A Refined Formulation and Rigorous Analysis of an Inverse Free Boundary Problem in Modeling Ductal Carcinoma In Situ
Applicable Nonlinear Analysis, Volume 4, Issue 1, April 2027, Pages: 1–9
Heng Li
Division of Science Mathematics and Technology, Governors State University, University Park, IL 60484, USA
Der-Chen Chang
Department of Mathematics and Statistics, Georgetown University, Washington D.C. 20057, USA and Graduate Institute of Business Administration, Fu Jen Catholic University, New Taipei City 242, Taiwan, ROC
Yongzhi Xu
Department of Mathematics, University of Louisville, Louisville, KY 40292, United States
Abstract
We revisit a mathematical model for ductal carcinoma in situ (DCIS) growth described by a free boundary problem. Specifically, we address an algebraic oversight in the fixed-domain transformation of an inverse problem aimed at recovering the tumor growth potential from terminal biopsy data. We derive the refined parabolic partial differential equation step-by-step, establish the exact boundary evaluation, and prove the uniqueness of the reconstructed potential. Furthermore, we construct the clinical data function via a spectral eigenfunction expansion, casting the inverse problem as a first-kind Fredholm integral equation to analyze its exact existence via Picard's criterion. Demonstrating the severe ill-posedness of the problem under clinical noise, we propose a Tikhonov regularization scheme. We explicitly connect these functional analytic proofs to the physical reality of the breast duct, providing rigorous guarantees for the existence of a unique regularized biological solution. Finally, we present comprehensive computational simulations, validating the algorithm's robustness and optimal convergence against noisy synthetic clinical data.
Cite this Article as
Heng Li, Der-Chen Chang, and Yongzhi Xu, A Refined Formulation and Rigorous Analysis of an Inverse Free Boundary Problem in Modeling Ductal Carcinoma In Situ, Applicable Nonlinear Analysis, 4(1), 1–9, 2027