Natural variation in HIV infection: Monte Carlo estimates that include CD8 effector cells
Introduction
Changes in CD4 lymphocyte counts and viral load (RNA copies) are widely used as indicators of HIV progression in infected patients. These measures inform decisions regarding the initiation or continuance of antiretroviral therapy and are likewise used to assess therapeutic effects in clinical trials. It is therefore critical to define the expected variance between consecutive measurements of T-cell count and plasma RNA levels, since random fluctuations may obscure clinically significant changes.
Total variability for sequential CD4 T-cell determinations from asymptomatic (clinically latent) HIV infected individuals has been examined clinically (Hughes et al., 1994, Paxton et al., 1997). Hughes et al. (1994) found that the within-patient mean standard deviation (SD) of T-cell count was , corresponding to a coefficient of variation (CoV) of , when derived from data at 0 and 8 weeks. Paxton et al. (1997) determined that the long-term within-individual variability for T-cell measurements had a CoV of , equivalent to a SD of .
Similar investigations measuring instead the variation among consecutive measurements of HIV RNA in plasma have revealed a total variability, expressed as SD, of 0.19– (Bartlett et al., 1998, Brambilla et al., 1999, Coombs et al., 1998, Deeks et al., 1997, Hughes et al., 1997, Paxton et al., 1997, Raboud et al., 1996, Yamashita et al., 2003). The wide range reported might be attributed to the sizes of the clinical cohorts—between 11 (Deeks et al., 1997) and 387 (Bartlett et al., 1998) individuals—or to variations in antiretroviral drug therapy among individuals participating in each study. Perhaps more importantly, the intervals between consecutive measurements in these studies vary from days to over a year, and thus the reported range may be an effect of differences in sampling intervals. For example, Deeks et al. (1997) reported the highest variability of RNA copies/mL for measurements one day apart. Bartlett et al. (1998) reported a slightly lower SD of RNA copies/mL for two consecutive measurement 2–4 weeks apart. Raboud et al. (1996) reported an even lower value still of RNA copies/mL derived from subjects seven days apart and Hughes et al. (1997) reported the lowest variability of RNA copies/mL based on paired baseline measurements with a median interval of 6 days apart (2–15 days).
A handful of mathematical models of HIV dynamics have attempted to quantify the inherent variability in HIV infection (Heffernan and Wahl, 2005, Kamina et al., 2001, Tan and Wu, 1998, Tuckwell and Le Corfec, 1998, Wu and Ding, 1999). Tuckwell and Le Corfec (1998) used SDEs to estimate the intrapatient variability in both the time to peak viremia and the viral load at this peak, two important characteristics of primary infection. Tan and Wu (1998) used a Monte Carlo simulation to estimate the probability of clearing an initial inoculum of virus. Kamina et al. (2001) later used the same Monte Carlo approach to estimate the time to peak and peak viremia. None of these investigations consider variability during the clinically latent stage of infection.
In a previous study, using a Monte Carlo model of in-host HIV infection, we investigated the variability in T-cell count and viral load that could be attributed to interactions between CD4 T-cells and free virus during both primary infection and clinical latency (Heffernan and Wahl, 2005). The simulation proceeded at the level of individual infected and uninfected CD4 T-cells and infectious and non-infectious virions. We also examined the effects of antiretrovirals, such as reverse transcriptase and protease inhibitors (RTIs and PIs), and the effects of more realistic lifetime distributions for T-cells. A limitation of this previous study was that many of the underlying processes contributing to variation in CD4 T-cell and virus concentrations in vivo were not included in the model.
In particular, the processes of activation and proliferation of CD4 T-cells directly affect cell and virus population dynamics. Naive CD4 T-cells, brought into the lymph tissues from the thymus, are stimulated by viral antigen to become activated CD4 T-cells. Activated CD4 T-cells, if they are uninfected, proliferate and aid the rest of the immune system in fighting the infection. If they are infected, activated CD4 T-cells bud virus.
