christoffer w tornoe oct 28 2008 population pk model building 1 q & a on session 1 what is...

Oct 28 2008 Population PK Model Building 1Christoffer W Tornoe

Q & A on Session 1

• What is naïve pooled analysis?– Definition

– One advantage/disadvantage

• What is naïve averaged analysis?– Definition


• What is a Two stage method?– Definition


• What is a One stage method?– Definition



Q & A on Session 1: 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

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Q & A on Session 1: Bayes theorem)|y(P)(P)y|(P

0 +-Prior

0 +-Current

0 +-Posterior

0 +-Prior

0 +-Current

0 +-Posterior


Q & A on Session 1: 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

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ijijijij CpCpCp ^^

True

SD


Q & A Homework Assignment 3

• Why is it S2=V/1000 for homework 3 and S1=V/1000 for homework 2?

• In homework 3, which of the following would not work? And if it works, what changes will have to made to the code?1. VC=THETA(2)*EXP(ETA(2))2. V=THETA(1)*EXP(ETA(1)) and CL=THETA(2)*EXP(ETA(2))3. CL=TVCL*EXP(ETCL)

• If the drug in homework 3 followed a two compartment model, what changes will you make to the code?

• Is it necessary to include $COVARIANCE block in every run?

• What do you specify in $OMEGA block?


Q & A Homework Assignment 3

• Where do the initial estimates of (theta), omega and sigma come from?

• Is there a difference between the omega and sigma estimates in the *.smr and *.lst output files?

• What is F and Y in $ERROR?• What does the NOAPPEND do?

Oct 28 2008 Population PK Model Building [email protected]

Population PK Model Building

Christoffer W TornoePharmacometrics

Office of Clinical Pharmacology

Food and Drug Administration


Agenda• Population PK Model Building

– Model based inference• Hypothesis testing• Likelihood ratio test

– Base model selection (not the focus of this session) – Covariate model building

• Continuous covariates• Discrete covariates• Covariate search methods

• Model Qualification and Assumption Checking– Likelihood profiling– Introduction and application of bootstrap to derive confidence

intervals• Parametric and non-parametric

– Posterior predictive check and predictive check– Internal and external validation– Sensitivity analysis


Hypothesis testing• Wikipedia definition - A method of making statistical decisions using

experimental data

• In population PK modeling building, hypothesis testing is used to choose between competing models

• Null-hypothesis

– Assuming the null hypothesis is true (H0: = 0), what is the probability of observing a value (c) for the test statistic ( that is at least as extreme as the value that was actually observed?

– Critical region of a hypothesis test is when the null hypothesis is rejected ( ≥ c, reject H0) and the alternative hypothesis (HA: = A) is accepted (< c accept (don’t reject) H0)


Likelihood Ratio Test• Likelihood Ratio Test (LRT) is used to compare goodness-of-fit for

nested models– Nested models: One model is a subset of the other, e.g. base model (without

covariates) is a subset of the full model (with covariates)• CL = CLpop + slope * WT ?

• First-order elimination [CL*C] vs. Michaelis-Menten [Vmax*C/(C+Km)] ?

• One-, two-, three-compartment model ?

• Combined residual error model Y = IPRED*(1+EPS(1)) + EPS(2) ?

• The ratio of likelihoods (L1/L2) can be used to test for significance

– Objective Function Value (OFV) = - 2 log-likelihood, i.e. sum of squared deviations between predictions and observations

• Distribution of -2 log(L1/L2) follows a 2 distribution

– -2 log(L1/L2) = -2 (log L1 – log L2) = 2 (LL2 – LL1)

– Difference in log likelihoods follows2 distribution


Likelihood Ratio Test• With a probability of 0.05, and 1 degree of freedom, the value of the 2

distribution is 3.84

Parameters -2LL

p 0.05 0.01 0.001

1

2

3

4

3.84

5.99

7.81

9.49

6.63

9.21

11.3

13.3

10.8

13.8

16.3

18.5


Other Information Criterions• Akaike Information Criterion (AIC) is another measure to compare

goodness-of-fit between competing models– Lower AIC = better model fit to the data– AIC = - 2LL + 2*k where k = no. of model parameters

• Bayesian Information Criterion (BIC or Schwarz) – Lower BIC = better model fit to the data

– BIC = - 2LL + k*ln(nobs) where nobs = number of observations

Which criterion penalizes the most for the number of parameters?


