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Optimal Control of Innate Immune Response

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Optimal Control of Innate Immune Response OPTIMAL CONTROL APPLICATIONS AND METHODS Optim. Control Appl. Meth., 2002; 23: 91–104 (DOI: 10.1002/oca.704) Optimal control of innate immune response Robert F. Stengel*, Raffaele Ghigliazza, Nilesh Kulkarni and Olivier Laplace School of Engineering and Applied Science, Princeton University, Princeton, NJ 08544, U.S.A SUMMARY Treatment of a pathogenic disease process is interpreted as the optimal control of a dynamic system. Evolution of the disease is characterized by a non-linear, fourth-order ordinary differential equation that describes concentrations of pathogens, plasma cells, and antibodies, as well as a numerical indication of patient health. Without control, the dynamic model evidences sub-clinical or clinical decay, chronic stabilization, or unrestrained lethal growth of the pathogen, depending on the initial conditions for the infection. The dynamic equations are controlled by therapeutic agents that affect the rate of change of system variables. Control histories that minimize a quadratic cost function are generated by numerical optimization over a fixed time interval, given otherwise lethal initial conditions. Tradeoffs between cost function weighting of pathogens, organ health, and use of therapeutics are evaluated. Optimal control solutions that defeat the pathogen and preserve organ health are demonstrated for four different approaches to therapy. It is shown that control theory can point the way toward new protocols for treatment and remediation of human diseases. Copyright # 2002 John Wiley & Sons, Ltd. KEY WORDS: optimal control; biological modelling; bioinformatics; optimization INTRODUCTION Immune-system response to invasion by infectious microbes is a dynamic process in which potentially uncontrolled growth of the invader (or pathogen) is countered by various protective mechanisms. There are both tactical and strategic responses once the outer perimeter of defense } the surface epithelial layers of the body, including the epidermal cells of the skin and the mucosal cells that line the respiratory, gastrointestinal, and genitourinary tracts } has been broached [1–3]. The innate immune system provides a humoral (or tactical) response, signalling the presence of ‘non-self ’ organisms and activating B cells to produce antibodies that bind to the intruders’ antigens. The antibodies identify targets for scavenging cells, such as neutrophils and macrophages, which engulf and consume the microbes, reducing them to non-functioning units. They also stimulate the production of molecules (e.g. cytokines, complement, and acute-phase proteins) that either damage an intruder’s plasma membrane directly (a process called lysis)or help trigger the second phase of immune response. The innate immune system protects against *Correspondence to: Robert F. Stengel, School of Engineering and Applied Science, Princeton University, D-202 Engineering Quadrangle, Princeton, NJ 08544, USA. E-mail: [email protected] Contract/grant sponsor: Alfred P. Sloan Foundation. Contract/grant sponsor: Burroughs-Wellcome Fund for Biological Dynamics. Received 6 November 2000 Copyright # 2002 John Wiley & Sons, Ltd. Revised 14 January 2002 92 R. F. STENGEL ET AL. many extracellular bacteria or free viruses that are found in blood plasma, lymph, tissue fluid, or interstitial space between cells (and, therefore, that are not hidden within the cells of the body). However, innate immunity normally is not adequate for defeating microbes that burrow into cells, such as viruses, intracellular bacteria, and protozoa. The adaptive immune system provides the cellular (or strategic) response to intracellular microbial assault, producing protective cells that remember specific antigens, that produce antibodies ‘customized’ to the pathogens, and that seek out epitopes (or defining regions) of antigens that are expressed on the surfaces of infected cells. Adaptive immune mechanisms are largely based on the actions of B and T lymphocyte cells that become dedicated to a single antibody type through clonal selection. In particular, killer T cells (or cytotoxic T lymphocytes) bind to infected cells and kill them by initiating apoptosis (programmed cell death). Helper T cells assist naive B cells in maturing into memory cells and plasma cells that produce the needed antibody type. The process of adaptation is slow, but it generates a cadre of immune cells with narrowly focused memory, cells that are ready to respond rapidly if invading microbes of the same type (i.e. with the same antigen epitopes) are encountered again. As inferred by this brief discussion, elements of the innate and adaptive immune systems are shared, and response mechanisms are coupled, even though separate modes of