Localization of tumor through a non-conventional numerical shape optimization technique

TL;DR

A method for estimating the shape and location of an embedded tumor using the coupled complex boundary method and an iterative algorithm to numerically determine the geometry of the tumor via the finite element method is introduced.

Abstract

This paper introduces a method for estimating the shape and location of an embedded tumor. The approach utilizes shape optimization techniques, applying the coupled complex boundary method. By rewriting the problem -- characterized by a measured temperature profile and corresponding flux (e.g., from infrared thermography) -- into a complex boundary value problem with a complex Robin boundary condition, the method simplifies the over-specified nature of the problem. The size and location of the tumor are identified by optimizing an objective function based on the imaginary part of the solution across the domain. Shape sensitivity analysis is conducted to compute the shape derivative of the functional. An iterative algorithm, which uses the Riesz representative of the gradient, is developed to numerically determine the geometry of the tumor via the finite element method. Additionally, we analyze the mesh sensitivity of the finite element solution of the associated state problem and derive a bound on its variation in terms of mesh deformation and its gradient. Numerical examples are provided to validate the theoretical findings and demonstrate the accuracy and effectiveness of the proposed method.