Identifying early tumor states in a Cahn–Hilliard-reaction–diffusion model

In this paper, we tackle the problem of reconstructing earlier tumor configurations starting from a single spatial measurement at a later time. We describe the tumor evolution through a diffuse interface model coupling a Cahn–Hilliard-type equation for the tumor phase field to a reaction–diffusion equation for the nutrient, also accounting for chemotaxis effects. Reconstructing earlier tumor states is essential for the model calibration and also to identify the tumor’s initial development areas. However, backward-in-time inverse problems are severely ill-posed, even for linear parabolic equations. Nonetheless, we can establish uniqueness by using logarithmic convexity methods under suitable a priori assumptions. To further address the ill-posedness of the inverse problem, we propose a Tikhonov-regularization approach that approximates the solution through a family of constrained minimization problems. For such problems, we analytically derive the first-order necessary optimality conditions. Finally, we develop a computationally efficient numerical approximation of the optimization problems by employing standard C0-conforming first-order finite elements. We conduct numerical experiments on several test cases and observe that the proposed algorithm meets expectations, delivering accurate reconstructions of the original ground truth.