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.
Anesthetized animals (NiAc Off and NiAc Stp-Dwn 12 h infusion groups)
Before the infusions began, lean and obese rats were fasted for 8 h. On the day of the acute study, at 01:00 (corresponding to time = 0 h), the implanted pre-programmed pump began infusing NiAc at a constant rate of 170 nmol min\(^{-1}\) kg\(^{-1}\) for 12 h. At 8.5 h animals were anesthetized (Na-thiobutabarbitol, Inactin®, 180 mg kg\(^{-1}\), i.p., RBI, Natick, MA, USA), underwent a tracheotomy with PE 240 tubing, and breathed spontaneously. One catheter (PE 50 tubing) was placed in the left carotid artery for blood sampling and for recording arterial blood pressure and heart rate. One catheter (PE 10 tubing) was placed in the right external jugular vein to infuse top-up doses of anesthetic. The arterial catheter patency was maintained by continuous infusion of sodium-citrate (20.6 mM in saline, 5 \(\upmu\)l min\(^{-1}\)) from shortly after carotid catheterization until the experiment ended. Body temperature was monitored using a rectal thermocouple and maintained at 37.5 \(^\circ\)C by means of servo controlled external heating. After surgery, animals were allowed a stabilization period of at least 1.5 h and blood sampling began at 11.0 h. At 12.0 h, NiAc infusion was either programmed to switch off (NiAc Off) or to decrease in a step-wise manner, with final switch-off at 15.5 h (NiAc Stp-Dwn). The step-down NiAc infusion rates were 88.9, 58.3, 43.7, 34.0, 24.3, 17.0, and 9.7 nmol min\(^{-1}\) kg\(^{-1}\). All NiAc protocols were matched with saline-infused controls. Blood samples (18/animal) were drawn during a 6 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 was stored at −80 \(^\circ\)C pending analysis for NiAc, FFA and insulin. All of the experimental groups are summarized in Table 1.
Table 1 Summary of experimental protocols—including conscious or anesthetized state, route of administration, duration of experiment, protocol name, and the number of lean and obese rats within each experiment (the number of saline infused controls is given in parenthesis)
Anesthetized animals (NiAc Off and NiAc Stp-Dwn 1 h infusion groups)
After an overnight fast, lean and obese rats were anesthetized and surgically prepared, as described above. They were allowed a stabilization period after surgery of at least 1.5 h. Two basal blood samples were obtained, after which an i.v. NiAc infusion was given at a constant rate (170 nmol min \(^{-1}\)kg\(^{-1}\)) for 1.0 h (the start of infusion was taken as time = 0 h). The NiAc infusion was then either switched off (NiAc-Off 1 h) or decreased in a step-wise manner, with final switch-off at 4.5 h (NiAc Stp-Dwn 1 h). The step-down NiAc infusion rates were: 31.1, 20.4, 15.3, 11.9, 8.50, 5.95 and 3.40 nmol min\(^{-1}\)kg\(^{-1}\). All NiAc protocols were matched with saline infused controls. Blood samples (13–18/animal) were drawn during a 6 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 was stored at −80 \(^\circ\)C pending analysis for NiAc, FFA, and insulin.
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.
Disease modeling and inter-study variability
The PK and PD were significantly different between lean (normal) and obese (diseased) rats and, consequently, these groups were modeled separately. Furthermore, the animal experiments were done under different conditions (separate time periods, anesthetized/conscious animals) which may have provoked different dynamic behaviors. To account for this, inter-study variability was included in the models in the form of fixed-study effects [25].
Notation conventions
To improve readability and enable the reader to differentiate between separate sub-model parameters, PD (insulin and FFA) model parameters are labeled with a subscript, indicating to which model they belong. For example, the turnover rate of FFA will be referred to as \(k_{\text{inF}}\) and the turnover rate of insulin is \(k_{\text{inI}}\) (i.e., F for FFA and I for insulin). Parameters that link NiAc, insulin, and FFA are labeled with both sub-model subscripts (e.g., potency of NiAc as an FFA inhibitor will be called \(IC_{\text{50NF}}\), whereby the N is for NiAc and the F is for FFA).
