# Forward and inverse optimality problems of bone adaptation at the homogenised RVE level

## TL;DR

This paper presents new methods to model how bone adapts to mechanical stress using density and fabric relationships.

## Contribution

The paper introduces analytical solutions for forward and inverse bone adaptation problems using homogenized RVE-level models.

## Key findings

- Analytical solutions for bone adaptation are derived for three mechanostat criteria.
- Forward and inverse solutions for density and fabric are formulated at the continuum level.
- The solutions are simplified to 2D and 1D for better understanding and comparison.

## Abstract

Bone was shown to adapt to mechanical loading through the concept of a mechanostat that regulates cell activity to maintain a specific strain signal within the tissue. Current computer models simulate bone resorption and formation in the presence of key biological agents, reproduce a realistic architecture of trabecular bone along principal stresses and estimate changes in bone strength related to immobilisation, overloading, metabolic diseases or drug therapies. However, clinical diagnostics of bone diseases in vivo rely primarily on X-ray-based densitometry and computer tomography that do not have the resolution to describe bone microarchitecture in full detail and evaluation of personalised bone strength is therefore based on a homogenised description of bone mechanical properties using density and fabric. Continuum-level bone adaptation theories rely primarily on bone density and do not involve local optimisation principles to predict fabric. The inverse problems of predicting applied loads from bone morphology typically exploit density but not fabric. Accordingly, this work formulates and provides analytical solutions for optimal bone adaptation at the homogeneous, anisotropic RVE level using bone density- and fabric-mechanical property relationships for three different mechanostat criteria. Two of these criteria elicit different adaptive responses for tensile and compressive strains. Forward solutions for density and fabric are provided at a continuum point for a given local stress, while inverse solutions for local stress are derived for given density and fabric for all three criteria. The 3D solutions are specialised to 2D and 1D for comprehension and compared among the different criteria. In the future work, the obtained solutions will enable simple forward simulation of personalised bone adaptation and inverse estimation of bone loading for clinical diagnostic tools such as high-resolution peripheral quantitative computed tomography (HR-pQCT) or photon counting computed tomography (PCCT).

The online version contains supplementary material available at 10.1007/s10237-025-02024-8.