Medical Engineering & Physics 26 (2004) 61–69www.elsevier.com/locate/medengphy

An improved method for the automatic mapping of computedtomography numbers onto finite element models

Fulvia Taddei∗, Alberto Pancanti, Marco VicecontiLaboratorio di Tecnologia Medica, Istituti Ortopedici Rizzoli, Via di Barbiano, 1/10, 40136 Bologna, Italy

Received 10 December 2002; received in revised form 5 May 2003; accepted 21 July 2003

Abstract

The assignment of bone tissue material properties is a fundamental step in the generation of subject-specific finite element modelsfrom computed tomography data. Aim of the present work is to investigate the influence of the material mapping algorithm on theresults predicted by the finite element analysis. Two models, a coarse and a refined one, of a human ileum, femur and tibia, weregenerated from CT data and used for the tests. In addition a convergence analysis was carried out for the femur model, using sixrefinement levels, to verify whether the inclusion of the material properties would significantly alter the convergence behaviour ofthe mesh. The results showed that the choice of the mapping algorithm influences the material distribution. However, this did notalways propagate into the finite element results. The difference between the maximum Von Mises stress remained always lowerthan 10%, apart one case when it reached the 13%. However, the global behaviour of the meshes showed more marked differencesbetween the two algorithms: in the finer meshes of the two long bones 20�30% of the bone volume showed differences in thepredicted Von Mises stresses greater than 10%. The convergence behaviour of the model was not worsened by the introduction ofinhomogeneous material properties. The software was made available in the public domain. 2003 IPEM. Published by Elsevier Ltd. All rights reserved.

Keywords: Computed tomography; Subject-specific finite element models; Mechanical properties

1. Introduction

The finite element method has been increasinglyadopted in the past few years to study the mechanicalbehaviour of biological structures. Computed tomogra-phy (CT) represents the method of choice for the gener-ation of the finite element models of bone segments. Itis well known that CT images can provide fairly accuratequantitative information on bone geometry since theattenuation coefficient of bone tissue is much higher thanthe one of the surrounding soft tissues, resulting in wellcontrasted edges. Moreover, it has been shown that thenumbers reported in the CT images can be related withthe mechanical properties of bone tissues. Once the finiteelement mesh has been generated, the analyst shoulddefine the material properties relative to each element.

∗ Corresponding author. Tel.:+39-051-6366864; fax:+39-051-6366863.

E-mail address: taddei@tecno.ior.it (F. Taddei).

1350-4533/$30.00 2003 IPEM. Published by Elsevier Ltd. All rights reserved.doi:10.1016/S1350-4533(03)00138-3

If a generic or average bone is modelled, then the mech-anical properties of the different bone tissues is usuallyderived from average values reported in publishedexperimental studies[1–3]. To the contrary, when a sub-ject-specific model is considered mechanical propertiesshould be derived from CT data. It has been shown thatthe stress distribution of a bone is strongly related to themechanical properties distribution in the bone tissue[4,5]. Hence it is of great importance to find an effectivemethod to derive the distribution of mechanical proper-ties in the bone tissue from CT data, and to properlymap it into subject-specific finite element models.

In X-ray CT, the function imaged is the distributionof the tissue linear attenuation coefficient. CT numbersare usually expressed in Hounsfild units (HU) that arenow recognised as a standard. This system of units rep-resents a transformation from the original linear attenu-ation coefficient measurements into one where water isassigned a value of zero and air is assigned a value of�1000. It has been demonstrated that CT numbers arealmost linearly correlated with apparent density of bio-

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logical tissues [6–8] over a wide range of density values.The process of translating the CT numbers into the den-sity of biological tissue is usually called calibration ofthe CT dataset and it will be so indicated through therest of the paper. Moreover, good empirical relationshipshave been established experimentally between densityand mechanical properties of bone tissues [9–11]. Thus,in principle is possible to derive the inhomogeneous dis-tribution of the tissue properties from the CT data, andto account somehow for such distribution in the subject-specific finite element model. Several approaches wereproposed in literature to perform this task, with differentlevel of automation and depending on the techniqueadopted for the mesh generation.

