Journal of Biomechanics 35 (2002) 1091–1099
Combination of topological parameters and bone volume fraction
better predicts the mechanical properties of trabecular bone
Laurent Pothuauda, Bert Van Rietbergenb, Lis Mosekildec, Olivier Beuf a, Pierre Levitzd,
Claude L. Benhamoue, Sharmila Majumdara,*
a
Magnetic Resonance Science Center, Department of Radiology, University of California, 1 Irving street, Room AC-109 San Francisco,
CA 94143-1290, USA
b
Biomechanics Section, Eindhoven University of Technology, Eindhoven, The Netherlands
c
Department of Cell Biology, Institute of Anatomy, University of Aarhus, Denmark
d
Centre de Recherche sur la Mati"ere Divis!ee, CNRS / Universit!e d’Orl!eans,Orl!eans, France
e
Institut de Pr!evention et de Recherche sur l’Ost!eoporose, CHR d’Orl!eans,Equipe ERIT-M INSERM, Orl!eans, France
Accepted 14 March 2002
Abstract
Trabecular bone structure may complement bone volume/total volume fraction (BV/TV) in the prediction of the mechanical
properties. Nonetheless, the direct in vivo use of information pertaining to trabecular bone structure necessitates some predictive
analytical model linking structure measures to mechanical properties. In this context, the purpose of this study was to combine BV/
TV and topological parameters so as to better estimate the mechanical properties of trabecular bone. Thirteen trabecular bone midsagittal sections were imaged by magnetic resonance (MR) imaging at the resolution of 117 117 300 mm3. Topological
parameters were evaluated in applying the 3D-line skeleton graph analysis (LSGA) technique to the binary MR images. The same
images were used to estimate the elastic moduli by finite element analysis (FEA). In addition to the mid-sagittal section, two
cylindrical samples were cored from each vertebra along vertical and horizontal directions. Monotonic compression tests were
applied to these samples to measure both vertical and horizontal ultimate stresses. BV/TV was found as a strong predictor of the
mechanical properties, accounting for 89–94% of the variability of the elastic moduli and for 69–86% of the variability of the
ultimate stresses. Topological parameters and BV/TV were combined following two analytical formulations, based on: (1)
the normalization of the topological parameters; and on (2) an exponential fit-model. The normalized parameters accounted for
96–98% of the variability of the elastic moduli, and the exponential model accounted for 80–95% of the variability of the
ultimate stresses. Such formulations could potentially be used to increase the prediction of the mechanical properties of trabecular
bone. r 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Trabecular bone structure; Topological parameters; Mechanical properties; 3D-LSGA; FEA
1. Introduction
The assessment of bone quality is important in the
diagnosis of osteoporosis and for studying the efficacy
of bone therapeutic interventions. Skeletal status is
currently evaluated by X-ray or ultrasound absorptiometry, primarily densitometric techniques. Although the
density of bone is a strong predictor of the strength of
*Corresponding author. Tel.: +1-415-476-6830; fax: +1-415-4768809.
E-mail address: [email protected]
(S. Majumdar).
bone and the propensity to fracture (Genant et al.,
1996), the structure of trabecular bone, is an important
contributor to bone strength (Ulrich et al., 1999). Over
the last several years, different imaging techniques have
been developed and optimized for the reconstruction of
trabecular bone structure both in vitro and in vivo (Link
et al., 1999). The development of image analysis
techniques for the characterization of the 3D-trabecular
bone structure remains a privileged research field
(Pothuaud et al., 2000a; Laib et al., 2001; Wehrli et al.,
2001), the main requirement of a structure parameter
being its complementarity with the density of bone, and
indirectly with the fraction of solid bone volume/total
0021-9290/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 1 - 9 2 9 0 ( 0 2 ) 0 0 0 6 0 - X
1092
L. Pothuaud et al. / Journal of Biomechanics 35 (2002) 1091–1099
volume (BV/TV). Furthermore, the definition of a
predictive analytical model linking trabecular bone
structure parameter to mechanical properties, could be
useful for the practical use of such parameter (Kabel
et al., 1999a). Thus in addition to bone mineral density,
the complementary role of an in vivo evaluation of the
trabecular bone structure for understanding the age
related skeletal changes or the role and mechanism
therapeutic interventions, would be invaluable.
