The release of tumor antigens during traditional cancer treatments such as radio- or chemotherapy leads to a stimulation of the immune response which provides synergistic effects these treatments have when …
The release of tumor antigens during traditional cancer treatments such as radio- or chemotherapy leads to a stimulation of the immune response which provides synergistic effects these treatments have when combined with immunotherapies. A low-dimensional mathematical model is formulated which, depending on the values of its parameters, encompasses the 3 E’s (elimination, equilibrium, escape) of tumor immune system interactions. For the escape situation, optimal control problems are formulated which aim to revert the process to the equilibrium scenario. Some numerical results are included.
A nonlinear mathematical model which includes synergistic effects of chemo- and immunotherapy is analyzed (both analytically and numerically) as an optimal control problem with free terminal time for the problem …
A nonlinear mathematical model which includes synergistic effects of chemo- and immunotherapy is analyzed (both analytically and numerically) as an optimal control problem with free terminal time for the problem of scheduling combination therapies. Side effects of the drugs are measured indirectly by including the total doses of the respective drugs with weights as penalty terms in the objective. The formulation allows us to judge the amounts of the agents required to achieve tumor eradication as well as the time it will take to do so. For various weights for the penalty terms, extremal controlled trajectories are computed numerically and their local optimality is verified with second-order conditions for optimality.
Abstract The release of tumor antigens during traditional cancer treatments leads to a stimulation of the immune response which provides synergistic effects these treatments have when combined with immunotherapies (e.g., …
Abstract The release of tumor antigens during traditional cancer treatments leads to a stimulation of the immune response which provides synergistic effects these treatments have when combined with immunotherapies (e.g., based on check‐point blockade). Based on a low‐dimensional mathematical model, the interactions between chemo‐ and immunotherapy are analyzed and optimal control problems for the combination therapy are considered.
We analyze a mathematical model for cancer chemotherapy which includes antiangiogenic effects of the cytotoxic agent. Assuming that the total amount of agents to be given has been determined a …
We analyze a mathematical model for cancer chemotherapy which includes antiangiogenic effects of the cytotoxic agent. Assuming that the total amount of agents to be given has been determined a priori based on a medical assessment of its side effects, we consider the problem how to best administer this amount. The model assumes a homogenous tumor and if the aim is to minimize the tumor volume, then optimal controls administer the total dose in a single maximum dose session. As, however, angiogenic effects of the agent are taken into account, this no longer is optimal. Lower dose strategies determined by an optimal singular arc with significantly reduced dose rates give a better response over time. In this paper, for the medically realistic domain of the mathematical model, the concatenation structure of optimal controlled trajectories as segments of bang and singular arcs is determined. This leads to simple numerical minimization procedures for the computation of globally optimal controls.
<p style='text-indent:20px;'>In recent years, an increasing number of papers have been published (and many more submitted for publication) in which optimal control theory is superficially applied to specific problems, especially …
<p style='text-indent:20px;'>In recent years, an increasing number of papers have been published (and many more submitted for publication) in which optimal control theory is superficially applied to specific problems, especially from the biological and health sciences, but also many other fields. A lack of understanding of what it actually means to solve an optimal control problem—complex infinite-dimensional optimization problems—often leads to heavily overblown claims about optimality of solutions. In this editorial, a critical assessment of these efforts is given.</p>
We analyze a mathematical model for the combination of chemotherapy with antiangiogenic treatment as a multi‐input optimal control problem. Assuming that the total amounts of both agents to be given …
We analyze a mathematical model for the combination of chemotherapy with antiangiogenic treatment as a multi‐input optimal control problem. Assuming that the total amounts of both agents to be given have been determined a priori based on a medical assessment of side effects, we consider the problem to minimize a weighted average of tumor volume and the carrying capacity of the tumor vasculature. For medically realistic initial conditions and data, based on a thorough theoretical analysis of the necessary conditions for optimality given by the Pontryagin maximum principle, in this paper, the optimal administration regimen for the antiangiogenic agent is determined.
Necessary and sufficient conditions for optimality in optimal control problems with state space constraints are reviewed with emphasis on geometric aspects.
Necessary and sufficient conditions for optimality in optimal control problems with state space constraints are reviewed with emphasis on geometric aspects.
<p style='text-indent:20px;'>We review and discuss various aspects that the modeling of pharmacometric properties has on the structure of optimal solutions in mathematical models for cancer treatment. These include (i) the …
<p style='text-indent:20px;'>We review and discuss various aspects that the modeling of pharmacometric properties has on the structure of optimal solutions in mathematical models for cancer treatment. These include (i) the changes in the interpretation of the solutions as pharmacokinetic (PK) models are added, respectively deleted from the modeling and (ii) qualitative changes in the structures of optimal controls that occur as pharmacodynamic (PD) models are varied. The results will be illustrated with a sample of models for cancer treatment.
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 …
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.
