The AAPS Journal, Vol. 16, No. 3, May 2014 ( # 2014)
DOI: 10.1208/s12248-014-9585-8
Research Article
A Network Modeling Approach for the Spatial Distribution and Structure
of Bone Mineral Content
Hui Li,1 Aidong Zhang,1 Lawrence Bone,2 Cathy Buyea,2 and Murali Ramanathan3,4,5
Received 22 November 2013; accepted 28 February 2014; published online 27 March 2014
ABSTRACT. This study aims to develop a spatial model of bone for quantitative assessments of bone
mineral density and microarchitecture. A spatially structured network model for bone microarchitecture
was systematically investigated. Bone mineral-forming foci were distributed radially according to the
cumulative normal distribution, and Voronoi tessellation was used to obtain edges representing bone
mineral lattice. Methods to simulate X-ray images were developed. The network model recapitulated key
features of real bone and contained spongy interior regions resembling trabecular bone that transitioned
seamlessly to densely mineralized, compact cortical bone-like microarchitecture. Model-simulated
imaging profiles were similar to patients’ X-ray images. The morphometric metrics were concordant
with microcomputed tomography results for real bone. Simulations comparing normal and diseased bone
of 20–30 to 70–80 year-olds demonstrated the method’s effectiveness for modeling osteoporosis. The
novel spatial model may be useful for pharmacodynamic simulations of bone drugs and for modeling
imaging data in clinical trials.
KEY WORDS: bone; imaging; modeling; osteoporosis.
INTRODUCTION
According to the International Osteoporosis Foundation,
osteoporosis is estimated to affect 200 million women and
cause approximately 9 million fractures worldwide (1,2).
About one in three women and one in five men, 50 years or
older, are at risk of having an osteoporotic fracture in their
lifetime (1,3,4). The prevalence and costs of osteoporotic
fractures will increase in many parts of the developed world
because of demographic shifts that have resulted in an
increasing proportion of older adults in the population (5).
Bone fractures and diseases of the musculoskeletal
system, which includes the bones and joints, frequently cause
death, permanent disability, and loss of quality of life and
independence in the geriatric population. Repairing damaged
bone and restoring bone homeostasis is a therapeutic goal in
diseases ranging from osteoporosis, many cancers including
multiple myeloma, and in autoimmune diseases such as
multiple sclerosis and rheumatoid arthritis (6,7).
1
Department of Computer Science and Engineering, State University
of New York, Buffalo, New York, USA.
2
Department of Orthopedics, State University of New York, Buffalo,
New York, USA.
3
Department of Pharmaceutical Sciences, State University of New
York, Buffalo, New York, USA.
4
Department of Neurology, State University of New York, Buffalo,
New York, USA.
5
To whom correspondence should be addressed. (e-mail:
murali@buffalo.edu)
1550-7416/14/0300-0478/0 # 2014 American Association of Pharmaceutical Scientists
Furthermore, many commonly prescribed drugs including the corticosteroids, anticancer agents, and certain antiepileptics increase bone loss. The molecular pathways
regulating bone remodeling are thus major targets in drug
discovery and development. A spatiotemporal model of bone
could be very useful in many drug development settings
because clinical trials for evaluating bone-modulating drugs
require very large sample sizes and long study durations to
enable meaningful assessment of fracture risk as a clinical
endpoint. Pharmacologically, bone mineral density can be
increased by either promoting osteoblast-mediated bone
mineral deposition or by inhibiting bone resorption by
osteoclasts. However, the increases in bone mineral density
that result from administration of some bone drugs does not
always correspond to decreased fracture risk.
Although the antiresorptive bisphosphonates are potent
and effective at increasing bone density and decreasing fracture
risk in the first 5 years of use, their long-term effects on bone are
not known (8). The orthopedics community is now increasingly
limiting the duration use of bisphosphonate drugs because of
reports of unexpected drug-related side effects including
temporomandibular (jaw) joint osteonecrolysis (9–11) and
increased risk of sudden femur fractures with unusual morphology (12,13) that are drawing increased regulatory scrutiny to this
important class of drug.
There are several pharmacokinetic-pharmacodynamic
models for drugs that target bone homeostasis, e.g.,
antiresorptive agents and parathyroid hormone (14,15).
These models are capable of describing drug kinetics and
478
Spatial Modeling of Bone
479
the time course of biomarkers of bone turnover (e.g., bone
resorption biomarkers such as telopeptides of collagen and
bone formation biomarkers such as bone alkaline phosphatase). However, it is currently very challenging to assess
temporally distal clinical outcomes such as bone mineral
density, bone microarchitecture changes, and fracture risk
from the kinetics of drugs and the dynamics of biomarkers
from these models, which change over shorter time scales.
