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Journal of Virology, September 2002, p. 8963-8965, Vol. 76, No. 17
0022-538X/02/$04.00+0     DOI: 10.1128/JVI.76.17.8963-8965.2002
Copyright © 2002, American Society for Microbiology. All Rights Reserved.

Decelerating Decay of Latently Infected Cells during Prolonged Therapy for Human Immunodeficiency Virus Type 1 Infection

Viktor Müller,1* Javier Flavio Vigueras-Gómez,2 and Sebastian Bonhoeffer1

Ecology and Evolution, ETH Zürich, 8092 Zürich, Switzerland,1 INRIA-Lorraine, 54600 Villers-les-Nancy, France2

Received 12 February 2002/ Accepted 20 May 2002


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ABSTRACT
 
Antiviral therapy induces a rapid drop in human immunodeficiency virus type 1 viremia, but the decline of virus levels decelerates over time. Mathematical modeling demonstrates that the source of residual virus production might be a single compartment of latently infected cells with an extended distribution of activation rates.


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TEXT
 
The rapid initial decline of virus levels after the initiation of highly active antiretroviral therapy (HAART) reflects the decline of productively infected CD4+ T cells (15, 18, 24). The slower second phase has been attributed to another cell population that produces fewer viruses initially but that has a slower turnover rate (17). Eventually, virus RNA becomes undetectable in patients with sustained suppression of virus replication. Occasional viral "blips" and the rebound of virus after discontinuation of therapy (5, 9), however, indicate the persistence of virus reservoirs even after 5 or more years of suppressive treatment (for a recent review on reservoirs see reference 21). As even the hypothetical persistently infected cell population should dwindle to a point of extinction in 2 to 3 years (17), the source of residual virus replication must be sought elsewhere. Finally, statistical analyses have shown that the observed decay of the virus level cannot be described in terms of a simple exponential decay (S. Holte, A. Melvin, J. I. Mullins, and L. Frenkel, Abstr. 8th Conf. Retroviruses Opportunistic Infect., abstr. 394, 2001; M. C. Strain, C. C. Ignacio, T. R. Macaranas, O. Daly, R. Lam, H. Levine, H. Günthard, D. V. Havlir, E. Daar, S. J. Little, D. D. Richman, and J. K. Wong, Abstr. 9th Int. Workshop HIV Dynam. Evol., abstr. 24, 2002).

Latently infected CD4+ T lymphocytes have been shown to harbor replication-competent integrated provirus even after years of suppressive therapy (8, 12, 25). The pool of latently infected cells is established early during primary human immunodeficiency virus type 1 (HIV-1) infection (6), presumably by the reversion of productively infected memory CD4+ T cells into the resting state (20). In vitro these cells can be induced to produce infectious virus, and activation has been shown to occur also in vivo upon encounter with the appropriate recall antigens (14) or by the administration of activating lymphokines (7). In patients with apparently complete suppression of HIV-1 replication by HAART, the source of rebounding virus after the cessation of therapy was identified as the latently infected cell pool (26). However, the exact contribution of this pool to the residual virus production during fully suppressive HAART has not been clarified. Here we present a mathematical model on the dynamics of latently infected cells and their impact on residual virus replication. We show that a decelerating decay might be an inherent characteristic of this system.

Our goal was to model virus production by reactivated latently infected T cells after prolonged fully suppressive therapy when the initial major virus-producing population has already dwindled. This restricted goal allowed us to neglect much of the complexity of HIV-1 infection. We considered latently infected cells, L, which die at rate {delta}LL and which are reactivated at rate {alpha}L, and cells actively producing virus, T*, which arise by the reactivation of latently infected cells and which die at rate {delta}T*. The generation of latently or productively infected cells by new rounds of infection is assumed to be fully suppressed by the antiviral therapy. The reversion of cells actively producing virus to the resting state can also be neglected, as this cell pool plummets after the start of therapy. Virus particles, V, are produced at rate pT* and decay at rate cV. Thus we write

((1a))

((1b))

((1c))

The clearance of latently infected cells reflects the sum of activation and death. Over time, one obtains a simple exponential decay as L(t) = L0, where L0 is the initial size of the latently infected cell pool. The dynamics of latently infected cells are much slower than those of cells actively producing virus, which allowed us to assume a quasi-steady state for the latter. By setting equation 1b to zero we derived T*(t) = ({alpha}/{delta})L(t). Similarly, as virus particles turn over much faster than virus-producing cells (16, 18, 22), the virus level will follow the latter at a constant ratio: V(t) = (p/c)T*(t). In all, the virus level maintained by production from reactivated latently infected cells can be obtained as

((2))

This form of the model predicts that virus production by this cell population declines parallel to the number of latently infected cells at a rate reflecting the sum of reactivation and death. However, the pool of latently infected cells is likely to be heterogeneous in the activation rate. Some cells are specific to common antigens and hence have a high probability to become activated, while other cells are specific for rare antigens and might persist for a long time without activation. This situation can be modeled by employing a continuous range of activation rates instead of a single fixed value. This transforms equation 2 into the form

where L0({alpha}) defines the initial distribution of the latently infected cell pool with respect to the activation rate, which has a maximum value of {alpha}max. The total initial pool size of around 106 cells (4) can be obtained as L(0) = L0({alpha})d{alpha}. It can be shown that for all biologically plausible distributions (L0[{alpha}] should be a smooth function of {alpha} and nonzero when {alpha} is close to zero; among others, uniform, normal, exponential, log normal, and gamma distributions fulfill these criteria) the behavior of the virus level is decelerating decay over time. The total number of latently infected cells declines according to the equation

Figure 1a shows an example with a normal distribution of activation rates for the initial cell pool. The decline of the virus level maintained by reactivated cells and that of the latently infected cell pool decelerate over time. This is consistent with previous observations on the apparent biphasic decay of latently infected cells (2, 19). Figure 1a also indicates the slopes reported for the first few weeks (half-life around 6 to 8 days [2, 17]) and the following months (half-life around 6 months [23, 27]).



