Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function

In this article, four different mathematical models of chemotherapy from the literature are investigated with respect to optimal control of drug treatment schedules. The various models are based on two different sets of ordinary differential equations and contain either chemotherapy, immunotherapy, anti-angiogenic therapy or combinations of these. Optimal control problem formulations based on these models are proposed, discussed and compared. For different parameter sets, scenarios, and objective functions optimal control problems are solved numerically with Bock’s direct multiple shooting method.In particular, we show that an optimally controlled therapy can be the reason for the difference between a growing and a totally vanishing tumor in comparison to standard treatment schemes and untreated or wrongly treated tumors. Furthermore, we compare different objective functions. Eventually, we propose an optimization-driven indicator for the potential gain of optimal controls. Based on this indicator, we show that there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optimal therapies into medical practice due to patient-, cancer-, and therapy-specific components.     

Optimal HIV treatment by maximising immune response

We present an optimal control model of drug treatment of the human immunodeficiency virus (HIV). Our model is based upon ordinary differential equations that describe the interaction between HIV and the specific immune response as measured by levels of natural killer cells. We establish stability results for the model. We approach the treatment problem by posing it as an optimal control problem in which we maximise the benefit based on levels of healthy CD4+ T cells and immune response cells, less the systemic cost of chemotherapy. We completely characterise the optimal control and compute a numerical solution of the optimality system via analytic continuation.Research supported by the Natural Science and Engineering Research Council (NSERC) and the Mathematics of Information Technology and Complex Systems (MITACS) of Canada     

一类分数阶微分方程的最优控制

主要研究了一类分数阶微分方程的最优控制问题.通过Oustaloup迭代逼近,可以将分数阶微分算子在频率域范围内进行近似.那么原先的分数阶微分方程就转换为一般的常微分方程.利用这个关系,可以得到关于分数阶微分方程的两个定理和一个引理.最后给出一个例子说明该方法的有效性.     

Optimal control for micro‐algae on a raceway model

We apply numerical optimal control methods to an existing algae growth model with the aim to determine the best performance of the model under known conditions using a variety of decision variables. We transform the system of differential algebraic equations in the existing model to a system of ordinary differential equations which introduces dynamics for average light intensity and chlorophyll. In addition, we allow for variable nitrogen concentration of the inflow as well as variable initial nitrogen concentration of the raceway. Our main focus is on optimizing of the production of lipids. We calculate both open and closed loop optimal controllers and test their robustness. Finally, we also consider raceway depth as a decision variable. © 2017 American Institute of Chemical Engineers Biotechnol. Prog., 34:107–119, 2018     

An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

In this paper we present a numerical method for the bidomain model, which describes the electrical activity in the heart. The model consists of two partial differential equations (PDEs), which are coupled to systems of ordinary differential equations (ODEs) describing electrochemical reactions in the cardiac cells. Many applications require coupling these equations to a third PDE, describing the electrical fields in the torso surrounding the heart. The resulting system is challenging to solve numerically, because of its complexity and very strict resolution requirements in time and space. We propose a method based on operator splitting and a fully coupled discretization of the three PDEs. Numerical experiments show that for simple simulation cases and fine discretizations, the algorithm is second-order accurate in space and time.     

Optimal enhancement of immune response

MOTIVATION: Therapeutic enhancement of innate immune response to microbial attack is addressed as the optimal control of a dynamic system. Interactions between an invading pathogen and the innate immune system are characterized by four non-linear, ordinary differential equations that describe rates of change of pathogen, plasma cell, and antibody concentrations, and of an indicator of organic health. Without therapy, the dynamic model evidences sub-clinical or clinical decay, chronic stabilization, or unrestrained lethal growth of the pathogen; the response pattern depends on the initial concentration of pathogens in the simulated attack. In the model, immune response can be augmented by therapeutic agents that kill the pathogen directly, that stimulate the production of plasma cells or antibodies, or that enhance organ health. A previous paper demonstrated open-loop optimal control solutions that defeat the pathogen and preserve organ health, given initial conditions that otherwise would be lethal (Stengel et al. (2002)). Therapies based on separate and combined application of the agents were derived by minimizing a quadratic cost function that weighted both system response and control usage, providing implicit control over harmful side effects. RESULTS: We demonstrate the ability of neighboring-optimal feedback control to account for a range of unknown initial conditions and persistent input of pathogens by adjusting the therapy to account for perturbations from the nominal-optimal response history. We examine therapies that combine open-loop control of one agent with closed-loop control of another. We show that optimal control theory points the way toward new protocols for treatment and cure of human diseases. CONTACT: stengel@princeton.edu; rghiglia@princeton.edu; nkulkarn@princeton.edu     

Self-propelled micro-swimmers in a Brinkman fluid

We prove an existence, uniqueness, and regularity result for the motion of a self-propelled micro-swimmer in a particulate viscous medium, modelled as a Brinkman fluid. A suitable functional setting is introduced to solve the Brinkman system for the velocity field and the pressure of the fluid by variational techniques. The equations of motion are written by imposing a self-propulsion constraint, thus allowing the viscous forces and torques to be the only ones acting on the swimmer. From an infinite-dimensional control on the shape of the swimmer, a system of six ordinary differential equations for the spatial position and the orientation of the swimmer is obtained. This is dealt with standard techniques for ordinary differential equations, once the coefficients are proved to be measurable and bounded. The main result turns out to extend an analogous result previously obtained for the Stokes system.     

