A model of HIV-1 pathogenesis that includes an intracellular delay
Introduction
In recent years, clinical research combined with mathematical modeling has enhanced progress in the understanding of HIV-1 infection. The introduction of a new class of drugs, protease inhibitors, used in combination with previously approved drugs such as zidovudine (AZT) and lamivudine (3TC) has shown much promise in reducing the detectable levels of viral RNA in the plasma. The introduction of these potent antiviral agents has opened the door to defining kinetic parameters associated with HIV-1 dynamics in infected individuals. Using mathematical models to interpret clinical data from HIV-1 infected patients treated with these potent antiviral agents has led to an estimate of the half-life of productively infected cells of 1.6 days or less and of the clearance rate of free virions from the plasma of six hours or less [1], [2], [3]. These initial estimates were claimed to be minimal estimates because the models used to analyze the patient data assumed the drug was completely effective. Refining these estimates will improve our quantitative description of viral dynamics and our understanding of the speed at which drug resistance emerges. Also, the dynamics of the models assumed that cells upon infection instantly begin producing virus. Both the assumption of 100% drug efficacy and of instantaneous production of virus upon infection are relaxed in this paper.
Ho et al. [4] and Wei et al. [3] developed models that were applied to data from an experiment where patients received an antiretroviral drug. Before drug therapy was initiated the patients were determined to have a quasi steady state concentration of virus in their plasma, suggesting that the rate of viral production equaled the rate of virus clearance. After administration of the drug, blood samples were taken for several days and plasma viral RNA levels were measured. The results showed that when virus production was interfered with by drugs the virus declined approximately exponentially. The data, when analyzed in the context of the model, allowed an estimate to be made of the quasi-steady state rate of viral production. The estimated production level was much higher then previously thought and provided an explanation for the ease by which HIV could mutate and escape therapy when only a single drug was used. Extensions of this work showed that after drug is applied there is a delay before virus begins to decline [1]. This delay is due to pharmacological events, the mechanism of protease inhibitor action that allows pre-existing virus particles to continue infecting cells and delays in the process of producing virus particles [1], [5], [6]. The viral decline is at first very rapid and then after about two weeks slows [2]. Thus there is a rapid first phase followed by a slower second phase of viral decline. Analysis of the data in the context of models showed that the viral decline during the first, rapid, phase could be used to estimate both c, the virus clearance rate constant and δ, the decay rate of productively infected T-cells [1]. In the model, the asymptotic rate of the exponential decline of virus concentration is e−δt. In order to get this result, it was assumed the drug was completely effective and that the virus clearance rate constant, c, was much larger than δ, and therefore associated with processes that decayed more quickly and consequently could be ignored after a brief initial period. It was later shown [7] that if the drug or combination of drugs was less then perfect, the exponential decline of the first phase could be represented by e−δnct where nc is the effectiveness of the drug combination with a value of one for complete efficacy and zero for no effect. In this paper we study the changes in these approximations when a constant delay is introduced in the model and the drug, a protease inhibitor, is assumed to have variable efficacy. Four models have been developed that attempt to generalize the early models by Wei et al. [3] and Perelson et al. [1] by including delays [5], [6], [8], [9]. Grossman et al. [8] assume that infected cells die after a delay, given by a gamma distribution, and allow drugs to be partially effective. Tam [9] assumes that infected cells produce virus after a fixed delay and shows that including this delay does not affect the stability properties of infected steady state of the model for reasonable parameter values. The effects of drug therapy are not studied. Herz et al. [5] examined the effect of including a constant delay in the source term for productively infected T-cells but their analysis was limited to the case where the drug treatment was assumed completely effective. Mittler et al. [6] also assumed that there was a delay between initial infection and the start of virus production, a period that virologists call the eclipse phase of the viral life cycle, but rather than using a fixed delay they assumed that the delay varied between cells according to a gamma distribution. They used this model to examine if the parameters that characterize viral dynamics could be identified from biological data and then compared the parameter estimates obtained in a previous model without delay and those obtained with their delay model [10]. These works extended the ability of models to describe the relevant biological processes and addressed the implications of ignoring the intracellular delays that are part of the viral life cycle. In the models of Herz et al. [5] and Mittler et al. [6], the drug was assumed to completely block viral production. This assumption, we will show, cancels out the effect the intracellular delay has on the decay rate of virus producing T-cells. Hence, while noticing a change in viral dynamics during the eclipse phase, it was assumed the delay had no effect during the first phase's exponential decline. This paper's main focus is to examine the change in dynamics when a delay of discrete type is considered and the drug, a protease inhibitor, is assumed less then perfect. We have already shown [6], [11] that when fitting a model to experimental data there are significant changes in the values of the infected T-cell loss rate and the viral clearance rate constant if the model includes a delay. Here, we derive a new approximation, e−δnpC(τ,np,δ)t, for the asymptotic first-phase viral decay when a constant intracellular delay τ is considered and protease inhibitor treatment is applied. Here the new factor is C(τ,np,δ)=[1+(1−np)δτ]−1, where np is the effectiveness of the protease inhibitor. We provide a detailed analytical study of the model equations and examine the effects of adding a delay on the stability of solutions, something that was neglected in the analysis of previous delay models (see Table 1).
Section snippets
General model
The models in this paper only deal with dynamics occurring after drug treatment. The models are not designed to examine the progression from time of infection until drug initiation [12], [13]. The study of HIV-1 dynamics has focused on the use of a general set of model equations that keep track of different cell populations and virus. In one case the model has been linear [4], while in most cases the model includes non-linear effects [14], [15], [16], [17]. In most cases the models, while
Intracellular delay model
In HIV-1 infection, the virus life cycle plays a crucial role in disease progression. The binding of a viral particle to a receptor on a target cell initiates a cascade of events that can ultimately lead to the target cell becoming productively infected, i.e. producing new virus. The previous model (1) assumed this process to occur instantaneously. In other words, it is assumed that as soon as virus contacts a target cell the cell begins producing virus. However, in reality there is a time
Delay effects on dominant eigenvalue
With the results of the previous section we can now present the main finding of this work: an asymptotic representation for the dominant eigenvalue and how the delay effects its value (see Fig. 1). Since we have shown that for μ∈(−3,0) there are no complex roots for a defined τ, we can let be the unique root of with the largest real part. Expanding in a Taylor series givesprovided that is sufficiently small.
Conclusions
The main goal of this paper was to show that the viral decay seen in HIV-1 infected patients put under antiretroviral therapy depends not only on the lifespan of viral producing cells, δ, but also the efficacy of the therapy, np and the length of the intracellular delay, τ, that measures the time between viral infection of a cell and the time the cell begins releasing virus. Here we have focused on the effects of protease inhibitor therapy because detailed data have been published about the
Acknowledgements
The authors thank John Mittler and Hans Weinberger for their help and Duncan Calloway for critically reading and commenting on the manuscript. Portions of this work were performed under the auspices of the US Department of Energy and supported by NIH grant RR06555 (ASP) and an Institute for Mathematics and Its Applications Postdoctoral Fellowship (PWN) with funds provided by the National Science Foundation.
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