How to Model HIV How to Model HIV Infection Infection Alan S. - PowerPoint PPT Presentation

How to Model HIV How to Model HIV Infection Infection Alan S. Perelson, PhD Alan S. Perelson, PhD Theoretical Biology & Biophysics Theoretical Biology & Biophysics Los Alamos National Laboratory Los Alamos National Laboratory Los Alamos, NM Los Alamos, NM asp@lanl.gov www.t10.lanl.gov/asp www.t10.lanl.gov/asp asp@lanl.gov

Progression to AIDS Progression to AIDS Improving HIV Therapy Bartlett and Moore, Scientific American, June 1998 http://www.sciam.com/1998/0798issue/0798bartlett.html

Unresolved Problems Unresolved Problems  What causes T cell depletion? What causes T cell depletion?   What determines the 10 year timescale? What determines the 10 year timescale?   What determines the viral What determines the viral setpoint setpoint? ?   Why does viral level increase late in Why does viral level increase late in  disease? disease? Long time scale is one of the features that led Long time scale is one of the features that led Peter Duesberg Duesberg, Berkeley, to argue that HIV , Berkeley, to argue that HIV Peter does not cause AIDS. does not cause AIDS.

What is HIV infection? What is HIV infection? The virus The host A retrovirus CD4+ T-cells (or helper T cells) Infects immune cells bearing: Macrophages and dendritic cells CD4 & CCR5/CXCR4

Model of HIV Infection Model of HIV Infection k Infection Rate p Virions/d T* T T* T Productively Target Cell Infected Cell δ c Death Clearance

Model of HIV Infection Model of HIV Infection dT t ( ) = T (0) T = l - - dT kTV 0 dt = * T (0) 0 * dT ( ) t = - d * kTV T = dt V (0) V 0 dV t ( ) = d - * N T cV Parameters Parameters dt l Supply of target cells Variables Variables Net loss rate of target cells d T Target Cell Density Infectivity rate constant k * T Infected Target Cell Density d Infected cell death rate V Virus Concentration d = Virion production rate N p Virion clearance rate constant c

Stafford et al. J Theoret Biol. 203: 285 (2000)

At longer times At longer times virus below virus below prediction -> prediction -> Immune response Immune response

Drug Therapy: Drug Therapy: Interferes with Viral Replication Interferes with Viral Replication  Medical: treat or cure disease Medical: treat or cure disease   Mathematical: a means of perturbing a Mathematical: a means of perturbing a  system and uncovering its dynamics system and uncovering its dynamics

Features in Data Features in Data  Before therapy virus level is constant Before therapy virus level is constant  – This implies a quasi-steady state This implies a quasi-steady state –  After therapy virus declines exponentially After therapy virus declines exponentially  – Simplest model: Simplest model: – – dV/dt dV/dt = P = P – –cV cV, , – – P = rate of viral production P = rate of viral production – – c = rate of virion clearance (per virion) c = rate of virion clearance (per virion) – -ct If drug causes P=0, then V=V 0 e -ct If drug causes P=0, then V=V 0 e

Model of HIV Infection Model of HIV Infection k Infection Rate p Virions/d T* T T* T Productively Target Cell Infected Cell δ c Death Clearance

Same experiment with more frequent sampling Perelson et al. Science 271, 1582 1996

Features in Data Features in Data  Decline is no longer a single exponential Decline is no longer a single exponential   Shoulder phase followed by an Shoulder phase followed by an  exponential decline exponential decline  Data suggests drug does not simply Data suggests drug does not simply  cause P=0 cause P=0

What If Drug What If Drug Blocks Infection? Blocks Infection? IFN k p Virions/d T* T T* T Infected Cell Target Cell δ c Death Clearance

Action of Antiretroviral Drugs Action of Antiretroviral Drugs Drug efficacy * dT t ( ) = - e - d * (1 ) kV T T ε RT ε PI RT I 0 dt dV t ( ) = - e d - * I (1 ) N T cV Subscripts: PI I dt “I”: infectious dV ( ) t “NI”: non-infectious = e d - * NI N T cV PI NI dt From HIV-Dynamics in Vivo: … , Perelson, et al , Science, 1996 Have assumed T=constant=T 0 Have assumed T=constant=T 0

Solution of Model Equations Assuming 100% Efficacy of Protease Inhibitor Therapy, Target Cells Constant. { c V c } 0 ) = V 0 exp ( - ct ) + exp ( - d t - exp - ct ) - ( d t exp ( - ct ) V ( t ) c - d c - d Solution has two parameters: c – clearance rate of virus δ – death rate of infected cells

