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MBW:Delayed Production and Viral Dynamics of the HIV-1 Virus

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This is an article is written by Sai'd Azzam and John Nelson Kane

Executive Summary

In this mathematical model we attempt to better understand the dynamics of the HIV-1 infection by incorporating an intracellular delay, which is the time period between an antiviral drug reaching a cell and the effectiveness of the drug. In the past, models have made the unrealistic assumption that the intracellular delay is constant or there is no delay after ingestion of the drug. This model introduces this delay in the form of a continuous distribution with the underlying principle that different cells will be associated with a different delay time. Following the solutions to the model, it will be shown that results are not highly dependent on the form of this initial delay distribution. It will be shown that a study of different parameters will allow one to solve for the viral clearance rate and to theoretically model the decay of the HIV virus in the human body following an antiviral drug ingestion.

Introduction

The Human immunodeficiency virus (HIV) is a replicating virus and the cause of acquired immunodeficiency syndrome (AIDS). HIV-1 is the most common and pathogenic strain of the virus and is typically associated with three stages. Once a patient is first infected, the amount of virus found in the blood stream is relatively high. This high concentration may last several weeks to a few months at which point the concentration decreases to some stable lower level. Following the beginning of this stable level of virus concentration, a typically long period follows where the CD4+ T-cell count slowly decreases. A CD4+ T-cell is also known as a T helper cell that is a type of white blood cell used to help the activity of other immune cells. This usually long period of a decrease in T-cells appears to be a quasi-steady state, because although it appears to be stable, the virus concentration is noticed to change slightly when analyzed on a long time scale. This quasi-steady state can last for up to a few years with little change. In the final stage of the disease, the CD4+ T-cell count drops to a point in which the immune system fails and the viral load increases rapidly, usually causing fatality.

Much research has been done to better understand this middle period in which the T-cells slowly decrease and the virus concentration stays at a relatively stable value. If one can provide the correct combination of antiretroviral drugs with treatment to better stabilize the white blood cells than one could dramatically prolong life and even essentially make the disease nonfatal to the human body.

Delay / Background

There are two different types of delays that will effect the infectious/treatment process: a pharmacological delay and an intracellular delay. A pharmacological delay is the time between the ingestion of the drug and it’s first appearance within the cells. An intracellular delay is the time between the initial infection of a cell and the release of new viruses. Studies show from Perelson et al. that the pharmacological delay is around two hours while the intracellular delay is of the magnitude of a few days. It is also must be considered that the pharmacological delay only occurs at the commencement of treatment, while intracellular delay occurs continuously. For this reason we can ignore pharmacological delay and we will do so for the following model.


Mathematical Model With Variable/Parameter Definitions

The main goal of the model is to consider decreasing the HIV-1 RNA by treating the virus with protease inhibitor. The protease inhibitors are drugs that prevent new viruses from becoming infectious without affecting the preexisting infectious viruses. The parameters associated with the model are: the density of productively infected cells, I, the concentration of infections virus, V(I), the viral clearance rate constant per day, c, and the concentration of virus has been rendered non-infections by the protease inhibitors, V(NI).

The total concentration of virus is V=V(I) + V(NI). We can also assume that the density of the non-infected cells, T, will remain constant. This is a fair assumption since the data are only examined for a short period of time following the initial ingestion of the drug.

In the absence of delay, Perelson et al. proposed the following model:

(1)\qquad \qquad {\frac  {dI}{dt}}=kTV_{t}-\delta I

(2)\qquad \qquad {\frac  {dV_{t}}{dt}}=(1-\eta )pI(t)-cV_{t}(t)

(3)\qquad \qquad {\frac  {V_{{NI}}}{dt}}=\eta pI(t)-cV_{{NI}},

We make note that k is the infection rate constant, p is the rate at which a productively infected cell release viruses. Productivity infected cells die at a rate \delta per cell. Plasma viruses are cleared at rate c per virus.\eta respresents the drug efficiency. If \eta =0 than the drug has no effective and if \eta =1 than the drug is 100% effective.

As noted above this model will only account for the intracellular delay, and the pharmacological delay will be ignored. With the addition of the delay the following model is proposed in the form of integro-differential equations as follows:


(4a)\qquad \qquad {\frac  {dI}{dt}}=\int _{0}^{\infty }kTf(t')V_{t}(t-t')e^{{-mt'}}dt-\delta I(t),

(4b)\qquad \qquad {\frac  {dV_{t}}{dt}}=[1-h(t)](1-\eta )pI(t)-cV_{t}(t),

(4c)\qquad \qquad {\frac  {V_{{NI}}}{dt}}=h(t)\eta pI(t)-cV_{{NI}},

Throughout the analysis we will assume that the system being treated is at steady state prior to being treated. We will also make note of t=0 at the time the drugs initially takes effect. This allows for the ignorance of the pharmacological delay. Noting that there are zero non-infectious virus cells at t=0 we can allow for initial conditions as follows:

(5)\qquad \qquad I(0)={\frac  {c}{p}}V_{t}(0)

(6)\qquad \qquad T=T(0)={\frac  {c\delta }{kp}}

The intracellular delay is not always known to an exact, nor is it the same value for every cell undergoing the therapy process. In fact it is unrealistic to represent this delay with a constant thus is will be represented with a gamma distribution known as the delay distribution. This distribution is defined as:

(7)\qquad \qquad f(t')=g_{{n,b}}(t')={\frac  {t'^{{(n-1)}}}{(n-1)!b^{n}}}e^{{\frac  {-t'}{b}}}

with a mean, \tau =nb, a variance, nb^{2} and a max peak, (n-1)b. This distribution is widely used in mathematical biology because it is a plausible delay and allows for analytic solutions to be obtained.

Converting time delay defferential equations into ordinary differential equation

To perform this conversion we must define be variables to be:

(8)\qquad \qquad {\hat  {b}}\equiv {\frac  {b}{1+mb}} and {\hat  {k}}\equiv \left({\frac  k{1+mb}}\right)^{n}

Substituting (7) and (8) into (4a) will then yield

(9)\qquad \qquad {\frac  {dI}{dt}}={\hat  {k}}T\int _{0}^{\infty }g_{{n,{\hat  {b}}}}(t')V_{I}(t-t')dt'-\delta I(t)

The integro-differential equations given in (4a), (4b) and (4c) are now equivalent to the following system of ordinary linear differential equations.

(10a)\qquad \qquad {\frac  {dI}{dt}}={\hat  {k}}TE_{n}(t)-\delta I(t)

(10b)\qquad \qquad {\frac  {dE_{i}}{dt}}={\frac  {[V_{I}(t)-E_{1}(t)]}{{\hat  {b}}}},


(10c)\qquad \qquad {\frac  {dE_{1}}{dt}}={\frac  {[E_{{i-1}}(t)-E_{i}(t)]}{{\hat  {b}}}},i=2,......,n

(10d)\qquad \qquad {\frac  {dV_{I}}{dt}}=[1-h(t)]pI(t)-cV_{t}(t)


(10e)\qquad \qquad {\frac  {V_{{NI}}}{dt}}=h(t)pI(t)-cV_{{NI}}(t),

Ei(t)\qquad for i=1,......,n are auxiliary functions and are related to the delay distribution as follows:

(11)\qquad \qquad E_{i}(t)={\hat  {k}}T\int _{0}^{\infty }g_{{i,{\hat  {b}}}}(t')V_{I}(t-t')dt'

It can be shown analytically from the n=1 case as well as the other n cases the ordinary differential equations are equivalent to the integro-differential equations although this will not be shown here.


Analysis/Interpretation

It has been assumed that the protease inhibitor is completely effective such that \eta =1. Following the administration of the drug all viruses produced after the delay will be introduced as non-infectious and the density of the infected cells will begin to decline. Here we will present the solution for the general case such that c\neq \delta ,c\neq {\frac  {1}{{\hat  {b}}}},\delta \neq {\frac  {1}{{\hat  {b}}}}\qquad and\qquad t(p)=0. We recall that t(p) represents the pharmacological delay, thus we have assumed this delay to be zero.

It is seen that the density of the infectious virus will decrease exponentially such that it can be represented by the solution

(12)\qquad \qquad V_{I}(t)=V_{I}(0)e^{{-ct}}

After the application of the gamma distribution delay, the density of the non-infectious virus will be represented by

(13)\qquad \qquad V_{{NI}}(t)=V_{I}(0)\left[G_{n}ct-{\frac  {c}{\delta -c}}H_{n}+{\frac  {c{\hat  {b}}}{1-c{\hat  {b}}}}\sum \limits _{{{\hat  {k}}=0}}^{{n-1}}{\frac  {1-G_{{n-{\hat  {k}}}}+H_{{n-{\hat  {k}}}}}{(1-c{\hat  {b}})^{{\hat  {k}}}}}\right]e^{{-ct}}

+V_{I}(0)\left[{\frac  {c}{\delta -c}}H_{n}e^{{-\delta }}-{\frac  {c{\hat  {b}}}{1-c{\hat  {b}}}}\sum \limits _{{{\hat  {k}}=0}}^{{n-1}}{\frac  {1-G_{{n-{\hat  {k}}}}+H_{{n-{\hat  {k}}}}}{(1-c{\hat  {b}})^{{\hat  {k}}}}}\}\sum \limits _{{l=0}}^{{{\hat  {k}}}}{\frac  {(1-c{\hat  {b}})^{l}t^{l}}{l!T^{l}}}\right]

such that

(14a)\qquad \qquad G_{i}\equiv \left({\frac  {\delta }{\delta -c}}\right)(1-c{\hat  {b}})^{{-i}}

(14b)\qquad \qquad H_{i}\equiv \left({\frac  {\delta }{\delta -c}}\right)(1-\delta {\hat  {b}})^{{-i}}


We remember that we have defined

(15)\qquad \qquad V(t)=V_{I}(t)+V_{{NI}}(t)

thus we now have a solution for the total density of the virus in the system. We can plot the total viral density vs. the amount of days the virus has been in the system. It is done so below by assuming the initial gamma distribution delay has a mean of 1 day. The first figure represents the delay distribution as the second figure represents the total virus density vs. days since the drug was introduced to the infected cells.

