design of individualized dosage regimes using a bayesian approach
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Design of Individualized Dosage Regimes using a Bayesian Approach. J. M. Laínez, G. Blau, L. Mockus, S. Or çun & G. V. Rekalitis. May 2011 . Statistical modeling framework. https://pharmahub.org/resources/145#series. Topics covered. Module I: Statistical modeling and design of experiments - PowerPoint PPT PresentationTRANSCRIPT
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)
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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:
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Dosage regimen optimization
• A special case – Fixed interval of administration:
Therapeutic window constraints
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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
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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