On simulating strongly interacting, stochastic population models. II. Multiple compartments

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.

On modeling the effects of T-cell vaccines on HIV acquisition and disease

A 'T-cell vaccine' aims at generating cytotoxic T-lymphocytes (CTLs; the so-called 'killer' T-cells) rather than antibodies (as for traditional vaccines). The first (phase IIb) trials of this concept against HIV/AIDS began in 2004. What can mechanistic modeling contribute to understanding the biological action of this class of vaccines, if any? Models are appropriate in any discussion of three potential vaccine effects: on acquisition of infection; on state of disease ('viral load', VL) after infection; and on preventing escape from immune control. Concerning the first two, P. Gilbert, S. Self and I introduced new stochastic models of early HIV infection and the CTL response, and, making use of recent estimates (derived in collaboration with O. Yang and L. Corey) of the rate that CTLs can kill HIV-infected cells, made the (surprising?) discovery that CTLs might prevent some infections--as the trial designers implicitly acknowledged when they chose the dual end points of the study. On sustaining control, we have derived a theoretical formula for the rate of escape by stepwise mutation and a new method of simulating HIV and CTL dynamics in vivo (permitting new mutant strains a stochastic evolution--essential, in our view). These quantitative models and simulation techniques can also prove useful to biostatisticians. For example, in preparation for the STEP trials, Gilbert, Bosch, and Hudgens developed a novel technique for estimating a causal effect of a vaccine on VL while accounting for post-randomization selection bias. By simulating thousands of trials, we demonstrated that GBH's method can correctly identify efficacy while protecting against falsely concluding that the vaccine exacerbates disease. When trial data becomes available, the models may also be exploited to make complementary analyses which, while not relevant to vaccine licensure, may suggest new biological hypotheses.

On modeling HIV and T cells in vivo: assessing causal estimators in vaccine trials

The first efficacy trials--named STEP--of a T cell vaccine against HIV/AIDS began in 2004. The unprecedented structure of these trials raised new modeling and statistical challenges. Is it plausible that memory T cells, as opposed to antibodies, can actually prevent infection? If they fail at prevention, to what extent can they ameliorate disease? And how do we estimate efficacy in a vaccine trial with two primary endpoints, one traditional, one entirely novel (viral load after infection), and where the latter may be influenced by selection bias due to the former? In preparation for the STEP trials, biostatisticians developed novel techniques for estimating a causal effect of a vaccine on viral load, while accounting for post-randomization selection bias. But these techniques have not been tested in biologically plausible scenarios. We introduce new stochastic models of T cell and HIV kinetics, making use of new estimates of the rate that cytotoxic T lymphocytes--CTLs; the so-called killer T cells--can kill HIV-infected cells. Based on these models, we make the surprising discovery that it is not entirely implausible that HIV-specific CTLs might prevent infection--as the designers explicitly acknowledged when they chose the endpoints of the STEP trials. By simulating thousands of trials, we demonstrate that the new statistical methods can correctly identify an efficacious vaccine, while protecting against a false conclusion that the vaccine exacerbates disease. In addition to uncovering a surprising immunological scenario, our results illustrate the utility of mechanistic modeling in biostatistics.

How many human immunodeficiency virus type 1-infected target cells can a cytotoxic T-lymphocyte kill?

The antiviral role of CD8+ cytotoxic T lymphocytes (CTLs) in human immunodeficiency virus type 1 (HIV-1) infection is poorly understood. Specifically, the degree to which CTLs reduce viral replication by killing HIV-1-infected cells in vivo is not known. Here we employ mathematical models of the infection process and CTL action to estimate the rate that CTLs can kill HIV-1-infected cells from in vitro and in vivo data. Our estimates, which are surprisingly consistent considering the disparities between the two experimental systems, demonstrate that on average CTLs can kill from 0.7 to 3 infected target cells per day, with the variability in this figure due to epitope specificity or other factors. These results are compatible with the observed decline in viremia after primary infection being primarily a consequence of CTL activity and have interesting implications for vaccine design.