The numerical method for the optimal supporting position and related optimal control for the catalytic reaction system

This paper considers the numerical approximation for the optimal supporting position and related optimal control of a catalytic reaction system with some control and state constraints, which is governed by a nonlinear partial differential equations with given initial and boundary conditions. By the Galerkin finite element method, the original problem is projected into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then the control parameterization method is applied to approximate the control and reduce the original system to an optimal parameter selection problem, in which both the position and related control are taken as decision variables to be optimized. This problem can be solved as a nonlinear optimization problem by a particle swarm optimization algorithm. The numerical simulations are given to illustrate the effectiveness of the proposed numerical approximation method.     

Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to derive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a function only of time. This mathematical model does not allow for spatial variations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordinary differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare them to the solution of the ordinary differential equations described above. It is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so the ordinary differential equation should not be used if an accurate calculation of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the density of the capillaries that supply blood to this tissue. As a result, techniques that are based on the ordinary differential equation to calculate the mixed-venous blood partial pressure may be in error.     

Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations

We develop and analyze a mathematical model, in the form of a system of ordinary differential equations (ODEs), governing cancer growth on a cell population level with combination immune, vaccine and chemotherapy treatments. We characterize the ODE system dynamics by locating equilibrium points, determining stability properties, performing a bifurcation analysis, and identifying basins of attraction. These system characteristics are useful not only to gain a broad understanding of the specific system dynamics, but also to help guide the development of combination therapies. Numerical simulations of mixed chemo-immuno and vaccine therapy using both mouse and human parameters are presented. We illustrate situations for which neither chemotherapy nor immunotherapy alone are sufficient to control tumor growth, but in combination the therapies are able to eliminate the entire tumor.     

A complete set of control equations for the human musculo-skeletal system

A mathematical model of the total human musculo-skeletal system is presented. The model comprises a link-mechanical and a musculo-mechanical set of ordinary first-order differential equations which describe the dynamics of the segment model and muscle model respectively. The interdependence of the two sets of equations is demonstrated. The set of musculo-mechanical equations contains the two neuromuscular control parameters motor unit recruitment and stimulation rate, and the significance of such a representation for a control-theoretical treatment of musculo-skeletal systems is discussed. Finally, after a short discussion of the successful application of the present model in the prediction of an optimal human motion, further possibilities are indicated of the use of the model for investigations into the control behaviour of musculo-skeletal systems.     

Complex dynamic behaviour of autonomous microbial food chains

 The dynamic behaviour of food chains under chemostat conditions is studied. The microbial food chain consists of substrate (non-growing resources), bacteria (prey), ciliates (predator) and carnivore (top predator). The governing equations are formulated at the population level. Yet these equations are derived from a dynamic energy budget model formulated at the individual level. The resulting model is an autonomous system of four first-order ordinary differential equations. These food chains resemble those occuring in ecosystems. Then the prey is generally assumed to grow logistically. Therefore the model of these systems is formed by three first-order ordinary differential equations. As with these ecosystems, there is chaotic behaviour of the autonomous microbial food chain under chemostat conditions with biologically relevant parameter values. It appears that the trajectories on the attractors consists of two superimposed oscillatory behaviours, a slow one for predator–top predator and a fast one for the prey–predator on one branch at which the top predator increases slowly. In some regions of the parameter space there are multiple attractors. Received 8 November 1995; received in revised form 7 January 1997     

Optimal control of a rabies epidemic model with a birth pulse

A system of ordinary differential equations describes the population dynamics of a rabies epidemic in raccoons. The model accounts for the dynamics of a vaccine, including loss of vaccine due to animal consumption and loss from factors other than racoon uptake. A control method to reduce the spread of disease is introduced through temporal distribution of vaccine packets. This work incorporates the effect of the seasonal birth pulse in the racoon population and the attendant increase in new-borns which are susceptible to the diseases, analysing the impact of the timing and length of this pulse on the optimal distribution of vaccine packets. The optimization criterion is to minimize the number of infected raccoons while minimizing the cost of distributing the vaccine. Using an optimal control setting, numerical results illustrate strategies for distributing the vaccine depending on the timing of the infection outbreak with respect to the birth pulse.     

Optimization of Running Strategies According to the Physiological Parameters for a Two-Runners Model

In order to describe the velocity and the anaerobic energy of two runners competing against each other for middle-distance races, we present a mathematical model relying on an optimal control problem for a system of ordinary differential equations. The model is based on energy conservation and on Newton’s second law: resistive forces, propulsive forces and variations in the maximal oxygen uptake are taken into account. The interaction between the runners provides a minimum for staying 1 m behind one’s competitor. We perform numerical simulations and show how a runner can win a race against someone stronger by taking advantage of staying behind, or how they can improve their personal record by running behind someone else. Our simulations show when it is the best time to overtake, depending on the difference between the athletes. Finally, we compare our numerical results with real data from the men’s 1500 m finals of different competitions.     