If the mesh is a so-called ‘voxel mesh’ [12–14], thenthe task is simple. The CT dataset represent a volumesampled at points of a regular grid. The elements of aCartesian mesh are automatically generated using athreshold criterion, and are perfectly shaped cubicelements, obtained from the union of a predefined num-ber of voxels. In this case it is natural to average theradiological density of the voxel composing eachelement, to obtain an average radiological density, fromwhich a Young’s modulus can be computed using oneof the different empirical relationships reported in litera-ture.

If a structured mesh is built, an accurate mapping ofthe material properties of different bone tissues is poss-ible since the facets of the elements may be properlydefined to follow the boundaries between different typesof bone tissues. In these cases the average density of asingle finite element is defined on the basis of the voxelsthat fall inside the element. The procedure can have adifferent level of automation, but always implicitly relieson an a priori knowledge of the mesh topology [5,15,16].

When the mesh is unstructured, the facets of theelements are not aligned with the CT axes nor with theboundaries between bone tissues, hence the manual map-ping of the material properties onto the finite elementsbecomes impossible and a more sophisticated approachis necessary. The CT data can be seen as a three-dimen-sional scalar field (related to the tissue density) sampledover a regular grid. If the unstructured mesh is generatedstarting form the same data, the mesh and the densitydistribution are perfectly registered in space. The onlyproblem is how to account for this inhomogeneous distri-bution of material properties into the finite elementmodel. Two distinct strategies are described in the litera-ture.

One approach is to include into the element formu-lation spatially variable mechanical properties. Thisapproach is the most general, as it can cope with a sub-stantial variability of the tissue properties over theelement volume. However, in authors’ knowledge, it hasbeen used so far only in one study [17], probablybecause of the complexity it involves. In general the

approach cannot be used with commercial general pur-pose FEM codes, as it requires a new finite element for-mulation and a substantial customisation of the solver.In addition, the need for such complexity has not beendemonstrated so far.

All other studies are based on the implicit assumptionthat the variability of the CT numbers within the volumeof each element can be neglected. Thus, they assign thetissue mechanical properties to each element by comput-ing the average value of the scalar field over eachelement volume. All differences rely on how thiselement-wise average value is computed.

The simplest approach proposed in the literature is tofind for each element’s node the value associated to thenearest point on the CT sampling grid, and then to assignto the element the weighted average of the node values[5]. Another variation proposed in the literature is toassign to each element the value obtained by averagingthe densities of the eight CT sampling points surround-ing the element centroid [18]. These methods are verysimple to implement, but may produce inaccurate resultswhen the element size is significantly larger that thespacing of the CT sampling grid.

A second approach firstly proposed in a limited way[19], and then in a more general implementation in thepublic domain code BONEMAT—V1 (http://www.cineca.it/hosted/LTM-IOR/back2net/SW/index.html, [20]),determines all CT sampling points that fall inside theelement volume and assigns to the element the averageof these values. In this case even if the size of theelement is significantly larger than that of the CT sam-pling grid, the result is fairly accurate. However, eventhis technique may not give satisfactory results when, onthe contrary, the elements are of comparable dimension,or smaller than the voxel size. For this reason it wouldbe important to have a method that can deal with bothsituations and that is fully automatic, so to be appliedalso to unstructured meshes.

The aim of the present work is to present an advancedmethod for the material properties assignment to finiteelement models starting from CT dataset, that overcomesthe aforementioned limits and that is fully automatic.The new algorithm was evaluated with respect to theresults obtained with the BONEMAT—V1 code, testingit over three different types of bone used for the com-parison. The software, called BONEMAT—V2, is pub-licly available at http://www.tecno.ior.it/back2net/.