The skeleton graph is a simplified thinned structure
that has some interesting properties for the characterization of 3D-structures. It is centered inside the initial
structure, it respects its connections, its elongations also,
and can be described as a network of vertices and
branches. Such representation has several applications
in image processing (Deseilligny et al., 1998; Lin and
Cohen, 1982; Krissian et al., 1998; Krass et al., 2000),
including trabecular bone structure characterization
(Garrahan et al., 1986; Wessels et al., 1997; Pothuaud
et al., 2000a; Saha et al., 2000). Pothuaud et al. (2000a)
have developed a technique able to quantify the 3D
topology of the trabecular bone using a 3D-line skeleton
graph analysis (LSGA) technique. The principle of this
technique is to associate to each branch of the skeleton
graph a trabecula of the bone network, and to each
vertex a connection between several trabeculae.
We hypothesized that in addition to bone mineral
density, skeleton graph based method would provide an
improved prediction of bone mechanical properties.
Specifically, the aim of this study was to combine both
BV/TV and 3D-LSGA-based topological parameters.
Two kinds of combination were used, based on: (1) a
novel normalization scheme of the topological parameters; and (2) the use of an exponential fit-model. The
potential of these combinations was tested for the
prediction of the elastic moduli and ultimate stress in
comparison to the prediction obtained with BV/TV
alone.
2. Materials and methods
Thirteen dried human lumbar vertebrae (L3) were
obtained from 13 individuals (6 women and 7 men, 22–
76 yr). A mid-sagittal section was sawed in each vertebra
with a diamond precision-parallel saw (Exakt, Apparatebau, Otto Herrmann, Norderstedt, Germany). The
sections were aligned along the main crano-caudial
vertical direction. The mean dimensions were 9.0 mm
thickness along the medial/lateral (M/L, horizontal)
direction, 22.2 mm height along the superior/inferior (S/
I, vertical) direction, and 29.1 mm width along the
anterior/posterior (A/P) direction. In each vertebra, two
additional cylindrical trabecular bone samples (7 mm
diameter, 5 mm height) were cored along S/I and M/L
directions, approximately 10 mm adjacent to each M/L
edge of the mid-sagittal section. The samples were sawed
(Exakt precision-parallel saw) carefully to obtain planoparallel ends (Mosekilde et al., 1985).
The cylindrical samples drilled in the neighborhood of
the mid-sagittal section were mechanically tested in
uniaxial compression using a material testing machine
(Alwetron TCT-5) (Mosekilde et al., 1985; Mosekilde,
1989) with a constant compression rate of 2 mm/min.
The samples were placed unsupported in the testing
machine, and small variations in the position of the
endplates of the cylinders during compression were
corrected for by an interposed steel ballbearing (Mosekilde et al., 1985). The bone ultimate load was
determined as the first maximum of the measured
force–displacement curve. The ultimate stress value
was calculated as the ultimate load divided by the
cross-sectional area of the sample. The vertical (VStress) and horizontal (H-Stress) ultimate stresses
(expressed in MPa unit) were determined from the
samples extracted along the S/I and M/L direction,
respectively.
For magnetic resonance (MR) imaging, the midsagittal sections were immersed in 0.3 volume-percent
gadolinium-DTPA-doped water solution to reduce
the T 1 relaxation time and obtain an equivalent
for fatty bone marrow. Then, the sections were
placed into a vacuum to remove air bubbles and
prevent artifact occurrence. MR images were acquired
on a 1.5 T scanner (General Electric Medical Systems,
Waukesha, WI) with a receive-only wrist coil (Medical
Advances, Milwaukee, WI). The sections were positioned inside the magnet, the S/I direction being aligned
along the static magnetic field. The data were acquired
using a 3D fast gradient-echo sequence with a partial
echo time (Majumdar et al., 1999): echo time=7 ms;
repetition time=50 ms; flip angle=301; band
width=715.6 kHz; scan-time=33 min. The resolution
was 117 117 mm2 in the (M/L–A/P) plan and 300 mm
along the S/I direction.
