*3.2. Parameter Analysis*

The parameters were searched in the intervals proposed in [21], bloated by one order of magnitude, when possible. The initial value of the concentration of readily releasable insulin was fitted for the control group at PND21 and kept the same for the other experiments. The initial concentration of fasting glucose and insulin in plasma were taken from [18]. We chose to take X(0) = 0 as initial condition of the withdrawal rate of glucose: this implies a lack of regulation effect at t = 0. To relax the parameter search, when fitting the parameters to the datasets corresponding to groups Cd1 and Cd2, we only considered a few hypotheses on the evolution of the parameters, starting from the ones fitting the control group, by only altering one parameter at a time.

We considered the following hypotheses for the possible effects of cadmium of plasma glucose regulation:

**Hypothesis 1.** *Modification of the sensitivity of insulin sensitive tissues.*


**Hypothesis 2.** *n varies: this models an effect on insulin degradation.*

**Hypothesis 3.** *Modification of the insulin release rate.*


The hypotheses on the effect of cadmium on the regulation of circulating glucose were tested in the following manner. We first looked for a starting parameter set **k***ctrl* minimizing Equation (5) for the

experimental results of the control group. Then, for experimental results of group Cd1 (respectively Cd2), we first took a hypothesis *Hi* and we refitted a parameter set **<sup>k</sup>***Hi* by varying only one parameter at a time as defined above; the values of the other parameters were taken equal to the ones in the **k***ctrl* parameter set.

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

For 21-day-old pups at weaning, the goodness of fit for each hypothesis and associated to group Cd1 dataset are shown in Table 3. Considering the initial conditions from Table 4, the associated best parameter sets are given in Table 5. The simulation corresponding to the best parameter fits are shown in Figure 4. It should be reminded here that the experimental points reported in Figure 4 showed significant differences between the animal groups for the areas under the arbitrary drawn curves (AUC) in previously published data [17,18]. Here, the goodness of fit associated to the control group dataset is 0.00311 (Table 6). The best fits for group Cd1 are obtained considering Hypothesis 3.1, i.e., decreased response of *β*-cells to glucose in the slow phase of insulin production. For the experiment associated to group Cd2 at PND21, the best fit is also obtained for Hypothesis 3.1 and yields a goodness of 0.00433: it shows a continuous decrease of *γ* as a function of increased cadmium burden of the dams (Table 5).

**Figure 4.** Simulations of the OGTT at PND21 for the control group, Cd1 and Cd2 groups.


**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.

**Table 4.** Initial condition determined for the control group at PND21. These initial conditions are conserved for groups Cd1 and Cd2. Note that1U= 0.0347 mg of insulin.


**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).


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


#### 3.2.2. Results at PND26

Concerning the experiments at PND26, after weaning at PND21 the pups shifted from a milk-based, i.e., lipid-dominated, diet to conventional rodent chow which is rich in carbohydrates. This change of diet induces important changes in the regulation mechanism: this translates into considerable changes of the parameter set fitting the control group dataset. In addition, the three groups of young animals were no longer differently exposed to cadmium after weaning, and the AUC as previously reported [18] did not show any statistically significant difference.

The initial conditions at PND26 are given in Table 7. The goodness of fit for each hypothesis on groups Cd1 and Cd2 are given in Table 8. The associated best parameters sets are given in Table 9. The simulation associated to the best fits are shown in Figure 5. The goodness of fit associated to the control group dataset is 0.00315 (see Table 6). We note that, unlike at PND21, there is no clear hypothesis that appears more likely than the others. Indeed, for group Cd1, although Hypothesis 2 yielded the best results, there is only a small difference with Hypotheses 1.1–1.3: thus, the present data cannot discriminate between changes of the insulin sensitivity of muscle and adipose tissue or of the insulin degradation rate.

**Figure 5.** Simulations of the OGTT at PND26 for the control group, Cd1 and Cd2 groups.

**Table 7.** Initial condition determined for the control group and groups Cd1 and Cd2 at PND26.


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


For group Cd2, we first observe that the higher variance on the experimental results leads to a lower error associated to the control parameter set (denoted no Hyp. in Table 8). In addition, even though Hypothesis 3.2 (reduction in the RRP release rate) yields the best goodness of fit, there is no clear distinction with the other hypotheses because of the increased variance. It follows that the previous interpretation given for group Cd1, namely that of a cadmium effect on the sensitivity of tissues to insulin or the hormone turnover, may still be valid.


**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).

#### 3.2.3. Results at PND60

Finally, considering the initial conditions from Table 10, the parameter set fitted to the datasets at PND60 are given in Table 11. The goodness of fit of the control group dataset is 0.00251 (see Table 6). It can be observed in Figure 6 that group Cd1 from mothers with medium cadmium burden and group Cd2 from the most intoxicated mothers show opposite behaviors with respect to the control curve. Whereas group Cd1 behaved in a similar fashion to the previous results at PND26 with a higher glucose peak than the control glucose response, the results of group Cd2 are characterized by a lower glucose peak.

In Table 12, we notice for group Cd1 that except Hypotheses 3.1 and 3.3, none of the other hypotheses have improved significantly the goodness of fit. However, Hypotheses 3.1 and 3.3 are not among the ones providing an important improvement of the fit for the results associated to group Cd1 at PND26. This raises the possibility of a delayed effect of cadmium on the function of *β*–cells after exposure during the perinatal period, in agreemen<sup>t</sup> with the variations of the C-peptide observed with the same animals [18].

For the results associated with group Cd2, the first three data points are below their control equivalent (see Figure 6). That means that, at PND60, group Cd2 (the one born of mothers with the highest cadmium burden) has a faster removal of glucose from the circulation in the first 40 min. In the simulation, it corresponds to an increase of the parameter *p*4 (Hypothesis 3.2) which best fits this evolution. All the other hypotheses demonstrate only marginal, or no, improvements.

**Table 10.** Initial condition determined for the control group and groups Cd1 and Cd2 at PND60.



**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).

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

**Figure 6.** Simulations of the OGTT at PND60 for the control group and groups Cd1 and Cd2.
