A Mathematical Model of the Oral Glucose Tolerance Test Illustrating the Effects of the Incretins

Abstract

Despite important empirical findings, current models of the oral glucose tolerance test (OGTT) do not incorporate the essential contributions of the incretin hormones, glucagon-like peptide-1 and glucose-dependent insulinotropic peptide, to glucose-stimulated insulin secretion. In order to address this deficiency, a model was, therefore, developed in which the incretins, as well as a term reflecting net hepatic glucose balance, were included. Equations modeling the changes in incretins, hepatic glucose balance, insulin and glucose were used to simulate the responses to 50 and 100 g oral glucose loads under normal conditions. The model successfully captures main trends in mean data from the literature using a simple ‘lumped-parameter,’ single-compartment approach in which the majority of the parameters were matched to known clinical data. The accuracy of the model and its applicability to understanding fundamental mechanisms was further assessed using a variety of glycemic and insulinemic challenges beyond those which the model was originally created to encompass, including hyper- and hypoinsulinemia, changes in insulin sensitivity, and the insulin infusion-modified intravenous glucose tolerance test.

Appendices

Appendix A

Determination of k ₁ and k ₂ using the Ratio NIMGU:IMGU

Let p = NIMGU/IMGU, the ratio of non-insulin mediated to insulin-mediated glucose uptake = 2 under basal conditions.4,27 Then, referring to Eq. (6), for basal steady state:

$$ p\; = \;k_1 {G}_B ^{1.3} /k_2 {I}_{B} $$

Setting dG/dt = dI/dt = 0 in Eq. (6), and setting Ra _GutG  = 0, we obtain:

$$ {Hepbal}_{{GB}} /V\; = \;k_1 {G}_{B} ^{1.3} \; + \;k_2 {I}_{B} $$

Then k ₁ and k ₂ can be obtained by solving the latter two equations:

$$ k_1 = \frac{p} {{p + 1}}\frac{{{Hepbal}_{{GB}} }} {{{G}_{B}^{{1}{.3}} V}} $$

$$ k_2 = \frac{1} {{p + 1}}\frac{{{Hepbal}_{{GB}} }} {{{I}_{B} V}} $$

Appendix B

Determination of the Fractional Turnover Rate of Glucose (K) Following an Intravenous Bolus Injection of Radioactive Tracer

With reference to Eq. (6), the steady state rate of disappearance of unlabeled glucose or tracee is equal to k₁ G _B ^1.3  + k ₂ I _B . Assuming steady state conditions for tracee, the rate of disappearance of labeled glucose or tracer (G*) is equal to the rate of disappearance of tracee (G) multiplied by the specific activity of the tracer (G*/G).43 That is,

$$ \frac{{d{G}^{*}(t)}} {{dt}} = - \frac{{{G}^{*}(t)}} {{{G}_{B} }}\left( {k_1 {G}_{B}^{{1}{.3}} + k_2 {I}_{B} } \right) = - \left( {k_1 {G}_{B}^{{\hbox{0}}{\hbox{.3}}} + k_2 {I}_{B} /{G}_{B} } \right){G}^{*}(t)$$

Since $\left( {k_1 {G}_{B}^{{\hbox{0}}{\hbox{.3}}} + k_2 {I}_{B} /{G}_{B} } \right)$ = 0.015 min⁻¹, when evaluated using the values for k ₁, k ₂, G _B and I _B (Tables 1, 2), the expected slope of the straight line obtained by plotting ln G*(t) against t is equal to 0.015 min⁻¹.