A region of interest (ROI) was defined avoiding the
cortical bone. Gray level images were segmented into
bone (solid) and marrow (assimilated-void) phases using
a dual bone/marrow reference binarization scheme.
Bone and marrow reference levels were evaluated from
the gray level images, and a global threshold value was
calculated in using a two phase histogram model
(Majumdar et al., 1997). A clustering analysis (Hoshen
and Kopelman, 1976) was applied to the binary images
in order to remove the small isolated solid and void
clusters that were assimilated to noise. Hence, it was
ensured only one cluster of solid (b0 ¼ 1; 0th-order Betti
number) and no internal holes (b2 ¼ 0; 2nd-order Betti
number), that corresponds to the physical state of the
bone sample. BV/TV was calculated as the ratio of the
number of pixels belonging to the solid phase and the
total number of pixels in the entire ROI.
L. Pothuaud et al. / Journal of Biomechanics 35 (2002) 1091–1099
3D-skeleton graph (Fig. 1) was obtained using an
iterative thinning algorithm constrained by both topological and morphological constraints (Pothuaud et al.,
1093
2000a). At each iteration, a set of ‘‘suppressible’’ pixels
was identified and removed, based on the conservation
of the topological invariants (b0 ¼ 1; b2 ¼ 0; and EPC:
Euler–Poincare! characteristic) before and after the
removing (topological constraints), as well as based on
the conservation of the elongation of the initial structure
(morphological constraints). At each iteration, b0 and b2
were evaluated from a clustering analysis algorithm
(Hoshen and Kopelman, 1976), and EPC was evaluated
following an algorithm described by Vogel (1997). The
thinning algorithm was stopped when no more pixels
were identified as ‘‘suppressible’’. Then, the 3D-digitized
skeleton graph was interpreted as a network of vertices
and branches. The numbers of ‘‘connection’’ and
‘‘termini’’ vertices (Vc ; Vt ) and branches (Bcc ; Bct ), the
total numbers of vertices (V ¼ Vc þ Vt ) and branches
(B ¼ Bcc þ Bct ), as well as the number of loops (L) were
calculated (Pothuaud et al., 2000a) (Fig. 2), using the
following non-dimensional equation:
L ¼ 1 ðV BÞ:
ð1Þ
A topological parameter (K) expresses a number of
topological event (loop, vertex, branch, etc.) appearing
in the TV of analysis. In order to compare several
samples together (with different volumes of analysis) the
most current normalization consists of dividing by the
total volume TV, giving ‘‘volumetric’’ topological
parameters ðKÞ1 :
Fig. 1. (a) 2D slice of a 3D-binary image obtained after thresholding
and cluster analysis (only one cluster of solid); and (b) the same slice of
the 3D skeleton graph. The three orientations M/L (medial/lateral,
horizontal), S/I (superior/inferior, vertical), and A/P (anterior/posterior) are reported in the figure, as well as the scales of the 2D slice. The
resolution was 117 117 mm2 in the plane of the figure, and 300 mm in
the perpendicular direction (S/I). The initial 3D images were enlarged
to ensure a 5-pixels void-border along each M/L, S/I, and A/P
directions.
ðKÞ1 ¼ K=TV mm3 :
ð2Þ
If S1 and S2 are two structures with a volume of analysis
TV1 and TV2, respectively, such as S2 is obtained from
S1 by homothety transform of ratio k (or ‘‘zoom’’ with a
constant magnification or reduction ratio k), we have
K½S1 ¼ K½S2 (conservation of the topology) and
TV2 ¼ k TV1 : Hence, the volumetric parameters
“termini” vertex (Vt)
“connection” vertex (Vc)
“termini” branch (Bct)
“connection” branch (Bcc)
Fig. 2. Simplified skeleton graph configuration, showing the classification of each point of the skeleton graph (reported in using different symbols
and gray levels). This particular configuration is constituted of B ¼ 11 branches (Bcc ¼ 7 ‘‘connection’’ branches þBct ¼ 4 ‘‘termini’’ branches) and
V ¼ 10 vertices (Vc ¼ 6 ‘‘connection’’ vertices þVt ¼ 4 ‘‘termini’’ vertices). The number of loops (or connectivity) is calculated as L ¼ 1 ðV BÞ ¼
2 (Eq. (1)).
