqTHORN1046

127ethqTHORN2 052ethqTHORN thorn 023

G

Es0093ethqTHORN307

126ethqTHORN3 133ethqTHORN2 thorn 051ethqTHORN 0065

21

sfrac14 12

s07ethqTHORN105

068ethqTHORN2 thorn 028ethqTHORN thorn 006

031004-6 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

for bone ingrowth [41] By comparing implants C and A weobserve that a raise of the implant porosity from point C to Aresults in an implant stiffness decrease which on one hand low-ers bone loss and on the other hand enhances the risk of interfacefailure When solution B is compared to C a reduction of 8 ofbone resorption is noted with a slight increase of the peak value ofthe interface failure On the other hand by contrasting solution Bto A we note a significant increase (60) of the peak value ofinterface failure which is below the Hoffman failure strength anda minor reduction (2) of the amount of bone resorption Themain benefit of solution A is the maximum porosity of the micro-structure that can promote bone ingrowth Solution B on the otherhand might be the preferred solution with respect to low boneresorption and interface failure These observations emerge onlyby inspection of the objective functions selected in this work weremark here that other parameters should be taken into accountfor the selection of the best implant These include patientrsquos bone

characteristics the range of activity age and desired level ofbone mass preservation after implantation

For prescribed geometric loading and constraint conditions wenow compare the metrics of resorbed bone mass (mr) and distribu-tion of interface stress (f ethrbTHORN) of the optimal solution B with thoseof (i) a currently-used fully dense titanium stem and (ii) a cellularimplant with a uniformly distributed relative density of 50 Fig-ures 7 and 8 illustrate the results of the comparison

For the solid titanium stem the amount of bone resorption cal-culated through Eq (7) is 67 and the interface failure indexFethbTHORN obtained from Eq (8) is 133 Using the distribution of f ethrTHORNgenerated around the titanium stem (Fig 8(a)) we observe thatthe maximum value of interface failure (051) occurs at the distalend of the implant As expected this implant is much stiffer thanthe surrounding bone thereby resulting in a higher amount ofbone resorption For the numerical validation the interface shearstress of the titanium implant at the proximal region is also

Fig 6 Trade-off distributions of relative density for the optimized cellular implant

Fig 7 Distribution of bone resorption around (a) fully dense titanium implant (b) cellularimplant with uniform relative density of 50 (c) graded cellular implant (solution B in Fig 6)

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-7

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

compared with the one obtained by Kowalczyk for a 3D model[4] The mean and the maximum values of interface shear stressfor the 3D titanium implant in the work by Kowalczyk [4] are057 and 28 MPa respectively These values are respectively031 and 215 MPa for the titanium implant in this paper The con-tribution to the higher level of shear stress in the 3D model ofKowalczyk is the distribution of shear force on a smaller area InKowalczykrsquos study [4] the implant and bone are bonded only atthe proximal region while in our work the whole bone-implantinterface is bonded which results in a decrease of the mean andthe maximum values of interface shear stress

The cellular implant with uniform relative density of 50 isapproximately three times more flexible than the titanium stemThis implant can qualitatively simulate the behavior of an implantmade out of tantalum foam For this stem the amount of boneresorption and the interface failure index are about 34 and 287respectively and the interface failure is maximum (071) at theedge of proximal region Compared to the solid titanium implantthe amount of bone resorption decreases by 50 whereas themaximum interface failure increases about 40 This shows that adecrease of the implant stiffness with uniform porosity distribu-tion aiming at reducing bone resorption has the undesirable effectof increasing the risk of interface failure at the proximal regionThis outcome confirms the findings of the previous work byKuiper and Huiskes [14]

Figure 7(c) and 8(c) show the results for the graded cellularimplant B Its bone resorption and interface failure index are 16and 115 respectively The peak value of the local interface fail-ure is 025 Compared to the titanium stem both the amount ofbone resorption and the peak of interface failure decrease of 76and 50 respectively With respect to the uniformly-distributedcellular implant the decrease of bone resorption and interface fail-ure peak is of 53 and 65 respectively A graded cellularimplant with optimized relative density distribution is thus capa-ble of reducing concurrently both the conflicting objective func-tions In particular bone resorption reduces as a result of thecellular material which makes the implant more compliant theinterface stress on the other hand is minimized by the optimizedgradients of cellular material

A proof-of-concept implant was built to verify the manufactur-ability of the optimum grade lattice material Figure 9 shows thepolypropylene prototype of solution B which was manufacturedwith a 3D printer Objet Connex500 [67] A uniform tessellationand a square unit cell of 18 mm size were assumed to draw themodel The cell geometry was calculated from the average rela-tive density obtained from the method described in this paper AnSTL file of the graded cellular implant solution B was finallyused for rapid prototyping

The limitations of the method proposed in this paper are herediscussed First the method tackles the design of an implant stati-cally loaded in the stance phase of walking Further research isrequired to extend it to variable loading and fatigue life design

Second the accuracy of the asymptotic homogenization needsto be investigated at the vicinity of the implant borders Althoughmacroscopic mean stress fields on the boundaries are satisfied inthe global solutions [4068] microscopic stresses do not satisfythe Y-periodicity assumption These boundary effects howevermay be captured by applying a boundary layer corrector [6869] aspatially decaying stress localization function [70] or an adaptivemultiscale methodology [71] To investigate the accuracy of thesetechniques it is required to construct a detailed finite elementmodel and compare the actual stressstrain distribution inside thecell walls with the microscopic stressesstrains estimated from thehomogenization method This task will be performed in future

Third to extend the proposed procedure to three dimensions a3D cell topology with high porosity interconnected pore struc-ture large surface area with suitable textures and good

Fig 8 Distribution of local interface failure f ethrTHORN around (a) fully dense titanium implant(b) cellular implant with uniform relative density of 50 (c) graded cellular implant (solu-tion B in Fig 6)

Fig 9 Polypropylene proof-of-concept of the optimal gradedcellular implant (solution B)

031004-8 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

mechanical properties should be selected [7273] The asymptotichomogenization will be applied to characterize tissue scaffoldingcell topologies for a wide range of relative density Their effectivemoduli yield and ultimate surfaces for multiaxial loading condi-tions will be obtained to create a standard library from which toselect suitable cell geometries for a 3D cellular implant design

Fourth bone has been here considered as consisting of a cancel-lous and a cortical part with isotropic mechanical properties Fur-ther study is necessary to improve the constitutive material modelof bone and to consider bone as a graded cellular material

Finally in the optimization procedure a simple formulation hasbeen used to quantitatively measure both bone resorption andimplant stability In a future study detailed simulations of implantstability [74ndash77] and bone remodeling process [6077ndash80] can beperformed to assess both the short and long term performance ofthe implant

5 Concluding Remarks

This work has presented a methodology integrating multiscaleanalysis and design optimization to design a novel hip implantmade of graded cellular material The method can contribute tothe development of a new generation of orthopedic implants witha graded cellular microstructure that will reduce the clinical con-sequences of current implants

In the first part of the paper the homogenization method hasbeen reviewed and then used to capture the mechanics of theimplant at the micro and macro scale In the second part multiob-jective optimization has been applied to find optimum gradients ofmaterial distribution that minimize concurrently bone resorptionand bone-implant interface stresses The results have shown thatthe optimized cellular implant exhibits a reduction of 76 ofbone resorption and 50 of interface stress with respect to a fullydense titanium implant

References[1] Kurtz S Ong K Lau E Mowat F and Halpern M 2007 ldquoProjections of

Primary and Revision Hip and Knee Arthroplasty in the United States from2005 to 2030rdquo J Bone Jt Surg Am Vol 89(4) pp 780ndash785

[2] Vichinsky E P Neumayr L D Haberkern C Earles A N Eckman JKoshy M and Black D M 1999 ldquoThe Perioperative Complication Rate ofOrthopedic Surgery in Sickle Cell Disease Report of the National Sickle CellSurgery Study Grouprdquo Am J Hematol 62(3) pp 129ndash138

[3] Kobayashi S Saito N Horiuchi H Iorio R and Takaoka K 2000 ldquoPoorBone Quality or Hip Structure as Risk Factors Affecting Survival of Total-HipArthroplastyrdquo The Lancet 355(9214) pp 1499ndash1504

[4] Kowalczyk P 2001 ldquoDesign Optimization of Cementless Femoral Hip Pros-theses Using Finite Element Analysisrdquo J Biomech Eng 123(5) pp 396ndash402

[5] Moen T C Ghate R Salaz N Ghodasra J and Stulberg S D 2011 ldquoAMonoblock Porous Tantalum Acetabular Cup Has No Osteolysis on Ct at 10Yearsrdquo Clin Orthop Relat Res 469(2) pp 382ndash386

