Study design and work-flow

The study design and work-flow is visualised in Fig 1 and detailed below. In summary, three existing models of postprandial adipose tissue glucose, fatty acid, and triglyceride dynamics were evaluated using arteriovenous measurements across the abdominal subcutaneous adipose tissue. Subsequently, a refined model was constructed by either selecting the best fitting model term of the three existing models or, in cases where none of the existing models allowed for an adequate fit to the data, new or modified physiological mechanisms were introduced. Finally, model parameters were estimated before and after weight loss in the Yoyo study [17] and compared, taking their confidence intervals into account.

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Fig 1. Work-flow for comparison of existing model terms and construction of refined model.

Jelic, Pratt, and Sips model terms are compared for each metabolite flux separately using the postprandial arteriovenous tracer measurements from the Yoyo study. Existing model terms are first evaluated based on their ability to describe the experimentally measured data. In addition, as the refined model is constructed with the aim that parameters could be estimated directly from measured data, existing model terms are also compared using identifiability analysis, using the Profile Likelihood, sensitivity analysis, and statistical evaluation using Akaike Information Criterion. The refined model of adipose tissue metabolism is constructed either by selecting the best fitting of the existing model terms, or by introducing modifications and, where necessary, novel physiological mechanisms. The resulting refined model is then parameterised using data from both the baseline and following weight stabilisation from the Yoyo study, and the values compared.

https://doi.org/10.1371/journal.pcbi.1007400.g001

Yoyo study and A-V measurements

The Yoyo study is a dietary intervention study designed to investigate the effect of the rate of weight loss (fast or slow) on subsequent weight regain [18]. Sixty-one overweight (BMI > 25kg/m²) and obese (BMI > 30kg/m²) individuals (BMI 28-35 kg/m²) were randomly assigned to one of two diet interventions designed to achieve a weight loss of approximately 10%, a low calorie diet of 1250 kcal/day over twelve weeks (slow weight loss) or a very low calorie diet of 500 kcal/day for five weeks (rapid weight loss). Following the diet intervention all participants underwent a four week weight stabilisation period with a diet based on the energy requirements of each participant. As part of this study, sixteen individuals underwent a high fat mixed-meal challenge test (milk shake) containing 100 mg of [U-¹³C] palmitate tracer at baseline and following the weight stabilisation period [17]. Fasting samples were collected at -30 and 0 minutes. The meal was consumed at 0 minutes, participants were asked to consume the shake within 10 minutes, and samples were collected at 60, 120, 180, 240, 300 minutes postprandially from an arterialised dorsal hand vein and the superficial epigastric vein draining the abdominal subcutaneous adipose tissue. Abdominal subcutaneous adipose tissue blood flow was measured using the ¹³³Xe washout technique [19, 20].

Ethics statement

All subjects gave their written informed consent before participation in the Yoyo Study. The Yoyo Study was performed according to the Declaration of Helsinki and was approved by the Medical Ethics Committee of Maastricht University Medical Centre (METC 11-3-066) with the approval number NL38099.068.11.

Calculations

Individual metabolite fluxes across the abdominal subcutaneous adipose tissue were computed for NEFA, triglyceride, glucose, and glycerol by multiplying the arterio-venous plasma concentration difference of each measured metabolite by the adipose tissue blood flow [17].

Plasma tracer concentrations in NEFA and TG fractions were computed as described in the original study [17]. The rate of fractional spill-over of NEFA derived from LPL lipolysis of chylomicron triglyceride was calculated as one minus the rate of fractional extraction as described by Bickerton et al. [5]

Two indices of insulin resistance were calculated using the arterial metabolite measurements. HOMA-IR is a measure of whole body insulin resistance, and is calculated as fasting plasma glucose(mmol/l) times the fasting plasma insulin (μ U/ml) divided by 22.4 [21]. Several studied have evaluated appropriate HOMA-IR cutt-off values for determining insulin resistance, with identified cut-off calues ranging from 1.85 to 2.01 depending on the study population [22]. ADIPO-IR is a surrogate measure of adipose tissue specific insulin resistance and is calculated as fasting plasma NEFA concentration (mmol/l) times the fasting plasma insulin concentration (pmol/l) [23].

