Modeling of free fatty acid dynamics: insulin and nicotinic acid resistance under acute and chronic treatments

Animals

Male Sprague Dawley (lean) and Zucker rats (fa/fa, obese) were purchased from (conscious groups) Harlan Laboratories B.V. (The Netherlands) or (anesthetized groups) Charles River Laboratories (USA). Experimental procedures were approved by the local Ethics Committee for Animal Experimentation (Gothenburg region, Sweden). Rats were housed in an Association for Assessment and Accreditation of Laboratory Animal Care accredited facility with environmental control: 20–22\(\,^\circ\)C, relative humidity 40–60\(\%\), and 12 h light-dark cycle. During acclimatization (\(\ge\)5 days), animals were housed in groups of 5 with free access to both water and standard rodent chow (R70, Laktamin AB, Stockholm, Sweden).

Surgical preparations

To prevent potential infections in conjunction with surgery, oral antibiotics were given 1 day before pump/catheter surgery and then once daily for 3 days (sulfamethoxazole and trimethoprim 40 + 8 mg mL\(^{-1}\); Bactrim ^®, 0.2mL /animal, Roche Ltd, Basel, Switzerland). Surgery was performed under isoflurane (Forene^®, Abbott Scandinavia AB, Solna, Sweden) anesthesia, with body temperature maintained at 37 \(^\circ\)C. For NiAc/saline administration, a programmable mini pump (iPrecio^® SMP200 Micro Infusion Pump, Primetech Corporation, Tokyo, Japan) was implanted subcutaneously, via a dorsal skin incision. To allow blood sampling during the terminal experiment (conscious animals only), a polyurethane catheter (Instech Laboratories Inc, Plymouth Meeting, PA USA) was placed in the right jugular vein via an incision in the neck. In order to maintain its patency up to the acute experiment, the jugular catheter was filled with sterile 45.5% (wt/wt) PVP (polyvinylpyrrolidone, K30, MW \(\sim\)40,000 Fluka, Sigma-Aldrich, Sweden) dissolved in a sodium-citrate solution (20.6 mmol), sealed and exteriorized at the nape of the neck. Each animal received a post-operative, subcutaneous analgesic injection (buprenorphine, Temgesic^®, 1.85 \(\upmu\)g kg\(^{-1}\), RB Pharmaceuticals Ltd, Berkshire, GB). Animals were then housed individually and allowed three days of recovery before the start of the pre-programmed pump infusion. Throughout the study, body weight and general health status were monitored and recorded daily.

Nicotinic acid exposure selection and formulation

A key aspect of the study design was to achieve plateau plasma nicotinic acid (NiAc) concentrations corresponding to therapeutically relevant levels in the rat (\(\sim\)1 \({\upmu {\text{M}}}\)), based on the relationship between plasma NiAc levels and FFA lowering [16]. For intravenous infusions (i.v.), NiAc (pyridine-3-carboxylic acid, Sigma-Aldrich, St. Louis, MO, USA) was dissolved in sterile saline. For subcutaneous (s.c.) infusions, NiAc was dissolved in sterile water and adjusted to physiological pH using sodium hydroxide. Vehicle, for control animals, consisted of sodium chloride solutions at equimolar concentrations. Freshly prepared formulations were loaded into the infusion pump (see below) via a 0.2 \(\upmu\)m sterile filter (Acrodisc^®, Pall Corporation, Ann Arbor, MI, USA) just before pump implantation.

Experimental protocols

Conscious animals (NiAc naïve, Cont. NiAc and Inter. NiAc groups)

Both lean and obese animals were divided into 3 dose groups and NiAc was given either acutely (NiAc naïve) or following 5 days of either continuous (Cont. NiAc) or intermittent (Inter. NiAc) administration. Each dose group was matched with corresponding saline infused controls. NiAc infusions were given subcutaneously at 170 nmol min\(^{-1}\)kg\(^{-1}\). The intermittent infusion protocol was programmed as a 12 h on-off cycle (infusion on at 13:00). Following overnight fast, in the morning of the acute experimental day, the jugular catheter was connected to a swivel system to enable blood sampling in unrestrained animals. Jugular catheter patency was maintained by continuous infusion (5 \(\upmu\)mol min\(^{-1}\)) of sodium-citrate solution (20.6 mM). After a 3–4 h adaptation period, at \(\sim\)12:00, the basal phase of the acute experiment commenced with 2–3 blood samples drawn between −60 and −5 min, relative to start of NiAc/saline infusion (note that, in the Cont. NiAc groups, infusion pumps were on throughout this sampling period). Blood samples (16–17/animal) were drawn under an 8 h experimental period. Samples, 30–150 \(\upmu\)l (with total loss less than 5% of blood volume), were collected in potassium-EDTA tubes, centrifuged and plasma stored at −80 \(^\circ\)C pending analysis for NiAc, FFA and insulin.

