Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor

A mathematical model of tumor cell population dynamics is considered. The tumor is assumed to consist of cells of two types: amenable and resistant to chemotherapeutic treatment. It is assumed that the growth of the cell populations of both types is governed by logistic equations. The effect of a chemotherapeutic drug on the tumor is specified by a therapy function. Two types of therapy functions are considered: a monotonically increasing function and a nonmonotone one with a threshold. In the former case, the effect of a drug on the tumor is stronger at a higher drug concentration. In the latter case, a threshold drug concentration exists above which the effect of the therapy reduces. The case when the total drug amount is subject to an integral constraint is also studied. A similar problem was previously studied in the case of a linear therapy function with no constraint imposed on the drug amount. By applying the Pontryagin maximum principle, necessary optimality conditions are found, which are used to draw important conclusions about the character of the optimal therapy strategy. The optimal control problem of minimizing the total number of tumor cells is solved numerically in the case of a monotone or threshold therapy function with allowance for the integral constraint on the drug amount.     

On strategies on a mathematical model for leukemia therapy

A mathematical model for leukemia therapy based on the Gompertzian law of cell growth is studied. It is assumed that the chemotherapeutic agents kill leukemic as well as normal cells.Effectiveness of the medicine is described in terms of a therapy function. Two types of therapy functions are considered: monotonic and non-monotonic. In the former case the level of the effect of the chemotherapy directly depends on the quantity of the chemotherapeutic agent. In the latter case the therapy function achieves its peak at a threshold value and then the effect of the therapy decreases. At any given moment the amount of the applied chemotherapeutic is regulated by a control function with a bounded maximum. Additionally, the total quantity of chemotherapeutic agent which can be used during the treatment process is bounded too.The problem is to find an optimal strategy of treatment to minimize the number of leukemic cells while at the same time retaining as many normal cells as possible.With the help of Pontryagin’s Maximum Principle it was proved that the optimal control function has at most one switch point in both monotonic and non-monotonic cases for most relevant parameter values.A control strategy called alternative is suggested. This strategy involves increasing the amount of the chemotherapeutical medicine up to a certain value within the shortest possible period of time, and holding this level until the end of the treatment.The comparison of the results from the numerical calculation using the Pontryagin’s Maximum Principle with the alternative control strategy shows that the difference between the values of cost functions is negligibly small.     

Multi-input Optimal Control Problems for Combined Tumor Anti-angiogenic and Radiotherapy Treatments

A cell-population-based model for tumor growth under anti-angiogenic treatment, with the tumor volume and its variable carrying capacity as variables, is combined with the linear-quadratic model for damage done by radiation ionization. The resulting multi-input system is analyzed as an optimal control problem with the objective of minimizing the tumor volume subject to isoperimetric constraints that limit the overall amounts of anti-angiogenic agents, respectively, the damage done to healthy tissue by radiotherapy. For various model formulations, explicit expressions for singular controls are derived for both the dosage of the anti-angiogenic therapeutic agent and the radiation dose schedule. Their role in the structure of optimal protocols is discussed.     

Optimal replacement policies for two-unit machines with increasing running costs 1

A machine consists of two stochastically failing units. Failure of either of the units causes a failure of the machine and the failed unit has to be replaced immediately. Associated with the units are running costs which increase with the age of the unit because of increasing maintenance costs, decreasing output, etc.A preventive replacement policy is proposed under which, at failure points, we also replace the second unit if its age exceeds a predetermined control limit. It is proved that, for two identical units with exponential life-time distributions and linear running costs, this policy is optimal and the optimal control limit is calculated. In an additional model we take into consideration the length of time it takes to replace one unit or both units.The method of solution is a variation of dynamic semi-Markov programming. Analytical results are obtained and the influence of the various parameters on them is investigated. Finally, we study the saving due to our policy in comparison with a policy in which only failed units are replaced.     

Optimal Control for a Mathematical Model of Glioma Treatment with Oncolytic Therapy and TNF-$$\alpha $$ Inhibitors

A mathematical model for combination therapy of glioma with oncolytic therapy and TNF-\(\alpha \) inhibitors is analyzed as an optimal control problem. In the objective, a weighted average between the tumor volume and the total amount of viruses given is minimized. It is shown that optimal controls representing the virus administration are generically of the bang-bang type, i.e., the virus should be applied at maximal allowed dose with possible rest periods. On the other hand, optimal controls representing the dosage of TNF-\(\alpha \) inhibitors follow a continuous regimen of concatenations between pieces that lie on the boundary and in the interior of the control set.     