[6] Kurtz S Gawel H and Patel J 2011 ldquoHistory and Systematic Review ofWear and Osteolysis Outcomes for First-Generation Highly Crosslinked Poly-ethylenerdquo Clin Orthop Relat Res 469(8) pp 2262ndash2277

[7] Grubl A Marker M Brodner W Giurea A Heinze G Meisinger VZehetgruber H and Kotz R 2007 ldquoLong Term Follow up of Metal on MetalTotal Hip Replacementrdquo J Orthop Res 25(7) pp 841ndash848

[8] Neumann D R P Thaler C Hitzl W Huber M Hofstadter T and DornU 2010 ldquoLong-Term Results of a Contemporary Metal-on-Metal Total HipArthroplasty A 10-Year Follow-up Studyrdquo J Arthroplasty 25(5) pp700ndash708

[9] Glassman A Bobyn J and Tanzer M 2006 ldquoNew Femoral Designs DoThey Influence Stress Shieldingrdquo Clin Orthop Relat Res 453(12) pp64ndash74

[10] Adam F Hammer D S Pfautsch S and Westermann K 2002 ldquoEarly Fail-ure of a Press-Fit Carbon Fiber Hip Prosthesis with a Smooth Surfacerdquo JAntroplasty 17(2) pp 217ndash223

[11] Trebse R Milosev I Kovac S Mikek M and Pisot V 2005 ldquoPoorResults from the Isoelastic Total Hip Replacementrdquo Acta Orthop 76(2) pp169ndash176

[12] Harvey E Bobyn J Tanzer M Stackpool G Krygier J and Hacking S1999 ldquoEffect of Flexibility of the Femoral Stem on Bone-Remodeling and Fix-ation of the Stem in a Canine Total Hip Arthroplasty Model without CementrdquoJ Bone Jt Surg 81(1) pp 93ndash107

[13] Huiskes R Weinans H and Rietbergen B 1992 ldquoThe Relationship betweenStress Shielding and Bone Resorption around Total Hip Stems and the Effectsof Flexible Materialsrdquo Clin Orthop Relat Res 274(1) pp 124ndash134

[14] Kuiper J and Huiskes R 1992 ldquoNumerical Optimization of Hip-ProstheticStem Materialrdquo Recent Advances in Computer Methods in Biomechanics andBiomedical Engineering J Middleton G N Pande and K R Williams edsBooks and Journals International Ltd Swansea pp 76ndash84

[15] Kuiper J H and Huiskes R 1997 ldquoMathematical Optimization of ElasticProperties Application to Cementless Hip Stem Designrdquo J Biomech Eng119(2) pp 166ndash174

[16] Hedia H Shabara M El-Midany T and Fouda N 2006 ldquoImproved Designof Cementless Hip Stems Using Two-Dimensional Functionally Graded Materi-alsrdquo J Biomed Mater Res Part B Appl Biomater 79(1) pp 42ndash49

[17] Hedia H S and Mahmoud N-A 2004 ldquoDesign Optimization of Function-ally Graded Dental Implantrdquo Biomed Mater Eng 14(2) pp 133ndash143

[18] Watari F Yokoyama A Saso F Uo M and Kawasaki T 1997ldquoFabrication and Properties of Functionally Graded Dental Implantrdquo Compo-sites Part B 28(1-2) pp 5ndash11

[19] Katti K S 2004 ldquoBiomaterials in Total Joint Replacementrdquo Colloids Surf B39(3) pp 133ndash142

[20] Thompson I and Hench L 1998 ldquoMechanical Properties of BioactiveGlasses Glass-Ceramics and Compositesrdquo Proc Inst Mech Eng Part H JEng Med 212(2) pp 127

[21] Fraldi M Esposito L Perrella G Cutolo A and Cowin S 2010ldquoTopological Optimization in Hip Prosthesis Designrdquo Biomech ModelMechanobiol 9(4) pp 389ndash402

[22] Bobyn J D Poggie R Krygier J Lewallen D Hanssen A Lewis RUnger A Orsquokeefe T Christie M and Nasser S 2004 ldquoClinical Validationof a Structural Porous Tantalum Biomaterial for Adult Reconstructionrdquo J BoneJt Surg 86(Supplement 2) pp 123ndash129

[23] Bobyn J Stackpool G Hacking S Tanzer M and Krygier J 1999ldquoCharacteristics of Bone Ingrowth and Interface Mechanics of a New PorousTantalum Biomaterialrdquo J Bone Jt Surg Br Vol 81(5) pp 907ndash914

[24] Parthasarathy J Starly B Raman S and Christensen A 2010 ldquoMechanicalEvaluation of Porous Titanium (Ti6al4v) Structures with Electron Beam Melt-ing (Ebm)rdquo J Mech Behav Biomed Mater 3(3) pp 249ndash259

[25] Heinl P Muller L Korner C Singer R F and Muller F A 2008ldquoCellular Ti-6al-4v Structures with Interconnected Macro Porosity for BoneImplants Fabricated by Selective Electron Beam Meltingrdquo Acta Biomater4(5) pp 1536ndash1544

[26] Stamp R Fox P Orsquoneill W Jones E and Sutcliffe C 2009 ldquoThe Devel-opment of a Scanning Strategy for the Manufacture of Porous Biomaterials bySelective Laser Meltingrdquo J Mater Sci Mater Med 20(9) pp 1839ndash1848

[27] Yang S Leong K Du Z and Chua C 2002 ldquoThe Design of Scaffolds forUse in Tissue Engineering Part Ii Rapid Prototyping Techniquesrdquo TissueEng 8(1) pp 1ndash11

[28] Murr L Gaytan S Medina F Lopez H Martinez E Machado B Hernan-dez D Martinez L Lopez M and Wicker R 2010 ldquoNext-Generation Biomed-ical Implants Using Additive Manufacturing of Complex Cellular and FunctionalMesh Arraysrdquo Philos Trans R Soc London Ser A 368(1917) pp 1999ndash2032

[29] Coelho P Fernandes P Guedes J and Rodrigues H 2008 ldquoA HierarchicalModel for Concurrent Material and Topology Optimisation of Three-Dimensional Structuresrdquo Struct Multidiscip Optim 35(2) pp 107ndash115

[30] Rodrigues H Guedes J and Bendsoe M 2002 ldquoHierarchical Optimizationof Material and Structurerdquo Struct Multidiscip Optim 24(1) pp 1ndash10

[31] Goncalves Coelho P Rui Fernandes P and Carrico Rodrigues H 2011ldquoMultiscale Modeling of Bone Tissue with Surface and Permeability ControlrdquoJ Biomech 44(2) pp 321ndash329

[32] Coelho P G Cardoso J B Fernandes P R and Rodrigues H C 2011ldquoParallel Computing Techniques Applied to the Simultaneous Design of Struc-ture and Materialrdquo Adv Eng Software 42(5) pp 219ndash227

[33] Gibson L J and Ashby M F 1999 Cellular Solids Structure and Proper-ties Cambridge University Press Cambridge UK

[34] Guedes J and Kikuchi N 1990 ldquoPreprocessing and Postprocessing for Mate-rials Based on the Homogenization Method with Adaptive Finite ElementMethodsrdquo Comput Methods Appl Mech Eng 83(2) pp 143ndash198

[35] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization I-Homogenization Theory for Media with PeriodicStructurerdquo Comput Struct 69(6) pp 707ndash717

[36] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Opimization I I-Analytical and Numerical Solution of HomogenizationEquationsrdquo Comput Struct 69(6) pp 719ndash738

[37] Fang Z Starly B and Sun W 2005 ldquoComputer-Aided Characterization forEffective Mechanical Properties of Porous Tissue Scaffoldsrdquo Comput-AidedDes 37(1) pp 65ndash72

[38] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics 6th ed Butterworth-Heinemann Burlington

[39] Hollister S and Kikuchi N 1992 ldquoA Comparison of Homogenization andStandard Mechanics Analyses for Periodic Porous Compositesrdquo ComputMech 10(2) pp 73ndash95

[40] Pellegrino C Galvanetto U and Schrefler B 1999 ldquoNumerical Homogeni-zation of Periodic Composite Materials with Non Linear MaterialComponentsrdquo Int J Numer Methods Eng 46(10) pp 1609ndash1637

[41] Bragdon C Jasty M Greene M Rubash H and Harris W 2004ldquoBiologic Fixation of Total Hip Implants Insights Gained from a Series of Ca-nine Studiesrdquo J Bone Jt Surg 86(Supplement 2) pp 105ndash117

[42] Harrysson O L A Cansizoglu O Marcellin-Little D J Cormier D R andWest Ii H A 2008 ldquoDirect Metal Fabrication of Titanium Implants with Tai-lored Materials and Mechanical Properties Using Electron Beam Melting Tech-nologyrdquo Mater Sci Eng C 28(3) pp 366ndash373

