May 23, 2013, Thursday

# MBW:HIV Dynamics

This wiki page summarizes the paper "HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time’’ by Perelson et al. (1996). A pdf of the paper is available here: http://www.tiem.utk.edu/~mikeg/courses/EID.S07/readings/Perelson.et.al.96.pdf

## Overview

• Mathematics used: Differential equations, steady-state analysis, quasi steady states, linear regressions, non-linear least squares fitting of data, virion production and clearance rates, and confidence intervals.
• Biological system studied: Interactions between the human immunodeficiency virus (HIV-1) and possible inhibitor drugs.

## Executive Summary

In the paper "HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time," Perelson et al. developed a mathematical model to further investigate the dynamics between HIV and an inhibitor drug known as ritonavir, which acts as an HIV-1 protease inhibitor. The authors used this model to predict the life-span of both infected cells and virions, the average total HIV-1 virions produced per day, the minimum duration of HIV-1 life-cycle in days, and the average generation time of a HIV-1 virion. This model and its conclusions about the dynamics between HIV-1 and an inhibitor drug increased the understanding of HIV and its interactions with potentially life-saving drugs. Hopefully this information can be used in the future to refine treatment strategies that maximize survival of those infected by this virus.

## Background and History of HIV-1 and Modeling it's Dynamics

Today, there are two known strains of HIV, known as HIV-1 and HIV-2. Scientists have classified HIV-1 to be the more dangerous of the two strains of virus, as it is usually proves to be deadlier and more contagious than HIV-2.

### The Spread and Detection of HIV

HIV spreads throughout a person’s body with the help of the individual’s own cells. Once a cell has been infected by the virus, the cell is used by the virus to produce more HIV, and the virus continues to spread. To measure a person’s level of infection with HIV-1, one draws blood and measures the concentration of CD4 lymphocyes. As the virus progresses, a person has less of these lymphocytes and more HIV in the blood. Two recent tests for HIV are known as HIV Elisa and HIV Western blood tests. These tests identify the HIV by detecting the specific antibodies the human body produces when HIV exists in their system. After what can be years of infection with this virus, both untreated and treated people may develop AIDs (acquired immunodeficiency syndrome), which eventually leads to the carrier’s death. A person with HIV-1 develops AIDs after the CD4lymphocyte count in their blood drops below a critical point. At this point, the body can no longer fight off infections, even those as common and harmless as a cold. After the CD4 lymphocyte count drops below a certain level, HIV has suppressed a person’s immune system and death is usually eminent.

### Treating HIV

Current medications for HIV are aimed at preventing any complications that may go hand in hand with the disease, as well as delaying the onset of AIDs. Medicines for HIV do one of two things to the virus: interfere with virus’s reverse transcriptase, or interfere with the virus’s protease. The main function of reverse transcriptase of HIV is to change the genetic material of the virus so it can enter cells and the nuclei undetected. If the HIV is able to enter the cell’s nucleus and be detected, it is then able to replicate itself using the cell. The main function of the enzyme protease in HIV is to cut long chains of proteins into more manageable lengths that will then produce new active copies of the virus. Protease inhibitor medicines help to stop the protease from cutting the proteins, so the new copies of the virus do not contain any protein chains, and are therefore noninfectious. For a visual representation of how protease inhibitors work, please see Figure 1.

Figure 1: How protease inhibitor medications work to produce noninfectious copies of HIV

Unfortunately, there is no cure for HIV, and people with the virus will eventually develop AIDs. With no miracle drug on the horizon that cures HIV, the more people know about how the dynamics of the virus work with the human body’s natural responses and current medicines, the better doctors can treat the disease and lengthen the amount of time until a person develops AIDs.

### Research and Models of HIV Dynamics

Since HIV became prevalent around the world in the 1980s, the amount of research on the subject has increased dramatically. Many mathematical models have been developed to study the dynamics between the virus and the drugs people use to combat the disease. The authors have done a previous study on the subject in addition to the study discussed here. Mathematical models of the disease are constantly being updated as more information becomes available and as more precise measuring techniques are developed.

## Experimental Methods

Five individuals at varying stages of HIV-1 were used in this study. Twice a day, 600mg of ritonavir, a known HIV drug, was administered to each of the five individuals. Following the drug regimen, the concentrations of HIV-1 RNA in each person’s blood was measured at set times. The concentrations were measured every 2 hours until the sixth hour, and every 6 hours until the second day, then every 24 hours until day 7. In drug treatments, there is usually a lag time between the first dose and when the first drug-induced change is seen. This lag was present in this study, and can be attributed to the lag in protease inhibitors. Protease inhibitors do not prevent the production of new virions and do not prevent the infection of new cells by infectious virions that were produced prior to the drug treatment. Protease inhibitors simply cause newly produced virions to be noninfectious. This accounts for the lag time between the administration of the drug and any noticeable effects of the drug. After the lag time, the decay of HIV-1 RNA in the blood of each individual followed the mathematical model presented in the next sections.

