**4. Discussion**

As shown in the previous section, we tested through simulations various hypotheses that may explain the cadmium influence on the glucose regulation mechanism by insulin. To this aim, we started from a model reproducing correctly the control experiment with pups from mothers that were not intentionally exposed to cadmium. Then, we tested each hypothesis by varying a single parameter until it best fitted the dataset associated with the studied group of cadmium exposed pups. For this reason, we can only provide insight on the effect of one mechanism at a time.

An apparent weakness of our approach might be the risk of overfitting a single curve with at least 10 parameters for four variables. However, once the initial conditions have been set, the procedure aims at testing a series of hypotheses by varying a single parameter at a time to optimize fitting of a complex curve. This way a clear trend may appear, as for the probed data at weaning (PND21), or not. Thus, it is possible to safely avoid over-interpreting the simulations. The main practical interest of the modeling effort is to readily help sorting out the most relevant effects of a perturbation of glucose homeostasis.

For the experiments at PND21, we observed that, for both groups Cd1 and Cd2, it is the hypothesis of a decreased response to glucose during the slow phase of production that best represents the changes between the control group and the more exposed groups. On the experimental datasets, this is noticeable by an increased plasma glucose concentration over the [30, 90] min interval, which previously led to increased AUC [18]. The fitted simulations under Hypothesis 3.1 show a good fitting especially on this interval.

However, a few noticeable points of the experimental datasets are not correctly represented by the simulation with Hypothesis 3.1. In particular for group Cd2, the points at 10 and 120 min are not correctly fitted by this single assumption on *γ*. The experimental point at 10 min, being higher than the simulation, suggests an additional effect in the first phase of insulin secretion. This is represented in our model by the parameter *p*4. However, this parameter should be affected together with *γ*, as Table 3 shows that Hypothesis 3.2 (*p*4 varies alone) does not correctly fit the data. Insights on the differences for the point at 120 min are more difficult to explain as they can be due to multiple reasons such as: a too coarse approximation of the glucose rate of appearance at 120 min, an effect on a mechanism not represented in the model such as gluconeogenesis, the role of other hormones than insulin, or a competition between multiple opposing effects that are not represented by our single hypothesis. In any case, the present results can be compared with the observations made in [18]. Few biochemical parameters were found to vary and consequently it was not possible to provide a robust explanation to the changes of the OGTT results and associated AUC. Here, simulations of these curves point to the decreased sensitivity of *β*-cells to glucose as the underlying factor. This decreased sensitivity is proportional to the dams' cadmium burden (Table 4), and it influences lipid metabolism [18]. The result applies to pups at weaning, i.e., at an age when the endocrine pancreas has ye<sup>t</sup> to fully mature.

The experimental results at PND26 and PND60 represent the evolution of the three groups without additional exposure to cadmium through feeding beyond weaning: they show the lasting effect on the metabolism of a previous cadmium exposure, even though the AUC derived from OGTT were no longer significantly different between animal groups [18].

At PND26, we can already observe a slight behavioral difference with the previous datasets at PND21: whereas at PND21 the glucose concentration of Cd2 is always significantly above Cd1 and control ([18], Figure 5), at PND26, all groups share similar results. Looking to the mean values, one observes that the dataset of group Cd2 is closer to control than Cd1, which is confirmed in the simulation by the lower fitting error for Cd2 compared to Cd1 in absence of hypothesis (see Table 8). Even with this bias in favor of mean points, it is hard to reach any strong conclusion. The experimental results of group Cd1 are best fitted under the hypotheses of a reduced sensitivity to insulin of the glucose withdrawing tissues, or a faster degradation rate of insulin (Hypotheses 1.1–2). The experimental results of group Cd2 are so close to the control experimental results that no hypothesis seems much better than another.

Finally, at PND60, the differences among the three groups are not statistically significant ([18], Figure 9). This is witnessed by the small error associated with the entry without hypothesis in Table 12. When looking to the mean values, it is surprising to note that they are lower for group Cd2, and larger than control for Cd1. The best fit for group Cd1 are Hypotheses 3.1 and 3.3 that are modeling a negative effect on the slow phase of insulin production. The best fit for group Cd2 is the Hypothesis 3.2 modeling a positive effect on the first phase of insulin production.
