Free Terminal Time Optimal Control Problem of an HIV Model Based on a Conjugate Gradient Method

The minimum duration of treatment periods and the optimal multidrug therapy for human immunodeficiency virus (HIV) type 1 infection are considered. We formulate an optimal tracking problem, attempting to drive the states of the model to a “healthy” steady state in which the viral load is low and the immune response is strong. We study an optimal time frame as well as HIV therapeutic strategies by analyzing the free terminal time optimal tracking control problem. The minimum duration of treatment periods and the optimal multidrug therapy are found by solving the corresponding optimality systems with the additional transversality condition for the terminal time. We demonstrate by numerical simulations that the optimal dynamic multidrug therapy can lead to the long-term control of HIV by the strong immune response after discontinuation of therapy.     

Pulse HIV Vaccination: Feasibility for Virus Eradication and Optimal Vaccination Schedule

We modify the classical virus dynamics model by incorporating an immune response with fixed or fluctuating vaccination frequencies and dosages to obtain a system of impulsive differential equations for the virus dynamics of both the wild-type and mutant strains. This model framework permits us to obtain precise conditions for the virus elimination, which are much more feasible compared with existing results, which require frequent vaccine administration with large dosage. We also consider the corresponding impulsive optimal control problem to describe when and how much of the vaccine should be administered in order to maximize levels of healthy CD4⁺ T cells and immune response cells. A gradient-based optimization method is applied to obtain the optimal schedule numerically. For a case study when the CTL vaccine is administered in a period of one year, our numerical studies support the optimal vaccination schedule consisting of vaccine administration three times, with the first dosage strong (to boost the immune system), followed by a second dosage shortly after (to strengthen the immune response) and then the third and final dosage long after (to ensure the immune system can handle viruses rebound).     

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     

A low T regulatory cell response may contribute to both viral control and generalized immune activation in HIV controllers

HIV-infected individuals maintaining undetectable viremia in the absence of therapy (HIV controllers) often maintain high HIV-specific T cell responses, which has spurred the development of vaccines eliciting HIV-specific T cell responses. However, controllers also often have abnormally high T cell activation levels, potentially contributing to T cell dysfunction, CD4+ T cell depletion, and non-AIDS morbidity. We hypothesized that a weak T regulatory cell (Treg) response might contribute to the control of viral replication in HIV controllers, but might also contribute to generalized immune activation, contributing to CD4+ T cell loss. To address these hypotheses, we measured frequencies of activated (CD38+ HLA-DR+), regulatory (CD4+CD25+CD127(dim)), HIV-specific, and CMV-specific T cells among HIV controllers and 3 control populations: HIV-infected individuals with treatment-mediated viral suppression (ART-suppressed), untreated HIV-infected "non-controllers" with high levels of viremia, and HIV-uninfected individuals. Despite abnormally high T cell activation levels, controllers had lower Treg frequencies than HIV-uninfected controls (P = 0.014). Supporting the propensity for an unusually low Treg response to viral infection in HIV controllers, we observed unusually high CMV-specific CD4+ T cell frequencies and a strong correlation between HIV-specific CD4+ T cell responses and generalized CD8+ T cell activation levels in HIV controllers (P ≤ 0.001). These data support a model in which low frequencies of Tregs in HIV controllers may contribute to an effective adaptive immune response, but may also contribute to generalized immune activation, potentially contributing to CD4 depletion.     

Adenosine signaling and adenosine deaminase regulation of immune responses: impact on the immunopathogenesis of HIV infection

Infection by human immunodeficiency virus (HIV) causes the acquired immune deficiency syndrome (AIDS), which has devastating effects on the host immune system. HIV entry into host cells and subsequent viral replication induce a proinflammatory response, hyperactivating immune cells and leading them to death, disfunction, and exhaustion. Adenosine is an immunomodulatory molecule that suppresses immune cell function to protect tissue integrity. The anti-inflammatory properties of adenosine modulate the chronic inflammation and immune activation caused by HIV. Lack of adenosine contributes to pathogenic events in HIV infection. However, immunosuppression by adenosine has its shortcomings, such as impairing the immune response, hindering the elimination of the virus and control of viral replication. By attempting to control inflammation, adenosine feeds a pathogenic cycle affecting immune cells. Deamination of adenosine by ADA (adenosine deaminase) counteracts the negative effects of adenosine in immune cells, boosting the immune response. This review comprises the connection between adenosinergic system and HIV immunopathogenesis, exploring defects in immune cell function and the role of ADA in protecting these cells against damage.     