Solution of an optimal control problem with phase constraints by the double variation method

An optimal control problem with an integral quality index specified in a finite time interval is formulated for a model of economic growth that leads to emission of greenhouse gases. The controlled system is linear with respect to control. The problem contains phase constraints that abandon emission of greenhouse gases above some predefined time-dependent limit. As is known, optimal control problems with phase constraints fall beyond the sphere of efficient application of the Pontryagin maximum principle because, for such problems, this principle is formulated in a complicated form difficult for analytic treatment in particular situations. In this study, the analytic structure of the optimal control and phase trajectories is constructed using the double variation method.     

Time-cost trade-off via optimal control theory in Markov PERT networks

We develop a new analytical model for the time-cost trade-off problem via optimal control theory in Markov PERT networks. It is assumed that the activity durations are independent random variables with generalized Erlang distributions, in which the mean duration of each activity is a non-increasing function of the amount of resource allocated to it. Then, we construct a multi-objective optimal control problem, in which the first objective is the minimization of the total direct costs of the project, in which the direct cost of each activity is a non-decreasing function of the resources allocated to it, the second objective is the minimization of the mean of project completion time and the third objective is the minimization of the variance of project completion time. Finally, two multi-objective decision techniques, viz, goal attainment and goal programming are applied to solve this multi-objective optimal control problem and obtain the optimal resources allocated to the activities or the control vector of the problem     

Homogenization of Optimal Control Problems for Functional Differential Equations

The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence.     

A replacement model for an additive damage model with restoration

A system existing in a random environment receives shocks at random points of time. Each shock causes a random amount of damage which accumulates over time. A breakdown can occur only upon the occurrence of a shock according to a known failure probability function. Upon failure the system is replaced by a new identical one with a given cost. When the system is replaced before failure, a smaller cost is incurred. Thus, there is an incentive to attempt to replace the system before failure. The damage process is controlled by means of a maintenance policy which causes the accumulated damage to decrease at a known restoration rate. We introduce sufficient conditions under which an optimal replacement policy which minimizes the total expected discounted cost is a control limit policy. The relationship between the undiscounted case and the discounted case is examined. Finally, an example is given illustrating computational procedures.     

Optimal control of assignment of jobs to processors under heavy traffic

The paper is concerned with the optimal control of the assignment of jobs from several arriving random streams to one of a bank of processors. Owing to the difficulty of the general problem, a heavy traffic approach is used. The required work depends on the processor to which it is assigned. The information that the assignment can be based on is quite flexible, and several information structures (data on which the control is based) are considered. The assignment can be made on arrival or when the job is to be processed. There can be bursty arrivals (the bursts depending on randomly varying environmental factors), rather general nonlinear cost functions and other complications. It is shown, under reasonably general conditions, that the optimal costs for the physical systems converge to the optimal cost for the heavy traffic limit problem, as the heavy traffic parameter goes to its limit. Numerical data is presented to illustrate some of the potential uses of the limit process for obtaining optimal contros, or controls satisfying optimal tradeoffs among competing criteria. The methods of proof are quite powerful tools for such optimal control problems     

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.     

最速反馈控制的不变性

变结构控制对系统模型和扰动具有一定的不变性是众所周知的事实。最速反馈控制是以其开关曲线为滑动曲线的变结构控制。本文用变结构控制理论来讨论修正了的最速反馈控制对一定范围的系统扰动具有完全的不变性,即完全能够抑制一定范围的扰动作用,而且闭环系统的所有轨线,在理论上,都以有限时间到达原点。这就为设计高效非线性反馈提供了一条有效途径,还给出了避免高频颤震来实现最速反馈控制的数字化办法。     

Limit cycles in an optimal control problem of diabetes

This article examines the behaviour of an individual diagnosed with diabetes. It is shown that the medical treatment of the disease creates incentives that make a diabetic's consumption, weight, and labour supply display cyclical patterns. The existence of a limit cycle is proved using an adaptation of the Hopf bifurcation theorem for optimal control problems.     

Admission control of a service station when the arrivals are internal

All studies in the admission control of a service station make decisions at arrival epochs. When arrivals are internal and are rejected from a queue, the rejected jobs have to be routed to other stations in the system. However the system will not know whether a job will be admitted to a queue or not until its arrival epoch to that queue. Thus, the system has to react dynamically and agilely to the decisions made at a specific queue and may try several queues before finding a queue that admits the job. This paper remedies these difficulties by changing the decision epochs of the admission control from arrival epochs to departure epochs with the actions of switching (keeping) the arrival stream on or off. Thus upstream stations will have information on the admission status of their downstream stations all the time. It is proved that the optimal policy for this revised admission control system is of control limit type for an M/G/1 queue. Comparisons of the optimal values and optimal policies for the admission controls made at arrival epochs and at departure epochs are included in the paper.     