NiAc exposure model
The pharmacokinetic properties of NiAc have been thoroughly characterized in previous studies [14, 16–19, 21–23, 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).
Lean rats
In lean rats, the NiAc disposition is given by
$$\begin{aligned} V_{\text{p}}\cdot \frac{{\text d{C_{\text{p}}(t)}}}{\text d{t}}&= {\text{Input}}(t) + {\text{Synt}} - \frac{V_{\text{max1}}\cdot C_{\text{p}}(t)}{K_{\text{m1}}+C_{\text{p}}(t)} \nonumber \\&- \frac{V_{\text{max2}}\cdot C_{\text{p}}(t)}{K_{\text{m2}}+C_{\text{p}}(t)} - Cl_{\text{d}} \cdot C_{\text{p}}(t) \nonumber \\&+ Cl_{\text{d}} \cdot C_{\text{t}}(t), \end{aligned}$$
(1)
$$\begin{aligned} V_{\text{t}}\cdot \frac{\text d{C_{\text{t}}(t)}}{\text d{t}}&= Cl_{\text{d}} \cdot C_{\text{p}}(t) - Cl_{\text{d}} \cdot C_{\text{t}}(t), \end{aligned}$$
(2)
where \(C_{\text{p}}(t)\) is the observed NiAc concentration in the central plasma compartment and \(C_{\text{t}}(t)\) is the concentration in the peripheral tissue compartment (derivations of the initial conditions for these compartments are given in Appendix 2), and \(V_{\text{p}}\) and \(V_{\text{t}}\) are, respectively, the volumes of distribution of the plasma and tissue compartments. The parameters \(V_{\text{max1}}\) and \(K_{\text{m1}}\) are the maximal elimination rate and the Michaelis constant of the first pathway, and \(V_{\text{max2}}\) and \(K_{\text{m2}}\) are the maximal elimination rate and the Michaelis constant of the second pathway (low and high affinity pathway, respectively). Furthermore, \(Cl_{\text{d}}\) is the inter-compartmental distribution, \({\text{Synt}}\) the endogenous NiAc synthesis, and \({\text{Input}}(t)\) is a time-dependent function determined by the route of administration according to
$$\begin{aligned} {\text{Input}}(t) = {\left\{ \begin{array}{ll} {\text{Inf. rate}} \quad &{}{\text{Intravenous infusion}} \\ k_{\text{a}}\cdot A_{\text{sc}}(t) \quad &{}\text{Subcutaneous infusion}, \end{array}\right. } \end{aligned}$$
(3)
where \({\text{Inf. rate}}\) is the infusion rate, \(A_{\text{sc}}(t)\) is the amount of drug in the subcutaneous compartment, and \(k_{\text{a}}\) is the absorption rate from the subcutaneous compartment to plasma. The rate of change of \(A_{\text{sc}}(t)\) is given by
$$\begin{aligned} \frac{\text d{A_{\text{sc}(t)}}}{\text d{t}}&= {\text{Pump rate}} - k_{\text{a}}\cdot A_{\text{sc}}(t), \end{aligned}$$
(4)
with initial condition \(A_{\text{sc}}(0)=0\). Here, Pump rate represents the infusion rate from a subcutaneous mini-pump. The mini-pump was surgically implanted seven days before the final acute experiment. During this period, when the pump is not infusing, interstitial tissue fluid may diffuse into the tip of the catheter, diluting the NiAc dosing solution, whilst the solution is leaking into the tissue. Consequently, a concentration gradient may form, resulting in an apparently lower initial infusion rate compared to the pre-programmed setting (particularly pronounced in lean NiAc naïve rats, see Fig. 7a). To capture this, the pump infusion rate is modeled as
$$\begin{aligned} {\text{Pump rate}} = {\text{Inf. rate}} \cdot {\text{erf}}\left( \frac{t\cdot \delta }{\sqrt{t_0}} \right) , \end{aligned}$$
(5)
where \({\text{Inf. rate}}\) is the programmed infusion rate of the pump, \(\delta\) is a lumped diffusion parameter, and \(t_0\) is the pump inactivation time (in this case 7 days). Here \({\text{erf}}\) is the error function [27]. The derivation of the Pump rate is given in Appendix 1. Given NiAc’s low molecular weight (123.11 g/mol), bioavailability from the subcutaneous compartment was assumed to be equal to unity.