Existing models

Models of postprandial lipid metabolism were assessed to identify models that described insulin mediated adipose tissue specific dynamics of glucose, triglyceride, NEFA, and/or glycerol. Three published models describing complementary aspects of adipose tissue metabolism were included in the analysis, namely the Jelic [11], Pratt [13], and Sips [10] models. The Pratt Model is a large, multi-compartmental, computational model describing the dynamics of several metabolite species including glucose, NEFA, glycerol, pyruvate, and both endogenous and dietary triglyceride in the fasting and postprandial states in plasma, skeletal muscle, adipose tissue, and liver [13]. The Jelic model is a two compartment, physiology-based, mathematical model describing NEFA dynamics across the plasma and a lumped interstitial adipose space in the postprandial state [11]. The Jelic model also accounts for two physiologically relevant delays in insulin signalling not present in the Pratt model. The model parameters for both the Jelic and Pratt models have been taken from literature and both models have been validated using oral glucose tolerance test (OGTT) and mix meal challenge test data in lean and abdominally obese individuals. The Sips model [10] is an extension of the Dalla-Man model [24] of postprandial glucose-insulin interplay to include NEFA kinetics. Unlike the previous two models, all parameters in the Sips model have been estimated from data. Where possible parameter values were maintained at the values provided for the lean healthy individual in the Dalla-Man model, as these parameters had been validated using gold standard triple-glucose tracer data [20]. The remaining 21 parameters were estimated from experimental data consisting of plasma measurements of several metabolites from individuals under various clamp conditions [25, 26] and also a frequently sampled oral glucose tolerance test (OGTT) and oral lipid tolerance test [10].

Model evaluation and refinement

The three existing models (Jelic, Pratt, Sips) were decomposed into relevant subunits describing each measured metabolite’s dynamics across and within the adipose tissue using a so-called divide and conquer approach [27] (Fig 1, S1 Table). These model terms were then evaluated by comparison to the calculated metabolite flux from the A-V data (average μ = (μ₁,…, μ₇) and standard deviation σ = (σ₁,…, σ₇)) according to the following error function C(p): (1) Here M(p, t_i) is the model prediction at time t_i, t = (−30, 0, 60, 120, 180, 240, 300), for a parameter set p.

Optimal parameter sets for each term were obtained through non-linear regression, by finding the parameter set which best described the measured flux, minimising the error term C(p). Optimisation of parameters was performed using lsqnonlin (MATLAB 2014b, The MathWorks Inc., Natick, Massachusetts, United States) a local, gradient-based least square solver. The parameter values supplied in the original model studies were used as initial values for the parameter search. Parameterisation of the final, refined model for comparison of baseline and following weight stabilisation was performed using a combination of a global and local search algorithm. Controlled Random Search [28] with 250 randomly selected initial parameter sets was utilised to search the parameter space in order to provide a good initial value for lsqnonlin.

Terms describing the uptake and release of metabolites from tissues other than the adipose tissue, for example appearance from the gut and insulin secretion, are not described in this model. Available measured arterial concentrations of insulin, glucose, NEFA, triglyceride, and glycerol are supplied as dependent inputs to the model terms [29].

In the cases when the existing models were not capable of describing the experimentally estimated fluxes, underlying assumptions and model terms were evaluated based on existing biological knowledge and modified accordingly to provide an improved description of postprandial metabolite dynamics across the adipose tissue.

Additionally, all model terms were examined statistically, using the Akaike Information Criterion (AIC_c) corrected for small sample sizes [30], to select the most parsimonious model that is both biologically sound and can adequately describe the measured flux data.

The above procedure was performed to evaluate terms describing the triglyceride, glucose, glycerol, and NEFA fluxes across the adipose tissue. Terms describing the fractional spill-over of NEFA by LPL lipolysis could also be assessed for the first time using the [U-¹³C] palmitate tracer data.

Parameter identifiability

Model terms were also evaluated for the identifiability of parameters, as a primary objective of this study was to provide a model which could be parameterised by experimental data and could therefore be used to quantify adipose tissue metabolism from meal challenge test data. Identifiability of model parameters was evaluated using Profile Likelihood Analysis [31], whereby one parameter was varied iteratively from its optimal value and the remaining parameters were re-estimated. For an identifiable parameter the error measure C(p) would be expected to increase as the parameter value deviated from its optimum [31, 32]. 95% confidence intervals for the estimated parameter values have been estimated using the parameter value covariance matrix approximated using the Jacobian matrix provided as output of the lsqnonlin algorithm used in the parameter estimation procedure.

