The outbreak of the Covid-19 pandemic has forced governments to impose restrictions on the individual liberty of people. Such containment measures have considerably reduced the number of infections but have …
The outbreak of the Covid-19 pandemic has forced governments to impose restrictions on the individual liberty of people. Such containment measures have considerably reduced the number of infections but have also caused substantial damage. In this context the following main issue arises: which policy is the best to contain fatalities and economic losses complying with the intensive care units capacity? This issue is investigated through the study of an optimal control problem based on a SEAIRD epidemic model referring to Covid-19. A state constraint is imposed on the number of infected individuals in order to maintain the infectious level under the health-facilities capacity threshold. The challenge is to find a control function that minimizes the total cost which represents a trade-off between economic losses and human deaths. After showing the existence of an optimal solution, the necessary optimality conditions provided by Pontryagin Minimum Principle are derived. Numerical solutions are obtained by discretizing the optimal control problem and applying nonlinear optimization methods. Various scenarios with different initial conditions representing different degrees of infection are studied and the solutions are compared.The COVID-19 control problem treated here may also serve as a prototypical example for solving an epidemiological control model with state constraints.
Abstract Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a …
Abstract Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.
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.
Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging …
Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.
In this paper a two-phase pandemic-economic model is proposed, with phase-specific modeling and policy variables – as suggested by the chronicle of pandemic and economic policy developments over the period …
In this paper a two-phase pandemic-economic model is proposed, with phase-specific modeling and policy variables – as suggested by the chronicle of pandemic and economic policy developments over the period 2020–2021. In a first phase, the spread of a pandemic disease is the primary concern of authorities, that still also pay attention to economic activity. A dynamic model is introduced, embedding a two-way interaction between an extended epidemic Susceptible-Infected-Recovered (SIR) model and output gap dynamics. In the second phase, posterior to lockdowns when waves fade away, monetary policy becomes the control variable, pursuing again a joint objective, of supporting a non-inflationary recovery without causing significant fatalities. We then use a standard stylized model for the macroeconomy with simplified infection dynamics, that also enter the policy objective. The two phases are thus studied in a regime change model where the control and state variables as well as the objective function are allowed to change across phases. We solve the model over a finite horizon and derive the optimal lockdown or monetary policy path that jointly minimizes pandemic and economic losses. The two-phase finite horizon decision model is empirically calibrated and numerically solved through AMPL, a new solution method for finite horizon dynamic models. In the first phase, albeit with lasting adverse effects on output, lockdown-based control can be effective in reducing infection rates, but less so when starting from a negative output gap. In the second phase, accommodative monetary policy appears to be effective on both fronts, with even an eventual need for a return to tightening as output gap closes and inflation resumes.
We improve a recent mathematical model for cholera by adding a time delay that represents the time between the instant at which an individual becomes infected and the instant at …
We improve a recent mathematical model for cholera by adding a time delay that represents the time between the instant at which an individual becomes infected and the instant at which he begins to have symptoms of cholera disease. We prove that the delayed cholera model is biologically meaningful and analyze the local asymptotic stability of the equilibrium points for positive time delays. An optimal control problem is proposed and analyzed, where the goal is to obtain optimal treatment strategies, through quarantine, that minimize the number of infective individuals and the bacterial concentration, as well as treatment costs. Necessary optimality conditions are applied to the delayed optimal control problem, with a $L^1$ type cost functional. We show that the delayed cholera model fits better the cholera outbreak that occurred in the Department of Artibonite -- Haiti, from 1 November 2010 to 1 May 2011, than the non-delayed model. Considering the data of the cholera outbreak in Haiti, we solve numerically the delayed optimal control problem and propose solutions for the outbreak control and eradication.
There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this …
There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [21] describing the tumour-immune response to a chemo-immuno-therapy. Assuming L¹-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [7]. For L¹-type objectives we obtain extremal controls of bang-bang type.
We propose a model for the human immunodeficiency virus type 1 (HIV-1) infection with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a …
We propose a model for the human immunodeficiency virus type 1 (HIV-1) infection with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a control function representing the efficiency of reverse transcriptase inhibitors and consider the pharmacological delay associated to the control. Finally, we propose and analyze an optimal control problem with state and control delays. Through numerical simulations, extremal solutions are proposed for minimization of the virus concentration and treatment costs.