Obese rats
For obese Zucker rats, the NiAc disposition is given by
$$\begin{aligned} V_{\text{p}}\cdot \frac{\text d{C_{\text{p}}(t)}}{\text d{t}}&= {\text{Input}}(t) + {\text{Synt}} - \frac{V_{\text{max1}}\cdot C_{\text{p}}(t)}{K_{\text{m1}}+C_{\text{p}}(t)}, \end{aligned}$$
(6)
where \(C_{\text{p}}(t)\) is the NiAc concentration in the central plasma compartment, \(V_{\text{c}}\) the volume of distribution, \(V_{\text{max1}}\) the maximal elimination rate, \(K_{\text{m1}}\) the Michaelis constant, and \({\text{Synt}}\) the endogenous synthesis. The term \({\text{Input}}(t)\) is the same as for the lean rats (the relations given in Eqs. 3, 4 and 5).
Between-subject and residual variability
The modeling was performed in an NLME framework to capture the between-subject variability seen in the exposure-time data. The parameters that varied within the population were \(k_{\text{a}}\), \(V_{\text{max1}}\), and \({\text{Synt}}\), though \({\text{Synt}}\) varied only in lean rats. These were assumed to be log-normally distributed in order to keep the parameter values positive. However, the five-day continuous infusion group of obese rats did not have exposure data. Consequently, these rats were assumed to behave like the estimated median individual. The residual variability was normally distributed and modeled using a proportional error model.
Estimated parameters
Because of sparse sampling, all parameter values could not be estimated from the data. By applying an a priori sensitivity analysis [28, 29], we identified the parameters that had the greatest influence on the output. These were then estimated from the data and the remaining parameters were obtained from the literature [23]. The population parameters estimated from the data were \(k_{\text{a}}\), \(\delta\), and \(V_{\text{max1}}\).
Insulin turnover model
The primary aim of the insulin model was to establish smooth trajectories that would accurately describe the insulin-time courses under various provocations of NiAc, rather than describe all of the mechanistic aspects of insulin dynamics. To this end, the model structure was kept as simple as possible. The insulin model could subsequently be used to provide an input to the FFA model, enabling a quantitative analysis of the antilipolytic effects of insulin. Given this premise, a phenomenologically based modeling approach was applied. Under the assumption that NiAc perturbs insulin, the characteristics seen in the data were used to establish an insulin model with NiAc as input. The characteristic behavior of the data for acute and long-term NiAc provocations in lean and obese rats is illustrated in Fig. 3. Attributes seen include indirect action, tolerance (drug resistance), rebound, and complete adaptation (insulin levels returning to pre-treatment levels). Data with similar properties as those seen in the acute experiments (Fig. 3a, c) were modeled using turnover equations with moderator feedback control [14, 30]. Furthermore, to capture the different long-term adaptive behaviors with (Fig. 3b), and without (Fig. 3d) rebound, a ’NiAc action compartment’ was included, as well as an integral feedback control. The insulin dynamics are given by
$$\begin{aligned} \frac{\text d{I(t)}}{\text d{t}}&= k_{\text{inI}}\cdot R_{\text{I}}(t) \cdot H_{\text{NI}}(C_{\text{p}}(t)) \cdot \frac{M_{\text{0I}}}{M_{\text{1I}}(t)} \nonumber \\&\quad- k_{\text{outI}} \cdot \frac{M_{\text{2I}}(t)}{M_{\text{0I}}} \cdot I(t), \end{aligned}$$
(7)
