Vol.2, No.6, 419-424 (2009)
SciRes Copyright © 2009 Openly accessible at http://www.scirp.org/journal/JBISE/
Inhomogeneous material property assignment and
orientation definition of transv erse isotropy of femur
Hai-Sheng Yang1,2, Tong-Tong Guo1, Jian-Huang Wu2, Xin Ma2,3
1Biomechanics Laboratory, Department of Mathematics and Natural Science, Harbin Institute of Technology Shenzhen Graduate
School, Shenzhen, China; 2Shenzhen Institute of Advanced Integration Technology, Chinese Academy of Sciences/The Chinese Uni-
versity of Hong Kong, Shenzhen, China; 3Harbin Institute of Technology, Harbin, China.
Email: xin.ma@siat.ac.cn
Received 28 April 2009; revised 10 June 2009; accepted 12 June 2009.
The finite element method has been increas-
ingly adopted to study the biomechanical be-
havior of biologic structures. Once the finite
element mesh has been generated from CT data
set, the assignment of bone tissue’s material
properties to each element is a fundamental
step in the generation of individualized or sub-
ject-specific finite element models. The aim of
this work is to simulate the inhomogeneous and
anisotropic material properties of femur using
the finite element method. A program is devel-
oped to read a CT data set as well as the finite
element mesh generated from it, and to assign
to each element of the mesh the material prop-
erties derived from the bone tissue density at
the element location. Moreover, for cancellous
bone in femoral neck and cortical bone in fe-
moral stem, the principal orientations of trans-
verse isotropy were defined based on the tra-
becular structures and the haversian system
Keywords: Finite Element; Material Property; In-
homogeneous; Transve rse Isotropy; Femur
The determination of the mechanical stresses in human
bones is of great importance in both research and clinical
practice. Since the stresses in bones cannot be measured
non-invasively in vivo, an effective way to estimate
them is through the finite element (FE) an alysis which is
widely used in academic research and clinical applica-
tions, such as the theory of bone remodeling [1], the
design of pro sthesis [2 ] and th e ev aluation of factur e risk
[3]. In early period the methods used to get bone geome-
try and mechanical properties were inaccurate and some-
times highly invasive and destructive. It is well known
that CT images can provide fairly accurate quantitative
information on bone geometry based on high contrast
between the bone tissue and the soft tissue around [4,5].
Moreover, it has been demonstrated that CT numbers are
almost linearly correlated with apparent density of bio-
logic tissues [6]. Good empirical relationships have been
established experimentally between density and me-
chanical properties of bone tissues [7,8].
In early studies, if a generic bone model was con-
structed, then the mechanical properties of the different
bone tissues were usually derived from average values
reported in published experimental studies [9]. For an
individualized or subject-specific model, however, me-
chanical properties should be derived from CT data. It
has been shown that the stress distribution of a bone is
strongly related to the mechanical p roperties distribution
in the bone tissue [10], and the mechanical properties of
bone have been showed to depend on the subject, anat-
omic location, orientation, biological processes and time.
Hence, it is important to find an effective method to
properly map the material properties derived from CT
data into finite element models.
The CT data can be regarded as a three-dimensional
scalar field (related to the tissue density) sampled over a
regular grid. If the finite element mesh is generated
starting from the same data, the mesh and the density
distribution are perfectly registered in space. The only
problem is how to account for this inhomogeneous dis-
tribution of material properties in to the FE model. Many
approaches were proposed in literature to perform this
task [11,12,13,14,15,16,17,18,19]. However, these algo-
rithms only simulated the inhomogeneity of bone, and
the isotropic material property assignment was adopted
without considering the orien tation of material. Since the
bone material is widely recognized as being anisotropic
rather than isotropic [8,20], the FE simulation of inho-
mogeneity and isotropy cannot reflect the actual charac-
teristics of bone structure. The aim of this study is to
simulate the inhomogeneous and anisotropic material
H. S. Yang et al. / J. Biomedical Science and Engineering 2 (2009) 419-424
SciRes Copyright © 2009 Openly accessible at http://www.scirp.org/journal/JBISE/
properties of femur with finite element method.
2.1. CT Data
The CT dataset of a man’s femur is obtained from the
public database which is created by VAKHUM project
(http://www.ulb.ac.b e/proj ect/vakhu m/index.h tml). The
use of the data is free for academic purposes. The CT
data are in standard DICOM formats. The slice thickness
is 1mm in the epiphysis and 3mm in the diaphysis.
2.2. Finite Element Mesh
The finite element mesh of the right femur (Figure 1)
generated from the corresponding CT dataset above is
also obtained from the VAKHUM project. It is in a
Patran Neutral file format. The mesh is made of linear
hexahedral elements and is generated using the HEXAR
(Cray Research, USA) automatic mesh generator that
implements a grid-based meshing algorithm. The model
mesh is spatially registered with the CT dataset. The
complete finite element mesh consisted of 9,294 nodes
and 7,934 elements.