Similarly, CD8 T-cells (cytotoxic T lymphocytes or CTL) also affect steady state variability. Activated CD8 T-cells (or effector cells) are produced by the activation of naive CD8 T-cells by viral antigen and CD4 T-cells. Effector cells kill infected cells displaying viral antigens (Janeway and Travers, 2005).
Analytical models based on differential equations (Culshaw and Ruan, 2000, De Boer and Perelson, 1995, Ho et al., 1995, Kirschner, 1996, McLean, 1994, Nelson and Perelson, 2002, Nelson et al., 2004, Nowak and May, 2000, Perelson, 2002, Stafford et al., 2000, Wei et al., 1995, Wodarz et al., 1998, Wodarz et al., 1999) have been used to predict the dynamics of in-host HIV infection as have numerous simulation studies (Chao et al., 2004, da Silva and Hughes, 2002, Kamina et al., 2001, Kousignian et al., 2003; Ruskin et al., 2002, Tan and Wu, 1998, Tuckwell and Le Corfec, 1998, Wu and Ding, 1999). Few of these studies, however, have incorporated the effects of the CD8 T-cell population and fewer still have included the effects of activation and proliferation in the CD4 or CD8 T-cell populations. This is sensible since the inclusion of numerous biological processes into analytical models can prove unwieldy, often requiring a numerical solution. Also, it has only recently become computationally feasible to incorporate realistic levels of biological detail in simulation models (da Silva and Hughes, 2002). Recently, Chao et al. (2004) developed an agent-based simulation to study cytotoxic T-cell responses at the individual level. This study included the effects of activation and proliferation of the CD8 T-cell population, however, it did not incorporate the activation and proliferation of CD4 T-cells. Also, the inherent variability in HIV dynamics was not examined.
In this study we further explore variability during the clinically latent phase of infection (steady state variability) by extending our Monte Carlo simulation to include the CD8 T-cell population, as well as the processes of activation and proliferation of CD4 and CD8 T-cells. Biologically motivated lifetime distributions describing cell death and CD8 T-cell activation are incorporated, and variability during drug therapy is examined using a pharmacokinetic model of drug dynamics. The correlations we observe between measures of variability separated by days or weeks may help explain some apparent discrepancies in the clinical literature.
Section snippets
Model
Our Monte Carlo simulation progresses at the level of individual uninfected CD4 T-cells (naive or activated), infected CD4 T-cells (naive or activated), virions (infectious or non-infectious), and CD8 T-cells (naive or activated). We may also include the effects of antiretroviral drugs, a RTI and a PI. The simulation moves forward in time using “event times” that are assigned to the individual cells and virions from the appropriate probability distributions i.e. death time, infection time or
Natural variability
Fig. 1 shows ten sample runs of the Monte Carlo simulation. This figure demonstrates that the viral dynamics can change dramatically from one run of the simulation to another, due to the inherent stochasticity of the model. To quantify the variability around the equilibrium for each population we determine the variance of T-cell count and viral load over the last ten months of a single run, well after equilibrium is reached. The variance is then divided by the mean uninfected T-cell count to
Discussion
We have developed a detailed Monte Carlo model of in-host HIV dynamics that includes both CD8 T-cells and T-cell activation and proliferation. The model allows us to quantify the contribution of these immune system components to the underlying variability (steady state variability) in CD4 T-cell count and viral load. These results follow from a previous study where we quantified this variability due solely to the CD4 T-cell and virus populations (Heffernan and Wahl, 2005).
The Monte Carlo
Acknowledgements
This work is supported by the Natural Sciences and Engineering Research Council of Canada, the Ontario Ministry of Science, Technology and Industry and by the SHARCNET parallel computing facility. We are indebted to Olga Krakovska for many helpful discussions and to Baolai Ge, Gary Molenkamp and Mark Hahn for their invaluable technical expertise. We also thank two anonymous referees for many insightful suggestions.
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