CADCS simulation of the closed-loop cardiovascular system

A pulsatile simulator of the closed-loop cardiovascular system, designed to solve simulation, identification and control problems in a research and education context, is presented. Its implimentation makes use of a command-driven interactive program for simulation of non-linear ordinary differential equations. The flexibility of the simulator is demonstrated by the results presented which refer to a basal steady-state circulatory condition as well as a transient induced by an abrupt change in peripheral resistance.     

A mathematical model for the transmission of Schistosoma japonicum in consideration of seasonal water level fluctuations of Poyang Lake in Jiangxi,China

BackgroundPoyang Lake, the largest fresh water lake in China, is the major transmission site of Schistosoma japonicum in China. Epidemics of schistosomiasis japonica have threatened the health of residents and stunted social–economic development there.ObjectiveThis article aims at evaluating the effect of various control measures against schistosomiasis: selective mass treatment (ST), targeted mass treatment (TT), mass treatment for animal reservoirs (MT), and health education (HE), on reduction of the prevalence through simulations based on a mathematical model.MethodsWe proposed a mathematical model, which is a system of ordinary differential equations for the transmission of S. japonicum among humans, bovines, and snails. The model takes into account the seasonal variation of the water level of Poyang Lake that is caused by the backflow of the Yangtze River and inflow from five small rivers, which influences the transmission of S. japonicum. For the purpose of dealing with the age-specific prevalence and intensity of infection, the human population was classified into four age categories in the model. We carried out several simulations resulting from the execution of ST and TT for elementary school children (E Sch), and combinations of ST, MT, and HE.ResultsThe simulations indicated that all of the control measures only for humans had a trend of revival after interruption, and a combination of ST and MT has a significant effect on reducing human infection. Although TT and HE had a significant effect on the prevalence in the E Sch group, it had little effect on the overall human population.ConclusionThe simulations indicate that measures targeted to bovines such as chemotherapy besides humans will be vital to eliminate the transmission of S. japonicum in the Poyang Lake region. Moreover, it is desirable to improve health education for fishermen and herdsmen.     

Recasting nonlinear differential equations as S-systems: a canonical nonlinear form

An enormous variety of nonlinear differential equations and functions have been recast exactly in the canonical form called an S-system. This is a system of nonlinear ordinary differential equations, each with the same structure: the change in a variable is equal to a difference of products of power-law functions. We review the development of S-systems, prove that the minimum for the range of equations that can be recast as S-systems consists of all equations composed of elementary functions and nested elementary functions of elementary functions, give a detailed example of the recasting process, and discuss the theoretical and practical implications. Among the latter is the ability to solve numerically nonlinear ordinary differential equations in their S-system form significantly faster than in their original form through utilization of a specially designed algorithm.     

A critical review of mathematical models and data used in diabetology

The literature dealing with mathematical modelling for diabetes is abundant. During the last decades, a variety of models have been devoted to different aspects of diabetes, including glucose and insulin dynamics, management and complications prevention, cost and cost-effectiveness of strategies and epidemiology of diabetes in general. Several reviews are published regularly on mathematical models used for specific aspects of diabetes. In the present paper we propose a global overview of mathematical models dealing with many aspects of diabetes and using various tools. The review includes, side by side, models which are simple and/or comprehensive; deterministic and/or stochastic; continuous and/or discrete; using ordinary differential equations, partial differential equations, optimal control theory, integral equations, matrix analysis and computer algorithms.     

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

The Lotka–Volterra model is a differential system of two coupled equations representing the interaction of two species: a prey one and a predator one. We formulate an optimal control problem adding the effect of hunting both species as the control variable. We analyse the optimal hunting problem paying special attention to the nature of the optimal state and control trajectories in long time intervals. To do that, we apply recent theoretical results on the frame to show that, when the time horizon is large enough, optimal strategies are nearly steady-state. Such path is known as turnpike property. Some experiments are performed to observe such turnpike phenomenon in the hunting problem. Based on the turnpike property, we implement a variant of the single shooting method to solve the previous optimisation problem, taking the middle of the time interval as starting point.     

Chemotherapeutic treatments: A study of the interplay among drug resistance, toxicity and recuperation from side effects

A system of differential equations for the control of tumor growth cells in a cycle nonspecific chemotherapy is analyzed. Spontaneously acquired drug resistance is taken into account, and a criterion for the selection of chemotherapeutic treatment is used. This criterion purports to describe the possibility of improvement of the patient's health when treatment is discontinued. Contrary to our early results which also take drug resistance into account, in this context strategies of continuous chemotherapy in which rest periods take part may be better than maximum drug concentration throughout the treatment (which appears to be in accordance with clinical practice). This bears out our previous conjecture that when drug resistance is accounted for, the imperfections in the usual modelling of treatment criteria, which in general do not allow for patient recuperation, ruled out the possibility of rest periods in optimal continuous chemotherapy.