Mathematical oncology: Models, methods, and clinical applications

Abstract

Mathematical oncology integrates mechanistic modeling, experimental biology, clinical medicine, and data science to improve cancer detection, treatment planning, and understanding of tumor dynamics. This work provides a systematic review of the field, tracing its historical development from early pop\-u\-lation-level growth models and the log-kill hypothesis to contemporary spatial, evolutionary, multiscale, and data-driven approaches, supported by a large-scale bibliometric analysis of the discipline's emergence and thematic clusters. The review surveys mechanistic descriptions of tumor growth, the tumor microenvironment, clonal evolution, and metastasis, before examining models of treatment response, optimal control of therapy, and evolutionary game theory as frameworks for understanding and counteracting drug resistance. Multiscale modeling approaches for coupling molecular, cellular, and tissue-level processes are discussed alongside data-driven methods, including genomics and bioinformatics, artificial intelligence, and mechanistic learning, which combines knowledge-driven mathematical models with data-driven machine learning. Particular attention is given to digital twins and their prospective clinical validation, and to the practical requirements for translating mathematical models into clinical implementation, including model validation, FAIR data principles, and shared benchmarking infrastructure. The review concludes by outlining future directions for the field, emphasizing that sustained interdisciplinary collaboration across mathematics, biology, data science, and clinical medicine, rather than any single modeling paradigm, will determine the extent to which mathematical oncology translates into improved patient care.