Feedback modeling of non-esterified fatty acids in obese Zucker rats after nicotinic acid infusions

Abstract

This study investigates the impact of disease on nicotinic acid (NiAc)-induced changes in plasma concentrations of non-esterified fatty acids (NEFA). NiAc was given by constant intravenous infusion to normal Sprague–Dawley and obese Zucker rats, and arterial blood samples were taken for analysis of NiAc, NEFA, insulin and glucose plasma concentrations. The intravenous route was intentionally selected to avoid confounding processes, such as absorption, following extravascular administration. Data were analyzed using nonlinear mixed effects modeling (NONMEM, version VI). The disposition of NiAc in the normal rats was described by a two-compartment model with endogenous synthesis of NiAc and two parallel capacity-limited elimination processes. In the obese rats disposition was described by a one-compartment model with endogenous synthesis of NiAc and one capacity-limited elimination process. The plasma concentration of NiAc drove NEFA (R) turnover via an inhibitory drug-mechanism function acting on the formation of NEFA. NEFA turnover was described by a feedback model with a moderator distributed over a series of transit compartments, where the first compartment (M ₁ ) inhibited the formation of R and the last compartment (M _N ) stimulated the loss of R. All processes regulating plasma NEFA concentrations were assumed to be captured by the moderator function. Differences in the pharmacodynamic response of the two strains included, in the obese animals, an increased NEFA baseline, diminished rebound and post-rebound oscillation, and a more pronounced slowly developing tolerance during the period of constant drug exposure. The feedback model captured the NiAc-induced changes in NEFA response in both the normal and obese rats. Differences in the parameter estimates between the obese and normal rats included, in the former group, increases in R ₀ , k _in and p by 44, 41 and 78 %, respectively, and decreases in k _out and γ by 64 and 84 %, respectively. The estimates of k _tol and IC ₅₀ were similar in both groups. The NiAc–NEFA concentration–response relationship at equilibrium was substantially different in the two groups, being shifted upwards and to the right, and being shallower in the obese rats. The extent of such shifts is important, as they demonstrate the impact of disease at equilibrium and, if ignored, will lead to erroneous dose predictions and, in consequence, poorly designed studies. The proposed models are primarily aimed at screening and selecting candidates with the highest potential of becoming a viable drug in man.

Appendix

Steady state response

The steady state values for the response R and the moderators M _i (i = 1,…, N) of the feedback model, formulated in Eq. (3) and (4), are given by:

$$ R = M_{1} = \cdots = M_{N} = R_{ss} $$

(10)

where R _ss is the unique solution of the equation:

$$ \frac{{{\text{d}}R}}{{{\text{d}}t}} = \frac{{k_{in} }}{{R_{ss}^{p} }} \cdot I(C_{p} ) + k_{cap} - k_{out} \cdot R_{ss}^{2} = 0 $$

(11)

in which

$$ k_{in} = (k_{out} \cdot R_{0}^{2} - k_{cap} ) \cdot R_{0}^{p} $$

(12)

and R ₀ is the baseline response. There is no explicit solution to Eq. (11).

For the obese animals, k _cap could not be estimated and was therefore fixed to zero throughout the analysis. Equation (11) can then be simplified and solved for R _SS :

$$ R_{SS,obese} = \left( {\frac{{k_{in} }}{{k_{out} }} \cdot I(C_{p} )} \right)^{1/(2 + p)} $$

(13)

Response at baseline

At baseline, I(C _p ) = 1, and Eq. (11) can be expressed as:

$$ \frac{{{\text{d}}R}}{{{\text{d}}t}} = k_{in} \cdot \frac{1}{{R_{0}^{p} }} + k_{cap} - k_{out} \cdot R_{0}^{2} = 0 $$

(14)

For the obese animals where k _cap  = 0, Eq. (14) can then be simplified and solved for R ₀ :

$$ R_{0} = \left( {\frac{{k_{in} }}{{k_{out} }}} \right)^{1/(2 + p)} $$

(15)

For the normal rats k _cap is assumed to be small compared to the other terms in Eq. (14), and the baseline can therefore be approximated according to Eq. (15).

Steady state response at IC ₅₀

Assuming that I _max  = 1, the inhibitory drug mechanism function at IC ₅₀ is equal to 0.5. For the obese rats, Eq. (13) can be expressed as:

$$ R_{SS,obese} = \left( {\frac{{k_{in} }}{{k_{out} }} \cdot 0.5} \right)^{1/(2 + p)} $$

(16)

If k _cap for the normal animals is assumed to be smaller than the other terms in Eq. (11), the steady state response at IC ₅₀ for the normal rats can be approximated by Eq. (16), although this will slightly underestimate the value of R _SS .

Steady state response at high NiAc concentrations

As I _max  = 1, it follows from the definition of I(C _p ) (Eq. 2) that $I(C_{p} ) \to 0$ as $C_{p} \to \infty$. This implies that:

$$ \mathop {\lim }\limits_{{C_{p} \to \infty }} R_{SS,normal} (C_{p} ) = \left( {\frac{{k_{cap} }}{{k_{out} }}} \right)^{1/2} $$

(17)

For the obese animals where k _cap  = 0 it follows that:

$$ \mathop {\lim }\limits_{{C_{p} \to \infty }} R_{SS,obese} (C_{p} ) = 0 $$

(18)

For the normal animals, the sigmoidicity factor γ is equal to 2.2. At NiAc concentrations higher than 0.5 μmol L⁻¹, $IC_{50}^{\gamma } \ll C_{p}^{\gamma }$ and I(C _p ) ≈ 0. Equation (17) is therefore valid at NiAc concentrations of 0.5 μmol L⁻¹ and higher.

For the obese animals, the sigmoidicity factor γ is equal to 0.35 and $IC_{50}^{\gamma }$ will then extensively affect I(C _p ) at NiAc concentrations far above 1,000 μmol L⁻¹. The steady state response $R_{SS,obese}$ will therefore not approach zero within a therapeutic concentration interval. The steady state response $R_{SS,obese}$ at high NiAc concentrations is still described by Eq. (13).