We review results about the influence tumor heterogeneity has on optimal chemotherapy protocols (relative to timing, dosing and sequencing of the agents) that can be inferred from mathematical models. If …
We review results about the influence tumor heterogeneity has on optimal chemotherapy protocols (relative to timing, dosing and sequencing of the agents) that can be inferred from mathematical models. If a tumor consists of a homogeneous population of chemotherapeutically sensitive cells, then optimal protocols consist of upfront dosing of cytotoxic agents at maximum tolerated doses (MTD) followed by rest periods. This structure agrees with the MTD paradigm in medical practice where drug holidays limit the overall toxicity. As tumor heterogeneity becomes prevalent and sub-populations with resistant traits emerge, this structure no longer needs to be optimal. Depending on conditions relating to the growth rates of the sub-populations and whether drug resistance is intrinsic or acquired, various mathematical models point to administrations at lower than maximum dose rates as being superior. Such results are mirrored in the medical literature in the emergence of adaptive chemotherapy strategies. If conditions are unfavorable, however, it becomes difficult, if not impossible, to limit a resistant population from eventually becoming dominant. On the other hand, increased heterogeneity of tumor cell populations increases a tumor's immunogenicity and immunotherapies may provide a viable and novel alternative for such cases.
We analyze mathematical models for cancer chemotherapy under tumor heterogeneity as optimal control problems. Tumor heterogeneity is incorporated into the model through a potentially large number of states which represent …
We analyze mathematical models for cancer chemotherapy under tumor heterogeneity as optimal control problems. Tumor heterogeneity is incorporated into the model through a potentially large number of states which represent sub-populations of tumor cells with different chemotherapeutic sensitivities. In this paper, a Michaelis-Menten type functional form is used to model the pharmacodynamic effects of the drug concentrations. In the objective, a weighted average of the tumor volume and the total amounts of drugs administered (taken as an indirect measurement for the side effects) is minimized. Mathematically, incorporating a Michaelis-Menten form creates a nonlinear structure in the controls with partial convexity properties in the Hamiltonian function for the optimal control problem. As a result, optimal controls are continuous and this fact can be utilized to set up an efficient numerical computation of extremals. Second order Jacobi type necessary and sufficient conditions for optimality are formulated that allow to verify the strong local optimality of numerically computed extremal controlled trajectories. Examples which illustrate the cases of both strong locally optimal and non-optimal extremals are given.
We analyze the structure of optimal protocols for a mathematical model of tumor anti-angiogenic treatment. The control represents the concentration of the agent and we consider the problem to administer …
We analyze the structure of optimal protocols for a mathematical model of tumor anti-angiogenic treatment. The control represents the concentration of the agent and we consider the problem to administer an a priori given total amount of agents in order to achieve a minimum tumor volume/maximum tumor reduction. In earlier work, this problem was studied with a log-kill type pharmacodynamic model for drug effects which does not account for saturation of the drug concentration. Here we study the effect of incorporating a Michaelis-Menten (MM) or $ E_{\max} $-type pharmacodynamic model, the most commonly used model in the field of pharmacometrics. We compare the formulations of both problems and the resulting solutions. The reformulated problem with $ E_{\max} $ pharmacodynamics is no longer linear in the control. This results in qualitative changes in the structure of optimal controls which, in line with an interpretation as concentrations, now are continuous while discontinuities exist if the log-kill model is used which is more in line with an interpretation of the control as dose rates. In spite of these qualitative differences, similarities in the structures of solutions can be observed. Both aspects are discussed theoretically and illustrated numerically.
We consider an optimal control problem for a general mathematical model of drug treatment with a single agent.The control represents the concentration of the agent and its effect (pharmacodynamics) is …
We consider an optimal control problem for a general mathematical model of drug treatment with a single agent.The control represents the concentration of the agent and its effect (pharmacodynamics) is modelled by a Hill function (i.e., Michaelis-Menten type kinetics).The aim is to minimize a cost functional consisting of a weighted average related to the state of the system (both at the end and during a fixed therapy horizon) and to the total amount of drugs given.The latter is an indirect measure for the side effects of treatment.It is shown that optimal controls are continuous functions of time that change between full or no dose segments with connecting pieces that take values in the interior of the control set.Sufficient conditions for the strong local optimality of an extremal controlled trajectory in terms of the existence of a solution to a piecewise defined Riccati differential equation are given.
Effects that tumor heterogeneity and drug resistance have on the structure of chemotherapy protocols are discussed from a mathematical modeling and optimal control point of view. In the case when …
Effects that tumor heterogeneity and drug resistance have on the structure of chemotherapy protocols are discussed from a mathematical modeling and optimal control point of view. In the case when two compartments consisting of sensitive and resistant cells are considered, optimal protocols consist of full dose chemotherapy as long as the relative proportion of sensitive cells is high. When resistant cells become more dominant, optimal controls switch to lower dose regimens defined by so-called singular controls. The role that singular controls play in the structure of optimal therapy protocols for cell populations with a large number of traits is explored in mathematical models.