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Analytical methods

Plasma FFA was analyzed using an enzymatic colorimetric method (Wako Chemicals GmbH, Neuss, Germany). Plasma insulin from obese rats was analyzed with a radioimmunoassay kit (rat insulin RIA kit, Millipore Corporation, St. Charles, Missouri, USA). Plasma insulin concentrations from lean rats were determined using a colorimetric ELISA kit (Ultra Sensitive Rat Insulin ELISA Kit, Crystal Chem INC, Downers Grove, IL, USA). The ELISA was used for lean rats to minimize blood sample volume (only 5 \(\upmu\)l plasma required vs. \(\sim\)50 \(\upmu\)l plasma for RIA). The RIA was used for the obese rats because their high lipid levels in plasma interfere with the ELISA but not the RIA measurement. Due to the hyperinsulinemia in the obese rats only 5 \(\upmu\)l of plasma was required. For lean-rat plasma (with low lipid levels) the absolute insulin measurements are equivalent for the RIA and ELISA assays, according to an in-house comparison. Plasma NiAc concentrations were analyzed using LC-MS/MS with a hydrophilic interaction liquid chromatography (HILIC) approach, separated on a \(50\times\)2.1 mm Biobasic AX column, with 5 \(\upmu\)m particles (Thermo Hypersil-Keystone, Runcorn, Cheshire, UK) as previously described [16].

Model development

The exposure (PK) and biomarker (PD) models were developed sequentially because of the interaction between the model components; the kinetics of NiAc are assumed to be unaffected by insulin and FFA, whereas NiAc inhibits the release of both insulin and FFA. Furthermore, due to its antilipolytic effect, insulin affects FFA release. The interactions between the three models (NiAc, insulin, and FFA) are illustrated in Fig. 1b, and model interactions of previously published NiAc-FFA models [14, 19, 20, 23] are illustrated in Fig. 1a for comparison. When a sub-model had been estimated, the random effects were fixed to the Empirical Bayes Estimates (EBE) and used as covariates in the subsequent sub-model.

NiAc exposure model

The pharmacokinetic properties of NiAc have been thoroughly characterized in previous studies [14, 16, 17, 18, 19, 21, 22, 23, 26]. Ahlström et al. [16] introduced a two-compartment disposition model with parallel nonlinear (Michaelis-Menten) elimination for lean Sprague-Dawley rats, and a one-compartmental model with a single nonlinear elimination for obese Zucker rats (a schematic illustration of the PK models is given in Fig. 2).

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Insulin turnover model

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The NiAc action compartment is initially at steady-state with the plasma NiAc compartment \(C_{\text{p}}\). As infusions begin, and the plasma compartment concentration increases, \(N_{\text{I}}(t)\) increases until it reaches the steady-state NiAc concentration \(N_{\text{ss}}(t)=C_{\text{pss}}\). With increasing levels in the NiAc action compartment, \(E(N_{\text{I}}(t))\) decreases to a minimum of \(1-S_{\text{NI}}\) and, consequently, the efficacy of NiAc as an insulin inhibitor is down-regulated. In other words, the system has developed tolerance to the drug. The turnover rate \(k_{\text{NI}}\) determines the rate at which tolerance develops. A schematic illustration of the insulin model is given in Fig. 4.

