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A Nonlinear Model Based Method for Steady-state Assessment

BASS XIV, November 5-9, 2007, Savannah, Georgia

Quan Hong, Eyas Abu-Raddad, Didier RenardEli Lilly and Company

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Outline

• Introduction– Pharmacokinetics and pharmaceutical research– Definition & Existing methods

• Method development– Pharmacokinetic Modeling– Decision rules to determine steady-state

• Is steady-state attained at the end of dosing?• At what time is steady-state reached?

• Simulation– Performance of the proposed method– Comparison to other existing methods

• Concluding remarks

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Introduction

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Pharmacokinetics (PK)what the body does to the drug

Pharmacodynamics (PD)what the drug does to the body

Pharmacokinetics : Font 40 Definition

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Pharmacokinetics : Font 14 Definition

Absorption Distribution Metabolism & Excretion

Pharmacokinetics: The quantitative study and characterization of the time course of drug absorption, distribution, metabolism, and excretion.

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Pharmacokinetics in Pharmaceutical Research

Preclinical Clinical

• Rate of absorption

• Extent of absorption

• Rate of elimination

• Metabolites

• Etc

What to expect

in humans

Phase I Design questions:

• Length of dosing period

• Length of washout period

• Time points of blood sample collections

• Doses to be tested

Phase I trials

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Pharmacokinetics in Pharmaceutical Research

Phase I Phase II, III, IV

Pharmacokinetic Parameters :

• Volume of distribution

• Clearance

• Cmax

• Tmax

• AUC

• Half life

• etc

Help Determine

Dose regimen

• Therapeutic dose

• Dosing frequency

• With/without food

• Dose adjustment in special subpopulations

• Drug-drug interaction

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Problem Background Pharmacokinetics in multiple dosing setting

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Example ProblemExample : 12 subjects, trough conc on day 3, 4, 5, 6 and 7.

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Why evaluating steady-state?

– Therapeutically relevant

– Certain PK concepts apply to SS (e.g, intrinsic clearance, protein binding, enzyme systems)

– Influence of internal/external factors on exposure of chronically administered drug is most relevant at SS; most importantly, in drug-drug interaction (DDI) studies.

– Constitute well controlled study

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Steady-state definition• Theoretical : the equilibrium

condition reached when the amount of drug admitted into the kinetic system over time exactly equals the amount of drug eliminated by the system over that same period of time. (rate in = rate out)

• Practical : “From a clinical perspective, 90% of the theoretical steady state value is often used as a practical definition”

- Rowland M, Tozer TN. Clinical pharmacokinetics: Concepts and applications.

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Existing methods in practice Analysis of Variance (ANOVA)

• Difference test :“each vs last” test

Day 6 vs Day 7 Day 5 vs Day 7

……..“Helmert test

Day 6 vs Day 7Day 5 vs (Day 6 + Day 7)/2

• Equivalence test : 90% CI of geometric mean ratio wholly contained in (0.8, 1.25).

Example : 12 subjects, trough conc on day 3, 4, 5, 6 and 7.

Results :

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Pros and Cons of ANOVA method

Advantages :• PK model free• Easily implemented

Disadvantages :• Qualitatively :

– Not related to practical definition (90% of asymptotic steady state)

– Estimated date restricted to actual sampling day– Yes/no responses. If no, no guidance on future studies– Can not determine individual steady state

• Quantitatively : – Will come back at Simulation

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Method Development 1. PK modeling 2. Decision rules

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PK modeling - Case I : constant IV infusion

0R

( )

( )( )ktSStP

ktttP

ktt

eCC

ekV

RVA

C

ekRA

kARdtdA

−

−

−

−=⇒

−⋅

==⇒

−=⇒

−=

1

1

1

)(

0)()(

0)(

0

Notations : amount

: volume

: elimination rate constant

: infusion rate

kVA, A

Vk

0R

Derivation :

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Suppose dosing interval is .

Immediately following first dose,

Immediately following the second dose,

….Following N-th dosing, by law of superpositioning :

τtkeAtA ⋅−⋅= 0)(

0)0( AA =

)1()( 000 qAeAAAk +=+= − ττ

)1()1()1(

))1((

1)1()1())1((

,0

,

010

ττ

τ

nktroughSS

ntroughP

nn

eCqqV

AnC

qqAqqAnA

⋅−

−

−=−−

=−

−−

=++=−

PK modeling - Case II : Multiple IV bolus

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PK modeling – Case III : multiple oral dosing

• When , the terminal elimination phase nearly identical to that of iv bolus dosing, the same model holds

• In general( )eaeaea kkkkkk ,,

( )tktkea

a ae eekkkAA −− −

−⋅

=)(

0

ea kk >>

)1())1((*

,,ττ nktroughSStroughP eCnC

⋅−−=−

),,( where * τea kkgk =

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Model Assumptions and Concepts

• Assumptions : if PK is linear with time and dose, then can be described as

• Measure of steady-state:

• as a function of half-life :

So, steady-state can be reasonably assumed at 5 to 7 half-life of time

99.0)7(97.0)5(

75.01)2(

5.01)(

2/1

2/1

/)2ln(22/1

2/12/1

==

=−=

=−=⋅−

⋅−

tftf

etf

etfkk

tk

)1()( ,,tk

troughSStroughP eCtC⋅−−=

troughPC ,

kt /)2ln(2/1 =

tk

troughSS

troughP eC

tCf ⋅−−== 1

)(

,

,

f

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Non-linear mixed effect Model expression

;,1 si = ;,1 lj =

ijii eeeCC jketroughSSjiετα β )1]([ ][,,

−−=

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛2

2

2,

,00

(~,

),0(~

βαβ

αβα

ε

σσσσ

βα

σε

N

N

ii

jiRandom error ~

Random subject effect ~

For individual dosing trough plasma concentration after jth dosing

Where

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Principles of Decision rules

To answer two central questions :

• Is steady-state attained at the end of dosing?

