On forward and inverse problems for a DCIS model with free boundaries in mathematical biology

We are concerned with the mathematical study of a DCIS model which arises in characterizing the biological development of breast cancer. A salient feature of a DCIS model is the presence of free boundaries for describing the tumor growth which is not known in advance. In this paper, we are particularly interested in the case with general free boundaries which may be asymmetric. We first propose an iterative finite difference method for the forward problem and show that the method is of 2nd order in both space and time. Then we propose to study an inverse problem of recovering the nutrient consumption rate by the incisional biopsy data. We establish the unique identifiability of the inverse problem and develop a novel reconstruction scheme based on a certain integral formulation. Extensive numerical experiments are conducted to corroborate the theoretical findings. Our study opens up a new direction of research in mathematical biology with many potential extensions and developments.