# Machine Learning in Bioequivalence: Towards Identifying an Appropriate Measure of Absorption Rate

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Strategy of the Analysis

#### 2.2. Bioequivalence Data-Noncompartmental Analysis

^{®}) from Pfizer, separated by a 21-day washout period. The pharmaceutical company (Verisfield SA) kindly provided the blind C-t data of the study participants in order to perform this computational study. According to the BE study protocol, the blood samples were obtained at 0.5, 1, 2, 3, 4, 6, 8, 12, 24, 48, 96, 144, and 192 h after the treatment. Overall, there were 14 observations per subject and 26 subjects per period of the study. Taking into consideration that the data come from a 2 × 2 crossover study and there was no discrimination between the two medicinal products, there are 52 subjects with 14 observations each. As a result, 52·14 = 728 measurement points.

^{2}. To confirm that they were healthy volunteers, a comprehensive medical history was collected, and a physical examination and laboratory tests were performed 21 days before enrollment in the study. After an overnight fast, the medication was taken orally with roughly 240 mL of water. Subjects fasted for 4 h, 8 h, 12 h, 24 h, 28 h, 32 h, 36 h, and 48 h following administration before eating regular meals. All the participants signed a written consent form, and the study was performed according to the ethical rules of the Helsinki Declaration.

^{TM}, MonolixSuite

^{TM}2021R2, and Simulation Plus) were used to calculate the PK parameters of donepezil. These parameters were AUC, Cmax, Tmax, lambda, and the area under the C-t curve extrapolated from time zero to infinity (AUCinf). The linear trapezoidal rule was used to determine AUC and AUCinf. The Cmax/AUC and Cmax/Tmax ratios were calculated for each subject based on these estimates. The term “lambda” refers to the apparent terminal elimination rate constant, which is found by applying a least squares regression analysis to the terminal log-linear phase of the C-t curve, in line with the regulatory guidelines [1,2].

#### 2.3. Principal Component Analysis

#### 2.4. Correlation Analysis

#### 2.5. Random Forest

#### 2.6. Non-Linear Mixed Effect Modeling

^{TM}2021R2 (Simulation Plus) was used for the population pharmacokinetic analysis.

#### 2.7. Simulated Bioequivalence Datasets

## 3. Results

#### 3.1. Relationships among the PK Variables

#### 3.1.1. PCA and Correlation Analysis

#### 3.1.2. Random Forest Analysis

#### 3.2. Relationships among the PK Variables, under Different Absorption Kinetic Conditions

#### 3.2.1. Development of a Population Pharmacokinetic Model for Donepezil

^{−1}, the mean apparent clearance was 14,466.77 mL/h, and the mean apparent intercompartmental clearance was 80,120.11 mL/h. Furthermore, the mean apparent volume of distribution of the central compartment was 44,386.59 mL, while it was 905,886.42 mL for the peripheral compartment. In addition, the between-subject variability estimates were reasonable for all parameters, and the highest estimate of the percent relative standard error did not exceed the value of 22.5%.

#### 3.2.2. Simulate Different Absorption Kinetics

#### 3.2.3. PCA Applied to the Simulated Bioequivalence Datasets

#### 3.2.4. Random Forest Applied to the Simulated Bioequivalence Datasets

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Quartiles of the distribution of the Cmax, Cmax/AUC, and Cmax/Tmax estimates from the original bioequivalence dataset of donepezil.

Quartile | Cmax | Cmax/AUC | Cmax/Tmax |
---|---|---|---|

Q1 | 13.08 | 0.02 | 3.85 |

Q2 | 17.20 | 0.03 | 5.30 |

Q3 | 20.18 | 0.03 | 8.43 |

**Table A2.**Quartiles of the distribution of the Cmax, Cmax/AUC, and Cmax/Tmax estimates from the three simulated datasets. Simulations were performed assuming three different (mean) absorption rate constant (Ka) values: (a) half of the true Ka estimated from the population modeling, (b) equal to the true Ka, (c) twice the value of the actual Ka.

Absorption Rate Constant: 0.5× | |||
---|---|---|---|

Quartile | Cmax | Cmax/AUC | Cmax/Tmax |

Q1 | 10.95 | 0.02 | 0.94 |

Q2 | 12.31 | 0.02 | 1.70 |

Q3 | 13.71 | 0.02 | 3.36 |

Absorption Rate Constant: 1× | |||

Quartile | Cmax | Cmax/AUC | Cmax/Tmax |

Q1 | 14.96 | 0.02 | 3.14 |

Q2 | 19.08 | 0.03 | 5.97 |

Q3 | 22.56 | 0.04 | 10.35 |

Absorption Rate Constant: 2× | |||

Quartile | Cmax | Cmax/AUC | Cmax/Tmax |

Q1 | 22.52 | 0.03 | 7.20 |

Q2 | 26.54 | 0.04 | 11.37 |

Q3 | 31.22 | 0.05 | 19.64 |

**Figure A2.**Variable importance scores for the feature parameters (AUC, AUCinf, lambda, Tmax) of the pharmacokinetic parameters in the case of the actual bioequivalence study. Three random forest models were developed each one referring to Cmax (

