Overview
Solid mechanics describes how materials deform under applied forces and boundary conditions. Biological soft tissues present a particularly challenging class of problems — they undergo large deformations, exhibit highly nonlinear stress-strain relationships, and are nearly incompressible. Understanding the mechanics of these materials at a mathematical depth is essential for building simulations that reflect physical reality rather than numerical artifacts.
Technical Formulation
We work in the framework of finite deformation continuum mechanics, where the governing equations are derived from the balance of linear momentum in the total Lagrangian setting. The constitutive behavior of soft biological tissues is modeled through hyperelastic strain energy density functions, which encode the anisotropic, nonlinear response of the material. Active stress contributions are incorporated additively to represent muscle contraction, independent of the electrophysiological trigger. The resulting system is a nonlinear elliptic PDE, solved through Newton’s method with consistent tangent stiffness derived analytically.
Clinical Application
The primary clinical targets are cardiac mechanics problems — recovering regional myocardial stiffness through inverse modeling and predicting ventricular deformation under physiological loading. The constitutive framework extends naturally to other soft tissue applications, including the breast tissue modeling work underlying recent publications.
References & Resources
- Software: CARDIAX
- Key Papers:
- High Speed Cardiac Simulations Using the JAX Framework — Thomas et al., engrxiv preprint
- A Novel Diffusion Tensor Based 3D Constitutive Model for Human Breast Tissue — Sacks, Thomas et al., JMBBM 2025