L. Pothuaud et al. / Journal of Biomechanics 35 (2002) 1091–1099
1094
ðKÞ1 ½S1 and ðKÞ1 ½S2 are linked by the equation
ðKÞ1 ½S1 ¼ k3 ðKÞ1 ½S2 : This equation shows that the
topology is not conserved with the volumetric parameters during homothety transform of ratio ka1; and
an inter-samples analysis could be debatable in this case.
Based on these considerations, we have defined another
normalization scheme. A volume of reference (VR),
characterizing the solid phase, was defined in using the
mean chord length of the solid phase (lS Þ (Levitz and
Tchoubar, 1992):
3
3
VR ¼ ðlS Þ mm :
ð3Þ
The ‘‘normalized’’ parameters were expressed as the
numbers of topological events appearing in the volume
VR by using the following non-dimensional equation:
ðKÞ2 ¼ VR K=TV;
ð4Þ
This normalization scheme conserves the topology
during homothety transform. We can easily demonstrate
the relation ðKÞ2 ½S1 ¼ ðKÞ2 ½S2 ; taking into account that
ðlS Þ1 ¼ k ðlS Þ2 :
The 3D-binary image was used to generate a FE model
by converting the pixels of the solid phase to equally
shape 8-node brick elements (van Rietbergen et al.,
1995a). Using a special-purpose FE-solver, compression
and shear tests were simulated in the three orthogonal
directions. For these simulations, the tissue elastic
properties were chosen linear elastic and isotropic with
a Young’s modulus of 5.33 GPa and a Poisson’s ratio of
0.3 (van Rietbergen et al., 1995b). The stiffness matrix
was calculated using the results of the six simulations
(van Rietbergen et al., 1996). Then, an optimization
procedure was used to find a new coordinate system
aligned with the best orthogonal symmetry-directions.
The stiffness matrix was rotated to this new orthogonal
coordinate system, and the Young’s moduli (E1 ; E2 ; E3 )
as well as shear moduli (G12 ; G13 ; G23 ) were calculated in
these principal directions. The matrix was sorted such
that the Young’s modulus in the primary direction (E1 )
represents the largest modulus and the third Young’s
modulus (E3 ) the smallest one: E1 > E2 > E3 : All analyses
were done using a special-purpose iterative solver
developed in-house (van Rietbergen et al., 1995a).
Univariate linear correlations were performed using
JMP software (SAS Institute, Inc.), and R2 -adjusted
determination coefficients were used. Results were
considered significant when they were in the 95%
confidence interval (po0:05).
An exponential fit-model combining both topological
and BV/TV parameters (Pothuaud et al., 2000b) was
used in this study to predict the mechanical properties:
M ¼ b0 þ b1 expðb2 BV=TVÞ ðb3 ðKÞ1;2 Þ MPa;
ð5Þ
where M relates the mechanical properties, ðKÞ1;2 the
volumetric (Eq. (2)) or normalized (Eq. (4)) parameters,
and (b0 ; b1 ; b2 ; b3 ) are the fit-coefficients. With (K)1 the
dimensionalities of the fit-coefficients b0, b1, b3 are
respectively MPa, mm3 MPa and mm3 and b2 is
dimensionless. With (K)2 both b0, b1, have dimension
MPa, b2 and b3 are dimensionless. All of the fit-models
were done using S-PLUS 2000 software (Mathsoft Inc.),
with a generalized non-linear least squares fit method.
3. Results
When compared to the volumetric topological parameters, BV/TV was the best predictor of the mechanical
properties (Table 1), accounting for 89–94% of the
variability of Young’s moduli, 91–94% of the variability
of shear moduli, 69% of the variability of V-Stress, and
86% of the variability of H-Stress. The ‘‘termini’’
topological parameters were non-significantly correlated
to all of the mechanical properties. Globally, the
predictions of vertical ultimate stress were lower than
those of horizontal ultimate stress. The volumetric
topological parameters (except for Vt and Bct ) were
significantly correlated with BV/TV, accounting between 91% and 96% of the variability of BV/TV.
