[email protected] september 30, 2008 introduction to population analysis 1 joga gobburu...

September 30, 2008 Introduction to Population Analysis [email protected]

Introduction to Population Analysis

Joga GobburuPharmacometrics

Office of Clinical Pharmacology

Food and Drug Administration

September 30, 2008 2Joga GobburuIntroduction to Population Analysis

Pharmacometrics Training


Agenda

• Introduction to population PK-PD– Application of population PK-PD in drug development and

regulatory decision making– Pharmacometrics @ FDA

• Introduction to population modeling– Linear and nonlinear regression– Introduction to mixed effects modeling

• Mixed effects modeling applied to population PK– Different methods of analysis– Bayesian theory– Maximum likelihood– Sources of variability– Variance (Error) models


Agenda






Definition: Modeling

Mathematical (conceptual) modeling is describing a physical phenomenon by logical principles characterized with quantitative relationships, e.g., formulas, whose parameters may be measured (or experimentally determined)

http://www.hawcc.hawaii.edu/math/Courses/Math100/Chapter0/Glossary/Glossary.htm


Uses of Models

Yates FE (1975) On the mathematical modeling of biological systems: a qualified “pro”, in Physiological Adaptation to the Environment (Vernberg FJ ed), Intext Educational Publishers, New York.

1. Conceptualize the system

2. Codify current facts

3. Test competing hypotheses

4. Identify controlling factors

5. Estimate inaccessible system variables

6. Predict system response under new conditions


Model and its parts• Parametric or Mechanistic model

parameters reflect biological processes

• Non-parametric or empiric model parameters do NOT reflect biological processes

• Deterministic models do not account for variability

• Stochastic models account for variability iji

i tV

CL

i

iip e

V

DoseC

,

),( PIDVfDV DV=Dependent variableIDV=Independent variablesP=Parameters

DV

P

IDV

kg

WTVV ipopi 70

DV?Whether a quantity is DV, IDVor P depends on the context


Model and its parts

iji

i tV

CL

i

iij e

V

DoseCp

^

ijijij CpCp ^

iCLpopi CLCL ,

Structural Model

Structural Model (covariate)Stochastic Model (BSV)

Stochastic Model(Residual Var)

iVi

popi kg

WTVV ,70


Variability versus Uncertainty

(Lower CI, Mean, Upper CI)

Point estimate

Confidence Interval

Confidence interval is a measure of the uncertainty on the point estimate. We obtain point estimates ofboth population means and variances.

(Lower CI, Variance, Upper CI)

Point estimate

Confidence Interval


Mixed-effects concept

0

0.25

0.5

0.75

1

0 5 10 15

Time

Cp

0 +-

(Individual-Pop Mean CL,V)

Between Subject Variability

0 +-

Pred-Obs Conc

Residual Variability

Between-occasion variability = zero

i (CL,i & V,i)

ij

Pop Avg

ith patient


Mixed-effects concept

ijijij CpCp ^

iVi

popi kg

WTVV ,70

Fixed effects

Random effects

Fixed effects

Random effectsRandom effects

Fixed effects

iCLpopi CLCL ,

iji

i tV

CL

i

iij e

V

DoseCp

^


Types of data

• Continuous– A variable can take any value (physically possible).– E.g.: concentrations, time, dose, glucose levels

• Discrete– A variable can take one of many pre-specified values– Binary, ordinal

• Binary – Yes or No type response (e.g.: death, pain/no pain)• Ordinal – Graded response (e.g.: mild/severe pain,

minor/major bleeding)– Frequency – how often does the event occur?

• E.g.: seizures, vomiting– Time to event – when does the event occur?

• E.g.: time to death, time to MI


PKPD Data

• Experimental– Rich data are collected under controlled

conditions, usually small– Best data for building structural models– Example: Dose-proportionality

• Observational– Sparse data are collected under ‘real’ life

conditions, usually large– Best data for building statistical models– Example: Pivotal or registration trials


Linear versus Nonlinear models

• Whether a model is linear or nonlinear will need to be determined relative to the parameters NOT the variables. For example:– Which of the two is linear?

• DV = a·IDV • DV = a·IDV + b·IDV2

• Linear models– Partial derivative of DV w.r.t parameters is

independent of parameters

– Estimate parameters using linear regression

• Nonlinear models– Partial derivative of DV w.r.t parameters is

NOT independent of parameters

– Estimate parameters using non-linear regression

IDV

DV

IDV

DV


Estimation via optimization

• Linear regression: Goal is to find a line that goes as close to the observations as possible.