Triglyceride flux

Circulating plasma triglyceride is hydrolysed by LPL at the endothelial wall releasing NEFA and glycerol. The Pratt model describes this reaction as being proportional to the concentration of triglyceride present in circulating lipoproteins (both chylomicron and VLDL) and occurring at a basal and insulin stimulated rate using linear terms [13], with the plasma insulin concentration having an instantaneous effect. The Jelic model accounts for the saturation of enzyme mediated LPL lipolysis using Michaelis-Menten kinetics, with the rate of hydrolysis of circulating triglyceride being dependent on the plasma triglyceride concentration and stimulated by a delayed insulin effect [11]. The transcription of LPL and subsequent secretion of the LPL protein to the endothelial wall are known to be stimulated by insulin, however, these processes take some time [4]. As a result, the Jelic model introduced a three-fold insulin delay term, with the time delay parameter set to 240 mins [11]. The Sips model makes use of the Jelic model term and parameters [10].

Using the parameter values as specified in the original publications, neither the Jelic nor Pratt models are capable of describing the mean measured triglyceride flux across the adipose tissue (Fig 3A). Allowing the parameters to be estimated from the data, the Jelic model provides an improved fit. A more detailed investigation demonstrated that the insulin delay is the essential component missing in the Pratt model (S2 and S3 Figs). As a result, the optimised triglyceride flux term ([TG]_flux) shown below was derived, making use of the insulin dependent linear approximation from the Pratt model while introducing the insulin delay term of the Jelic model. (2) Where K_ad is the rate parameter for LPL lipolysis, [TG_art] is the arterial triglyceride concentration and [I_LPL] is the delayed insulin signal, modelled with a three compartmental delay (Eq 12) with a time delay parameter τ_LPL also estimated from data.(Fig 4)

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Fig 3. Comparison of the Jelic, Pratt, Sips, and refined model fit to baseline flux data from the Yoyo study.

The fit of the refined model to the baseline adipose tissue flux data from the Yoyo Study is shown along with the terms from the Jelic, Pratt, and Sips models simulated using the parameter values provided in the respective publications. (A) Model terms from the Pratt (green), Jelic (blue), Sips (yellow), and refined model (red) describing the postprandial LPL mediated lipolysis of circulating triglycerides are shown. (B) Terms describing the fractional spill over of LPL derived NEFA from the Pratt, Sips, and refined adipose tissuse models are shown. The mean fractional spill over values, calculated using the postprandial palimate [U-¹³C] tracer included in the meal (black crosses ± standard error of mean). (C) Model terms describing the uptake of glucose into the adipose tissue from the Pratt and refined model are shown. (D) Depicts terms describing the postprandial efflux of glucose from the Pratt and refined models. (E) Model terms describing the efflux NEFA from the abdominal subcutaneous adipose tissue from the Pratt, Sips, Jelic, and refined models are shown. Metabolite fluxes are calculated as the arteriovenous difference in a metabolite across the adipose tissue multiplied by the rate of postprandial adipose tissue blood flow. The mean calculated adipose tissue fluxes are shown with the black crosses ± the standard error of the mean. Negative flux values indicate a net release of the metabolite from the adipose space, positive values indicate a net uptake.

https://doi.org/10.1371/journal.pcbi.1007400.g003

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Fig 4. Fitting of the refined model to adipose tissue flux data measured at baseline and following caloric restriction from the Yoyo study.

The refined model was fit to postprandial adipose tissue flux measurements of (A) LPL mediated lipolysis of circulating triglycerides, (B) the fractional spill over of LPL derived NEFA, (C) infflux of glucose, and efflux of both (D) glycerol, and (E) NEFA to the adipose tissue at baseline and following a period of caloric restriction. Blue crosses indicate the mean adipose tissue flux at baseline, the red crosses show the mean flux following caloric restriction, error bars indicate the standard error of the respective means. Refined model prediction at baseline is shown in blue, and the model fit following the diet intervention is shown in red. Negative flux values indicate a net release of the metabolite from the adipose space, positive values indicate a net uptake.