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  		FIG. 1. Total body load of virus and latently infected cells during HAART. (a) Number of latently infected cells (dashed line) and the level of virus maintained by these cells (heavy solid line). Light lines, decays with half-lives reported for the first few weeks of treatment (6 to 8 days) and for the following months (6 months). (b) Overall virus level (heavy solid line) including virus produced by the initial virus-producing population (dash-dotted line) and by long-lived persistently infected cells (dashed line). The first two phases of decay are dominated by these two cell populations, while the third phase is dominated by production from reactivated latently infected cells (light line). In both panels the initial population of latently infected cells is normally distributed, with parameters µ = 0.1d-1 and {sigma} = 0.1d-1, and is scaled to yield a total of 106 cells. The distribution has been truncated at zero and renormalized to exclude negative activation rates. Further parameters and initial values were {alpha}max = 1d-1, p/c = 100, {delta}L = 0.001d-1, pMM0 = 3 x 109d-1, {delta}M = 0.066d-1, T0 = 5 x 108, and {delta} = 0.7d-1.

Contrary to previous hypotheses, however, our model explains a decelerating decay without assuming a labile, preintegration form of latency. Deceleration here is rather a natural consequence of the heterogeneity in the activation rate as latently infected cells with a higher probability of activation (specific for common antigens) are depleted fast and early, while cells with a lower activation probability (specific for rare antigens) can persist for a long time. Eventually, the decay rate of latently infected cells approaches the innate death rate, {delta}L, which might be extremely slow (half-life of 43.9 months [11]). Note that our model predicts that the virus level decays somewhat faster than the latently infected cell pool (Fig. 1). The reason for this is illustrated by equation 2, which describes the simple case with a single fixed activation rate. The steady-state ratio of the virus level to the number of latently infected cells is linearly proportional to the activation rate. With a distributed activation rate, this implies that as the distribution shifts toward lower values, the ratio of virus level to the number of latently infected cells also declines over time, which results in a faster decay of the former.

To model the total virus level from the initiation of therapy, we needed to incorporate also the initial fast-producing cell populations into the model. In this we adopted the results of Perelson et al. (17) to obtain the total virus level as V(t) = VL(t) + (p/c)T0e-{delta}t + (pM/c)M0, where T0 is the initial population of cells actively producing virus, M0 is the initial population of long-lived persistently infected cells, and pM and {delta}M are the virus production rate and death rate, respectively, of the latter cells. We set the initial values and the additional parameters to correspond approximately to the averages obtained by Perelson et al. (17). The graph of overall virus production was plotted in Fig. 1b. The result is an apparently triphasic decline in which the third phase reflects the long tail of the decelerating production by reactivated latently infected cells. Deceleration at this stage is already very slow, which results in an apparently linear third phase. However, the slope of third-phase virus decay always remains steeper than the innate death rate of latently infected cells due to the decreasing ratio of virus to cells.

A decelerating decay of virus production by latently infected cells is not the only possible explanation for the observed persistence and long-term dynamics of HIV-1 under HAART. It has been shown that low steady-state viral loads can be explained in terms of density-dependent infected-cell death rates or compartment models including a drug sanctuary or a privileged cell type where drugs have a reduced effect (3). The waning of the HIV-specific cytotoxic immune response may also result in a decelerating virus decay if the response has a strong effect on the life span of virus-producing cells (1). A predecessor to our model is the work by Ferguson et al. (10), who assumed a distributed activation state (in terms of division times and susceptibility to infection) of the primary virus-producing cell population. Finally, Grossman et al. (13) attributed the slower second phase to the decreasing frequency of local virus bursts and also showed that under imperfect therapy new infectious bursts initiated by reactivated latently infected cells may continuously reseed the reservoir and maintain a stable steady-state virus level.

To summarize, our model predicts that the source of residual virus production after long-term sustained suppressive treatment might indeed be the pool of latently infected cells. By its decelerating decay, virus production from this compartment eventually dominates production from any other compartment with a fixed turnover faster than the innate decay of latently infected cells. Importantly, however, the deceleration of virus decay can be explained by production from this single compartment without invoking multiple virus-producing cell types with graded turnover rates. We have also demonstrated that heterogeneity in the activation rate can explain the observed behavior of the latently infected cells themselves: fast initial decay, which eventually decelerates to the slow innate turnover of this compartment. Finally, testable predictions of our model are that (i) during treatment the distribution of the specificity of latently infected cells is gradually shifted toward rare antigens as the cells specific for common antigens are expected to be depleted faster and (ii) the ratio of virus level to latently infected cell level declines over time.


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ACKNOWLEDGMENTS
 
We are grateful to Matt Strain for useful discussions.

V.M. was partially supported by the Hungarian Scientific Research Fund (OTKA), and S.B. gratefully acknowledges financial support from the Swiss National Science Foundation.


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FOOTNOTES
 
* Corresponding author. Mailing address: Ecology and Evolution, ETH Zürich, Clausiusstr. 25, ETH Zentrum NW, 8092 Zürich, Switzerland. Phone: (41) 1 6327105. Fax: (41) 1 6321271. E-mail: mueller{at}eco.umnw.ethz.ch. Back


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Journal of Virology, September 2002, p. 8963-8965, Vol. 76, No. 17
0022-538X/02/$04.00+0     DOI: 10.1128/JVI.76.17.8963-8965.2002
Copyright © 2002, American Society for Microbiology. All Rights Reserved.



This article has been cited by other articles:

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