All of the normalized parameters were significantly
correlated with the mechanical properties (Table 2) with
higher R2 values than those obtained using the
corresponding volumetric topological parameters, or
BV/TV (except for the prediction of H-Stress). For
example, the normalized parameters were able to
account for 95–97% of the variability of E1 (48% for
Vt and Bct ), while the volumetric topological parameters
accounted for 79–88% of the variability of E1 (nonsignificant correlation for Vt and Bct ), and BV/TV
accounted for 94% of the same variability. In parallel to
an increase of the predictions of the mechanical
properties, it is noticed that the normalization of the
topological parameters induced an increase of the
correlations with BV/TV as well.
For the prediction of the elastic moduli, exponential
fit-models gave better predictions than the volumetric
topological parameters, and equivalent predictions with
BV/TV (Table 1). These models were particularly
beneficial for the prediction of the ultimate stresses.
Exponential fit-models combining BV/TV and ðLÞ1 were
able to account for 72–80% (72% for ‘‘termini’’
parameters) of the variability of V-Stress, and for 86–
95% (86% for ‘‘termini’’ parameters) of the variability
of H-Stress. All of the exponential fit models combining
BV/TV and normalized parameters gave the same, or
moderately higher, predictions than the linear correlations (Table 2).
In the following, we examine the case of the
exponential fit-model combining BV/TV and ðLÞ1 for
the prediction of H-Stress. The linear correlation
between H-Stress and its exponential fit-model
L. Pothuaud et al. / Journal of Biomechanics 35 (2002) 1091–1099
1095
Table 1
Relationship between volumetric topological parameters and mechanical properties
Linear correlation
BV/TV
E1
E2
E3
G23
G13
G12
V-Stress
H-Stress
BV/TV
(V)1
(B)1
(Vc )1
(Vt )1
(Bcc )1
(Bct )1
(L)1
1.00
0.91
0.94
0.95
ns
0.95
ns
0.96
0.94
0.79
0.84
0.84
ns
0.86
ns
0.88
0.92
0.81
0.86
0.86
ns
0.89
ns
0.91
0.89
0.75
0.82
0.82
ns
0.85
ns
0.88
0.91
0.78
0.85
0.85
ns
0.87
ns
0.90
0.92
0.75
0.81
0.82
ns
0.84
ns
0.87
0.94
0.80
0.85
0.86
ns
0.88
ns
0.90
0.69
0.47
0.53
0.53
ns
0.56
ns
0.58
0.86
0.66
0.70
0.71
ns
0.72
ns
0.73
0.92
0.92
0.92
0.89
0.92
0.89
0.92
0.87
0.87
0.88
0.85
0.88
0.85
0.89
0.86
0.86
0.87
0.84
0.87
0.84
0.89
0.88
0.88
0.89
0.87
0.89
0.87
0.89
0.91
0.92
0.92
0.88
0.92
0.88
0.93
0.91
0.93
0.92
0.90
0.93
0.90
0.93
0.76
0.78
0.78
0.72
0.79
0.72
0.80
0.91
0.92
0.92
0.86
0.94
0.86
0.95
Exponential fit-R2
f (BV/TV, (V )1 )
f (BV/TV, (B)1 )
f (BV/TV, (Vc )1 )
f (BV/TV, (Vt )1 )
f (BV/TV, (Bcc )1 )
f ( BV/TV, (Bct )1 )
f ( BV/TV, (L)1 )
The upper part of the table reports the linear correlations (R2 coefficients) between volumetric topological parameters, mechanical properties, and
BV/TV. Non-significant correlations (p > 0:05) are designed by ‘‘ns’’. For each of the mechanical properties, the best of the predictions is in bold. The
lower part of the table reports the predictions (R2 coefficients) obtained with exponential fit-models f (BV/TV, (K)1) combining both BV/TV and
volumetric topological parameter (K)1. For each of the mechanical properties, the best of these predictions are written in bold.
