Cancer detection via electrical impedance tomography and optimal control of elliptic PDEs

We pursue a computational analysis of the biomedical problem on the identification of cancerous tumors 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 that of healthy tissue, we consider an inverse EIT problem for identifying the conductivity map in the complete electrode model based on $ 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 at the expense of an increase in the number of PDE constraints. We apply a gradient projection method (GPM) based on the Fréchet differentiability in Besov-Hilbert spaces. Numerical simulations of 2D and 3D model examples demonstrate the sharp increase in the resolution of the cancerous tumor by increasing the number of measurements from $ m $ to $ m^2 $.