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Fractional spill-over

NEFA released by LPL lipolysis of circulating triglyceride is taken up by the adipose tissue. However, this process is inherently leaky with a proportion of the released NEFA spilling-over directly into the plasma [33, 34]. Arteriovenous studies combined with NEFA stable isotope tracers have previously demonstrated that the fractional spill-over of NEFA from LPL lipolysis increases later in the postprandial period [5, 33].

The Jelic model does not describe the spill-over of NEFA derived from LPL lipolysis [11]. The Pratt model describes fractional spill-over at a constant rate of 25% [13]. The Sips model describes the fractional spill-over of LPL derived NEFA as occurring at a basal and insulin inhibited rate, using the same delayed insulin signal as LPL lipolysis with a 240 min time delay parameter [10].

The Sips term could not describe the increasing postprandial fractional spill-over measured in the Yoyo Study data (Fig 3B). The 240 min insulin time delay served to over-damp the insulin signal. As a result, the insulin delay was removed. Further analysis of the Sips term with use of the Akaike Information Criterion led to the removal of the basal fractional spill-over rate, yielding the following optimised fractional spill-over term. (3) Here I_B is the basal plasma insulin concentration, [I_art] the arterial insulin concentration and D_spill an estimated constant (Fig 4B).

Glucose flux

Glucose enters the adipose tissue along the concentration gradient facilitated by insulin dependent (GLUT4) and insulin independent (GLUT1) transporters [35]. Within the adipocyte, glucose is quickly converted to glucose-6-phosphate (G-6-P) which serves to trap the glucose within the adipocyte. Glucose is the primary source of the glyceraldehyde-3-phosphate (G-3-P) backbone needed for re-esterification of adipocyte NEFA [36].

Both the Pratt [13] and the Sips model [10] describes the uptake of glucose by the adipose tissue as occurring at an insulin dependent and independent rate. However, the Pratt model uses direct plasma insulin stimulation rather than accounting for a delay in the effect of insulin signalling which induces the translocation of GLUT4 transporters from the transport vesicles of the cell to the membrane. The Jelic model does not account for glucose dynamics.

With the introduction of a time delay in insulin stimulation the linear term from the Pratt model was capable of describing the glucose flux. In addition, it was necessary to explicitly account for the conversion of glucose to G-6-P in order to remedy an erroneous prediction of a postprandial glucose efflux from the interstitial adipose space (S3C Fig). (4) Where [G_art] is the arterial glucose concentration, [I_AT] is the adipose tissue delayed insulin signal described using a threefold delay with a time delay parameter τ_AT(Eq 12), GLUT1 and GLUT4 are constants estimated from the data (Fig 4C).

Glycerol flux

Glycerol is released into the plasma by LPL lipolysis of circulating triglyceride at the endothelial wall [35]. Glycerol is also released by lipolysis of triglyceride stored within the adipose tissue (denoted as ATL lipolysis [37]). It is commonly assumed tht due to the inactivity of glycerol kinase in the adipose tissue [38], all glycerol released by ATL lipolysis enters the plasma for transportation to the liver. Postprandial glycerol dynamics are not accounted for in the Jelic [11] nor Sips models [10]. The Pratt model accounts for the release of glycerol by ATL lipolysis within the adipose tissue which appears directly in the liver compartment [13]. Glycerol release into plasma by LPL lipolysis is not described in this model.

Analysis of the postprandial glycerol and triglyceride fluxes from the YoYo Study indicate the uptake of glycerol into the interstitial adipose space in the postprandial state (Fig 5). As a result, the Pratt glycerol term was extended to account for the concentration gradient diffusion of plasma glycerol into and from the interstitial adipose compartment. (5) (6) Where ATL lipolysis is described as occurring at a basal (B_ATL) and saturable insulin inhibited rate using terms from the Jelic model (ATL_max, K_ATL) [11]. Where [I_AT] is the delayed interstitial adipose compartment insulin signal as described for the glucose flux above. (Figs 3D and 4D)

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Fig 5. Postprandial uptake of glycerol by adipose space.

