Iterative Algorithms for the Reconstruction of Early Stages of Prostate Cancer Growth

Abstract The development of mathematical models of cancer informed by time-resolved measurements has enabled personalised predictions of tumour growth and treatment response. However, frequent cancer monitoring is rare, and many tumours are treated soon after diagnosis with limited data. To improve the predictive capabilities of cancer models, we investigate the problem of recovering earlier tumour states from a single spatial measurement at a later time. Focusing on prostate cancer, we describe tumour dynamics using a phase-field model coupled with two reaction–diffusion equations for a nutrient and the local prostate-specific antigen. We generate synthetic data using a discretisation based on Isogeometric Analysis. Then, building on our previous analytical work (Beretta et al. in SIAM J Appl Math 84:2000–2027, 2024), we propose an iterative reconstruction algorithm based on the Landweber scheme, showing local convergence with quantitative rates and exploring an adaptive step size that leads to faster reconstruction algorithms. Finally, we run simulations demonstrating high-quality reconstructions even with long time horizons and noisy data.