# Theoretical Modeling of Oral Glucose Tolerance Tests Guides the Interpretation of the Impact of Perinatal Cadmium Exposure on the Offspring’s Glucose Homeostasis

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Summary of the Animal Study on Which Modeling Was Applied

#### 2.2. The Minimal Model (MINMOD)

#### 2.3. Glucose Tolerance Test Simulation Procedures

## 3. Results

#### 3.1. OGTT Modeling

#### 3.2. Parameter Analysis

**Hypothesis**

**1.**

- -
**Hypothesis 1.1:**${r}_{Cd}$ varies: this shows the effect of cadmium on the glucose removal by the tissues. If ${r}_{Cd}<1$, then the system has developed insulin resistance.- -
**Hypothesis 1.2:**${p}_{3}$ varies: this represents the effect of insulin on the rate of glucose withdrawal from the circulation.- -
**Hypothesis 1.3:**${p}_{2}$ varies: this affects the decrease rate of $\mathrm{X}\left(t\right)$, which is the glucose withdrawal rate.

**Hypothesis**

**2.**

**Hypothesis**

**3.**

- -
**Hypothesis 3.1:**γ varies: this models an evolution of the insulin release rate, in response to glucose, in the slow phase of insulin production.- -
**Hypothesis 3.2:**${p}_{4}$ varies: this models the stored insulin release rate.- -
**Hypothesis 3.3:**h varies: this affects the glucose response threshold.

#### 3.2.1. Results at PND21 (Weaning)

#### 3.2.2. Results at PND26

#### 3.2.3. Results at PND60

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PND21 | Post-Natal Day 21 |

PND26 | Post-Natal Day 26 |

PND60 | Post-Natal Day 60 |

RRP | Readily Releasable Pool |

OGTT | Oral Glucose Tolerance Test |

AUC | Area Under the Curve |

MINMOD | Minimal Model |

## References

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**Figure 3.**Modeling method to test hypotheses on the possible targets of cadmium. In the diagram, the index i spans hypotheses 1.1–3.3.

PND21 | PND26 | PND60 | |
---|---|---|---|

Control | 48 | 13 | 12 |

Cd1 | 38 | 18 | 17 |

Cd2 | 35 | 10 | 12 |

PND21 | PND26 | PND60 | Unit | |
---|---|---|---|---|

Body mass | 45 | 65 | 205 | g |

Blood volume | $0.03$ | $0.04$ | $0.128$ | dL |

K | 3729 | 4040 | 3982 | mg/(dL$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$min) |

**Table 3.**Goodness of fit from Equation (5) of each hypothesis applied to the Cd1 and Cd2 datasets at PND21. The symbol − denotes a value no better than the control parameter set.

Hypothesis | Cd1 | Cd2 |
---|---|---|

No Hyp. | $0.0188$ | $0.0144$ |

Hyp 1.1 | $0.0144$ | $0.00820$ |

Hyp 1.2 | $0.0144$ | $0.00821$ |

Hyp 1.3 | $0.0140$ | $0.00789$ |

Hyp 2 | $0.0145$ | $0.00852$ |

Hyp 3.1 | $0.00566$ | $0.00433$ |

Hyp 3.2 | − | $0.0137$ |

Hyp 3.3 | $0.0102$ | $0.00740$ |

**Table 4.**Initial condition determined for the control group at PND21. These initial conditions are conserved for groups Cd1 and Cd2. Note that 1 U = 0.0347 mg of insulin.

Variable | Value | Unit |
---|---|---|

$\mathrm{G}\left(0\right)$ | $110.0$ | mg/dL |

$\mathrm{X}\left(0\right)$ | $0.0$ | min${}^{-1}$ |

$\mathrm{I}\left(0\right)$ | $16.0$ | nU/dL |

${\mathrm{I}}_{s}\left(0\right)$ | $5950.0$ | nU/dL |

**Table 5.**Parameters values fitted for the control group as well as groups Cd1 and Cd2 at PND21 (considering Hypothesis 3.1 for both groups).

