Bayesian inversion for electrical-impedance tomography in medical imaging using the nonlinear Poisson–Boltzmann equation

Abstract

We develop an electrical-impedance tomography (EIT) inverse model problem in an infinite-dimensional setting by introducing a nonlinear elliptic PDE as a new EIT forward model. The new model completes the standard linear model by taking the transport of ionic charge into account, which was ignored in the standard equation. We propose Bayesian inversion methods to extract electrical properties of inhomogeneities in the main body, which is essential in medicine to screen the interior body and detect tumors or determine body composition. We also prove well-definedness of the posterior measure and well-posedness of the Bayesian inversion for the presented nonlinear model. The new model is able to distinguish between liquid and tissues and the state-of-the-art delayed-rejection adaptive-Metropolis (DRAM) algorithm is capable of analyzing the statistical variability in the measured data in various EIT experimental designs. This leads to design a reliable device with higher resolution images which is crucial in medicine for diagnostic purposes. We first test the validation of the presented nonlinear model and the proposed inverse method using synthetic data on a simple square computational domain with an inclusion. Then we establish the new model and robustness of the proposed inversion method in solving the ill-posed and nonlinear EIT inverse problem by presenting numerical results of the corresponding forward and inverse problems on a real-world application in medicine and healthcare. The results include the extraction of electrical properties of human leg tissues using measurement data.