We consider cancer chemotherapy as an optimal control problem with the aim to minimize a combination of the tumor volume and side effects over an a priori specified therapy horizon …
We consider cancer chemotherapy as an optimal control problem with the aim to minimize a combination of the tumor volume and side effects over an a priori specified therapy horizon when the tumor consists of a heterogeneous agglomeration of many subpopulations. The mathematical model, which accounts for different growth and apoptosis rates in the presence of cell densities, is a finite-dimensional approximation of a model originally formulated by Lorz et al. [18,19] and Greene et al. [10,11] with a continuum of possible traits. In spite of an arbitrarily high dimension, for this problem singular controls (which correspond to time-varying administration schedules at less than maximum doses) can be computed explicitly in feedback form. Interestingly, these controls have the property to keep the entire tumor population constant. Numerical computations and simulations that explore the optimality of bang-bang and singular controls are given. These point to the optimality of protocols that combine a full dose therapy segment with a period of lower dose drug administration.
A $3$-compartment model for metronomic chemotherapy that takes intoaccount cancerous cells, the tumor vasculature and tumorimmune-system interactions is considered as an optimal controlproblem. Metronomic chemo-therapy is the regular, almost continuousadministration …
A $3$-compartment model for metronomic chemotherapy that takes intoaccount cancerous cells, the tumor vasculature and tumorimmune-system interactions is considered as an optimal controlproblem. Metronomic chemo-therapy is the regular, almost continuousadministration of chemotherapeutic agents at low dose, possibly withsmall interruptions to increase the efficacy of the drugs. Thereexists medical evidence that such administrations of specificcytotoxic agents (e.g., cyclophosphamide) have both antiangiogenicand immune stimulatory effects. A mathematical model for angiogenicsignaling formulated by Hahnfeldt et al. is combined with theclassical equations for tumor immune system interactions byStepanova to form a minimally parameterized model to capture theseeffects of low dose chemotherapy. The model exhibits bistablebehavior with the existence of both benign and malignant locallyasymptotically stable equilibrium points. In this paper, thetransfer of states from the malignant into the benign regions isused as a motivation for the construction of an objective functionalthat induces this process and the analysis of the correspondingoptimal control problem is initiated.
Optimal control problems with fixed terminal time are considered formulti-input bilinear systems with the control set given by a compact intervaland the objective function affine in the controls. Systems of …
Optimal control problems with fixed terminal time are considered formulti-input bilinear systems with the control set given by a compact intervaland the objective function affine in the controls. Systems of this type havebeen widely used in the modeling of cell-cycle specific cancer chemotherapyover a prescribed therapy horizon for both homogeneous and heterogeneous tumorpopulations. Necessary conditions for optimality lead to concatenations ofbang and singular controls as prime candidates for optimality. In this paper,the method of characteristics will be formulated as a general procedure toembed such a controlled reference extremal into a field of broken extremals.Sufficient conditions for the strong local optimality of a controlledreference bang-bang trajectory will be formulated in terms of solutions to associatedsensitivity equations. These results will be applied to a model for cell cycle specificcancer chemotherapy with cytotoxic and cytostatic agents.
A mathematical model for cancer chemotherapy of heterogeneous tumor populations is considered as an optimal control problem with the objective to minimize the tumor burden over a prescribed therapy horizon. …
A mathematical model for cancer chemotherapy of heterogeneous tumor populations is considered as an optimal control problem with the objective to minimize the tumor burden over a prescribed therapy horizon. While an upfront maximum tolerated dose (MTD) regimen with rest-period has been confirmed as mathematically optimal for models when the tumor population is homogeneous, in the presence of partially sensitive or even resistant cells, protocols that administer the therapeutic agents at lower dose rates described by so-called singular controls become a viable alternative. In this paper, the structure of protocols that follow an initial upfront maximum dose treatment with reduced dose rate singular controls is investigated. Such protocols reflect structures which in the medical literature sometimes are called chemo-switch protocols.
We review mathematical results about the qualitative structure of chemotherapy protocols that were obtained with the methods of optimal control. As increasingly more complex features are incorporated into the mathematical …
We review mathematical results about the qualitative structure of chemotherapy protocols that were obtained with the methods of optimal control. As increasingly more complex features are incorporated into the mathematical model—progressing from models for homogeneous, chemotherapeutically sensitive tumor cell populations to models for heterogeneous agglomerations of subpopulations of various sensitivities to models that include tumor immune-system interactions—the structures of optimal controls change from bang-bang solutions (which correspond to maximum dose rate chemotherapy with restperiods) to solutions that favor singular controls (representing reduced dose rates). Medically, this corresponds to a transition from standard MTD (maximum tolerated dose) type protocols to chemo-switch strategies towards metronomic dosing.