We study a model based on the so called SIR model to control the spreading of a disease in a varying population via vaccination and treatment. Since we assume that …
We study a model based on the so called SIR model to control the spreading of a disease in a varying population via vaccination and treatment. Since we assume that medical treatment is not immediate we add a new compartment, $M$, to the SIR model. We work with the normalized version of the proposed model. For such model we consider the problem of steering the system to a specified target. We consider both a fixed time optimal control problem with $L^1$ cost and the minimum time problem to drive the system to the target. In contrast to the literature, we apply different techniques of optimal control to our problems of interest.Using the direct method, we first solve the fixed time problem and then proceed to validate the computed solutions using both necessary conditions and second order sufficient conditions. Noteworthy, we perform a sensitivity analysis of the solutions with respect to some parameters in the model. We also use the Hamiltonian Jacobi approach to study how the minimum time function varies with respect to perturbations of the initial conditions. Additionally, we consider a multi-objective approach to study the trade off between the minimum time and the social costs of the control of diseases. Finally, we propose the application of Model Predictive Control to deal with uncertainties of the model.
We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease …
We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied. Although it is well-known that there is a delay between two to eight weeks between TB infection and reaction of body's immune system to tuberculin, delays for the active infected to be detected and treated, and delays on the treatment of persistent latent individuals due to clinical and patient reasons, which clearly justifies the introduction of time delays on state and control measures, our work seems to be the first to consider such time-delays for TB and apply time-delay optimal control to carry out the optimality analysis.
We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We …
We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.
We discuss the optimization of chemotherapy treatment for low-grade gliomas using a mathematical model. We analyze the dynamics of the model and study the stability of solutions. The dynamical model …
We discuss the optimization of chemotherapy treatment for low-grade gliomas using a mathematical model. We analyze the dynamics of the model and study the stability of solutions. The dynamical model is incorporated into an optimal control problem for which different objective functionals are considered. We establish the existence of optimal controls and give a detailed discussion of the necessary optimality conditions. Since the control variable appears linearly in the control problem, optimal controls are concatenations of bang-bang and singular arcs.We derive a formula of the singular control in terms of state and adjoint variables.Using discretization and optimization methods we compute optimal drug protocols in a number of scenarios.For small treatment periods, the optimal control is bang-bang, whereas for larger treatment periods we obtain both bang-bang and singular arcs. In particular, singular controls illustrate the metronomic chemotherapy.
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.
Optimal control is an important tool to determine vaccination poli-cies for infectious diseases. For diseases transmitted horizontally, SEIR com-partment models have been used. Most of the literature on SEIR models …
Optimal control is an important tool to determine vaccination poli-cies for infectious diseases. For diseases transmitted horizontally, SEIR com-partment models have been used. Most of the literature on SEIR models deals with cost functions that are quadratic with respect to the control variable, the rate of vaccination. In this paper, we consider L 1 –type objectives that are linear with respect to the control variable. Various pure control, mixed control–state and pure state constraints are imposed. For all constraints, we discuss the necessary optimality conditions of the Maximum Principle and determine optimal control strategies that satisfy the necessary optimality con-ditions with high accuracy. Since the control variable appears linearly in the Hamiltonian, the optimal control is a concatenation of bang-bang arcs, singu-lar arcs and boundary arcs. For pure bang-bang controls, we are able to check second-order sufficient conditions.
We survey the results on no-gap second order optimality conditions (both necessary and sufficient) in the Calculus of Variations and Optimal Control, that were obtained in the monographs [31] and …
We survey the results on no-gap second order optimality conditions (both necessary and sufficient) in the Calculus of Variations and Optimal Control, that were obtained in the monographs [31] and [40], and discuss their further develop-ment. First, we formulate such conditions for broken extremals in the simplest prob-lem of the Calculus of Variations and then, we consider them for discontinuous con-trols in optimal control problems with endpoint and mixed state-control constraints, considered on a variable time interval. Further, we discuss such conditions for bang-bang controls in optimal control problems, where the control appears linearly in the Pontryagin-Hamilton function with control constraints given in the form of a con-vex polyhedron. Bang-bang controls induce an optimization problem with respect to the switching times of the control, the so-called Induced Optimization Problem. We show that second-order sufficient condition for the Induced Optimization Problem together with the so-called strict bang-bang property ensure second-order sufficient conditions for the bang-bang control problem. Finally, we discuss optimal control problems with mixed control-state constraints and control appearing linearly. Tak-ing the mixed constraint as a new control variable we convert such problems to bang-bang control problems. The numerical verification of second-order conditions is illustrated on three examples.
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 …
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.