Optimal control for resistance and suboptimal response in CML

The mathematical modelling of hematopoiesis received a significant attention in the last few years. However, the treatment of hematological diseases is less investigated by optimal control tools. In this paper, we consider the dynamics of chronic myeloid leukemia based on a four dimensional model. First we analyze the global dynamics of normal and cancer hematopoietic stem cells and differentiated cells using the principle of Bendixson-Dulac. Then we introduce some nonlinear effects of imatinib treatment over a fixed period of time. We represent therapy effects as an optimal control problem to minimize the cost of treatment and the level of cancer cells. The influence of imatinib onto the division and the mortality rates of cancer cells produces the suboptimal response, resistance and recovery forms.     

Monocyte-derived cultured dendritic cells are susceptible to human immunodeficiency virus infection and transmit virus to resting T cells in the process of nominal antigen presentation.

The susceptibility of monocyte-derived cultured dendritic cells (DCs) to human immunodeficiency virus (HIV) infection and their role in viral transmission in the immune response were studied in detail. We observed that highly purified cultured DCs were infected with the T-tropic Lai strain of HIV type 1 (HIV-1Lai) via the CD4 receptor, and this was followed by formation of the complete provirus as detected by PCR. HIV mRNAs were transcribed at only low levels, and virus production was undectable; however, the addition of the purified protein derivative antigen of tuberculin and of autologous resting T cells to HIV-1Lai-infected DCs but not to HIV-1Lai-infected macrophages led to massive HIV transmission and production. These data suggest that the interaction of infected DCs with T cells during the normal immune response could play an important role in the activation and expansion of HIV.     

Modelling and optimal control of immune response of renal transplant recipients

We consider the increasingly important and highly complex immunological control problem: control of the dynamics of immunosuppression for organ transplant recipients. The goal in this problem is to maintain the delicate balance between over-suppression (where opportunistic latent viruses threaten the patient) and under-suppression (where rejection of the transplanted organ is probable). First, a mathematical model is formulated to describe the immune response to both viral infection and introduction of a donor kidney in a renal transplant recipient. Some numerical results are given to qualitatively validate and demonstrate that this initial model exhibits appropriate characteristics of primary infection and reactivation for immunosuppressed transplant recipients. In addition, we develop a computational framework for designing adaptive optimal treatment regimes with partial observations and low-frequency sampling, where the state estimates are obtained by solving a second deterministic optimal tracking problem. Numerical results are given to illustrate the feasibility of this method in obtaining optimal treatment regimes with a balance between under-suppression and over-suppression of the immune system.     

Modelling and optimal control of immune response of renal transplant recipients

We consider the increasingly important and highly complex immunological control problem: control of the dynamics of immunosuppression for organ transplant recipients. The goal in this problem is to maintain the delicate balance between over-suppression (where opportunistic latent viruses threaten the patient) and under-suppression (where rejection of the transplanted organ is probable). First, a mathematical model is formulated to describe the immune response to both viral infection and introduction of a donor kidney in a renal transplant recipient. Some numerical results are given to qualitatively validate and demonstrate that this initial model exhibits appropriate characteristics of primary infection and reactivation for immunosuppressed transplant recipients. In addition, we develop a computational framework for designing adaptive optimal treatment regimes with partial observations and low-frequency sampling, where the state estimates are obtained by solving a second deterministic optimal tracking problem. Numerical results are given to illustrate the feasibility of this method in obtaining optimal treatment regimes with a balance between under-suppression and over-suppression of the immune system.     

Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection

Primary human immunodeficiency virus (HIV) infection is characterized by an initial exponential increase of viral load in peripheral blood reaching a peak, followed by a rapid decline to the viral setpoint. Although the target-cell-limited model can account for part of the viral kinetics observed early in infection [Phillips, 1996. Reduction of HIV concentration during acute infection: independence from a specific immune response. Science 271 (5248), 497-499], it frequently predicts highly oscillatory kinetics after peak viremia, which is not typically observed in clinical data. Furthermore, the target-cell-limited model is unable to predict long-term viral kinetics, unless a delayed immune effect is assumed [Stafford et al., 2000. Modeling plasma virus concentration during primary HIV infection. J. Theor. Biol. 203 (3), 285-301]. We show here that extending the target-cell-limited model, by implementing a saturation term for HIV-infected cell loss dependent upon infected cell levels, is able to reproduce the diverse observed viral kinetic patterns without the assumption of a delayed immune response. Our results suggest that the immune response may have significant effect on the control of the virus during primary infection and may support experimental observations that an anti-HIV immune response is already functional during peak viremia.     

Optimal antigens for HIV vaccines based on CD8+ T response,protein length,and sequence variability

The identification of optimal antigens to include in an HIV vaccine designed to elicit a cellular immune response requires careful consideration of protein length, variability, and immunogenicity. Here, we have examined the relationship of these parameters in a cohort of HIV-infected subjects. We find that HIV Gag and Nef represent optimal antigens for the CD8+ T cell response, based on size, variability, and immunogenicity. Although the Env and Pol proteins have a poorer response to length (Pol) or variability (Env) ratio than Gag or Nef, they are still strong candidates for inclusion into an HIV vaccine, based on overall response frequency in the cohort. The accessory proteins Tat, Rev, Vif, Vpr, and Vpu in general all elicit very low CD8+ T cell responses, and this, in combination with their high variability, makes them less attractive as vaccine antigen.     

Vpu mediates depletion of interferon regulatory factor 3 during HIV infection by a lysosome-dependent mechanism

HIV has evolved sophisticated mechanisms to avoid restriction by intracellular innate immune defenses that otherwise serve to control acute viral infection and virus dissemination. Innate defenses are triggered when pattern recognition receptor (PRR) proteins of the host cell engage pathogen-associated molecule patterns (PAMPs) present in viral products. Interferon regulatory factor 3 (IRF3) plays a central role in PRR signaling of innate immunity to drive the expression of type I interferon (IFN) and interferon-stimulated genes (ISGs), including a variety of HIV restriction factors, that serve to limit viral replication directly and/or program adaptive immunity. Productive infection of T cells by HIV is dependent upon the targeted proteolysis of IRF3 that occurs through a virus-directed mechanism that results in suppression of innate immune defenses. However, the mechanisms by which HIV controls innate immune signaling and IRF3 function are not defined. Here, we examined the innate immune response induced by HIV strains identified through their differential control of PRR signaling. We identified viruses that, unlike typical circulating HIV strains, lack the ability to degrade IRF3. Our studies show that IRF3 regulation maps specifically to the HIV accessory protein Vpu. We define a molecular interaction between Vpu and IRF3 that redirects IRF3 to the endolysosome for proteolytic degradation, thus allowing HIV to avoid the innate antiviral immune response. Our studies reveal that Vpu is an important IRF3 regulator that supports acute HIV infection through innate immune suppression. These observations define the Vpu-IRF3 interface as a novel target for therapeutic strategies aimed at enhancing the immune response to HIV.     

Macrophages exposed to lymphotropic and monocytotropic HIV induce similar CTL responses despite differences in productive infection

Macrophages are accessory cells that are vulnerable to infection by HIV-1. HTLV-IIIB, a lymphotropic strain of HIV, infects macrophages poorly resulting in either no or low levels of virus expression compared to high levels of productive infection after exposure of macrophages to the monocytotropic HIV strain Ada-M. Whether this results in an impaired ability of HTLV-IIIB-exposed macrophages to initiate protective cytotoxic T lymphocyte (CTL) immune responses against these strains is not well defined. We investigated the ability of monocyte-derived macrophages (MDM) exposed to lymphotropic and monocytotropic HIV strains to initiate primary CTL responses in vitro. MDM exposed to HTLV-IIIB induced a specific primary CTL response that was comparable to MDM exposed to the monocytotropic strain Ada-M despite marked differences in productive HIV infection in MDM between the two strains. CTL generated in this model were MHC-restricted, strain-specific, and CD8+. These data demonstrate that high levels of productive HIV infection in accessory cells are not a prerequisite for the generation of a primary CTL response, suggesting a novel immunologic interaction between MDM and lymphotropic HIV strains.     