We consider the optimal control problem of minimizing an objective function that is quadratic in the control over a fixed intervalfor a multi-input bilinear dynamical system in the presence of …
We consider the optimal control problem of minimizing an objective function that is quadratic in the control over a fixed intervalfor a multi-input bilinear dynamical system in the presence of control constraints.Such models are motivated by and applied to mathematical models for cancer chemotherapy over an a priori specified fixed therapy horizon.The necessary conditions for optimality of the Pontryagin maximum principle are easily evaluatedand give a functional description of optimal controls as continuous functions of states and multipliers.However, there is no a priori guarantee that a numerically computed extremal controlled trajectory is locally optimal.In this paper, we formulate sufficient conditions for strong local optimality that are based on the existence of a bounded solutionto a matrix Riccati differential equation. The theory is applied to a $3$-compartment model for multi-drugcancer chemotherapy with cytotoxic and cytostatic agents. The numerical results are compared with those for a correspondingoptimal control problem when the objective is taken linear in the controls.
In this paper, results about the structure of cancer treatment protocols that can be inferred from an analysis of mathematical models with the methods and tools of optimal control are …
In this paper, results about the structure of cancer treatment protocols that can be inferred from an analysis of mathematical models with the methods and tools of optimal control are reviewed. For homogeneous tumor populations of chemotherapeutically sensitive cells, optimal controls are bang-bang corresponding to the medical paradigm of maximum tolerated doses (MTD). But as more aspects of the tumor microenvironment are taken into account, such as heterogeneity of the tumor cell population, tumor angiogenesis and tumor-immune system interactions, singular controls which administer agents at specific time-varying reduced dose rates become optimal and give an indication of what might be the biologically optimal dose (BOD).
An optimal control problem for combination of cancer chemotherapy withimmunotherapy in form of a boost to the immune system is considered as amulti-input optimal control problem. The objective to be …
An optimal control problem for combination of cancer chemotherapy withimmunotherapy in form of a boost to the immune system is considered as amulti-input optimal control problem. The objective to be minimized is chosenas a weighted average of (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), the overall amounts of (iii) cytotoxic agentsand (iv) immune boost given as a measure for the side effects oftreatment and (v) a small penalty on the free terminal time that limits theoverall therapy horizon. This last term is essential in obtaining a mathematically well-posedproblem formulation. Both analytical and numerical results about thestructures of optimal controls will be presented that give some insightsinto the structure of optimal protocols, i.e., the dose rates and sequencing ofdrugs in these combination treatments.
In this paper, a mathematical model for chemotherapy that takestumor immune-system interactions into account is considered for astrongly targeted agent. We use a classical model originallyformulated by Stepanova, but replace …
In this paper, a mathematical model for chemotherapy that takestumor immune-system interactions into account is considered for astrongly targeted agent. We use a classical model originallyformulated by Stepanova, but replace exponential tumor growth with ageneralised logistic growth model function depending on a parameter$\nu$. This growth function interpolates between a Gompertzian model(in the limit $\nu\rightarrow0$) and an exponential model (in thelimit $\nu\rightarrow\infty$). The dynamics is multi-stable andequilibria and their stability will be investigated depending on theparameter $\nu$. Except for small values of $\nu$, the system hasboth an asymptotically stable microscopic (benign) equilibrium pointand an asymptotically stable macroscopic (malignant) equilibriumpoint. The corresponding regions of attraction are separated by thestable manifold of a saddle. The optimal control problem of movingan initial condition that lies in the malignant region into thebenign region is formulated and the structure of optimal singularcontrols is determined.
In standard chemotherapy protocols, drugs are given at maximum tolerated doses(MTD) with rest periods in between. In this paper, we briefly discuss therationale behind this therapy approach and, using as …
In standard chemotherapy protocols, drugs are given at maximum tolerated doses(MTD) with rest periods in between. In this paper, we briefly discuss therationale behind this therapy approach and, using as examplemulti-drug cancer chemotherapy with a cytotoxic and cytostatic agent, show thatthese types of protocols are optimal in the sense of minimizing a weightedaverage of the number of tumor cells (taken both at the end of therapy and atintermediate times) and the total dose given if it is assumed that the tumorconsists of a homogeneous population of chemotherapeutically sensitive cells.A $2$-compartment linear model is used to model the pharmacokinetic equations for the drugs.
Perturbation feedback control is a classical procedure in control engineering that is based on linearizing a nonlinear system around some locally optimal nominal trajectory. In the presence of terminal constraints …
Perturbation feedback control is a classical procedure in control engineering that is based on linearizing a nonlinear system around some locally optimal nominal trajectory. In the presence of terminal constraints defined by a $k$-dimensional embedded submanifold, the corresponding flow of extremals for the underlying system gives rise to a canonical foliation in the $(t,x)$-space consisting of $(n-k+1)$-dimensional leaves and $k$-dimensional cross sections. In this paper, the connections between the formal computations in the engineering literature and the geometric meaning underlying these constructions are described.
