Modeling energy intake by adding homeostatic feedback and drug intervention

Abstract

Energy intake (EI) is a pivotal biomarker used in quantification approaches to metabolic disease processes such as obesity, diabetes, and growth disorders. Eating behavior is however under both short-term and long-term control. This control system manifests itself as tolerance and rebound phenomena in EI, when challenged by drug treatment or diet restriction. The paper describes a model with the capability to capture physiological counter-regulatory feedback actions triggered by energy imbalances. This feedback is general as it handles tolerance to both increases and decreases in EI, and works in both acute and chronic settings. A drug mechanism function inhibits (or stimulates) EI. The deviation of EI relative to a reference level (set-point) serves as input to a non-linear appetite control signal which in turn impacts EI in parallel to the drug intervention. Three examples demonstrate the potential usefulness of the model in both acute and chronic dosing situations. The model shifts the predicted concentration–response relationship rightwardly at lower concentrations, in contrast to models that do not handle functional adaptation. A fourth example further shows that the model may qualitatively explain differences in rate and extent of adaptation in observed EI and its concomitants in both rodents and humans.

Appendix

Basic model, step 2

Several alternatives to Eq. 1 can be considered. In case a basal level $p_0$ is anticipated one can alternatively use

$$\begin{aligned} EI_{veh^{\prime}} = p_1 \times e^{-p_2 \times t} + p_0. \end{aligned}$$

(13)

Another possibility is to delay the peak EI from $t=0$ to $t=1/p_2$ by the function

$$\begin{aligned} EI_{veh^{\prime\prime}} = p_1 \times t \times e^{-p_2 \times t}, \end{aligned}$$

(14)

or by the Bateman function

$$\begin{aligned} EI_{veh^{\prime\prime\prime}} = p_1 \times \left( e^{-p_2 \times t} - e^{-p_3 \times t} \right) . \end{aligned}$$

(15)

Extended model, step 6

We model the appetite control signal $h(Y)$ as a function of the cumulative energy imbalance function $Y$ by the Gompertz function $h(Y)$ defined as

$$\begin{aligned} h(Y)= h_{\rm min} + \left( h_{\rm max}-h_{\rm min} \right) \times \exp \left( h_{disp} \times \exp (-h_{slope} \times Y) \right) \end{aligned}$$

(16)

where the parameter $h_{disp}$ is a negative number that sets the displacement on the y-axis, and the parameter $h_{slope}$ sets the rate to steady-state, both during treatment and of the overshoot.

When the cumulative energy imbalance function is zero, $Y(t)=0$, there is no modulation of EI, $h(Y)=1$. The constraint $Y=0 \rightarrow h(Y)=1$ gives

$$\begin{aligned} h_{disp} = \log \left( \frac{1-h_{\rm min}}{h_{\rm max}-h_{\rm min}} \right) \end{aligned}$$

(17)

and $h(Y)$ is defined with the three parameters $h_{\rm min},\,h_{\rm max}$, and $h_{slope}$ as

$$\begin{aligned} h(Y)= h_{\rm min} + \left( h_{\rm max}-h_{\rm min} \right) \times \left( \frac{1-h_{\rm min}}{h_{\rm max}-h_{\rm min}} \right) ^{\exp (-h_{slope} \times Y)}. \end{aligned}$$

(18)

Steady-state calculation

For the perfect adaptation case, when $k=0$, the utility during treatment is defined as follows. First, from Eq. 10 we see that the relative change in EI rate is defined by the product

$$\begin{aligned} I(C) \times h(Y) \end{aligned}$$

(19)

that expands to

$$\begin{aligned} \left( 1 - I_{\rm max} \times \frac{C^n}{C^n+IC_{50}^n} \right) \times h_{\rm min} + \left( h_{\rm max}-h_{\rm min} \right) \times \left( \frac{1-h_{\rm min}}{h_{\rm max}-h_{\rm min}} \right) ^{\exp (-h_{slope} \times Y)} \end{aligned}$$

(20)

At steady-state EI during treatment, $Y$, representing the cumulative appetite goes to a large number, and Eq. 20 reduces to

$$\begin{aligned} \left( 1 - I_{\rm max} \times \frac{C^n}{C^n+IC_{50}^n} \right) \times h_{\rm max} \end{aligned}$$

(21)

If this value is equal to (or greater than) one, the drug provocation is matched by the feedback and the system reaches a steady-state where EI is the same as for the reference vehicle. If the value is less than one, the drug provocation is stronger than the maximum feedback, and steady-state EI is less than that of the vehicle. For example, if $I(C)=0.25$ and $h_{\rm max}=3$, then EI is initially reduced to 25 % (as $I(C)=0.25$) and then gradually reaches 75 % (the product of $I(C)$ and $h(Y)$ equals 0.75) at steady-state during treatment.

The utility of a drug treatment, defined as treatment effect area divided by overshoot area, mainly depends on the parameter $k$. A large $k$ value results in high utility. The special case $k=0$ gives perfect adaptation with no utility of the drug treatment (utility = 1). Utility is also influenced by e.g. $h_{slope}$ and treatment length.

At steady-state, Eq. 8 gives

$$\begin{aligned} 0 = EI_{veh} \times \left( 1 - I(C) \times h(Y) \right) - k \times Y. \end{aligned}$$

(22)

Considering average $EI_{veh}$ and steady-state $C$, we can numerically approximate $h(Y)$ versus steady-state concentration from the model.

Parameter sensitivity

Sensitivity of the error function (sum-of-squared residuals) to variations in the regressed parameters of Example 2–4 are reported in Table 6. In particular, the parameter $h_{slope}$ is insensitive using available data.

Table 6 Sensitivity of the error function (sum-of-squared residuals) to variations in the parameters of Example 2–4