Consequently, regression techniques and Hill-type transduction functions are often used to empirically relate
pharmacodynamic outputs to these important clinical
outcomes.
Our goal is to develop a novel spatiotemporal model and an
effective mechanism-based in silico platform for evaluating the
efficacy and pharmacodynamics of drugs and drug dosing
regimens in diseases that affect bone health. A good spatiotemporal model can be useful in clinical and drug development
settings because: (1) model-derived measures of drug effects on
bone mineral density, bone mineral distribution, and
microarchitecture can be investigated in silico; (ii) bone imaging
results from dual-energy X-ray absorptiometry (DEXA) can be
visualized; (3) the differential impact of drugs on different body
sites that have different proportions of trabecular and cortical
bone can be evaluated; and (4) a mechanistic understanding of
the relationship of bone mineral distribution and
microarchitecture to fracture risk can be obtained.
In this research, we focus on developing the spatial
characteristics of such a modeling strategy. We demonstrate
how bone mineral distribution and microarchitecture can be
incorporated to obtain spatially informative models that can
be individualized from imaging data obtained in clinical
settings.
MODELING AND METHODS
Spatial Model
Geometry
We modeled a cylindrical region of bone with outside
radius R O and length L because we had dual X-ray
absorptiometry data for the femur available. For ease of
understanding, a schematic of spatial approach in its simplest
two-dimensional form is shown in Fig. 1. The three-dimensional extension of the schematic approach represented in
Fig. 1 was used for modeling. The steps in the spatial model
are as follows:
Step 1. Distribution of Bone-Forming Foci for Voronoi
Tessellation The radial distribution of the mean
number of bone-forming foci c(r) followed a
cumulative normal distribution:
c0
cðrÞ ¼ pffiffiffiffiffi
σ 2π
Zr
−ðu−uÞ2 =2σ2
e
du
−∞
r is the radial coordinate, the mean and standard
deviation of the normal distribution are denoted
by μ and σ, respectively, c0 is a constant of
proportionality, and u is a dummy integration
variable for the radial coordinate.
Within a differential region dA=2πr dr, the
actual number of bone mineral-forming foci
was obtained as Poisson variate, Poisson(λ(r)),
whose rate parameter λ(r) was set to c(r). The
axial and angular distribution of the foci was
assumed to follow a uniform distribution. These
foci were used as the centers or sites for the
Voronoi tessellation in step 2.
Step 2. Voronoi Tessellation Voronoi tessellation is an
algorithmic procedure that takes as input a set of
points called Voronoi centers or sites and
generates a closed lattice called Voronoi regions
or cells that are bounded by straight lines or
edges.
In the two-dimensional case, the boundaries of
Voronoi regions contain polygons and in the threedimensional case, the regions are closed polyhedra
with polygonal faces. Every point on the polygonal
face of a polyhedral Voronoi region is equidistant
from the two nearest Voronoi sites. Thus, every
point inside of a given Voronoi region is closer to
its own site than to adjoining sites.
The field of bone mineral-forming foci created in
step 1 provided the set of Voronoi centers or
sites that were subject to Voronoi tessellation.
The edges of Voronoi cells were interpreted as
the structural matrix upon which bone mineral
was laid down.
Step 3. Distribution of Bone Mineral Weights were
assigned to the edges of the Voronoi lattice.
Operationally, the weights were interpreted as
the cross-sectional area of the bone mineral in
the edge, which was interpreted to be cylindrical. The distribution of the circular cross-sectional area of bone mineral w(r) was also varied
in the radial direction r according to a cumulative normal distribution:
w0
wðrÞ ¼ pffiffiffiffiffi
σ 2π
Zr
e−ðu−μÞ
2
=2σ2
du
−∞
For calculations, the radial distance of the center
of the edge was used to determine the weights,
which were obtained by sampling random variates from the cumulative normal distribution.
The mean, μ, and standard deviation, σ, of this
cumulative normal distribution were fixed at the
values used for the distribution of bone forming
foci; w0 is a constant of proportionality and u is a
dummy integration variable for the radial
coordinate.
Step 4. Stochastic Edge Pruning To generate a lattice with
spatial characteristics that more closely resembled
the real bone mineral lattices, the edges in the closed
Voronoi lattice were randomly deleted with a low
probability p. Random variates from the Bernoulli
distribution, Bernoulli(p), were used to identify
edges that were deleted or pruned. This created a
lattice with containing both open and closed Voronoi
regions. We refer to the lattice remaining after this
Li et al.
480
a
b
the edges in the PVL:
m¼ρ
X
wi l i
edges
The wi denotes the cross-sectional area of the edge, and li
is the length of the edge. The true density of the bone mineral
is denoted by ρ.