In this paper, we use the method of characteristics to studysingularities in the flow of a parameterized family of extremals foran optimal control problem. By means of the Lyapunov--Schmidtreduction a …
In this paper, we use the method of characteristics to studysingularities in the flow of a parameterized family of extremals foran optimal control problem. By means of the Lyapunov--Schmidtreduction a characterization of fold and cusp pointsis given. Examples illustrate the local behaviors of the flow nearthese singular points. Singularities of fold type correspond to thetypical conjugate points as they arise for the classical problem ofminimum surfaces of revolution in the calculus of variations andlocal optimality of trajectories ceases at fold points. Simple cusppoints, on the other hand, generate a cut-locus that limits theoptimality of close-by trajectories globally to times prior to theconjugate points.
Tumor-immune system interplay is extremely complex, and, as such, it represents a big challenge for mathematical oncology. Here we investigate a simple general family of models for this important interplay …
Tumor-immune system interplay is extremely complex, and, as such, it represents a big challenge for mathematical oncology. Here we investigate a simple general family of models for this important interplay by considering both the delivery of a cytotoxic chemotherapy and of immunotherapy. Then methods of geometrical optimal control are applied to a special case (the Stepanova model) in order to infer (under suitable constraints) the best combination of drugs scheduling to transfer — through therapy —the system from an initial condition in the malignant region of the state space into a benign region. Our findings suggest that chemotherapy is always needed first to reduce a large tumor volume before the immune system can become effective.
Linear Systems and the Time Optimal Control Problem. - Exercises. - Optimal Control for Nonlinear Systems. - Exercises. - Geometric Optimal Control. - Exercises. - Singular Trajectories and Feedback Classification. …
Linear Systems and the Time Optimal Control Problem. - Exercises. - Optimal Control for Nonlinear Systems. - Exercises. - Geometric Optimal Control. - Exercises. - Singular Trajectories and Feedback Classification. - Exercises. - Controllability, Higher Order Maximum Principle, Legendre-Clebsch and Goh Necessary Optimality Conditions. - Exercises. - The Concept of Conjugate Points in the Time Minimal Control Problem for Singular Trajectories, C0-Optimality. - Time Minimal Control of Chemical Batch Reactors and Singular Trajectories. - Generic Properties of Singular Trajectories. - Exercises. - Singular Trajectories in Sub-Riemannian Geometry. - Exercises. - Micro-Local Resolution of the Singularity near a Singular Trajectory, Lagrangian Manifolds and Symplectic Stratifications. - Exercises. - Numerical Computations. - Conclusion and Perspectives. - Exercises. - References. - Index
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 …
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.
The effects of the angiogenic inhibitors endostatin, angiostatin, and TNP-470 on tumor growth dynamics are experimentally and theoretically investigated. On the basis of the data, we pose a quantitative theory …
The effects of the angiogenic inhibitors endostatin, angiostatin, and TNP-470 on tumor growth dynamics are experimentally and theoretically investigated. On the basis of the data, we pose a quantitative theory for tumor growth under angiogenic stimulator/inhibitor control that is both explanatory and clinically implementable. Our analysis offers a ranking of the relative effectiveness of these inhibitors. Additionally, it reveals the existence of an ultimate limitation to tumor size under angiogenic control, where opposing angiogenic stimuli come into dynamic balance, which can be modulated by antiangiogenic therapy. The competitive influences of angiogenically driven growth and inhibition underlying this framework may have ramifications for tissue size regulation in general.
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. …
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.
This paper is a review of mathematical models of control of cell cycle dynamics using external agents which take into account cell-cycle-phase specificity of the drugs. Although still far from …
This paper is a review of mathematical models of control of cell cycle dynamics using external agents which take into account cell-cycle-phase specificity of the drugs. Although still far from being ready for clinical usage the results of these investigations give a new insight into difficulties in introduction of effective strategies in the war against cancer.
We consider a general class of mathematical models P for cancer chemotherapy described as optimal control problems over a fixed horizon with dynamics given by a bilinear system and an …
We consider a general class of mathematical models P for cancer chemotherapy described as optimal control problems over a fixed horizon with dynamics given by a bilinear system and an objective which is linear in the control. Several twoand three-compartment models considered earlier fall into this class. While a killing agent which is active during cell division constitutes the only control considered in the two-compartment model, Model A, also two three-compartment models, Models B and C, are analyzed, which consider a blocking agent and a recruiting agent, respectively. In Model B a blocking agent which slows down cell growth during the synthesis allowing in consequence the synchronization of the neoplastic population is added. In Model C the recruitment of dormant cells from the quiescent phase to enable their efficient treatment by a cytotoxic drug is included. In all models the cumulative effect of the killing agent is used to model the negative effect of the treatment on healthy cells. For each model it is shown that singular controls are not optimal. Then sharp necessary and sufficient optimality conditions for bang-bang controls are given for the general class of models P and illustrated with numerical examples.
