1

Design evaluation and Design evaluation and optimization foroptimization for

models of hepatitis C viral models of hepatitis C viral dynamicsdynamics

Jeremie GuedjJeremie Guedj1,21,2

Caroline BazzoliCaroline Bazzoli33

Avidan NeumannAvidan Neumann22

France MentrFrance Mentréé33

1Los Alamos National Laboratory, New Mexico, USA.

2Bar-Ilan University, Ramat-Gan, Israel.3UMR 738 INSERM and University Paris Diderot,

Paris, France.

Background (1)Background (1)

Chronic hepatitis C virus (HCV) infection Chronic hepatitis C virus (HCV) infection is one of the most common causes of is one of the most common causes of chronic liver disease, with as many as chronic liver disease, with as many as 170 million people infected worldwide170 million people infected worldwide

The standard of care is weekly injections The standard of care is weekly injections of pegylated interferon + daily oral of pegylated interferon + daily oral ribavirin ribavirin

After a year treatment viral eradication is After a year treatment viral eradication is achieved in 50% in HCV genotype 1 achieved in 50% in HCV genotype 1 patientspatients

2

Background (2)Background (2)

Mathematical modeling of HCV RNA (viral Mathematical modeling of HCV RNA (viral load) decay after treatment initiation has load) decay after treatment initiation has brought critical insights for the brought critical insights for the understanding of the virus pathogenesisunderstanding of the virus pathogenesis

The parameters of this model are crucial The parameters of this model are crucial for early predicting treatment outcome for early predicting treatment outcome (<W4)(<W4)

How to rationalize the sampling of the How to rationalize the sampling of the measurements to increase the precision of measurements to increase the precision of the parameter estimates ? the parameter estimates ?

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4

Modeling HCV RNA decay with Modeling HCV RNA decay with

daily IFN daily IFN Neumann et al. (Science 1998)

V

I

death / loss

clearance

c

T0

infection

(1-p)p

production

Target cells (T), infected cells (I)

and free virus (V)

cVIpdt

dV

IVTdt

dI

dTVTsdt

dT

1

)1(

)1(

HCV RNA with weekly HCV RNA with weekly injections of peg-IFNinjections of peg-IFN

5

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 7 14 21 28

Days

HC

V-R

NA

(l

og

cp

/ml)

0

300

600

900

1,200

1,500

1,800

IFN

(p

g/m

l)

The changes in viral load are inversely correlated with the changes in treatment peg-IFNA more complicate model is needed to describe the viral kinetics

HCV-RNA

peg-IFN

δ

δ new infections less blocking viral production

Modeling viral dynamics Modeling viral dynamics with weekly peg-IFNwith weekly peg-IFN

6

cVIECtC

tCp

dt

dV

IVTdt

dI

dTVTsdt

dT

V

tAtC

AkXkdt

dA

XkDdt

dX

nn

n

d

ea

a

50)(

)(1

)1(

)1(

)()(

• 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, (1- η )=10-7 mL.d-1

Population model

Population parameters values of fixed effects

Random effect model: Normal distribution of all log-parameters (CV =50%)

additive error model for concentrations and log10 viral load

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

22

1101

),(

),(log

iijij

iijij

tCY

tVY

Population designs

Five popular designs of the Five popular designs of the literatureliterature

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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}

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D5 Talal (2006) {0, 0.25, 0.5, 1, 2, 3, 5, 6, 7, 7.25, 7.5, 8, 9, 14, 15, 16}

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Simulation with the median values Simulation with the median values for the parametersfor the parameters

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D1 ▲D3D2 D4▼ D5♦D1D1 ▲D3▲D3D2D2 D4▼D4▼ D5♦D5♦

Fisher Information Fisher Information MatrixMatrix

10

)(),(

2

1exp

2

1),(log

2

1exp

2

1),(

2

2

2

2

2

1

101

,...,1 1

iiijijiijij

njii bdp

tCYtVYYl

i

The likelihood is given by:

and hence the FIM is:)),(log(),( YlYL

iiii

Tii

iF

YLEdM

),(

),(2

By independence between the patients, the FIM for the whole sample is simply

ni

iFF dMDM,...,1

),(),(

Where D is the design for the whole population D={di}i=1,…,n

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

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Fisher Information Fisher Information MatrixMatrix

Fisher Information Fisher Information MatrixMatrix

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

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The PFIM softwareThe 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 ?