A mathematical model for the scheduling of a combination of anti-angiogenic and chemotherapeutic agents is considered as a multi-input optimal control problem. Numerical results that are based on an explicit …
A mathematical model for the scheduling of a combination of anti-angiogenic and chemotherapeutic agents is considered as a multi-input optimal control problem. Numerical results that are based on an explicit equation for a singular control confirm as optimal a structure of protocols that administer the anti-angiogenic agent according to the optimal monotherapy control and the cytotoxic agent at maximum dose at the end of therapy.
Second order sufficient conditions (SSC) for control problems with control-state constraints and free final time are presented. Instead of deriving such SSC from first principles, we transform the control problem …
Second order sufficient conditions (SSC) for control problems with control-state constraints and free final time are presented. Instead of deriving such SSC from first principles, we transform the control problem with free final time into an augmented control problem with fixed final time for which well-known SSC exist. SSC are then expressed as a condition on the positive definiteness of the second variation. A convenient numerical tool for verifying this condition is based on the Riccati approach, where one has to find a bounded solution of an associated Riccati equation satisfying specific boundary conditions. The augmented Riccati equations for the augmented control problem are derived, and their modifications on the boundary of the control-state constraint are discussed. Two numerical examples, (1) the classical Earth-Mars orbit transfer in minimal time and (2) the Rayleigh problem in electrical engineering, demonstrate that the Riccati equation approach provides a viable numerical test of SSC.
Perturbed nonlinear control problems with data depending on a vector parameter are considered. Using second-order sufficient optimality conditions, it is shown that the optimal solution and the adjoint multipliers are …
Perturbed nonlinear control problems with data depending on a vector parameter are considered. Using second-order sufficient optimality conditions, it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof exploits the close connections between solutions of a Riccati differential equation and shooting methods for solving the associated boundary value problem. Solution differentiability provides a firm theoretical basis for numerical feedback schemes that have been developed for computing neighbouring extremals. The results are illustrated by an example that admits two extremal solutions. Second-order sufficient conditions single out one optimal solution for which a sensitivity analysis is carried out.
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.
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
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.
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 …
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.
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.
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.
In [13] a new sufficiency criterion for strong local minimality in multidimensional non-convex control problems with pure state constraint was developed. In this paper we use a similar method to …
In [13] a new sufficiency criterion for strong local minimality in multidimensional non-convex control problems with pure state constraint was developed. In this paper we use a similar method to obtain sufficient conditions for weak local minimality in multidimensional control problems with mixed state-control restrictions. The result is obtained by applying duality theory for control problems of Klötzler [11] as well as first and second order optimality conditions for optimization problems described by C^1 -functions having a locally Lipschitzian gradient mapping. The main theorem contains the result of Zeidan [17] for one-dimensional problems withoutstate restrictions.
Abstract A numerical method is developed for the real‐time computation of neighbouring optimal feedback controls for constrained optimal control problems. The first part of this paper presents the theory of …
Abstract A numerical method is developed for the real‐time computation of neighbouring optimal feedback controls for constrained optimal control problems. The first part of this paper presents the theory of neighbouring extremals. Besides a survey of the theory of neighbouring extremals, special emphasis is laid on the inclusion of complex constraints, e.g. state and control variable inequality constraints and discontinuities of the system equations at interior points. The numerical treatment of these constraints is particularly emphasized. The linearization of all necessary conditions of optimal control theory leads to a linear, mulitpoint, boundary value problem with linear jump conditions that is especially well suited for numerical treatment.
Necessary conditions for the switching function, holding at junction points of optimal interior and boundary arcs or at contact points with the boundary, are given. These conditions are used to …
Necessary conditions for the switching function, holding at junction points of optimal interior and boundary arcs or at contact points with the boundary, are given. These conditions are used to derive necessary conditions for the optimality of junctions between interior and boundary arcs. The junction theorems obtained are similar to those developed for singular control problems in [1] and establish a duality between singular control problems and control problems with bounded state variables and control appearing linearly. The transition from unconstrained to constrained extremals is discussed with respect to the order p of the state constraint. A numerical example is given where the adjoins variables are not unique but form a convex set which is determined numerically.