2. Materials and methods

The materials properties of a human ileum, femur andtibia were derived from CT data and mapped onto therelative unstructured finite element mesh. To evaluatethe influence of the material-mapping algorithm on theresults of a finite element analysis, the operation was

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repeated using two different mapping softwares calledBONEMAT—V1 and BONEMAT—V2, and the resultsobtained from each finite element model compared.

2.1. The materials mapping softwares

The overall structure of the software was not changedin this second version. A detailed description of the firstalgorithm could be found in [20], however its fundamen-tal steps are here summarised for sake of clarity. Thesoftware could be logically divided into three steps:

1. Evaluation of a uniform HU number: for each elementof the mesh, the software computes a uniform HUvalue, on the basis of the CT data. This step was com-pletely rewritten in the new software, implementinga new algorithm, that constitutes the major innovationproposed in the new software. For this reason thisstep will be discussed in grater details at the end ofthis section.

2. Calibration of the CT dataset: CT numbers are depen-dent on many factors related to the specific exam,among with the physical parameters set by the radiol-ogist such as the KVP or the tube current. A cali-bration, usually performed with the use of a densityphantom, is then necessary to correctly transformthese values into meaningful bone tissue densityvalues. It is assumed that the calibration relationshipis linear and the user is allowed to adopt his ownrelationship. The calibration equation is then:

rn � a � b HUn, (1)

where rn is the uniform density assigned to theelement n of the mesh, HUn is the uniform CT numberand a and b are the calibration coefficients providedby the user.

3. Definition of a uniform Young’s modulus: many arethe relationships that have been published in theliterature that express the Young’s modulus E as afunction of bone tissue density. The user is allowedto use his own equation, that can be either linear orexpress E as a power function of rn. It is assumedthat a single equation is valid through the whole den-sity range. The equation is then

En � a � b· rcn, (2)

where En is the uniform Young’s modulus assignedto the element n of the mesh, rn is its density and(a,b,c) are the coefficients provided by the user. Thisprocedure may, theoretically, lead to a differentmaterial card for each element of the mesh, whichmay give computational problems with those codesthat can handle a limited number of materials. Theuser can reduce the number of materials groupingthem with a strategy designed to assigns to eachelement a value of E that is always equal or greater

than the one calculated with Eq. (2). The control onthis procedure is left to the user that can choose a �Ethreshold. Then the maximum computed value of theelastic modulus, Emax, is assigned to the materialMat—1. All the elements with E�(Emax - �E) areassigned the material Mat—1. Material Mat—2 ischaracterised by the maximum E of the remainingelements and so on until the whole set of the modelmaterial is defined.

As anticipated the fundamental difference between thetwo algorithm is the calculation of the HU uniform valuethat is assigned to each finite element of the mesh, onthe basis of the CT data. In the BONEMAT V1 algor-ithm this value was determined as

HUn �1r �

r

i � 1

HUi (3)

where HUn is the uniform value of CT number assignedto the element n, and r are the CT lattice vertices thatfall inside the element. If the element is smaller than thevoxel, and no CT lattice vertices are found inside it, thealgorithm computes this average considering the eightCT lattice vertices that surround the element centroid.

In the BONEMAT V2 software, to the contrary, thiscalculation is performed with a numerical integration ofthe HU field as follows:

HUn �

�Vn

HU(x,y,z)dV

�Vn

dV

(4)

�

�Vn’

HU(r,s,t)detJ(r,s,t)dV’

Vn

,

where Vn indicates the volume of the element n, (x,y,z)are the co-ordinates in the CT reference system, (r,s,t)are the local co-ordinates in the element reference sys-tem, and J represents the Jacobian of the transformation.The integrals in Eq. (4) are evaluated numerically, andthe user can choose the order of the numerical inte-gration. The value of HU(x,y,z) in a generic point of theCT domain is determined by a tri-linear interpolationbetween the eight adjacent grid points’ values. This newmethod introduces a higher computational effort withrespect to the older one, but estimates the uniform HUvalue for each element in a more accurate way, usingall the available information to interpolate the HU overthe CT domain. In this case the dimension of the finiteelement is not influent on the accuracy of the uniformHU estimation since even in the single voxel the descrip-tion of the HU field is not uniform and takes into account

64 F. Taddei et al. / Medical Engineering & Physics 26 (2004) 61–69

the spatial distribution of the HU value in the eight sur-rounding CT lattice vertices.