**a**), Cmax/AUC (

**b**), and Cmax/Tmax (

**c**).

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**Figure 1.**Strategy of the analysis. The basic purposes, actions, and software utilized for each phase of the analysis. Actual (steps: A1 and A2) and simulated (steps: B1–B4) bioequivalence data are used.

**Figure 2.**Principal component analysis of the pharmacokinetic parameters from the actual bioequivalence study. Left panel: Biplot of the two principal components showing the individual scores and the loadings (blue lines) of the pharmacokinetic parameters. Right panel: Loading values for the two initial principal components.

**Figure 3.**Variable importance scores for the feature parameters (AUC, lambda, Tmax) of the actual bioequivalence study. Three random forest models were developed each one referring to Cmax (

**a**), Cmax/AUC (

**b**), and Cmax/Tmax (

**c**).

**Figure 4.**Goodness of fit plots for the finally selected population pharmacokinetic model for donepezil. (

**a**) Visual predictive check plot where the blue lines show the 10th, 50th, and 90th percentiles of the empirical data, while the shaded areas represent the anticipated 90% confidence intervals. A number of 1000 Monte Carlo simulations were utilized. (

**b**) Individual observed vs. predicted individual concentrations of donepezil. Closed circles refer to (predicted, observed) pairs, solid lines denote the ideal condition of unity (i.e., y = x), and dotted lines represent the 90% prediction interval.

**Figure 5.**Simulated concentration vs. time profiles of donepezil using three different absorption rate constant average (Ka) values: (

**a**) half of the true Ka estimated from the population modeling, (

**b**) equal to the true Ka, (

**c**) twice the value of the actual Ka. In each group, 26 subjects were generated in a 2 × 2 crossover bioequivalence study, while the sampling scheme was assumed to take place every 15 min. With the exception of Ka, all other pharmacokinetic parameters were those derived from the population modeling.

**Figure 6.**Principal component analysis of the three simulated bioequivalence studies. Depending on the assumed absorption rate constant (Ka), PCA was applied separately the following three cases: (

**a**) Setting Ka equal to the half of the true Ka (i.e., 0.5x-), (

**b**) equal to the true Ka (i.e., 1x-), (

**c**) twice the value of the actual Ka (i.e., 2x-). The left panels refer to the biplot of the two principal components (PC), where the individual scores and the loadings (blue lines) of the pharmacokinetic parameters are shown. The right panels denote the “loadings” for the two first PCs.

**Figure 7.**Variable importance scores for the feature parameters for the simulated bioequivalence studies. The response variable was either Cmax (

**a**), Cmax/AUC (

**b**), or Cmax/Tmax (

**c**). Three levels of the simulated absorption rate constant were assumed: 0.5x, 1x, and 2x. Thus, in total, nine Random Forest models were developed.

**Table 1.**Pearson correlation coefficients for the bivariate relationships between the pharmacokinetic parameter estimates of the actual donepezil bioequivalence study.

Correlation Coefficient | |||||||
---|---|---|---|---|---|---|---|

AUCinf | AUC | Cmax | Tmax | Cmax/AUC | Cmax/Tmax | Lambda | |

AUCinf | 1.000 | 0.986 | 0.588 | 0.106 | −0.432 | 0.209 | −0.478 |

AUC | 0.986 | 1.000 | 0.632 | 0.114 | −0.425 | 0.236 | −0.375 |

Cmax | 0.588 | 0.632 | 1.000 | −0.142 | 0.38 | 0.617 | −0.213 |

Tmax | 0.106 | 0.114 | −0.142 | 1.000 | −0.341 | −0.719 | −0.137 |

Cmax/AUC | −0.432 | −0.425 | 0.38 | −0.341 | 1.000 | 0.418 | 0.201 |

Cmax/Tmax | 0.209 | 0.236 | 0.617 | −0.719 | 0.418 | 1.000 | −0.011 |

Lambda | −0.478 | −0.375 | −0.213 | −0.137 | 0.201 | −0.011 | 1.000 |

Parameters (Units) | Estimate | Standard Error | Relative Standard Error (%) |
---|---|---|---|

