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. Accuracy of a Bayesian forecasting program is largely determined by the robustness and generalizability of the population model used. One compartment models can be constructed with with a large number of patients (~100) with varying degrees of renal when a post dose peak and trough are drawn for each patient. (Shingde RV 2019).
Common quoted statements about Bayesian Dosing Methods.
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"Bayesian methods can use serum levels obtained at anytime." This is not related to the Bayesian Method, but is related to software design.
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"Bayesian methods can incorporate changes in renal function and volume of distribution." This is not related to the Bayesian Method, but is related to software design."
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"Bayesian methods allow accurate and reliable AUC estimates with trough-only data." This is only possible when richly sampled data is used as a Bayesian prior. Richly sampled models have eight or more levels drawn through out the dosage interval to capture the distribution and elimination phases for each patient during model development and a large number of patients are included in the analysis. Most published models are based on small populations with limited peaks and trough sampling and usually with only trough sampling. It is advisable to review the original publications and vendor documentation for verification of the sampling used during model development. Otherwise a peak and trough are required to adequately calculate the AUC in a Bayesian or non-Bayesian model. First-order analytic equations are as accurate as Bayesian methods of AUC calculations.
The first two are a function of software design and not Bayesian analysis. Any computerized pharmacokinetic program can perform these functions if designed with appropriate nonlinear least squares regression routines. Most commonly the method of superposition is apply to allow for changes in doses, dosage intervals, lengths of infusions, and time of serum levels, and serum creatinines during data fitting. Data fitting is possible with steady state or non-steady state levels at any point in time.
Bayesian Calculations are as follows:
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 applied Bayesian model needs to match the 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.
If the patient's kinetic parameter are different than the population mean values predicted levels will move closer to the drawn values as more levels are collected and the calculated kinetic parameter will move towards the individual's actual parameters. An analysis was performed using a one compartment open model for vancomycin. The hypothetical patient, 70 kg, 70 years old , and 70 inches had a clearance of 63% of the population expected value and a Vd set at the population mean. Serial non steady state trough levels were calculated for the hypothetical patient after doses 1 through 5 (15.9, 23.4, 26.9, 28.5, and 29.3 mcg/ml) for 1500 mg q24H with a 2 hour infusion period. The expected steady state level was ~ 30 mcg/ml for the hypothetical patient versus 14.9 mcg/ml for population mean kinetic parameters. The expected AUC was 1043 for the hypothetical patient and 658 mg*hour/L per day for the population mean. The Bayesian coefficients of variation were set at 0.3 for volume of distribution and clearance. The coefficient of variation for serum levels was set at 0.1, 0.2, and 0.3 and data fittings were performed for the cumulative serial trough levels 1 through 5 (trough 1, trough 1 & 2, trough 1, 2, & 3, trough 1, 2, 3, & 4, and troughs 1, 2, 3, 4, & 5). With each added trough the predicted steady state AUC moved closer to the true steady state AUC.
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.
Timing of Serum Levels for AUC Monitoring and Dosing - A Bayesian Analysis for Vancomycin
Peak serum levels are mainly determined by 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 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 demonstrated 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 very inaccurate. It is also very dependent on accurate timing of the level, administration time, and infusion duration.
In order to maximum the utility of levels for AUC dosing a peak and trough after the same dose at steady state is recommended.
A more robust analysis was performed using the Monte Carlo method and the findings were the same. Please see the link below for details.