Table 2
Relationship between normalized parameters and mechanical properties
linear correlation
BV/TV
E1
E2
E3
G23
G13
G12
V-Stress
H-Stress
BV/TV
(V)2
(B)2
(V c)2
(V t)2
(Bcc)2
(Bct)2
(L)2
1.00
0.96
0.97
0.97
0.47
0.97
0.47
0.96
0.94
0.97
0.97
0.97
0.48
0.96
0.48
0.95
0.92
0.95
0.97
0.97
0.39
0.97
0.39
0.97
0.89
0.93
0.95
0.95
0.36
0.96
0.36
0.96
0.91
0.95
0.96
0.96
0.35
0.97
0.35
0.98
0.92
0.94
0.95
0.95
0.39
0.95
0.39
0.96
0.94
0.97
0.98
0.98
0.41
0.98
0.41
0.98
0.69
0.70
0.70
0.70
0.33
0.70
0.33
0.69
0.86
0.82
0.82
0.82
0.38
0.82
0.38
0.80
0.97
0.97
0.97
0.81
0.97
0.81
0.96
0.98
0.98
0.97
0.92
0.98
0.92
0.97
0.97
0.97
0.97
0.89
0.97
0.89
0.97
0.98
0.98
0.98
0.90
0.98
0.90
0.98
0.96
0.96
0.96
0.91
0.96
0.91
0.96
0.98
0.98
0.98
0.82
0.98
0.82
0.98
0.69
0.70
0.70
0.63
0.70
0.63
0.68
0.86
0.84
0.86
0.81
0.84
0.81
0.82
Exponential fit-R2
f ( BV/TV, (V )2 )
f ( BV/TV, (B)2 )
f ( BV/TV, (V c)2 )
f ( BV/TV, (V t)2 )
f ( BV/TV, (Bcc)2 )
f ( BV/TV, (Bct)2 )
f ( BV/TV, (L)2 )
The upper part of the table reports the linear correlations (R2 coefficients) between normalized parameters, mechanical properties, and BV/TV. Nonsignificant correlations (p > 0:05) are designed by ‘‘ns’’. For each of the mechanical properties, the best of the predictions is in bold. The lower part of
the table reports the predictions (R2 coefficients) obtained with exponential fit-models f (BV/TV, (K)1) combining both BV/TV and normalized
parameter (K)1. For each of the mechanical properties, the best of these predictions are written in bold.
(H-Stress’) was characterized by a linear relation HStress’=0.99 H-Stress (Fig. 3a). Experimental data were
plotted on a (ðLÞ1 ; H-Stress) coordinate system (Fig. 3b).
The relationship between ðLÞ1 and H-Stress was
characterized by a linear correlation with R2 ¼ 0:73
(Table 1) and a positive slope regression line. Hence,
when the connectivity increased, the ultimate stress
increased as well, and conversely. The particularity of
the exponential fit-model (Eq. (5)) is that when we fix
BV/TV at a particular value, the relationship between
H-Stress and ðLÞ1 becomes linear. Using this, several
lines with BV/TV=constant (10%, 15%, 20%, 25%,
30%) are superimposed in the (ðLÞ1 ; H-Stress) representation (Fig. 3b). It is noted that these lines have a
negative slope, meaning that when BV/TV is maintained
constant (BV/TV=10%, for example), an increase of
L. Pothuaud et al. / Journal of Biomechanics 35 (2002) 1091–1099
1096
H-Stress' [MPa]
3
y = 0,99x
R2 = 0,95
2
1
0
0
1
2
(a)
3
H-Stress [MPa]
5
H-Stress [MPa]
4
BV/TV=30%
BV/TV=25%
3
BV/TV=20%
(D)
BV/TV=15%
2
(B)
BV/TV=10%
1
(E)
(A)
(C)
0
0
1
3
2
(b)
4
5
(χ)1 [mm ]
-3
Fig. 3. (a) Linear regression between horizontal ultimate stress (H-Stress) and its exponential fit-model H-Stress’=f (BV/TV, ðLÞ1 ), combining BV/
TV and volumetric topological parameter (number of loops, or connectivity) ðLÞ1 : This relationship is characterized by the Eq. H-Stress’=0.99 HStress, and a correlation coefficient R2 ¼ 0:95: (b) Two-dimensional representation of the exponential fit-model. The experimental data are plotted on
the (ðLÞ1 ; H-Stress) coordinate system, showing a linear regression with positive slope (characterized by R2 ¼ 0:73; cf. Table 1). Some lines with BV/
TV constant are modeled from the exponential model equation (Eq. (5)), and reported in the figure with BV/TV between 10% and 30%. These lines
are characterized by a negative slope. Using a K-means clustering analysis, the experimental data were clustered following BV/TV parameter in five
different clusters (A–D). A line BV=TV ¼ constant is associated to each cluster, corresponding to the mean BV/TV value of the cluster.
