Inverse Problems Seminar - On the identification of early tumour states in some nonlinear reaction-diffusion models

The availability of longitudinal cancer measurements enables personalized assessments of tumor growth and therapeutic response dynamics. However, many tumours are treated upon diagnosis without the collection of such data, and cancer monitoring protocols often involve infrequent measurements. To facilitate the estimation of disease dynamics and better inform subsequent clinical decisions, we investigate an inverse problem that reconstructs earlier tumour states using a single spatial tumour dataset and a biomathematical model describing disease progression. Initially, we focus on prostate cancer, modeling tumour evolution with a phase-field approach driven by a generic nutrient governed by reaction-diffusion dynamics. The model is further enhanced by incorporating a reaction-diffusion equation for the local production of prostate-specific antigen, a key biomarker for prostate cancer. We extend previous well-posedness results by demonstrating that the solution operator is continuously Frechet differentiable. Subsequently, we analyze the backward inverse problem of reconstructing earlier tumor states based on measurements of the model variables at the final time. Due to the severe ill-posedness of this problem, only very weak conditional stability of logarithmic type can be achieved from the terminal data. However, by restricting the unknowns to a compact subset of a finite-dimensional subspace, we derive an optimal quantitative Lipschitz stability estimate. In the final part of this talk, we review recent results on reconstructing early tumour states using a Cahn–Hilliard-reaction-diffusion model, which is well-suited for describing more aggressive tumors.