Design of Individualized Dosage Regimes using a Bayesian Approach

J. M. Laínez, G. Blau, L. Mockus, S. Orçun & G. V. Rekalitis

May 2011

Statistical modeling frameworkhttps://pharmahub.org/resources/145#series

Define theproblem

Postulatemodel

candidates

Design ofexperiments

Experimentaldata

Parameterestimation

One goodmodel?

Goodparameters?

0

>1

YES

NO

Final model

Topics covered• Module I: Statistical modeling

and design of experiments• Probability theory• Multilinear regression• Design of experiments

• Module II: Mathematical modeling• When to use non-linear models• Design and analysis of experiments

with non-linear models• Likelihood estimation• Bayesian estimation

– Markov Chain Monte Carlo methods (MCMC)

• Discrimination of rival models• Statistical properties of estimators• Properties of predictors

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Design of Individualized Dosage Regimens

Previous work

• Vast amount of data from clinical trials• “One fits all” dosing regimen

• Individuals vary significantly in their response to drugs• Over/undermedication additional

costs• Exploit clinical data for individualized

dosing

• Population pharmacokinetics• Naïve approaches• Two stage approach• NONMEM• Nonparametric approaches

• Dosage regimen individualization• Average concentration at

steady state Target (Mehvar, Am. J. Pharm. Educ., 1998)

• Target AUC/ maximum posterior distribution fitting (McCune et al., Clin. Pharm. Ther., 2009)

4

Proposed Bayesian approachSTAGE II

STAGE I

Individuals PK parameters

Define the problem

Postulate PK model

Experimental data

PK parameters estimation

Population Prior

estimation

Design of experiments

Experimental data

New individual parameter estimation

Good PK parameters?

NO

YES

Population prior

Individualized distribution of PK parameters

STAGE III

Dosage regimen

optimization

Dosage regimen (dose amount & dosing interval)

“Offline“ “Online“

Sampling schedule

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Stage I – An “off-line” process• Assuming:

• Structure of PK model is the same for all individuals

• PK parameters () vary among individuals

• Application of Bayes’ theorem to each patient in the clinical trials

1. Population prior ()• A multivariate probability

distribution• Build a population parameters

distribution by mixing the parameters distribution of each subject (j)

2. Sampling schedule • Select samples from new subject to

provide meaningful information• New data is to reduce variability

• PK parameters Serum concentration• Response controlled input variable:

time• Select serum sampling times which

have potential for reducing variability

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Stage-II & III – “On-line” processNew patientsPKP estimation

• Application of Bayes’ theorem for the new subject k• Prior knowledge: Prior

population ( )• Experimental outcomes:

sampling schedule

• Probability distribution for drug concentration

Dose regimen optimization

• Components• Dose amount (Dose)• Interval of Administratios ()

• Optimal dose regimen drug level remains in the desired therapeutic window given a confidence level

• Most multi-dose PK models:

7

Dosage regimen optimization

• A special case – Fixed interval of administration:

Therapeutic window constraints

8

Obtaining the posterior distributionsMCMC vs. Variational Bayes’Markov Chain Morte Carlo (MCMC)

• Stochastic approximation – sampling method

• High accuracy – convergence• Simple implementation –

large number of samples converge

• Computational costs – model complexity/prior evaluation

• Metropolis algorithm • R and MCMCpack package

Variational Bayes’ (VB)

• Optimization based deterministic approximation

• Propose a family of distributions (q)

• Accuracy depends on how well that assumption holds

• Widely used in signal processing – Statistical Physics• Linear models • Disregard covariance – Product of

marginal distributions

Phase 1Classical Variational

Inferenceq-variance fixed

Pre-processingFinding significant region

for evaluation

Phase 2Expectation propagation

problemq-means*

quadratureevaluation

points

9

Case study - GabapentinGeneralities• Anticonvulsant for epilepsy

and neuropathic disorders• Proposed therapeutic window

is 2-10 g/mL• Oral administration• Clinical study (Urban et al.,

2008)• 36 h study• 19 individuals completed the

study• A single dose – 400 mg• 14 serial blood collections (6 ml)

Predictive model1. System model:

One compartment – Single dose – Oral administration

Unknown parameters:

2. Error model Homoscedastic data

3. Lack of fit test • 95% -HPD for concentration –

0.014%

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Stage I Parameter estimation Population prior

___ VB----- MCMC

CPU Time (Intel i5 at 2.66GHZ)

MCMC: 225.0 s (3E5 samples)VB: 9.4 s

log(F/V)

log(ka) log(to)

log(ke)

Sampling scheduleParameter estimation Population prior

___ VB----- MCMC

CPU Time (Intel i5 at 2.66GHZ)

MCMC: 225.0 s (3E5 samples)VB: 9.4 s

log(F/V)

log(ka) log(to)

log(ke)

Population prior

log(F/V)

log(ka) log(to)

log(ke)

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Stage II - Distributions for new patientsPatient P01 Patient P06

95% HPD bands for the predicted concentration

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Stage III- Individualized dosage regimensFeasible dosing intervals (mg) for a 95% confidence level

A 95% concentration confidence band at steady state for P06 (500mg, 4h)

Patient

Dosing interval

(h)

Population prior (2 data pts.)

MCMCCovariance

VBP01 4 [270,574] [242,597]

6 [560,798] [460,830]8 n/d n/d

P06 4 [346, 619] [360,570]6 [530,798] n/d8 n/d n/d

P10 4 [268,572] [236,587]6 [530,798] [447,869]8 n/d [806,993]

CPU time (s) 4828.4 81

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Nominal dosageRecommended therapy: 300mg every 8h – 600mg every 8h

Dosing interval

Dose amount

Probability

Patient P018h 300mg n/d8h 600mg n/dPatient P068h 300mg n/d8h 600mg n/dPatient P108h 300mg n/d8h 600mg 54%

A 95% concentration confidence band at steady state for P06 (300mg, 8h)

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Final remarks • A Bayesian approach for individualized dosage

regimen for drug whose PK varies widely among patients, severe adverse reactions• Formally definition of the optimal dosage regimen problem• Few samples are needed to characterize a new patient• Nominal dosages may not be the most adequate therapy for all

patients• The individualized regimen provides a safer and more effective

therapy• Variational Bayes’ as an alternative to reduce the

computational cost• Sequential approach• Applicability to other domains

• Kinetic models for catalytic and polymerization applications• Demand forecasting

Further reading• Bishop, C., 2006. Pattern recognition and machine

learning, Ch. 10.• Blau, G., Lasinski, M., Orçun, S., Hsu, S., Caruthers, J.,

Delgass, N. , Venkatasubramanian, V., 2008. Computers & Chemical Engineering 32, 971.

• Ette, E., Williams, P., Ahmad, A., 2007. Population pharmacokinetic estimation methods. In: Pharmacometrics: The Science of Quantitative Pharmacology, Ch. 1, 265.

• Gilks,W., Richardson, S., Spiegelhalter, D., 1996. Markov chain Monte Carlo in practice. Chapman & Hall/CRC.

• Laínez, J.M., Blau, G., Mockus, L., Orçun, S., Reklaitis, G., 2011. Industrial & Engineering Chemistry Research, 50, 5114.

Acknowledgements

• This work was supported by the US National Science Foundation (Grant NSF-CBET-0941302).

• We would like to thank University of California, San Francisco for providing the data that was used in this study.

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Thank you for your attention!

Design of Individualized Dosage Regimes using a Bayesian Approach

J. M. Laínez, G. Blau, L. Mockus, S. Orçun & G. V. Rekalitis

New Jersey, May 11th 2011