# Predicting the Area under the Plasma Concentration-Time Curve (AUC) for First Dose Vancomycin Using First-Order Pharmacokinetic Equations

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## Abstract

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## 1. Introduction

## 2. Results

^{2}; the initial vancomycin dose of 15 mg/kg was infused over 60 min. Detailed characteristics of these studies are described in Table S1. Details about the building of pharmacokinetic models for the reference standard and two first-order pharmacokinetic equations are provided in the Materials and Methods section.

_{1}sampled at 0, 40, 60, and 90 min after the completed infusion combined with C

_{2}at 240 min after the completed infusion gave an acceptable −4.6 to 4.1 percent mean difference. The correlation and agreement were better using C

_{1}obtained 60 and 90 min after the end of the infusion (Pearson’s r = 0.976 and Lin’s correlation coefficient = 0.971 and 0.967, respectively) than those obtained at 0 and 40 min (Pearson’s r = 0.899, 0.929 and Lin’s correlation coefficient = 0.869 and 0.923, respectively).

_{2}at 240 min after the completed infusion, only C

_{1}sampled immediately after completing the infusion showed an acceptable −2.7 percent mean difference, with Pearson’s r = 0.974 and Lin’s correlation coefficient = 0.940. Model 1 using C

_{2}sampled 300 min after the completed infusion and C

_{1}at 0 and 15 min gave 3.4 and −4.8 percent mean difference with Pearson’s r = 0.977, 0.98 and Lin’s correlation coefficient = 0.965, 0.960, respectively.

_{1}between samples 60 and 180 min after the infusion ended resulted in a −4.2 to 2.8 percent mean difference when C

_{2}was taken either at 240 or 300 min after the completed infusion. Alternatively, C

_{1}sampled 30 min after the completed infusion gave an acceptable percent mean difference only with C

_{2}at 240 min after the infusion. In terms of correlation and agreement, all mentioned paired concentration time points had Pearson’s r > 0.96 and Lin’s correlation coefficient >0.95, with the exception of C

_{1}at 180 min paired with C

_{2}at 240 min after the completed infusion and C

_{1}at 180 min paired with C

_{2}at 300 min after the completed infusion, both of which had lower correlation and agreement (Pearson’s r = 0.883 and Lin’s correlation coefficient = 0.881; Pearson’s r = 0.939 and Lin’s correlation coefficient = 0.933, respectively).

_{1}sampled in a time window from 60 to 90 min after the completed infusion, and C

_{2}sampled in a time window from 240 to 300 min after the completed infusion (Figure 1). Bland–Altman plots for the selected equation-paired concentration time points are shown in Figure 2.

_{1}sampled from 60 to 90 min, and C

_{2}from 240 to 300 min after the completed infusion, showed a strong correlation and goodness of fit between the reference standard and the proposed selected equation-paired concentration time point combinations. Figure 3 shows the regression line of best fit between the two methods. The regression line had a slope of 1.03 and an intercept of 0.15, with a correlation coefficient between the two methods of r = 0.994 (p < 0.001).

_{f}, full area under the plasma concentration-time curve.

_{f}) calculation, the first-order pharmacokinetic equations were also used for calculating other pharmacokinetic parameters including vancomycin clearance (Cl), volume of distribution (Vd), and half-life (t

_{1/2}). The corresponding mean and standard deviation for these parameters, as well as the reference standards, are shown in Table 2 and Table 3.

## 3. Discussion

_{f}. We found that Model 2 with a compensation area for the alpha-phase, vancomycin serum concentrations collected from 60 to 90 min after the completed infusion as C

_{1}, and those collected 240–300 min after the completed infusion as C

_{2}produced a calculated AUC

_{f}with less than 5% mean difference compared to the full linear-log trapezoid method. Pearson’s correlation coefficient and Lin’s agreement coefficient for these combinations were >0.96. The mean difference and 95% limit of agreement for this equation-paired concentration time points combination were sufficiently narrow for clinical purposes. These findings were consistent with whether the infusion period was one or two hours, and regardless of the difference in kidney function in the two data sets. As shown in Table 2 and Table 3, other pharmacokinetic parameters, including vancomycin Cl, Vd, and t

_{1/2}, showed comparable results with the log-linear trapezoid method. These findings suggested that Model 2 is highly relevant for determining pharmacokinetic parameters after the first dose of vancomycin.

