alan s. perelson, - los alamos national...

Viral DynamicsViral Dynamics

Alan S. Perelson, Alan S. Perelson, PhDPhD

Theoretical Biology & BiophysicsTheoretical Biology & Biophysics

Los Alamos National LaboratoryLos Alamos National Laboratory

Los Alamos, NMLos Alamos, NM

[email protected]@lanl.gov

00003-E-2 July 2004

People living with HIV (2005)

TOTAL: 40.3 (36.745.3) million

00003-E-3 July 2004

Deaths resulting from HIV (2005)


00003-E-4 July 2004

New infections with HIV (2005)


00003-E-5 July 2004

Mathematics

entered the field

What is HIV infection?What is HIV infection?The virus The host

A retrovirus

Infects immune cells bearing:

CD4 & CCR5/CXCR4

CD4+ T-cells (or helper T cells)

Macrophages and dendritic cells

No treatment

Drug TherapyDrug Therapy

Medical: a means of interfering with viralMedical: a means of interfering with viral

replication replication treat or cure disease treat or cure disease

Mathematical: a means of perturbing aMathematical: a means of perturbing a

system and uncovering its dynamicssystem and uncovering its dynamics

T*T* TT

k

Infection Rate

c

Clearance

p

Virions/d

Target CellProductively

Infected

Cell

Death

Model of HIV InfectionModel of HIV Infection

Model of HIV InfectionModel of HIV Infection

**

*

( )

( )

( )

dT tdT kTV

dt

dT tkTV T

dt

dV tN T cV

dt

=

=

=

Supply of target cells

Net loss rate of target cells

Infectivity rate constant

Infected cell death rate

Virion production rate

Virion clearance rate constant

d

k

N p

c

=

0

*

0

(0)

(0) 0

(0)

T T

T

V V

=

=

=

*

Target Cell Density

Infected Target Cell Density

Virus Concentration

T

T

V

ParametersParameters

VariablesVariables

Model Used for Drug Perturbation StudiesModel Used for Drug Perturbation Studies

**

0

*

*

( )(1 )

( )(1 )

( )

RT I

I

PI I

NI

PI NI

dT tkV T T

dt

dV tN T cV

dt

dV tN T cV

dt

=

=

=

Drug efficacy

RT PI

Subscripts:

I: infectious

NI: non-infectious

From HIV-Dynamics in Vivo: ,

Perelson, et al, Science, 1996

V ( t ) = V 0 exp ( ct ) + c V 0

c

c

c exp ( t exp ct ) t exp ( ct ){ ) ( }

Solution of Model Equations Assuming 100%

Efficacy of Protease Inhibitor Therapy; Target

Cells Assumed Constant

Solution has two parameters:

c clearance rate of virus

death rate of infected cells

HIV-1: First Phase Kinetics

Perelson et al.

Science 271, 1582

1996

1010 to 1012 virions/d

from

107 to 109 T cells

- 1 hr

ImplicationsImplications

HIV infection is not a slow processHIV infection is not a slow process

Virus replicates rapidly and is clearedVirus replicates rapidly and is cleared

rapidly rapidly can compute to maintain set can compute to maintain set

point level > 10point level > 101010 virions produced/day virions produced/day

Cells infected by HIV are killed rapidlyCells infected by HIV are killed rapidly

Rapid replication implies HIV canRapid replication implies HIV can

mutate and become drug resistantmutate and become drug resistant

Combination

therapy

HIV-1: Two Phase KineticsHIV-1: Two Phase Kinetics

Combination TherapyCombination Therapy

Perelson et al. Nature 387, 186 (1997)

Perelson & Ho, Nature 1997

HIV-1: Two Phase KineticsHIV-1: Two Phase Kinetics

Perelson et al. Nature 387, 186 (1997)

Decay of latent reservoir on HAART

Finzi et al. Nat Med 1999

Infe

cti

ou

s u

nit

s p

er

millio

n

0.001

4 8 12 16 20 24 28 32 36 40 44

Time on combination therapy (months)

0.01

0.1

1

10

100

1,000

10,000

0.0001

0.00001

0

Limit of detection

Eradication?

t? = 43.9 months

60.8 years to eradicate 105 cells

Basic Biology of HIV-1 In Vivo Revealed by Modeling

Virions:

Infected T cells:

Infected long-lived cells:

t1/2

< 1 hr

0.7 d

14 d

Contribution

to viral load

>1010/day

93-99%

1-7%

Latently infected T cells: months

- years

< 1 %

ImplicationsImplications

Due to long-lived infected cellDue to long-lived infected cell

populations, would need to treat HIVpopulations, would need to treat HIV

infected individuals for many years withinfected individuals for many years with

100% effective drugs to eradicate the100% effective drugs to eradicate the

virus. Initial estimates were 3-4 years ofvirus. Initial estimates were 3-4 years of

treatment, new estimates at least 10treatment, new estimates at least 10

years.years.

But, do not have 100% effective therapyBut, do not have 100% effective therapy

Limit of detection

1st phase (t1/2 ~ days)

2nd phase (t1/2 ~ weeks)

Low steady state ?

3rd phase (t1/2 ~ months)?

Treatment time

HIV

-1 R

NA

/ml

What happens after the limit of detection is reached?

t1/2

= 8

months

t1/2

= 5 months t1/2

= 5 months

Low steady state

Low steady state

Pomerantz supersensitive RT-PCR

Di Mascio, M. et al., J. Virol., 2003

How to explain low steadyHow to explain low steady

state?state?

For the standard model Bonhoeffer et al. JV 71:3275 1997 showed that there

was a sensitive dependence of steady state VL on drug efficacy

Critical efficacy

1 1 1 1 1

2 2 2 2

* *

1 1 1 1

* *

2 2 2 2

* *

1 1 1 1

* *

2 2 2 2

* *

1 1 1 1 1 2 1

* *

2 2 2 2 2 1 2

(1 )

(1 )

(1 )(1 )

(1 )(1 )

(1 )

(1 )

( )

( )

T C

T C

T dT kVT

T dT f kV T

T kVT T

T f kV T T

C kVT C

C f kV T C

V N T N C cV D V V

V N T N C cV D V V

=

=

=

=

=

=

= + +

= + +

Two-Compartment Drug Sanctuary Model(Callaway & Perelson, Bull. Math. Biol. 64:29 2002)

Main compartmentsanctuaryV

drug

efficacy f , f< 1efficacy

Drug sanctuary solves theDrug sanctuary solves the

problemproblem

Two compartment model does not have sensitive dependence on

Part IIPart II

Hepatitis C Virus ModelingHepatitis C Virus Modeling

Viral Hepatitis - Overview

B C D ESource ofvirus

feces blood/blood-derived

body fluids

blood/blood-derived

body fluids

blood/blood-derivedbody fluids

feces

Route oftransmission

fecal-oral percutaneouspermucosal

percutaneouspermucosal

percutaneouspermucosal

fecal-oral

Chronicinfection

no yes yes yes no

Prevention pre/post-exposure

immunization

pre/post-exposure

immunization

blood donorscreening;

risk behaviormodification

pre/post-exposure

immunization;risk behaviormodification

ensure safedrinkingwater

Type of HepatitisType of Hepatitis

A

Estimates of Acute and Chronic DiseaseEstimates of Acute and Chronic Disease

Burden for Viral Hepatitis, United StatesBurden for Viral Hepatitis, United States

HAV HBV HCV HDV

Acute infections(x 1000)/year* 125-200 140-320 35-180 6-13

Fulminant deaths/year 100 150 ? 35

Chronicinfections

0 1-1.25 million

3.5million 70,000

Chronic liver disease deaths/year 0 5,000 8-10,000 1,000

* Range based on estimated annual incidence, 1984-1994.

