Age-related changes in bone fundamentally occur at the surface. Understanding and modeling these changes is the primary means of understanding and preventing age-related fractures. However, this is a challenging task, as the bone microarchitecture changes topology during adaption when rods disconnect and plates form holes. The primary objective is to handle topological changes mathematically and develop computational methods for the simulation of bone adapta- tion.

This thesis develops a model of age-related bone loss based on the axioms that the bone surface is orientable and smooth. First, a novel artifact is discovered and described for the distance transform of sampled signals that limits their applicability in simulation and morphometry. Second, a new transform is defined termed the “high-order signed distance transform” that is better than the so-called exact signed distance transform in the sense that it has an order of accuracy greater than one. However, this transform does not permit a unique solution on sampled binary images, and another method is needed. Third, an algorithm is presented for computing the unique, high-order signed distance transform of biphasic materials from computed tomography data. Fourth, a method of performing morphometry on closed surfaces is described that relates existing global bone morphometric techniques to local curvature values. This method works on binary images without the need for signed distance transforms when small changes in the bone volume are permitted. Finally, the morphometry and high-order signed distance transform are integrated into a model of age-related bone loss. Principally, this work establishes bone adaptation as a geometric flow, simulated using level set methods that are efficient and naturally handle topological changes.

The contribution of this thesis is the establishment of a strong mathematical foundation for modeling bone adaptation. High accuracy computational methods are defined to integrate the theory into practice. The theory and methods form a rigorous basis for biological theories of bone adaptation and provide techniques for measuring and falsifying theories.