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Section 2 - Applied Pharmacokinetics

Bayesian analysis

Introduction

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 gentamicin 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.

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.

Pharmacokinetic formulas

  1. The Bayesian approach estimates pharmacokinetic parameters (e.g., CL, Kel 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:

    Bayesian formula

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

    CPss = (MD / tinf x Vd x kel ) x (1 - e-kel x tinf /1 - e-kel x tau ) x e-kel x t
    where

    • MD = maintenance dose
    • tinf = infusion time
    • Vd = Volume of distribution
    • kel = 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-kd) (1 - ekd x tinf) ekd x t)] / [Vc x kd (kd-kel) (1 - ekd x tau)] +
    [k0 (kel-k21) (1 - ekel x tinf) ekel x t)] / [Vc x kel (kd-kel) (1 - ekel x tau)]
    where

    • k0 = infusion rate (mg/hour)
    • tau = dosing interval (hours)
    • tinf = 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
    • Vc = Volume of central compartment
    • kd = hybrid distribution rate constant
    • kel = hybrid elimination rate constant

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Section 2 - Applied Pharmacokinetics

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