Traction force microscopy for linear and nonlinear elastic materials as a parameter identification inverse problem

TL;DR

This work formulates the traction force microscopy inverse problem for linear 2.5D and nonlinear pure 2D substrate models within a rigorous elasticity framework, defining forward maps $\hat{A}$ and $\hat{S}$ that relate traction stresses and force densities to substrate displacements. It proves well-posedness, derives the Fréchet derivative and its adjoint via energy methods and Gelfand triples, and introduces a polyconvex, isotropic material law to guarantee existence and stability of solutions in the nonlinear case. Numerical results on simulated and experimental data demonstrate stable, regularized reconstructions using CGNE for linear problems and truncated Newton-CG for nonlinear problems, with artifacts reduced by appropriate $H^1$-type penalties and homotopy strategies. The approach provides a robust mathematical basis for extending TFM to nonlinear substrate materials and for systematic comparison with standard physics-based methods like FTTC, with ready-to-use software tools for practitioners.

Abstract

Traction force microscopy is a method widely used in biophysics and cell biology to determine forces that biological cells apply to their environment. In the experiment, the cells adhere to a soft elastic substrate, which is then deformed in response to cellular traction forces. The inverse problem consists in computing the traction stress applied by the cell from microscopy measurements of the substrate deformations. In this work, we consider a linear model, in which 3D forces are applied at a 2D interface, called 2.5D traction force microscopy, and a nonlinear pure 2D model, from which we directly obtain a linear pure 2D model. All models lead to a linear resp. nonlinear parameter identification problem for a boundary value problem of elasticity. We analyze the respective forward operators and conclude with some numerical experiments for simulated and experimental data.

Figures (9)

• Figure 1: (a) Experimental setup for TFM. Small fiducial markers (dark green) are embedded in a soft elastic substrate (light green). The cell (blue; nucleus in dark blue) adheres to the substrate and applies traction stresses. For 2D TFM one often only uses markers in a thin layer just underneath the substrate surface, as only tangential deformations are being considered. For 2.5D TFM the markers have to be distributed in the whole substrate, because one also has to track perpendicular deformations. In practice, the substrate is much thicker than the force generating structures in the cell (about $50$$\mu$m vs. a few $\mu$m). The markers are typically imaged with an inverse optical microscope from below. (b) Microscopy image of a cell plated on a planar substrate. (c) Image of the corresponding deformation.
• Figure 2: Substrate $\Omega$ with boundary $\partial \Omega = \Gamma_T \cup \Gamma_{T_0} \cup \Gamma_D$ and outer normal vectors $n$.
• Figure 3: Change of TFM setting from 3D to 2D substrate.
• Figure 4: 2.5D linear TFM: simulated traction force via \ref{['eq:force1']}. The color code shows the magnitude of the traction stress (force per area), which is physically measured in Pa.
• Figure 5: 2.5D linear TFM: simulated traction force as in \ref{['eq:force2']}.

Theorems & Definitions (16)

• Remark 2.1
• Lemma 2.2
• proof
• Definition 3.1: Polyconvexity
• Definition 3.2: Coercivity
• Theorem 3.3
• proof
• Theorem 3.4
• Theorem 3.5: Constitutive Equation
• proof