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Mechanistic FFA model

The model suggested in this study (schematically illustrated in Fig. 5) is founded on preceding approaches [14, 19, 20, 23]; however, insulin has been included as the main endogenous regulator of FFA as insulin provides a homeostatic force on the system—thereby keeping FFA levels in the vicinity of its baseline concentration. Furthermore, the NiAc efficacy is dynamic in that it is decreasing during long-term infusions, which allows for complete systemic adaptation - a feature apparent in the data [32, 33]. The characteristic behavior of the data, for acute and chronic NiAc provocations in lean and obese rats, is illustrated in Fig. 6. Attributes observed include indirect response, tolerance (drug resistance), rebound, and complete adaptation (FFA concentrations returning to pre-treatment levels). The behavior observed in the acute experiment (Fig. 6a, c) has been described by turnover equations with moderator feedback (as described for the insulin system). The long-term behavior, and in particular the adaptations with, and without, rebound, is captured by dynamic NiAc efficacy and an insulin-controlled regulator. The FFA model is given by

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$$\begin{aligned} \frac{\text d{F(t)}}{\text d{t}}&= k_{\text{inF}} \cdot R(t) \cdot H_{\text{NF}}(C_{\text{p}}(t)) \cdot \frac{M_{\text{0F}}}{M_{\text{F}}(t)} \nonumber \\&- k_{\text{outF}}\cdot F(t), \end{aligned}$$

(20)

with initial condition

$$\begin{aligned} F(0)=F_0. \end{aligned}$$

(21)

Here, F(t) denotes the observed FFA level, \(k_{\text{inF}}\) the turnover rate, and \(k_{\text{outF}}\) the fractional turnover rate. The moderator compartment \(M_{\text{F}}\) is given by

$$\begin{aligned} \frac{\text d{M_{\text{F}}(t)}}{\text d{t}} =k_{\text{tolF}}\cdot \left( F(t)-M_{\text{F}}(t)\right) , \end{aligned}$$

(22)

with initial condition

$$\begin{aligned} M_{\text{F}}(0)= M_{\text{0F}}=F_0, \end{aligned}$$

(23)

where the parameter \(k_{\text{tolF}}\) represents the turnover rate of the moderator compartment. The moderator compartment provides a feedback mechanism for the turnover of FFA, that strives to dampen deviations from the baseline response. The regulator compartment \(R_{\text{F}}(t)\), that links insulin dynamics to FFA release, is similar to that of the insulin model (Eq. 12) and is given by

$$\begin{aligned} \frac{\text d{R(t)}}{\text d{t}} =k_{\text{inRF}} - k_{\text{outRF}}\cdot I(t),\quad R(0)=1, \end{aligned}$$

(24)

where \(k_{\text{inRF}}\) is the turnover rate, \(k_{\text{outRF}}\) the fractional turnover rate, and I(t) the insulin concentration. As for the insulin regulator, \(R_{\text{F}}(t)\) represents the output of an insulin-driven integral controller with \(I_0\) as the set-point and \(k_{\text{outRF}}\) as the integral gain parameter. The contribution of this integral controller during acute and chronic NiAc treatments in lean and obese rats is illustrated in Fig. 9b. The inhibitory NiAc function on FFA (similar to that for the insulin model, Eq. 15), is given by

$$\begin{aligned} H_{\text{NF}}(C_{\text{p}}(t))=1-E_{\text{NF}}(N_{\text{F}}(t))\cdot \frac{ C_{\text{p}}^m(t)}{ IC _{\text{50NF}}^m+C_{\text{p}}^m(t)}, \end{aligned}$$

(25)

where \(IC _{\text{50NF}}\) is the potency of NiAc as an inhibitor of FFA release and m is the Hill coefficient. The drug efficacy is dynamic and changes (down-regulates) during long-term infusions of NiAc. The efficacy is given by

$$\begin{aligned} E_{\text{NF}}(N_{\text{F}}(t))= I_\text{maxNF}\cdot \left( 1-\frac{S_{\text{NF}}\cdot N_{\text{F}}^\phi (t)}{ N _{\text{50F}}^\phi +N_{\text{F}}^\phi (t)}\right) , \end{aligned}$$

(26)

where \(I_{\text{maxNF}}\) is the initial efficacy of NiAc on FFA, \(N _{\text{50F}}\) the potency of the NiAc action compartment, \(S_{\text{NF}}\) the long-term NiAc efficacy loss, \(\phi\) the Hill coefficient, and \(N_{\text{F}}(t)\) the concentration in the NiAc action compartment. The dynamics of the NiAc action compartment are in turn described by

$$\begin{aligned} \frac{\text d{N_\text{F}(t)}}{\text d{t}} = k_{\text{NF}}\cdot (C_{\text{p}}(t)-N_\text{F}(t)), \end{aligned}$$

(27)

with initial condition \(N_{\text{F}}(0)=C_{\text{p}}(0)\). Here, the parameter \(k_{\text{NF}}\) is the turnover rate of the NiAc action state.