Population PK Model Building – Base Model• Base Model

– Structural

• Input (IV bolus, first-order absorption, zero-order input)

• Distribution (one-, two-, three-compartment model)

• Elimination (linear or non-linear)

• Single/multiple dose

– Between-subject variability

• Individual PK estimates should be positive (i.e. CLi=CLpop*exp(i))

– Residual variability

• Additive (Constant residual error (LLOQ))

• Proportional (Increasing variability with increasing concentrations, CCV)

• Combined


Methods for Assessing Goodness-of-Fit

• Hypothesis Testing– Likelihood-ratio test (Compare OFV)

– AIC, BIC

• Precision of parameter estimates– Large standard errors indicate over-parameterization

• Diagnostic plots– Observed and predicted concentration vs. time

– Observed vs. predicted concentration

– Residuals vs. time

– Residuals vs. predictions


Covariate Model Building

• Why build covariate models?– Explain between-subject variability in parameters and response

using patient covariates

– Improve predictive performance

– Understand causes of variability

• Patient covariates– Demographic (weight, age, height, gender, ethnicity)

– Biomarkers (renal/hepatic function)

– Concomitant medication (beta-blocker, CYP inhibitors)

– Comorbidity (other diseases)


Different Ways to Implement Covariate Models

• Continuous covariates

– Linear• CL = CLpop + slope * WT

• CL = CLpop + slope * (WT-WTpop) (Centered around population mean)

– Piecewise linear• CL = CLpop + (WT<40)*slope1 * (WT) + (WT≥40)*slope2 * (WT)

– Power• CLi = CLpop * WTi

exponent (Allometric model: exponent=0.75)

• CLi = CLpop * (WTi/WTpop)exponent (Normalized by population mean)

– Exponential• CLi = CLpop * exp (slope*WTi)


Different Ways to Implement Covariate Models

• Categorical covariates

– Linear• CL = CLpop,female + Male_diff * SEX

– Proportional• CL = CLpop,female * (1 + Male_diff * SEX)

– Power• CL = CLpop,female * Male_diff

SEX

– Exponential• CL = CLpop,female * exp(Male_diff * SEX)

(SEX = gender, 0 = Female, 1 = Male)


Covariate Model Building Essentials• Visualize the range and distribution of the covariate data

• Identify strong correlations or co-linearities between covariates

• Apply prior knowledge about the PK of the drug– Renally cleared drug (e.g. CL~CrCL)– Fix covariate parameters to literature value if they can’t be estimated

(CL ~ WT 0.75, V ~ WT 1.0)

• Keep clinical utility in mind when incorporating covariates– Use body weight instead of BSA as covariate for clearance when

dosed mg/kg– Limit to clinical important covariates, e.g cause >20% difference

• Consider study design before ruling out a covariate effect– Too narrow covariate range– Insufficient information to estimate effect (e.g. 95% CI includes 0)


Example

• One-compartment model with 1-order absorption– 100 subjects

– Samples at t=1, 2, 6, 8, 12, 16, and 24 hours postdose

– Single dose of 50 mg


Visualization of Covariate Data

Continuous Covariates

Categorical Covariates


Identify Covariate Correlations or Co-Linearities• Body weight and age a co-linear• Body weight and sex are correlated


Clearance Model Building

• Base Model Clearance ( CLi = CLpop * exp(i) ) vs Body Weight

– OFV: 8277

– Try linear model: CLi = (CLpop + slope * WTi ) * exp(i)


Clearance Model Building• Covariate Model 1: CLi = (CLpop + slope * WTi ) * exp(i)

– OFV = -30 (Base OFV = 8277, Cov1 OFV = 8247)

– Correlation between CLpop and slope = -0.984


Clearance Model Building• Covariate Model 2: CLi = (CLpop + slope * (WTi -70) * exp(i)

– Try centering around median body weight

– OFV = 0 (Cov1 OFV = 8247, Cov2 OFV = 8247)

– Corr(CLpop, slope) = 0.307


Clearance Model Building• Covariate Model 3: CLi = (CLpop* (WTi /70)exponent * exp(i)

– Try power model to avoid problems for WT = 0

– OFV = 0 (Cov2 OFV = 8247, Cov3 OFV = 8247)



– Look for other potential continuous clearance covariates

– Clearance appears correlated with Age due to co-linearity with WT

– IIV Clearance does not show a trend with Age



– Look for other potential categorical clearance covariates

– Higher clearance in males compared to females – Why?