• Comment on the goodness-of-fit of red, blue and black lines shown on the right.

• Linear models can be analytically solved for intercept and slope estimates.

2^

)(Re ijij YYsidualsSquaredofSum

IDV

DV

Ideal value of the SSR is zero


Estimation via optimization

• Nonlinear models do not have analytical solutions, so we need to solve them numerically.

Obj Fn

CL 0

Maximum Likelihood Estimate

2^

)(Re ijij YYsidualsSquaredofSum


Maximum Likelihood Estimation• Non-linear mixed effects model

• Likelihood for individual i

i i i

2i

2

Y =f( , ,X )+

(0, )

(0, )

i

i

N

N

2i i

i 22

2i

i22

( f( , ,X ))1( ;X ) exp( )

22

1 exp( )

22

ii

YL

d


Technical goals of Population analyses

• Estimate population mean and variance– Population mean CL, V

– Between subject variability of CL, V

– Residual variability of concentrations

• Explain between subject variability using patient covariates such as body size, age, organ function

• Estimate individual CL and V to impute concentrations to perform PKPD analyses– Sometimes PK and PD measurements are not performed at the

same time

– PD change could be delayed from PK

– Modeling PD using differential equations mandates a functional form (model) for PK


Population mean versus Typical value

• Population mean is the naïve overall mean of a parameter– For example, the population mean CL is 10 L/h.

• When there are influential covariates that explain meaningful variability in PK parameters, then Typical value is the mean of a group of similar subjects. – For example, the typical value of CL for a 70 kg subject is 10 L/h.

Similarly, for a 35 kg it is 5 L/h.

iCLpopi CLCL ,

iCLi

popi kg

WTCLCL ,70


Methods of Population analyses

• Naïve averaged

• Naïve pooled

• Two-Stage

• One-Stage


Naïve Averaged

• Average concentration at each time point is calculated using all subjects’ observed concentrations.

• Average calculation does not take into the number of observations at each time point are equal or not; also subjects’ characteristics (heavy/light) are not considered – hence called ‘naïve’.

• Average time course of concentrations is then modelled to obtain naïve average PK parameters.

Time

Cp

Cp

Time


Naïve Pooled

• Individual observations from each subject are ‘pooled’ to obtain average PK parameters.

• Estimation does not take into the number of observations at each time point are equal or not; also subjects’ characteristics (heavy/light) are not considered – hence called ‘naïve’.

Time

Cp

Cp

Time


Two-Stage

• Individual observations from each subject are modelled separately to obtain average PK parameters for each subject.

• Uncertainty in individual parameter estimates is ignored.

• Each subject’s covariates and PK parameters are correlated to explain BSV.

• Population mean (or typical value) and variance are calculated.

Time

Cp

Cp

Time

Cp

Time


Two-Stage

Uncertainty in individual parameter

estimates is ignored.

Cp

Time

Cp

Time

Which subject’s PK parameters are estimated with more certainty -Red or Blue? Say, CL = 10±5 L/h and 10±1 L/h. When calculating the meanonly the point estimate is considered, the two-stage analysis does not accountfor the different uncertainty


One-Stage

• Data from all subjects are simultaneously modeled. Population mean and variance are estimated simultaneously, including covariate modeling.

• Individual subject’s PK parameters are calculated subsequent to ‘one-stage’ estimation. There is no model ‘optimization’ in this step – hence called ‘post-hoc’ step.

Time

Cp

Cp

TimeTime

Cp


Methods of Population analyses

Feature Naïve Averaged

Naïve Pooled Two-Stage One-stage

Uncertainty at each obs level

(also missing obs)

Ignores; So, mean will be close to extreme obs

Ignores; So, mean will be close to extreme obs

Accounts; Will not be influenced by extreme obs

Accounts; Will not be influenced by extreme obs.

Uncertainty at each subject level

Ignores Ignores Ignores; Subjects with more or few obs are weighed equal.

