Curvature-based schedule ordering is preserved in homogeneous and heterogeneous tumors

During treatment, dosing protocols with identical total dose may favor one schedule over another depending on how exposure is distributed over time. In homogeneous cell populations, Jensen's inequality has been shown to predict whether continuous or intermittent high-dose followed by low-dose is favored, based on the curvature of the concentration-effect function. Scheduling ordering can be performed based on curvature only: concave functions favor continuous dosing while convex functions favor intermittent dosing schedules with a high time-variance. However, tumors exhibit heterogeneous drug sensitivities, and treatment preferentially eliminates drug-sensitive cells, progressively shifting the surviving population toward greater drug resistance. Whether this treatment-induced selection can reverse curvature-based schedule ordering remains unclear. We develop a mathematical framework in which phenotypes share a common concentration-effect function but differ by fixed, nonnegative sensitivity scales. We prove that the ordering of equal-mean schedules is administrated solely by the scalar cumulative treatment effect, that is, by the curvature of the effect function alone. Consequently, baseline heterogeneity and treatment-induced selection can alter the magnitude of treatment response and the composition of surviving cells, but cannot reverse the ordering of equal-mean schedules predicted by the curvature of the single-cell concentration-effect function. This preservation holds across different temporal dosing profiles and repeated treatment cycles. Our work uncovers a general principle that curvature-based schedule ordering is preserved in heterogeneous tumors despite treatment-induced selection toward resistance.