_{f}calculation. As shown in Figure S2, Model 2 captures additional area that at least partially compensates for the alpha-phase AUC, which is not captured by Model 1, resulting in lower bias and improved prediction.

_{0–24}) for each individual patient. Calculating pharmacokinetic parameters based on the first vancomycin dose can be used for optimizing the subsequent doses using the method described by Sawchuk-Zaske [17]. Several studies using this method to individualize vancomycin regimens showed that first-dose TDM resulted in faster target AUC attainment than routine steady-state TDM in various group of patients, including critically ill adults and neonates [18,19,20]. Second, our study aimed to identify the best-performing first-order pharmacokinetic equation and optimal sampling times that reliably reproducibly estimated first dose vancomycin AUC and other pharmacokinetic parameters using two independent data sets. The data sets we used to compare the actual AUC against the AUC predicted by each equation were from a group of adults with septic shock and a group of children with severe infection (Table S1). Our findings may therefore need further evaluation before generalization to other groups of patients, such as those who are overweight/obese, and those with renal impairment. Lastly, the data sets used in our study were comparatively small, and resulted in wide standard deviations around the mean difference in Bland-Altman analysis. Although conducting a new similar study with a larger sample size would likely lower this variability, accessing intensive pharmacokinetic data sets to make these calculations is difficult and designing a new study is an ethically challenging because the subjects are vulnerable. Furthermore, the 95% limit of agreements of selected equation-paired concentration time point combinations in our study was not expected to be clinically significant. Therefore, despite the limitations mentioned and the small sample size, our current study results provide information which can guide clinical practice in some groups of patients, and pharmacokinetic parameters based on the first vancomycin dose are useful for further AUC

_{0–24}research study (TCTR20210617001).

## 4. Materials and Methods

#### 4.1. Pharmacokinetic Model Building

#### 4.1.1. The Reference Standard

_{f}of the individuals in both studies were calculated using all plasma concentrations available (10 time points for subjects in the adult data set and 9 time points for subjects in the pediatric data set, Table S1) and the linear-log trapezoid rule. The pharmacokinetic analysis of the serum concentration-time data was performed using PKSolver (version 2.0, China Pharmaceutical University, Nanjing, China) [21].

#### 4.1.2. The First-Order Pharmacokinetic Equations [9]

_{1}is the first concentration measured after the infusion has been completed, C

_{2}is the second concentration collected toward the end of the dosing interval, and t is the difference in time between C

_{1}and C

_{2}. Once the Ke is computed, it can be used to compute theoretical concentrations through forward- and backward-extrapolation.

_{1}was sequentially chosen from 0 to 120 min after the completed infusion and C

_{2}was the last sample, collected 240 min after the completed infusion. In the pediatric data set, vancomycin was infused over 60 min and levels were sampled at 0, 15, 30, 60, 120, 180, 240, and 300 min after the completed infusion. C

_{1}was sequentially chosen from 0 to 240 min after the completed infusion and C

_{2}was chosen from 240 or 300 min after the completed infusion.

- Model 1 (Figure S1)

_{eoi′}) regardless effect of the alpha-phase of the 2-compartment distribution model. The area between the start and the end of the infusion time (t

_{eoi}) can be measured as the area of the triangle:

_{eoi}to the end of dose (t

_{∞}) is:

_{f}for first dose vancomycin in Model 1 can be simplified to:

- Model 2 (Figure S2)

_{0′}), aiming to compensate for the unmeasured alpha-phase. Under this model, the equation can be simplified to:

#### 4.2. Statistical Analysis

_{f}using all available samples from the full pharmacokinetic studies, and served as the reference standard. Bland-Altman analysis was used to assess agreement and bias [23] between the reference standard and each combination of the first-order pharmacokinetic equations and selected paired concentration time points. To facilitate the comparison of the equation-derived and gold standard AUC from the adult and pediatric studies, results were expressed as percent change. An a priori acceptable mean difference was set at 5%. Pearson’s correlation and linear regression were used to assess the linear correlation of estimates. Lin’s correlation coefficient was also used to assess agreement. Statistical significance was defined as p < 0.05. Furthermore, a summary of pharmacokinetic parameters derived from selected equation-paired concentration time points combination including vancomycin Cl, Vd, and t

_{1/2}were calculated and compared with the parameters derived from the linear-log trapezoid rule, which served as the reference standard. All statistical analyses were performed using Stata version 15.1 (Stata Corp LCC, College Station, TX, USA).