A model for speculation in a dynamic economy

The paper deals with speculation strategies in a dynamic economy, where “speculation” means participating in a market with the intention to gain a reward by first buying an item and thereafter selling it at a possibly higher price. By assuming that the states of the economy form a Markov chain the problem is modeled as a discrete time Markov decision process. The optimal strategies (which are pairs of stopping times) are identified. Under quite general conditions the optimal rule for the selling process turns out to be a control limit policy in both state of economy and time. Techniques for the computation of optimal strategies are presented; some numerical examples are also discussed. For a static economy closed-form solutions are given     

Heavy traffic analysis of controlled multiplexing systems

The paper develops the mathematics of the heavy traffic approach to the control and optimal control problem for multiplexing systems, where there are many mutually independent sources which feed into a single channel via a multiplexer (or of networks composed of such subsystems). Due to the widely varying bit rates over all sources, control over admission, bandwidth, etc., is needed to assure good performance. Optimal control and heavy traffic analysis has been shown to yield systems with greatly improved performance. Indeed, the heavy traffic approach covers many cases of great current interest, and provides a useful and practical approach to problems of analysis and control arising in modern high speed telecommunications. Past works on the heavy traffic approach to the multiplexing problem concentrated on the uncontrolled system or on the use of the heavy traffic limit control problem for applications, and did not provide details of the proofs. This is done in the current paper. The basic control problem for the physical system is hard, and the heavy traffic approach provides much simplification. Owing to the presence of the control, as well as to the fact that the cost function of main interest is “ergodic”, the problem cannot be fully treated with “classical” methods of heavy traffic analysis for queueing networks. A basic result is that the optimal average costs per unit time for the physical problem converge to the optimal cost per unit time for the limit stationary process as the number of sources and the time interval goes to infinity. This convergence is both in the mean and pathwise senses. Furthermore, a “nice” nearly optimal control for the limit system provides nearly optimal values for the physical system, under heavy traffic, in both a mean and pathwise sense. This revised version was published online in June 2006 with corrections to the Cover Date.     

Oligopoly with Hyperbolic Demand: A Differential Game Approach

Convex demand functions, although commonly used in consumer theory and in accordance with a large amount of empirical evidence, are known to be problematic in the analysis of firms’ behavior; therefore, they are rarely used in oligopoly theory, due to the possible lack of concavity of the firms’ profit functions and the indeterminacy arising in the limit as marginal costs tend to zero. We investigate a dynamic oligopoly model with hyperbolic demand and sticky price, characterizing the open-loop optimal control and the related steady-state equilibrium, to show that the indeterminacy associated with the limit of the static model is indeed confined to the steady state of the dynamic model, while the latter allows for a well-behaved solution at any time during the game. Although the feedback solution cannot be analytically attained since the model is not built in linear-quadratic form, we show that analogous considerations also apply to the Bellman equation of the individual firm.     

Optimal control of a dam under seasonal electricity prices

This study concerns the optimal control of a hydroelectric dam under seasonal electricity prices: high in winter, low in summer. The goal is to maximize the expected discounted infinite-horizon return through a policy that determines the amount of electricity to be produced in each time period, depending on the water level at the beginning of the period under consideration. The prices are assumed to be deterministic, and the flows into the reservoir are seasonal, stochastic, but independent from one period to another. The electric power generated is proportional to the amount of water flow through the turbines. There exist seepage and evaporation losses.It is shown that in the simplest price structure, the optimal policy is entirely determined by a single critical water level in each period of time, at which one starts producing. An example shows that the discretization of the reservoir levels can destroy this property. A method is proposed to avoid this difficulty. Another way of defining a policy is through goal-levels. This approach is shown to give higher returns than the standard approach.     

Optimal Control Problem for Cancer Invasion Reaction–Diffusion System

Abstract

An optimal control problem constrained by a reaction–diffusion mathematical model which incorporates the cancer invasion and its treatment is considered. The state equations consisting of three unknown variables namely tumor cell density, normal cell density, and drug concentration. The main goal of the considered optimal control problem is to minimize the density of cancer cells and decreasing the side effects of treatment. Moreover, existence of a weak solution of brain tumor reaction–diffusion system and the corresponding adjoint system of optimal control problem is also investigated. Further, existence of minimizer for the optimal control problem is established and also the first-order optimality conditions are derived.     

复合Poisson 模型带比例与固定交易费用的最优分红与注资

研究了复合Poisson 模型带比例与固定费用的最优分红与注资问题. 每次分红与注资时, 存在比例及固定的交易费用. 通过控制分红与注资的时刻以及分红及注资量,实现破产前分红减注资的折现期望的最大化. 由于存在固定交易费用, 问题为一个脉冲控制问题. 根据问题的参数不同, 问题的解可分为两大类. 一类解为只进行最优分红不需要注资, 而另一类情况需要注资. 需要注资时, 最优注资策略由最优注资上界以及最优注资下界描述. 当赤字小于最优注资下界的绝对值时, 进行注资. 最后, 在理赔为指数分布时明确地给出了两类共七种最优策略以及值函数的形式. 从而彻底地解决了该问题.