📝 SummarXiv

Abstract

We pursue a computational analysis of the biomedical problem on the identification of the cancerous tumor at an early stage of development based on the Electrical Impedance Tomography (EIT) and optimal control of elliptic partial differential equations. Relying on the fact that the electrical conductivity of the cancerous tumor is significantly higher than the conductivity of the healthy tissue, we consider an inverse EIT problem on the identification of the conductivity map in the complete electrode model based on the $m$ current-to-voltage measurements on the boundary electrodes. A variational formulation as a PDE-constrained optimal control problem is introduced based on the novel idea of increasing the size of the input data by adding "voltage-to-current" measurements through various permutations of the single "current-to-voltage" measurement. The idea of permutation preserves the size of the unknown parameters on the expense of increase of the number of PDE constraints. We apply a gradient projection (GPM) method based on the Fr\'echet differentiability in Besov-Hilbert spaces. Numerical simulations of 2D and 3D model examples demonstrate the sharp increase of the resolution of the cancerous tumor by increasing the number of measurements from $m$ to $m^2$