Modeling Glycogen Depletion and Glucose Regulation

Liver glycogen stores may deplete during prolonged PA and GP cannot be maintained by gluconeogenesis alone, causing an accelerated drop in glucose levels [7 (link), 38 (link)]. We follow Roy et al. [17 (link)] and assume that glycogen stores deplete in proportion to exercise intensity and duration. The time t_depl [min] to depletion determined from the integrated AC count and PA duration is then given by:
$$\begin{array}{c}\hfill t_{depl}\left(t\right)=-a_{depl}·\frac{P{A}_{int}\left(t\right)}{t_{PA}\left(t\right)}+b_{depl}.\end{array}$$
After depletion sets in, we allow a drop in GP rate, r_depl [1/min], defined by
$$\begin{array}{c}\hfill \begin{array}{ccc}\frac{d{r}_{depl}\left(t\right)}{dt}\hfill & =& q_6·[f(t_{PA}\left(t\right);t_{depl},n_1)·r_m\left(t\right)-r_{depl}\left(t\right)]\hfill \\ r_m\left(t\right)\hfill & =& \beta ·(\frac{q_3}{q_4}·Y\left(t\right)+r_{GPb}),\hfill \end{array}\end{array}$$
where the transfer function f(t_PA; t_depl, n₁) indicates whether exercise time exceeds t_depl and q₆ is a rate parameter. The maximum decrease r_m [1/min] in GP is the sum of the basal resting GP rate, r_GPb, and the PA-driven GP rate at steady state, q₃/q₄ ⋅ Y(t), scaled by the proportion of net hepatic glucose production attributed to glycogenolysis, β.