Abstract We explore mathematical properties of models of cancer chemotherapy including cell‐cycle dependence. Using the mathematical methods of control theory, we demonstrate two assertions of interest for the biomedical community: …
Abstract We explore mathematical properties of models of cancer chemotherapy including cell‐cycle dependence. Using the mathematical methods of control theory, we demonstrate two assertions of interest for the biomedical community: 1 Periodic chemotherapy protocols are close to the optimum for a wide class of models and have additional favourable properties. 2 Two possible approaches, ( a ) to minimize the final count of malignant cells and the cumulative effect of the drug on normal cells, or ( b ) to maximize the final count of normal cells and the cumulative effect of the drug on malignant cells, lead to similar principles of optimization. From the mathematical viewpoint, the paper provides a catalogue of simplest mathematical models of cell‐cycle dependent chemotherapy. They can be classified based on the number of compartments and types of drug action modelled. In all these models the optimal controls are complicated by the singular and periodic trajectories and multiple solutions. However, efficient numerical methods have been developed. In simpler cases, it is also possible to provide an exhaustive classification of solutions. We also discuss developments in estimation of cell cycle parameters and cell‐cycle dependent drug action.
We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four‐population model that includes tumor cells, host cells, …
We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four‐population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug‐free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor‐drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy regimens. Using optimal control theory with constraints and numerical simulations, we obtain new therapy protocols that we then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations.
We propose a mathematical model for the growth of cell-cycle-specific dose limiting bone marrow. In an attempt to determine effective methods of treatment without overdestruction of the bone marrow we …
We propose a mathematical model for the growth of cell-cycle-specific dose limiting bone marrow. In an attempt to determine effective methods of treatment without overdestruction of the bone marrow we implement optimal control theory. We design the control functional to maximize both the bone marrow mass and the dose over the treatment interval. Next we show that an optimal control exists for this problem, and then we characterize our optimal control in terms of the solutions to the optimality system, which is the state system coupled with the adjoint system. We show that the optimality system is unique for suitably small time intervals. Finally, we analyze the optimal control and the optimality system using numerical techniques. This allows us to suggest optimal methods of treatment that prevent excessive destruction of the bone marrow based on the specific weights in our objective functional.
This paper presents a brief survey of our research in which we have used control theoretic methods in modelling and control of cancer populations. We focus our attention on two …
This paper presents a brief survey of our research in which we have used control theoretic methods in modelling and control of cancer populations. We focus our attention on two classes of problems: optimization of anticancer chemotherapy taking into account both phase specificity and drug resistance, and modelling, and optimization of antiangiogenic therapy. In the case of chemotherapy the control action is directly aimed against the cancer cells while in the case of antiangiogenic therapy it is directed against normal cells building blood vessels and only indirectly it controls cancer growth. We discuss models (both finite and infinite dimensional) which are used to find conditions for tumour eradication and to optimize chemotherapy protocols treating cell cycle as an object of control. In the case of antiangiogenic therapy we follow the line of reasoning presented by Hahnfeldt et al. who proposed to use classical models of self-limiting tumour growth with variable carrying capacity defined by the dynamics of the vascular network induced by the tumour in the process of angiogenesis. In this case antiangiogenic protocols are understood as control strategies and their optimization leads to new recommendations for anticancer therapy.
Mathematical models for cancer chemotherapy as optimal control problems are considered. Results on scheduling optimal therapies when the controls represent the effectiveness of chemotherapeutic agents, or, equivalently, when the simplifying …
Mathematical models for cancer chemotherapy as optimal control problems are considered. Results on scheduling optimal therapies when the controls represent the effectiveness of chemotherapeutic agents, or, equivalently, when the simplifying assumption is made that drugs act instantaneously, are compared with more realistic models that include pharmacokinetic (PK) equations modelling the drug's plasma concentration and various pharmacodynamic (PD) models for the effect the concentrations have on cells.
We consider an optimal control problem of the Mayer- type for a single-input, control affine, nonlinear system in s di- mension. In this paper, we analyze effects that a modeling …
We consider an optimal control problem of the Mayer- type for a single-input, control affine, nonlinear system in s di- mension. In this paper, we analyze effects that a modeling extension has on the optimality of singular controls when the control is re- placed with the output of a first-order, time-invariant linear system driven by a new control. This analysis is motivated by an opti- mal control problem for a novel cancer treatment method, tumor anti-angiogenesis, when such a linear differential equation, which represents the pharmacokinetics of the therapeutic agent, is added to the model. We show that formulas that define a singular control of order 1 and its associated singular arc carry over verbatim under this model extension, albeit with a different interpretation. But the intrinsic order of the singular control increases to 2. As a conse- quence, optimal concatenation sequences with the singular control change and the possibility of optimal chattering arcs arises.