Numerical analysis

The NLME modelling and simulations and the identifiability analysis were performed using Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 10.3, Champaign, IL (2014).

Parameter estimation

Parameter estimates for the NLME models were computed by maximizing the first-order conditional estimation (FOCE) approximation of the population likelihood. This was done using a method developed and implemented in Mathematica 10 (Wolfram Research) at the Fraunhofer-Chalmers Research Centre for Industrial Mathematics (Gothenburg, Sweden) [38], which combines exact gradients of the FOCE likelihood based on the so-called sensitivity equations with the Boyden-Fletcher-Goldfarb-Shanno optimization algorithm [39]. Parameter standard errors were derived using the Hessian of the approximate population likelihood with respect to the parameters, evaluated at the point estimate. The Hessian was computed using finite differences of the exact gradients.

From the steady-state relations in the insulin and FFA models, dependencies were derived which enabled the parameters \(k_{\text{inI}}\), \(k_{\text{inF}}\), \(k_{\text{inRI}}\), and \(k_{\text{inRF}}\) to be expressed in terms of other model parameters (derivation given in appendix 2). Consequently, these parameters were redundant and could be replaced in the parameter estimation. Furthermore, some parameters were initially estimated to be very close to their physiological limit (e.g. \(I_{\text{maxNI}}=0.9999\approx 1\) for obese rats) and were consequently fixed for numerical stability. Finally, to simplify the parameter estimation, some parameters were fixed (e.g., \(S_{\text{NI}}=1\) for obese rats). This is motivated by the complete systemic adaptation apparent in the long-term insulin-time data (obese rats), implying that \(S_{\text{NI}}\) must be 1 (The fixed parameters are given in Table 2)). The long-term NiAc efficacy loss for lean rats was initially estimated to be \(\approx0\), whereby this part was omitted in the final model.

Table 2

Estimates of parameter median values and between-subject variabilities with corresponding relative standard errors (RSE%) for normal Sprague-Dawley rats and obese Zucker rats. Estimates highlighted in blue were taken from the literature (Tapani et al. [23]) while the remaining parameters were estimated in this study

