Bayesian 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. 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 one-compartment open model, the mean volume of distribution and mean clearance are determined with their standard deviations or coefficient of variations and the equations used to estimate an individual patient's values.

Once serum levels are determined after a known drug dosage history for an individual these values (means and standard deviations of PK parameters) are included in the data fitting minimization calculation of 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]

Coefficient of variation (CV) = Standard Deviation / Mean

Standard Deviation (SD) = Coefficient of variation * Mean

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 SSE (Sum of Square of Errors) becomes more heavily weighted towards the actual serum levels as more levels are obtained.

Population Mean = calculated pharmacokinetic parameter mean for the patient based on the derived population pharmacokinetic equations. For example, if vancomycin's one-compartment open model population mean Vd is 0.65 L/kg of total body weight and the patient weighs 100 kg then 65 liters is the population mean value for the patient.

The Coefficient of Variation for parameters may be found in the literature and is 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

The derived Bayesian model needs to match the patient population or patient to which it is later applied. 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.

The example below helps to visualize what Bayesian kinetics calculation achieves. Bayesian Simulation Download Excel spreadsheet.

Note the mean level below is just an example of a level in the middle of the desired therapeutic range for the hypothetical medication.

Bayesian Calculations Simulation


Timing of Serum Levels for AUC Monitoring and Dosing - A Bayesian Analysis for Vancomycin

Peak serum levels are mainly determined by the volume of distribution, Cp=Dose/Vd + Trough*exp(-K*T'), and to some extend by the prior trough with typical doses and intervals. Trough levels are mainly determined by the clearance. Levels are usually drawn after a loading dose and one or two days of therapy. Some software vendors state monitoring of levels may be drawn at any time using a single level. The analysis below will demonstrate this is not advisable. 

A hypothetical individual with a clearance of 3.3 L/hr and Vd of  45.5 Liters was analyzed using a one-compartment open model. Clearance was varied incrementally to -2SD and + 1 SD from the population mean. Steady-state peaks, troughs, and AUC values were calculated for a dose of 1000 mg Q12H infused over 1 hour for each increment. The volume of distribution was not varied. The data set was then analyzed using a Bayesian serum fitting analysis to calculate estimates of AUC and clearance using the data sets peak, peak and trough, and trough for each increment. Clearance and VD were set at the population mean during data fitting. CV for Vd and Cl was set at 0.3 in the Bayesian model. The Bayesian fit values of clearance and AUC were compared to the actual clearance and AUC values. The analysis demonstrates that calculation accuracy for AUC and clearance is best when a peak and trough are used, followed by a trough, and lastly a peak. Monitoring using a peak value is not suggested as it is the least accurate and is very dependent ofnaccurate timing of the level, infusion, and infusion duration.

 In order to maximum the utility of levels for AUC dosing a peak and trough after the same dose or trough are recommended at steady state.

A more robust analysis was performed using the Monte Carlo method and the findings were the same. Please see the link below for details.

Monte Carlo Analysis of Vancomycin Bayesian Pharmacokinetic Modeling Using a Peak & Trough, Trough or Peak Serum Level For AUC Calculations

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