2.2. Finite element models

In order to compare the differences in the results ofa finite element analysis induced by the two algorithms,three different human skeletal segments of the lowerlimb were considered: an ileum, a femur and a tibia.The dataset of the human femur was taken in vivo, fordiagnostic reasons, with a clinical CT scanner (HiSpeed,General Electric Co., USA). The patient was a 60 yearsold woman with no particular bone disease affecting theconsidered femur. The pixel size was 0.7 mm, and thespacing of the slices was 1 mm in the epiphysis and 2mm in the diaphysis. The other two datasets were takenex vivo for research purposes, with a clinical CT scanner(CT Twin, Elscint Ltd., Israel), and belonged to a 99years old man: the majority of the soft tissues were stillsurrounding the bone, but their integrity was not perfect.Metallic pins were inserted in the bone for further regis-tration. The pixel size was 0.84 mm. The maximumspacing between slices was 3 mm, in the tibial shaft, andthe minimum one was 1 mm.

Two meshes were generated for each bone segment(Fig. 1): a coarser mesh with elements significantly big-ger, on average, than the spacing of the CT samplinggrid, and a refined one, with elements whose dimensionswere comparable to such CT spacing. The meshes weregenerated using a validated technique described in Vice-conti et al. [21], that can be summarised in four steps:

1. semi-automatic segmentation of the bone using an in-house developed software that implements a bordertracing algorithm [22];

2. reconstruction of the bone surface using a Delaunaytriangulation algorithm [23];

3. automatic generation of the mesh using the HEXAR[24] software that implements a grid-based algorithmand generates 8 node hexahedral meshes; and

4. mapping the material properties on the finite elementmeshes using the two versions of the BONEMATsoftware.

Since the aim of the present study is the comparisonbetween different material-mapping algorithms, a cali-bration of the CT dataset with a density phantom wasnot considered necessary, but reference values forcortical bone tissue density and Young’s modulus werederived from literature. For both algorithms the cali-bration of the dataset was performed using a linearrelationship between HU and apparent bone densityassuming the density of the hardest cortical bone was1.8 g/cm3, corresponding to a radiological density of1650 HU, while the density of water was assumed 1g/cm3 corresponding to a radiological density of 0 HU.

Fig. 1. The coarse and refined meshes for the three bone segments:femur (5779 and 93081 nodes), ileum (8015 and 124231 nodes) andtibia (4616 and 76406 nodes).

The relationship between apparent density and Young’smodulus was derived from Carter and Heyes [9] for thewhole density range. A Young’s modulus of 22 GPa wasassociated to a density of 1.8 g/cm3 [20].

The loading conditions, rather than being representa-tive of the loads applied to that bone segment underphysiological conditions, were designed to produced thehighest possible stress gradient:

� femur: a vertical force (100 N) applied on the femoralhead, constrained distal region;

� tibia: two vertical forces (100 N) applied on the tibialplateau, constrained distal region; and

� ileum: a force (100 N) applied to the acetabulum, con-strained iliac crest and symmetry conditions on thepubic synphisis

For each mesh, the distribution of the Young’s modu-lus values assigned to each finite element by two algor-

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ithms was synthetically represented in terms of numberof different materials as well as in terms of average andminimum Young’s modulus. The maximum modulus ofelasticity was not considered a significant indicatorbecause the presence of metallic pins implanted in theileum and in the tibia for other research purposes, wouldmake this value abnormally high. The distributions pro-duced by the two algorithms were statistically comparedusing a non-parametric Kolomogorov–Smirnov statisti-cal test [25].

