Design of Individualized Dosage Regimes using a Bayesian Approach

1. Design of Individualized Dosage Regimes using a Bayesian Approach J. M. Laínez, G. Blau, L. Mockus, S. Orçun & G. V. Rekalitis May 2011

2. Statistical modeling framework https://pharmahub.org/resources/145#series Topics covered Module I: Statistical modeling and design of experiments Probability theory Multilinearregression 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

3. Design of Individualized Dosage Regimens Previous work 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) • 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

4. Proposed Bayesian approach

5. 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 • Population prior () • A multivariate probability distribution • Build a population parameters distribution by mixing the parameters distribution of each subject (j) • 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

6. Stage-II & III – “On-line” processNew patients PKP estimation Dose regimen optimization • Application of Bayes’ theorem for the new subject k • Prior knowledge: Prior population () • Experimental outcomes: sampling schedule • Probability distribution for drug concentration • 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 Therapeutic window constraints • A special case – Fixed interval of administration:

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

9. Case study - Gabapentin Generalities Predictive model System model: One compartment – Single dose – Oral administration Unknown parameters: Error model Homoscedastic data Lack of fit test 95% -HPD for concentration – 0.014% • 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)

10. Stage I Parameter estimation Population prior Sampling schedule Population prior Parameter estimation Population prior log(F/V) log(F/V) log(F/V) log(ke) log(ke) log(ke) CPU Time (Intel i5 at 2.66GHZ) MCMC: 225.0 s (3E5 samples) VB: 9.4 s CPU Time (Intel i5 at 2.66GHZ) MCMC: 225.0 s (3E5 samples) VB: 9.4 s ___ VB ----- MCMC ___ VB ----- MCMC log(ka) log(ka) log(ka) log(to) log(to) log(to)

11. Stage II - Distributions for new patients Patient P01 Patient P06 95% HPD bands for the predicted concentration

12. 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)

13. Nominal dosage Recommended therapy: 300mg every 8h – 600mg every 8h A 95% concentration confidence band at steady state for P06 (300mg, 8h)

14. 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

15. 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.

16. 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.

17. Thank you for your attention!

18. 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