HIV-1: First Phase Kinetics Perelson et al. Science 271, 1582 1996

Infectious virions decay Infectious virions decay

An experiment to measure clearance An experiment to measure clearance dV/dt = P = P – – cV cV = 0 = 0 dV/dt ε V ε dV/dt = P - = P - cV cV – – V dV/dt ε V – ε = cV 0 – –cV cV – V, V(0) = , V(0) = = cV 0

- 1 hr 10 10 to 10 12 virions/d from 10 7 to 10 9 T cells

Implications Implications  HIV infection is not a slow process HIV infection is not a slow process   Virus replicates rapidly and is cleared Virus replicates rapidly and is cleared  rapidly – – can compute to maintain set can compute to maintain set rapidly point level > 10 10 10 virions produced/day virions produced/day point level > 10  Cells infected by HIV are killed rapidly Cells infected by HIV are killed rapidly   Rapid replication implies HIV can Rapid replication implies HIV can  mutate and become drug resistant mutate and become drug resistant

Combination therapy

HIV-1: Two Phase Kinetics HIV-1: Two Phase Kinetics (Combination Therapy) (Combination Therapy) Perelson et al. Nature 387, 186 (1997)

Perelson & Ho, Nature 1997

Decay of latent reservoir on HAART 10,000 1,000 100 Infectious units per million 10 1 0.1 0.01 Limit of detection t? = 43.9 months 60.8 years to eradicate 10 5 cells 0.001 0.0001 Eradication ? 0.00001 4 8 12 16 20 28 32 36 40 44 0 24 Time on combination therapy (months) Finzi et al. Nat Med 1999

Basic Biology of HIV-1 In Vivo Revealed by Patient Studies Contribution Generations T 1/2 to viral load per year ~45 min Virions: 0.7 d Infected T cells: 93-99% ~180 14 d Infected long-lived cells: 1-7% ~20 few Latently infected T cells: months

Problems with Standard Model Problems with Standard Model T cell kinetic equation and parameters not known Labeling studies BrdU, d-glucose have provided some insights What are target cells? - Most assume target cells = activated (Ki67+) cells - Haase et al. suggest resting cells are also targets No good estimates of the infection rate k. Is mass-action correct? - find correlation between N and k; at steady state NkT 0 = c. - solution very sensitive to value of k - value of k may vary between isolates No good estimates of the burst size N - Haase Science 1996 N ~ 100 based on # HIV-1 RNA/ cell - Hockett et al. J Exp Med 1999, N ~ 4,000 - Yuen et al. PNAS 2007, N ~ 50,000 (SIV) No good estimates of drug efficacy – generally assumed high

What is the magnitude of HIV-1 residual replication on standard HAART ? Models discussed so far have assumed drugs Models discussed so far have assumed drugs are 100% effective are 100% effective

Viral Dynamics and Drug Efficacy 10 6 Plasma viremia (copies/ml) δ ε , where δ is the 1st phase slope ~ 5 death rate of productively infected 10 1st phase CD4 T cells, and ε is the efficacy of 10 4 the antiretroviral regimen. 2nd phase 10 3 Recent impression: 10 2 ε approaching 100% δ yields t 1/2 of ~1 day 10 0 7 14 21 Days on HAART

Study 377 (Louie, Hurley, Markowitz, Sun) Drugs: lopinavir/ritonavir, tenofovir, lamivudine & efavirenz Patients: drug-naïve or drug-sensitive Objectives: measure the increased potency of the regimen based on sharper 1st phase decline in plasma viremia

Mean Mean Relative T 1/2 (d) Slope (/d) Efficacy Study 377 0.7 0.99 1.00 <0.80 Standard HAART ~0.45-0.80 ~0.9-1.5 Slope = death rate of infected T cells x relative efficacy

Estimating Burst Size Estimating Burst Size How many viruses does an How many viruses does an infected cell produce in its infected cell produce in its lifetime? lifetime?

Experimental Procedure

SIV RNA vs. Days Post-inoculation 1.E+08 1.E+08 T696 T118 1.E+07 1.E+07 1.E+06 1.E+06 1.E+05 1.E+05 SIV RNA (copies/ml) 1.E+04 1.E+04 1.E+03 1.E+03 1.E+02 1.E+02 0 10 20 30 40 50 60 0 10 20 30 40 50 60 1.E+08 1.E+08 T599 T646 1.E+07 1.E+07 1.E+06 1.E+06 1.E+05 1.E+05 1.E+04 1.E+04 1.E+03 1.E+03 1.E+02 1.E+02 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Days post-inoculation

Method 1: Area Under the Curve dV/dt =N δ T* - cV V( ∞ )-V(0)=N ∫ δ T*(0)e - δ t - c ∫ Vdt = 0 NT*(0) = c ∫ Vdt total production = total clearance viral RNA time c ∫ Vdt [total virions produced] N = T* 0 [total number of cells infected]