		Normal Sprague-Dawley rats		Obese Zucker rats
Parameter	Definition	Estimate (RSE\(\%\))	BSV\(^{a}\) (RSE\(\%\))	Estimate (RSE\(\%\))	BSV\(^{a}\) (RSE\(\%\))
Pharmacokinetic model parameters
\(k_{\text{a}}\) (h\(^{-1}\))	First order absorption rate	4.27 (13)	80.1 (51)	5.54 (16)	80.2 (47)
\(\delta\) (h\(^{1/2}\))	Lumped diffusion coeff. catheter	77.4 (15)	–	62.4 (17)	–
\(V_{\text{max1}}\) (\(\upmu\) mol kg\(^{-1}\)h\(^{-1}\))	Max. elimination - pathway 1	2.64 (12)	93.5 (51)	164 (5.1)	22.4 (13)
\(K_{\text{m1}}\) (\({\upmu {\text{M}}}\))	Michaelis constant - pathway 1	0.235 (29.2)	–	18.9 (21.5)	–
\(V_{\text{max2}}\) (\(\upmu\)mol kg\(^{-1}\)h\(^{-1}\))	Max. elimination - pathway 2	425 (39.6)	–	–	–
\(K_{\text{m2}}\) (\({\upmu \text{M}}\))	Michaelis constant - pathway 2	74.5 (43.4)	–	–	–
\(V_{\text{p}}\) (L kg\(^{-1}\))	Volume of distribution - plasma	0.393 (5.29)	–	0.323 (12.4)	–
\(V_{\text{t}}\) (L kg\(^{-1}\))	Volume of distribution - tissue	0.172 (35.2)	–	–	–
\(Cl_{\text{d}}\) (L kg\(^{-1}\)h\(^{-1}\))	Inter-compartmental distribution	0.0511 (27.8)	–	–	–
\({\text{Synt}}\) (\(\upmu\)mol kg\(^{-1}\)h\(^{-1}\))	Endogenous NiAc synthesis	0.213 (23.3)	66.7 (57)	0.168 (10.1)	95 (110)
\(\sigma _{\text{propN}}\)	Residual proportional error	0.313 (5.1)	–	0.483 (5.3)	–
Insulin model parameters
\(I_{\text{0}}\) (nM)	Baseline insulin conc.	0.188 (9.7)	49.3 (5.5)	3.26 (12)	10.3 (21)
\(k_{\text{outI}}\) (h\(^{-1}\))	Fractional turnover rate insulin	6.58 (14)	–	10.8 (17)	–
\(I_{\text{maxNI}}\)	Efficacy - NiAc on insulin	0.793 (11)	–	1\(^b\)	–
\(IC _{\text{50NI}}\) (\({\upmu {\text{M}}}\))	Potency - NiAc on insulin	0.338 (15)	111 (67)	0.175 (27)	190 (160)
n	Hill coefficient - NiAc on insulin	3.54 (6.6)	–	0.840 (6.0)	–
\(k_{\text{tolI}}\) (h\(^{-1}\))	Turnover rate moderator	0.646 (28)	93.9 (20)	0.125 (48)	310 (9.4)
\(k_{\text{outRI}}\) (nM\(^{-1}\)h\(^{-1}\))	Integral gain parameter	3.94 (17)	–	0.0612 (27)	–
\(k_{\text{NI}}\) (h\(^{-1}\))	Turnover rate NiAc action comp.	–	–	0.0242 (35)	–
\(N _\text{50I}\) (\({\upmu \text{M}}\))	Potency NiAc action compartment	–	–	0.897 (4.9)	–
\(\gamma\)	Hill coefficient	–	–	18.9 (44)	–
\(S_{\text{NI}}\)	Long-term NiAc effect loss	–	–	1\(^b\)	–
\(\sigma _{\text{addI}} ({\text{nM}})\)	Residual additive error	0.0699 (3.3)	–	0.748 (3.0)	–
Free fatty acid model parameters
\(F_{\text{0}}\) (mM)	Baseline FFA conc.	0.707 (5.0)	17.8 (26)	1.14 (3.1)	0.874 (25)
\(k_{\text{outF}}\) (h\(^{-1}\))	Fractional turnover rate FFA	428 (140)	–	173 (120)	–
\(I_{\text{maxNF}}\)	Efficacy - NiAc on FFA	1\(^b\)	–	1\(^b\)	–
\(IC _\text{50NF}\) (\({\upmu \text{M}}\))	Potency - NiAc on FFA	0.436 (12)	41.8 (28)	0.456 (14)	41.8 (26)
m	Hill coefficient - NiAc on FFA	1.24 (11)	–	0.731 (9.0)	–
\(k_{\text{tolF}}\) (h\(^{-1}\))	Turnover rate moderator	1.21 (67)	58.4 (9.6)	0.708 (24)	34.2 (15)
\(k_{\text{outRF}}\) (nM\(^{-1}\)h\(^{-1}\))	Integral gain parameter	0.965 (29)	–	0.0165 (38)	–
\(k_{\text{NF}}\) (h\(^{-1}\))	Turnover rate NiAc action comp.	0.00654 (65)	–	0.0377 (14)	–
\(N _\text{50F}\) (\({\upmu \text{M}}\))	Potency NiAc action compartment	3.05 (160)	–	0.854 (4.5)	–
\(\phi\)	Hill coefficient	1\(^b\)	–	8.83 (33)	–
\(S_{\text{NF}}\)	Long-term NiAc effect loss	0.807 (190)	–	1\(^b\)	–
\(\sigma _\text{addF} ({\text{mM}})\)	Residual additive error	0.130 (3.5)	–	0.135 (3.0)	–

\(^{a}\) Between-subject variability expressed in CV%, calculated as \(100\times \sqrt{\omega ^2}\).

\(^{b}\) Fixed in the estimations.

