by Merry Maisel




Osteoporosis
is a disease characterized by low bone mass
and structural deterioration of bone tissue,
leading to bone fragility. Left untreated,
osteoporosis can progress silently until a
bone breaks. Sufferers have an increased susceptibility
to fractures of the hip, spine, and wrist,
according to the National Osteoporosis Foundation.
Osteoporosis is a major public health threat
for an estimated 44 million Americans.
• Of the 10 million individuals estimated
to have osteoporosis, eight million are women.
• About 34 million Americans, or 55 percent
of the people 50 years of age and older, have
low bone mass, which puts them at increased
risk of developing osteoporosis and related
fractures.
• Significant risk has been reported
in people of all ethnic backgrounds.
• While osteoporosis primarily affects
postmenopausal women because hormonal changes
accelerate bone resorption, it can strike
women or men at any age.
The estimated national direct expenditures
(hospitals and nursing homes) for osteoporotic
and associated fractures were $17 billion
in 2001 ($47 million per day), and the cost
is rising. 



To biomechanical engineers,
bones are exquisitely complex structures, more breathtaking
than Gothic cathedrals or the world’s longest bridges.
While bone shapes seem strange and irregular, it is their
internal structure that fascinates and that many scientists
are attempting to model computationally. "From a micromechanical
point of view, bone architecture is wonderfully various, responding
to compression and tension in a nonlinear fashion, and presenting
very subtle challenges to the modeler," said Tony Keaveny,
a professor of mechanical engineering and bioengineering who
directs the Orthopaedic Biomechanics Laboratory at UC Berkeley.
Keaveny and his group have been using the resources of the
National Partnership for Advanced Computational Infrastructure
(NPACI) to develop accurate, threedimensional, finiteelement
models of human bones. Their models are uniquely efficient,
and their work will aid researchers, clinicians, and physicians
seeking ways to prevent and treat a panoply of bone diseases.
Among these, the most serious in economic and social terms
is osteoporosis.
LIVING TISSUE
Bones have been affected by many diseases throughout human
history, and orthopaedic medical practices are first recorded
in ancient Egypt. Prehistoric human fossil bones show that
their bearers suffered from diseases still seen today.
A misconception about bones is that, once grown, they become
inert or "dead" tissue. Actually, bones are living
organs. Bone cells are of three types. Osteoblasts come from
the bone marrow and form new bone. Some osteoblasts turn into
osteocytes after new bone is formed and become surrounded
by the new bone tissue. These trapped osteocytes send out
long branches that connect to other osteocytes and are thought
to sense fluid stresses or cracks in the bone and help direct
where the third type of bone cell, the osteoclasts, will dissolve
bone.
Like all other cells in the human body, bone cells are born,
live, and die, and the whole collection of cells in our bones
participates in growth and degeneration.
"These complex physiological facts demand a change in
the usual paradigm of biomechanical engineering research,"
said Keaveny. "In place of testing the assumption that
one or another engineering metric–density, porosity,
elasticity–may govern or explain bone failure mechanisms,
we look instead at the micromechanical behavior of the tissue
as a whole and model it to derive appropriate metrics as they
emerge from our simulations. Scientists have long recognized
that bone density alone is not the sole indicator of risk
for fracture, or even bone strength itself, but the multiple
factors involved in both health and disease need to be examined
on the microscopic level." Since risk prediction depends
on the simultaneous interactions of these factors, the group’s
longterm research goal is to develop techniques that allow
precise individual predictions–and precise individual
interventions–to prevent bone failures.
The group studies the two main types of bone tissue: cortical
and trabecular. Cortical bone is the more compact tissue that
forms the outer shell of bones; trabecular bone is the porous,
spongy tissue found within the spine, hips, knees and other
joints. Trabecular bone is the primary loadbearing biological
tissue in these key spots. It has an intricate and irregular
structure with high porosity–up to 90 percent in healthy
vertebral bone, for example, and more in osteoporosis.
In Keaveny’s lab at Berkeley, the structural properties
of trabecular bone are measured on standard, rectangular or
cylindrical samples (about three to five cubic centimeters).
These are subjected to pulling or stretching (tensile stress),
compression, or a combination of stresses applied along various
axes through the samples. But such measurements conducted
on small samples cannot reflect the effects of a tensile stress
or compressive loading transmitted from afar through the length
of an entire bone.
"Our computational approach makes contact with this
body of measurement because we model the micromechanics of
small samples like those we use in the lab," said Keaveny
"But as computers have become more powerful, we are reaching
beyond the limitations of small samples and discovering ways
to scale up from the micromechanical sample. We must generalize
and extend our results to the scale of the entire bone and
model the continuum behavior of entire bones under largescale
deformation."
FINITEELEMENT MODEL
The modeling method used by Keaveny and his group is finiteelement
analysis. A "finite element" is a regular or irregular
polygon or a polyhedron, a solid bounded by polygons. A model
structure is typically built from many thousands or millions
of these. Structural properties can be assigned and forces
applied to edges and vertices. Finiteelement methods are
preferred to modeling on a regular grid by practitioners of
structural engineering, because the methods can deal better
with the highly irregular geometries of artifacts or structures.
Where a grid point might only receive and transmit information
from its eight nearestneighbor points, a finiteelement edge
or vertex can receive and transmit information from and to
an arbitrary number of neighbors, depending on the geometry
of the element and the system.