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Summary of intersected C1 and C2 windows as inputs of the first-order pharmacokinetic equation using Model 2.

**Figure 2.**Bland-Altman plots of the reference standard and each selected equation-paired concentration time points combination. (

**A**) Adult data set using Model 2 with C1 at 60 min and C2 at 240 after the completed infusion had mean bias of 13.0 (3.0%) and 95% limits of agreement of −45.0 to 70.9. (

**B**) Adult data set using Model 2 with C1 at 90 min and C2 at 240 after the completed infusion had mean bias of 18.0 (4.1%) and 95% limits of agreement of −40.8 to 76.7. (

**C**) Pediatric data set using Model 2 with C1 at 60 min and C2 at 240 after the completed infusion had bias of −0.1 (−0.1%) and 95% limits of agreement of −22.6 to 22.3. (

**D**) Pediatric data set using Model 2 with C1 at 60 min and C2 at 300 after the completed infusion had bias of 2.1 (1.9%) and 95% limits of agreement of −14.4 to 18.7.

**Figure 3.**The regression line between the reference standard AUC

_{f}and pooled selected equation-paired concentration time point combinations. The regression equation was y = 0.15 + 1.03x (R

^{2}= 0.988, p < 0.001). Pearson’s correlation coefficient between the two methods was r = 0.994 (p < 0.001).

**Figure 4.**Study flow diagram. Abbreviations: AUC

_{f}, full area under the plasma concentration-time curve.

**Table 1.**Selected equation-paired concentration time points with an acceptable mean difference from the Bland–Altman analysis from adult and pediatric data sets.

Data Set | Time Points | Bland–Altman Analysis | Correlation | Lin’s Coefficients | ||||||
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C_{1} | C_{2} | Mean | Difference | 95% Limits of Agreement | Pearson’s r | p-Value | Rho_c | p-Value | ||

Mean (%) | SD | |||||||||

Model 1 | ||||||||||

Pediatric | 0 | 240 | 111.9 | −3.0 (−2.7) | 13.4 | (−29.1, 23.2) | 0.974 | <0.001 | 0.940 | <0.001 |

0 | 300 | 115.3 | 3.9 (3.4) | 10.2 | (−16.2, 24.0) | 0.977 | <0.001 | 0.965 | <0.001 | |

15 | 300 | 109.3 | −5.3 (−4.8) | 9.9 | (−24.7, 14.2) | 0.98 | <0.001 | 0.960 | <0.001 | |

Model 2 | ||||||||||

Adult | 0 | 240 | 417.5 | −19.3 (−4.6) | 60.2 | (−137.3, 98.8) | 0.899 | <0.001 | 0.869 | <0.001 |

40 | 240 | 422.2 | −9.9 (−2.3) | 50.0 | (−107.9, 88.0) | 0.929 | <0.001 | 0.923 | <0.001 | |

60 | 240 | 433.6 | 13.0 (3.0) | 30.0 | (−45.0, 70.9) | 0.976 | <0.001 | 0.971 | <0.001 | |

90 | 240 | 436.1 | 18.0 (4.1) | 30.0 | (−40.8, 76.7) | 0.976 | <0.001 | 0.967 | <0.001 | |

Pediatric | 30 | 240 | 114.1 | 4.3 (3.8) | 11.3 | (−17.8, 26.4) | 0.969 | <0.001 | 0.955 | <0.001 |

60 | 240 | 111.8 | −0.1 (−0.1) | 11.4 | (−22.6, 22.3) | 0.964 | <0.001 | 0.962 | <0.001 | |

