# Taking Kinetic Evaluations of Degradation Data to the Next Level with Nonlinear Mixed-Effects Models

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## Abstract

**:**

## 1. Introduction

## 2. Theory and Methods

#### 2.1. Models

#### 2.1.1. Degradation Models

#### 2.1.2. Residual Error Models

#### 2.1.3. Parameter Distribution Models

#### 2.2. Methods

#### 2.2.1. Degradation Model Solutions

#### 2.2.2. Separate Evaluations by Nonlinear Regression

#### 2.2.3. Simultaneous Evaluations Using Nonlinear Mixed-Effects Models

#### 2.2.4. Model Selection

#### 2.3. Datasets

#### 2.3.1. Synthetic Datasets

#### 2.3.2. Experimental Datasets

#### 2.4. Evaluation Software and Variants

#### 2.4.1. Synthetic Datasets

#### 2.4.2. Experimental Datasets

## 3. Results

#### 3.1. Synthetic Data

#### 3.1.1. Synthetic Data for Simple Exponential Decline

#### 3.1.2. Synthetic Data for Biphasic Decline with One Transformation Product

#### 3.2. Experimental Data

#### 3.2.1. Experimental Data for 2,4-D and Two Transformation Products

_{2}, non-extractable residues or other, unknown transformation products). The acronym used here for this simplified degradation model is SFO-SFO(ns)-SFO, where the ‘ns’ in parentheses stands for ‘no sink’. The degradation rate constants obtained in the separate evaluations were normalised to reference conditions and their mean values were calculated as an estimate for the population of agricultural soils in the EU.

#### 3.2.2. Experimental Data for DMTA and Three Transformation Products

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Separate fits and mean population curve for one synthetic dataset with half-lives around 15 days assuming two-component error.

**Figure 2.**Boxplots of the 100 population half-lives derived from the synthetic grouped datasets with half-lives around 15 days.

**Figure 3.**Separate fits with mean population curve and simultaneous fit for one synthetic dataset with half-lives around 800 days assuming constant variance.

**Figure 4.**Boxplots of the 32 population half-lives derived from the grouped datasets with half-lives around 800 days. The largest half-lives obtained in the separate fits exceed the range of the y axis (compare Table 4).

**Figure 6.**Boxplots of input sample means and four different estimations of two selected parameters for the synthetic DFOP-SFO data.

**Figure 7.**SFO-SFO(ns)-SFO model with two-component error fitted with saemix to the normalised data for 2,4-D with an additional degradation curve derived from separate evaluations for comparison.

**Figure 8.**Separate fits of the DFOP-SFO3+ model to the normalised DMTA data assuming two-component error, with an average curve only based on rate constants that can be distinguished from zero.

**Figure 9.**DFOP-SFO3+ model with two-component error fitted with saemix to the DMTA data, with overlayed degradation curve from Figure 8.

Parameter | Unit | Mode of Parameter Distribution | Standard Deviation | |||
---|---|---|---|---|---|---|

Variant 1 | Variant 2 | Variant 3 | Variant 4 | Variants 1–4 | ||

${p}_{0}$ | [%] ${}^{a}$ | 100 | 100 | 100 | 100 | 2 |

Half-life | [days] | 15 days | 120 days | 500 days | 800 days | |

$\lambda $ | [days${}^{-1}$] | 4.62 × 10^{−2} | 5.78 × 10^{−3} | 1.39 × 10^{−3} | 8.66 × 10^{−4} | |

