Within-host HIV dynamics: estimation of parameters

Basic Problem

Infection with human immunodeficiency virus type 1 (HIV-1) may go unnoticed for a very long time: symptoms first appear after several years of "latency" of the virus. However, the apparent tranquility of this asymptomatic period is deceiving and conceals the steady state of highly dynamic processes, in which fast production of the virus keeps almost perfect balance with its fast decay. The dynamic nature of this steady state was revealed by the application of the first effective antiviral drugs, which blocked the production of the virus and thereby perturbed the equilibrium. Mathematical analysis of the decline of virus levels after the start of treatment allowed for the estimation of the key rate parameters of these processes.

General approach

You will learn how to fit a model to empirical data in order to estimate biologically relevant parameters. We will model the within-host dynamics of HIV with simple sets of differential equations. We will fit the model to both computer-generated and real data to understand the procedure and interpretation of parameter estimation.

What can be learned

Concepts:

Model fitting
Perturbation of steady state

Methods:

Numerical simulation of ordinary differential equations
Parameter estimation by non-linear minimization

Starting point

Download Download handout (PDF, 707 KB) and the script Download treat.r (R, 2 KB) describing the equations for the treatment model. Generate simulated time series, then estimate parameters from the data points (see Download estimate.r (R, 2 KB)). Download Download original clinical data (TXT, 567 Bytes), and estimate parameters.

Interesting questions that you can investigate

How can the quality of parameter estimation be improved?
What happens if you use the logarithm of the virus load in the minimization?
What happens if you vary your initial guess for the parameters?

Advanced questions:

Generate datasets with additive and multiplicative random noise, and repeat the estimation procedure.
How is the estimation affected if the drugs are not 100% effective in blocking new infections?
How is the estimation affected if a small fraction of the virus is produced from long-lived cells?

Glossary

ODE: Ordinary differential equations are used to describe the dynamics of homogeneous systems in continuous time.
Model fitting: Finding the set of parameters for which the prediction of the model is closest to the observations, typically by minimizing the sum of squared distances between the measured data and the values predicted by the model (least squares method).
Linear regression vs. non-linear minimization: If a model is linear, the best fit parameters can be obtained analytically by performing a linear regression. If the model depends nonlinearly on the parameters, error minimization must be performed by numerical methods, and it can no longer be guaranteed that the best fit parameters are actually obtained.
HIV: Human immunodeficiency virus. A retrovirus that infects mainly T cells. HIV infection is a lethal, incurable but treatable disease with three stages: primary infection, asymptomatic infection and AIDS.
Virus load: The level (concentration) of virus in an infected individual, typically expressed as viral RNA copies per ml.
Half-life: The time that an exponential decaying population requires to fall to half its initial value.

Literature & Weblinks

A.S. Perelson et al. (1996). Download HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. (PDF, 628 KB) Science, 271:1582-1586.
A.S. Perelson et al. (1997). external page Decay characteristics of HIV-1-infected compartments during combination therapy. Nature, 387:188-91.
A.S. Perelson. (2002). external page Modelling viral and immune system dynamics. Nat Rev Immunol. 2:28-36.
A recent general review on HIV infection: J.A. Levy. (2009). external page HIV pathogenesis: 25 years of progress and persistent challenges. AIDS, 23:147–160.

Some papers on HCV for the advanced exercise:

A.U. Neumann et al. (1998). external page Hepatitis C Viral Dynamics in Vivo and the Antiviral Efficacy of Interferon-α Therapy. Science, 282: 103-107. (An earlier estimation paper)
M. Gao et al. (2010). external page Chemical genetics strategy identifies an HCV NS5A inhibitor with a potent clinical effect. Nature, 465: 96-100. (New data that you can use to do your own estimation).