120 | 240 | 109.6 | −4.6 (−4.2) | 11.6 | (−27.4, 18.2) | 0.963 | <0.001 | 0.957 | <0.001 | |

180 | 240 | 113.3 | 2.8 (2.5) | 20.5 | (−37.4, 43.0) | 0.883 | <0.001 | 0.881 | <0.001 | |

60 | 300 | 113.0 | 2.1 (1.9) | 8.4 | (−14.4, 18.7) | 0.981 | <0.001 | 0.979 | <0.001 | |

120 | 300 | 109.7 | −4.4 (−4.01) | 10.6 | (−25.1, 16.2) | 0.974 | <0.001 | 0.966 | <0.001 | |

180 | 300 | 113.5 | 3.2 (2.8) | 16.2 | (−28.5, 34.9) | 0.939 | <0.001 | 0.933 | <0.001 |

**Table 2.**Summary of pharmacokinetic parameters of the adult data set derived from selected equation-paired concentration time point combinations compared to the reference standard.

Adult Data Set | Reference Standard | Model 2 | |
---|---|---|---|

C_{1} at 60 and C_{2} at 240 min after the Completed Infusion | C_{1} at 90 and C_{2} at 240 min after the Completed Infusion | ||

AUC, mean ± SD (mg/L × h) | 427.14 ± 135.26 | 440.11 ± 132.85 | 445.09 ± 138.21 |

Vancomycin Cl, mean ± SD (L/h) | 4.62 ± 1.45 | 4.47 ± 1.38 | 4.43 ± 1.40 |

Vd, mean ± SD (L) | 39.35 ± 8.95 | 39.88 ± 8.92 | 41.33 ± 7.76 |

Half-life ± SD (h) | 6.28 ± 1.97 | 6.56 ± 2.02 | 6.83 ± 1.69 |

**Table 3.**Summary of pharmacokinetic parameters of the pediatric data set derived from selected equation-paired concentration time point combinations compared to the reference standard.

Pediatric Data Set | Reference Standard | Model 2 | |
---|---|---|---|

C_{1} at 60 and C_{2} at 240 min after the Completed Infusion | C_{1} at 60 and C_{2} at 300 min after the Completed Infusion | ||

AUC, mean ± SD (mg/L × h) | 111.91 ± 42.73 | 111.78 ± 39.83 | 114.03 ± 43.39 |

Vancomycin Cl, mean ± SD (L/kg/h) | 0.16 ± 0.04 | 0.16 ± 0.04 | 0.15 ± 0.04 |

Vd, mean ± SD (L/kg) | 0.55 ± 0.10 | 0.51 ± 0.06 | 0.52 ± 0.07 |

Half-life ± SD (h) | 2.58 ± 0.91 | 2.38 ± 0.69 | 2.47 ± 0.84 |

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## Share and Cite

**MDPI and ACS Style**

Sujjavorakul, K.; Katip, W.; Kerr, S.J.; Wacharachaisurapol, N.; Puthanakit, T.
Predicting the Area under the Plasma Concentration-Time Curve (AUC) for First Dose Vancomycin Using First-Order Pharmacokinetic Equations. *Antibiotics* **2023**, *12*, 630.
https://doi.org/10.3390/antibiotics12040630

**AMA Style**

Sujjavorakul K, Katip W, Kerr SJ, Wacharachaisurapol N, Puthanakit T.
Predicting the Area under the Plasma Concentration-Time Curve (AUC) for First Dose Vancomycin Using First-Order Pharmacokinetic Equations. *Antibiotics*. 2023; 12(4):630.
https://doi.org/10.3390/antibiotics12040630

**Chicago/Turabian Style**

Sujjavorakul, Kritsaporn, Wasan Katip, Stephen J. Kerr, Noppadol Wacharachaisurapol, and Thanyawee Puthanakit.
2023. "Predicting the Area under the Plasma Concentration-Time Curve (AUC) for First Dose Vancomycin Using First-Order Pharmacokinetic Equations" *Antibiotics* 12, no. 4: 630.
https://doi.org/10.3390/antibiotics12040630