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-9

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

for bone ingrowth [41] By comparing implants C and A weobserve that a raise of the implant porosity from point C to Aresults in an implant stiffness decrease which on one hand low-ers bone loss and on the other hand enhances the risk of interfacefailure When solution B is compared to C a reduction of 8 ofbone resorption is noted with a slight increase of the peak value ofthe interface failure On the other hand by contrasting solution Bto A we note a significant increase (60) of the peak value ofinterface failure which is below the Hoffman failure strength anda minor reduction (2) of the amount of bone resorption Themain benefit of solution A is the maximum porosity of the micro-structure that can promote bone ingrowth Solution B on the otherhand might be the preferred solution with respect to low boneresorption and interface failure These observations emerge onlyby inspection of the objective functions selected in this work weremark here that other parameters should be taken into accountfor the selection of the best implant These include patientrsquos bone

characteristics the range of activity age and desired level ofbone mass preservation after implantation

For prescribed geometric loading and constraint conditions wenow compare the metrics of resorbed bone mass (mr) and distribu-tion of interface stress (f ethrbTHORN) of the optimal solution B with thoseof (i) a currently-used fully dense titanium stem and (ii) a cellularimplant with a uniformly distributed relative density of 50 Fig-ures 7 and 8 illustrate the results of the comparison

For the solid titanium stem the amount of bone resorption cal-culated through Eq (7) is 67 and the interface failure indexFethbTHORN obtained from Eq (8) is 133 Using the distribution of f ethrTHORNgenerated around the titanium stem (Fig 8(a)) we observe thatthe maximum value of interface failure (051) occurs at the distalend of the implant As expected this implant is much stiffer thanthe surrounding bone thereby resulting in a higher amount ofbone resorption For the numerical validation the interface shearstress of the titanium implant at the proximal region is also

Fig 6 Trade-off distributions of relative density for the optimized cellular implant

Fig 7 Distribution of bone resorption around (a) fully dense titanium implant (b) cellularimplant with uniform relative density of 50 (c) graded cellular implant (solution B in Fig 6)

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-7

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

compared with the one obtained by Kowalczyk for a 3D model[4] The mean and the maximum values of interface shear stressfor the 3D titanium implant in the work by Kowalczyk [4] are057 and 28 MPa respectively These values are respectively031 and 215 MPa for the titanium implant in this paper The con-tribution to the higher level of shear stress in the 3D model ofKowalczyk is the distribution of shear force on a smaller area InKowalczykrsquos study [4] the implant and bone are bonded only atthe proximal region while in our work the whole bone-implantinterface is bonded which results in a decrease of the mean andthe maximum values of interface shear stress

The cellular implant with uniform relative density of 50 isapproximately three times more flexible than the titanium stemThis implant can qualitatively simulate the behavior of an implantmade out of tantalum foam For this stem the amount of boneresorption and the interface failure index are about 34 and 287respectively and the interface failure is maximum (071) at theedge of proximal region Compared to the solid titanium implantthe amount of bone resorption decreases by 50 whereas themaximum interface failure increases about 40 This shows that adecrease of the implant stiffness with uniform porosity distribu-tion aiming at reducing bone resorption has the undesirable effectof increasing the risk of interface failure at the proximal regionThis outcome confirms the findings of the previous work byKuiper and Huiskes [14]

Figure 7(c) and 8(c) show the results for the graded cellularimplant B Its bone resorption and interface failure index are 16and 115 respectively The peak value of the local interface fail-ure is 025 Compared to the titanium stem both the amount ofbone resorption and the peak of interface failure decrease of 76and 50 respectively With respect to the uniformly-distributedcellular implant the decrease of bone resorption and interface fail-ure peak is of 53 and 65 respectively A graded cellularimplant with optimized relative density distribution is thus capa-ble of reducing concurrently both the conflicting objective func-tions In particular bone resorption reduces as a result of thecellular material which makes the implant more compliant theinterface stress on the other hand is minimized by the optimizedgradients of cellular material

A proof-of-concept implant was built to verify the manufactur-ability of the optimum grade lattice material Figure 9 shows thepolypropylene prototype of solution B which was manufacturedwith a 3D printer Objet Connex500 [67] A uniform tessellationand a square unit cell of 18 mm size were assumed to draw themodel The cell geometry was calculated from the average rela-tive density obtained from the method described in this paper AnSTL file of the graded cellular implant solution B was finallyused for rapid prototyping

The limitations of the method proposed in this paper are herediscussed First the method tackles the design of an implant stati-cally loaded in the stance phase of walking Further research isrequired to extend it to variable loading and fatigue life design

Second the accuracy of the asymptotic homogenization needsto be investigated at the vicinity of the implant borders Althoughmacroscopic mean stress fields on the boundaries are satisfied inthe global solutions [4068] microscopic stresses do not satisfythe Y-periodicity assumption These boundary effects howevermay be captured by applying a boundary layer corrector [6869] aspatially decaying stress localization function [70] or an adaptivemultiscale methodology [71] To investigate the accuracy of thesetechniques it is required to construct a detailed finite elementmodel and compare the actual stressstrain distribution inside thecell walls with the microscopic stressesstrains estimated from thehomogenization method This task will be performed in future

Third to extend the proposed procedure to three dimensions a3D cell topology with high porosity interconnected pore struc-ture large surface area with suitable textures and good

Fig 8 Distribution of local interface failure f ethrTHORN around (a) fully dense titanium implant(b) cellular implant with uniform relative density of 50 (c) graded cellular implant (solu-tion B in Fig 6)

Fig 9 Polypropylene proof-of-concept of the optimal gradedcellular implant (solution B)

031004-8 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

mechanical properties should be selected [7273] The asymptotichomogenization will be applied to characterize tissue scaffoldingcell topologies for a wide range of relative density Their effectivemoduli yield and ultimate surfaces for multiaxial loading condi-tions will be obtained to create a standard library from which toselect suitable cell geometries for a 3D cellular implant design

Fourth bone has been here considered as consisting of a cancel-lous and a cortical part with isotropic mechanical properties Fur-ther study is necessary to improve the constitutive material modelof bone and to consider bone as a graded cellular material

Finally in the optimization procedure a simple formulation hasbeen used to quantitatively measure both bone resorption andimplant stability In a future study detailed simulations of implantstability [74ndash77] and bone remodeling process [6077ndash80] can beperformed to assess both the short and long term performance ofthe implant

5 Concluding Remarks

This work has presented a methodology integrating multiscaleanalysis and design optimization to design a novel hip implantmade of graded cellular material The method can contribute tothe development of a new generation of orthopedic implants witha graded cellular microstructure that will reduce the clinical con-sequences of current implants

In the first part of the paper the homogenization method hasbeen reviewed and then used to capture the mechanics of theimplant at the micro and macro scale In the second part multiob-jective optimization has been applied to find optimum gradients ofmaterial distribution that minimize concurrently bone resorptionand bone-implant interface stresses The results have shown thatthe optimized cellular implant exhibits a reduction of 76 ofbone resorption and 50 of interface stress with respect to a fullydense titanium implant

References[1] Kurtz S Ong K Lau E Mowat F and Halpern M 2007 ldquoProjections of

Primary and Revision Hip and Knee Arthroplasty in the United States from2005 to 2030rdquo J Bone Jt Surg Am Vol 89(4) pp 780ndash785

[2] Vichinsky E P Neumayr L D Haberkern C Earles A N Eckman JKoshy M and Black D M 1999 ldquoThe Perioperative Complication Rate ofOrthopedic Surgery in Sickle Cell Disease Report of the National Sickle CellSurgery Study Grouprdquo Am J Hematol 62(3) pp 129ndash138

[3] Kobayashi S Saito N Horiuchi H Iorio R and Takaoka K 2000 ldquoPoorBone Quality or Hip Structure as Risk Factors Affecting Survival of Total-HipArthroplastyrdquo The Lancet 355(9214) pp 1499ndash1504

[4] Kowalczyk P 2001 ldquoDesign Optimization of Cementless Femoral Hip Pros-theses Using Finite Element Analysisrdquo J Biomech Eng 123(5) pp 396ndash402

[5] Moen T C Ghate R Salaz N Ghodasra J and Stulberg S D 2011 ldquoAMonoblock Porous Tantalum Acetabular Cup Has No Osteolysis on Ct at 10Yearsrdquo Clin Orthop Relat Res 469(2) pp 382ndash386

[6] Kurtz S Gawel H and Patel J 2011 ldquoHistory and Systematic Review ofWear and Osteolysis Outcomes for First-Generation Highly Crosslinked Poly-ethylenerdquo Clin Orthop Relat Res 469(8) pp 2262ndash2277