Hepatitis C and B VirusHepatitis C and B Virus

HCV is a positive strand RNA virusHCV is a positive strand RNA virus

Genome is about 9.3kb,Genome is about 9.3kb,approximately the same size as HIVapproximately the same size as HIV

No vaccine; therapy successful inNo vaccine; therapy successful in50% of people treated50% of people treated

HBV is a DNA virusHBV is a DNA virus

Genome is very small, ~ 3.2kb,Genome is very small, ~ 3.2kb,

Takes the form of a partially closedTakes the form of a partially closedcirclecircle

Vaccine; therapy to control not cureVaccine; therapy to control not cure

Treatment of HCVTreatment of HCV

Two drugs are currently used to treatTwo drugs are currently used to treat

HCV infection HCV infection

Interferon Interferon (IFN), which is (IFN), which is

naturally made cytokine involved innaturally made cytokine involved in

protection against viral infectionsprotection against viral infections

Ribavirin (RBV), which is aRibavirin (RBV), which is a

nucleoside analog of nucleoside analog of guanosineguanosine. Its. Its

mechanism of action is controversialmechanism of action is controversial

but it may act as a mutagenbut it may act as a mutagen

Effects of TreatmentEffects of TreatmentVirus particles (called virions) are madeVirus particles (called virions) are madein the liver but are transportedin the liver but are transportedthroughout the body via the bloodthroughout the body via the blood

Each virion contains one HCV RNAEach virion contains one HCV RNAmolecule that encodes the genome formolecule that encodes the genome forthe virus.the virus.

Experimentalists can accuratelyExperimentalists can accuratelymeasure the amount of HCV RNA permeasure the amount of HCV RNA perml of blood (plasma or serum).ml of blood (plasma or serum).

Treatment should lower the amount ofTreatment should lower the amount ofHCV RNAHCV RNA

Acute Changes in HCV RNA LevelAcute Changes in HCV RNA Level

Following First Dose of IFN-Following First Dose of IFN-

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

5 mIU

10 mIU

15 mIU

Me

an

Ch

an

ge

in

HC

V R

NA

Le

ve

l (L

og

10,

eq

/ml)

*

****

* significantly different compared to 5

mIU

Time after

1st dose of

IFN

Mean Decrease in HCV RNA Levels OverMean Decrease in HCV RNA Levels OverFirst 14 Days of QD IFN-First 14 Days of QD IFN- Treatment Treatment

Lam N. DDW. 1998 (abstract L0346).

Mean D

ecre

ase H

CV

RN

A(L

og

10 C

opie

s/m

L)

Days

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 7 14

5MU

10MU

15MU

HCV Genotype 1

II TT

Infection Rate

c

Clearance

p

Virions/d

Target Cell

Infected

Cell

Loss

Model of HCV InfectionModel of HCV Infection

What if IFN blocks infection?What if IFN blocks infection?

I T

c

clearanc

e

p

virions/d

Target cellInfected

Cell

death

IFN

What if IFNWhat if IFNBlocks Production?Blocks Production?

II TT

Clearance

p

Virions/d

Target Cell

Infected

Cell

Death/Loss

IFN

c

Effectiveness

What if IFN blocksWhat if IFN blocks

production?production?

If IFN treatment If IFN treatment totallytotally blocks virus blocks virus

production, thenproduction, then

dV/dtdV/dt = - = - cVcV => V(t)=V => V(t)=V00 e e- c t- c t

Viral load should fall exponentially withViral load should fall exponentially with

slope c. However, data shows an acuteslope c. However, data shows an acute

exponential fall followed by slower fall.exponential fall followed by slower fall.

IFN Effectiveness inIFN Effectiveness inBlocking ProductionBlocking Production

Let Let = = effectivenesseffectiveness of IFN in of IFN inblocking production of virusblocking production of virus

= 1 is 100% effectiveness= 1 is 100% effectiveness

= 0 is 0% effectiveness= 0 is 0% effectiveness

dV/dtdV/dt = (1 = (1 )pI)pI cVcV

Early Kinetic AnalysisEarly Kinetic AnalysisBefore therapy, assume steady state so that pIBefore therapy, assume steady state so that pI00=cV=cV00. Also, assume at short times, I=constant=I0,so that

dV/dt= (1- )pI cV =(1- )cV0 cV, V(0)=V0

Model predicts that after therapy is initiated, theModel predicts that after therapy is initiated, theviral load will initially change according to:viral load will initially change according to:

V(t) = VV(t) = V00[1 [1 + + exp(-ct)]exp(-ct)]

This equation can be fit to data and This equation can be fit to data and ccand and estimated. estimated.

Thus drug effectiveness can be determined withinThus drug effectiveness can be determined withinthe first few days!the first few days!

