Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy

Abstract

We study a collection of problems associated with the optimization of cancer chemotherapy treatments, under the assumptions of Gomperztiantype tumor growth and that the drug killing effect is proportional to the rate of growth for the untreated tumor (Norton-Simon hypothesis). Classical pharmacokinetics and different pharmacodynamics (Skipper and Emax) are considered, together with a toxicity limit or the penalization of the accumulated drug effect. Existence and uniqueness of the optimal control is proved in some cases, while in others the total amount of drug is the unique relevant aspect to take into account and the existence of an infinite number of optimal controls is shown. In all cases, explicit expressions for the solutions are derived in terms of the problem data. Finally, numerical results of illustrative examples and some conclusions are presented.

Optimal control for a mathematical model for chemotherapy with pharmacometrics

“…Whereas a solution to equation (7) always exists on the full interval [0, T ] on which the control is defined, this need not be the case for the dynamics (6). It rather depends on the vector fields f and g whether this will be the case.…”

A mathematical justification for metronomic chemotherapy in oncology

“…Here the term Ψ(L(t))ρ(t) represents the growth-inhibitory influence due to the cytotoxic chemotherapy effect and it reflects both, the level of the therapy at time t, ρ(t), and the tumor's sensitivity to therapy. First let us present a general result for the existence and uniqueness of solution for the Cauchy problem associated with (2) (see [4] for the proof). As usual, we will denote by W 1,∞ (0, T ) the Sobolev space of all functions in L ∞ (0, T ) having first order weak derivative (in the distributional sense) also belonging to L ∞ (0, T ).…”

“…iii) We have studied a related problem for continuous (non-discrete) drug infusion in [4] (labeled (OP 2 ) with G = G 2 ). There, it was proved the appearance of a constant maintenance infusion rate (during a fairly long time interval) in the expression of the optimal control.…”

Hybrid modeling frameworks of tumor development and treatment

“…The dose schedules were selected considering three time-dependent objective functions (average tumor volume, tumor mass variance, and average T cell population density), which were minimized using a gradient-based interior-point optimization algorithm. Fernández and Pola (2019) posted a goal of limiting the systemic drug toxicity while simultaneously minimizing tumor volume. The authors applied an optimal control technique to a continuous model of tumor growth and considered different growth laws, such as Gomperzian growth, the Norton-Simon hypothesis, and Skipper pharmacodynamics.…”