  This paper presents bone adaptation as a geometric flow. The proposed method
is based on two assumptions: first, that the bone surface is smooth (not
fractal) permitting the definition of a tangent plane and, second, that the
interface between marrow and bone tissue phases is orientable. This permits the
analysis of bone adaptation using the well-developed mathematics of geometric
flows and the numerical techniques of the level set method. Most importantly,
topological changes such as holes forming in plates and rods disconnecting can
be treated formally and simulated naturally. First, the relationship between
biological theories of bone adaptation and the mathematical object describing
geometric flow is described. This is termed the adaptation function, $F$, and
is the multi-scale link described by Frost's Utah paradigm between cellular
dynamics and bone structure. Second, a model of age-related bone loss termed
curvature-based bone adaptation is presented. Using previous literature, it is
shown that curvature-based bone adaptation is the limiting continuous equation
of simulated bone atrophy, a discrete model of bone aging. Interestingly, the
parameters of the model can be defined in such a way that the flow is
volume-preserving. This implies that bone health can in principle change in
ways that fundamentally cannot be measured by areal or volumetric bone mineral
density, requiring structure-level imaging. Third, a numerical method is
described and two in silico experiments are performed demonstrating the
non-volume-preserving and volume-preserving cases. Taken together, recognition
of bone adaptation as a geometric flow permits the recruitment of mathematical
and numerical developments over the last 50 years to understanding and
describing the complex surface of bone.