The problem of scheduling a given amount of angiogenic inhibitors is consideredas an optimal control problem with the objective of maximizing the achievable tumor reduction.For a dynamical model for the …
The problem of scheduling a given amount of angiogenic inhibitors is consideredas an optimal control problem with the objective of maximizing the achievable tumor reduction.For a dynamical model for the evolution of the carrying capacity of the vasculature formulatedin [15] optimal controls are computed for both a Gompertzian and logistic model of tumor growth.While optimal controls for the Gompertzian model typically contain a segment along whichthe control is singular, for the logistic model optimal controls are bang-bang with atmost two switchings.
We analyze non cell-cycle specific mathematical models for drug resistance incancer chemotherapy. In each model developing drug resistance is inevitableand the issue is how to prolong its onset. Distinguishing between …
We analyze non cell-cycle specific mathematical models for drug resistance incancer chemotherapy. In each model developing drug resistance is inevitableand the issue is how to prolong its onset. Distinguishing between sensitiveand resistant cells we consider a model which includes interactions of twokilling agents which generate separate resistant populations. We formulate anassociated optimal control problem for chemotherapy and analyze thequalitative structure of corresponding optimal controls.
We perform critical-point analysis for three-variable systems that represent essential processes of the growth of the angiogenic tumor, namely, tumor growth, vascularization, and generation of angiogenic factor (protein) as a …
We perform critical-point analysis for three-variable systems that represent essential processes of the growth of the angiogenic tumor, namely, tumor growth, vascularization, and generation of angiogenic factor (protein) as a function of effective vessel density. Two models that describe tumor growth depending on vascular mass and regulation of new vessel formation through a key angiogenic factor are explored. The first model is formulated in terms of ODEs, while the second assumes delays in this regulation, thus leading to a system of DDEs. In both models, the only nontrivial critical point is always unstable, while one of the trivial critical points is always stable. The models predict unlimited growth, if the initial condition is close enough to the nontrivial critical point, and this growth may be characterized by oscillations in tumor and vascular mass. We suggest that angiogenesis per se does not suffice for explaining the observed stabilization of vascular tumor size.
Cancer research deals with all aspects of malignant transformation, tumour growth and the effects of treatment. Mathematical models enable quantitative representations of the changes affecting cell state and cell number. …
Cancer research deals with all aspects of malignant transformation, tumour growth and the effects of treatment. Mathematical models enable quantitative representations of the changes affecting cell state and cell number. This book provides a review of the scope of mathematical modelling in cancer research, bringing together for the first time a group of related mathematical topics including multistage carcinogenesis, tumour growth kinetics, growth control, radiotherapy, chemotherapy and biological targeting in cancer treatment. Physicists and mathematicians interested in medical research, biomathematicians, biostatisticians, radiation and medical oncologists and experimental and theoretical biologists will welcome this critical review of mathematical modelling in cancer research. This book will also be of interest to clinicians, basic cancer scientists and physicists working in radiotherapy departments, and to postgraduate students on courses in oncology and subjects.
This paper derives the conditions guaranteeing that a singular extremal that joins fixed endpoints provides a strong minimum for the independent time variable. For the nonsingular case, Weierstrass has shown …
This paper derives the conditions guaranteeing that a singular extremal that joins fixed endpoints provides a strong minimum for the independent time variable. For the nonsingular case, Weierstrass has shown that the extremal must be embedded in a field. The principal conditions that imply the existence of a field for a nonsingular problem with an n-dimensional state vector ${\bf x}$ and a scalar control variable u are that ${{\partial ^2 H} / {\partial u^2 }}$ is unequal to zero and that the $n \times (n + 1)$ matrix $[{{\partial {\bf x}(t)} / {\partial \lambda (t_0 )}},\dot {\bf x}(t)]$ has rank n. Here t is the time, H is the generalized Hamiltonian, and $\lambda $ is the adjoint vector. This paper shows that under proper assumptions the field concept can be extended to the singular case. The condition on ${{\partial ^2 H} / {\partial u^2 }}$ is replaced by \[ \frac{\partial } {{\partial u}}\frac{{d^2 }} {{dt^2}}\frac{{\partial H}} {{\partial u}} \ne 0. \] The above matrix, whose first n columns are obtained in a uniform manner, is replaced by a matrix whose vectors are obtained by diverse procedures. Weirstrass' analysis is carried out in detail, using a singular nominal extremal and its field.
An optimal control problem for combination of cancer chemotherapy withimmunotherapy in form of a boost to the immune system is considered as amulti-input optimal control problem. The objective to be …
An optimal control problem for combination of cancer chemotherapy withimmunotherapy in form of a boost to the immune system is considered as amulti-input optimal control problem. The objective to be minimized is chosenas a weighted average of (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), the overall amounts of (iii) cytotoxic agentsand (iv) immune boost given as a measure for the side effects oftreatment and (v) a small penalty on the free terminal time that limits theoverall therapy horizon. This last term is essential in obtaining a mathematically well-posedproblem formulation. Both analytical and numerical results about thestructures of optimal controls will be presented that give some insightsinto the structure of optimal protocols, i.e., the dose rates and sequencing ofdrugs in these combination treatments.
