Modeling, Analysis, and Optimal Control of Leukemic Cell Population Dynamics Under Therapy

Building upon the ODE model describing the dynamics of healthy and leukemic cells introduced in Kumar et al. (2024); Stiehl and Marciniak-Czochra (2012), we propose an extended framework that incorporates a control variable representing the effects of chemotherapy. This extension aims to provide a more refined mathematical basis for investigating anti-cancer strategies. First, we perform a stability analysis of the equilibria associated with healthy and leukemic states, partly estimated from clinical data. This analysis reveals a complex structure, including the emergence of a continuum of coexistence states and bifurcation thresholds that play a key role in the subsequent optimization stage. Based on these findings, we investigate an optimal control problem to minimize leukemia stem cells while limiting drug toxicity. Pontryagin's Maximum Principle provides necessary conditions for optimality, and direct numerical optimization confirms the predicted structures, motivating the study of the static problem. This static formulation reveals an unconventional feature: the cost functional becomes set-valued due to the continuum of equilibria, placing the problem outside the scope of standard methods. Simulations reveal a turnpike phenomenon, where over long time horizons the dynamic trajectories closely approximate the ideal static structure. Finally, a sensitivity analysis of the performance criterion with respect to key parameters complements the study, providing preliminary insights into which biological mechanisms may influence the optimal therapeutic outcomes. We conclude with a discussion of these findings.