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 framework 5#series 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

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) 3

Proposed Bayesian approach 4

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 5

Stage-II & III – “On-line” process New patients PKP 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 6

Dosage regimen optimization 7 A special case – Fixed interval of administration: Therapeutic window constraints

Obtaining the posterior distributions MCMC 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 8

Case study - Gabapentin Generalities 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 model 1.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% 9

Stage I Parameter estimation Population prior 10 ___ VB MCMC CPU Time CPU Time (Intel i5 at 2.66GHZ) MCMC: s (3E5 samples) VB: 9.4 s CPU Time CPU Time (Intel i5 at 2.66GHZ) MCMC: s (3E5 samples) VB: 9.4 s log(F/V) log(k a )log(t o ) log(k e ) Sampling schedule Parameter estimation Population prior ___ VB MCMC CPU Time CPU Time (Intel i5 at 2.66GHZ) MCMC: s (3E5 samples) VB: 9.4 s CPU Time CPU Time (Intel i5 at 2.66GHZ) MCMC: s (3E5 samples) VB: 9.4 s log(F/V) log(k a )log(t o ) log(k e ) Population prior log(F/V) log(k a )log(t o ) log(k e )

Stage II - Distributions for new patients Patient P01Patient P % HPD bands for the predicted concentration

Stage III- Individualized dosage regimens Feasible dosing intervals (mg) for a 95% confidence level 12 A 95% concentration confidence band at steady state for P06 (500mg, 4h) Patien t Dosing interv al (h) Population prior (2 data pts.) MCMC Covariance VB P014[270,574][242,597] 6[560,798][460,830] 8n/d P064[346, 619][360,570] 6[530,798]n/d 8 P104[268,572][236,587] 6[530,798][447,869] 8n/d[806,993] CPU time (s)

Nominal dosage Recommended therapy: 300mg every 8h – 600mg every 8h 13 Dosing interval Dose amount Probabil ity Patient P01 8h300mgn/d 8h600mgn/d Patient P06 8h300mgn/d 8h600mgn/d Patient P10 8h300mgn/d 8h600mg54% A 95% concentration confidence band at steady state for P06 (300mg, 8h)

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 14

Further reading Bishop, C., Pattern recognition and machine learning, Ch. 10. Blau, G., Lasinski, M., Orçun, S., Hsu, S., Caruthers, J., Delgass, N., Venkatasubramanian, V., Computers & Chemical Engineering 32, 971. Ette, E., Williams, P., Ahmad, A., Population pharmacokinetic estimation methods. In: Pharmacometrics: The Science of Quantitative Pharmacology, Ch. 1, 265. Gilks,W., Richardson, S., Spiegelhalter, D., Markov chain Monte Carlo in practice. Chapman & Hall/CRC. Laínez, J.M., Blau, G., Mockus, L., Orçun, S., Reklaitis, G., Industrial & Engineering Chemistry Research, 50, 5114.

Acknowledgements This work was supported by the US National Science Foundation (Grant NSF-CBET ). We would like to thank University of California, San Francisco for providing the data that was used in this study.

Thank you for your attention! 17

Design of Individualized Dosage Regimes using a Bayesian Approach J. M. Laínez, G. Blau, L. Mockus, S. Orçun & G. V. Rekalitis New Jersey, May 11 th 2011