Bayesian dosing methods are recommended for vancomycin dosing. An analysis of Bayesian pharmacokinetic modeling of vancomycin was conducted using the Monte Carlo method in Excel. The Monte Carlo method is a technique of random sampling employed to approximate solutions to quantitative problems. It is a mathematical procedure or algorithm in which random numbers are run through a model or simulation to observe the properties of large sets of results. The Monte Carlo method was used to determine the effects of Bayesian optimization on pharmacokinetic data fittings using either a peak and trough set, a trough, or a peak. A data set of 10,000 pharmacokinetic values for both Vd and Cl was created. The values followed a normal distribution with a mean and standard deviation of the pharmacokinetic parameters. The Vd and Cl values were independently calculated. This was done by setting 10,000 rows in two columns in Excel = NORMINV(rand(), mean of kinetic parameter, standard deviation of kinetic parameter) for each parameter which produced random values that fit a normal distribution described by the input mean and standard deviation of the parameter. Rand() is an Excel function that produces random numbers greater than or equal to 0 and less than 1. It does not require an argument. Excel's NORMINV(probability, mean, standard deviation) function requires the arguments in parenthesis. It returns a value described by its probability in a normal distribution for the input mean and standard deviation. If all 10,000 values were plotted on a histogram a normal curve would result with the input mean and standard deviation. Standard one-compartment open model equations were used to calculate the peak, trough, and AUC at steady state for each Vd and Cl data pair. The calculated peak and troughs were used as theoretical "measured levels" in the following Bayesian analyses as they represent the possible values that would be obtained in a large population analysis.
Vd fit and Cl fit were determined by minimizing the following Bayesian equations for each row of simulated data using Excel's Solver. The full data set took about 8 hours of run time when a peak and trough were used and over 24 hours when a single was used.
Peak and Trough Bayesian Equation
Minimize the sum of the following = (Peak measured - Peak Predicted)2 / (0.1 * Peak Measured)2 + (Trough Measured - Tough Predicted)2 / (0.1*Trough Measured)2 + (Vd population Mean - Vd fit)2 / (0.3*Vd population mean)2 + (Cl population mean -Cl fit)2 / (0.3*Cl population mean)2
Trough Bayesian Equation
Minimize the sum of the following = (Trough Measured - Tough Predicted)2 / (0.1*Trough Measured)2 + (Vd population Mean - Vd fit)2 / (0.3*Vd population mean)2 + (Cl population mean -Cl fit)2 / (0.3*Cl population mean)2
Peak Bayesian Equation
Minimize the sum of the following = (Peak measured - Peak Predicted)2 / (0.1 * Peak Measured)2 + (Vd population Mean - Vd fit)2 / (0.3*Vd population mean)2 + (Cl population mean -Cl fit)2 / (0.3*Cl population mean)2
Cl population mean = 7.87 L/hr (Standard Deviation 5.51), Vd population mean = 45.5 liters (Standard Deviation 13.65), Coefficient of variation for both Vd and Cl were set at 0.3. Coefficient of variation of lab assay was set at 0.1.
Other Equations and Values Used in the Analysis
CV = Standard Deviation of Parameter / Mean
Dose (mg) = 1000, Tau = 8, Infusion Period = 1 hour
The predicted peak and trough are the values calculated for the fit Vd and Cl using standard one-compartment open model equations.
Peak(mcg/ml) = Dose(mg)*(1-exp(-K*T'))/(Vd*K*T'*(1-exp(-K*Tau)))
Trough(mcg/ml) = Peak(mcg/ml)*(exp(-K*(Tau-T')))
K(1/hours) =Cl(L/hr) / Vd(L)
AUC (mg*hour/liter per day) = Dose(mg) * (24/Tau) / Cl(L/hr), Note AUC is determined by clearance.
Linear Regression Analysis
The formula for linear regression was coded in Excel as Excel has a maximum of 255 data points for linear fitting. Then linear regression was performed on the Monte Carlo values of Cl and Vd versus the Bayesian fit values.
N= number of data points pairs, xi are the individual x values, yi are the individual y values to be fit.
