Mathematical and numerical challenges in optical screening of female breast.

Diffuse optical tomography (DOT) is an emerging imaging technique which uses light for diagnostic purposes in a non-invasive and non-ionizing way. In this paper, we focus on DOT application to female breast screening, where the surface of the breast is illuminated by light sources and the outgoing light is collected on the surface. The comparison of measured light data with the equivalent field obtained from a relevant mathematical model yields the DOT inverse problem whose solution provides an estimate of the optical coefficients of the tissue. These latter, in turn, can be related to clinical markers for cancer detection. The goal of this work is to propose a mathematical and computational approach tailored to the concept of a DOT imaging device able to perform fast and accurate screenings at an affordable cost. Namely, we address two original points about the crucial issue of the solution of the severely ill-conditioned DOT inverse problem: (a) a computational approach based on Green's functions which do not require the exact knowledge of the tissue geometry, proposed here in the declination of the Method of Fundamental Solutions, which allows to enforce correct boundary conditions; (b) the elastic net regularization technique that shares the desirable properties of both the ℓ2 - and ℓ1 -norm penalization approaches and opens the possibility for sparsity recognition in the optical coefficients field and refinement procedures.