Bayesian Two-Compartment Pharmacokinetics

The Bayesian pharmacokinetic approach maximizes prior information about a population in pharmacokinetic calculations to improve pharmacokinetic parameter estimations and dosing predictions when drug levels are available. Traditionally a large data set is used to extract the mean pharmacokinetic parameters and their standard deviations for a population. Equations are derived to calculate parameter values for an individual patient. For a two-compartment open model the population means for volume of central distribution (Vc), volume of peripheral distribution (Vp), clearance (Cl), and interdepartmental clearance (Q) are determined with their standard deviations or coefficient of variations, and the equations used to estimate an individual patient's values. The large data set contains multiple serum levels during the distribution and elimination phases for each individual to adequately describe the model.

Data Fitting of serum levels for an individual using the Bayesian Method:

Once serum levels are determined after a known drug dosage history for an individual the population mean values and standard deviations of PK parameters are included in the data fitting when minimizing the sum of the calculated prediction errors.

Sum of the Square of Errors to Minimize  = [Sum for all levels (Level Measured - Level Predicted)2 / (SD for assay)2] + [Sum for all PK parameters (Population Mean - Fit Value)2/(SD of parameter)2]

Sum of the Square of Errors to Minimize = [Sum of all levels (Level Measured - Level Predicted)2 / (CV of Assay*Level Measured)2] + [Sum for all parameters (Population Mean - Fit Value)2/(CV*Population Mean)2]

The Coefficient of Variation for parameters may be found in the literature and are typically 30-40%. 

Fit value = the fit value for the parameter that minimizes the sum of the square of the errors for the above equation.

Level Predicted = the level calculated for the fit values of the pharmacokinetic parameters.

Standard Deviation (SD) = Coefficient of variation * Mean

Coefficient of variation (CV) = Standard Deviation / Mean

In clinical practice one or two levels are drawn, usually a peak and a trough, to monitor the patient. These levels are then fit in a non-linear regression analysis to determine the Bayesian estimate of patient's Vc and Cl. The other parameters Q, and Vp are not fit as too few levels have been drawn to characterize these parameters and they are held constant at the population mean. Then the patient's projected levels and AUC are calculated. A post-dose peak and trough are recommended as studies have shown that calculations are more accurate than using a single level.  

Numerous two-compartment models have been published for vancomycin. The AUC calculated is model-dependent as the models have different population mean values for Vc, Vp, Q, and Cl and have different equations to calculate these values.

Bayesian pharmacokinetic models should be based on large patient data sets with rich data sampling before being applied for general use. Commercial Bayesian software packages should update their models as more patient information is acquired to improve dosing prediction accuracy. There should be pre-built functionality in the software to accomplish this. End users may find that published models or commercial software models do not fit their local site data well requiring the model to be revised. Separate Bayesian models are needed for non-homogenous populations as their population pharmacokinetic parameters are different. These groups include critical care, non-critical care, obese, non-obese, paraplegia/quadriplegia, malnourished, hepatic dysfunction, and amputees.

The Bayesian model needs to match the patient population in which the patient resides. If it doesn't Bayesian calculations will be erroneous and may negatively impact patient outcomes. If actual levels are consistently higher or lower than the predicted levels the model does not fit the patient and more emphasis should be placed on the actual measured levels when making dosing adjustments.

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