operation can be identified. We distinguish between pre-exposure vaccination (or immunization) that is intended to prevent clinical disease from ever occurring and post-exposure therapy for a clinically diagnosed condition. The options available for clinical treatment of infection once it has been recognized focus on killing the invading microbes directly, neutralizing their deleterious effects, enhancing the efficacy of immune response, and providing palliative or healing care to other organs of the body. Few biological or chemical agents have just a single effect; for example, an agent that kills a pathogen also may damage healthy ‘self ’ cells. Nevertheless, such agents normally are introduced with one particular goal in mind, and it is a critical function of drug discovery and development to identify new compounds that have maximum intended efficacy with minimal side effects in the general population. Within the pharmacopoeia, we find examples of antibiotics as microbe killers; interferons as microbe neutralizers; interleukins, antigens from killed (i.e. non-toxic) pathogens, and pre-formed and monoclonal antibodies as immunity enhancers (each of very different nature); and anti-inflammatory and anti-histamine compounds as palliative drugs. Numerous models of immune response to infection have been postulated (e.g. [4–7]), with recent emphasis on the human-immunodeficiency pathogen (HIV) [8–13], but there have been few applications of control theory to the healing process. Norbert Wiener [14] and Richard Bellman [15] appreciated and anticipated that possibility, and References [16–19] advance optimization as a model for immune system dynamics. References [20, 21] survey early optimal control applications to biomedical problems, while References [22–26] investigate the role of optimization in cancer therapy. References [27, 28] offer an optimal control approach to HIV treatment, and intuitive control approaches are presented in References [29–32]. A goal of the present analysis is to show how regimens for applying drugs in a manner that maximizes efficacy while minimizing side effects and cost can be designed. In the current context, that means killing, neutralizing, or limiting growth of the pathogen with small impact on the patient’s health and pocketbook. Optimal control theory can help discriminate between options for attacking the infection directly, stimulating the immune system, suppressing or amplifying ancillary biological processes, prescribing drug dosage and timing, and combining therapies for a ‘cocktail’ of effects. Copyright # 2002 John Wiley & Sons, Ltd. Optim. Control Appl. Meth. 2002; 23: 91–104 OPTIMAL CONTROL OF INNATE IMMUNE RESPONSE 93 Here, we focus on the control of a simple mathematical model for the evolution of a generic pathogen in a human. The model contains essential features of disease dynamics, including a pathogen, the innate immune system, and an organ whose function is affected by the pathogen. The model does not represent a specific pathogen, and the control variables represent hypothetical therapeutic agents that have idealized effect on the system. We describe the disease model, define a treatment cost function to be minimized by application of the therapeutic agents, present results of the optimization for a range of treatment parameters, and draw conclusions about the method and results. The simulated results show not only the progression of the patient’s state from an initially life-threatening state to a controlled or cured condition but the optimal history of therapeutic agents that produces that condition. Extension of the method to more explicit models of specific diseases, e.g. HIV and cancer, is the topic of future papers. The present paper demonstrates the power of the optimization approach to treatment using a representative model. A MODEL OF DISEASE DYNAMICS We consider a simple model for a pathogenic attack on an organism and the organism’s immunological defense. There are four components to the dynamic state [4]: x1=concentration of a pathogen (=concentration of associated antigen) x2=concentration of plasma cells, which are carriers and producers of antibodies x3=concentration of antibodies, which kill the pathogen x4=relative characteristic of a damaged organ [0=healthy, 1=dead] The mathematical model presented in Reference [4] does not allow for the effects of therapy. Consequently, we modify the model, adding the following idealized therapeutic control agents: u1=pathogen killer u2=plasma cell enhancer u3=antibody enhancer u4=organ healing factor The four scalar, non-linear, ordinary differential equations of the modified dynamic model are: x’1 ¼ða11 À a12x3Þx1 þ b1u1 x’2 ¼ a21ðx4Þa22x1ðt À tÞx3ðt À tÞa23ðx2 À x2Þ þ b2u2 x’3 ¼ a31x2 ða32 þ a33x1Þx3 þ b3u3 x’4 ¼ a41x1 À a42x4 þ b4u4 ð124Þ The aij and bi are constants, except a21ðx4Þ;
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