Inverse modeling of heterogeneous ECM mechanical properties in nonlinear 3DTFM

Accurate characterization of cellular tractions is crucial for understanding cell-extracellular matrix (ECM) mechanical interactions and their implications in pathology-related situations, yet their direct measurement in experimental setups remains challenging. Traction Force Microscopy (TFM) has emerged as a key methodology to reconstruct traction fields from displacement data obtained via microscopic imaging techniques. While traditional TFM methods assume homogeneous and static ECM properties, the dynamic nature of the ECM through processes such as enzyme-induced collagen degradation or cellmediated collagen deposition i.e. ECM remodeling, requires approaches that account for spatio-temporal evolution of ECM stiffness heterogeneity and other mechanical properties. In this context, we present a novel inverse methodology for 3DTFM, capable of reconstructing spatially heterogeneous distributions of the ECMs stiffness. Our approach formulates the problem as a PDE-constrained inverse method which searches for both displacement and the stiffness map featured in the selected constitutive law. The elaborated numerical algorithm is integrated then into an iterative Newton-Raphson/Finite Element Method (NR/FEM) framework, bypassing the need for external iterative solvers. We validate our methodology using in silico 3DTFM cases based on real cell geometries, modeled within a nonlinear hyperelastic framework suitable for collagen hydrogels. The performance of our approach is evaluated across different noise levels, and compared versus the commonly used iterative L-BFGS algorithm. Besides the novelty of our formulation, we demonstrate the efficacy of our approach both in terms of accuracy and CPU time efficiency.