Bayesian principles in pharmacokinetics

I. Introduction

The selection of a drug dosage regimen in the absence of measured drug levels (ie., a priori drug dosing) is based on estimates of the patient's pharmacokinetic parameters adjusted for patient characteristics (ie., weight, age, sex, serum creatinine). This is also referred to as population kinetics. An example of a priori or population kinetics is the Hull and Sarrubi nomogram for aminoglycoside dosing.

The traditional use of measured drug levels (ie., a posteriori drug dosing) is to estimate the patient's pharmacokinetic parameters from the measured drug levels without relying in any way on the population model. An example of a posteriori dosing is the Sawchuk and Zaske method of aminoglycoside serum level analysis.

The Bayesian approach incorporates both sets of data for estimating the patient's pharmacokinetic parameters. It uses the a priori pharmacokinetic parameters of the population model as the starting estimate for an individual; it then adjusts these estimates based on the patient's measured drug levels, taking into consideration the variability of the population parameters and the variability of the serum level measurement. Our population model is not discarded, rather it is incorporated into the estimation procedure. The serum level data is interepreted in light of both the variability of the population model the variability of the serum level measurement itself.

The appeal of the Bayesian approach is that it mimics human thinking. Before deciding what to do, our intuition first tries to forsee all the possibilities that might arise. Then, we judge how likely each is, based on both what we see and all our past experiences. Human intuition is good at judging what pieces of information are relevant to a question, but very unreliable in judging the relative cogency of different pieces of information. Bayes' theorem tells us quantitatively just how cogent every piece of information is, however it is not always able to discern whether a piece of information is relevant. Let's look at a practical example.

II. Practical application

To illustrate the practical application of Bayesian methods, suppose our patient is a 73 y/o F, 65 in, 62 kg, SCr=1. Around the 5th dose of a gentimicin 80mg Q 8 hour regimen, the trough level is reported as 4.5 and the peak as 5.4. You are then called in to consult by the patient's worried physician.

If we plug those levels into the traditional Sawchuk and Zaske equations, we get a Vd of 1.13 L/kg and half-life of 22.8 hours, giving an ideal dose of 340mg every 61 hours. Obviously this is not the proper dosage recommendation.

If we select the Bayesian algorithm for this set of levels, we get a Vd of 0.36 L/kg and half-life of 7.5 hours, giving an ideal dose of 112mg every 20 hours. Okay, that's a more reasonable dose than before, but let's dig a little deeper into this scenario.

An experienced pharmacokineticist would realize at first glance that these levels are just not right, they don't jive with what one would expect from this patient. Because any number of things could have gone wrong, your first step is to find out what when wrong from the nursing and lab staffs. You find out that the over-worked nurse hung the dose late. The hurried phlebotomist drew the trough "on time", but didn't notice that the infusion was already in progress.

Now that we know the "trough" is not a trough at all, what is the next step? If we throw out the trough level that we know is wrong, and use only the peak, we get a Vd of 0.32 L/kg and half-life of 5 hours, giving an ideal dose of 103mg every 14 hours. This is obviously the most reasonable of the three alternatives.

The lesson to take from this example is to never use bad data. Even a sophisticated Bayesian algorithm can not completely overcome bad data. The software engineer's cliche, "garbage in = garbage out", still applies. We must look at all "unusual" serum level data in a critical light.

III. Precautions

In general, the Bayesian approach to the determination of individual drug-dosage requirements performs better than other approaches. However, it should be emphasized that the population model must be appropriate for the patient. It is wrong to use a drug model derived from a dissimilar patient population. For example, you should never use a model based on data from otherwise healthy adults in a frail elderly patient.

Likewise, outlying patients in a population (ie, those patients whose pharmacokinetic parameters lie outside of the 95th percentile of the population) may be put at risk.

And, as shown in the example above, bad data will corrupt the analysis. As is always the case, the computerized algorithms outlined below can only assist in the decision-making process and should never become a substitute for rational clinical judgement.

IV. Pharmacokinetic formulas

1. The Bayesian approach estimates pharmacokinetic parameters (e.g., k_el and Vd) that will be most consistent with serum levels predicted by both the population model and the actual measured serum levels. To achieve that end, the least squares method based on the Bayesian algorithm estimates the parameters which minimize the following function:

   

2. For one compartment drugs, the following equation is used to estimate serum levels:

   CPss = (MD / t_inf x Vd x k_el ) x (1 - e^{-k_el x t_inf} /1 - e^{-k_el x tau} ) x e^{-k_el x t}
   where

   • MD = maintenance dose
   • t_inf = infusion time
   • Vd = Volume of distribution
   • k_el = elimination rate constant
   • tau = dosing interval
   • t = time at which to predict serum concentration

3. For two compartment drugs, the following equation is used to estimate serum levels:

   CPss = [k₀ (k₁₂-k_d) (1 - e^{k_d x t_inf}) e^{k_d x t})] / [V_c x k_d (k_d-k_el) (1 - e^{k_d x tau})] +
   [k₀ (k_el-k₂₁) (1 - e^{k_el x t_inf}) e^{k_el x t})] / [V_c x k_el (k_d-k_el) (1 - e^{k_el x tau})]
   where

   • k₀ = infusion rate (mg/hour)
   • tau = dosing interval (hours)
   • t_inf = infusion time (hours)
   • t = time at which to predict serum concentration
   • k₁₂ = rate constant for distribution from central to peripheral compartment
   • k₂₁ = rate constant for distribution from peripheral to central compartment
   • V_c = Volume of central compartment
   • k_d = hybrid distribution rate constant
   • k_el = hybrid elimination rate constant

V. Bibliography

1. Sarubbi FA, Hull JW. Gentamicin serum concentrations:pharmacokinetic predicitions. Ann Intern Med 1976;85:183-189.
2. Sawchuk RJ, Zaske DE, et al. Kinetic model for gentamicin dosing. Clin Pharmacol Ther 1977;21;3:362-369.
   
3. Peck CC, D'Argenio, Rodman JH. "Analysis of pharmacokinetic data for individualizing drug dosage regimens", in Evans W, Schentag J, Jusko J (eds): Applied Pharmacokinetics, 3rd edition. San Francisco. Applied Therapeutics, 1992; pp 3-2 to 3-26.
   
4. Sheiner LB, Beal S, Rosenberg B, et al: Forecasting individual pharmacokinetics. Clin Pharmacol Ther 1979;26:294.
5. Sheiner LB, Beal S. Bayesian individualization of pharmacokinetics: simple implementation and comparison with non-Bayesian methods. J Pharm Sci 1982 71:1344-1348.
6. Schumacher GE, Barr JT. Bayesian approaches in pharmacokinetic decision making. Clin Pharm 1984 3:525-30.
7. Yamaoka K, Nakagawa T, et al. A nonlinear multiple regression program based on Bayesian algorithm for microcomputers. J. Pharmacobio-Dyn., 8, 246-256 1985.
8. Ito M, Duren L, et al. Computer program for the initiation of vancomycin therapy. Clin Pharm 1993 12:126-30.
9. Burton ME, Brater DC, et al. A Bayesian feedback method of aminoglycoside dosing. Clin Pharmacol Ther 37:349-357, 1985.

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