Precision medicine has recently seen both extensive theoretical advancements and wide-ranging biomedical applications. After a brief presentation of the problem and the perspectives, we will evoke one of the major limits of the approach on the clinical frontier, the validation of treatments, which illustrates the imperious need for a massive implication of mathematics in this field.
Precision medicine algorithms can be generated and applied on multiple levels of human health, from cellular subtyping to estimating individualized dynamic treatment rules (DTRs), which are decision rules mapping patient states to treatments that are optimized for desired health outcomes. We detail the development of methodologies for two main biomedical domains: (1) the identification of novel trajectories and signatures of fractionally treatment resistant tumor cells, and (2) learning optimal when-to-treat DTRs under dynamic resource constraints. For domain (1), we first establish a framework of identifying unique high-dimensional proteomic signatures of resistance to CDK4/6 inhibitors for ER+/HER2- breast cancer in preclinical cell lines and in primary tumor samples using manifold estimation for the visualization and validation of treatment perturbations. In domain (2), we propose an extension to existing work to obtain optimal DTRs under a when-to-treat setting. We expand this policy learning into indefinite time horizons and seek an optimal constrained DTR by jointly managing the issue of treatment timing and limited resource allocation.