Transcript Understanding HCV kinetics in-vitro vs. in-vivo

Design evaluation and optimization
for
models of hepatitis C viral dynamics
Jeremie Guedj1,2
Caroline Bazzoli3
Avidan Neumann2
France Mentré3
1Los
Alamos National Laboratory, New Mexico, USA.
2Bar-Ilan University, Ramat-Gan, Israel.
3UMR 738 INSERM and University Paris Diderot, Paris, France.
1
Background (1)



Chronic hepatitis C virus (HCV) infection is one
of the most common causes of chronic liver
disease, with as many as 170 million people
infected worldwide
The standard of care is weekly injections of
pegylated interferon + daily oral ribavirin
After a year treatment viral eradication is
achieved in 50% in HCV genotype 1 patients
2
Background (2)



Mathematical modeling of HCV RNA (viral
load) decay after treatment initiation has brought
critical insights for the understanding of the
virus pathogenesis
The parameters of this model are crucial for
early predicting treatment outcome (<W4)
How to rationalize the sampling of the
measurements to increase the precision of the
parameter estimates ?
3
Modeling HCV RNA decay with daily IFN
T0
Neumann et al. (Science 1998)
(1-h)b
infection
I
V
(1-ep)p
production
c
clearance
d
death / loss
 dT
 dt  s - b (1 - h )VT - dT

 dI
 b (1 - h )VT - dI

 dt
 dV
 dt  p (1 - e )I - cV

Target cells (T), infected cells (I)
and free virus (V)
4
new infections
6.0
δ
1,800
less blocking viral production
1,500
5.0
HCV-RNA
δ
4.0
1,200
3.0
900
IFN (pg/ml)
7.0
HCV-RNA (log cp/ml)
Data from: Formann et al (Ferenci), JVH 2003
HCV RNA with weekly injections
of peg-IFN
2.0
peg-IFN
600
1.0
300
0.0
-1.0
0
0
7
14
21
28
Days
The changes in viral load are inversely correlated with the
changes in treatment peg-IFN
A more complicate model is needed to describe the viral
kinetics
5
Modeling viral dynamics with
weekly peg-IFN
 dX
 dt  D - k a X

 dA  k X - k A
a
e
 dt

A(t )
C (t ) 
Vd

 dT

 s - b (1 - h )VT - dT
 dt
 dI
 b (1 - h )VT - dI

 dt
 dV


C (t ) n
 I - cV
 p1 
n
n 
dt
C
(
t
)

EC

50 

• D = dose of injection (weekly basis)
= 180 μg
• No closed-form solution to this
system
• This model describes the changes in
drug concentrations (C) and in HCV
RNA (V)
• As only C and V are measured,
some parameters are fixed:
F=1 (apparent volume)
p=10, s=20,000 mL-1.d-1, d=0.001 d-1,
b(1- η )=10-7 mL.d-1
6
Population model

Population parameters

values of fixed effects
EC50(μg. L-1)
n
δ (d-1)
C (d-1)
ka (d-1)
ke (d-1)
Vd (L)
0.20
0.12
0.10
0.13
0.12
0.10
0.10
Random effect model: Normal distribution of all
log-parameters (CV =50%)
 additive error model for concentrations and
log10 viral load
Y  log V (t , b )  e

 ij1
10
ij
i

Yij 2  C (t ij , b i )  e 2
1
Population designs

Five popular designs of the literature
Design
Reference
Measurement times (in days after first injection)
Number of
samples
D1
Zeuzem (2005)
{0, 1, 4, 7, 8, 15, 22, 29}
8
D2
Sherman (2005)
{0, 0.25, 0.5, 1, 2, 3, 7, 10, 14, 28}
10
D3
Herrmann (2003)
{0, 0.25, 0.5, 1, 2, 3, 4, 7, 10, 14, 21, 28}
12
D4
Zeuzem (2001)
{0, 0.040, 0.080, 0.12, 0.20, 0.33, 1, 2, 3, 4, 7, 14, 21, 28}
14
D5
Talal (2006)
{0, 0.25, 0.5, 1, 2, 3, 5, 6, 7, 7.25, 7.5, 8, 9, 14, 15, 16}
16
8
Simulation with the median values for the
parameters
D1
D2
▲ D3
▼ D4
♦D
5
9
Fisher Information Matrix
The likelihood is given by:
li ( , Yi )  

j 1,...,ni
 1  Yij1 - log10 V (t ij , b i )  2 
 1  Yij 2 - C (t ij , b i )  2 
1
  
  dp(b i )
exp- 
exp- 
1
2
2
 2 
 2 
   2 2
 
1
1
Li ( , Yi )  log( li ( i , Y ))
and hence the FIM is:
  2 Li ( , Yi ) 

M F ( , d i )  E  T
   


By independence between the patients, the FIM for the whole
sample is simply M ( , D)   M ( , d )
F
i 1,...,n
F
i
Where D is the design for the whole population D={di}i=1,…,n
10
Fisher Information Matrix



