May 20, 2018, Sunday

# MBW:Modelling Viral Dynamics and Immune System Dynamics

This is a summary of the paper “Modelling Viral and Immune System Dynamics” by Alan S. Perelson published in Nature Reviews Immunology in 2002.

## Executive Summary

This article focuses on mathematical models of HIV and T-lymphocyte dynamics. It also briefly discusses mathematical models used to study hepatitis C, hepatitis B, and cytomegalovirus and their interactions with the immune system. It highlights the insights that have been gained through mathematical models as well as clinical studies. It uses a system of basic equations as a starting point to study these infections, and then goes over several other systems of equations that expand on the basic equations.

Biological phenomenon under consideration: Immunology

## Insights Gained from Modeling HIV Dynamics

The plasma viral load for someone suffering from chronic HIV generally remains somewhat constant--on the surface this seems to suggest that HIV’s replication rate is very slow. However, clinical trials of one of the first HIV protease inhibitors showed plasma levels of HIV RNA dropping one to two orders of magnitude in the first two weeks of treatment. This suggests that the virus replicates rapidly. The analysis of basic mathematical models of HIV dynamics provides evidence of the latter--namely it suggests two things:

-The clearance rate for HIV in chronically infected patients is very rapid--the models suggested a half life of six hours or less. -Because the viral load remains somewhat constant, the reproduction rate must be very high to counteract the rapid clearance rate. (REF 1)

Experimental data has also suggested that HIV is a rapidly reproducing virus (REF 1).

## Basic Model of Viral Dynamics

Below is a basic model of viral infection.

These are the equations for this basic model.

Here T is susceptible target cells, I is productively infected cells, and V is virus particles. T cells are produced at a rate of $\lambda$ from a source, die at at rate of d, and become infected by the virus at a constant rate of k. Infected cells produce new virions at rate p and die at rate $\delta$ per cell, which gives an average virion lifespan of 1/$\delta$. c is the clearance rate of virions per virion.

Making the assumption that an uninfected person becomes infected with a small number of viruses, the solution to the above equations fits the actual kinetics of primary HIV infection.

## Basic Model Incorporating the Effects of Antiretroviral Drugs

The following equations build off of the basic equations above, incorporating the effects of two types of antiretroviral drugs, reverse transcriptase (RT) and protease inhibitors (PI). RT’s affect a virus’s ability to infect a cell, and PI’s cause newly reproduced virus particles to be non-infectious.

Here $\epsilon _{{RT}}$ and $\epsilon _{{PI}}$ are the efficacies of the two drugs, with a range from 0 to 1 where 1 is a perfectly effective drug. $V_{I}$ and $V_{{NI}}$ represent the concentration of infectious and non-infectious virus, with the total amount of virus being equal to the sum of $V_{I}$ and $V_{{NI}}$.

## Modelling Lymphocyte Dynamics

The model below is a simple representation of T Cell dynamics.

In mathematical form, this model looks like:

Information about lymphocyte kinetics can be gathered by labeling cells with bromodeoxyuridine (BrdU) or deuterated glucose (2H-glucose). Cells take up the labelling agents into newly synthesized DNA when dividing. This separates T cells into either labelled cells, L, or unlabelled cells, U. In order to gain any quantitative information using labeled lymphocytes, it’s necessary to have a mathematical model representing the labeling system. The equations below model the dynamics of labelled and unlabelled T cells.

## Analysis/interpretation

Looking at Equation 3 of the basic model, it’s obvious that at the viral set point, where a quasi steady state exists, the viral production must be equal to the viral clearance. This gives us

pI = cV

Unfortunately this doesn’t really give us any information as to the values of the parameters of interest, p and c. In order to gain information about p and c we have to perturb the system. The system can be perturbed by giving a patient antiretroviral drugs.

Looking at equation 3, if we were able to make P go to zero through the use of a 100% effective protease inhibitor, V(t) would decay exponentially. The equation below is the solution to Equation 7 with a 100% effective PI.

The parameter c was estimated using this model and laboratory tests measuring the loss of viral infectivity in a patient with a high baseline viral load. Below is a graph of data gathered from an HIV infected patient beginning antiretroviral therapy.

The solid line represents the best fit theoretical curve, and this is where the parameter c was estimated from. The parameter $\delta$ was estimated from this as well.

The parameter c was also estimated using models of plasma apheresis combined with data from apheresis experiments. This latter method supported a high clearance rate for HIV, with a mean half life of about 1 hour (REF 2). Estimates for c and V were plugged into P = cV to give an estimate for P. This gave an estimated minimum value of $10^{{10}}$ virions produced daily for P (REF 1). Furthermore, this estimated value of P along with the mutation rate and genome size of the HIV can be used to compute the number of point mutations that occur per day in the HIV genome. These computations showed that on average, every possible point mutation will occur multiple times per day, as well as a significant number of double mutations (REF 3). This is the mathematical basis for the multi drug approach to treating HIV infected patients.

Clinical studies backed this up further--when three or more drugs were given to HIV infected patients, the plasma viral load decayed exponentially in two phases, first at a rapid rate, and then at a slower rate until the viral load fell below detectable levels.

The original model didn’t explain the two phase decline, so a new model was introduced by Perelson, et al which hypothesized that there is a secondary source of HIV not incorporated into the basic model (REF 4). Data obtained from an HIV infected patient beginning combination antiretroviral therapy is shown below.

The solid line in the graph is the best fit curve, from which model parameters and half life of the virus were estimated.