Said1.jpg

As seen from the figure, there is an initial part such that the virus density stays relatively constant. Following that there is a middle period known as the transition stage where the virus density begins to decrease. After the transition period there is a final stage where the density virus decreases at a steady rate of \delta .

Results

To theoretically test the ability of the model to estimate actual viral declines following drug treatment, V(t), defined as the total virus density, was fit for different values of V(0), c, \delta , m, n and b. Fifteen different cases were examined. V(0), c, \delta , m and n were all estimated to be within 0.1% of there simulated values. The only parameter that was not simulated as precise was b. This is because equation (15) uses {\hat  {b}} instead of b. The error between b and {\hat  {b}} is still relatively small, less than 0.3%. This means that it is possible, in theory, to find good estimates for the virus clearance rate constant, c, the death rate of infected cells,\delta , and the intracellular delay, \tau =nb, from changes in the decline rate of HIV – 1 following drug therapy.

These results obtained thus far have been fit and compared to the idealized viral decay curves. These can still deviate from clinical data under actual circumstances for several reasons. For one, we have assumed that the intracellular delay is perfectly represented by a gamma distribution. This might not be the case and these delays may not conform to a gamma distribution. Second, due to noise in experimental data, it is possible that computational methods within the software may not pick up on slight changes in the rate of decline of virus during the first few days of drug treatment.

To test the level of dependency of the solution to the initial delay distribution, several different initial delay distributions have been tested. The model was fit to simulated data sets that have both uniformly and triangularly distributed delays. The models were then placed over those models found with the gamma distribution for comparison.

In the figure below, one can see results from gamma distributions fit to noise-free simulated viral declines in which the underlying delay distributions are not gamma distributions.

Said2.jpg

and the values that are used to generate these plots are:

Said3.jpg

(a) Uniform distribution between 0.2 and 1.8 days.

(b) Uniform distribution between 0.5 and 1.5 days.

(c) Uniform distribution between 0.8 and 1.2 days.

(d) Triangle distribution between 0.5 and 1.5 days with peak at 1.0

To be able to test the model and to take into an account the noisy data, we used random normal gaussian varieties with standard deviations of 15% to see the simulation of viral decline.

Said4.jpg

Estimated values for the clearance rate of free virus, the death rate of infected cells, and the mean delay were relatively close to those found using the gamma distribution thus it can be concluded that the ability of the model to provide reasonable parameter estimates is not entirely dependent on the type of delay distribution initially assumed.


Discussion

In past studies, existing models for viral decline have made the unrealistic assumption that there is either no time delay or a fixed delay between the time of drug ingestion and time of effectiveness. This paper attempts to better understand the effects of this delay by using a model of a more realistic stature. Due to the fact that a pharmacological delay only plays effect at the initial ingestion of the drug, it has still been assumed that this delay is zero. This is believed to still be accurate because the pharmacological delay is much shorter than the delay incorporated in this model, the intracellular delay. This mathematical model incorporates this delay in the form of a gamma distribution, as well as investigates the dependency on the form of the initial delay distribution. The gamma distribution is used for two reasons. For one, it is believed that it is a fitting distribution for the delay in a biological sense. For two, the gamma distribution allows for analytical solutions to be found whereas various other delays can only be solved for numerically.

The goal of this paper has been to model HIV-1 dynamics including continuously distributed intracellular delays. Nonlinear least square fits for this model suggest that intracellular delays cannot be greater than roughly 1.5 days. If new experimental data were to show that a delay appears to be longer than this, the model would simply be rejected. In the case of this happening, a new explanation for this would need to be considered.

This model attempts to make past models of HIV-1 dynamics more realistic although one can always introduce additional realism. Doing so would not detract from the usefulness of clinical research; it may in fact provide insight to better understand the kinetics of HIV infection.

External Links

1. J. E. Mittler, B. Sulzer, A. U. Neumann, and A. S. Perelson. In􏰃uence of delayed viral production on viral dynamics in HIV-1 infected patients. Math. Biosciences, 152:143-163, 1998. [1]

2. D.D. Ho, A.U. Neumann, A.S. Perelson, W. Chen, J.M. Leonard, M. Markowitz Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection Nature, 373 (1995), p. 123. [2]