Cramer-Rao: the inverse of the FIM is the lower
bound of the variance-covariance matrix of any
unbiased estimator.
The precision attainable by a design D and
parameter set-up ψ is given by MF(ψ,D)-1
If parameters are estimated on their log-scale the
square-root of the diagonal elements of
MF(ψ,D)jj-1 are the (expected) relative errors of
the parameters
11
Fisher Information Matrix





The likelihood has no closed-form solution
The complexity of the biological model still
increases the complexity of the FIM
The FIM can be computed by simulations but
cumbersome (not possible to optimize the FIM)
By using a first order approximation around the
expectation of the random effects, an analytical
expression for the FIM can be obtained
Here the block-diagonal matrix was used
12
The PFIM software

PFIM uses the first-order linearization and has
been shown to provide very good
approximation for “standard“ PK model

Recently extended to address multi-response
models (www.pfim.biostat.fr)

However how does it work in such a complex
ODE model ?
13
PFIM vs simulated FIM (design D3)
•The empirical SE was computed by simulating 500 samples
of N=30, estimating the parameters using the SAEM
algorithm and taking the empirical standard deviation of the
estimates
•PFIM works pretty well with negligible computation time
(1 min of computation versus 5 days)!
14
PFIM vs simulated FIM (design D3)
option 1 = block diagonal
option 2 = full matrix

PFIM works well in this challenging context

Designs of the literature can be compared in
their ability to provide precise estimations of the
parameters

Optimal designs can be found
16
Expected standard errors of the
fixed effects (N=30)
Design
Number of
sampling times
per patient
log(EC50)
log(n)
log(δ) log(c) log(ka)
log(ke)
log(Vd)
D1
8
0.20
0.12
0.10
0.13
0.12
0.10
0.10
D2
10
0.16
0.10
0.095
0.11
0.13
0.11
0.10
D3
12
0.16
0.10
0.094
0.11
0.12
0.10
0.10
D4
14
0.16
0.11
0.094
0.10
0.12
0.10
0.10
D5
16
0.14
0.10
0.10
0.11
0.11
0.10
0.10
•RSE(δ) ≈ SE(log(δ ))= 0.10; here δ=0.20 → CI95%=[0.16;0.24]
•Designs with few but long-term data (W2, W3 W4) make as
good as rich design focusing on the early kinetics (D5) for δ
•D5 can precisely estimate IFN effectiveness (EC50 & n)
17
Optimal design

The total number of samples allowed was fixed
N*n=240 (idem D1)

The potential sampling times are in {D1-D5}

t=0 is observed

What is the balance between N and n ?
18
Optimal design according to M
Number
of samples
N
Optimal Design
{(sampling times), n}
(0 ,7 ,9 ), 3 


(0 ,10 ,28), 11


(0 ,1,28), 16 
(0 ,4 ,29 ), 19 




(
)
0
,
1
,
4
,
31


log(EC50)
log(n)
log(δ) log(c) log(ka)
0.21
0.12
0.081
0.096
log(ke)
log(Vd)
Information
criterion
0.14
0.11
0.084
193.2
3
80
4
60
(0,1,4,28), 38 


(0,1,10 ,28), 22
0.17
0.090
0.070
0.090
0.12
0.083
0.08
230.2
5
48
(0 ,1,4 ,16 ,28), 14 


(0 ,1,7 ,10 ,29 ), 34
0.14
0.061
0.057
0.075
0.087
0.068
0.061
224.0
6
40
(0,1,4,7,16,28), 40
0.15
0.095
0.084
0.10
0.11
0.095
0.090
208.3
7
34
0.15
0.070
0.065
0.081
0.094
0.075
0.070
193.0
(0,0.040 ,1,4,7 ,9,29 ), 4 


(0,1,4,7 ,9,28,29), 10

(0,0.040 ,1,4,7 ,16 ,28), 20


19
Optimal Design




n=4 gives the best design
Gives the same precision than the D5 while the
number of samples has been reduced by 2
CI95%(δ)=[0.18;0.22]
Importance of sampling times spread out over
the 4 weeks to distinguish the PK-related viral
rebound from virologic non-response
20
Conclusion




PFIM provides a very good approximation of
the FIM with a negligible computation burden
The total number of sampling measurements
could be reduced by half with an appropriate
design
Design should not neglect long-term kinetics
(W3 & W4)
The antiviral effectiveness of ribavirin & the
kinetics of the hepatocytes cannot be estimated.
21
Future works



Predictions are done at the individual level: how
to find an optimal design both at the population
and at the individual level ?
To increase the number of patients is more
expensive: how to include the cost in the design
optimization ?
New direct-acting antivirals have a more
profound effect on viral load. How to take into
account the information brought by data under
the level of detection ?
22