Domain of Interest
c
Distribution of Centers
Expression for Bone Mineral Density
The bone mineral density (BMD) is the mass of bone
mineral per unit of projected area. The BMD was calculated
from the bone mineral mass, m, and the projected area, Ap,
using:
d
BMD ¼
m
m
ρ X
¼
¼
wi l i
Ap 2R0 L 2R0 L edges
Relationship to Imaging Data
Voronoi Tessellation
e
Bounded Vornoi Tessellation
We used Beer’s law to quantitatively relate the model to
imaging findings from DEXA and X-ray imaging methods.
According to Beer’s law, the logarithm of the fraction of
electromagnetic energy absorbed is a linear function of the
concentration of the absorbing species and the path length.
Because the concentration of bone mineral varies along
the direction y of the imaging beam, integration (summation)
of Beer’s law over every differential element δy is needed for
calculating the image intensity at position x from the center:
X
ln AðxÞ ¼ − αδyi
f
Weighted Graph
Edge Pruning
Fig. 1. A simple two-dimensional schematic of the Voronoi tessellation modeling procedure used to creating a spatial model for bone. a
An annular region representing bone. b The random distribution of
Voronoi centers from a cumulative normal distribution function with
μ=0.3, σ=0.02, and c0 =1,500. c The result from Voronoi tessellation.
d The Voronoi tessellation bounded by its outer boundary. e The
result after the weights were assigned to the edges. f The weighted
Voronoi tessellation after stochastic pruning
stochastic edge-pruning step as the pruned Voronoi
lattice (PVL).
The probability of deletion was an exponential
function of the edge weight wi, which represents
the level of bone mineral.
−βwi =wmax
p¼e
The wmax is the maximum value of wi and β is a
nondimensional parameter akin to a rate
constant that determines the extent of edge
pruning.
Expression for Bone Mineral Mass
The mean mass of bone mineral m for a given realization
of the network model was obtained as the summation over all
j
where A(x) is the fraction of the input intensity that is
absorbed by bone mineral at position x from the center, α is
the extinction coefficient, and δy is the differential path length
of bone mineral along the path of the imaging beam. The
summation is taken over every element of bone j traversed by
the imaging beam. The procedure is equivalent to a Radon
transform, which was computed using a MATLAB routine
(MathWorks, Natick, MA, USA).
The transmitted energy T(x) reaching the imaging device
produces the intensity variations responsible for image
formation:
T ðxÞ ¼ 1−AðxÞ
This equation can be extended to complex tissue
containing for example, soft tissue, fat, and blood by
incorporating different extinction coefficients for each material type in the tissue.
Image Analysis
We conducted image analysis of DEXA scans of the
femur from patients to identify macroscopic radial and axial
variations in BMD. The imaging data were analyzed using
ImageJ (16,17) and BoneJ software tools (18). The values of
μ and σ were estimated iteratively from imaging data.
Spatial Modeling of Bone
481
computed using CTAn (v.1.13) software (SkyScan Software, http://
www.skyscan.be/products/downloads.htm).
(R2, )
(R1, )
Image Processing
(r2 = r1 R2/R1 , )
(r1, )
C1
C2
The MATLAB Image Processing Toolbox (MathWorks)
was used. The bwboundaries routine was used to trace
exterior boundaries in images from published papers. The
boundary points were connected with a closed cubic B-spline
using the bspline_deboor subroutine.
Simulations of Diseased Bone
Circle
Arbitrary Shape
Fig. 2. A schematic of the procedures used to model arbitrary shapes. Each
vertex of the Voronoi polyhedral on the circle (left) is mapped to the
corresponding position on the arbitrary shape (right). The centroids of the
circle and the arbitrary shape are denoted by C1 and C2, respectively. The
points (R1, θ) and (R2, θ) are the radial coordinates on the boundary. The
radial coordinates of the Voronoi vertex at (r1, θ) in the circle are mapped to
the corresponding point (r2, θ) on the arbitrary shape using the constant of
proportionality R2/R1
Extension to Irregular Geometries
We extended our approach to irregular geometries by
mapping the vertices of the Voronoi polyhedra from a
cylindrical geometry to the irregular three-dimensional geometry of interest. Consider the irregular geometry of
interest summarized in Fig. 2.
The coordinates of centroid of the shape at each value of
distance z along the axis were computed. The outer perimeter of
the irregular geometry was obtained using an alpha shape edge
detection algorithm (alphavol, written for MATLAB by Jonas
Lundgren) and fitted to a cubic spline to obtain a continuous
function.