[7] Grubl A Marker M Brodner W Giurea A Heinze G Meisinger VZehetgruber H and Kotz R 2007 ldquoLong Term Follow up of Metal on MetalTotal Hip Replacementrdquo J Orthop Res 25(7) pp 841ndash848

[8] Neumann D R P Thaler C Hitzl W Huber M Hofstadter T and DornU 2010 ldquoLong-Term Results of a Contemporary Metal-on-Metal Total HipArthroplasty A 10-Year Follow-up Studyrdquo J Arthroplasty 25(5) pp700ndash708

[9] Glassman A Bobyn J and Tanzer M 2006 ldquoNew Femoral Designs DoThey Influence Stress Shieldingrdquo Clin Orthop Relat Res 453(12) pp64ndash74

[10] Adam F Hammer D S Pfautsch S and Westermann K 2002 ldquoEarly Fail-ure of a Press-Fit Carbon Fiber Hip Prosthesis with a Smooth Surfacerdquo JAntroplasty 17(2) pp 217ndash223

[11] Trebse R Milosev I Kovac S Mikek M and Pisot V 2005 ldquoPoorResults from the Isoelastic Total Hip Replacementrdquo Acta Orthop 76(2) pp169ndash176

[12] Harvey E Bobyn J Tanzer M Stackpool G Krygier J and Hacking S1999 ldquoEffect of Flexibility of the Femoral Stem on Bone-Remodeling and Fix-ation of the Stem in a Canine Total Hip Arthroplasty Model without CementrdquoJ Bone Jt Surg 81(1) pp 93ndash107

[13] Huiskes R Weinans H and Rietbergen B 1992 ldquoThe Relationship betweenStress Shielding and Bone Resorption around Total Hip Stems and the Effectsof Flexible Materialsrdquo Clin Orthop Relat Res 274(1) pp 124ndash134

[14] Kuiper J and Huiskes R 1992 ldquoNumerical Optimization of Hip-ProstheticStem Materialrdquo Recent Advances in Computer Methods in Biomechanics andBiomedical Engineering J Middleton G N Pande and K R Williams edsBooks and Journals International Ltd Swansea pp 76ndash84

[15] Kuiper J H and Huiskes R 1997 ldquoMathematical Optimization of ElasticProperties Application to Cementless Hip Stem Designrdquo J Biomech Eng119(2) pp 166ndash174

[16] Hedia H Shabara M El-Midany T and Fouda N 2006 ldquoImproved Designof Cementless Hip Stems Using Two-Dimensional Functionally Graded Materi-alsrdquo J Biomed Mater Res Part B Appl Biomater 79(1) pp 42ndash49

[17] Hedia H S and Mahmoud N-A 2004 ldquoDesign Optimization of Function-ally Graded Dental Implantrdquo Biomed Mater Eng 14(2) pp 133ndash143

[18] Watari F Yokoyama A Saso F Uo M and Kawasaki T 1997ldquoFabrication and Properties of Functionally Graded Dental Implantrdquo Compo-sites Part B 28(1-2) pp 5ndash11

[19] Katti K S 2004 ldquoBiomaterials in Total Joint Replacementrdquo Colloids Surf B39(3) pp 133ndash142

[20] Thompson I and Hench L 1998 ldquoMechanical Properties of BioactiveGlasses Glass-Ceramics and Compositesrdquo Proc Inst Mech Eng Part H JEng Med 212(2) pp 127

[21] Fraldi M Esposito L Perrella G Cutolo A and Cowin S 2010ldquoTopological Optimization in Hip Prosthesis Designrdquo Biomech ModelMechanobiol 9(4) pp 389ndash402

[22] Bobyn J D Poggie R Krygier J Lewallen D Hanssen A Lewis RUnger A Orsquokeefe T Christie M and Nasser S 2004 ldquoClinical Validationof a Structural Porous Tantalum Biomaterial for Adult Reconstructionrdquo J BoneJt Surg 86(Supplement 2) pp 123ndash129

[23] Bobyn J Stackpool G Hacking S Tanzer M and Krygier J 1999ldquoCharacteristics of Bone Ingrowth and Interface Mechanics of a New PorousTantalum Biomaterialrdquo J Bone Jt Surg Br Vol 81(5) pp 907ndash914

[24] Parthasarathy J Starly B Raman S and Christensen A 2010 ldquoMechanicalEvaluation of Porous Titanium (Ti6al4v) Structures with Electron Beam Melt-ing (Ebm)rdquo J Mech Behav Biomed Mater 3(3) pp 249ndash259

[25] Heinl P Muller L Korner C Singer R F and Muller F A 2008ldquoCellular Ti-6al-4v Structures with Interconnected Macro Porosity for BoneImplants Fabricated by Selective Electron Beam Meltingrdquo Acta Biomater4(5) pp 1536ndash1544

[26] Stamp R Fox P Orsquoneill W Jones E and Sutcliffe C 2009 ldquoThe Devel-opment of a Scanning Strategy for the Manufacture of Porous Biomaterials bySelective Laser Meltingrdquo J Mater Sci Mater Med 20(9) pp 1839ndash1848

[27] Yang S Leong K Du Z and Chua C 2002 ldquoThe Design of Scaffolds forUse in Tissue Engineering Part Ii Rapid Prototyping Techniquesrdquo TissueEng 8(1) pp 1ndash11

[28] Murr L Gaytan S Medina F Lopez H Martinez E Machado B Hernan-dez D Martinez L Lopez M and Wicker R 2010 ldquoNext-Generation Biomed-ical Implants Using Additive Manufacturing of Complex Cellular and FunctionalMesh Arraysrdquo Philos Trans R Soc London Ser A 368(1917) pp 1999ndash2032

[29] Coelho P Fernandes P Guedes J and Rodrigues H 2008 ldquoA HierarchicalModel for Concurrent Material and Topology Optimisation of Three-Dimensional Structuresrdquo Struct Multidiscip Optim 35(2) pp 107ndash115

[30] Rodrigues H Guedes J and Bendsoe M 2002 ldquoHierarchical Optimizationof Material and Structurerdquo Struct Multidiscip Optim 24(1) pp 1ndash10

[31] Goncalves Coelho P Rui Fernandes P and Carrico Rodrigues H 2011ldquoMultiscale Modeling of Bone Tissue with Surface and Permeability ControlrdquoJ Biomech 44(2) pp 321ndash329

[32] Coelho P G Cardoso J B Fernandes P R and Rodrigues H C 2011ldquoParallel Computing Techniques Applied to the Simultaneous Design of Struc-ture and Materialrdquo Adv Eng Software 42(5) pp 219ndash227

[33] Gibson L J and Ashby M F 1999 Cellular Solids Structure and Proper-ties Cambridge University Press Cambridge UK

[34] Guedes J and Kikuchi N 1990 ldquoPreprocessing and Postprocessing for Mate-rials Based on the Homogenization Method with Adaptive Finite ElementMethodsrdquo Comput Methods Appl Mech Eng 83(2) pp 143ndash198

[35] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization I-Homogenization Theory for Media with PeriodicStructurerdquo Comput Struct 69(6) pp 707ndash717

[36] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Opimization I I-Analytical and Numerical Solution of HomogenizationEquationsrdquo Comput Struct 69(6) pp 719ndash738

[37] Fang Z Starly B and Sun W 2005 ldquoComputer-Aided Characterization forEffective Mechanical Properties of Porous Tissue Scaffoldsrdquo Comput-AidedDes 37(1) pp 65ndash72

[38] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics 6th ed Butterworth-Heinemann Burlington

[39] Hollister S and Kikuchi N 1992 ldquoA Comparison of Homogenization andStandard Mechanics Analyses for Periodic Porous Compositesrdquo ComputMech 10(2) pp 73ndash95

[40] Pellegrino C Galvanetto U and Schrefler B 1999 ldquoNumerical Homogeni-zation of Periodic Composite Materials with Non Linear MaterialComponentsrdquo Int J Numer Methods Eng 46(10) pp 1609ndash1637

[41] Bragdon C Jasty M Greene M Rubash H and Harris W 2004ldquoBiologic Fixation of Total Hip Implants Insights Gained from a Series of Ca-nine Studiesrdquo J Bone Jt Surg 86(Supplement 2) pp 105ndash117

[42] Harrysson O L A Cansizoglu O Marcellin-Little D J Cormier D R andWest Ii H A 2008 ldquoDirect Metal Fabrication of Titanium Implants with Tai-lored Materials and Mechanical Properties Using Electron Beam Melting Tech-nologyrdquo Mater Sci Eng C 28(3) pp 366ndash373