$log(\lambda $ days) | [-] | −3.07 | −5.15 | −6.58 | −7.05 | 0.5 |

Parameter | Variable | Unit | Mode | Standard Deviation |
---|---|---|---|---|

${p}_{0}$ | parent_0 | [%] ${}^{a}$ | 100 | 2 |

${\lambda}_{1}$ | k1 | [days${}^{-1}$] | 0.05 | |

$log({\lambda}_{1}$ days) | [-] | −3 | 0.5 | |

${\lambda}_{2}$ | k2 | [days${}^{-1}$] | 0.01 | |

$log({\lambda}_{2}$ days) | [-] | −4.61 | 0.5 | |

${\gamma}_{1}$ | g | [-] | 0.5 | |

logit ${\gamma}_{1}$ | [-] | 0 | 0.5 | |

${\gamma}_{2}$ | f_parent_to_m1 | [-] | 0.5 | |

logit ${\gamma}_{2}$ | [-] | 0 | 0.5 | |

${\lambda}_{3}$ | k_m1 | [days${}^{-1}$] | 0.002 | |

$log({\lambda}_{3}$ days) | [-] | −6.21 | 0.5 |

**Table 3.**Statistics for half-lives derived from 100 synthetic grouped datasets with an input half-life of 15 days.

Input | Separate Fits | Nlme | Saemix | |||||
---|---|---|---|---|---|---|---|---|

Distribution | Samples | const | tc | const | tc | const | tc | |

Number of results | 500 | 500 | 100 | 100 | 100 | 100 | ||

Fits with lower AIC | 11 | 489 | 0 | 100 | 0 | 100 | ||

Minimum | 8.749 | 9.019 | 9.056 | 8.986 | 9.017 | 9.000 | 9.048 | |

25th percentile | 12.90 | 13.555 | 13.524 | 13.691 | 13.549 | 13.705 | 13.566 | 13.724 |

Median | 15.00 | 15.053 | 15.530 | 15.312 | 15.563 | 15.272 | 15.544 | 15.303 |

75th percentile | 17.44 | 17.358 | 17.326 | 17.382 | 17.336 | 17.416 | 17.346 | 17.438 |

Maximum | 28.600 | 27.775 | 27.964 | 27.863 | 27.796 | 27.860 | 27.801 |

**Table 4.**Statistics for half-lives derived from 100 synthetic grouped datasets, with an input half-life of 800 days.

Input | Separate Fits | Nlme | Saemix | |||||
---|---|---|---|---|---|---|---|---|

Distribution | Samples | const | tc | const | tc | const | tc | |

Number of results | 500 | 500 | 94 | 88 | 100 | 99 | ||

Fits with lower AIC | 499 | 1 | 76 | 11 | 88 | 11 | ||

Minimum | 480.4 | 438.1 | 424.4 | 408.6 | 407.6 | 440.3 | 430.0 | |

25th percentile | 688.0 | 657.3 | 683.5 | 678.9 | 604.2 | 595.7 | 648.8 | 647.0 |

Median | 800.0 | 772.9 | 1029.7 | 852.7 | 690.7 | 701.4 | 761.3 | 756.5 |

75th percentile | 930.2 | 897.8 | 6483.5 | 1381.7 | 841.7 | 865.2 | 1075.9 | 1029.2 |

Maximum | 1401.2 | 856,749.0 | 444,191.3 | 4246.1 | 4260.1 | 57,459.9 | 2829.1 |

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**MDPI and ACS Style**

Ranke, J.; Wöltjen, J.; Schmidt, J.; Comets, E.
Taking Kinetic Evaluations of Degradation Data to the Next Level with Nonlinear Mixed-Effects Models. *Environments* **2021**, *8*, 71.
https://doi.org/10.3390/environments8080071

**AMA Style**

Ranke J, Wöltjen J, Schmidt J, Comets E.
Taking Kinetic Evaluations of Degradation Data to the Next Level with Nonlinear Mixed-Effects Models. *Environments*. 2021; 8(8):71.
https://doi.org/10.3390/environments8080071

**Chicago/Turabian Style**

Ranke, Johannes, Janina Wöltjen, Jana Schmidt, and Emmanuelle Comets.
2021. "Taking Kinetic Evaluations of Degradation Data to the Next Level with Nonlinear Mixed-Effects Models" *Environments* 8, no. 8: 71.
https://doi.org/10.3390/environments8080071