# Uncertainty Evaluation for the Quantification of Urinary Amphetamine and 4-Hydroxyamphetamine Using Liquid Chromatography–Tandem Mass Spectrometry: Comparison of the Guide to the Expression of Uncertainty in Measurement Approach and the Monte Carlo Method with R

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. GUM Approach for Evaluating the Measurement Uncertainty

#### 2.1.1. Urine Sample Dilution

#### 2.1.2. Preparation of Calibrators

#### 2.1.3. Calibration Curve

#### 2.1.4. Method Repeatability

#### 2.1.5. Calculating the Combined and Expanded Uncertainty

#### 2.2. MCM for the Evaluation of MU

#### 2.2.1. Mathematical Modeling of the Measurand

#### 2.2.2. MCM Simulation

^{6}simulation trials (M = 10

^{6}). The calibrators were first generated by utilizing the serial dilution described in Supplementary Data S1. The uncertainty of response y

_{i}was obtained from the standard deviations of four repeated measured peak area ratios $\left(\frac{{A}_{a}}{{A}_{is}}\right)$ at each calibrator, while response y

_{i}was drawn from a normal distribution. The calibration curve was fitted using the weighted linear regression model with the generated calibrators. The responses of the peak area ratios and predicted measurand were obtained for each trial. The additive factors accounting for method repeatability and urine sample dilution were drawn from normal distributions with a mean of 0 and standard deviations obtained through the GUM approach, respectively. The R code to implement the MCM is available at Supplementary Data S2.

#### 2.2.3. Uncertainty with MCM and Comparison with GUM Approach

## 3. Materials and Methods

#### 3.1. Chemicals and Reagents

_{8}were obtained as solutions from Cerilliant (Austin, TX, USA). HPLC-grade methanol was purchased from J.T. Baker/Avantor (Center Valley, PA, USA). Water (LiChrosolv-grade) was supplied by Merck (Darmstadt, Germany). Formic acid (LC–MS LiChropur-grade) was purchased from Sigma Aldrich (St. Louis, MO, USA). All other chemicals were of analytical grade or higher. The AP and 4HA working standard solutions (10 µg/mL) were prepared by dilution with methanol. The IS working solution was prepared in methanol to obtain an AP-d

_{8}solution (0.1 µg/mL). These solutions were stored at −20 °C in amber bottles before use.

#### 3.2. Preparation of Urine Sample

#### 3.3. Preparation of Calibrators and QC Samples

^{2}, was used to generate a calibration curve. QC samples for AP (30, 150, and 300 ng/mL) and 4HA (6, 30, and 150 ng/mL) were prepared by spiking the pooled blank urine samples with known amounts of each compound to ensure the repeatability of the LC–MS/MS method. The inter-day precision of the method was established via six independent determinations with the same QC samples on four independent experimental assays of the abovementioned replicates (n = 24).

#### 3.4. LC–MS/MS Conditions

_{8}.

#### 3.5. Identification of the Uncertainty Sources

#### 3.6. Uncertainty Estimation Method

#### 3.7. Specifying the Measurand

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**Relative contribution of the four different uncertainty sources to the overall combined uncertainty of amphetamine (AP) and 4-hydroxyamphetamine (4HA) in the Guide to the Expression of Uncertainty in Measurement (GUM) approach.

**Figure 2.**Probability density function of the concentration measurand for amphetamine (AP) and 4-hydroxyamphetamine (4HA).

Quantity | Unit | Value | Standard Uncertainty | Probability Density Function |
---|---|---|---|---|

${C}_{R}$ | mg/mL | 1.00 | 0.003 | Normal (1.00, ${\left(0.003\right)}^{2}$) |

${V}_{f10}$ | mL | 10 | 0.003 | Normal (10, ${\left(0.003\right)}^{2})$ |

${V}_{Tf}$ | °C | - | 0.04301 | Uniform (−0.0745, 0.0745) |

${V}_{p1000}$ | μL | 1000 | 0.75 | Normal (1000, ${\left(0.75\right)}^{2}$) |

${V}_{p500}$ | μL | 500 | 0.75 | Normal (500, ${\left(0.75\right)}^{2}$) |

${V}_{p100}$ | μL | 100 | 0.1 | Normal (100, ${\left(0.1\right)}^{2}$) |

${V}_{p50}$ | μL | 50 | 0.1 | Normal (50, ${\left(0.1\right)}^{2}$) |

**Table 2.**Results of the evaluation of method repeatability from low-, middle-, and high-quality control samples.

Analyte | Nominal Concentration (ng/mL) | Experiments | |||||||
---|---|---|---|---|---|---|---|---|---|

Assay 1 | Assay 2 | Assay 3 | Assay 4 | ||||||

Mean (n = 6) | Standard Deviation | Mean (n = 6) | Standard Deviation | Mean (n = 6) | Standard Deviation | Mean (n = 6) | Standard Deviation | ||