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-9

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

compared with the one obtained by Kowalczyk for a 3D model[4] The mean and the maximum values of interface shear stressfor the 3D titanium implant in the work by Kowalczyk [4] are057 and 28 MPa respectively These values are respectively031 and 215 MPa for the titanium implant in this paper The con-tribution to the higher level of shear stress in the 3D model ofKowalczyk is the distribution of shear force on a smaller area InKowalczykrsquos study [4] the implant and bone are bonded only atthe proximal region while in our work the whole bone-implantinterface is bonded which results in a decrease of the mean andthe maximum values of interface shear stress

The cellular implant with uniform relative density of 50 isapproximately three times more flexible than the titanium stemThis implant can qualitatively simulate the behavior of an implantmade out of tantalum foam For this stem the amount of boneresorption and the interface failure index are about 34 and 287respectively and the interface failure is maximum (071) at theedge of proximal region Compared to the solid titanium implantthe amount of bone resorption decreases by 50 whereas themaximum interface failure increases about 40 This shows that adecrease of the implant stiffness with uniform porosity distribu-tion aiming at reducing bone resorption has the undesirable effectof increasing the risk of interface failure at the proximal regionThis outcome confirms the findings of the previous work byKuiper and Huiskes [14]

Figure 7(c) and 8(c) show the results for the graded cellularimplant B Its bone resorption and interface failure index are 16and 115 respectively The peak value of the local interface fail-ure is 025 Compared to the titanium stem both the amount ofbone resorption and the peak of interface failure decrease of 76and 50 respectively With respect to the uniformly-distributedcellular implant the decrease of bone resorption and interface fail-ure peak is of 53 and 65 respectively A graded cellularimplant with optimized relative density distribution is thus capa-ble of reducing concurrently both the conflicting objective func-tions In particular bone resorption reduces as a result of thecellular material which makes the implant more compliant theinterface stress on the other hand is minimized by the optimizedgradients of cellular material

A proof-of-concept implant was built to verify the manufactur-ability of the optimum grade lattice material Figure 9 shows thepolypropylene prototype of solution B which was manufacturedwith a 3D printer Objet Connex500 [67] A uniform tessellationand a square unit cell of 18 mm size were assumed to draw themodel The cell geometry was calculated from the average rela-tive density obtained from the method described in this paper AnSTL file of the graded cellular implant solution B was finallyused for rapid prototyping

The limitations of the method proposed in this paper are herediscussed First the method tackles the design of an implant stati-cally loaded in the stance phase of walking Further research isrequired to extend it to variable loading and fatigue life design

Second the accuracy of the asymptotic homogenization needsto be investigated at the vicinity of the implant borders Althoughmacroscopic mean stress fields on the boundaries are satisfied inthe global solutions [4068] microscopic stresses do not satisfythe Y-periodicity assumption These boundary effects howevermay be captured by applying a boundary layer corrector [6869] aspatially decaying stress localization function [70] or an adaptivemultiscale methodology [71] To investigate the accuracy of thesetechniques it is required to construct a detailed finite elementmodel and compare the actual stressstrain distribution inside thecell walls with the microscopic stressesstrains estimated from thehomogenization method This task will be performed in future

Third to extend the proposed procedure to three dimensions a3D cell topology with high porosity interconnected pore struc-ture large surface area with suitable textures and good

Fig 8 Distribution of local interface failure f ethrTHORN around (a) fully dense titanium implant(b) cellular implant with uniform relative density of 50 (c) graded cellular implant (solu-tion B in Fig 6)

Fig 9 Polypropylene proof-of-concept of the optimal gradedcellular implant (solution B)

031004-8 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

mechanical properties should be selected [7273] The asymptotichomogenization will be applied to characterize tissue scaffoldingcell topologies for a wide range of relative density Their effectivemoduli yield and ultimate surfaces for multiaxial loading condi-tions will be obtained to create a standard library from which toselect suitable cell geometries for a 3D cellular implant design

Fourth bone has been here considered as consisting of a cancel-lous and a cortical part with isotropic mechanical properties Fur-ther study is necessary to improve the constitutive material modelof bone and to consider bone as a graded cellular material

Finally in the optimization procedure a simple formulation hasbeen used to quantitatively measure both bone resorption andimplant stability In a future study detailed simulations of implantstability [74ndash77] and bone remodeling process [6077ndash80] can beperformed to assess both the short and long term performance ofthe implant

5 Concluding Remarks

This work has presented a methodology integrating multiscaleanalysis and design optimization to design a novel hip implantmade of graded cellular material The method can contribute tothe development of a new generation of orthopedic implants witha graded cellular microstructure that will reduce the clinical con-sequences of current implants

In the first part of the paper the homogenization method hasbeen reviewed and then used to capture the mechanics of theimplant at the micro and macro scale In the second part multiob-jective optimization has been applied to find optimum gradients ofmaterial distribution that minimize concurrently bone resorptionand bone-implant interface stresses The results have shown thatthe optimized cellular implant exhibits a reduction of 76 ofbone resorption and 50 of interface stress with respect to a fullydense titanium implant

References[1] Kurtz S Ong K Lau E Mowat F and Halpern M 2007 ldquoProjections of

Primary and Revision Hip and Knee Arthroplasty in the United States from2005 to 2030rdquo J Bone Jt Surg Am Vol 89(4) pp 780ndash785

[2] Vichinsky E P Neumayr L D Haberkern C Earles A N Eckman JKoshy M and Black D M 1999 ldquoThe Perioperative Complication Rate ofOrthopedic Surgery in Sickle Cell Disease Report of the National Sickle CellSurgery Study Grouprdquo Am J Hematol 62(3) pp 129ndash138

[3] Kobayashi S Saito N Horiuchi H Iorio R and Takaoka K 2000 ldquoPoorBone Quality or Hip Structure as Risk Factors Affecting Survival of Total-HipArthroplastyrdquo The Lancet 355(9214) pp 1499ndash1504

[4] Kowalczyk P 2001 ldquoDesign Optimization of Cementless Femoral Hip Pros-theses Using Finite Element Analysisrdquo J Biomech Eng 123(5) pp 396ndash402

[5] Moen T C Ghate R Salaz N Ghodasra J and Stulberg S D 2011 ldquoAMonoblock Porous Tantalum Acetabular Cup Has No Osteolysis on Ct at 10Yearsrdquo Clin Orthop Relat Res 469(2) pp 382ndash386

[6] Kurtz S Gawel H and Patel J 2011 ldquoHistory and Systematic Review ofWear and Osteolysis Outcomes for First-Generation Highly Crosslinked Poly-ethylenerdquo Clin Orthop Relat Res 469(8) pp 2262ndash2277

[7] Grubl A Marker M Brodner W Giurea A Heinze G Meisinger VZehetgruber H and Kotz R 2007 ldquoLong Term Follow up of Metal on MetalTotal Hip Replacementrdquo J Orthop Res 25(7) pp 841ndash848

[8] Neumann D R P Thaler C Hitzl W Huber M Hofstadter T and DornU 2010 ldquoLong-Term Results of a Contemporary Metal-on-Metal Total HipArthroplasty A 10-Year Follow-up Studyrdquo J Arthroplasty 25(5) pp700ndash708

[9] Glassman A Bobyn J and Tanzer M 2006 ldquoNew Femoral Designs DoThey Influence Stress Shieldingrdquo Clin Orthop Relat Res 453(12) pp64ndash74

[10] Adam F Hammer D S Pfautsch S and Westermann K 2002 ldquoEarly Fail-ure of a Press-Fit Carbon Fiber Hip Prosthesis with a Smooth Surfacerdquo JAntroplasty 17(2) pp 217ndash223

[11] Trebse R Milosev I Kovac S Mikek M and Pisot V 2005 ldquoPoorResults from the Isoelastic Total Hip Replacementrdquo Acta Orthop 76(2) pp169ndash176

[12] Harvey E Bobyn J Tanzer M Stackpool G Krygier J and Hacking S1999 ldquoEffect of Flexibility of the Femoral Stem on Bone-Remodeling and Fix-ation of the Stem in a Canine Total Hip Arthroplasty Model without CementrdquoJ Bone Jt Surg 81(1) pp 93ndash107

[13] Huiskes R Weinans H and Rietbergen B 1992 ldquoThe Relationship betweenStress Shielding and Bone Resorption around Total Hip Stems and the Effectsof Flexible Materialsrdquo Clin Orthop Relat Res 274(1) pp 124ndash134

[14] Kuiper J and Huiskes R 1992 ldquoNumerical Optimization of Hip-ProstheticStem Materialrdquo Recent Advances in Computer Methods in Biomechanics andBiomedical Engineering J Middleton G N Pande and K R Williams edsBooks and Journals International Ltd Swansea pp 76ndash84

[15] Kuiper J H and Huiskes R 1997 ldquoMathematical Optimization of ElasticProperties Application to Cementless Hip Stem Designrdquo J Biomech Eng119(2) pp 166ndash174