In this paper we give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we …
In this paper we give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we consider the C0 topology in the state space. The controls appear linearly and take values in a polyhedron, and the state space and the end point constraints are finite-dimensional smooth manifolds. In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial, and hence the usual second variation, which is defined on the kernel of the first one, does not give any information. We consider the finite-dimensional subproblem generated by perturbing the switching times, and we prove that the sufficient second order optimality conditions for this finite-dimensional subproblem yield local strong optimality. We give an explicit algorithm to check the positivity of the second variation which is based on the properties of the Hamiltonian fields.
The administration schedule appears to be a particularly relevant factor in determining the effectiveness of an antiangiogenic drug. A better quantitative knowledge of the interactions between tumour growth and the …
The administration schedule appears to be a particularly relevant factor in determining the effectiveness of an antiangiogenic drug. A better quantitative knowledge of the interactions between tumour growth and the development of its vasculature could help to design effective therapies.Biological and clinical inferences were derived from the analysis of a mathematical model proposed by Hahnfeldt et al. (1999), and some of its variants. In particular, we compared the effect of constant continuous infusion of an anti-angiogenic drug that induces vascular loss, to the effect of periodic, bolus-based therapy.The role of drug elimination rate and of dose fractionation was investigated, and we show that different schedulings, guaranteeing the same mean value of drug concentration, may exhibit very different long-term responses according to their concentration vs. time profile. For a large class of tumour growth laws, the profiles that approach the constant one are the most effective. This behaviour appears to depend on the 'cooperativity' of the tumour-vasculature interaction and on the functional form of the relationship between tumour growth and vasculature extent. Moreover, we suggest that a therapy approaching constant drug infusion might be advantageous also in the case of cytostatic anti-angiogenic drugs.
Anti-angiogenesis is a novel cancer treatment that targets thevasculature of a growing tumor. In this paper a metasystem isformulated and analyzed that describes the dynamics of the primarytumor volume and …
Anti-angiogenesis is a novel cancer treatment that targets thevasculature of a growing tumor. In this paper a metasystem isformulated and analyzed that describes the dynamics of the primarytumor volume and its vascular support under anti-angiogenictreatment. The system is based on a biologically validated model byHahnfeldt et al. and encompasses several versions of this modelconsidered in the literature. The problem how to schedule an apriori given amount of angiogenic inhibitors in order to achieve themaximum tumor reduction possible is formulated as an optimal controlproblem with the dosage of inhibitors playing the role of thecontrol. It is investigated how properties of the functions definingthe growth of the tumor and the vasculature in the general systemaffect the qualitative structure of the solution of the problem. Inparticular, the presence and optimality of singular controls isdetermined for various special cases. If optimal, singular arcs arethe central part of a regular synthesis of optimal trajectoriesproviding a full solution to the problem. Two specific examples of aregular synthesis including optimal singular arcs are given.
We analyze the effect which a fold and simple cusp singularity in the flow of a parametrized family of extremal trajectories of an optimal control problem has on the corresponding …
We analyze the effect which a fold and simple cusp singularity in the flow of a parametrized family of extremal trajectories of an optimal control problem has on the corresponding parametrized cost or value function. A fold singularity in the flow of extremals generates an edge of regression of the value implying the well-known results that trajectories stay strongly locally optimal until the fold-locus is reached, but lose optimality beyond. Thus fold points correspond to conjugate points. A simple cusp point in the parametrized flow of extremals generates a swallow-tail in the parametrized value. More specifically, there exists a region in the state space which is covered 3:1 with both locally minimizing and maximizing branches. The changes from the locally minimizing to the maximizing branch occur at the fold-loci and there trajectories lose strong local optimality. However, the branches intersect and generate a cut-locus which limits the optimality of close-by trajectories and eliminates these trajectories from optimality near the cusp point prior to the conjugate point. In the language of partial differential equations, a simple cusp point generates a shock in the solutions to the Hamilton--Jacobi--Bellman equation while fold points will not be part of the synthesis of optimal controls near the simple cusp point.
We deal with a biophysical description of antitumor antiangiogenic therapies. In particular, by means of some simple models, we study the possible effects of the delay between the drug consumption …
We deal with a biophysical description of antitumor antiangiogenic therapies. In particular, by means of some simple models, we study the possible effects of the delay between the drug consumption by endothelial cells and their death on the outcome of the therapy. We have found that this time lag implies an increase in the minimal dose guaranteeing tumor eradication and, if the delay is greater than a meaningful threshold, it may preclude the total regression. These results might be of interest in better understanding the causes underlying the contradictory literature on the clinical trials of antiangiogenic therapies.