A Phase Field Approach to Trabecular Bone Remodeling

Sebastian Aland,Florian Stenger,Axel Voigt,Robert Müller,Andreas Deutsch

doi:10.3389/fams.2020.00012

Abstract

We introduce a continuous modeling approach which combines elastic responds of the trabecular bone structure, the concentration of signaling molecules within the bone and a mechanism how this concentration at the bone surface is used for local bone formation and resorption. In an abstract setting bone can be considered as a shape changing structure. For similar problems in materials science phase field approximations have been established as an efficient computational tool. We adapt such an approach for trabecular bone remodeling. It allows for a smooth representation of the trabecular bone structure and drastically reduces computational costs if compared with traditional micro finite element approaches. We demonstrate the advantage of the approach within a minimal model. We quantitatively compare the results with established micro finite element approaches on simple geometries and consider the bone morphology within a bone segment obtained from $\mu$CT data of a sheep vertebra with realistic parameters.

Highlights

Bone undergoes a continuous renewal process, which helps to maintain its mechanical performance and allows for adaptation to changes in mechanical requirements
We briefly review the role of the different cells which are involved in the remodeling process and describe the mechanical properties of bone on the level required for our continuous modeling approach
We apply our model to a segment of a trabecular bone, which is obtained from tomography data of a sheep vertebra

Summary

Introduction

Bone undergoes a continuous renewal process, which helps to maintain its mechanical performance and allows for adaptation to changes in mechanical requirements. In an abstract setting we consider bone as a shape changing structure, with concentrations of mechanosensing cells within the bone and resorbing and depositing cells on the bone surface. In contrast to previous modeling approaches, using micro finite element analysis [10,11,12,13,14], we describe the structure implicitly using a time-dependent phase field function. This leads to a more accurate model, as the artificial voxel-roughness of the bone surface can be avoided, and to a drastic reduction of system size and required computing time. In the last decade these models were extended to be used as a general numerical tool to solve problems in complex time-evolving geometries [17] and have since been established for two-phase flow [18,19,20,21], biomembranes [22, 23], single cell mechanics [24], and fluid-structure interaction [25]

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