• At what time is steady-state reached?

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Decision Rule #1 Hypothesis testing – SS attained?

))(,( 2ˆ qnnnk kk eneNe δτ τμτμτ ⋅⋅∈ ⋅−⋅−⋅−

9.0:9.0: ≤ ⋅−⋅− ττ nkank

o eHvseH

1.0=⋅− τnke )1,0(NT ∈

knk

nk

eseneT

ˆ1.0

ˆ

ˆ

⋅⋅

−=

⋅−

⋅−

τ

τ

τ

9.01 ≥−= ⋅− ττnk

n ef

k̂k

αZT > αZ

δ ),(ˆ 2kkNk δμ∈

th-)1( α−

SS is reached, if, after the n-th dosing

Thus, statistical hypothesis of SS attainment is set at :

Equivalently,

Suppose can be estimated by . By -method, when , then

Test statistics can be formed

which, when , ,reject H0 when , where is the quantile of standard normal distribution

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Decision Rule #2 estimating time to reach steady-state

9.0 when ˆ)1ln(ˆ =

−−

= fk

ftSS

),(ˆ 2kkNk δμ∈

)))1ln((,)1ln((ˆ)1ln(ˆ 2

2 kkk

SSffN

kft δ

μμ−

−−

∈−−

=

τα leskfzt kSS <

−+ − ˆˆ

)1ln(ˆ21

%)1(100 α−

SSt̂

tk

troughSS

troughP eC

tCf ⋅−−== 1

)(

,

,

SSt̂

As

Time to reach steady-state can be estimated as

By delta-method, if

Then

CI of can be calculated.

Corollary : SS can be determine alternatively when upper boundary of is less than the time of the last dosing

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A similar method by Hoffman et al Hoffman et al, Pharmaceut. Statist. 2005; 4: 15–24

Modeling is the same but decision rule is different

Decision rule is different : “ We suggest that both the 50th percentile (for average steady state) and the 90th percentile (for individual steady state) of the subject-specific predicted values be reported.”

k)fln(t f −

−=

1i

if k)fln(t

−−

=1

ijii eeeCC jketroughSSjiετα β )1]([ ][,,

−−=

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Example : By Hoffman’s method• Design: 4 dose groups, 6 subject each, • Length of study : QD for 20 days; • Measurements : Ctrough on D1,2,3,4,6,13,19,20

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Simulation

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Simulation : single dataset

Dataset 1 : day 2,3,4,5

Dataset 2 : day 7,8,9,10

Dataset 3 : day 2, 3, 9 ,10

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Dataset 1 : day 2,3,4,5 Dataset 2 : day 7,8,9,10 Dataset 3 : day 2, 3, 9 ,10

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Simulation – Performance of the proposed method

=0.461(reaching steady state on Day 5); =100, , multiple dosing up to 7 days. ek SSC

Parameter Parameter estimators Decision rules Hoffman’s

7PSS is the percentage of concluding steady state among simulation replications.

1=τ

50HP90HP

is percentage of steady state conclusion under Hoffman’s rule, using 50% percentile criteria, using 90% percentile criteria

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Comparison to existing methods

=0.231(reaching steady state on Day 10)ek

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Comparison to existing methods

=0.231(reaching steady state on Day 10) ek

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Sample Size Selection

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Concluding Remarks

• Proposed method yields unbiased estimators of nonlinear model parameters, and both decision rules perform satisfactorily in determining SS and estimating tss, given sufficient data.

• ANOVA-based method declares steady-state prematurely (when the fraction f is far below 0.9), while Hoffman’s method is on the conservative side.

• Proposed method is less sensitive to the influence of sample size and variability than other existing methods.

• When datasets include only days toward steady-state(the plateau part of the curve), estimation of k and tss can become less precise. (Caveat : when designing the study, plan on obtaining trough plasma concentration on both early and late days relative to steady state)

• A monogram can be constructed to guide the sample size selection to ensure power and type I error.

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Thank you!

Questions?

A Nonlinear Model Based Method for Steady-state AssessmentOutlineIntroductionPharmacokinetics : Font 40 DefinitionPharmacokinetics : Font 14 DefinitionPharmacokinetics in Pharmaceutical ResearchPharmacokinetics in Pharmaceutical Research Problem Background�Pharmacokinetics in multiple dosing settingExample ProblemWhy evaluating steady-state?Steady-state definitionExisting methods in practice �Analysis of Variance (ANOVA)Pros and Cons of ANOVA methodMethod Development�1. PK modeling� 2. Decision rulesPK modeling - Case I : constant IV infusionPK modeling - Case II : Multiple IV bolusPK modeling – Case III : multiple oral dosingModel Assumptions and ConceptsNon-linear mixed effect Model expressionPrinciples of Decision rulesDecision Rule #1 � Hypothesis testing – SS attained?Decision Rule #2 �estimating time to reach steady-stateA similar method by Hoffman et al� Hoffman et al, Pharmaceut. Statist. 2005; 4: 15–24Example : By Hoffman’s methodSimulationSimulation : single dataset Simulation – Performance of the proposed methodComparison to existing methodsComparison to existing methodsSample Size SelectionSlide Number 32Thank you!