Urszula Ledzewicz

AntiAngiogenic Therapy in Cancer Treatment as an Optimal Control Problem

Antiangiogenic therapy is a novel treatment approach in cancer therapy that aims at preventing a tumor from developing its own blood supply system that it needs for growth. In this paper a mathematical model for antiangiogenic treatments based on a biologically validated model by Hahnfeldt et al. is analyzed as an optimal control problem and a full solution of the problem is given. Geometric methods from optimal control theory are utilized to arrive at the solution.

On optimal delivery of combination therapy for tumors

A mathematical model for the scheduling of angiogenic inhibitors in combination with a chemotherapeutic agent is formulated. Conditions are given that allow tumor eradication under constant infusion therapies. Then the optimal scheduling of a vessel disruptive agent in combination with a cytotoxic drug is considered as an optimal control problem. Both theoretical and numerical results on the structure of optimal controls are presented.

Optimal bang-bang controls for a two-compartment model in cancer chemotherapy

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 two-dimensional 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.

Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis

Tumor anti-angiogenesis is a cancer treatment approach that aims at preventing the primary tumor from developing its own vascular network needed for further growth. In this paper the problem of how to schedule an a priori given amount of angiogenic inhibitors in order to minimize the tumor volume is considered for three related mathematical formulations of a biologically validated model developed by Hahnfeldt et al. [1999. Tumor development under angiogenic signalling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res. 59, 4770-4775]. Easily implementable piecewise constant protocols are compared with the mathematically optimal solutions. It is shown that a constant dosage protocol with rate given by the averaged optimal control is an excellent suboptimal protocol for the original model that achieves tumor values that lie within 1% of the theoretically optimal values. It is also observed that the averaged optimal dose is decreasing as a function of the initial tumor volume.

ANALYSIS OF A CELL-CYCLE SPECIFIC MODEL FOR CANCER CHEMOTHERAPY

A class of mathematical models for cancer chemotherapy which has been described in the literature takes the form of an optimal control problem with dynamics given by a bilinear system. In this paper we analyze a three-dimensional model in which the cell-cycle is broken into three compartments. The cytostatic agent used as control to kill the cancer cells is active in a compartment which combines the second growth phase and mitosis where cell-division occurs. A blocking agent is used as a second control to slow down the transit of cells during synthesis, but does not kill cells. The cumulative effect of the killing agent is used to model the negative effect of the treatment on healthy cells. It is shown that singular controls are not optimal. This eliminates treatments where during some time only a portion of the full drug dose is administered. Consequently only treatments which alternate between a full and no dose, i.e., so-called bang-bang controls, canbe optimal for this model. Both necessary and sufficient conditions for optimality of treatment schedules of this type are given.

A Synthesis of Optimal Controls for a Model of Tumor Growth under Angiogenic Inhibitors

A mathematical model for the scheduling of angiogenic inhibitors to control a vascularized tumor is considered as an optimal control problem. A complete synthesis of optimal solutions is given.

Optimal response to chemotherapy for a mathematical model of tumor–immune dynamics

An optimal control problem for cancer chemotherapy is considered that includes immunological activity. In the objective a weighted average of several quantities that describe the effectiveness of treatment is minimized. These terms include (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), (iii) the overall amount of cytotoxic agents given as a measure for the side effects of treatment and (iv) a small penalty on the terminal time that limits the overall therapy horizon which is assumed to be free. This last term is essential in obtaining a well-posed problem formulation. Employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated. Solutions initially follow a full dose treatment, but then at one point switch to a singular regimen that only applies partial dosages. This structure is consistent with protocols that apply an initial burst to reduce the tumor volume and then maintain a small volume through lower dosages. Optimal controls end with either a prolonged period of no dose treatment or, in a small number of scenarios, this no dose interval is still followed by one more short burst of full dose treatment.

Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy.

We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given for four versions of this problem: the case when only anti-angiogenic agents are administered, combination treatment with a cytotoxic agent, and when a standard linear pharmacokinetic equation for the anti-angiogenic agent is added to each of these models. It is shown that the solutions to the more complex models naturally build upon the simplified versions. This gives credence to a modeling approach that starts with the analysis of simplified models and then adds increasingly more complex and medically relevant features. Furthermore, for each of the problem formulations considered here, there exist excellent simple piecewise constant controls with a small number of switchings that virtually replicate the optimal values for the objective.

Second-order conditions for extremum problems with nonregular equality constraints

Combining results of Avakov about tangent directions to equality constraints given by smooth operators with results of Ben-Tal and Zowe, we formulate a second-order theory for optimality in the sense of Dubovitskii-Milyutin which gives nontrivial conditions also in the case of equality constraints given by nonregular operators. Secondorder feasible and tangent directions are defined to construct conical approximations to inequality and equality constraints which within a single construction lead to first- and second-order conditions of optimality for the problem also in the nonregular case. The definitions of secondorder feasible and tangent directions given in this paper allow for reparametrizations of the approximating curves and give approximating sets which form cones. The main results of the paper are a theorem which states second-order necessary condition of optimality and several corollaries which treat special cases. In particular, the paper generalizes the Avakov result in the smooth case.

On optimal controls for a general mathematical model for chemotherapy of HIV

We describe a class of optimal control problems which arise as mathematical models for biological systems in the chemotherapy of diseases which have a strong cell proliferation aspect such as cancer or AIDS. Although individually these problems are very different in their specifics, yet due to. the underlying mechanisms of cell dynamics, they also have many aspects in common and can be put into one general abstract mathematical model which encompasses them all. While on one side there is a need to consider-these problems individually to gain insight into implications for the underlying disease, on the other side there are also simplifications and insights to be gained by looking at the general properties common to all these models. In this paper we will develop and analyze such a structure in models for HIV-infection and anti-viral treatment of AIDS which have been proposed in the literature. Specifically, we give general sufficient conditions for strong local optimality of reference trajectories.