Parameters | Ctrl | Cd1 | Cd2 | Unit |
---|---|---|---|---|

${p}_{1}$ | $0.01$ | − | − | min${}^{-1}$ |

${\mathrm{G}}_{b}$ | $100.0$ | − | − | mg/dL |

${p}_{2}$ | $0.56$ | − | − | min${}^{-1}$ |

${p}_{3}$ | $0.0155$ | − | − | (dL/nU)min${}^{-2}$ |

${\mathrm{I}}_{b}$ | $10.0$ | − | − | nU/dL |

n | $10.53$ | − | − | min${}^{-1}$ |

$\gamma $ | $0.0310$ | $0.0258$ | $0.0215$ | (nU/dL)min${}^{-2}$ |

h | $85.0$ | − | − | mg/dL |

${p}_{4}$ | $0.033$ | − | − | min${}^{-1}$ |

${r}_{Cd}$ | $1.0$ | − | − | N.U. |

**Table 6.**Goodness of fit from Equation (5) for each dataset of the control group at PND21, -26 and -60.

Control Groups | Goodness of Fit |
---|---|

PND21 | $0.00311$ |

PND26 | $0.00315$ |

PND60 | $0.00251$ |

Variable | Value | Unit |
---|---|---|

$\mathrm{G}\left(0\right)$ | $76.0$ | mg/dL |

$\mathrm{X}\left(0\right)$ | $0.0$ | min${}^{-1}$ |

$\mathrm{I}\left(0\right)$ | $34.0$ | nU/dL |

${\mathrm{I}}_{s}\left(0\right)$ | $5950.0$ | nU/dL |

**Table 8.**Goodness of fit from Equation (5) of each hypothesis applied to the datasets of groups Cd1 and Cd2 at PND26.

Hypothesis | Cd1 | Cd2 |
---|---|---|

No Hyp. | $0.00972$ | $0.00600$ |

Hyp 1.1 | $0.00411$ | $0.00444$ |

Hyp 1.2 | $0.00410$ | $0.00435$ |

Hyp 1.3 | $0.00398$ | $0.00434$ |

Hyp 2 | $0.00395$ | $0.00442$ |

Hyp 3.1 | $0.00762$ | $0.00584$ |

Hyp 3.2 | $0.00638$ | $0.00388$ |

Hyp 3.3 | $0.00819$ | $0.00465$ |

**Table 9.**Parameters values fitted for the control group as well as groups Cd1 and Cd2 at PND26 (considering Hypothesis 3.2 for Cd2 and Hypothesis 2 for Cd1).

Parameters | Ctrl | Cd1 | Cd2 | Unit |
---|---|---|---|---|

${p}_{1}$ | $0.01$ | − | − | min${}^{-1}$ |

${\mathrm{G}}_{b}$ | $100.0$ | − | − | mg/dL |

${p}_{2}$ | $0.61$ | − | − | min${}^{-1}$ |

${p}_{3}$ | $0.0245$ | − | − | (dL/nU)min${}^{-2}$ |

${\mathrm{I}}_{b}$ | $05.0$ | − | − | nU/dL |

n | $09.44$ | $9.93$ | − | min${}^{-1}$ |

$\gamma $ | $0.0110$ | − | − | (nU/dL)min${}^{-2}$ |

h | $79.0$ | − | − | mg/dL |

${p}_{4}$ | $0.0215$ | − | $0.201$ | min${}^{-1}$ |

${r}_{Cd}$ | $1.0$ | − | − | N.U. |

Variable | Value | Unit |
---|---|---|

$\mathrm{G}\left(0\right)$ | $95.0$ | mg/dL |

$\mathrm{X}\left(0\right)$ | $0.0$ | min${}^{-1}$ |

$\mathrm{I}\left(0\right)$ | $34.0$ | nU/dL |

${\mathrm{I}}_{s}\left(0\right)$ | $5950.0$ | nU/dL |

**Table 11.**Parameter values fitted for the control group as well as groups Cd1 and Cd2 at PND60 (considering Hypothesis 3.3 for Cd2 and Hypothesis 3.2 for Cd1).

Parameters | Ctrl | Cd1 | Cd2 | Unit |
---|---|---|---|---|

${p}_{1}$ | $0.01$ | − | − | min${}^{-1}$ |

${\mathrm{G}}_{b}$ | $100.0$ | − | − | mg/dL |

${p}_{2}$ | $0.79$ | − | − | min${}^{-1}$ |

${p}_{3}$ | $0.0335$ | − | − | (dL/nU)min${}^{-2}$ |

${\mathrm{I}}_{b}$ | $06.0$ | − | − | nU/dL |

n | $8.35$ | − | − | min${}^{-1}$ |

$\gamma $ | $0.0078$ | − | − | (nU/dL)min${}^{-2}$ |

h | $65.0$ | $73.0$ | − | mg/dL |

${p}_{4}$ | $0.0170$ | − | $0.0180$ | min${}^{-1}$ |

${r}_{Cd}$ | $1.0$ | − | − | N.U. |

**Table 12.**Goodness of fit from Equation (5) of each hypothesis applied to the datasets of groups Cd1 and Cd2 at PND60.

Hypothesis | Cd1 | Cd2 |
---|---|---|

No Hyp. | $0.00440$ | $0.00403$ |

Hyp 1.1 | $0.00395$ | $0.00304$ |

Hyp 1.2 | $0.00395$ | $0.00304$ |

Hyp 1.3 | $0.00396$ | $0.00312$ |

Hyp 2 | $0.00374$ | $0.00325$ |

Hyp 3.1 | $0.00267$ | − |

Hyp 3.2 | $0.00439$ | $0.00194$ |

Hyp 3.3 | $0.00204$ | $0.00402$ |

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## Share and Cite

**MDPI and ACS Style**

Rocca, A.; Fanchon, E.; Moulis, J.-M.
Theoretical Modeling of Oral Glucose Tolerance Tests Guides the Interpretation of the Impact of Perinatal Cadmium Exposure on the Offspring’s Glucose Homeostasis. *Toxics* **2020**, *8*, 30.
https://doi.org/10.3390/toxics8020030

**AMA Style**

Rocca A, Fanchon E, Moulis J-M.
Theoretical Modeling of Oral Glucose Tolerance Tests Guides the Interpretation of the Impact of Perinatal Cadmium Exposure on the Offspring’s Glucose Homeostasis. *Toxics*. 2020; 8(2):30.
https://doi.org/10.3390/toxics8020030

**Chicago/Turabian Style**

Rocca, Alexandre, Eric Fanchon, and Jean-Marc Moulis.
2020. "Theoretical Modeling of Oral Glucose Tolerance Tests Guides the Interpretation of the Impact of Perinatal Cadmium Exposure on the Offspring’s Glucose Homeostasis" *Toxics* 8, no. 2: 30.
https://doi.org/10.3390/toxics8020030