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PFIM vs simulated FIM PFIM vs simulated FIM (design D3)(design D3)

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•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)!

PFIM vs simulated FIM PFIM vs simulated FIM (design D3)(design D3)

option 1 = block diagonaloption 1 = block diagonaloption 2 = full matrixoption 2 = full matrix

PFIM works well in this challenging PFIM works well in this challenging contextcontext

Designs of the literature can be Designs of the literature can be compared in their ability to provide compared in their ability to provide precise estimations of the parametersprecise estimations of the parameters

Optimal designs can be found Optimal designs can be found

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Expected standard errors Expected standard errors of the fixed effects of the fixed effects

(N=30)(N=30)

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•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)

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

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

Optimal designOptimal design

The total number of samples allowed was The total number of samples allowed was fixed N*n=240 (idem Dfixed N*n=240 (idem D11))

The potential sampling times are in {DThe potential sampling times are in {D11--DD55}}

t=0 is observedt=0 is observed

What is the balance between N and n ?What is the balance between N and n ?18

Optimal design Optimal design according to Maccording to M

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0.094

0.11

0.087

0.12

0.14

log(ka)

0.075

0.095

0.068

0.083

0.11

log(ke)

0.070

0.090

0.061

0.08

0.084

log(Vd)Numberof samples

N Optimal Design{(sampling times), n}

log(EC50) log(n) log(δ) log(c) Informationcriterion

3 80 0.21 0.12 0.081 0.096 193.2

4 60 0.17 0.090 0.070 0.090 230.2

5 48 0.14 0.061 0.057 0.075 224.0

6 40 0.15 0.095 0.084 0.10 208.3

7 34 0.15 0.070 0.065 0.081 193.00.094

0.11

0.087

0.12

0.14

log(ka)

0.075

0.095

0.068

0.083

0.11

log(ke)

0.070

0.090

0.061

0.08

0.084

log(Vd)Numberof samples

N Optimal Design{(sampling times), n}

log(EC50) log(n) log(δ) log(c) Informationcriterion

3 80 0.21 0.12 0.081 0.096 193.2

4 60 0.17 0.090 0.070 0.090 230.2

5 48 0.14 0.061 0.057 0.075 224.0

6 40 0.15 0.095 0.084 0.10 208.3

7 34 0.15 0.070 0.065 0.081 193.0

31410

192940

162810

1128100

3970

,,,

,,,

,,,

,,,

,,,

22281010

3828410

,,,,

,,,,

342910710

142816410

,,,,,

,,,,,

4028167410 ,,,,,,

20281674104000

10292897410

429974104000

,,,,,,.,

,,,,,,,

,,,,,,.,

Optimal DesignOptimal Design

n=4 gives the best designn=4 gives the best design Gives the same precision than the DGives the same precision than the D5 5

while the number of samples has been while the number of samples has been reduced by 2 reduced by 2

CICI9595%(δ)=[0.18;0.22]0.18;0.22] Importance of sampling times spread Importance of sampling times spread

out over the 4 weeks to distinguish the out over the 4 weeks to distinguish the PK-related viral rebound from virologic PK-related viral rebound from virologic non-responsenon-response

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ConclusionConclusion

PFIM provides a very good approximation of PFIM provides a very good approximation of the FIM with a negligible computation burden the FIM with a negligible computation burden

The total number of sampling measurements The total number of sampling measurements could be reduced by half with an appropriate could be reduced by half with an appropriate designdesign

Design should not neglect long-term kinetics Design should not neglect long-term kinetics (W3 & W4)(W3 & W4)

The antiviral effectiveness of ribavirin & the The antiviral effectiveness of ribavirin & the kinetics of the hepatocytes cannot be kinetics of the hepatocytes cannot be estimated.estimated.

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Future worksFuture works

Predictions are done at the individual Predictions are done at the individual level: how to find an optimal design both at level: how to find an optimal design both at the population and at the individual level ?the population and at the individual level ?

To increase the number of patients is more To increase the number of patients is more expensive: how to include the cost in the expensive: how to include the cost in the design optimization ?design optimization ?

New direct-acting antivirals have a more New direct-acting antivirals have a more profound effect on viral load. How to take profound effect on viral load. How to take into account the information brought by into account the information brought by data under the level of detection ?data under the level of detection ?

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