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Pharmacokinetic model

The pharmacokinetic system reached a steady-state concentration of about \({1}{\upmu {\text{M}}}\) for all protocols both in lean and obese rats (first column in Fig. 7 and first column in Fig. 8). The steady-state was attained faster with intravenous than with subcutaneous administration. When infusions were terminated, the drug was cleared from the system within minutes and the NiAc concentration approached the endogenous level.

The absorption from the subcutaneous compartment had a half-lives of 0.16 and 0.13 h for lean and obese rats, respectively. At steady-state, the elimination of NiAc from the plasma compartment in lean rats was approximately three times faster for the high affinity pathway than the low affinity one. Moreover, the drug elimination rate from the plasma at steady-state was \(\sim {20}\) and \(\sim {25}{\upmu } {\text{mol}} {\text{kg}}^{-1}{\text{h}}^{-1}\) for lean and obese rats, respectively. The lumped diffusion coefficient was estimated to be 77 and 62 h\(^{-1/2}\) for lean and obese rats, respectively, implying that the NiAc dosing solution was diluted during the first \(\sim\)1.5 h.

Insulin model

The median baseline concentrations across groups were 0.233 and 3.51 nM for lean and obese rats, respectively. The estimates of the individual groups ranged between 0.151 and 0.264 nM (inter-study variability of 33%¹) for lean rats and 2.69–4.82 nM (inter-study variability of 33%\(^{1}\)) for obese rats. The inter-study variability was not correlated to the anesthetic condition of the rats. The median turnover half-lives of insulin, for the moderator, the integral controller, and the NiAc action level for lean and obese rats are given in Table 3. For lean rats, the efficacy of NiAc on insulin inhibition, \(I_{\text{maxNI}}\), was estimated to be 0.793; consequently, NiAc cannot completely inhibit insulin release. The established NiAc exposure was about 1\({\upmu {\text{M}}}\) which is approximately three times the NiAc potency related to inhibition of insulin (\(IC _{\text{50NI}}=\)0.338 \({\upmu {\text{M}}}\)). This implies that the inhibitory function was saturated at steady-state. The estimated Hill coefficient n indicates a steep NiAc concentration-insulin response relationship at steady-state. Furthermore, for obese rats, the efficacy was fixed to 1 (described in the parameter estimation section) and the corresponding potency was high since the \(IC _{\text{50NI}}\) was low in comparison to the NiAc steady-state exposure. However, since the estimated Hill coefficient was \(0.84<1\), indicating a gentle NiAc-concentration insulin-response relationship at steady state, the NiAc concentrations never reached levels high enough to saturate the inhibitory function. The estimated \(N_{50I}\) of the NiAc action compartment was lower than the steady-state NiAc concentration (0.897<1) and the Hill coefficient of the dynamic efficacy was estimated to be 18.9 (suggesting an all or non-response). This implies that the efficacy was completely down-regulated at the end of the long-term experiments in obese rats, implying no NiAc inhibition on insulin release.

FFA model

The FFA concentration was suppressed below its baseline value for all provocations of NiAc. Suppression was more pronounced initially during NiAc infusions. FFA concentrations then drifted back towards their baseline values (cf. Figs. 7c–f or 8c–i). After the infusions were terminated, the FFA concentrations rebounded before reaching their baseline values. The step-down protocols resulted in less rebound. The FFA concentrations returned to their baselines during extended exposure of NiAc in lean and obese rats (Figs. 7f, 8i, respectively). However, as the long-term exposure was terminated, rebound occured in lean, but not in obese rats (Figs. 7f, 8i). The median baseline concentrations across groups were 0.707 and 1.14 mM with corresponding ranges of 0.652–0.801 mM (inter-study variability of 11%²) in lean and 0.789–1.22 mM (inter-study variability of 20%\(^2\)) obese rats, respectively. The inter-study variability was not correlated to the anesthetic condition of the rats. The median turnover half-lives of the FFA, the moderator, the integral controller, and the NiAc action level are given in Table 3.