The radial coordinates of the Voronoi polyhedra vertices
from the center of the circle were mapped proportionately
from the circle to the corresponding radial distances from
centroid of the irregular geometry for each z. The proportionality factor was based on the ratio of the radius of the
circle to the radial distance of the perimeter of the irregular
shape from the centroid. The angular and axial coordinates
and edges of the Voronoi polyhedra were identity mapped.
For simulating examples of interindividual variability
in BMD, we obtained the transverse dimensions and
shape of the midline of right femur (23 cm from the
proximal end) of a 52-year-old woman from Fig. 3d of the
report by Locke (20). The bone mineral density and
projected bone areas were obtained from statistical
analyses of the National Health and Nutrition Examination Survey (NHANES) (21). The simulations were
conducted with 1,000 centers and the modeling procedure
for irregular bone shape. To allow direct comparisons of
the microarchitecture of normal and diseased bone, we
conducted analysis from the same Voronoi tessellation.
RESULTS
Schematic Representation of Bone Microarchitecture by Spatial
Model
Figure 1a schematically shows the procedure in a twodimensional form for obtaining the spatial model of bone
architecture. The procedure involves (1) distributing points
corresponding to the Voronoi centers in a circular regency. The
density of the Voronoi centers was varied in the radial direction
according to the cumulative normal distribution, (2) identification
of the external boundary, (3) Voronoi tessellation, (4) the edges of
the Voronoi polyhedra were randomly pruned, and (5) the edges
remaining were assigned weights.
Figure 2 shows how the approach of Fig. 1 can be extended to
irregular geometry, and Fig. 3 is a schematic of the threedimensional version of the approach. The details of the simulations
are provided in the “MODELING AND METHODS” sections.
Concordance of Model with X-ray Images of Bone
Quantitative Bone Morphometry
Computed tomography (CT) enables the three-dimensional
characteristics of bone to be obtained from imaging. We compared
the results from our modeling to CT findings from using multiple
bone morphometric metrics that have been established to assess
structural properties of cortical and trabecular bone from CT data
(19): (1) bone surface density, BS/TV, where BS is the bone surface
area and TV is the total volume of the sample; (2) bone volume
density, BV/TV, where BV is the bone mineral volume; (3)
trabecular thickness, Tb.Th, is computed by filling maximal spheres
into the bone mineral structure; (4) trabecular separation, Tb.Sp, is
calculated by filling maximal spheres into the nonbone voxels; (5)
trabecular number, Tb.N, is the inverse of the mean distance
between the mid-axes of the structure. These parameters were
Figure 4 shows the results from a three-dimensional model
simulation that includes both top and side views. Figure 4e shows
that the resulting representation visually resembles real bone in
several key characteristics. For example, the regions closer to the
outer circumference of the region are densely connected and
compact like cortical bone, and the interior regions are more
loosely connected and less dense like trabecular bone. The results
suggest that our model provides an intuitive and parameterefficient approach for simulating the geometric appearance of
bone tissue.
Figure 4 shows the extensive concordance between the
imaging results from spatial model and X-ray imaging of
patient bone. Figure 4a is a representative region of interest
in the femur on an X-ray image obtained from a
Li et al.
482
a
b
c
d
e
f
g
h
i
Fig. 3. The results from a three-dimensional implementation of the Voronoi tessellation modeling procedure. The six figures
in the top two rows are the top-down views of the three-dimensional spatial structures obtained from steps shown in Fig. 1a–g. The
lowest row shows the front views of the three-dimensional structures corresponding to Fig. 1b, e, and f. The outer surface was
removed so that the interior can be seen h and i. The simulation parameters μ=0.25, σ=0.05, β=0.5, and c0 =3,000 were used
postmenopausal patient with hip replacement. The red line in
Fig. 4b represents the cross-section along the femur at which
the observed variation of X-ray absorption plotted is shown
in Fig. 4a. The corresponding results from the spatial model
along a typical line, overlaid in Fig. 4b, demonstrate that the
spatial model provides simulated X-ray profiles that are
concordant with observed data. We computed the model Xray absorption profile across 20 transverse lines at equally
spaced axial positions. The profiles were scaled to obtain the
grayscale images in Fig. 4e. Figure 4e is granular because the
grayscale images were obtained from a low-resolution axial
scan with only 20 sections. Figure 4f was computed from
Fig. 4e with an elliptical Gaussian filter and provides a two-
dimensional grayscale image that is visually concordant with t
he original bone image.
Modeling Irregular Bone Shape
To assess the usefulness of our method to irregular bone,
we used the images and imaging results of the mandible from
Moon et al. who provided imaging data and detailed
information on multiple bone morphometric metrics (22).