Accounts; Subjects with more are weighted more

Covariate exploration

Not easy; can average subjects by groups

Not easy; can force model with covariates

Possible Possible

Complexity Low Low Low High; Needs training


Bayes Theorem

Future = Past ·Present

Posterior = Prior · Likelihood

)(

)|()()|(

Yp

YPPYP

P( ) Probability

Model Parameter

y Data


Bayes Theorem – Uninformative Prior

)|()(~)|( yPPyP

0 +-

Prior0 +-

Current0 +-

Posterior


Bayes Theorem – Informative Prior

)|()(~)|( yPPyP

0 +-

Prior0 +-

Current0 +-

Posterior


Bayes Theorem – One-Stage analysis

• ML estimation (such as that in NONMEM) uses an empirical approach in obtaining the individual PK estimates. It uses the maximum likelihood estimates (population parameters: mean and variance) as PRIOR and the individual observations as LIKELIHOOD (CURRENT) to calculate POSTERIOR. For this reason, these individual estimates are called – ‘post hoc’, ‘empiric bayesian’ estimates.

• According to pure Bayesian estimation, POSTERIOR is a distribution. ML only estimates the MODE (central tendency) of that POSTERIOR distribution. Newer versions of NONMEM are able to estimate the POSTERIOR distribution (never used it myself). WinBUGS is a full fledged bayesian estimation program.


Bayes Theorem – One-Stage analysis

Posterior = Prior · Likelihood

PopulationParameters

IndividualObservations

Individual‘post-hoc’

Parameters

Rich obs/subjectPop EstimatesIndv estimatesclose to indv

Few obs/subjectPop EstimatesIndv estimatesclose to pop


Sources of ‘random’ variability

• Between subject variability (BSV)– Signifies deviance among subjects

– For example, CL varies between two ‘clones’

• Between occasion variability (BOV)– Signifies deviance between occasions within a subject

– For example, CL varies between day 1 and 14 for subject#1

• Residual (or within subject) variability (WSV)– Signifies deviance between predicted and observed in each subject.

This is at the observation level. Usually not assumed to be different at the subject level also.

– For example, predicted Cp at time=0 is 10 ug/L, obs Cp=12 ug/L.


Sources of ‘random’ variability

• All variability is typically assumed to be centered at zero. This is so because if the deviation from mean is truly random, then when the experiment is performed enough number of times, observations will be some times above mean, sometimes below mean with equal probability.

• Random variability is also ‘modeled’. Variability models also need to be carefully considered. Differences between individual and mean are generally described using normal or lognormal distribution models.


BSV, BOV Variability modelsResiduals are normally

distributed with a mean of zero

0

CL

0CL

Residuals are log-normally distributedwith a mean of one

1

ln(CL)

iCLpopi CLCL ,

iCLeCLCL popi,

Normal GFR = 120 mL/minIs GFR=60 mL/min possible?Is GFR=240 mL/min possible?


True

Mea

sure

d

True

Mea

sure

dResidual variability models

-Spread of ‘measured’ values is constant acrosstrue value range

-Spread of ‘measured values is higher at highertrue values

What would be the SD at eachtrue value for both scenarios?


Residual variability models

True

CV

-Variability (SD) is same at low and high true values-Called “additive” model

True

SD

-Variability (SD) increases with true values-Called “proportional” or “constant CV” model

ijijij CpCp ^

ijeCpCp ijij

^

ijijijij CpCpCp ^^

True

SD


Residual variability models

-Variability (SD) is constant at low true values, butincreases with true values at higher values-Called “combined additive-prop” model

addpropijijij ijij

CpCpCp ^^

True

SD

addpropijij ijij

CpCp ^


Agenda






Pharmacometrics

Pharmacometrics is the science that deals with quantifying pharmacology and disease to influence drug development and regulatory decisions

• Includes– Population PK– Exposure-Response (or PKPD) for

effectiveness, safety– Clinical trial simulations– Disease-drug-trial modeling


Regulatory Initiatives Dictating Pharmacometrics

• Guidances for Industry– Population PK– Exposure-Response– Dose-Response– Evidence for Effectiveness– Pediatrics Clinical Pharmacology– EOP2A Meetings (draft)