Model simulations of the acute and chronic action of NiAc (\(H_{\text{NF}}(C_\text{p}(t))\)), the insulin-driven integral controller (\(R_{\text{F}}(t)\)—also referred to as the insulin-controlled regulator), and the moderator feedback (\(M_{\text{0F}}/M_{\text{F}}(t)\)) on the FFA turnover are illustrated in Fig. 9 with the corresponding acute and chronic FFA responses. The simulated NiAc concentrations were set to be around 1\({\upmu {\text{M}}}\) (corresponding to the experimental exposures). The turnover rate of FFA was initially inhibited about 80% (lean) and 70% (obese) by the NiAc infusion (Fig. 9a). Upon the extended exposure to NiAc (120 h) the inhibitory action on the turnover rate was decreased by approximately 13% due to intrinsic tolerance mechanisms (Eq. 26). In obese rats, the NiAc action vanished completely (Fig. 9a). The insulin-driven controller (Eq. 14) provides a stimulatory action on the FFA turnover rate as insulin concentrations fall below the baseline (Fig.9b). The positive (stimulatory) action increases from 100% (at baseline) to about 200% after extended (120 h) exposure to NiAc in lean rats. The insulin action is totally abolished at equilibrium (120 h) in obese rats (Fig. 9b). The positive impact of the moderator is seen acutely both in lean and obese rats, whereas the moderator action has receded in obese rats at equilibrium (Fig. 9c). In lean rats, the combined inhibitory (NiAc) and stimulatory (insulin, moderator) action on the FFA turnover rate causes a rebound in FFA response when the NiAc infusion is stopped (Fig. 9d). This is not seen in obese rats since all of the NiAc, insulin, and moderator actions are back to baseline at equilibrium (120 h).

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The potency of the NiAc action compartment was low for lean rats, since \(N _{\text{50F}}>>N(t)\) at equilibrium (120 h), and high for obese rats, since \(N _{\text{50F}}<N(t)\) at equilibrium (120 h). Consequently, the loss of efficacy was low in lean rats and high in obese ones. Furthermore, the estimate of Hill coefficient \(\phi =8.37\) for obese rats suggests an all-or-none efficacy loss in obese rats.

Model predictions

The resulting model was used to predict 24 h FFA lowering (AUC\(_{24}\)) for a range of protocols at steady-state (i.e., after multiple dosing). Here, the lowering over a period T is given by

$$\begin{aligned} {\text{AUC}}_{\text{T}}=F_0\cdot T - \int _0^T F(\tau )\,{\text{d}}\tau . \end{aligned}$$

(28)

The protocol design consisted of a 0.25–12 h NiAc exposure period followed by a 0–12 h washout period. The NiAc infusions were designed to generate concentrations around the therapeutically relevant level (\(\sim\)1\({\upmu {\text{M}}}\)) that was used in the experiments [16]. The predicted AUC\(_{24}\) and proportional reduction, in comparison to baseline levels, on a median obese rat are given in Fig. 10. The model predicted an optimal dosing strategy of \(\sim\)2 h longer washout period than the exposure period and the maximal AUC reduction is 5.60 mM h. These results were consistent when AUC\(_{24}\) was predicted for outliers with high/low potencies (\(IC _{\text{50NF}}\)), baseline responses (\(F_0\)), and/or moderator turnover rates (\(k_{\text{tolF}}\)).

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Furthermore, the NiAc-concentration FFA-response relationship at steady-state, for obese rats, is illustrated in Fig. 11, with the corresponding NiAc-, insulin-, and moderator actions. The largest reductions in FFA exposure occurs within a ‘window of opportunity’, with NiAc concentrations between the \(IC_{\text{50NF}}\) and the \(N_{\text{50F}}\). The predicted AUC\(_{24}\) at steady-state, for the optimal NiAc exposure of 0.500\({\upmu {\text{M}}}\), was 7.40 mM h.

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Model characteristics

A key observation of the FFA-time course data in our study is that lean and obese rats both acquire complete adaptation, with FFA levels returning towards baseline at equilibrium (Fig. 6). The post-infusion FFA rebound in lean rats implies a NiAc-sensitive system. However, the rebound was less pronounced than that for the 1–12 h experiments. Consequently, the inhibiting effect of NiAc has been down-regulated during the long-term infusions. For obese rats, the rebound completely vanished, which implies a NiAc-insensitive system. This suggests two separate mechanisms that result in complete adaptation. To describe adaptation with drug effect (seen in lean rats), an insulin-driven integral feedback control was incorporated into the model [39]. Control system techniques have previously been applied in glucose-insulin models in form of PID (proportional-integral-derivative) controllers [42]. The set-point of the insulin-driven integral controller is the insulin baseline and, hence, its action reflects the deviations from baseline insulin concentrations. For the FFA model, the controller represents the antilipolytic effect of insulin [24] via a regulator; as the insulin level increases, the antilipolysis will be more pronounced since the elimination rate of \(R_{\text{F}}\) will increase and, consequently, \(R_{\text{F}}\)’s stimulation of FFA release will be lowered (and vice versa). The traditional \(I_{\text{max}}\)-equation was modified with a dynamic efficacy function in order to capture the phenomena of NiAc resistance. The dynamic efficacy may represent the down-regulation of the PDE3B gene expression [7].