2.3. The Procedure of Material Property
2.3.1. Calculation of the Average CT Number
(HU )
For each element of the mesh, an average HU value is
calculated with a numerical integration as follows [17]:
(,,)det (,,)
Urst JrstdV
Figure 1. (1) The finite element mesh of
femur (2) The geometrical model of femur.
where 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 Jacob ian of the transformation.
The integrals in (1) are evaluated numerically, and the
order of the numerical integration can be chosen by us.
The value of HU(x,y,z) in a generic point of the CT do-
main is determined by a tri-linear interpolation between
the eight adjacent grid points’ values.
2.3.2. Calibration of the CT Dataset and
Simulation of the Inhomogeneity
CT numbers are dependent on many factors related to
the specific exam. It is assumed that the relationship
between CT numbers and apparent density is linear. A
calibration phantom with bone-equivalent (solution of
hydroxyapatite) insertions of different densities, embed-
ded in a water-equivalent resin-based plastic (The Euro-
pean Spine Phantom, [21]) was used to obtain the pa-
rameters of the linear regression. The calibration equa-
tion is then:
 (2)
where n
is the average density assigned to the ele-
ment n of the mesh, n
HU is the average CT number
are the coefficients provided by calibra-
Referenced values for calibration are selected for ap-
proximate calibration from [16]: Radiographic and ap-
parent density of water (0 HU, 1 g/cm3); Average radio-
graphic density in the cortical region and the apparent
density value for cortical bone (1840 HU, 1.73 g/cm3).
With the steps above, each element of the mesh is as-
signed different apparent density. Therefore, this proce-
dure makes the simulation of inhomogeneity of femoral
material come true.
2.3.3 The Density-Modulus Relationship and the
Simulation of Anisotropy
Various empirical models of the relationship between
Young’s modulus and the apparent bone density can be
found in the literatures [7,8]. The relations are generally
 (3)
where n
E is the average Young’s modulus assigned to
the element n of the mesh, n
is its apparent density
and a, b and c are the coefficients.
In most of these works [11,12,13,14,15,16,17,18,19],
the isotropic material property assignment was adopted
without considering th e direction of material property as
its simplicity. Many prev ious studies had clearly demon-
strated the anisotropic behavior of bone, and the deter-
mination of bone material properties as a function of
H. S. Yang et al. / J. Biomedical Science and Engineering 2 (2009) 419-424
SciRes Copyright © 2009 http://www.scirp.org/journal/JBISE/
direction is the essential requirement [22]. The axial
direction is defined according to the Haversian osteons
of the cortical bone and according to the spatial main
direction of the trabecular structures of cancellous bone.
The relationship between bone material properties and
apparent density is given by [8]:
coincide with the structure of bone.
As we know, bone structure is customarily recognized
as confirming to ‘wolff’s law’ which is essentially the
observation that bone changes its external shape and
internal architecture in response to stresses acting on it.
Thus, the structure of bone (or material orientation)
strongly coincides with the principal stress track. The
directions of the femoral neck and stem can be approxi-
mately regarded as stress track in femur.
Cortical bone:
1.57 3.09
12 3
1213 23
12 1121323
2314 ,2065 ,
0.58, 0.32,
/ 2(1),3300;
 
 
(4) Femoral bone can be recognized as transversely iso-
tropic material. According to the cortical bone structure
in femoral stem and cancellous bone structure in femoral
neck, the principal material orientation of cancellous
bone is defined by the direction of the trabecular struc-
tures and the principal material orientation of cortical
bone by the direction of the haversian system.
Cancellous bone:
1.78 1.64
12 3
1213 23
12 1121323
1157 ,1094 ,
0.58, 0.32,
/ 2(1),110;
 
 
where E is the Young’s modulus (MPa), G the shear
modulus (MPa),
is the apparent density (g/cm3),
the Poisson’s ratio. All parameters are set in elemental
Cartesian coordinate system of Patran (MSC, USA).
3.1. Material Properties Distribution
This material assignment procedure produces 165 dif-
ferent material cards. The distribution of the material
properties in femur are shown in Figure 2. The maxi-
mum and minimum values for apparent density and elas-
tic modulus are listed in Ta b l e 1. The maximum values
are corresponding to the Mat-1 and the minimum to
Mat-165 as a result of the definitio n in the program. The
numbers of elements for each material property card are
showed in Figure 3.
This procedure may, theoretically, lead to a different
material card (Mat) for each element of the mesh, which
may result in computational problems with those codes
that can handle a limited number of materials. We can
reduce the number of materials by choosing a
threshold. Then the maximum computed value of the
elastic modulus, is assigned to the material Mat-1.
All the elements with are assigned the
material Mat-1. Mat-2 is characterized by the maximum
E of the remaining elements and so on until the whole
set of the model material is defined. In this paper, the
)( max EEE 
is taken as 50 MPa.
3.2. The FE Simulation of Transversely
Isotropic Material Property
Figure 2 only showed the inhomogeneous distribution
of material properties. After separating the femoral stem
and neck, th e prin cipa l material orientations for elements
in neck and stem are defined respectively. Figure 4 il-
lustrates the vector form of the principal orientations in
femoral neck. Figure 5 presents the vector form of the
principal orientations in femoral stem.