Notably, the morphometric metrics included multiple regions
of interest and both regional measures as well as metrics that
were representative of the microarchitecture. Figure 5a shows
Spatial Modeling of Bone
483
a
b
1
Image Intensity
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
Dimensionless Radial Distance
Original X-ray Image
Radial Profiles: Image vs. Model
c
d
Front View from Model
e
z-Stack from Model
f
Top View from Model
g
Gaussian Filterd z-Stack
from Model
Tilted and Scaled
Fig. 4. Modeling bone images. a A section of the X-ray image of femur from a
postmenopausal hip transplantation patient with BMD of 0.90 g/cm2. b The distribution
of image intensity across the radial section of bone shown as red line in a; also shows the
normalized imaging intensity and the corresponding fit using the image modeling
procedure. c, d The front and top view from a simulation with parameters μ=0.25, σ=
0.05, β=0.5, and c0 =6000 containing 20 z slices, respectively. e A z stack obtained by
simulating an X-ray image in transmission mode using the Radon transform on the model.
f Obtained with a Gaussian filter to “smooth” the boundary of the z stack image of Fig. 4e.
g Simply rotates Fig. 4e by a 3° angle so that it can be compared more effectively with the
original X-ray image of Fig. 4a, which is tilted relative to the vertical
Li et al.
484
a
b
Model BV/TV (%)
40
30
20
10
0
0
10
20
30
40
50
Reported BV/TV (%)
Model BS/BV (1/pixel)
16
c
14
12
10
8
8
10
12
14
16
Reported BS/BV (1/pixel)
d
1.5
Model Tb.Sp (pixel)
Model Tb.Th (pixel)
0.35
0.3
0.25
e
1
0.5
0.2
0.2
0.25
0.3
0.35
Reported Tb.Th (pixel)
0.5
1
1.5
Reported Tb.Sp (pixel)
Fig. 5. Modeling results for an irregular geometry. a An image of the mandible that was analyzed in detail
by Moon et al (22). These authors derived quantitative morphometric measures in the alveolar region
(marked with Ts and Ti), the basal bone superior to the mandibular canal (marked M) and the basal bone
inferior to the mandibular canal (B). b–e Comparison of the results for bone volume fraction (BV/TV),
bone surface density (BS/BV), trabecular thickness (Tb.Th), and trabecular separation (Tb.Sp),
respectively, from the modeling to the measurements obtained by Moon et al (22). The simulation
parameters were μ=0.25, σ=0.05, β=1 and c0 =4,000. The slopes for the best-fit lines in b–e were 0.924,
0.84, 1.02, and 1.21, respectively. The Pearson correlation coefficients were greater than 0.99 in each case
the irregular-shaped bone analyzed by Moon et al. and the
three bone regions of interest for which they reported
morphometric data. Figure 5b–d demonstrate the concordance between the morphometric metrics obtained from our
modeling approach to those reported by Moon et al. These
results indicate that our modeling approach can be generalized to bones with more complex geometries.
Simulations of Diseased Bone
Figure 6 summarizes our simulations of healthy and diseased
femurs from 20- to 30- and 70- to 80-year-old women that were
based on BMD statistics from the National Health and Nutrition
Survey. The simulations were done for women because osteoporosis is more common in women after the onset of menopause.
Spatial Modeling of Bone
485
20 to 30-year old
70 to 80-year old
b
95th Percentile
a
d
50th Percentile
c
f
5th Percentile
e
Fig. 6. Top views of simulations of normal and diseased bone. a, c, and e The bone mineral
density values at the 95th percentile, 50th percentile (median), and 5th percentile for 20- to
30-year-old women, respectively. b, d, f Bone mineral density values at the 95th percentile,
50th percentile (median), and 5th percentile for 70- to 80-year-old women, respectively.
The shape and dimensions were obtained from the report by Locke, and the mean bone
mineral density values for these age groups were obtained from Looker et al (21). The 5th,
50th (median), and 95th percentiles for BMD of the 20- to 30-year-old groups were 0.870,
1.078, and 1.292 g/cm2, respectively. The corresponding values for the 70- to 80-year-old
groups were 0.731, 0.960, and 1.210 g/cm2. The values of μ were 0.5 for a and b, and 0.25
for c–f. The values of σ were 0.05 for a–d and 0.025 for the e–f. The β values for a–f were
1,000, 1, 10, 1, 5, and 1, respectively. The value of c0 =3,000 for a–f
Figure 6a, c, e shows that at all of the three percentiles, the bone
network lattice of the younger group was visually denser and more
extensively connected than those of the older 70- to 80-year-old
group. The 5th, 50th (median), 95th percentiles for BMD of the
20- to 30-year-old group were 0.870, 1.078, and 1.292 g/cm2,
respectively. The corresponding values for the 70- to 80-year-old
group were 0.731, 0.960, and 1.210 g/cm2. The loss of bone mineral
is most apparent in Fig. 6f, which represents the 5th percentile of
BMD values for the 70- to 80-year-old group. These findings
suggest that the model may be sensitive to and capable of
identifying changes in bone structure associated with age-related
bone loss and osteoporosis.