Three specific analysis results were used to comparethe resulting finite element models: the maximum VonMises stress, the maximum nodal displacement, themean strain energy density. In order to avoid localeffects, the comparisons were limited to a control regiondefined excluding the portions of mesh too close to theregions where the boundaries conditions are applied.These indicators highlight intensive differences. A fourthindicator was used to make evident the extensive differ-ences between the stress fields predicted by the twomodels. For each element n it was computed a relativestress difference en on the Von Mises stress as:

en �sn

EQ—1�snEQ—2

snEQ—2

� 100, (5)

where snEQ—1 indicates the Von Mises stress calculated

for element n with the BONEMAT—V1 software andsn

EQ—2 indicates the one calculated with the BONE-MAT—V2 software. The global difference wasexpressed as the fraction of the mesh volume where therelative stress difference was greater than 10%.

2.3. Convergence test

In addition a convergence test was performed on thefemur model to verify whether the inclusion of thematerial properties would significantly alter the conver-gence behaviour of the mesh. Bone material propertieswere assigned to six meshes (Fig. 2) of the femur ofincreasing refinement levels using the BONEMAT—V2software. For each mesh the results obtained with theinhomogeneous material model were compared to thoseproduced by a homogenous material model. TheYoung’s modulus of the homogeneous model was selec-ted in order to obtain, in the mesh with the highestrefinement, the same maximum nodal displacement pre-dicted by the most refined inhomogeneous model. Themaximum Von Mises nodal stress and the magnitude ofthe maximum displacement in a control region spanningall the diaphysis of the femur were used to assess theconvergence.

3. Results

3.1. Material properties distribution

The V2 algorithm always assigned a bigger numberof material properties than the V1 one. The minimumand average values of the Young’s modulus computedfor each mesh presented in some cases appreciable dif-ferences between the two algorithms (Table 1). This ismade evident by the distributions of the assignedmaterial properties (Fig. 3) where is also reported thelevel of significance of the differences observed in eachplot. In all coarse meshes the two material propertiesdistributions were not significantly different, while theywere for all fine meshes.

3.2. Stress distribution

The difference in the material distributions did not sig-nificantly affect intensive indicators. For the maximumVon Mises stress predicted in the control region the dif-ferences between the two material-mapping methodswere always lower than 10%, apart from the ileumcoarse mesh where a difference of the 13% was reached(Table 2). Differences in the other intensive indicatorswere always less that 2%.

However, the two algorithms produce ‘global’ differ-ences. The volume of the mesh characterised by a differ-ence in the Von Mises element stress greater than 10%was found to be nearly 30% in the finest mesh of thefemur and almost 20% in that of the tibia (Table 3).

3.3. Convergence test

The convergence behaviour of the model was notworsened by the introduction of inhomogeneous materialproperties. The behaviour of the maximum Von Misesnodal stress and of the maximum nodal displacement isnot significantly different from the one obtained using asingle material property, as can be seen in Fig. 4.

The inhomogeneous model showed a convergencecurve on the displacements very similar to that obtainedwith the homogeneous model. Increasing the meshrefinement from 56 073 to 110 733 degrees of freedomchanged the peak displacement of 0.115 mm (1.51%) inthe homogenous model and 0.114 mm (1.50%) in theinhomogeneous.

Similar results were found for the convergence of thepeak stress. Increasing the mesh refinement from 56 073to 110 733 degrees of freedom changed the peak VonMises stress of 2.53 MPa (4.38%) in the homogenousmodel and of 2.79 MPa (4.66%) in the inhomo-geneous one.

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Fig. 2. The six femur meshes used for the convergence analysis.

