introduction in virus

7/30/2019 Introduction in virus

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Estimation of parameters in viral dynamics models

Viral decay after treatment and infected cell turnover rates

Perelson et.al. Science 1996

Model equations for Pre-treatment viral dynamics

dT

dt= kT V T

dV

dt= N T cV

T is the population of infected cells

V is the population of (infectious) viral RNA

T is the population of uninfected cells - remains constant

Prior to treatment - assume system is in steady state

Analysis can be conducted in some cases without this assumption

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Treatment perturbs this steady state allowing decay rates to be estimated

Treatment with protease inhibitors does not halt the production

of viral RNA, but stops virion formation so that viral RNA

produced after treatment is non-infectious.

Treatment perturbs viral steady state by halting production of

infectious virus.

Using measurements of viral loads after treatment, viral

clearance and infected cell turnover rates are estimated using amodel.

The model provides the relationship between viral loads and

infected cell turnover rate.

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Model equations for Pre-treatment viral dynamics

dT

dt= kT VI T

dVI

dt= N T cVI

Model equations for viral decay after treatment

dT

dt= kT VI T

dVIdt

= cVI

dVNI

dt= N T cVNI

T

is the population of infected cells

VI (VNI) is the population of infectious (non-infectious) virus

V = VI + VNI is the total observed viral load

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Example: simulated viral load up to 7 days post treatment

Days post RX

ViralLoad

-2 0 2 4 6

5000

1000

0

50000

100000

estimate C

estimate delta

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Decay rates CANNOT be estimated from steady state data

Days post RX

ViralLoad

-2 0 2 4 6

5000

10000

50000

100000

delta=0.5delta=2.5delta=0.05

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Conclusions based on estimation of viral clearance and infected cell turnover.

Estimates of (infected cell turnover rate) and c (viral RNA

clearance rate) showed that infected cells and viral RNA are

turning over rapidly and continuously during the long latent

stage of HIV infection prior to AIDS

Previously, it was thought that HIV was relatively inactive

during the latent stage prior to development of AIDS

Estimates of number of viral particles produced per day havebeen obtained using these estimates and explain why HIV

escapes immune response and can easily becomes resistant to

suboptimal treatment.

Homework Find something very unusual in the presentation of

the results in Perelson Science 1996 paper.

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Adjustment to single infected cell compartment model

As data from longer periods of time post treatment became

available it becomes apparent that the initial model did not

accurately describe the data.

Data collected up to 2 months post infection suggest bi-phasic

decay.

A new model with two infected cell compartments is proposed:

One compartment of infected cells is short-lived, Activated

CD4 lymphocytes?

Second compartment is longer-lived, tissue macrophages,

virus bound to dendritic cells, etc?

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Bi-phasic viral decay, more than one infected cell compartment produces virus

Perelson et.al. Nature 1997

Pre-Treatment

dXdt

= kTT V X

dY

dt= kMM V Y

dVdt

= pxX +pyY cV

T (M) is the population of uninfected short (long) lived cells

Xis the population of short lived infected cells

Y is the population of longer lived infected cells

V is cell free viral RNA

mu


9/24

Bi-phasic viral decay, more than one infected cell compartment produces virus

Perelson et.al. Nature 1997

Post-Treatment

dXdt

= kTT V X

dY

dt= kMM V Y

dVdt

= pxX +pyY cV

T (M) is the population of uninfected short (long) lived cells

Xis the population of short lived infected cells



mu


10/24

Viral decay after treatment in children

Constant decay model

0 50 150 250

1

e+05

1

e+07

1

e+09

Days past treatment

ViralLoad

delta = 0.27mu = 0.032

0 50 150 250

1

e+05

1

e+07

1

e+09

Days past treatment

ViralLoad

delta = 0.8mu = 0.02

0 50 150 250

1

e+05

1

e+07

1

e+09

Days past treatment

ViralLoad

delta = 0.32mu = 0.001

0 50 150 250

1

e+05

1

e+07

1

e+09

Days past treatment

ViralLoad

delta = 0.12mu = 0.01

0 50 150 250

1

e+05

1

e+07

1

e+09

Days past treatment

ViralLoad

delta = 0.14mu = 0.006

0 50 150 250

1

e+05

1

e+07

1

e+09

Days past treatment

ViralLoad

delta = 1.28mu = 0.095

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Conclusions based on Constant Decay model

Estimates of and using plasma viral load are obtained and

used to estimate time on treatment of approximately 2 years to

eradicate virus

Second phase rates, significantly different than zero in most

children.

The model provides a relationship between the rates of interest,

and and the observed viral load data.

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Is the decay of infected cells the same in plasma and female genital tract?

Vaginal, cervical, and plasma viral loads from 21 women collected

after RX start

Susan Graham, Scott McClelland, Julie Overbaugh CROI 2006

Days post treatment

Plasmaviralload

0 5 10 15 20 25 30

10^2

10^3

10^4

10^5

10^6

delta = 0.605

mu = 0.035

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Days post treatment

Cervicalviralload

0 5 10 15 20 25 30

10^2

10^3

10^4

10^5

10^6

delta = 0.867 p : cervix to plasma = 0.26

mu = 0.053 p : cervix to plasma = 0.65

Days post treatment

Vaginalviralload

0 5 10 15 20 25 30

10^2

10^3

10^4

10^5

10^6

delta = 1.295 p : vagina to plasma = 0.02

mu = 0.061 p : vagina to plasma = 0.46

First phase decay (of productively infected cells) is significantly

faster in the vaginal compartment than in the plasma

compartment.