DISCUSSION
We have systematically investigated a novel approach for
modeling bone structure, the distribution of bone mineral
content, and microarchitecture. The network-based threedimensional spatial model utilizes Voronoi tessellation and it
yields structures that contain cortical and trabecular bone-like
regions. We demonstrated that the modeling approach can be
used to compute simulated X-ray images and that the
resulting imaging profiles are concordant with the profiles
obtained from patient bone.
Li et al.
486
One of the unique aspects of our model is that structures
similar to loosely organized, spongy trabecular bone and
densely mineralized, compact cortical bone emerge spontaneously and transition seamlessly as a result of the modeling
strategy. There is therefore no necessity for an artificially
enforced boundary between cortical and trabecular bone in
our model. The density distribution of the Voronoi centers
and the pruning steps are the primary determinants of the
trabecular-cortical structural variation that emerges. The
radial location over which the transition between cortical
and bone occurs is determined by the standard deviation term
of the density distribution function. Another feature of our
model is that interindividual variability in bone mineral lattice
structure also emerges naturally in the modeling process. This
is because of the stochastic nature of the initial distribution of
Voronoi centers and the pruning steps.
Delaunay triangulation, a mathematical method closely
related to the Voronoi tessellation, is widely used in finite
element modeling and has been used in biomechanical
analyses of bone (23,24). However, our use of Voronoi
tessellation in bone modeling differs substantially from the
use of Delaunay triangulation in finite element modeling. In
our model, Voronoi tessellation is used to create the
microarchitecture of bone mineral. In finite element modeling, Delaunay triangulation is used to create a grid of points
across which partial differential equations can be solved
numerically.
A limitation of our current approach is that it focuses
principally on the spatial aspects of the bone mineral
distribution. In the next step of our approach, the spatial
pruned Voronoi lattice-based model will be elaborated to
obtain a structural model through segmentation into a threecomponent composite material consisting of bone mineral,
extracellular matrix, and blood. Recently, “universal” composition, hydration, and mineralization rules for bone tissues
have been proposed that describe bone tissues of different
ages, anatomical locations, and species (25–27). There is a linear
relationship between mineral and organic content of bone.
These rules can be leveraged as constraints that will provide a
powerful strategy for parsimoniously elaborating our model
from a spatial description of bone mineral to a richer model of
bone tissue. The composition, microarchitecture, and dimensions of the bone would be needed as inputs for modeling. Such
a structural model could be used to compute the mechanical
properties of bone and simulate the risks of different types of
fracture. However, the development of structural models to
satisfactorily describe the loading patterns, complex biomechanics, and the fracture risk present many challenges that will
require further research. It may be possible to apply approaches
similar to those proposed by Fritsch et al. for this purpose (28).
Additionally, further improvements to the method have to be
made so that nonlinear regression techniques can be used more
efficiently for parameter estimation.
Many morphometric metrics have been developed to
analyze bone images, particularly those from microcomputed
tomography. A 2D box-counting algorithm has been used to
obtain the fractal index, which is a geometric measure of bone
(29) that takes on a value of 1 for hollow and a value of 2 for
solid structures. Measurements of normal, osteopetrotic, and
osteoporotic bone in clinical settings have fractal indices that
are intermediate in magnitude. The fractal index is strongly
correlated with trabecular spacing and number of trabeculae
(30). However, our approach is not focused on the development and validation of individual morphometric measures for
image analysis but instead on the development of comprehensive modeling framework. In Fig. 5, we demonstrated how
the morphometric methods from images of real bone were
concordant with those obtained from simulations of bone
from our model.
Compartmental, noncompartmental, and physiologically
based compartmental models have been widely used in
pharmacokinetic and pharmacodynamics modeling with great
success. Some of these approaches have also been used to
describe bone-modulating drugs (14,31). However, spatial
information is lost within compartmental models because
each compartment is treated as a well-mixed homogeneous
region. Separate compartments are needed to describe
interacting organs and spatial distributions. Furthermore,
although plasma bisphosphonate concentrations decrease
rapidly (within days) after dosing, bisphosphonate disposition
is complex because of their avid binding to bone tissue.
Bisphosphonate binding to bone is heterogeneous and
exhibits preference for regions that have a high remodeling
rate. The half-life of tissue-bound bisphosphonates has been
reported to be as long as 1–10 years (32,33). Numerous
modeling challenges are still unresolved in bisphosphonate
pharmacokinetics and pharmacodynamics.