– Females have lower body weight compared to males

– No trend in ETA CL


Covariate Search Methods

• Generalized Additive Modeling (GAM)– Multiple linear regression to quickly screen for linear and non-linear

covariate-parameters relationships

– Based on empirical Bayes parameter estimates from NONMEM

– Does not account for correlation between model parameters

• Stepwise Covariate Modeling (SCM)– Forward addition

– Backward elimination

– Forward/backward stepwise


Generalized Additive Modeling (GAM)• Implemented in Xpose4 in R

– Clearance covariate model (Revisited)• xpose.gam(xp0, parnam="CL", covnams = xvardef("covariates", xp0))

– Initial Model: CL ~ 1

– Final Model: CL ~ BW

– Call: gam(formula = CL ~ BW, data = gamdata, trace = FALSE)• Deviance Residuals:

– Min 1Q Median 3Q Max – -0.99254 -0.37352 -0.02943 0.32620 1.32247

• (Dispersion Parameter for gaussian family taken to be 0.293)– Null Deviance: 38.0509 on 99 degrees of freedom– Residual Deviance: 28.7098 on 98 degrees of freedom– AIC: 164.9947

• Coefficients– (Intercept) BW – 0.08806173 0.02200286

http://xpose.sourceforge.nethttp://cran.r-project.org


Generalized Additive Modeling (GAM)


Stepwise Covariate Modeling (SCM)

• Implemented in Perl-Speaks-NONMEM

– Forward Inclusion Step• Includes covariates one step at a time using LRT (typically p<0.05)• Univariate analysis of all specified covariate-parameter relationships• Adds best covariate and repeats univariate analysis with remaining

covariates• Continue until no more significant covariates are left

– Backward Elimination Step• Starts with final model in forward inclusion step and removes covariates

one at a time in a stepwise manner using LRT (typically p<0.01 or p<0.001)

• Remove covariate that has the smallest increase in OFV when fixed to 0• Continues until all remaining covariates are significant

http://psn.sourceforge.net


Stepwise Covariate Modeling (SCM)– Forward inclusion (p<0.05), Backward eliminition (p<0.001)– Continuous (Age, BW, Exponential=4) and Categorical (Sex, Linear=2)

Model TestBase OFV

New OFV

Test Value Goal Significant?

CLAGE-4 OFV 8277 8268 9.54 > 3.84 YES!

CLBW-4 OFV 8277 8247 30.28 > 3.84 YES!

CLSEX-2 OFV 8277 8266 11.73 > 3.84 YES!

VAGE-4 OFV 8277 8272 5.17 > 3.84 YES!

VBW-4 OFV 8277 8251 26.20 > 3.84 YES!

VSEX-2 OFV 8277 8260 16.78 > 3.84 YES!

Parameter-covariate relation chosen in this forward step: CL-BW

1. Forward Step



Model TestBase OFV

New OFV


CLAGE-4 OFV 8247 8247 0.342 > 3.84

CLSEX-2 OFV 8247 8246 0.556 > 3.84

VAGE-4 OFV 8247 8242 5.01 > 3.84 YES!

VBW-4 OFV 8247 8222 24.87 > 3.84 YES!

VSEX-2 OFV 8247 8231 16.00 > 3.84 YES!

Parameter-covariate relation chosen in this forward step: V-BW

2. Forward Step



3. Forward Step

Model TestBase OFV

New OFV


CLAGE-4 OFV 8222 8222 0.379 > 3.84

CLSEX-2 OFV 8222 8222 0.583 > 3.84

VAGE-4 OFV 8222 8222 0.034 > 3.84

VSEX-2 OFV 8222 8222 0.373 > 3.84

Parameter-covariate relation chosen in this forward step: -



1. Backward Step

Model TestBase OFV

New OFV


CLBW-1 OFV 8222 8251 -28.96 > -10.83

VBW-1 OFV 8222 8247 -24.87 > -10.83

Parameter-covariate relation chosen in this backward step: -


Summary of Covariate Model Building

• Why build covariate models?– Explain between-subject variability in parameters and response

using patient covariates

– Improve predictive performance

– Understand causes of variability

• Before building covariate models– Apply prior knowledge about the PK of the drug when deciding on

which covariates to test– Keep clinical utility in mind when incorporating covariates– Consider whether the available data and design is adequate to

detect covariate effect

• Covariate search methods– Generalized additive modeling– Stepwise covariate modeling

christoffer w tornoe oct 28 2008 population pk model building 1 q & a on session 1 what is...

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