ðLÞ1 is linked to a decrease of H-Stress. A K-means
clustering analysis was performed (S-PLUS 2000,
Mathsoft Inc.) following BV/TV parameter and in
setting five clusters (A, B, C, D, E) (Fig. 3b). The
centers (mean values) of these clusters were 14.1% (A),
23.4% (B), 18.5% (C), 27.6% (D), and 20.4% (E),
respectively. For each of these mean values, the
corresponding line BV=TV ¼ constant stemmed from
the exponential relation was plotted (close to the
corresponding cluster of data).
4. Discussion
In this study, we have shown that BV/TV was a strong
predictor of the mechanical properties of the trabecular
bone structure. A normalization of the 3D-LSGA based
topological parameters was defined. In combining both
morphological and topological explanatory powers, the
normalized parameters were able to increase the
prediction of the elastic moduli compared to the
prediction of BV/TV alone. Furthermore, BV/TV and
volumetric topological parameters were associated in an
exponential fit-model. As hypothesized, such a model
was able to increase the prediction of the ultimate
stresses compared to the prediction of BV/TV alone.
The resolution of the images used in this study was
117 117 300 mm3, which is comparable to the resolution achievable in vivo with MR imaging (Link et al.,
1999). Although the ‘‘apparent’’ trabecular bone network depicted at such resolution is affected by resolution effects (Majumdar et al., 1997; Kothari et al., 1998),
it has been demonstrated that the evaluation of the
‘‘apparent’’ trabecular network is well correlated to the
evaluation performed at higher resolution (van Rietbergen et al., 1998).
L. Pothuaud et al. / Journal of Biomechanics 35 (2002) 1091–1099
The specimens used in this study constituted a large
study called ‘‘Danish in vitro bone study (DAVIBO)’’
(Thomsen et al., 1998) that preceded the time period
during which the present study was conceived. Hence,
the design of our study was limited to the use of midsagittal sections from which experimental elastic property measurements were not available. That is why such
elastic properties were estimated using FE analysis. On
the contrary, ultimate stresses were experimentally
evaluated from adjacent cylindrical samples, which one
could speculate was the reason for better predictions of
the elastic moduli compared to those of the ultimate
stresses (Tables 1 and 2). The protocol strategy of this
study was to evaluate the ultimate stress in two
perpendicular directions (Beuf et al., 2001). Hence this
cannot be performed on the same specimen, it has been
imposed to apply mechanical tests on two different
specimens.
For MR imaging, the specimens were aligned with the
S/I direction parallel to the magnetic field. Hence, the
contrast of the MR images was more dependent on the
trabeculae oriented approximately perpendicular to the
magnetic field (Beuf et al., 2001). This could explain that
the predictions of horizontal ultimate stress were always
greater than those of vertical ultimate stress. Furthermore, there was a projection effect along the S/I
direction due to the anisotropic resolution, and this fact
could attenuate the S/I dependence of the structure
evaluation.
The tissue modulus used for the FE analysis was
constant, and variations in the tissue quality (gender,
age) were not accounted for. The value chosen
(5.33 Gpa) was obtained in an earlier study (van
Rietbergen et al., 1995a; van Rietbergen et al., 1995b).
It should be noted, however, that the FE results can be
easily scaled for any other value without affecting the
calculation of the correlation coefficients.
