Identification of the Time-Dependent Proliferation Coefficient for a Brain Tumor Model

In this paper, we consider an inverse problem that arises in a brain tumor model. The reaction-diffusion model is used to describe the time evolution for the net rate of growth of glioma cells, one of the most common types of brain tumor. The inverse problem is to reconstruct the concentration of glioma cells and the proliferation coefficient based on some given data. Under certain consistency conditions for the data, we prove the existence and uniqueness of the inverse problem. When further assumptions are made, we can show the stability of the inverse problem. We also develop a numerical method to solve the inverse problem and demonstrate its performance using three numerical examples.