The impact of the dynamic efficacy and the integral controller (Fig. 9) show that both adaptive actions push FFA concentrations back towards baseline (at equilibrium) despite ongoing NiAc exposure. The insulin-driven controller has less effect in obese as compared to lean rats in spite of 10-fold higher insulin concentrations in the former group. This reflects the insulin resistance of the obese rats [7].

Model evaluation

The VPC’s (Figs. 7, 8) demonstrate the flexibility of the insulin and FFA models in that response-time courses were captured in both acute and chronic settings. The fractional turnover rate is operating on a significantly shorter time-scale than the feedback and adaptive mechanisms. This resulted in low precision of \(k_{\text{outF}}\) (RSE% of 140 and 120, respectively, in lean and obese rats). To achieve higher precision a denser sampling of the FFA time-course is needed during the initial infusion phases. On the other hand, the drug resistance in lean rats was a slower process (half-life of the NiAc action of more than 100 h). This resulted in low precision of the parameters linked to efficacy loss (RSE% of 160 for \(k_{\text{NF}}\) and 190 for \(S_{\text{NF}}\)).

Some estimated median responses of the insulin and FFA models were under- or over-predicted (cf. Fig. 7l or Fig. 7k). This is most likely due to the low number of individuals per trial (5–10), implying that every 4-9th population median will be estimated below or above the trends seen in the individual data [43]. Furthermore, some predicted 90% population spans also under- or over-predicted the response for the insulin and FFA model (cf. Fig. 7l or Fig. 7k). This is most likely due to correlations of the between-subject parameter variabilities, which were not captured because diagonal covariance matrices were used in the FFA model [44]. When sampling from the resulting distributions to generate the VPC’s, non-feasible parameter combinations may occur which render a skewed population [44]. Diagonal covariance matrices were chosen in order to simplify the parameter estimation.

The turnover processes in the insulin and FFA models operate over completely different time scales (Table 3). The general trend in both models is that the moderator and integral controller processes are slower in obese rats. For example, the insulin-driven integral controller has a half-life that is 50 times longer in obese as compared to lean rats. These control processes are tightly controlled in lean animals, and are probably elongated by the tremendous hyperinsulinemia (nearly 10-fold greater pathological concentrations), and corresponding insulin resistance, in obese rats. In the FFA model, turnover of FFA is more than 100 times faster than moderator feedback and integral control. Furthermore, turnover of the NiAc action compartment had a half-life of more than 100 h for lean rats, thus spanning the entire duration of the experiment. The corresponding half-life in obese rats was 18 h. Hence, obese rats reach complete adaptation much faster than lean rats.

Due to the nature of the FFA dynamics, with tolerance development and rebound post-infusions, constructing an optimal dosing protocol is challenging. By selecting an inappropriate dosing regimen, the NiAc provocation can yield an increased FFA exposure in comparison to controls (Fig. 10—the negative AUC area). Given a NiAc exposure of \(\sim\)1\({\upmu {\text{M}}}\) (the exposure that was used in the experiments), there is an optimal strategy whereby washout periods are 2 h longer than infusion periods; this is illustrated by Fig. 10. These strategies were consistent in that they were tested on the median individual and on 90% quantiles (i.e., individuals that had the 5 and 95% quantiles of the parameters that varied in the population: \(IC_{\text{50NF}}\), \(F_0\), and \(k_{\text{tolF}}\)). However, rather surprisingly, higher reductions in AUC were attained at constant NiAc exposure (without washout periods) at lower concentrations, where the maximal reduction was 7.40 mM h (in comparison to the maximal reduction of 5.60 mM attained from the exposure/washout protocols).