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. Accuracy of a Bayesian forecasting program is largely determined by the robustness and generalizability of the population model used.
Common quoted statements about Bayesian Dosing Methods.
"Bayesian methods can use serum levels obtained at anytime." This is not related to the Bayesian Method, but is related to software design.
"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.
"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 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 model that aren't based on data rich sampling are over parameterized and use equations with too many paramaters that have not be adequately resolved. Calculation accuracy will be limited and unreliable.
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.
Accurate and reliable estimation of the AUC with trough-only data is only possible when richly sampled data is used as a Bayesian prior. Richly sampled two compartment 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. Two compartment models built from trough only or peak and trough data sets underestimate the AUC when trough levels are measured by 23% and 15% respectively (Neely MN 2014). Most published models are based on small populations with limited peaks and trough sampling and usually with only trough sampling. Two compartment models derived from trough only data are not as accurate as one compartment models in AUC calculations. One compartment models derived from either trough or peak and trough data are accurate in AUC calculations (Maung NH 2022).
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.
Coming soon a Monte Carlo Analysis of AUC calculation errors using a peak and trough versus a trough for a two compartment model.
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