Table 1Number of materials, minimum and average values (in MPa) of the Young’s modulus assigned by the two algorithms to the various meshes

Bone algorithm Femur Ileum Tibia

V1 V2 V1 V2 V1 V2

Coarse meshesN. of mat 1113 1154 463 531 939 1749E minimum 3427 3448 1381 1575 2251 2552E average 9020 9033 5219 5203 7490 7429Fine meshesN. of mat. 1540 2016 1135 1210 929 2109E minimum 3142 3249 751 816 770 1035E average 9537 9480 5396 5355 7668 7601

The maximum value of E was not considered for comparison since unrealistic high values were originated by small metal pins in the tibia andthe ileum.

4. Discussion

When subject-specific finite element models are usedfor the analysis of the biomechanical behaviour of bones,the information on bone morphology and materialproperties are usually derived from CT data. It is of greatimportance to find an effective method to account forthe different types of tissue from the CT data, since ithas been proved that the materials distribution stronglyinfluences the predicted mechanical behaviour of bone[4,5]. The majority of works reported in literature assigna uniform elastic module to each element of the mesh,averaging, with more or less automatic procedures, theradiological densities of the voxels that fall inside theelement. When the element size is comparable with thatof the CT voxels, however, this procedure may induceerror in the assignment of the material properties. Aimof the present work was to propose an improved, fullyautomatic method to map the material properties onto a

finite element mesh starting form the information pro-vided by the CT dataset that performs correctly evenwhen the elements are small. The software does notrequire any a priori information on the mesh topology,and hence can be applied to unstructured meshes as wellas structured ones.

The effect on the materials distribution, and on thefinite element analysis induced by the introduction of thenew algorithm was evaluated with respect to the pre-vious version of the software (BONEMAT—V1). In itsfirst version the algorithm computed the average CTnumber of an element, averaging the values of the CTlattice vertices that fall inside it.

The results showed that the algorithm chosen to per-form the material mapping influences the material distri-bution. The new algorithm generally introduced a biggernumber of materials, probably due to the more accuratedescription of the HU field that induces a more uniformdistribution of the HU values among the elements, and

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Fig. 3. The distribution of the Young’s modulus (dark grey V1, light grey V2) and the results of the Kolmogorov–Smirnov test is reported foreach mesh.

Table 2Comparison between analyses results in the control regions of all meshes with the two mapping algorithms

Bone algorithm Femur Ileum Tibia

V1 V2 V1 V2 V1 V2

Coarse meshesMax U (mm) 1.278 1.258 0.049 0.050 0.97 0.97Max sEQ 6.228 6.017 1.613 1.423 5.112 4.942Mean SED 5.62 5.56 0.31 0.31 4.53 4.57Fine meshesMax U (mm) 1.135 1.136 0.054 0.054 0.91 0.92Max sEQ 6.841 6.465 2.193 2.018 6.041 5.952Mean SED 4.90 4.89 0.31 0.32 4.49 4.51

the two distributions resulted to be statistically differentfor all fine meshes. The mean value of the Young’smodulus in all the models was not significantly differentbetween the two algorithms, but the standard deviationwas in the finer meshes, hence resulting in a differentdistribution of the material properties. This difference,however, did not always propagate into the finite

element results. The difference between the maximumVon Mises stress remained always lower than 10%, apartin the coarse mesh of the iliac bone were it reachedthe 13%.

However, the global behaviour of the meshes showedmore marked differences between the two algorithms. Inthe finer meshes of the two long bones 20–30% of the

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Table 3The percent volume fraction where a difference in the Von Mises stressgreater than 10% is found

Coarse (%) Fine (%)

Femur 21 28Ileum 0.6 1.4Tibia 8 18

Fig. 4. Convergence of peak displacement due to mesh refinementfor two finite element models one with homogenous materials proper-ties and one with inhomogeneous material properties derived from theCT radiological density field by means of the BONEMAT—V2 algor-ithm.