Fixed Effects | |||

Ka (h^{−1}) | 0.18 | 0.01 | 6.2 |

Cl/F (mL/h) | 14,466.77 | 785.84 | 5.43 |

V1/F (mL) | 44,386.59 | 9454.34 | 21.3 |

Q/F (mL/h) | 80,120.11 | 5805.65 | 7.25 |

V2/F (mL) | 905,886.42 | 59,624.17 | 6.58 |

Random Effects | |||

omega_Tlag | 0.01 | 0 | 22.5 |

omega_ka | 0.07 | 0.01 | 18.2 |

omega_Cl | 0.27 | 0.04 | 14.6 |

omega_V1 | 1.16 | 0.18 | 15.5 |

omega_Q | 0.25 | 0.05 | 20.1 |

omega_V2 | 0.31 | 0.05 | 16.4 |

Error Model Parameters | |||

a | 0.17 | 0.02 | 12.5 |

b | 0.22 | 0.01 | 3.77 |

**Table 3.**An overview of the analyses took place in this study (see Figure 1) and the main findings.

Purpose | Action | Main Findings |
---|---|---|

A. Identify Relationships Among the PK Variables | ||

A1. Identify relationships among the PK variables | NCA to the dataset of Donepezil BE study | - Estimate the PK parameters (Cmax, AUC, AUCinf, Tmax, lambda, Cmax/AUC, Cmax/Tmax) from the C-t data |

PCA to the calculated PK parameters | - AUC and AUCinf show almost identical behavior - Cmax is strongly related to AUC (and AUCinf) - Cmax/AUC and Cmax/Tmax are not much related to AUC (or AUCinf) - Cmax/AUC and Cmax/Tmax have an opposite behavior compared to Tmax | |

Correlation analysis of the PK parameters | - Bivariate correlations verify the abovementioned findings | |

A2. Contribution of PK variables to Cmax, Cmax/AUC, Cmax/Tmax | Random forest to Cmax, Cmax/AUC, Cmax/Tmax | The contribution of Tmax properties into the parameters is: - the least for Cmax - slightly higher for Cmax/AUC - the most for Cmax/Tmax - It appears that Cmax/Tmax reflects better the kinetic properties of absorption rate |

B. Identify relationships among the PK variables, under different absorption kinetic conditions | ||

B1. Develop a population pharmacokinetic model for donepezil | Apply non-linear mixed effect modeling to the actual C-t data of donepezil (from the BE study) | A robust population pharmacokinetic model is developed for donepezil |

B2. Simulate different absorption kinetics | Simulate three 2 × 2 BE datasets, by setting the absorption rate constant at 0.5x-, 1x-, 2x- the observed value | - Three different BE datasets, with N = 26 for each study, were simulated - All pharmacokinetic properties of donepezil were kept unaltered, except from the absorption rate constant - The kinetics of slower (0.5x-), the same (1x-), and faster (2x-) absorption were simulated - The impact of absorption kinetics, on the choice of the most suitable PK metric, was assessed |

B3. Identify relationships among the PK variables | NCA to each simulated dataset | - Estimate the PK parameters (Cmax, AUC, AUCinf, Tmax, lambda, Cmax/AUC, Cmax/Tmax) for each simulated dataset |

PCA to each of the three simulated datasets | - The findings identified in A1 are verified - As absorption kinetics becomes faster, the relationship between AUC (or AUCinf) and the three other metrics (Cmax, Cmax/AUC, Cmax/Tmax) becomes less strong | |

B4. Contribution of PK variables to Cmax, Cmax/AUC, Cmax/Tmax | Random forest to Cmax, Cmax/AUC, Cmax/Tmax for each of the 3 simulated datasets (0.5x-, 1x-, 2x-) | - Verified the findings identified in A2 - As absorption kinetics becomes faster, the contribution of Tmax into Cmax or Cmax/AUC becomes less - As absorption kinetics becomes faster, the contribution of Tmax into Cmax/Tmax remains the predominant characteristic - Cmax/Tmax reflects better the kinetic properties of absorption rate, regardless the kinetic properties of absorption |

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## Share and Cite

**MDPI and ACS Style**

Karalis, V.D.
Machine Learning in Bioequivalence: Towards Identifying an Appropriate Measure of Absorption Rate. *Appl. Sci.* **2023**, *13*, 418.
https://doi.org/10.3390/app13010418

**AMA Style**

Karalis VD.
Machine Learning in Bioequivalence: Towards Identifying an Appropriate Measure of Absorption Rate. *Applied Sciences*. 2023; 13(1):418.
https://doi.org/10.3390/app13010418

**Chicago/Turabian Style**

Karalis, Vangelis D.
2023. "Machine Learning in Bioequivalence: Towards Identifying an Appropriate Measure of Absorption Rate" *Applied Sciences* 13, no. 1: 418.
https://doi.org/10.3390/app13010418