None of the topological parameters classified as
‘‘termini’’ were significantly correlated to the mechanical
properties. This could be explained by the fact that the
‘‘termini’’ parameters are related to some trabeculae
that are connected from one side only (Fig. 1). On the
contrary, the ‘‘connection’’ parameters (Vc ; Bcc and L)
were found as good predictors of the mechanical
properties. The connectivity (or number of loops) is
commonly evaluated in a global way in using the EPC
characteristic: w ¼ 1EPC. It is noted, however, that the
3D-LSGA technique has been developed and optimized
following topological considerations, ensuring a very
low relative error between the parameter L evaluated
from the analysis of the digitized skeleton graph, and the
parameter w evaluated in a global way. In this study, the
mean relative error between L and w was 0.98%.
Based on the hypothesis of the conservation of the
topology with homothety transform (or ‘‘zoom’’), we
have defined normalized parameters (Eq. (4)) as the
1097
numbers of topological events appearing in a particular
volume of reference (VR) defined from the mean chord
length of the solid phase (lS ) (Levitz and Tchoubar,
1992) which is a morphological parameter. The normalized parameters gave higher predictions of the mechanical properties than volumetric topological parameters,
and higher than BV/TV for the elastic moduli (Tables 1
and 2). It must be noted that the correlation between
BV/TV and the normalized parameters were higher than
those between BV/TV and the volumetric topological
parameters (Tables 1 and 2). Hence, the higher
correlations obtained with the normalized parameters
could be explained by the fact that some explanatory
power of BV/TV was transferred to the normalized
parameters via the normalization scheme with lS (the
coefficient of correlation between BV/TV and lS was
R2 ¼ 0:65). The normalization of the topological parameters (Eq. (4)) could be considered as an analytical
model combining both morphological (or BV/TV) and
topological parameters which was particularly efficient
for the prediction of the elastic moduli.
Exponential fit-model was particularly efficient for the
prediction of the ultimate stresses. In particular, the
model combining both BV/TV and the connectivity (or
number of loops) ðLÞ1 was able to account for 95% of
the variability of H-Stress. The interpretation of this
model observed on a 2D-plot combining both connectivity and ultimate stress axes gave an account of two
opposite trends (Fig. 3b). First, when both connectivity
and BV/TV evolved together, an increase of connectivity
was associated to an increase of ultimate stress. Second,
when BV/TV was maintained constant, an increase of
connectivity was associated to a decrease of ultimate
stress. The dispersion of the experimental data on both
sides of the regression line between ðLÞ1 and H-Stress
(Fig. 3b) could be justified in taking into account the
inversion of the relationship between ðLÞ1 and H-Stress
for some data having the same BV/TV (see in particular
the clusters C, E, and D). Connectivity has been studied
by several authors, and different trends have been
reported (Parisien et al., 1992; Parisien et al., 1995; Lane
et al., 1998; Kabel et al., 1999b; Thomas et al., 1999;
Kinney et al., 2000). In a discussion concerning
connectivity and its potential relationship to strength,
it was concluded that the role of connectivity remained
understood, and it was questioned whether connectivity,
in addition to bone mass, can be used as a predictor of
bone strength (Kinney, 1999). This is exactly the
objective of the exponential model proposed in this
study. The two opposite trends deduced from this
exponential model are surprising. Nevertheless, we need
to interpret BV/TV constant as a situation where the
bone solid volume is redistributed in the 3D space
(Jensen et al., 1990; Yeh and Keaveny, 1999), and some
additional morphological analysis (TbTh, y) are
needed at this stage to better explain such behavior.
1098
L. Pothuaud et al. / Journal of Biomechanics 35 (2002) 1091–1099
This study has shown the potential of the combination of BV/TV with 3D-LSGA based topological
parameters in the prediction of the mechanical properties of trabecular bone. Further studies will investigate
the potential of such combination for the in vivo
evaluation of the quality of bone. In particular, the
exponential analytical fit-model could be extended, in
relating the constant BV/TV (Fig. 3b) to the overlap of
bone mineral density measurements from patients with
and without osteoporotic fractures (Cann et al., 1985;
Ott et al., 1987; Kimmel et al., 1990; Pothuaud et al.,
1998; Ciarelli et al., 2000).
Acknowledgements
The authors would like to thank Dr. D.C. Newitt for
helpful discussions. This work was supported by Grant
NIH-RO1-AG17762. We acknowledge the Regional
Council of Region Centre (France) for its financial help
during this work.
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