Optimal Stochastic Control Of Drug Dosage Regimens
Roger Jelliffe MD1, Alan Schumitzky PhD1,2, David Bayard PhD1,3, Robert Leary1,4, PhD, Alison Thomson PhD5, Andreas Botnen, MS1, Maurice Khayat BS1, Michael Neely MD1
Laboratory of Applied Pharmacokinetics, CSC 134-B, USC School of Medicine, 2250 Alcazar St, Los Angeles CA, 90033; www.lapk.org, jelliffe@usc.edu
Objectives: Maximally precise dosage regimens.
Methods and Results:
1. Nonparametric (NP) population PK/PD modeling. NP models estimate the entire most likely joint parameter distribution [1]. The distribution is supported at multiple discrete points, up to one for each subject, each with an estimated probability [1-3].
2. Determining assay noise as Fisher information, not CV%. CV% provides no method for weighting data. Fisher information [4] does, and avoids censoring low data.
3. Estimating environmental noise. This can then be estimated, as a separate term, quantifying the contribution of both noise sources.
4. Evaluating Changing Renal Function in Clinical Settings. Most methods for estimating creatinine clearance (CCr) assume stable renal function and use only a single serum creatinine (SCr). We use pairs of SCr's and calculate the CCr that would make SCr change from the first to the second value over a stated time in a acutely ill patient of stated age, gender, height, and weight [5].
5. Multiple Model (MM) dosage design. The many NP model support points provide multiple predictions of future responses to a dosage regimen. Each prediction is weighted by the probability of its support point. One can calculate the weighted squared error of the failure of any regimen to hit the target, and find the regimen specifically minimizing this error [6,7].
6. MM Bayesian Analysis. This computes the posterior probability of each support point given the population model and an individual patient's data. Usually a few or one point remain. Most become negligible. That distribution is used to develop the next MM dosage regimen.
7. Hybrid Bayesian (HB) Analysis. As an unusual patient may be outside the population parameter range, a MAP Bayesian estimate is first made. Extra support points are added in that area. This "hybrid" population model is then used for MM Bayesian analysis.
8. Interacting MM Bayesian (IMM) Analysis. An unstable patient's parameter values may change. Current Bayesian methods assume fixed values. We implemented a sequential interacting MM (IMM) Bayesian method which permits a patient's posterior support points to change to others with each new dose or serum concentration if more likely [8]. In over 130 post cardiac surgery patients on gentamicin and over 130 on vancomycin, IMM tracked drugs better than other methods [9].
Conclusions: Maximally precise therapy with toxic drugs requires specific methods. The above methods now provide this [10].
References:
[1]. Mallet A: A Maximum Likelihood Estimation Method for Random Coefficient Regression Models. Biometrika. 73: 645-656, 1986.
[2]. Schumitzky A: Nonparametric EM Algorithms for Estimating Prior Distributions. App. Math and Computation. 45: 143-157, 1991.
[3]. Leary R, Jelliffe R, Schumitzky A, and Van Guilder M: Nonparametric Pharmacokinetic/Dynamic Population Modeling with Adaptive Grids. Presented at the Annual Meeting of the American Society for Clinical Pharmacology and Therapeutics, Orlando, FL, March 6-10, 2001, Proceedings, p. P58.
[4]. DeGroot M: Probability and Statistics, 2nd ed., Addison-Wesley, 1989, p423.
[5]. Jelliffe R: Estimation of Creatinine Clearance in Patients with Unstable Renal Function, without a Urine Specimen. Am. J. Nephrology, 22: 320-324, 2002.
[6]. Jelliffe R, Schumitzky A, Bayard D, Milman M, Van Guilder M, Wang X, Jiang F, Barbaut X, and Maire P: Model-Based, Goal-Oriented, Individualized Drug Therapy: Linkage of Population Modeling, New "Multiple Model" Dosage Design, Bayesian Feedback, and Individualized Target Goals. Clin. Pharmacokinet. 34: 57-77, 1998.
[7]. Bayard D, Jelliffe R, Schumitzky A, Milman M, Van Guilder M, "Precision drug dosage regimens using multiple model adaptive control: Theory and application to simulated Vancomycin therapy," in Selected Topics in Mathematical Physics, Prof. R. Vasudevan Memorial Volume, Ed. by R. Sridhar, K. S. Rao, V. Lakshminarayanan, World Scientific Publishing Co., Madras, 1995.
[8]. Bayard D, and Jelliffe R: A Bayesian Approach to Tracking Patients having Changing Pharmacokinetic Parameters. J. Pharmacokin. Pharmacodyn. 31 (1): 75-107, 2004.
[9]. Macdonald I, Staatz C, Jelliffe R, and Thomson A: Evaluation and Comparison of Simple Multiple Model, Richer Data Multiple Model, and Sequential Interacting Multiple Model (IMM) Bayesian Analyses of Gentamicin and Vancomycin Data Collected From Patients Undergoing Cardiothoracic Surgery. Ther. Drug Monit. 30:67-74, 2008.
[10]. Jelliffe R, Schumitzky A, Bayard D, Leary R, Van Guilder M, Gandhi A, Neely M, and Bustad A: The USC*PACK BigWinPops and MM-USCPACK Software. A software demonstration at the PAGE 15 (2006) abstract 1037, Population Approach Group Europe, Bruges, Belgium, June 14-16, 2006. (Available by license from the first author.)
