How to position sensors in thermo-acoustic tomography

Thermo-acoustic tomography is a non-invasive medical imaging technique: the object to be reconstructed is excited by an impulse, inducing inhomogeneous heating and thus tissue expansion. This creates an acoustic wave pressure that can be measured by sensors. The reconstruction of internal heterogeneities can then be achieved by solving an inverse problem, once the sound wave measurements are known outside the body. As the measured pressure intensity is expected to be low, a difficult problem is to position the sensors properly. This article is devoted to determining the positions of the sensors in order to carry out the reconstruction procedure in an optimal way. We first introduce a model of optimal sensor position that involves a deviation function between the theoretical pressure and the measured pressure for a first series of sensor measurements, and a constant observable type function that describes the quality of reconstruction. We use it to determine an appropriate position of the sensors for a second set of measurements. Far from providing an intrinsic solution to the general issue of positioning sensors, solving this problem makes it possible to obtain a new position of the sensors improving the quality of the reconstruction before obtaining a new series of measurements. This model is analyzed mathematically: we study the existence problems and introduce a numerical algorithm to solve them. Finally, several 2D numerical simulations illustrate our approach.