**1. Introduction**

The diagnostic and monitoring of diabetes mellitus rely on the experimental assessment of glucose homeostasis. Various tests and indices have been developed over time with the aim of identifying the sources of dysfunction among two main categories, deficient insulin secretion and insulin resistance. The former reflects the failure of *β*-cells in pancreatic islets of Langerhans to respond to increased glucose concentrations, whereas the latter corresponds to the increased inability of insulin-sensitive tissues, such as liver, muscles, and adipose tissue, to internalize glucose upon insulin stimulation together with deficient repression by insulin of hepatic glucose synthesis. Such conditions progressively develop in pre-diabetic states, and they are the hallmark of diabetes. However, discriminating between *β*-cells failure and insulin resistance is a challenge and involves invasive assays [1].

Among the environmental contributors to diabetes, particularly type 2 diabetes, exposure to the widespread metal cadmium has been regularly proposed. Epidemiological data promoted the idea at the beginning of the century [2], and, since then, conflicting results have bolstered debate on the issue [3–10] without reaching any clear consensus. The same statement applies to epidemiological studies considering the influence of the cadmium burden of the mother on the glucose homeostasis of both mother and child [11,12]. Besides, mechanistic investigations on the effects of cadmium on the function of the *β*-cells often focused on large, short-term, exposure (see [13,14] for the latest examples of such approaches) that bear little relevance to environmental conditions. Comparatively, very few studies investigated the relationship between low-level cadmium burden and impaired glucose homeostasis in relatively well defined conditions that can be implemented in the laboratory [15].

Among currently applied assays probing glucose homeostasis, oral glucose tolerance tests (OGTT) gather several advantages such as low-end staff and patient burden, integrated physiological behavior of the main contributors to glucose homeostasis, and clinically valuable information. In clinical practice, many parameters, such as 1-h or 2-h post-load glucose values that are extracted during OGTT, fasting glucose, or more indirect markers such as glycated haemoglobin, are considered in their ability to provide sensitive and cheap ways to diagnose dysglycemia and predict the development of diabetes. However, none gathers a consensus for application to all populations, or in the presence of the many confounding conditions [16]. By contrast, in research settings as with laboratory animals, the high information content of OGTT is more readily accessible since the kinetic data relative to blood glucose increase after the bolus and glucose disposal up to 2–3 h can be obtained.

The OGTT include a wealth of quantitative information that cannot always be extracted by mere examination of the curves presenting the variations of circulating glucose over time after a bolus, or even integration in the form of the area under the curve. We propose here a simple modeling and parameter analysis of such curves after cadmium exposure. The experimental data on which the present work is based were all reported before [17,18] for groups of rat pups exposed to low-level cadmium contamination through their mothers during gestation and lactation. All experimental details are available in these [17,18] and other [19] publications. In our hands [18], one of the frequently emphasized disadvantages of OGTT, namely its variability as compared to intravenous methods, has not been encountered as witnessed by the narrow spreading of measured values observed within experimental groups. This experimental advantage sets a strong basis for detailed analysis, which should allow us to focus on the variations of different parameters of glucose homeostasis obtained for these animals [18].

The purpose of the present study was to first build a simple kinetic model describing the evolution of the glucose concentration in the context of OGTT. Then, numerical simulations were run on previously obtained experimental results [17,18]. The process allowed us to test three groups of hypothesis via the sensitivity of the associated parameters as a function of cadmium exposure at three ages of pups post-weaning. These hypotheses were grouped as: (1) insulin sensitivity of glucose withdrawing tissues such as liver and muscles; (2) insulin turnover; and (3) insulin secretion by *β*-cells.

#### **2. Materials and Methods**

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

The model built in the present study was applied to previously published data on pups born from cadmium-exposed dams [18]. The animal study was approved by the ethics committee (224\_LBFA-U1055, 7 April 2015) affiliated to the animal facility (D3842110001) and agreed by the French Ministry of research (approval number 02397.02, 8 January 2018). A summary of the experimental protocol is shown in Figure 1. Shortly, dams were separated into three different groups and offered ad libitum doses of cadmium (CdCl2) in drinking water adjusted to 0, 50, and 500 <sup>μ</sup>g·(kg body mass)−1·day−<sup>1</sup> above the diet baseline [19]. The groups were reorganized post hoc as 'control', Cd1, and Cd2 according to the increasing Cd concentrations of the dam's kidneys [18]. This way, the OGTT measured for the respective pups are more representative of the cadmium exposure of the progeny via their mothers. The oral glucose tolerance tests (OGTT) measure, after overnight fasting, the evolution of plasma glucose concentration during the 2 h following force-feeding glucose intake at 2 mg per g of body mass. The tests were performed on the pups 21 days after birth, i.e., at weaning at Post-Natal Day 21 (PND21), at PND26, and at PND60. It has to be emphasized that the groups of pups were not exposed to different cadmium concentrations after weaning (>PND21) as they were all put on the same, not-intentionally cadmium-supplemented, diet. The population of the groups Control, Cd1 and Cd2 at the time points PND21, PND26, and PND60 are recalled in Table 1 to appreciate the statistical power of the studied data.


**Figure 1.** Protocol for indirect exposure of rat litters to cadmium through their mothers.

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

As we plan in future work to apply formal or computationally expensive methods on our model, we built the simplest model possible while retaining important and meaningful variables for experimentalists, namely the plasma glucose concentration and the plasma insulin concentration. For this purpose, we used the minimal model (MINMOD) [20,21] as a starting point. The MINMOD model [21] is a small ODE model describing the evolution of glucose concentration after an initial intravenous injection of a glucose bolus.

$$\begin{aligned} \dot{\mathbf{G}} &= -p\_1(\mathbf{G}(t) - \mathbf{G}\_b) - \mathbf{X}(t)\mathbf{G}(t) \\ \dot{\mathbf{X}} &= -p\_2\mathbf{X}(t) + p\_3(\mathbf{I}(t) - \mathbf{I}\_b) \\ \mathbf{I} &= -n\mathbf{I}(t) + \gamma\left(\mathbf{G}(t) - h\right)t \end{aligned} \tag{1}$$

The MINMOD model in Equation (1) has three variables: G is the glucose concentration in circulating blood, X is the rate of glucose withdrawal by muscles and adipocytes due to insulin, and I is the insulin concentration in circulating blood. This dynamic is modulated by seven parameters *p*1, *p*2, *p*3, *n*, *γ*, G*b*, I*b* and *h*. Parameters *p*1 is a control rate on the glucose G(*t*) to maintain the threshold concentration G*b* in absence of insulin regulation and glucose intake. Parameter *p*2 is the decrease rate of the variable glucose absorption rate <sup>X</sup>(*t*). Parameter *p*3 is the increase rate of <sup>X</sup>(*t*), and is associated to the insulin threshold I*b*. The parameters associated to insulin modeling in the MINMOD model are as follows: *n* is the degradation rate of insulin and *γ* is the long-term insulin production rate when glucose is above threshold *h*.

#### *2.3. Glucose Tolerance Test Simulation Procedures*

Numerical simulations were performed in JULIA using the DIFFERENTIALEQUATIONS library [22]. The process of fitting the parameter sets by minimizing Equation (5) for each dataset was performed manually. The code associated to these simulations can be found at https://github.com/roccaa/OGTT\_Simulations.
