Mathematical Analysis of a Model-Constrained Inverse Problem For the Reconstruction of Early States of Prostate Cancer Growth

The availability of cancer measurements over time enables the personalized assessment of tumor growth and therapeutic response dynamics. However, many tumors are treated after diagnosis without collecting longitudinal data, and cancer monitoring protocols may include infrequent measurements. To facilitate the estimation of disease dynamics and better guide ensuing clinical decisions, we investigate an inverse problem enabling the reconstruction of earlier tumor states by using a single spatial tumor dataset and a biomathematical model describing disease dynamics. We focus on prostate cancer, since aggressive cases of this disease are usually treated after diagnosis. We describe tumor dynamics with a phase field model driven by a generic nutrient ruled by reaction-diffusion dynamics. The model is completed with another reaction-diffusion equation for the local production of prostate-specific antigen, which is a key prostate cancer biomarker. We first improve previous well-posedness results by further showing that the solution operator is continuously Fr\'echet differentiable. We then analyze the backward inverse problem concerning the reconstruction of earlier tumor states starting from measurements of the model variables at the final time. Since this problem is severely ill-posed, only very weak conditional stability of logarithmic type can be recovered from the terminal data. However, by restricting the unknowns to a compact subset of a finite-dimensional subspace, we can derive an optimal Lipschitz stability estimate.