10MU 10MU

0 1 2

Log

10 H

CV

RN

A/m

L

Days

8

7

6

5Log1

0H

CV

RN

A/m

L

Days

0 1 2

7

6

5

4

15MU15MU

Viral Kinetics of HCV Genotype 1Viral Kinetics of HCV Genotype 1

5MU5MU

10MU10MU

15MU15MU

DrugDrug

EfficacyEfficacy

81 4%81 4%

95 4%95 4%

96 4%96 4%

ViralViral

ClearanceClearance

ConstantConstant

(1/d)(1/d)

6.2 0.86.2 0.8

6.3 2.46.3 2.4

6.1 1.96.1 1.9

Half-lifeHalf-life

ofof

VirionsVirions

(Hours)(Hours)

2.72.7

2.62.6

2.72.7

ProductionProduction

& Clearance& Clearance

RatesRates

(10(101212 Virions/d) Virions/d)

0.4 0.2 0.4 0.2

2.3 42.3 4

0.6 0.8 0.6 0.8

Standard Model of HCV DynamicsStandard Model of HCV Dynamics

T Target Cell Density

I Infected Cell Density

V Virus Concentration

Equations Supply of target cells

Net loss rate of target cells

Infectivity rate constant

Infected cell death rate

Drug efficacy

p Virion production rate

c Virion clearance rate constant

Initial Conditions

T(0) = T0I (0) = I0

V(0) = V0

Variables

Parameters

(1 )

dTdT VT

dt

dIVT I

dt

dVpI cV

dt

=

=

=

Assume Steady StateAssume Steady State

Before treatment patient is generally inBefore treatment patient is generally in

a steady state.a steady state.

( )

0 0 0

0 0

0 0 0 0

0

0

0

/ /

/

dIV T I

dt

dVpI cV

dt

Hence

V T cV p or T c p

and

dIVT I c p V I

dt

dVpI cV

dt

= =

= =

= =

= =

=

Solution: Change in Viral LoadSolution: Change in Viral Load

Assuming Assuming TT = = TT0 0 =constant,=constant,

When c>> , 1 c and 2

where

1

1( )

2c= + +

2

1( )

2c= + 2( ) 4(1 )c c= +

1 0 2 0( ) ( )

0

1 2 2( ) [(1 ) (1 ) ]

2

t t t tc c c cV t V e e

+ += + +

t0 = delay between treatment commencement and onset of effect

10MU10MU

Days

Log

10 H

CV

RN

A/m

L

0 7 14

8

7

6

5

4

15MU15MU

Days

Log1

0H

CV

RN

A/m

L

0 7 14

7

6

5

4

3

Viral Kinetics of HCV Genotype 1Viral Kinetics of HCV Genotype 1

5MU5MU

10MU10MU

15MU15MU

DrugDrug

EfficacyEfficacy

81 4%81 4%

95 4%95 4%

96 4%96 4%

SecondSecond

Phase DecayPhase Decay

Constant, Constant,

(1/d)(1/d)

0.09 0.140.09 0.14

0.10 0.050.10 0.05

0.24 0.150.24 0.15

Half-life ofHalf-life of

Infected CellsInfected Cells

(Days)(Days)

2.22.269.369.3

4.34.317.317.3

1.71.76.36.3

HCV RNA (per RT-PCR) at 12 wk

5 MU

10 MU

15 MU

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Positive Negative

0.13

Se

co

nd

-Ph

ase

Slo

pe

(D

ays 2

1

4)

High Second-Phase Slope IsHigh Second-Phase Slope IsPredictive of HCV BeingPredictive of HCV Being

Undetectable at 12 WeeksUndetectable at 12 Weeks

HCV Viral Kinetics : SummaryHCV Viral Kinetics : Summary

Biphasic clearance of serum HCV RNABiphasic clearance of serum HCV RNA

1st phase rapid; depends on IFN- 1st phase rapid; depends on IFN-

dosedose

This appears to be due to dose-This appears to be due to dose-

dependent efficacy in blocking HCVdependent efficacy in blocking HCV

productionproduction

2nd phase slower. Slope appears to be2nd phase slower. Slope appears to be

a measure of rate of infected cell lossa measure of rate of infected cell loss

alan s. perelson, - los alamos national...

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