Bayesian dosing methods are recommended for vancomycin dosing. An analysis of Bayesian and non-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. 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 non-Bayesian analysis was conducted using a trough with the volume of distribution set a population mean, as this is commonly done when Bayesian methods are not employed.
Completion of the Monte Carlo analysis required the following:
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A predictive model was created with independent and dependent variables. The dependent variables in this analysis were vancomycin's calculated AUC, peak, and trough. The independent variables were vancomycin's clearance and volume of distribution. The predictive model was a pharmacokinetic one compartment open model with the equations described below.
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The mean and standard deviation of the independents variables in the model were known.
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A normal distribution of possible values of the independent variables were created using Excel's functions Rand() and NORMINV() based on the variables' means and standard deviations. A larger number of values were created to increase the accuracy of the results. Ten thousand values for vancomycin's volume of distribution and clearance were created each fitting a normal distribution.
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The created values for volume of distribution and clearance in the normal distributions created in step three were used to populate the predictive model to calculated the dependent values of AUC, peak, and trough. These calculated values describe the possible values that would be obtain in a large pharmacokinetic population study.
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Excel's solver was then used with the one compartment open model equations to determine the clearance and volume of distribution which best predicted the peaks and troughs created in step 4 while minimizing the values for the Bayesian and non-Bayesian equations described below. Four independent analysis were performed, one for each equation to minimize.
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The determined Vd and clearance in step 5 along with the predicted values for peak, trough, and AUC were then compared to the values calculated in step 4. Thus a comparison was made of the possible values that would be obtained in a large population study with those determined using Bayesian and non-Bayesian fitting methods for the independent and dependent variables (volume of distribution, clearance, peak, trough, and AUC).
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The results of the analysis were summarized using linear regression analysis, clinical decision agreement tables for AUC, and scatter plots of the Z-score of the Monte Carlo assigned clearance versus the percent error in AUC calculated.
A data set of 10,000 pharmacokinetic values for both volume of distribution (Vd) and clearance (Cl) was created. The values followed a normal distribution with the mean and standard deviation for each respective pharmacokinetic parameter. The values were restricted to a range of + 3 standard deviations of the population mean. The Vd and Cl values were independently calculated. This was done by setting 10,000 rows in two columns in Excel equal to the function 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 each parameter. Rand() is an Excel function that produces a random six digit decimal number between 0 and 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 are plotted on a histogram a normal curve would result with a mean and standard deviation equivalent to the input values in the NORMINV function. 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 AUC, peak, and troughs were used as theoretical "true or measured values" in the following Bayesian and non-Bayesian analyses as they represent the possible values that would be obtained in a large population.
Volume of distribution fit and clearance fit were determined by minimizing the following Bayesian equations for each row of simulated data using Excel's Solver. In the non-Bayesian analysis volume of distribution was held at the population mean and only clearance was fit. Solver is a built in non-linear fitting routine that determines the optimal value of parameter(s), to minimum, maximize, or set to 0, the results of an user defined equation. The full data set for each equation below took approximately 24 hours of run time to complete.
Peak and Trough Bayesian Equation for Determination of Vd and Cl
Minimize 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 for Determination of Vd and Cl
Minimize 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 for Determination of Vd and Cl
Minimize 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
Trough Non-Bayesian Equation for Determination of Clearance (With Vd Set at the Population Mean)
Minimize the following = (Trough Measured - Trough Predicted)2 / (0.1*Trough Measured)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
Z score = (X-Mean)/SD, X is the Monte Carlo Assigned Value
Percent Error in AUC Fit = (AUC Fit - AUC Monte Carlo) *100 / AUC Monte Carlo
Coefficient of Variation (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)
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 (fit or predicted).
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)
Recommendations
The AUC is calculated most accurately with a peak and trough, less accurately with a trough, and least accurately with a peak. The chance of inappropriate dosage adjustment is lowest in 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 and highly error prone. Bayesian AUC analysis with only trough levels misclassifies AUCs 18.8% of the time which would negate the benefits of AUC dosing in a significant percent of patients.
The sensitivity to accurately classify the AUC calculated was 96.7% with a peak and trough Bayesian analysis, 81.2 % with a trough Bayesian analysis, 80.4% with a trough non-Bayesian analysis, and 62.5% with a peak Bayesian analysis.
Misclassification of the calculated AUC can lead to overdosing and under-dosing and is lowest with a Bayesian analysis of a peak and trough at 3.3%, 18.8% for Bayesian analysis of a trough, 19.6% with non-Bayesian analysis of a trough, and highest with a Bayesian analysis of a peak at 37.5%.
Linear regression analysis of AUC calculated with Bayesian optimization versus the Monte Carlo generated values demonstrated the highest coefficient of determination when a peak and trough (R2 0.997) were used, lower when a trough was used (R2 0.934), and the lowest when a peak (R2 0.624) was used.
Linear regression analysis of clearance calculated with Bayesian optimization versus the Monte Carlo generated values demonstrated the highest coefficient of determination when a peak and trough (R2 0.95) were used, intermediate when a trough was used (R2 0.73), and the lowest when a peak (R2 0.5) was used.
Linear regression analysis of Vd calculated with Bayesian optimization versus the Monte Carlo generated values demonstrated the highest coefficient of determination when a peak and trough (R2 0.96) were used and were much low when a peak (R2 0.35), or trough (R2 0.27) were used.