• Critical Path Initiative• OCP Strategic Plan• Internal CDER Deliverables


Pharmacometrics Scope

• NDA Reviews• Protocols

– Dose-Finding trials– Registration trials

• QT Reviews• Central QT team • EOP2A Meetings• Disease Models

– Knowledge Management

• Evidence of Effectiveness

• Labeling• Quantify benefit/risk

– Dose optimization– Dose adjustments

• Trial design

Tasks Decisions Influenced

1. Bhattaram et al. AAPS Journal. 20052. Bhattaram et al. CPT. Feb 2007

3. Garnett et al. JCP. Jan 2008 4. Wang et al. JCP. 2008 (in press)


FDA PharmacometricsDemand Increasing, Focus Expanding

0

5

10

15

FTE

s

1995 2000 2005 2006 2008 0

50

100

150

200

250

Rev

iew

s

1995 2000 2005 2006 2007

QTResources

Demand

Focus


Integration of Knowledge

DoseRangingStudies

BridgingStudies

Model

EffectivenessSafety

DDI, AgeGender, DiseaseSmoking, Food

Effectiveness

Safety

Gobburu, Sekar, Int.J.Clin.Pharm., 2002


Argatroban• Synthetic Direct Thrombin Inhibitor• Approved in Adults

– prophylaxis or treatment of thrombosis in patients with heparin‑induced thrombocytopenia (HIT)

– Anticoagulant in PCI patiets with HIT or at risk for HIT

• Dosing– Initial dose in HIT: 2 mcg/kg/min– Titrated to 1.5 – 3 times baseline aPTT (aPTT not to

exceed 100s) at steady-state (1 – 3 hrs)


PKPD in Adults

• Mainly distributed in ECF• Predominantly hepatically (CYP3A4/5) metabolized• Elimination half-life is 39 – 15 min• Direct relationship between argatroban plasma

concentration and anticoagulant effects.• Steady-state reached in 1-3 hrs


Age group Total

Birth – 6 months 76 months – 8 years 4

8 years -16 years 5

Total 16

Pediatric PKPD Data


• 15 of the 16 patients received 6-10 doses of argatroban over 14 days.

• Serial concentration and aPTT measurements were available in each patient. In total, about 166 concentration and 329 aPTT measurements were available over a concentration range of 100 to 10,000 ng/mL.

PKPD Data

• Argatroban plasma concentration and aPTT data from 5 healthy adult studies (N=52) were used for model development.

• Infusion doses range from 1µg/kg/min – 40µg/kg/min


Body weight reduces the between-patient variability from 70% to 41%


Patients with elevated bilirubin exhibit 75% lower CL than normalsVariability reduces further to 30% upon adjusting for hepatic status, after body weight

Patients with normal bilirubin

(N=11)

Patients with elevated bilirubin

(N=4)CL, L/hr/kg 0.17 0.04

Elevated bilirubin was manifested by cardiac complications


Effect on aPTT is concentration dependentConcentration-aPTT relationship is similar between adults (healthy) and pediatrics (patients)


Simulations to explore optimal dosing regimen

ModelsPKPD

DemographicsBaseline aPTT

Dosing0.25-10 ug/kg/minin increments of0.25 ug/kg/min

Starting Dose Simulations

Generate conc. &aPTT data in

10000 peds ateach dose

AnalysisCount % patients:

< TargetAchieving TargetExceeding Target

Target: 1.5-3 times baseline aPTT and < 100 seconds.

Titration SchemeSimulations

Patients < Target ateach dose are giventhe next higher dose


0.75 µg/kg/min in pediatrics is a reasonable starting dose

In adults, the approved starting dose is 2 µg/kg/min and the max dose is 10 µg/kg/min. This starting dose results in 1.92% exceeding & 66.9% reaching target aPTT.

0.75 µg/kg/min - 1.15% exceeds target and 58.86% reach target

Target: 1.5-3 times baseline aPTT and < 100

seconds.


0.25 µg/kg/min is a reasonable incremental dose No additional advantage beyond 3 ug/kg/min

20 of the 39/100 non-respondersat 0.75 ug/kg/min respond whentitrated to 1.0 ug/kg/min.


Summary

0.75 ug/kg/min is a reasonable starting dose in pediatrics

0.25 ug/kg/min is a reasonable incremental dose

What other approaches can you think of for optimizing dosing?


Knowledge, Skills Requirement

• What knowledge and skills do you need to perform the previous analysis?

Knowledge-Clinical Pharmacology

-Population Analysis (PKPD, Stats)

Skills-Data formatting

-Modeling software usage-Graphics

-Communication


REST AREA

[email protected] september 30, 2008 introduction to population analysis 1 joga gobburu...

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