Journal Article Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment Get access Urszula Ledzewicz, Urszula Ledzewicz † Department of Mathematics and Statistics, Southern Illinois University …
Journal Article Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment Get access Urszula Ledzewicz, Urszula Ledzewicz † Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA †Corresponding author. Email: [email protected] Search for other works by this author on: Oxford Academic Google Scholar John Marriott, John Marriott Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA Search for other works by this author on: Oxford Academic Google Scholar Helmut Maurer, Helmut Maurer Department of Mathematics and Statistics, Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms Universität Münster, Einsteinstrasse 62, D-48149 Münster, Germany Search for other works by this author on: Oxford Academic Google Scholar Heinz Schättler Heinz Schättler Department of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130-4899, USA Search for other works by this author on: Oxford Academic Google Scholar Mathematical Medicine and Biology: A Journal of the IMA, Volume 27, Issue 2, June 2010, Pages 157–179, https://doi.org/10.1093/imammb/dqp012 Published: 01 June 2010 Article history Received: 23 October 2008 Revision received: 06 February 2009 Accepted: 27 April 2009 Published: 01 June 2010
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.
A model of antiangiogenic cancer chemotherapy is proposed and analyzed. The model is a modified Hahnfeldt model and the analysis has two goals: first of all we check stability of …
A model of antiangiogenic cancer chemotherapy is proposed and analyzed. The model is a modified Hahnfeldt model and the analysis has two goals: first of all we check stability of the equilibrium point of the model to find conditions leading to tumour eradication then we propose an optimization problem and give necessary conditions of its solution
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.
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.
In this paper we propose a class of models that describe the mutual interaction between tumour growth and the development of tumour vasculature and that generalize existing models. The study …
In this paper we propose a class of models that describe the mutual interaction between tumour growth and the development of tumour vasculature and that generalize existing models. The study is mainly focused on the effect of a therapy that induces tumour vessel loss (anti-angiogenic therapy), with the aim of finding conditions that asymptotically guarantee the eradication of the disease under constant infusion or periodic administration of the drug. Furthermore, if tumour and/or vessel dynamics exhibit time delays, we derive conditions for the existence of Hopf bifurcations. The destabilizing effect of delays on achieving the tumour eradication is also investigated. Finally, global conditions for stability and eradication in the presence of delays are given for some particular cases.
This paper serves as an introduction to the theory of optimal control applied to systems of ordinary differential equations with emphasis on disease models. We outline the steps in formulating …
This paper serves as an introduction to the theory of optimal control applied to systems of ordinary differential equations with emphasis on disease models. We outline the steps in formulating an optimal control problem and derive necessary conditions. Several simple examples provide detailed methodology in characterizing the optimal control through use of Pontryagin’s Maximum Principle. An SEIR (Susceptible, Exposed, Infected, Recovered) model with control acting as a rate of vaccination is presented and an optimal control problem is formulated to include an isoperimetric constraint on the vaccine supply. Numerical results illustrate how such a constraint alters the optimal vaccination schedule and its effect on the population.
Optimal control can be of help to test and compare different vaccination strategies of a certain disease.In this paper we propose the introduction ofconstraints involving state variables on an optimal …
Optimal control can be of help to test and compare different vaccination strategies of a certain disease.In this paper we propose the introduction ofconstraints involving state variables on an optimal control problem applied to a compartmental SEIR (Susceptible. Exposed, Infectious and Recovered) model. We study the solution of such problems when mixed state control constraints are used to impose upper bounds on the available vaccines at each instant of time. We also explore the possibility of imposing upper bounds on the number of susceptible individuals with and without limitations on the number of vaccines available. In the case of mere mixed constraints a numerical and analytical study is conducted while in the other two situations only numerical results are presented.
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.
We present sufficient conditions of local controllability for a class of models of treatment response to combined anticancer therapies which include delays in control strategies. The combined therapy is understood …
We present sufficient conditions of local controllability for a class of models of treatment response to combined anticancer therapies which include delays in control strategies. The combined therapy is understood as combination of direct anticancer strategy e.g. chemotherapy and indirect modality (in this case antiangiogenic therapy). Controllability of the models in the form of semilinear second order dynamic systems with delays in control enables to answer the questions of realizability of different objectives of multimodal therapy in the presence of PK/PD effects. We compare results for the models without delays and conditions for relative local controllability of models with delays.
Several simple ordinary differential equation (ODE) models of tumor growth taking into account the development of its vascular network are discussed. Different biological aspects are considered from the simplest model …
Several simple ordinary differential equation (ODE) models of tumor growth taking into account the development of its vascular network are discussed. Different biological aspects are considered from the simplest model of Hahnfeldt et al. proposed in 1999 to a model which includes drug resistance of cancer cells to chemotherapy. Some of these models can be used in clinical oncology to optimize antiangiogenic and cytostatic drugs delivery so as to ensure maximum efficacy. Simple models of continuous and periodic protocols of combined therapy are implemented. Discussion on the dynamics of the models and their complexity is presented.
Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number …
Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number R are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.
We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We …
We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.
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.
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.