Immune responses and the emergence of drug-resistant virus strains in vivo

The treatment of viral infections using antiviral drugs has had a significant public health benefit in the setting of human immunodeficiency virus (HIV) infection, and newly developed drugs offer potential benefits in the management of other viral infections, including acute self-limiting infections such as influenza and picornaviruses (including the rhinoviruses that are responsible for a large proportion of 'common colds'). A serious concern with such treatments is that they may lead to the selection of drug-resistant strains. This has been a significant problem in the case of HIV infection. Existing mathematical-modelling studies of drug resistance have focused on the interactions between virus, target cells and infected cells, ignoring the impact of immune responses. Here, we present a model that explores the role of immune responses in the rise of drug-resistant mutants in vivo. We find that drug resistance is unlikely to be a problem if immune responses are maintained above a threshold level during therapy. Alternatively, if immune responses decline at a fast rate and fall below a threshold level during treatment (indicating impaired immunity), the rise of drug-resistant mutants is more likely. This indicates an important difference between HIV, which impairs immunity and for which immune responses have been observed to vanish during treatment, and viral infections such as influenza and rhinoviruses, for which such immune impairment is not present. Drug resistance is much more likely to be a problem in HIV than in acute and self-limiting infections.     

Augmented designs to assess immune response in vaccine trials

This article introduces methods for use in vaccine clinical trials to help determine whether the immune response to a vaccine is actually causing a reduction in the infection rate. This is not easy because immune response to the (say HIV) vaccine is only observed in the HIV vaccine arm. If we knew what the HIV-specific immune response in placebo recipients would have been, had they been vaccinated, this immune response could be treated essentially like a baseline covariate and an interaction with treatment could be evaluated. Relatedly, the rate of infection by this baseline covariate could be compared between the two groups and a causative role of immune response would be supported if infection risk decreased with increasing HIV immune response only in the vaccine group. We introduce two methods for inferring this HIV-specific immune response. The first involves vaccinating everyone before baseline with an irrelevant vaccine, for example, rabies. Randomization ensures that the relationship between the immune responses to the rabies and HIV vaccines observed in the vaccine group is the same as what would have been seen in the placebo group. We infer a placebo volunteer's response to the HIV vaccine using their rabies response and a prediction model from the vaccine group. The second method entails vaccinating all uninfected placebo patients at the closeout of the trial with the HIV vaccine and recording immune response. We pretend this immune response at closeout is what they would have had at baseline. We can then infer what the distribution of immune response among placebo infecteds would have been. Such designs may help elucidate the role of immune response in preventing infections. More pointedly, they could be helpful in the decision to improve or abandon an HIV vaccine with mediocre performance in a phase III trial.     

Evolutionary game theoretic strategy for optimal drug delivery to influence selection pressure in treatment of HIV-1

Cytotoxic T-lymphocyte (CTL) escape mutation is associated with long-term behaviors of human immunodeficiency virus type 1 (HIV-1). Recent studies indicate heterogeneous behaviors of reversible and conservative mutants while the selection pressure changes. The purpose of this study is to optimize the selection pressure to minimize the long-term virus load. The results can be used to assist in delivery of highly loaded cognate peptide-pulsed dendritic cells (DC) into lymph nodes that could change the selection pressure. This mechanism may be employed for controlled drug delivery. A mathematical model is proposed in this paper to describe the evolutionary dynamics involving viruses and T cells. We formulate the optimization problem into the framework of evolutionary game theory, and solve for the optimal control of the selection pressure as a neighborhood invader strategy. The strategy dynamics can be obtained to evolve the immune system to the best controlled state. The study may shed light on optimal design of HIV-1 therapy based on optimization of adaptive CTL immune response.     