Imaging has been an integral part of orthopedics
diagnosis and treatment since the discovery of X-rays. With
the advent of advanced imaging techniques, there is an even
greater need of more effective and innovative modeling
techniques that can better describe spatiotemporal processes
parsimoniously. There is, however, a dearth of an effective
repertoire of spatial models for pharmaceutical and clinical
applications. Therefore, the development of conceptually
novel spatial modeling techniques that have imaging capabilities may prove useful.
ACKNOWLEDGMENTS AND DISCLOSURES
Support from the National Multiple Sclerosis Society
(RG4836-A-5) to the Ramanathan laboratory is gratefully
acknowledged.
CONFLICT OF INTEREST There are no conflicts of interest
related to the work in the manuscript.
CONFIDENTIALITY Use of the information in this manuscript
for commercial, noncommercial, research, or purposes other than
peer review not permitted prior to publication without the
expressed written permission from the author.
REFERENCES
1. Foundation IO. Facts and statistics: International Osteoporosis
Foundation 2013. http://www.iofbonehealth.org/facts-statisticscategory-14.
2. Johnell O, Kanis JA. An estimate of the worldwide prevalence
and disability associated with osteoporotic fractures. Osteoporos
Int J Established Result Cooperation Eur Found Osteoporos
Spatial Modeling of Bone
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Nat Osteoporos Found USA. 2006;17(12):1726–33. PubMed
PMID: 16983459.
Kanis JA. Assessment of osteoporosis at the primary health care
level: report of a WHO Scientific Group. Sheffield, UK: World
Health Organization; 2007.
Dennison E, Mohamed MA, Cooper C. Epidemiology of
osteoporosis. Rheum Dis Clin N Am. 2006;32(4):617–29.
Burge R, Dawson-Hughes B, Solomon DH, Wong JB, King A,
Tosteson A. Incidence and economic burden of osteoporosisrelated fractures in the United States, 2005–2025. J Bone Miner
Res Off J Am Soc Bone Miner Res. 2007;22(3):465–75.
Seeman E, Delmas PD. Bone quality—the material and structural basis of bone strength and fragility. Engl J Med.
2006;354(21):2250–61. PubMed PMID: 16723616.
Weinstock-Guttman B, Gallagher E, Baier M, Green L, Feichter
J, Patrick K, et al. Risk of bone loss in men with multiple
sclerosis. Mult Scler. 2004;10(2):170–5. PubMed PMID:
15124763.
Pazianas M, Abrahamsen B. Safety of bisphosphonates. Bone.
2011;49(1):103–10. PubMed PMID: 21236370.
Almazrooa SA, Woo SB. Bisphosphonate and nonbisphosphonateassociated osteonecrosis of the jaw: a review. J Am Dent Assoc.
2009;140(7):864–75. PubMed PMID: 19571050.
Khosla S, Burr D, Cauley J, Dempster DW, Ebeling PR,
Felsenberg D, et al. Bisphosphonate-associated osteonecrosis of
the jaw: report of a task force of the American Society for Bone
and Mineral Research. J Bone Miner Res Off J Am Soc Bone
Miner Res. 2007;22(10):1479–91. PubMed PMID: 17663640.
Ruggiero SL, Dodson TB, Assael LA, Landesberg R, Marx RE,
Mehrotra B, et al. American Association of Oral and
Maxillofacial Surgeons position paper on bisphosphonate-related
osteonecrosis of the jaws—2009 update. J Oral Maxillofac Surg
Off J Am Assoc Oral Maxillofac Surg. 2009;67(5 Suppl):2–12.
PubMed PMID: 19371809.
Meier RP, Perneger TV, Stern R, Rizzoli R, Peter RE. Increasing
occurrence of atypical femoral fractures associated with bisphosphonate use. Arch Intern Med. 2012;172(12):930–6. PubMed
PMID: 22732749.
Park-Wyllie LY, Mamdani MM, Juurlink DN, Hawker GA, Gunraj
N, Austin PC, et al. Bisphosphonate use and the risk of
subtrochanteric or femoral shaft fractures in older women. JAMA
J Am Med Assoc. 2011;305(8):783–9. PubMed PMID: 21343577.
Cremers SC, Pillai G, Papapoulos SE. Pharmacokinetics/pharmacodynamics of bisphosphonates: use for optimisation of
intermittent therapy for osteoporosis. Clin Pharmacokinet.
2005;44(6):551–70. PubMed PMID: 15932344.
Pillai G, Gieschke R, Goggin T, Barrett J, Worth E, Steimer JL.