bone volume showed differences in the predicted VonMises stresses greater than 10%. As expected, the differ-ences between the two methods were always more evi-dent in the finer meshes, were the material mappingstrategy becomes more critical, however not negligibledifferences were found also for the coarser meshes. Itcan be noticed, in addition, that the behaviour of thethree bone segments is not similar, for what regards theabsolute values of the indicator reported in . This canbe related to the nature of the considered bones and tothe specific meshing algorithm adopted. The smaller dif-ferences between the two methods are reported for theiliac bone. The iliac bone is actually principally madeof trabecular bone surrounded with a thin layer ofcortical bone [15]. When an automatic mesh generatoris used to build the finite element mesh, it is unlikelythat the cortical layer is perfectly represented and partialvolume effects are supposed to be present, loweringsomehow the density of the external elements, even inthe finer meshes. Apart from the external surface, dueto the lack of other abrupt density changes inside thebone, it is expected that no other ‘partial volume’ effectswill be present. This holds true in the present case where,apart from some high density elements due to the pres-ence of small metal pins, the majority of the elementshave a elastic modulus in the range from 2500 to 10 000MPa (Fig. 3). To the contrary, the differences between

the global behaviour of the models become moreimportant for the two long bones considered. In thesecase the partial volume effects are more important andthe differences between the two mapping algorithmsbecome more evident. This effect is probably enhancedby the choice of the meshing algorithms. In the presentcase a fully automatic grid-based unstructured hexa-hedral algorithm was used, that builds a full mesh, thatextends also in the inside of the bone. In this case thedistinction between different kinds of bone tissues, andeven between the shaft and the endosteal canal is left tothe material mapping algorithms. For this reason thechoice of the mapping algorithms influences the behav-iour of the coarser meshes as well, even if the effect isless evident.

From the results of the experiments performed in thiswork, it seems that the choice of the mapping algorithmcan be critical, especially when the size of the elementsis comparable with that of the CT voxel and if anunstructured mesh generator is used.

For what regards convergence of the level of meshrefinement, it seems that the introduction of multiplematerial properties does not affect the convergencebehaviour of the finite element model. Hence the meshrefinement needs not to be increased to obtain the samenumerical accuracy when multiple materials are con-sidered.

A direct comparison of the obtained results with litera-ture is not possible, since to the authors’ knowledge, nowork has been published so far specifically addressingthe influence of the material mapping algorithm on theresults of a finite element analysis. However an approachsimilar to the one developed in this work was alreadyproposed by Edidin in 1991 [4]. The results obtained inthe present work confirm the need of a material mappingalgorithm that computes a more sophisticated element’sYoung’s modulus than simply averaging the value of theCT lattice vertices that fall inside each element.

The major limit of the present work is that the algor-ithms have been tested onto meshes obtained with anunstructured mesh generator, that cannot producemeshes that takes into account the endosteal surface.This choice surely enhances the effect of the material-mapping algorithm on the results of the finite elementanalysis. However to the author’s opinion this limit donot invalidate the conclusion of this work. If a moresophisticated meshing algorithm is used, eventually evena mapped mesh generator, the proposed algorithm willnot introduce a drastic improvement of the materials’description, at least if the elements are not too small, butit will not worsen it neither. On the other hand, in clini-cal contexts, when subject specific finite element modelsof bones should be developed, within times that are com-patible with the clinical practice, unstructured automaticmesh generators are the only choice and in these cases

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the proposed algorithm may introduce significantimprovement in the materials’ description.

4.1. Software availability

The software is available on public domain athttp://www.cineca.it/hosted/LTM-IOR/back2net/SW/index.html. The C++ source code isprovided, together with a GUI implemented in Tcl/Tk.A limited support for the software usage is also providedvia e-mail. At present the software can handle TET4,TET10, HEXA8, and WEDGES finite element top-ologies.

Acknowledgements

The authors would like to thank Luigi Lena for theillustrations and Mauro Ansaloni for the support duringthe experiments. This work was partially funded by theEuropean projects VAKHUM (IST�1999-10954) andPRE-HIP (IST–1999–56408).

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