Several secondary sources were proposed--a population of infected cells with a longer half life, latently infected cells that become activated, and virions that are released into the blood that had been trapped in tissue reservoirs. One study showed CD4+ cells could represent a population of infected cells that decay very slowly, with a half life anywhere from 6 months to 43.9 months (REF 5). In addition, mathematical models showed that virions could remain in some tissue reservoirs, namely follicular dendritic cells (FDCs), for over a decade and that their release rate would be consistent with a two phase decay (REF 6, 7, 8). Experimental data in mice also gave evidence to the theory that virions could remain in tissue reservoirs (REF 9).

One of the surprising aspects of the basic model is that even though it doesn’t contain a term taking the immune system’s response into account, it still can account for the kinetics seen in the early stages of infection. More specifically, the basic model shows an initial rapid rise, followed by a peak, decline, and leveling off of viral loads, similar to what’s seen in the initial stages of individuals infected with HIV. In fact, clinical data shows that the basic model fits the data for the first 100 days of HIV infection (REF 10).

In the basic model the rate of infected cell death ($\delta$) is assumed to be a constant. This means that if the immune response does affect this process it’s effect is either relatively small or happening at a constant rate. In chronic infections, it’s reasonable to assume that the effects of the immune response would be constant since the viral load remains constant. However, this doesn’t make sense in the initial stages of infection, where the immune system generates a response to infection that has been shown to have a temporary effect on viraemia (REF 11). Analysis of the basic model provides an explanation of how it can demonstrate a rise, peak, and fall of viral load even though it can only account for either a small or constant effect from the immune response--the decline in viral load is due to running out of target cells (REF 12).

Although the basic model often accounts for the kinetics seen in the early stages of HIV infection, some clinical data showed a sharper decline in viral load after the initial peak than the basic model predicted. Interestingly enough, after their unpredicted rapid decline their set point viral load evened back out to a level consistent with the basic model (REF 10). The fact that the basic system, without a term to account for the effects of the immune system, is mostly consistent with the early stages of infection except for the period between the viral load peak and the flattening out to the set point points to an immune response that only affects viral loads during the time between the peak and flattening out. Speculations to the cause of this loss of the immune response’s effectiveness include the loss of HIV specific helper T cells and the dysfunction of effector CD8+ T cells (REF 13, 14). These speculations are based on insights gained from modeling as well as limited data.

One final insight that this paper reviews deals with the transition from HIV to AIDS. Looking at Equation (9) above, it’s obvious that once the viral load has evened out to its set point, and the system is in a quasi steady state, the production of T cells would decrease if the s or p parameters decrease or if the d parameter increases. This means that for HIV to transition into AIDS either the production of T cells has to decrease or the cellular destruction of T cells has to increase. Data from HIV infected patients has shown a threefold increase in proliferation and cell death rates of CD4+ T cells that then reduce to nearly normal levels after a year of antiretroviral drug therapy (REF 15). This suggests that the CD4+ T cell depletion seen in AIDS patients is due to an increase in cellular death.

For a thorough investigation of another study focusing on HIV/AIDS parameter estimation see [MBW:Estimation Of HIV/AIDS Parameters]

## Viral Set Point Analysis

High viral set points for patients with HIV lead to a faster progression to AIDS, while a low viral set point leads to a slower progression to AIDS. Several studies have set out to find which parameters determine the viral load of the set point. Data shows that patients with different viral set points have slight differences in almost all of their estimated parameters (REF 10). This along with other analysis (REF 16) leads researchers to believe that the set point is influenced by multiple parameters.

One of the factors believed to influence the set point viral load is the role of CD8+ T Cells. Several experiments in SIV infecteted rhesus macaques in which CD8+ T cells where depleted showed a significant rise in plasma SIV virons (REF 17, 18). By mimicking this depletion of CD8+ T cells through decreasing the $\delta$ term in the basic model’s equations, the basic model was shown to exhibit a tenfold increase in viral load over one week (REF 18). The experiment with rhesus macaques showed a similar rise, although much higher, with some macaques exhibiting a 3 or 4 order of magnitude increase.

## Modelling Other Viruses

In addition to HIV, the basic model previously discussed has been used to understand hepatitis C (HCV), hepatitis B (HBV), and cytomegalovirus (CMV). These viruses, similar to HIV, can all lead to chronic infection.

HCV

When treated with Interferon-alpha, a protein naturally secreted by many cell types, the viral load of individuals infected with HCV can drop by two orders of magnitude. Mathematical models showed that this could be because interferon alpha causes a drastic reduction in viral production. Clinical trials gave supported this, showing that interferon alpha confers a dose dependant reduction in viral production (REF 19, 20). Mathematical models also gave evidence that the steady state P of HCV was even higher than HIV, with about 10^12 virions per day being produced and cleared by the immune system. The half life of an HC virion was also shown to be about 3 hours.

HBV

HBV was modelled using the basic equation used for HIV as well as a model that included cells being cured from the infected state (REF 21-25). HBV was also shown to be rapidly produced, with steady state viral production rate of $10^{{11}}$-$10^{{12}}$ virions being produced per day.

CMV

The basic viral dynamic model was also applied in studying CMV, a virus that was originally thought to have a slow viral production rate. By using models and perturbing steady states through drug therapy, it was shown that CMV replicates rapidly, doubling its viral load in about a day (REF 26). It was also shown that the primary drug used in treating CMV is 91.5% effective when given intravenously, while only 46.5% effective when taken orally (REF 27). This probably explains the emergence of drug resistant strains of CMV in infected individuals taking the drug orally for long periods of time.

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