Optimal Control for Mathematical Models of Cancer Therapies

In this introductory chapter, we briefly describe the more important aspects of the medical and biological background for the mathematical models of cancer development and treatment that will be considered and analyzed in this text. Obviously, this merely constitutes a nonexperts’ attempt to summarize the major structural features that motivate these models. We focus on the “big picture,” with at times full disregard for the myriad and complex details. Yet, it is precisely this highly simplified overall understanding that has motivated much of the historical developments of cancer research and it still defines most current activities in the “search for a cure.”

In this chapter, we analyze a class of cell cycle specific compartmental models for cancer chemotherapy. Besides drug resistance, cell cycle specificity of drugs is viewed as one of the major obstacles against successful chemotherapy [83, 52]. By considering the phases of the cell cycle separately, it is possible to appropriately model the different actions of various drugs involved. A first such model was introduced for leukemia in the work of Kimmel and Swierniak [150] and later has been expanded greatly in the work by Swierniak and his co-workers (e.g., see [313, 321, 322, 323, 324] and many more).

The results of the previous chapter are consistent with the classical MTD paradigm in medicine: give as much of the drug as possible immediately. This makes perfect sense in many situations: cancer is a widely symptomless disease which, once finally detected, often is in an advanced stage where immediate action is required. Then the aim simply is to be as toxic as possible to the cancerous cells. However, this presumes that cells can be killed, i.e., that the tumor population consists of chemotherapeutically sensitive cells. Malignant cancer cell populations on the other hand are often highly genetically unstable and coupled with fast proliferation rates; this leads to a great variety in the structure of the cells within one tumor—the number of genetic errors present within one cancer cell can lie in the thousands [220]. Consequently, many tumors consist of heterogeneous agglomerations of subpopulations of cells that show widely varying sensitivities toward the actions of a particular chemotherapeutic agent [104, 107]. Coupled with the fact that growing tumors also exhibit considerable evolutionary ability to enhance cell survival in an environment that is becoming hostile, this leads to multi-drug resistance of some strains of the cells. Naturally, it makes sense to combine drugs with different activation mechanisms to reach a larger population of the tumor cells—and this is what is being done—but the sad fact remains that some cells develop multi-drug resistance to a wide variety of even structurally unrelated drugs. There may even exist subpopulations of cells that are not sensitive to the treatment from the beginning (ab initio, intrinsic resistance). For certain types of cancer cells, there are simply no effective agents known.

In this chapter, we give optimal solutions for systems with a control-affine dynamics (c.f., Section A.3 in Appendix A) when the dependence on the control in the objective is taken as a positive definite quadratic function. The mathematical advantages of such a formulation are obvious: the Hamiltonian H for the optimal control problem becomes strictly convex in the control u and thus has a unique minimizer, albeit only in the state-multiplier space (cotangent bundle). While this does not guarantee that controls found by an analysis of these necessary conditions are necessarily optimal, it considerably simplifies the analysis. However, as already mentioned, a quadratic functional form often is somewhat questionable and may be difficult to justify for biomedical problems. The prevalence of such models has its origin in an abundance of classical problems related to mechanical or electro-dynamical systems when such a term has a clear and justified connection with the kinetic energy of the system. If such a connection is not there—as it is lacking in the case of drug treatments—usually other, and often arbitrary “systemic” cost arguments are put forward to justify the choice. But such reasoning rarely is based on the underlying biology of the problem.

In the models considered so far, the focus was on the cancerous cells progressing from mathematical models for homogeneous tumor populations of chemotherapeutically sensitive cells to heterogeneous structures of cell populations with varying sensitivities or even resistance. From an optimal control point of view, optimal treatment schedules change from bang-bang solutions with upfront dosing (that correspond to classical MTD approaches in medicine) to administrations that also include singular controls (which correspond to time-varying dosing schedules at less than maximum rates) as heterogeneity of the tumor population becomes more prevalent. In this chapter, we begin to analyze mathematical models that also take into account a tumor’s microenvironment.

As was shown in the previous chapter, singular controls play an essential role in determining the overall structure of optimal controlled trajectories for the class of mathematical models for antiangiogenic treatments based on the model by Hahnfeldt et al. Lie algebraic computations provide an elegant framework in which the singular controls and corresponding arcs can be determined analytically, but the resulting formulas define feedback controls that administer time varying partial doses depending on the current state of the system, that is, on the tumor volume p and its carrying capacity q.

Antiangiogenic treatment discussed in Chapters 5 and 6 is an indirect approach to cancer therapy that aims at limiting a tumor’s ability to grow by depriving it of the required vasculature. It initially provided a new hope in cancer treatment since targeting the healthy and genetically stable endothelial cells of the lining for the blood vessels showed no drug resistance, the curse of chemotherapy [141, 142]. However, since the treatment is only limiting the tumor’s support mechanism without actually killing the cancer cells, antiangiogenic therapy by itself only achieves a temporary, “pseudo-therapeutic effect” that goes away with time. In some cases, once treatment is halted, the tumor grows back even more vigorously than before. While antiangiogenic monotherapy thus is not considered a viable treatment option, it has become a staple of anticancer treatments in combination with radio- and chemotherapy. In this way, simultaneously two separate mechanisms that support cancer are targeted, the cancerous cells and the vasculature that supports them [283]. The idea simply is that antiangiogenic therapy can enhance the efficacy of traditional approaches by normalizing a tumor’s vasculature. For example, Jain and Munn argue that a normalization of a tumor’s irregular and dysfunctional vasculature [131, 132] through prior antiangiogenic treatment enhances the delivery of chemotherapeutic agents and thus improves the effectiveness of chemotherapy.

In this chapter, we consider the second major feature of the tumor microenvironment: interactions between the tumor and the immune system. Fundamental principles that have already been outlined in the introduction (see Section 1.3.4) will be expanded upon in this chapter. As a vehicle for the analysis we use the classical model by Stepanova [303] and some of its modifications that have been developed in the literature. This model captures the main features that we want to discuss here—immune surveillance and tumor dormancy—and, at the same time, being low-dimensional and minimally parameterized, has the advantage of allowing us to easily visualize associated geometric features (regions of attractions, stability boundaries, etc.). We formulate an optimal control problem whose objective to be minimized is tailored to the inherent multi-stable structure that these systems have. These problems are considered under chemotherapy and under combinations of chemotherapy with a rudimentary form of an immune boost.

The question how therapeutic agents should be administered in order to maximize their potential effects is of fundamental importance in medical treatments. In the administration of cancer treatments, these questions are still far from being answered conclusively. In this text, we have explored what can be said about this topic from an analysis of minimally parameterized models described by ordinary differential equations using an optimal control approach. Clearly, more precise and in this sense more realistic models exist. These range from the inclusion of spatial aspects in partial differential equations to the incorporation of random features in stochastic models to complex agent-based models. Becoming increasingly more precise, such models, however, are prone to the pitfalls of Borges’s cartographers guild [33]. While current computer technologies enable large-scale computations and simulations, the number of parameters involved automatically carries with it uncertainty. Also, no matter how precise the data are that are available, the values of the parameters are for a particular case and numerical results pertain to a specific situation. On the other hand, the models considered here are all highly aggregated population based models, which, having small dimensions, allows to examine the underlying models analytically, not just numerically. Indeed, fairly robust qualitative features emerge from our analysis that we still would like to summarize.