The finiteelement model is built on information derived
directly from the laboratory analyses of bone tissue. Until
recently, the group was using special microtome slicing and
staining techniques, then scanning thin sections with a microscope
and computationally reconstructing the threedimensional object
using tiny cubic volume elements. But advances over the past
few years in the resolution of imaging methods now allow the
group to derive their models directly from microcomputed
tomographic scans of the samples. Eventually, they hope that
additional computing power will allow them to use the method
on wholebone scans. "At the sample sizes we are able
to compute today, our models are highprecision structural
representations of bone," Keaveny said.
"Some of our models contain as many as 57 million finite
elements," said Harun Bayraktar, a mechanical engineering
graduate student whose recent research in the group is the
basis for his doctoral dissertation. "The large number
of voxels allows us to capture the very complex geometries
of trabecular bone, but it immediately requires largescale
computational resources to calculate the response of the model
to applied stresses."
Bayraktar and the Keaveny group, which includes Panayiotis
Papadopoulos, a professor of mechanical engineering, have
worked with a Berkeley alumnus, collaborator Mark Adams of
Sandia National Laboratories at Livermore, CA, to make the
finiteelement models computationally tractable. Adams concentrates
on methods for improving the performance of finiteelement
calculations on parallel architectures. "The scale and
complexity of these microfinite element bone calculations
is unprecedented on distributedmemory computers, and thus
it pushes common methods for both the finiteelement parallelization
and the linear solver past their limits," said Adams.
The implementation used on the IBM Blue Horizon supercomputer
at the San Diego Supercomputer Center (SDSC) combines a string
of codes. "We use a custom code, Bobcat, to translate
the digital images obtained from microcomputed tomography
into a finiteelement mesh," Adams said. "Our parallel
finiteelement code, called Athena, partitions this finiteelement
mesh in parallel and constructs a local finiteelement problem
on each processor. Athena uses ParMetis from the University
of Minnesota to construct highquality partitions that minimize
the amount of interprocessor communication required in the
linear solution process. The processorlocalized finiteelement
problem is solved by our serial finiteelement analysis program,
FEAP."
Adams explained that the most computationally expensive aspect
of a finiteelement simulation is the solution of a linear
algebra problem (Ax = b), where b is
a vector of applied forces, A is the "stiffness
matrix," and x is a vector of displacements that
need to be computed with the linear solver (FEAP). FEAP constructs
the stiffness matrices and the output is given to Adams’s
parallel algebraic multigrid solver, called Prometheus.
The large scale of problem to be solved in accurate finiteelement
models of bone tissue requires highly optimal solution methods,
and Adams said, "Multigrid methods are the most optimal.
The multigrid method used by Prometheus in this work, smoothed
aggregation, uses the principles of graph theory to aggregate
vertices in the finiteelement graph. Each aggregate becomes
one vertex in a new, coarser graph (known as a ‘grid’
for historical reasons). This process is applied recursively
by Prometheus to construct a succession of coarsegrid representations
of the finegrid problem to be solved." Prometheus was
built on ParMetis and PETSc from Argonne National Laboratory.
The combination of codes gives the bone modelers enormous
power to solve their problems despite the wellknown difficulties
associated with speeding up finiteelement codes.
MATRIX LOADING
"As an example, we did a scaled speedup study of one
finiteelement model of a vertebral body," Bayraktar
reports. "Even though the problem had more than 107 million
degrees of freedom, we were able to solve it in less than
eight minutes on 140 nodes of Blue Horizon." The speed
of the method let them test the relative effects of two kinds
of nonlinearity in the stress problem: nonlinear response
caused by the nonlinear structure of trabecular bone on the
microscale, and the nonlinear response of an entire sample
of bone to a large geometrical deformation.
"The model results show much greater sensitivity than
linear models to the direction and kind of stress loading
applied to a bone," Bayraktar said. Using samples from
four different sites, he found that the stresses under tension
were greater and those under compression lesser than the same
stresses calculated in linear models. "The combination
of material and geometric nonlinearity in the models enabled
them to exhibit more realistic behavior," Keaveny said,
"and that will be the key to consistent prediction of
bone behavior under various degrees of stress."
The group presented its results at the 2003 American Society
of Mechanical Engineers Summer Bioengineering Conference in
June 2003, and will present them a month later at the U.S.
Association for Computational Mechanics conference. "Our
results are consistent with those of a group in the Netherlands
that is also employing microfiniteelement modeling, which
is encouraging," Keaveny said. "But our techniques
enable us to perform the computations much more efficiently
and thus will enable us to scale up to the wholebone level
more accurately."
In addition to the highresolution visualizations made from
the modeling runs, the group has produced a number of animations.
"We’re hoping to put together a good demonstration
of all this at the Supercomputing conference in Phoenix in
November," said Bayraktar. "We have submitted a
paper to the conference."
"We haven’t included all the subtleties of the
complex bone tissue, but we’ve come a lot closer to realism,
and we’re looking forward to further improvements in
both the methods and the computational platforms," said
Keaveny. Using wholebone models subjected to stresses like
those encountered in accidents or disease, doctors will eventually
be able to plan operations or prescribe drugs to rebuild healthy
bone of the correct composition. "We’d like to be
able to do this on demand in the clinical setting and prevent
the enormous social costs of bone failure by intervening earlier
and more effectively," Keaveny said. "That will
take a much greater amount of computing power."
Merry Maisel is a senior science writer
at the San Diego Supercomputer Center.