A program was developed to read the original data in-
cluding CT data, finite element mesh data and some pa-
rameters and then to implement the procedure described
above. In this way, each element is assigned different
density, elastic modulus and Poisson’s ratio. At last, a
finite element mesh provided with the assigned material
properties is written out in a Patran Neutral file format. 4. DISCUSSION
Every element of the finite element model has its own
element coordinate system used to determine the mate-
rial’s principal axe of this element. As the element coor-
dinate systems vary from element to element, the prin-
cipal material orientations need to be defined so as to
As is well-known, it is of significance to explore the
biomechanical behavior of bone. FE analysis has been
adopted by many researchers based on CT data that can
provide useful information on the geometrical topology
Table 1. Material properties of femur (the unit for density is g/cm3, and for elastic modulus is MPa).
Material properties Maximum Minimum
1.787 0.686
E1 (E2) 5755.799 591.512
E3 12410.846 1026.157
Openly accessible at
H. S. Yang et al. / J. Biomedical Science and Engineering 2 (2009) 419-424
SciRes Copyright © 2009 Openly accessible at http://www.scirp.org/journal/JBISE/
Figure 2. Right femur with different material properties
mapped on it, observed from different angles of view.
Figure 3. The number of elements for each material property card.
H. S. Yang et al. / J. Biomedical Science and Engineering 2 (2009) 419-424
SciRes Copyright © 2009 http://www.scirp.org/journal/JBISE/
Figure 4. The principal material orientations
in femoral neck showed in vector form (the
purple arrows).
Figure 5. The principal material orientations in
femoral stem showed in vector form (the pur-
ple arrows).
and material properties of bone. However, how to con-
struct accurate FE models that reflect the real material
properties of bone remains a problem. This work is
aimed to simulate the inhomogeneity and anisotropy of
the femur in order to make the FE material model more
As the relationship between density and Young’s
modulus is power law, different results will be produced
when firstly averaging on each element the HU field and
then calculating th e element Young’s modulus, or, on the
contrary, averaging Young’s modulus derived from HU
value. It has been demonstrated that the latter can
achieve more accurate results [18]. In the present work,
CT number over each element volume is averaged first
and then the Young’s modulus is calculated with the ex-
periential relationships. However, it has no influences on
the FE simulation of anisotropy.
Different algorithms are adopted to assign material
properties into elements of the mesh model [17]. Most of
these algorithms are based on the assumption that the
variability of the CT numbers within the volume of each
element can be neglected. All differences rely on how
the average value of each element is computed. As we know, inner of the femur is marrow cavity that
contains marrow or air. Howev er, the mesh model in this
study does not take into account the endosteal surface.
Thus, the elements that belong to marrow cavity are as-
signed density and elastic modulus as well.
Some simple methods are introduced to assign to each
element the value by averaging the CT numbers of the
CT sampling grids which are nearest to corresponding
nodes of element or by averaging the CT numbers of the
eight CT sampling points surrounding the element cen-
troid [10,15]. These methods may produce inaccurate
results when the element size is significantly larger than
the spacing of the CT sampling grid.
FE simulation of complete anisotropy of femoral bone
is impossible at present. Thus, femoral material is sim-
plified as being transv ersely isotrop ic in this study. It has
been demonstrated that the structure of femur is highly
variable, especially to cancellous bone. Thus, a clear and
exact definition of the principal axes of transversely
isotropy is impossible. In this study, we only separated
the femoral neck and stem. Then, the principal orienta-
tions of neck were defined on the basis of the direction
of trabecular structure and the principal orientations of
stem on the basis of the direction of harversian system.
As the structure of femur (or material orientation) coin-
cides with the track of principal stress, the orientation
definition based on pass of stress is reasonable.
A second approach that can avoid those shortcomings
determines all CT sampling points that fall inside the
element volume and assigns to the element the average
of these values [16]. However, this method may not
produce satisfactory results when the elements are of
comparable dimension or smaller than the voxel size.
An advanced method adopted in our study estimates
the average CT numbers by numerical integration over
the element volume [17]. In this case, the dimension of
the finite element is not influent on the accuracy of the
average CT number estimation since the calculation of
CT numbers takes into account the spatial distribution of
the CT number in the eight surrounding CT lattice verti-
Muscle, as well as bone, is one of the most important
parts of the biologic body. Musculoskeletal loading in-
fluences the stresses and strains within the human bone
and thereby affects the processes of bone modeling and
Openly accessible at
H. S. Yang et al. / J. Biomedical Science and Engineering 2 (2009) 419-424
SciRes Copyright © 2009 Openly accessible at http://www.scirp.org/journal/JBISE/
remodeling. Therefore, three-dimensional FEA for mus-
cle is also of great importance. Assignment of the mate-
rial properties to muscle FE mesh using the method in
this paper may encounter some difficulties as the muscle
belong to soft tissue that do not have steady shape. Thus,
further research may be focused on the assignment of
muscle material property.
Although the inhomogen eous and transv ersely iso tropic
material properties simulated in this work are theoreti-
cally close to that of real femur, experimental validation
need be performed in the future.
This work was supported by National Natural Science Founda-
tion of China (No. 30700165 and No. 60803108).
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