Costs versus benefits: best possible and best practical treatment regimens for HIV

Current HIV therapy, although highly effective, may cause very serious side effects, making adherence to the prescribed regimen difficult. Mathematical modeling may be used to evaluate alternative treatment regimens by weighing the positive results of treatment, such as higher levels of helper T cells, against the negative consequences, such as side effects and the possibility of resistance mutations. Although estimating the weights assigned to these factors is difficult, current clinical practice offers insight by defining situations in which therapy is considered “worthwhile”. We therefore use clinical practice, along with the probability that a drug-resistant mutation is present at the start of therapy, to suggest methods of rationally estimating these weights. In our underlying model, we use ordinary differential equations to describe the time course of in-host HIV infection, and include populations of both activated CD4⁺ T cells and CD8⁺ T cells. We then determine the best possible treatment regimen, assuming that the effectiveness of the drug can be continually adjusted, and the best practical treatment regimen, evaluating all patterns of a block of days “on” therapy followed by a block of days “off” therapy. We find that when the tolerance for drug-resistant mutations is low, high drug concentrations which maintain low infected cell populations are optimal. In contrast, if the tolerance for drug-resistant mutations is fairly high, the optimal treatment involves periods of reduced drug exposure which consequently boost the immune response through increased antigen exposure. We elucidate the dependence of the optimal treatment regimen on the pharmacokinetic parameters of specific antiviral agents.     

Dynamics and potential impact of the immune response to chronic myelogenous leukemia

Recent mathematical models have been developed to study the dynamics of chronic myelogenous leukemia (CML) under imatinib treatment. None of these models incorporates the anti-leukemia immune response. Recent experimental data show that imatinib treatment may promote the development of anti-leukemia immune responses as patients enter remission. Using these experimental data we develop a mathematical model to gain insights into the dynamics and potential impact of the resulting anti-leukemia immune response on CML. We model the immune response using a system of delay differential equations, where the delay term accounts for the duration of cell division. The mathematical model suggests that anti-leukemia T cell responses may play a critical role in maintaining CML patients in remission under imatinib therapy. Furthermore, it proposes a novel concept of an “optimal load zone” for leukemic cells in which the anti-leukemia immune response is most effective. Imatinib therapy may drive leukemic cell populations to enter and fall below this optimal load zone too rapidly to sustain the anti-leukemia T cell response. As a potential therapeutic strategy, the model shows that vaccination approaches in combination with imatinib therapy may optimally sustain the anti-leukemia T cell response to potentially eradicate residual leukemic cells for a durable cure of CML. The approach presented in this paper accounts for the role of the anti-leukemia specific immune response in the dynamics of CML. By combining experimental data and mathematical models, we demonstrate that persistence of anti-leukemia T cells even at low levels seems to prevent the leukemia from relapsing (for at least 50 months). As a consequence, we hypothesize that anti-leukemia T cell responses may help maintain remission under imatinib therapy. The mathematical model together with the new experimental data imply that there may be a feasible, low-risk, clinical approach to enhancing the effects of imatinib treatment.     

A model predictive control based scheduling method for HIV therapy

Recently developed models of the interaction of the human immune system and the human immunodeficiency virus (HIV) suggest the possibility of using interruptions of highly active anti-retroviral therapy (HAART) to simulate a therapeutic vaccine and induce cytotoxic lymphocyte (CTL) mediated control of HIV infection. We have developed a model predictive control (MPC) based method for determining optimal treatment interruption schedules for this purpose. This method provides a clinically implementable framework for calculating interruption schedules that are robust to errors due to measurement and patient variations. In this paper, we discuss the medical motivation for this work, introduce the MPC-based method, show simulation results, and discuss future work necessary to implement the method.     

Optimal control of the chemotherapy of HIV

 Using an existing ordinary differential equation model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The control represents the percentage of effect the chemotherapy has on the viral production. Using an objective function based on a combination of maximizing benefit based on T cell counts and minimizing the systemic cost of chemotherapy (based on high drug dose/strength), we solve for the optimal control in the optimality system composed of four ordinary differential equations and four adjoint ordinary differential equations. Received 5 July 1995; received in revised form 3 June 1996