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7/30/2019 Introduction in virus
1/24
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
7/30/2019 Introduction in virus
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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
Y is the population of longer lived infected cells
V is cell free viral RNA
mu
7/30/2019 Introduction in virus
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
Y is the population of longer lived infected cells
V is cell free viral RNA
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Alternative to the constant decay model
Holte et.al. JAIDS 2006
Density Dependant Decay Model
dXdt
= Xr
dY
dt= Yr
dVdt
= pxX +pyY cV
X is the population of short lived infected cells
Y is the population of longer lived infected cells
V is cell free viral RNA
Null hypothesis: Constant decay model is correct, r = 1
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Alternative to the constant decay model
Holte et.al. JAIDS 2006
Density Dependant Decay Model
dXdt
= Xr1X
dY
dt= Yr1Y
dVdt
= pxX +pyY cV
X is the population of short lived infected cells
Y is the population of longer lived infected cells
V is cell free viral RNA
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
dens dep decay delta = 0.24
dens dep decay mu = 0.007
constant decay delta = 0.8
constant decay mu = 0.02
r = 1.08
0 50 150 250
1
e+0
5
1
e+07
1
e+09
Days past treatment
ViralLoad
dens dep decay delta = 0.01
dens dep decay mu = 0
constant decay delta = 0.32
constant decay mu = 0.001
r = 1.43
0 50 150 250
1
e+05
1
e+07
1
e
+09
Days past treatment
ViralLoad
dens dep decay delta = 0.01
dens dep decay mu = 0
constant decay delta = 0.12
constant decay mu = 0.01
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
dens dep decay mu = 0
constant decay delta = 0.14
constant decay mu = 0.006
r = 1.34
0 50 150 250
1
e+05
1
e+07
1
e
+09
Days past treatment
ViralLoad
dens dep decay delta = 0.25
dens dep decay mu = 0.026
constant decay delta = 1.28
constant decay mu = 0.095
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
time to eradication 2.1 years
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
time to eradication 0.4 years
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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 5.2 years
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
time to eradication 3.6 years
time to eradication 2.1 years
0 1 2 3 4 5 6 71
e+00
1
e+06
Days post treatment
Longlived
infected
cells
time to eradication 38 years
time to eradication 4 years
0 1 2 3 4 5 6 71
e+00
1
e+06
Days post treatment
Longlived
infected
cells
time to eradication 0.6 years
time to eradication 0.4 years
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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
Density dependant decayConstant decay
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
time to eradication 5.2 years
time to eradication 1.2 years
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
Density dependant decayConstant decay
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
time to eradication 5.2 years
time to eradication 1.2 years
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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