## Mathematical Models

For this mathematical model, Perelson et al. defined many variables and parameters to be used in differential equations (for a discussion on how some of these parameters are estimated see [MBW:Estimation Of HIV/AIDS Parameters]). For a discussion of additional developments in these parameters see Perelson 2002. These variables and parameters include:

$T$ = target cells

$k$ = rate constant at which HIV-1 infects target cells

$T*$ = productively infected cells created by the rate constant k

$V$ = concentration of viral particles in plasma

$\delta$ = rate of loss of virus-producing cells

$N$ = number of new virons produced by each infected cell across its lifetime

$c$ = rate constant for virion clearance

$V_{1}$ = plasma concentration of virions before the drug treatment (still infectious)

$V_{N1}$ = concentration of virions after the drug treatment (noninfectious)

$t$ = 0 is the time of onset of the drug effect

Before the treatment with ritonavir, the model of the dynamics of cell infection and virion production is given by:

$dT^{*}/dt = kVT - \delta T^{*}$

$dV/dt = N\delta T^{*} - cV$

The authors attributed the loss of infected cells to viral cytopathicity, immune elimination, or processes such as apoptosis (cell death). Also, the authors explained that virion clearance may be a consequence of virions binding and entering into cells or immune elimination. Many reasonable assumptions were made about the model. These assumptions include: the drug ritonavir does not change the rate that virions are produced in infected cells, or the survivability of these virions. Two more assumptions were that after the lag time, all the freshly constructed virions are noninfectious, and that ritonavir completely inhibits infectious viral production. So, after the ritonavir treatment, the model is given by:

$dT^{*}/dt = kV_{1}T - \delta T^{*}$

$dV_{1}/dt = -cV_{1}$

$dV_{N1}/dt = N \delta T^{*} - cV_{N1}$

For a description of the full mathematical model, please see the section below.

## Full Model and Results

To further the above model, it was assumed that the system was at quasi steady state before drug treatment. Also, it was assumed that T, the uninfected cell concentration, remained approximately at $T_{0}$, its steady-state value, for one week after a dose of ritonavir was given. It was then concluded, from the above equations, the total concentration of plasma virions, $V = V_{1} + V_{N1}$, varies according to:

$V(t) = V_{0}e^{-ct} + cV_{0}/(c- \delta)$

If T is allowed to increase, numerical methods must be used to predict V(t). However, this does not significantly change the conclusions of the following analysis.

Nonlinear regression analysis was used to estimate both c and δ for each patient. This was accomplished by fitting the above equation to the plasma HIV-1 RNA measurements (Table 1). With the use of the best-fit values of c and δ, the theoretical curves obtained from the above equation provided an exceptional fit to the data for all patients. It was found that the clearance of free virions is the faster process. Values of c ranged from 2.06 to 3.81 per day, while the corresponding $t_{1/2}$ values for free virions ranged from 0.18 to 0.34 days. An independent experiment was used to confirm the virion clearance rate. It was found that the loss of infectious virions exhibited a first-order decay rate. The rate constant was determined to be 3.0 per day. This is within the 68% confidence interval of the estimated value of c.

Table 1: Summary of HIV-1 clearance rate, infected cell loss rate, and virion production rate for the five patients.

An estimate for the rate of virion production before ritonavir was administered was obtained by using the estimated c value and $V_{0}$, the pretreatment viral concentration. It was also taken into account that the production rate of the virus must equal its clearance rate, cV. The total virion production and clearance rates were found to range from 0.4 x 10^9 to 32.1 x 10^9 virions per day, with a mean of 10.3 x 10^9 virions per day released into the extracellular fluid. It was also found that the rate of loss of virus-producing cells was much slower than that of free virions. Values of δ were determined to range between 0.26 to 0.68 per day, while the corresponding $t_{1/2}$ values ranged from 1.02 to 2.67 days. Please see Table 1.