$$\begin{aligned} \frac{\text d{M_{\text{1I}}(t)}}{\text d{t}}&= k_{\text{tolI}}\cdot \left( I(t) - M_{\text{1I}}(t) \right) , \end{aligned}$$
(8)
$$\begin{aligned} \frac{\text d{M_{\text{2I}}(t)}}{\text d{t}}&= k_{\text{tolI}}\cdot \left( M_{\text{1I}}(t) - M_{\text{2I}}(t) \right) , \end{aligned}$$
(9)
with initial conditions
$$\begin{aligned} I(0) = I_0, \end{aligned}$$
(10)
and
$$\begin{aligned} M_{\text{1I}}(0)=M_{\text{2I}}(0)=M_{\text{0I}} = I_0, \end{aligned}$$
(11)
where I(t) denotes the observed insulin level, and \(M_{\text{1I}}(t)\) and \(M_{\text{2I}}(t)\) the first and second moderator compartments, respectively. The parameters \(k_{\text{inI}}\) and \(k_{\text{outI}}\) are the turnover rate and fractional turnover rate of insulin, respectively, and \(k_{\text{tolI}}\) is the fractional turnover rate of the moderators. The regulator compartment \(R_{\text{I}}(t)\) is given by
$$\begin{aligned} \frac{\text d{R_{\text{I}}(t)}}{\text d{t}} =k_{\text{inRI}} - k_{\text{outRI}}\cdot I(t),\quad R_{\text{I}}(0)=1, \end{aligned}$$
(12)
where \(k_{\text{inRI}}\) is the turnover rate, \(k_{\text{outRI}}\) the fractional turnover rate, and I(t) the insulin concentration. The regulator compartment is initially at steady-state with
$$\begin{aligned} \frac{\text d{R_{\text{I}}(0)}}{\text d{t}} =k_{\text{inRI}}-k_{\text{outRI}}\cdot I_0 = 0 \iff I_0 = \frac{k_{\text{inRI}}}{k_{\text{outRI}}}. \end{aligned}$$
(13)
By integrating Eq. 12, the dynamics of \(R_{\text{I}}(t)\) can be expressed as
$$\begin{aligned} R_{\text{I}}(t)=1+\int _0^t k_{\text{inRI}} - k_{\text{outRI}}\cdot I(\tau )\,{\text{d}}\tau . \end{aligned}$$
(14)
Hence, by construction, \(R_{\text{I}}(t)\) represents the output of an insulin-driven integral feedback controller [31] with \(I_0\) as the set-point and \(k_{\text{outRI}}\) as the integral gain parameter (\(k_{\text{outRI}}\) will from here on be referred to as the integral gain parameter). The integral feedback controller will ensure that insulin levels return to the baseline \(I_0\), despite persistent external effects on insulin turnover and fractional turnover. The inhibitory NiAc function on insulin is given by
$$\begin{aligned} H_{\text{NI}}(C_{\text{p}}(t)) = 1 - E_{\text{NI}}(N_{\text{I}}(t))\cdot \frac{ C_{\text{p}}^{n}(t)}{ IC _{\text{50NI}}^{n}+C_{\text{p}}^n(t)}, \end{aligned}$$
(15)
where \(IC _{\text{50NI}}\) is the potency of NiAc on insulin and n the Hill coefficient of the inhibitory function. The term \(E_\text{NI}(N_{\text{I}}(t))\) represents the drug efficacy, which is fixed for lean rats and dependent on the concentration in a hypothetical NiAc action compartment, \(N_{\text{I}}(t)\), for obese rats, according to
$$\begin{aligned} E_{\text{NI}}(N_{\text{I}}(t)) = {\left\{ \begin{array}{ll} I_{\text{maxNI}} &{\text{lean}}\\ I_{\text{maxNI}}\left( 1-\frac{S_{\text{NI}}\cdot N_{\text{I}}^{\gamma }(t)}{ N _{\text{50I}}^\gamma +N_{\text{I}}^\gamma (t)}\right) &{\text{obese}}, \end{array}\right. } \end{aligned}$$
(16)
where \(I_{\text{maxNI}}\) is the initial efficacy of NiAc on insulin, \(N _{\text{50I}}\) the potency of the NiAc action compartment, \(S_{\text{NI}}\) the long-term NiAc efficacy loss, and \(\gamma\) the corresponding Hill coefficient of the efficacy relation. The dynamics of \(N_{\text{I}}\) are in turn given by
$$\begin{aligned} \frac{\text d{N_{\text{I}}(t)}}{\text d{t}} = k_{\text{NI}}\cdot (C_{\text{p}}(t)-N_{\text{I}}(t)), \end{aligned}$$
(17)
with \(N_{\text{I}}(0)=C_{\text{p}}(0)\). Here \(k_{\text{NI}}\) is the turnover rate of the NiAc action concentration.
