Abstract:
While physiological loading on lower long bones changes during bone development, the bone cross section either remains
circular or slowly changes from nearly circular to other shapes such as oval and roughly triangular. Bone is said to be an
optimal structure, where strength is maximized using the optimal distribution of bone mass (also called Wolf’s law). One
of the most appropriate mathematical validations of this law would be a structural optimization-based formulation where
total strain energy is minimized against a mass and a space constraint. Assuming that the change in cross section during bone
development and homeostasis after adulthood is direct result of the change in physiological loading, this work investigates
what optimization problem formulation (collectively, design variables, objective function, constraints, loading conditions,
etc.) results in mathematically optimal solutions that resemble bones under actual physiological loading. For this purpose, an
advanced structural optimization-based computational model for cortical bone development and defect repair is presented.
In the optimization problem, overall bone stifness is maximized frst against a mass constraint, and then also against a polar
frst moment of area constraint that simultaneously constrains both mass and space. The investigation is completed in two
stages. The frst stage is developmental stage when physiological loading on lower long bones (tibia) is a random combination of axial, bending and torsion. The topology optimization applied to this case with the area moment constraint results
into circular and elliptical cross sections similar to that found in growing mouse or human. The second investigation stage is
bone homeostasis reached in adulthood when the physiological loading has a fxed pattern. A drill hole defect is applied to
the adult mouse bone, which would disrupt the homeostasis. The optimization applied after the defect interestingly brings
the damaged section back to the original intact geometry. The results, however, show that cortical bone geometry is optimal
for the physiological loading only when there is also a constraint on polar moment of area. Further numerical experiments
show that application of torsion along with the gait-analysis-based physiological loading improves the results, which seems
to indicate that the cortical bone geometry is optimal for some amount of torsion in addition to the gait-based physiological
loading. This work has a potential to be extended to bone growth/development models and fracture healing models, where
topology optimization and polar moment of area constraint have not been introduced earlier.