A scatter plot of the Z-score of the Monte Carlo assigned clearance versus the percent error in AUC calculated demonstrated an error range between -5 to +10 percent for a Bayesian peak and trough analysis. Outliers were noted at extremely low volume of distributions (Z scores for Vd of < -2.5). The error range was -45% to + 45% for a Bayesian trough analysis. Outliers were also noted. These plots demonstrated the superior accuracy of AUC calculation with the Bayesian peak and trough analysis.
This analysis demonstrates that the Bayesian method does not overcome the need for a peak and trough set during AUC calculation. Peak levels should not be used alone in Bayesian analysis. Bayesian analysis using only a trough level appears to be similar to non-Bayesian AUC analysis when the Vd is held at the population mean. Bayesian analysis using trough levels has an AUC misclassification rate that would negate the benefits of AUC dosing in a significant percentage of patients and is not recommended.
Data Analysis & Graphics
Z score of Monte Carlo assigned clearance versus the Percent Error in AUC calculated = (AUC calculated - AUC Monte Carlo)*100/AUC Monte Carlo. Listed in order of accuracy, highest to lowest.
Clinical Decision Agreement
The green color denotes concordance of AUC calculation for Bayesian or non-Bayesian fitting and Monte Carlo values. Red color denotes Bayesian or non-Bayesian fitting calculated AUC values lower than the Monte Carlo values with the potential to cause the user to increase the dose in error. The yellow color denotes Bayesian or non-Bayesian calculated AUC values higher than the Monte Carlo values with the potential to cause the user to decrease the dose in error.
| Peak and Trough Bayesian Fitting: % of Calculated AUCs in Stated Range | ||||
| Monte Carlo AUC | <=400 | 401-500 | 501-600 | >600 |
| <=400 | 98.7 | 1.3 | 0.0 | 0.0 |
| 401-500 | 3.7 | 95.7 | 0.6 | 0.0 |
| 501-600 | 0.0 | 8.7 | 91.1 | 0.2 |
| >600 | 0.1 | 0.0 | 6.4 | 93.6 |
| Total %
Potential To Under Dose |
0.9 | |||
| Total % Potential To Over Dose |
2.4 | |||
| Total % Congruent Results | 96.7 | |||
| Trough Only Bayesian Fitting: % of Calculated AUCs in Stated Range | ||||
| Monte Carlo AUC | <=400 | 401-500 | 501-600 | >600 |
| <=400 | 91.9 | 8.1 | 0.0 | 0.0 |
| 401-500 | 27.5 | 64.4 | 8.1 | 0.0 |
| 501-600 | 5.7 | 28.1 | 60.0 | 6.3 |
| >600 | 1.3 | 3.5 | 15.8 | 79.6 |
| Total %
Potential To Under Dose |
7.0 | |||
| Total % Potential To Over Dose |
11.8 | |||
| Total % Congruent Results | 81.2 | |||
| Peak Only Bayesian Fitting: % of Calculated AUCs in Stated Range | ||||
| Monte Carlo AUC | <=400 | 401-500 | 501-600 | >600 |
| <=400 | 75.5 | 17.7 | 3.2 | 3.6 |
| 401-500 | 39.1 | 46.4 | 7.8 | 6.7 |
| 501-600 | 2.3 | 64.0 | 22.0 | 11.7 |
| >600 | 0.0 | 9.2 | 24.5 | 66.3 |
| Total % Potential To
Under Dose |
18.3 | |||
| Total % Potential To Over Dose |
19.2 | |||
| Total % Congruent Results | 62.5 | |||
| Trough Only Non-Bayesian Fitting (Vd held at the Population Mean): % of Calculated AUCs in Stated Range | ||||
| Monte Carlo AUC | <=400 | 401-500 | 501-600 | >600 |
| <=400 | 91.0 | 9.0 | 0.0 | 0.0 |
| 401-500 | 25.4 | 60.8 | 13.8 | 0.0 |
| 501-600 | 5.3 | 22.3 | 61.1 | 11.3 |
| >600 | 1.0 | 2.6 | 12.5 | 83.9 |
| Total % Potential To
Under Dose |
9.4 | |||
| Total % Potential To Over Dose |
10.3 | |||
| Total % Congruent Results | 80.4 | |||
| Linear Regression Analysis | AUC = Slope*AUC Monte Carlo + Intercept | ||
| AUC Calculated | Slope | Intercept | R2 |
| Peak & Trough Bayesian Fitting | 0.973 | 11.12 | 0.997 |
| Trough Bayesian Fitting | 0.8991 | 42.387 | 0.9334 |
| Trough Non-Bayesian Fitting | 0.9857 | -6.2076 | 0.9156 |
| Peak Bayesian Fitting | 0.8413 | 94.78 | 0.6238 |
| Linear Regression Analysis | Cl = Slope*Cl Monte Carlo + Intercept | ||
| Clearance Calculated | Slope | Intercept | R2 |
| Peak & Trough Bayesian Fitting | 0.9438 | 0.3960 | 0.952 |
| Trough Bayesian Fitting | 0.7183 | 2.0441 | 0.7323 |
| Trough Non-Bayesian Fitting | 1.141 | -0.4867 | 0.4369 |
| Peak Bayesian Fitting | 0.56194 | 2.8808 | 0.505 |
| Linear Regression Analysis | Vd = Slope*Vd Monte Carlo + Intercept | ||
| Vd Calculated | Slope | Intercept | R2 |
| Peak & Trough Bayesian Fitting | 0.7576 | 9.31 | 0.9567 |
| Trough Bayesian Fitting | 0.2721 | 30.14 | 0.2691 |
| Peak Bayesian Fitting | 0.2493 | 33.807 | 0.35 |
This page was updated 8/25/2024