Population pharmacokinetics of ibandronate in Caucasian and
Japanese healthy males and postmenopausal females. Int J Clin
Pharmacol Ther. 2006;44(12):655–67.
Girish V, Vijayalakshmi A. Affordable image analysis using NIH
Image/ImageJ. Indian J Cancer. 2004;41(1):47. PubMed PMID:
15105580.
Hartig SM. Chapter 14: Unit 14.15. Basic image analysis and
manipulation in ImageJ. In: Frederick M Ausubel [et al] (eds.).
Current protocols in molecular biology. 2013. PubMed PMID:
23547012.
Doube M, Klosowski MM, Arganda-Carreras I, Cordelieres FP,
Dougherty RP, Jackson JS, et al. BoneJ: Free and extensible
bone image analysis in ImageJ. Bone. 2010;47(6):1076–9.
487
19. Hildebrand T, Laib A, Muller R, Dequeker J, Ruegsegger P.
Direct three-dimensional morphometric analysis of human
cancellous bone: microstructural data from spine, femur, iliac
crest, and calcaneus. J Bone Miner Res Off J Am Soc Bone
Miner Res. 1999;14(7):1167–74. PubMed PMID: 10404017.
20. Locke M. Structure of long bones in mammals. J Morphol.
2004;262(2):546–65. PubMed PMID: 15376271.
21. Looker AC, Borrud LG, Hughes JP, Fan B, Shepherd JA,
Melton LJ, III. Lumbar spine and proximal femur bone mineral
density, bone mineral content, and bone area: United States,
2005–2008. Data from the National Health and Nutrition
Examination Survey (NHANES). In: Statistics NCfH, editor.
Vital Health Stat Hyattsville, MD: Centers for Disease Control
and Prevention; 2012.
22. Moon HS, Won YY, Kim KD, Ruprecht A, Kim HJ, Kook HK,
et al. The three-dimensional microstructure of the trabecular
bone in the mandible. Surg Radiol Anat SRA. 2004;26(6):466–
73. PubMed PMID: 15146293.
23. Keyak JH, Fourkas MG, Meagher JM, Skinner HB. Validation
of an automated method of three-dimensional finite element
modelling of bone. J Biomed Eng. 1993;15(6):505–9. PubMed
PMID: 8277756.
24. Viceconti M, Davinelli M, Taddei F, Cappello A. Automatic
generation of accurate subject-specific bone finite element
models to be used in clinical studies. J Biomech.
2004;37(10):1597–605. PubMed PMID: 15336935.
25. Morin C, Hellmich C. Mineralization-driven bone tissue evolution follows from fluid-to-solid phase transformations in closed
thermodynamic systems. J Theor Biol. 2013;335:185–97. PubMed
PMID: 23810933.
26. Morin C, Hellmich C, Henits P. Fibrillar structure and elasticity
of hydrating collagen: a quantitative multiscale approach. J
Theor Biol. 2013;317:384–93. PubMed PMID: 23032219.
27. Vuong J, Hellmich C. Bone fibrillogenesis and mineralization:
quantitative analysis and implications for tissue elasticity. J
Theor Biol. 2011;287:115–30. PubMed PMID: 21835186.
28. Fritsch A, Hellmich C, Dormieux L. Ductile sliding between
mineral crystals followed by rupture of collagen crosslinks:
experimentally supported micromechanical explanation of bone
strength. J Theor Biol. 2009;260(2):230–52. PubMed PMID:
19497330.
29. Chappard D, Legrand E, Haettich B, Chales G, Auvinet B,
Eschard JP, et al. Fractal dimension of trabecular bone:
comparison of three histomorphometric computed techniques
for measuring the architectural two-dimensional complexity. J
Pathol. 2001;195(4):515–21. PubMed PMID: 11745685.
30. Lespessailles E, Roux JP, Benhamou CL, Arlot ME, Eynard E,
Harba R, et al. Fractal analysis of bone texture on os calcis
radiographs compared with trabecular microarchitecture analyzed by histomorphometry. Calcif Tissue Int. 1998;63(2):121–5.
PubMed PMID: 9685516.
31. Pillai G, Gieschke R, Goggin T, Jacqmin P, Schimmer RC,
Steimer JL. A semimechanistic and mechanistic population PKPD model for biomarker response to ibandronate, a new
bisphosphonate for the treatment of osteoporosis. Br J Clin
Pharmacol. 2004;58(6):618–31.
32. Cremers S, Papapoulos S. Pharmacology of bisphosphonates.
Bone. 2011;49(1):42–9. PubMed PMID: 21281748.
33. Lin JH. Bisphosphonates: a review of their pharmacokinetic
properties. Bone. 1996;18(2):75–85. PubMed PMID: 8833200.