[16] Hedia H Shabara M El-Midany T and Fouda N 2006 ldquoImproved Designof Cementless Hip Stems Using Two-Dimensional Functionally Graded Materi-alsrdquo J Biomed Mater Res Part B Appl Biomater 79(1) pp 42ndash49

[17] Hedia H S and Mahmoud N-A 2004 ldquoDesign Optimization of Function-ally Graded Dental Implantrdquo Biomed Mater Eng 14(2) pp 133ndash143

[18] Watari F Yokoyama A Saso F Uo M and Kawasaki T 1997ldquoFabrication and Properties of Functionally Graded Dental Implantrdquo Compo-sites Part B 28(1-2) pp 5ndash11

[19] Katti K S 2004 ldquoBiomaterials in Total Joint Replacementrdquo Colloids Surf B39(3) pp 133ndash142

[20] Thompson I and Hench L 1998 ldquoMechanical Properties of BioactiveGlasses Glass-Ceramics and Compositesrdquo Proc Inst Mech Eng Part H JEng Med 212(2) pp 127

[21] Fraldi M Esposito L Perrella G Cutolo A and Cowin S 2010ldquoTopological Optimization in Hip Prosthesis Designrdquo Biomech ModelMechanobiol 9(4) pp 389ndash402

[22] Bobyn J D Poggie R Krygier J Lewallen D Hanssen A Lewis RUnger A Orsquokeefe T Christie M and Nasser S 2004 ldquoClinical Validationof a Structural Porous Tantalum Biomaterial for Adult Reconstructionrdquo J BoneJt Surg 86(Supplement 2) pp 123ndash129

[23] Bobyn J Stackpool G Hacking S Tanzer M and Krygier J 1999ldquoCharacteristics of Bone Ingrowth and Interface Mechanics of a New PorousTantalum Biomaterialrdquo J Bone Jt Surg Br Vol 81(5) pp 907ndash914

[24] Parthasarathy J Starly B Raman S and Christensen A 2010 ldquoMechanicalEvaluation of Porous Titanium (Ti6al4v) Structures with Electron Beam Melt-ing (Ebm)rdquo J Mech Behav Biomed Mater 3(3) pp 249ndash259

[25] Heinl P Muller L Korner C Singer R F and Muller F A 2008ldquoCellular Ti-6al-4v Structures with Interconnected Macro Porosity for BoneImplants Fabricated by Selective Electron Beam Meltingrdquo Acta Biomater4(5) pp 1536ndash1544

[26] Stamp R Fox P Orsquoneill W Jones E and Sutcliffe C 2009 ldquoThe Devel-opment of a Scanning Strategy for the Manufacture of Porous Biomaterials bySelective Laser Meltingrdquo J Mater Sci Mater Med 20(9) pp 1839ndash1848

[27] Yang S Leong K Du Z and Chua C 2002 ldquoThe Design of Scaffolds forUse in Tissue Engineering Part Ii Rapid Prototyping Techniquesrdquo TissueEng 8(1) pp 1ndash11

[28] Murr L Gaytan S Medina F Lopez H Martinez E Machado B Hernan-dez D Martinez L Lopez M and Wicker R 2010 ldquoNext-Generation Biomed-ical Implants Using Additive Manufacturing of Complex Cellular and FunctionalMesh Arraysrdquo Philos Trans R Soc London Ser A 368(1917) pp 1999ndash2032

[29] Coelho P Fernandes P Guedes J and Rodrigues H 2008 ldquoA HierarchicalModel for Concurrent Material and Topology Optimisation of Three-Dimensional Structuresrdquo Struct Multidiscip Optim 35(2) pp 107ndash115

[30] Rodrigues H Guedes J and Bendsoe M 2002 ldquoHierarchical Optimizationof Material and Structurerdquo Struct Multidiscip Optim 24(1) pp 1ndash10

[31] Goncalves Coelho P Rui Fernandes P and Carrico Rodrigues H 2011ldquoMultiscale Modeling of Bone Tissue with Surface and Permeability ControlrdquoJ Biomech 44(2) pp 321ndash329

[32] Coelho P G Cardoso J B Fernandes P R and Rodrigues H C 2011ldquoParallel Computing Techniques Applied to the Simultaneous Design of Struc-ture and Materialrdquo Adv Eng Software 42(5) pp 219ndash227

[33] Gibson L J and Ashby M F 1999 Cellular Solids Structure and Proper-ties Cambridge University Press Cambridge UK

[34] Guedes J and Kikuchi N 1990 ldquoPreprocessing and Postprocessing for Mate-rials Based on the Homogenization Method with Adaptive Finite ElementMethodsrdquo Comput Methods Appl Mech Eng 83(2) pp 143ndash198

[35] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization I-Homogenization Theory for Media with PeriodicStructurerdquo Comput Struct 69(6) pp 707ndash717

[36] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Opimization I I-Analytical and Numerical Solution of HomogenizationEquationsrdquo Comput Struct 69(6) pp 719ndash738

[37] Fang Z Starly B and Sun W 2005 ldquoComputer-Aided Characterization forEffective Mechanical Properties of Porous Tissue Scaffoldsrdquo Comput-AidedDes 37(1) pp 65ndash72

[38] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics 6th ed Butterworth-Heinemann Burlington

[39] Hollister S and Kikuchi N 1992 ldquoA Comparison of Homogenization andStandard Mechanics Analyses for Periodic Porous Compositesrdquo ComputMech 10(2) pp 73ndash95

[40] Pellegrino C Galvanetto U and Schrefler B 1999 ldquoNumerical Homogeni-zation of Periodic Composite Materials with Non Linear MaterialComponentsrdquo Int J Numer Methods Eng 46(10) pp 1609ndash1637

[41] Bragdon C Jasty M Greene M Rubash H and Harris W 2004ldquoBiologic Fixation of Total Hip Implants Insights Gained from a Series of Ca-nine Studiesrdquo J Bone Jt Surg 86(Supplement 2) pp 105ndash117

[42] Harrysson O L A Cansizoglu O Marcellin-Little D J Cormier D R andWest Ii H A 2008 ldquoDirect Metal Fabrication of Titanium Implants with Tai-lored Materials and Mechanical Properties Using Electron Beam Melting Tech-nologyrdquo Mater Sci Eng C 28(3) pp 366ndash373

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-9

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

mechanical properties should be selected [7273] The asymptotichomogenization will be applied to characterize tissue scaffoldingcell topologies for a wide range of relative density Their effectivemoduli yield and ultimate surfaces for multiaxial loading condi-tions will be obtained to create a standard library from which toselect suitable cell geometries for a 3D cellular implant design

Fourth bone has been here considered as consisting of a cancel-lous and a cortical part with isotropic mechanical properties Fur-ther study is necessary to improve the constitutive material modelof bone and to consider bone as a graded cellular material

Finally in the optimization procedure a simple formulation hasbeen used to quantitatively measure both bone resorption andimplant stability In a future study detailed simulations of implantstability [74ndash77] and bone remodeling process [6077ndash80] can beperformed to assess both the short and long term performance ofthe implant

5 Concluding Remarks

This work has presented a methodology integrating multiscaleanalysis and design optimization to design a novel hip implantmade of graded cellular material The method can contribute tothe development of a new generation of orthopedic implants witha graded cellular microstructure that will reduce the clinical con-sequences of current implants

In the first part of the paper the homogenization method hasbeen reviewed and then used to capture the mechanics of theimplant at the micro and macro scale In the second part multiob-jective optimization has been applied to find optimum gradients ofmaterial distribution that minimize concurrently bone resorptionand bone-implant interface stresses The results have shown thatthe optimized cellular implant exhibits a reduction of 76 ofbone resorption and 50 of interface stress with respect to a fullydense titanium implant

References[1] Kurtz S Ong K Lau E Mowat F and Halpern M 2007 ldquoProjections of

Primary and Revision Hip and Knee Arthroplasty in the United States from2005 to 2030rdquo J Bone Jt Surg Am Vol 89(4) pp 780ndash785

[2] Vichinsky E P Neumayr L D Haberkern C Earles A N Eckman JKoshy M and Black D M 1999 ldquoThe Perioperative Complication Rate ofOrthopedic Surgery in Sickle Cell Disease Report of the National Sickle CellSurgery Study Grouprdquo Am J Hematol 62(3) pp 129ndash138

[3] Kobayashi S Saito N Horiuchi H Iorio R and Takaoka K 2000 ldquoPoorBone Quality or Hip Structure as Risk Factors Affecting Survival of Total-HipArthroplastyrdquo The Lancet 355(9214) pp 1499ndash1504

[4] Kowalczyk P 2001 ldquoDesign Optimization of Cementless Femoral Hip Pros-theses Using Finite Element Analysisrdquo J Biomech Eng 123(5) pp 396ndash402