## Analysis and Interpretation

As a result of the model described above, many characteristics of the replication cycle of HIV-1 in vivo can be determined. The parameters c and δ correspond to the decay rate constants of plasma virions and productively infected cells, respectively. Therefore, 1/c and 1/δ correspond to the average life-spans of plasma virions and productively infected cells, respectively. The average life-span of a virion is approximately 0.3 days. The average life-span of a productively infected cell is approximately 2.2 days (Table 2). In addition, the average viral generation time $\tau$ is defined to be the time from the release of a virion until the virion infects another cell and initiates the release of a new wave of viral particles. Therefore, $\tau$ is equal to the sum of the average life-spans of a free virion and a productively infected cell. The average value of $\tau$ for the patients in the experiment is approximately 2.6 days (Table 2).

Table 2: Summary of virion life-span, infected cell life-span, duration of the viral life cycle and of the intracellular phase, and average viral generation time for the five patients.

Minimal estimates for the average duration of the life cycle of HIV-1 and its intracellular phase (from binding to the release of the first progeny) were determined using a heuristic procedure. The parameter S was defined as the life cycle of HIV-1. The lag in the decay of HIV-1 RNA in plasma after the pharmacological delay is subtracted was used to estimate S. The values of S estimated for each of the patients were quite consistent. The mean of S was approximately 1.2 days. In addition, the average time for infection is given by 1/c. If this average is assumed to be larger than the minimal time for infection, then a minimal estimate of the average length of the intracellular phase of the HIV-1 life cycle is determined by S – (1/c) and found to be 0.9 days.

In previous studies, a crude estimate of the $t_{1/2}$ of viral decay was given without the life-span of productively infected cells being separated from the life-span of plasma virions. However, with the results described above, it was calculated that productively infected cells are lost with an average $t_{1/2}$ of approximately 1.6 days. Between the five patients, the life-spans of productively infected cells were not significantly different. This is interesting as a person with a low CD4 lymphocyte count typically has a decreased number of virus-specific, major histocompatibility complex class I-restricted cytotoxic T lymphocytes.

Figure 2: Schematic summary of the dynamics of HIV-1 infection in vivo.

In addition, the mean life-expectancy of a virion in the blood was found to be 0.3 days. As a result, a population of plasma virions is cleared with a $t_{1/2}$ of 0.24 days. In other words, on average, every six hours a population of plasma virions is turned over. The estimates for the virion clearance rate and infected cell loss rate are minimal estimates, as a result of the assumption that the antiviral effect of ritonavir was complete and that the target cells did not recover during treatment. Hence, the true value of $t_{1/2}$ for a virion may be less than six hours. As a consequence, the total number of virions that are produced and released into the extracellular fluid is at least 10.3 x 10^9 virions per day. This is approximately 15 times the previous minimum estimate. Furthermore, at least 99% of the substantial group of virus is produced by recently infected cells (Fig. 2). Due to the fact that c has similar values for each patient involved in the study, the degree of plasma viremia is a representation of the total virion production, which is proportional to the number of productively infected cells T* and their viral burst size N. As the average generation time of HIV-1 is about 2.6 days, the authors suggest that approximately 140 viral replication cycles occur each year. This is about half the number of viral replication cycles estimated by Coffin. With this conclusion, it becomes clear that the repetitive replication of HIV-1, shown on the left side of Fig. 2, is responsible for at least 99% of the plasma viruses in infected individuals and causes the high destruction rate of CD4 lymphocytes.

## Conclusion

This new insight into the highly dynamic nature of the cyclic process of HIV-1 replication provides many theoretical ideas to direct the advancement of treatment. The authors strongly suggest that if an antiviral agent is to be effective, it should detectably lower the viral load in plasma after a few days of treatment. Next, one must take into account the large turnover rate of HIV-1 described above. In fact, the failure of the antiviral agents used around 1996 in monotherapy is the result of the HIV-1 replication dynamics. The authors argue that an effective treatment must force the virus to mutate simultaneously at multiple positions in one viral genome by means of a combination of multiple, potent antiretroviral agents. Early and aggressive therapeutic attention is necessary to make a clinical impact, especially as the production of mutant viruses is repeated for approximately 140 generations each year. Lastly, although it is apparent that the “raging fire” of HIV-1 replication could be extinguished using potent antiretroviral treatments in two to three weeks, the dynamics of other viral compartments must be studied. Even though these viral compartments cause 1% or less of the plasma virus, they have the ability to restart a high rate of viral replication after the therapeutic treatment has concluded. In the future, the decay rate of long-lived, virus-producing groups of cells must be determined. The activation rate of latent cell compartments carrying infectious proviruses is also essential to determine. The authors hope that this information will allow a new type of treatment to block de novo HIV-1 replication for a long enough period of time for each latent cell compartment to extinguish.