13.01.2026 | Research | Biomechanics
Solutions for Optimality Problems of Bone Adaptation
Philippe Zysset, head of the MSB group at University of Bern, investigated bone adaptation from a mathematical perspective and proposed analytical solutions for optimal forward and inverse adaptation problems at the homogenised continuum level.
Bone is a remarkable hierarchical composite biomaterial found in compact (outer dense shell) and trabecular (internal porous core) forms, capable of adaptation to its mechanical environment. This adaptation is understood through the concept of a mechanostat, which hypothesizes that mechanical loading influences bone structure by scaling its mass (amount of bone) and optimizing its architecture (arrangement) to provide a structure able to maintain a specific strain signal within the tissue.
The 1D mechanostat concept (Frost, 2003) with a resorption branch (cyan), a lazy zone (violet), an anabolic zone (green) and an overloading or damage zone (red). The vertical axis corresponds to the change in bone mass, while the horizontal axis stands for the mechanical stimulus.
© Philippe Zysset
Numerical models based on ex vivo, high-resolution computer tomography, can simulate bone resorption and formation in the presence of key biological agents, reproduce a realistic architecture of trabecular bone along principal stress axes and estimate changes in bone strength that are related to immobilization, overloading, metabolic diseases or drug therapy.
However, these models require large computational resources and current clinical diagnostics of bone diseases rely on in vivo medical imaging such as X-ray-based densitometry or computer tomography, which do not have the image resolution necessary to describe bone microarchitecture with full accuracy. The evaluation of personalized bone stiffness and strength are therefore based on a homogenized or continuum description of bone mechanical properties using density and fabric, an approximation of trabecular architecture.
The fabric ratios versus principal stress ratios for the admissible butterfly-shaped domain that minimise the complementary free energy density. As observed in the skeleton, the larger fabric is usually oriented along the larger stress amplitude, but the solutions differ with the signs of the principal stress ratios.
© Philippe Zysset
In this recent study, Philippe Zysset fills a gap in the current state of the art in bone adaptation. Specifically, he investigated the forward and inverse optimality problems of bone adaptation at the homogenized representative volume (RVE) level. He used bone density- and fabric-mechanical property relationships and derived analytical solutions for three different mechanostat criteria: strain energy density, a yield/damage criterion and a principal strain criterion.
The forward solutions are provided for density and fabric at a continuum point for a given local stress. The inverse solutions for local stress are derived for given density and fabric. The solutions for the fabric tensor were computed in 3D and specialized to the 2D case.
In future work, the results from this study will allow for efficient computation of bone adaptation, enabling forward simulation and inverse estimation of bone loading for clinical diagnostic tools such as high-resolution peripheral quantitative computer tomography (HR-pQCT) or photon counting computed tomography (PCCT), which provide both density and fabric in vivo.