[5] Moen T C Ghate R Salaz N Ghodasra J and Stulberg S D 2011 ldquoAMonoblock Porous Tantalum Acetabular Cup Has No Osteolysis on Ct at 10Yearsrdquo Clin Orthop Relat Res 469(2) pp 382ndash386

[6] Kurtz S Gawel H and Patel J 2011 ldquoHistory and Systematic Review ofWear and Osteolysis Outcomes for First-Generation Highly Crosslinked Poly-ethylenerdquo Clin Orthop Relat Res 469(8) pp 2262ndash2277

[7] Grubl A Marker M Brodner W Giurea A Heinze G Meisinger VZehetgruber H and Kotz R 2007 ldquoLong Term Follow up of Metal on MetalTotal Hip Replacementrdquo J Orthop Res 25(7) pp 841ndash848

[8] Neumann D R P Thaler C Hitzl W Huber M Hofstadter T and DornU 2010 ldquoLong-Term Results of a Contemporary Metal-on-Metal Total HipArthroplasty A 10-Year Follow-up Studyrdquo J Arthroplasty 25(5) pp700ndash708

[9] Glassman A Bobyn J and Tanzer M 2006 ldquoNew Femoral Designs DoThey Influence Stress Shieldingrdquo Clin Orthop Relat Res 453(12) pp64ndash74

[10] Adam F Hammer D S Pfautsch S and Westermann K 2002 ldquoEarly Fail-ure of a Press-Fit Carbon Fiber Hip Prosthesis with a Smooth Surfacerdquo JAntroplasty 17(2) pp 217ndash223

[11] Trebse R Milosev I Kovac S Mikek M and Pisot V 2005 ldquoPoorResults from the Isoelastic Total Hip Replacementrdquo Acta Orthop 76(2) pp169ndash176

[12] Harvey E Bobyn J Tanzer M Stackpool G Krygier J and Hacking S1999 ldquoEffect of Flexibility of the Femoral Stem on Bone-Remodeling and Fix-ation of the Stem in a Canine Total Hip Arthroplasty Model without CementrdquoJ Bone Jt Surg 81(1) pp 93ndash107

[13] Huiskes R Weinans H and Rietbergen B 1992 ldquoThe Relationship betweenStress Shielding and Bone Resorption around Total Hip Stems and the Effectsof Flexible Materialsrdquo Clin Orthop Relat Res 274(1) pp 124ndash134

[14] Kuiper J and Huiskes R 1992 ldquoNumerical Optimization of Hip-ProstheticStem Materialrdquo Recent Advances in Computer Methods in Biomechanics andBiomedical Engineering J Middleton G N Pande and K R Williams edsBooks and Journals International Ltd Swansea pp 76ndash84

[15] Kuiper J H and Huiskes R 1997 ldquoMathematical Optimization of ElasticProperties Application to Cementless Hip Stem Designrdquo J Biomech Eng119(2) pp 166ndash174

[16] Hedia H Shabara M El-Midany T and Fouda N 2006 ldquoImproved Designof Cementless Hip Stems Using Two-Dimensional Functionally Graded Materi-alsrdquo J Biomed Mater Res Part B Appl Biomater 79(1) pp 42ndash49

[17] Hedia H S and Mahmoud N-A 2004 ldquoDesign Optimization of Function-ally Graded Dental Implantrdquo Biomed Mater Eng 14(2) pp 133ndash143

[18] Watari F Yokoyama A Saso F Uo M and Kawasaki T 1997ldquoFabrication and Properties of Functionally Graded Dental Implantrdquo Compo-sites Part B 28(1-2) pp 5ndash11

[19] Katti K S 2004 ldquoBiomaterials in Total Joint Replacementrdquo Colloids Surf B39(3) pp 133ndash142

[20] Thompson I and Hench L 1998 ldquoMechanical Properties of BioactiveGlasses Glass-Ceramics and Compositesrdquo Proc Inst Mech Eng Part H JEng Med 212(2) pp 127

[21] Fraldi M Esposito L Perrella G Cutolo A and Cowin S 2010ldquoTopological Optimization in Hip Prosthesis Designrdquo Biomech ModelMechanobiol 9(4) pp 389ndash402

[22] Bobyn J D Poggie R Krygier J Lewallen D Hanssen A Lewis RUnger A Orsquokeefe T Christie M and Nasser S 2004 ldquoClinical Validationof a Structural Porous Tantalum Biomaterial for Adult Reconstructionrdquo J BoneJt Surg 86(Supplement 2) pp 123ndash129

[23] Bobyn J Stackpool G Hacking S Tanzer M and Krygier J 1999ldquoCharacteristics of Bone Ingrowth and Interface Mechanics of a New PorousTantalum Biomaterialrdquo J Bone Jt Surg Br Vol 81(5) pp 907ndash914

[24] Parthasarathy J Starly B Raman S and Christensen A 2010 ldquoMechanicalEvaluation of Porous Titanium (Ti6al4v) Structures with Electron Beam Melt-ing (Ebm)rdquo J Mech Behav Biomed Mater 3(3) pp 249ndash259

[25] Heinl P Muller L Korner C Singer R F and Muller F A 2008ldquoCellular Ti-6al-4v Structures with Interconnected Macro Porosity for BoneImplants Fabricated by Selective Electron Beam Meltingrdquo Acta Biomater4(5) pp 1536ndash1544

[26] Stamp R Fox P Orsquoneill W Jones E and Sutcliffe C 2009 ldquoThe Devel-opment of a Scanning Strategy for the Manufacture of Porous Biomaterials bySelective Laser Meltingrdquo J Mater Sci Mater Med 20(9) pp 1839ndash1848

[27] Yang S Leong K Du Z and Chua C 2002 ldquoThe Design of Scaffolds forUse in Tissue Engineering Part Ii Rapid Prototyping Techniquesrdquo TissueEng 8(1) pp 1ndash11

[28] Murr L Gaytan S Medina F Lopez H Martinez E Machado B Hernan-dez D Martinez L Lopez M and Wicker R 2010 ldquoNext-Generation Biomed-ical Implants Using Additive Manufacturing of Complex Cellular and FunctionalMesh Arraysrdquo Philos Trans R Soc London Ser A 368(1917) pp 1999ndash2032

[29] Coelho P Fernandes P Guedes J and Rodrigues H 2008 ldquoA HierarchicalModel for Concurrent Material and Topology Optimisation of Three-Dimensional Structuresrdquo Struct Multidiscip Optim 35(2) pp 107ndash115

[30] Rodrigues H Guedes J and Bendsoe M 2002 ldquoHierarchical Optimizationof Material and Structurerdquo Struct Multidiscip Optim 24(1) pp 1ndash10

[31] Goncalves Coelho P Rui Fernandes P and Carrico Rodrigues H 2011ldquoMultiscale Modeling of Bone Tissue with Surface and Permeability ControlrdquoJ Biomech 44(2) pp 321ndash329

[32] Coelho P G Cardoso J B Fernandes P R and Rodrigues H C 2011ldquoParallel Computing Techniques Applied to the Simultaneous Design of Struc-ture and Materialrdquo Adv Eng Software 42(5) pp 219ndash227

[33] Gibson L J and Ashby M F 1999 Cellular Solids Structure and Proper-ties Cambridge University Press Cambridge UK

[34] Guedes J and Kikuchi N 1990 ldquoPreprocessing and Postprocessing for Mate-rials Based on the Homogenization Method with Adaptive Finite ElementMethodsrdquo Comput Methods Appl Mech Eng 83(2) pp 143ndash198

[35] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization I-Homogenization Theory for Media with PeriodicStructurerdquo Comput Struct 69(6) pp 707ndash717

[36] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Opimization I I-Analytical and Numerical Solution of HomogenizationEquationsrdquo Comput Struct 69(6) pp 719ndash738

[37] Fang Z Starly B and Sun W 2005 ldquoComputer-Aided Characterization forEffective Mechanical Properties of Porous Tissue Scaffoldsrdquo Comput-AidedDes 37(1) pp 65ndash72

[38] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics 6th ed Butterworth-Heinemann Burlington

[39] Hollister S and Kikuchi N 1992 ldquoA Comparison of Homogenization andStandard Mechanics Analyses for Periodic Porous Compositesrdquo ComputMech 10(2) pp 73ndash95

[40] Pellegrino C Galvanetto U and Schrefler B 1999 ldquoNumerical Homogeni-zation of Periodic Composite Materials with Non Linear MaterialComponentsrdquo Int J Numer Methods Eng 46(10) pp 1609ndash1637

[41] Bragdon C Jasty M Greene M Rubash H and Harris W 2004ldquoBiologic Fixation of Total Hip Implants Insights Gained from a Series of Ca-nine Studiesrdquo J Bone Jt Surg 86(Supplement 2) pp 105ndash117