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Alternative to the constant decay model

Holte et.al. JAIDS 2006

Density Dependant Decay Model

dXdt

= X

dY

dt= Y

dVdt

= pxX +pyY cV

X is the population of short lived infected cells



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dXdt

= Xr

dY

dt= Yr

dVdt

= pxX +pyY cV




Null hypothesis: Constant decay model is correct, r = 1

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dXdt

= Xr1X

dY

dt= Yr1Y

dVdt

= pxX +pyY cV




Null hypothesis: Constant decay model is correct, r = 1

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Density Dependant Decay model results

0 50 150 250

1

e+0

5

1

e+07

1

e+09

Days past treatment

ViralLoad

dens dep decay delta = 0.02

dens dep decay mu = 0.002

constant decay delta = 0.27

constant decay mu = 0.032

r = 1.21

0 50 150 250

1

e+0

5

1

e+07

1

e+09

Days past treatment

ViralLoad





r = 1.08

0 50 150 250

1

e+0

5

1

e+07

1

e+09

Days past treatment

ViralLoad


dens dep decay mu = 0



r = 1.43

0 50 150 250

1

e+05

1

e+07

1

e

+09

Days past treatment

ViralLoad





r = 1.31

0 50 150 250

1

e+05

1

e+07

1

e

+09

Days past treatment

ViralLoad

dens dep decay delta = 0




r = 1.34

0 50 150 250

1

e+05

1

e+07

1

e

+09

Days past treatment

ViralLoad





r = 1.13

Density dependant decayConstant decay

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Density Dependant Decay model results - Continued

The parameter r is significantly greater than 1 for all but one

child suggesting that that the constant decay model is not

appropriate for the observed data.

Estimated second phase decay, , is significantly different than 0for all children using the constant decay model, but only for one

child using the density dependant decay model.

Very different conclusions about the long term dynamics of viralload after treatment depending on which model is used for

prediction and inference.

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Density dependant vs constant decay model - time to eradication

0 1 2 3 4 5 6 71

e+00

1

e+06

Days post treatment

Longlived

inf

ected

cells

time to eradication 1.2 years

0 1 2 3 4 5 6 71

e+00

1

e+06

Days post treatment

Longlived

inf

ected

cells


0 1 2 3 4 5 6 71

e+00

1

e+06

Days post treatment

Longlived

infected

cells

time to eradication 4 years

0 1 2 3 4 5 6 71

e+00

1

e+06

Days post treatment

Longlived

infected

cells


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Density dependant vs constant decay model - time to eradication

0 1 2 3 4 5 6 71

e+00

1

e+06

Days post treatment

Longlived

inf

ected

cells



0 1 2 3 4 5 6 71

e+00

1

e+06

Days post treatment

Longlived

inf

ected

cells



0 1 2 3 4 5 6 71

e+00

1

e+06

Days post treatment

Longlived

infected

cells



0 1 2 3 4 5 6 71

e+00

1

e+06

Days post treatment

Longlived

infected

cells



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Conclusions based on models for viral decay after treatment

0 20 40 60 80 100

2

e+05

1

e+06

5

e+06

5

e+07

Days past treatment

ViralLoad


0 1 2 3 4 5 6 71

e+00

1

e+02

1

e+04

1

e+06

Years post treatment

Lon

glivedi

nfectedc

ells



Using models to make predictions is subject to the dangers of the

potential for incorrect mathematical models....

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Conclusions based on models for viral decay after treatment

0 20 40 60 80 100

2

e+05

1

e+06

5

e+06

5

e+07

Days past treatment

ViralLoad


0 1 2 3 4 5 6 71

e+00

1

e+02

1

e+04

1

e+06

Years post treatment

Lon

glivedi

nfectedc

ells



Using models to make predictions is subject to the dangers of the

potential for incorrect mathematical models....

... in addition to extrapolating beyond the range of observed

data

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When/why should mathematical models be used in HIV research?

Can and should be used to generate conjectures and predictions

that can be tested in laboratory or clinical populations

Can and should be used to estimate dynamic parameters which

have prognostic value.

Viral set point after infection is prognostic for disease

outcome, Mellors et.al Annals Int. Med. 1997.

Similar studies for infected cell decay rates would be useful.

Can be used to explore time varying phenomena within a fixed

interval of time. Care is needed when interpreting the results

When modelling results are treated as just that: Modellingresults. To be differentiated from observed data.

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Summary

Viral dynamics models have been used successfully to describe

disease mechanisms.

Caution is needed in interpretation of model predictions since

models can be incorrect and extrapolation is always risky.

Ongoing collaboration between modelers and clinicians and lab

scientists is essential.

Viral dynamics research and modelling needs to be more

transparent.

Viral dynamics studies and analysis require the same rigorous

design and analysis as any other type of study.

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introduction in virus

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