Optimal control for a mathematical model for chemotherapy with pharmacometrics

Abstract

An optimal control problem for an abstract mathematical model for cancer chemotherapy is considered. The dynamics is for a single drug and includes pharmacodynamic (PD) and pharmacokinetic (PK) models. The aim is to point out qualitative changes in the structures of optimal controls that occur as these pharmacometric models are varied. This concerns (i) changes in the PD-model for the effectiveness of the drug (e.g., between a linear log-kill term and a non-linear Michaelis-Menten type $E_{\max}$-model) and (ii) the question how the incorporation of a mathematical model for the pharmacokinetics of the drug effects optimal controls. The general results will be illustrated and discussed in the framework of a mathematical model for anti-angiogenic therapy.

On the role of pharmacometrics in mathematical models for cancer treatments

“…We note that this formula also exhibits the removable singularity on the diagonal (p = q) alluded to above. As expected, it is shown in [22] that the dynamics of the system [CC1] along the singular control agrees with the dynamics of the model [CC0] along the interior control. This confirms that it is possible to consider the pharmacokinetic model as an afterthought once the problem formulation [CC0] has been solved.…”

“…It is easy to see [22] that the concentration c is well defined over [0, T ] for any admissible control and that 0 ≤ c(t) < umax ρ+ηumax holds for all times. In particular, ηc(t) < 1.…”

“…It is not difficult to see that extremals are normal unless the optimal control u * vanishes identically [22]. We ignore this degenerate case and only consider normal extremals.…”

“…The Legendre-Clebsch condition for singular controls for problem [CC1] has been analyzed in [22]. It follows from the necessary conditions for optimality that the multiplier ν is constant and non-negative.…”

“…If a standard linear model for PK is incorporated into the dynamics, then the optimal control problem [CC1] once more is control affine and optimal controls (which now represent the dose rates) are again concatenations of bang and singular controls [22]. It is shown in that paper that the singular control is given as a feedback function of the state variables (p, q, c) † in the form…”