Optimal Bang-Bang Controls for a Two-Compartment Model in Cancer Chemotherapy

Abstract

Abstract.A class of mathematical models for cancer chemotherapy which have been described in the literature take the form of an optimal control problem over a finite horizon with control constraints and dynamics given by a bilinear system. In this paper, we analyze a twodimensional model in which the cell cycle is broken into two compartments. The cytostatic agent used as control to kill the cancer cells is active only in the second compartment where cell division occurs and the cumulative effect of the drug is used to model the negative effect of the treatment on healthy cells. It is shown that singular controls are not optimal for this model and the optimality properties of bang-bang controls are established. Specifically, transversality conditions at the switching surfaces are derived. In a nondegenerate setting, these conditions guarantee the local optimality of the flow if satisfied, while trajectories will be no longer optimal if they are violated.

A Dynamic Programming Approach for Approximate Optimal Control for Cancer Therapy

“…In spite thatĤ is linear in u, the control may not be determined by the minimum condition (detailed discussions are in [8]). Following [8], to exclude discussions about the structure of optimal controls in regions where the model does not represent the underlying biological problem, we confine ourselves to the biologically realistic domain …”

“…In the last 15 years, tumor anti-angiogenesis became an active area of research not only in medicine [3,4], but also in mathematical biology [5][6][7], and several models of dynamics of angiogenesis have been described, e.g., by Hahnfeldt et al [5] and d'Onofrio [6,7]. In a sequence of papers [8][9][10][11], Ledzewicz and Schaettler completely described and solved corresponding to them, optimal control problems from a geometrical optimal control theory point of view. In most of the mentioned papers, the numerical calculations of approximate solutions are presented.…”

Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy

“…In [15] we have formulated an algorithm which allows to determine the local optimality of the corresponding bang-bang controls for the problem P 0 . This algorithm is based on earlier work (see, for example [21,13,25]) and needs to be modified slightly for problem P pk to allow for the incorporation of the pharmacokinetic equation (15). The argument itself is a straightforward application of the general results derived in [21] to the specific equations of this model.…”

“…9-12 below give the graphs for steady-state initial conditions and Figs. [13][14][15][16] give the graphs for (p 0 , q 0 ) = (.9, .1). Figs.…”

“…Most models [6,18,20,26] put cancer cells as central and differentiate various stages of the cell cycle to create the dynamics of so-called compartmental models. Various models of this type were formulated and analyzed including, for example, three-compartment models with two-drug treatments where the actions of a killing drug were coupled with either blocking or recruitment agents [26,13,14,25] or various models with developing stages of drug resistance. In all these models the dynamics represents the effect of the drug on the cancer cells exclusively and the damage done to bone marrow or other healthy cells are not included in the dynamics at all.…”