[42] Harrysson O L A Cansizoglu O Marcellin-Little D J Cormier D R andWest Ii H A 2008 ldquoDirect Metal Fabrication of Titanium Implants with Tai-lored Materials and Mechanical Properties Using Electron Beam Melting Tech-nologyrdquo Mater Sci Eng C 28(3) pp 366ndash373

Journal of Biomechanical Engineering MARCH 2012 Vol 134 031004-9

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm

[43] Wang H V 2005 ldquoA Unit Cell Approach for Lightweight Structure and Com-pliant Mechanismrdquo PhD thesis Georgia Institute Of Technology Atlanta GA

[44] Matsui K Terada K and Yuge K 2004 ldquoTwo-Scale Finite Element Analy-sis of Heterogeneous Solids with Periodic Microstructuresrdquo Comput Struct82(7-8) pp 593ndash606

[45] Masters I and Evans K 1996 ldquoModels for the Elastic Deformation of Hon-eycombsrdquo Compos Struct 35(4) pp 403ndash422

[46] Christensen R M 2000 ldquoMechanics of Cellular and Other Low-DensityMaterialsrdquo Int J Solids Struct 37(1-2) pp 93ndash104

[47] Wang A and Mcdowell D 2004 ldquoIn-Plane Stiffness and Yield Strength ofPeriodic Metal Honeycombsrdquo J Eng Mater Technol 126(2) pp 137ndash156

[48] Kumar R and Mcdowell D 2004 ldquoGeneralized Continuum Modeling of 2-DPeriodic Cellular Solidsrdquo Int J Solids Struct 41(26) pp 7399ndash7422

[49] Warren W and Byskov E 2002 ldquoThree-Fold Symmetry Restrictions onTwo-Dimensional Micropolar Materialsrdquo Eur J Mech ASolids 21(5) pp779ndash792

[50] Chen J and Huang M 1998 ldquoFracture Analysis of Cellular Materials AStrain Gradient Modelrdquo J Mech Phys Solids 46(5) pp 789ndash828

[51] Wang W-X Luo D Takao Y and Kakimoto K 2006 ldquoNew SolutionMethod for Homogenization Analysis and Its Application to the Prediction ofMacroscopic Elastic Constants of Materials with Periodic MicrostructuresrdquoComput Struct 84(15-16) pp 991ndash1001

[52] Andrews E Gioux G Onck P and Gibson L 2001 ldquoSize Effects in Duc-tile Cellular Solids Part II Experimental Resultsrdquo Int J Mech Sci 43(3) pp701ndash713

[53] Simone A and Gibson L 1998 ldquoEffects of Solid Distribution on the Stiff-ness and Strength of Metallic Foamsrdquo Acta Mater 46(6) pp 2139ndash2150

[54] Bendsoslashe M P and Sigmund O 2003 Topology Optimization Theory Meth-ods and Applications Springer-Verlag Berlin

[55] Hassani B and Hinton E 1998 ldquoA Review of Homogenization and Topol-ogy Optimization III-Topology Optimization Using Optimality Criteriardquo Com-put Struct 69(6) pp 739ndash756

[56] Dıaaz A and Kikuchi N 1992 ldquoSolutions to Shape and Topology Eigen-value Optimization Problems Using a Homogenization Methodrdquo Int J NumerMethods Eng 35(7) pp 1487ndash1502

[57] Suzuki K and Kikuchi N 1991 ldquoA Homogenization Method for Shape andTopology Optimizationrdquo Comput Methods Appl Mech Eng 93(3) pp291ndash318

[58] Bendsoslashe M P and Kikuchi N 1988 ldquoGenerating Optimal Topologies inStructural Design Using a Homogenization Methodrdquo Comput Methods ApplMech Eng 71(2) pp 197ndash224

[59] Hassani B 1996 ldquoA Direct Method to Derive the Boundary Conditions of theHomogenization Equation for Symmetric Cellsrdquo Commun Numer MethodsEng 12(3) pp 185ndash196

[60] Weinans H Huiskes R and Grootenboer H 1992 ldquoEffects of MaterialProperties of Femoral Hip Components on Bone Remodelingrdquo J Orthop Res10(6) pp 845ndash853

[61] Hoffman O 1967 ldquoThe Brittle Strength of Orthotropic Materialrdquo J ComposMater 1(3) pp 200ndash206

[62] Pal B Gupta S and New A 2009 ldquoA Numerical Study of Failure Mecha-nisms in the Cemented Resurfaced Femur Effects of Interface Characteristicsand Bone Remodellingrdquo Proc Inst Mech Eng Part H J Eng Med 223(4)pp 471ndash484

[63] Carter D Orr T and Fyhrie D 1989 ldquoRelationships between Loading His-tory and Femoral Cancellous Bone Architecturerdquo J Biomech 22(3) pp231ndash244

[64] Bidanda B and Bartolo P 2008 Virtual Prototyping amp Bio Manufacturing inMedical Applications Springer-Verlag Berlin Chap 5

[65] Deb K Pratap A Agarwal S and Meyarivan T 2002 ldquoA Fast and ElitistMultiobjective Genetic Algorithm Nsga-Iirdquo IEEE Trans Evol Comput 6(2)pp 182ndash197

[66] Zienkiewicz O C and Taylor R L 2005 The Finite Element Method forSolid and Structural Mechanics Butterworth-Heinemann Burlington

[67] Objet-Geometries I 2011 httpwwwobjetcom[68] Lefik M and Schrefler B 1996 ldquoFE Modelling of a Boundary Layer Correc-

tor for Composites Using the Homogenization Theoryrdquo Eng Comput 13(6)pp 31ndash42

[69] Dumontet H 1986 ldquoStudy of a Boundary Layer Problem in Elastic CompositeMaterialsrdquo RAIRO Model Math Anal Numer 20(2) pp 265ndash286

[70] Kruch S 2007 ldquoHomogenized and Relocalized Mechanical Fieldsrdquo J StrainAnal Eng Des 42(4) pp 215ndash226

[71] Ghosh S Lee K and Raghavan P 2001 ldquoA Multi-Level ComputationalModel for Multi-Scale Damage Analysis in Composite and Porous MaterialsrdquoInt J Solids Struct 38(14) pp 2335ndash2385

[72] Cheah C Chua C Leong K and Chua S 2003 ldquoDevelopment of a TissueEngineering Scaffold Structure Library for Rapid Prototyping Part 2 Paramet-ric Library and Assembly Programrdquo Int J Adv Manuf Technol 21(4) pp302ndash312

[73] Cheah C Chua C Leong K and Chua S 2003 ldquoDevelopment of aTissue Engineering Scaffold Structure Library for Rapid Prototyping Part 1Investigation and Classificationrdquo Int J Adv Manuf Technol 21(4) pp291ndash301

[74] Viceconti M Monti L Muccini R Bernakiewicz M and Toni A 2001ldquoEven a Thin Layer of Soft Tissue May Compromise the Primary Stability ofCementless Hip Stemsrdquo Clin Biomech 16(9) pp 765ndash775

[75] Viceconti M Brusi G Pancanti A and Cristofolini L 2006 ldquoPrimary Sta-bility of an Anatomical Cementless Hip Stem A Statistical Analysisrdquo J Bio-mech 39(7) pp 1169ndash1179

[76] Abdul-Kadir M R Hansen U Klabunde R Lucas D and Amis A 2008ldquoFinite Element Modelling of Primary Hip Stem Stability The Effect of Inter-ference Fitrdquo J Biomech 41(3) pp 587ndash594

[77] Viceconti M Muccini R Bernakiewicz M Baleani M and Cristofolini L2000 ldquoLarge-Sliding Contact Elements Accurately Predict Levels of Bone-Implant Micromotion Relevant to Osseointegrationrdquo J Biomech 33(12) pp1611ndash1618

[78] Weinans H Huiskes R and Grootenboer H J 1992 ldquoThe Behavior ofAdaptive Bone-Remodeling Simulation Modelsrdquo J Biomech 25(12) pp1425ndash1441

[79] Boyle C and Kim I Y 2011 ldquoThree-Dimensional Micro-Level Computa-tional Study of Wolffrsquos Law Via Trabecular Bone Remodeling in the HumanProximal Femur Using Design Space Topology Optimizationrdquo J Biomech44(5) pp 935ndash942

[80] Boyle C and Kim I Y 2011 ldquoComparison of Different Hip ProsthesisShapes Considering Micro-Level Bone Remodeling and Stress-Shielding Crite-ria Using Three-Dimensional Design Space Topology Optimizationrdquo J Bio-mech 44(9) pp 1722ndash1728

031004-10 Vol 134 MARCH 2012 Transactions of the ASME

Downloaded 03 Apr 2012 to 13221688112 Redistribution subject to ASME license or copyright see httpwwwasmeorgtermsTerms_Usecfm