Recovery of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation

We study the inverse boundary value problem of detecting a non-uniform conductivity motivated by pacing-guided ablation in cardiac electrophysiology. At the stationary level, the transmembrane potential $u$ in a region $\mathrm{\Omega }\subset {\mathbb{R}}^3$ of cardiac tissue satisfies $$-\mathrm{\nabla }\text{\hspace{0.17em}⁣}\cdot (\gamma \mathrm{\nabla }u)+\alpha u^3=0\text{in }\mathrm{\Omega },\gamma \mathrm{\nabla }u\cdot \nu =g\text{on }\mathrm{\partial }\mathrm{\Omega },$$ where $\gamma$ is an anisotropic conductivity tensor and $\alpha$ a nonlinear ionic response coefficient. The Neumann data $g$ represent pacing currents, and the boundary values $u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ correspond to invasive voltage measurements. Ischemic regions are modeled by a subdomain $D\subset \mathrm{\Omega }$ where $\gamma$ is piecewise constant. We address the inverse problem of determining $\gamma$ from the Neumann-to-Dirichlet (NtD) map, assuming that $\alpha$ and $D$ are known. To our knowledge, uniqueness in the case of NtD data with anisotropic conductivities in this nonlinear setting has not been analyzed in previous work. Using a first-order linearization around a nontrivial pacing current, we prove uniqueness for $\gamma$.