Simultaneous reconstruction of birth condition and mortality rate in an age-structured tumor model

Abstract This paper proposes a computational method to reconstruct both the birth and mortality coefficients in an age-structured population diffusive model. In mathematical oncology, solving this inverse problem is crucial for assessing the effectiveness of anti-cancer treatments and thus, gaining insights into the post-treatment dynamics of tumors. Through some linear and nonlinear transformations, the targeted inverse model is transformed into an auxiliary third-order nonlinear PDE. Subsequently, a coupled age-dependent quasi-linear parabolic PDE system is derived using the Fourier–Klibanov basis. The resulting PDE system is then approximated through the minimization of a cost functional, weighted by a suitable Carleman function. Ultimately, an analysis of the minimization problem is studied through a new Carleman estimate, and some computational results are presented to show how the proposed method works.