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.
Between-subject, inter-study, and residual variability
Individual variations seen in the insulin data were incorporated in the model by allowing the parameters \(I_0\), \(k_{\text{tolI}}\), and \(IC_{\text{50NI}}\) to vary in the population. As in the PK model, these parameters were assumed to be log-normally distributed. The choice of these parameters was guided by an a priori sensitivity analysis. Moreover, the parameters \(I_0\) and \(k_{\text{tolI}}\) varied over study groups according to fixed-study effects on both the mean and individual parameter distributions [25]. In other words, for S, the number of groups, the parameter \(I_0\) for an individual j was modeled as
$$\begin{aligned} I_{0j} = (I_{01}\cdot {\text{Study}}_1 + \ldots + I_{0S}\cdot {\text{Study}}_S)\cdot \end{aligned}$$
(18)
$$\begin{aligned} {\text{exp}}(\eta _1\cdot {\text{Study}}_1 + \ldots + \eta _S\cdot {\text{Study}}_S), \end{aligned}$$
(19)
where \({\text{Study}}_k=1\) if individual j is in group k and 0 otherwise. The residual variability was modeled using an additive model (with normally distributed errors).
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
$$\begin{aligned} \frac{\text d{F(t)}}{\text d{t}}&= k_{\text{inF}} \cdot R(t) \cdot H_{\text{NF}}(C_{\text{p}}(t)) \cdot \frac{M_{\text{0F}}}{M_{\text{F}}(t)} \nonumber \\&- k_{\text{outF}}\cdot F(t), \end{aligned}$$
(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.
Between-subject, inter-study, and residual variability
Random effects were again selected using an a priori sensitivity analysis. The parameters that varied in the population were \(F_0\), \(k_{\text{tolF}}\), and \(IC _{\text{50NF}}\) (according to a log-normal distribution). Moreover, inter-study variability was incorporated in the model according to a fixed-study effect (as described for the insulin model). The parameters that varied between experimental groups were \(F_0\) and \(k_{\text{tolF}}\). The residual variability was modeled using an additive model (with normally distributed errors).
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).
Identifiability analysis
All population model structures analyzed in this study were proven to be structurally locally identifiable in a fixed effects setting (identifiability of the population model (fixed effects) implies identifiability of the statistical model (random effects) [34]). The identifiability analysis was performed using the Exact Arithmetic Rank (EAR) approach [35–37]—implemented in the IdentifiabilityAnalysis Wolfram Mathematica package, developed by the Fraunhofer-Chalmers Centre. The EAR algorithm requires that all states and system parameters are rational functions of their arguments. This requirement is not fulfilled in the insulin and FFA systems (for example, the state space variable \(C_{\text{p}}\) is raised to the power of n in Eq. 15). An illustrative example of how this requirement can be achieved is provided in the Appendix 3.
Selection of random effect parameters
An a priori sensitivity analysis was used to guide selection of the random parameters [28, 29]. The system output sensitivity, with respect to the parameters, was analyzed and the parameters were ranked accordingly. The parameters with the highest sensitivity, given by the absolute value of the partial derivative of the system output with respect to a specific parameter evaluated at a given point in the parameter space, were considered random in the model.
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