AP | 30 | 32.3 | 0.4176 | 32.9 | 0.4840 | 29.3 | 0.3014 | 31.5 | 0.8831 |

150 | 165.4 | 2.7589 | 160.4 | 1.5858 | 144.5 | 4.3782 | 155.9 | 3.3410 | |

300 | 308.7 | 3.6577 | 314.9 | 5.2759 | 288.8 | 4.2205 | 303.4 | 2.7928 | |

4HA | 6 | 6.7 | 0.1797 | 6.6 | 0.1135 | 5.8 | 0.1940 | 6.5 | 0.2496 |

30 | 32.9 | 0.4256 | 33.6 | 0.6306 | 30.2 | 0.6749 | 31.4 | 0.9422 | |

150 | 163.2 | 3.5016 | 163.2 | 2.2802 | 145.3 | 2.5725 | 151.3 | 3.1556 |

**Table 3.**Estimation of uncertainty contributions in quantitative LC–MS/MS analysis results of amphetamine (AP) and 4-hydroxyamphetamine (4HA) in urine.

Source of Uncertainty | AP | 4HA | ||||
---|---|---|---|---|---|---|

Relative Standard Uncertainty | Effective Degrees of Freedom | Degree of Contribution (%) | Relative Standard Uncertainty | Effective Degrees of Freedom | Degree of Contribution (%) | |

Urine sample dilution (${u}_{r}\left({D}_{S}\right)$) | 0.000866 | ∞ | 0.05 | 0.000866 | ∞ | 0.03 |

Preparation of calibrators (${u}_{r}\left({C}_{CS}\right)$) | 0.018953 | ∞ | 25.98 | 0.019036 | ∞ | 16.05 |

Calibration curve (${u}_{r}\left({C}_{C}\right)$) | 0.025458 | 5 | 46.87 | 0.032891 | 5 | 47.92 |

Method repeatability (${u}_{r}\left({R}_{qc}\right)$) | 0.019355 | 55 | 27.09 | 0.028505 | 51 | 35.99 |

Relative combined standard uncertainty (${u}_{rc}\left(C\right)$) | 0.037184 | 22 | - | 0.047512 | 21 | - |

Relative expanded uncertainty (${U}_{r}\left(C\right)$) | 0.076971 | - | - | 0.098826 | - | - |

**Table 4.**Statistical parameters obtained for the Guide to the Expression of Uncertainty in Measurement (GUM) approach and the Monte Carlo method (MCM) for the determination of urinary concentrations of amphetamine (AP) and 4-hydroxyamphetamine (4HA).

Statistical Data | AP | 4HA | ||
---|---|---|---|---|

GUM | MCM | GUM | MCM | |

Mean (ng/mL) | 99.74 | 99.75 | 11.33 | 11.33 |

Standard deviation (ng/mL) | 3.71 | 2.53 | 0.54 | 0.41 |

95% confidence interval (ng/mL) | (92.06, 107.41) | (94.83, 104.74) | (10.21, 12.45) | (10.52, 12.14) |

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**MDPI and ACS Style**

Kim, S.Y.; Shin, D.W.; Hyun, J.; Kwon, N.H.; Cheong, J.C.; Paeng, K.-J.; Lee, J.; Kim, J.Y.
Uncertainty Evaluation for the Quantification of Urinary Amphetamine and 4-Hydroxyamphetamine Using Liquid Chromatography–Tandem Mass Spectrometry: Comparison of the Guide to the Expression of Uncertainty in Measurement Approach and the Monte Carlo Method with R. *Molecules* **2023**, *28*, 6803.
https://doi.org/10.3390/molecules28196803

**AMA Style**

Kim SY, Shin DW, Hyun J, Kwon NH, Cheong JC, Paeng K-J, Lee J, Kim JY.
Uncertainty Evaluation for the Quantification of Urinary Amphetamine and 4-Hydroxyamphetamine Using Liquid Chromatography–Tandem Mass Spectrometry: Comparison of the Guide to the Expression of Uncertainty in Measurement Approach and the Monte Carlo Method with R. *Molecules*. 2023; 28(19):6803.
https://doi.org/10.3390/molecules28196803

**Chicago/Turabian Style**

Kim, Seon Yeong, Dong Won Shin, Jihye Hyun, Nam Hee Kwon, Jae Chul Cheong, Ki-Jung Paeng, Jooyoung Lee, and Jin Young Kim.
2023. "Uncertainty Evaluation for the Quantification of Urinary Amphetamine and 4-Hydroxyamphetamine Using Liquid Chromatography–Tandem Mass Spectrometry: Comparison of the Guide to the Expression of Uncertainty in Measurement Approach and the Monte Carlo Method with R" *Molecules* 28, no. 19: 6803.
https://doi.org/10.3390/molecules28196803