Sxx = Sum(xi2) - (Sum (xi))2/N
Syy = Sum(yi2) - (Sum (yi))2/N
Sxy = Sum(xi*yi) - (Sum (xi) * Sum (yi))/N
Slope = Sxy/Sxx
Intercept = (Sum(yi) -Slope*Sum(xi))/N
R2 = (Sxy)2 /(Sxx*Syy)
Results of Analysis and Recommendations
Calculating the AUC is the most accurate with a peak and trough, less accurate with a trough level, and least accuracy with a peak. The chance of inappropriate dosage adjustments is lowest with a peak and trough analysis and highest with a peak analysis. Using a peak for AUC dosing is not recommended as the information supplied is of low quality. If the desired AUC range is 500-600 using a trough is not recommended as the chance of inappropriate dosage adjustment is higher than using a peak and trough.
Linear regression analysis of clearance calculated with Bayesian optimization versus the Monte Carlo value demonstrated the highest coefficient of determination when a peak and trough (R2 0.81) were used and the lowest when a peak ((R2 0.5) was used. Note AUC is determined by clearance.
Linear regression analysis of Vd calculated with Bayesian optimization versus the Monte Carlo value demonstrated the highest coefficient of determination when a peak and trough (R2 0.68) were used and the lowest when a trough (R2 0.22) was used.
The sensitivity of the AUC calculation was 88.7% with a peak and trough, 81.1 % with a trough, and 62.4% with a peak.
Misclassification of the calculated AUC can lead the overdosing and under-dosing, see the tables below, and is lowest with a peak and trough and highest with a peak. The further the calculated parameters are from the population mean values the greater the need to draw both a peak and trough level when determining AUC.
As both a peak and trough are required to calculate Vd and Cl using standard one-compartment equations the above findings are expected. Otherwise, Vd has to be fixed or both parameters have to be restricted using a Bayesian method. This analysis demonstrates that the Bayesian method does not overcome the need for a peak and tough during AUC calculation.
| Peak and Trough Bayesian Fitting: % of Calculated AUCs in Stated Range | ||||
| Monte Carlo AUC | <=400 | 401-500 | 501-600 | >600 |
| <=400 | 96.5 | 3.5 | 0.0 | 0.0 |
| 401-500 | 20.1 | 78.4 | 1.4 | 0.0 |
| 501-600 | 3.3 | 23.4 | 73.3 | 0.1 |
| >600 | 1.0 | 1.8 | 13.7 | 83.5 |
| Total %
Potential To Under Dose |
5.0 | |||
| Total % Potential To Over Dose |
63.3 | |||
| Trough Only Bayesian Fitting: % of Calculated AUCs in Stated Range | ||||
| Monte Carlo AUC | <=400 | 401-500 | 501-600 | >600 |
| <=400 | 91.6 | 8.4 | 0.0 | 0.0 |
| 401-500 | 26.7 | 64.5 | 8.8 | 0.0 |
| 501-600 | 5.4 | 27.3 | 62.0 | 5.3 |
| >600 | 1.3 | 3.5 | 16.8 | 78.4 |
| Total %
Potential To Under Dose |
22.5 | |||
| Total % Potential To Over Dose |
81.0 | |||
| Peak Only Bayesian Fitting: % of Calculated AUCs in Stated Range | ||||
| Monte Carlo AUC | <=400 | 401-500 | 501-600 | >600 |
| <=400 | 75.2 | 17.6 | 3.6 | 3.6 |
| 401-500 | 38.7 | 45.5 | 9.3 | 6.4 |
| 501-600 | 2.2 | 60.1 | 27.0 | 10.6 |
| >600 | 0.1 | 8.1 | 28.4 | 63.5 |
| Total % Potential To
Under Dose |
51.2 | |||
| Total % Potential To Over Dose |
137.5 | |||
| Linear Regression Analysis | Cl = Slope*Cl Monte Carlo + Intercept | ||
| Clearance Calculated | Slope | Intercept | R2 |
| Peak & Trough Bayesian Fitting | 0.9302 | 0.7553 | 0.81 |
| Trough Bayesian Fitting | 0.9152 | 0.9368 | 0.67 |
| Peak Bayesian Fitting | 0.5467 | 2.9936 | 0.50 |
| Linear Regression Analysis | Vd = Slope*Vd Monte Carlo + Intercept | ||
| Vd Calculated | Slope | Intercept | R2 |
| Peak & Trough Bayesian Fitting | 0.2948 | 30.81 | 0.68 |
| Trough Bayesian Fitting | 0.1133 | 39.31 | 0.22 |
| Peak Bayesian Fitting | 0.0474 | 43.49 | 0.34 |
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