A) Mean postprandial triglyceride influx (blue) (equivalent to the release of glycerol by LPL lipolysis (red)) and total glycerol efflux (yellow) across the abdominal subcutaneous adipose tissue at baseline in the Yoyo study (± standard error of the mean). B) Mean (± standard error of the mean) of glycerol flux minus triglyceride flux in postprandial phase measured at baseline in the Yoyo study indicating a net uptake of glycerol by the abdominal subcutaneous adipose tissue in the postprandial condition (green shaded region). This is in contradiction to the commonly held assumption that the glycerol efflux equals the direct sum of glycerol released by LPL lipolysis of circulating triglyceride and glycerol released by ATL lipolysis within the adipose tissue.

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NEFA flux

In the fasting state NEFA is released by the lipolysis of stored triglyceride within the adipocyte and passes into the plasma for delivery to other tissues. The lipolysis of stored triglyceride is inhibited by increasing insulin concentration in the postprandial state [39, 40] which also stimulates the rate of re-esterification of free NEFA within the adipocyte for storage as triglyceride [41]. In addition, insulin stimulates the LPL lipolysis of circulating triglyceride, removing excess dietary triglyceride from the plasma. While the majority of NEFA released by LPL lipolysis passes into the adipocyte for re-esterification a fraction spills-over directly into the plasma [33, 34].

Terms describing the fractional spill-over, LPL and ATL lipolysis have previously been derived. Terms describing the rate of re-esterification were compared using the Yoyo study NEFA flux. The Pratt model describes the rate of re-esterification as a linear term, directly stimulated by plasma insulin and adipose glucose concentration [13]. The Jelic model uses saturable kinetics coupled with a delayed insulin stimulation [11]. Both models used passive diffusion of NEFA between the plasma and interstitial adipose compartment. The Sips model does not explicitly describe the adipose compartment, instead using a linear term which accounts for net lipolysis and re-esterification of NEFA within the adipose tissue [10].

While the Jelic term could produce a good describtion of the NEFA flux when fit to the data (S2 Fig), it assumed an unlimited supply of G-3-P and required the estimation of five parameters from the data (S1 Table). The linear re-esterification term of the Pratt model was modified to include the delayed interstitial adipose compartment insulin signal. In addition, terms explicitly accounting for the production of the G-3-P backbone necessary for re-esterification from glucose taken into the adipose tissue were introduced, providing a sufficient description of the measured postprandial NEFA flux (Figs 3E and 4E). (7) (8) With .

Where the adipose tissue concentration of G-3-P ([G-3-P_AT]) calculated using the following equations. (9) (10) (11) Re-esterification occurs at an insulin stimulated linear rate K_reester dependent on the available concentrations of G-3-P and NEFA. A portion of the glucose taken up by the adipose space (frac_use) is converted to G-6-P and then to adipose tissue G-3-P ([G-3-P_AT]) which will be available for use in re-esterification, the remaining glucose leaves the system for use by other cellular functions which are not explicitly described here. The production of adipose tissue G-3-P is described by the term [G-3-P_pro] with a two compartmental delay governed by the delay constant τ_G-3-P.

Insulin delays

Insulin stimulates the translocation, secretion, and in some cases transcription of several enzymes and transport proteins involved in adipose tissue metabolism. These processes occur over time spans of several minutes to several hours. The effectiveness of the measured plasma insulin signal may be further damped by the presence of insulin resistance, a condition where the responsiveness of tissues to plasma insulin is reduced. Rather than describing the full sequence of reactions involved the dampening of the plasma insulin signal is approximated using a three compartmental delay, as in the Jelic model [11]. (12) (13) (14) Where [I_art] is the measured arterial insulin concentration and τ is the respective time delay parameter. Two insulin time delays are simulated, a long time delay for LPL lipolysis (τ = τ_LPL) and a shorter interstitial adipose compartment time delay (τ = τ_AT). The structure of the insulin delay terms are not altered from the original model [11]. However, the time delay parameters are not fixed, as in the Jelic and Sips models, and are estimated from the data.

Application of adipose model—Quantification of caloric restriction

The resulting refined model was then used to quantify the effect of caloric restriction using meal challenge test data collected before and after a diet intervention as part of the Yoyo study.

Parameter values before and after caloric restriction.

Parameter sets were estimated from data collected at baseline and after a weight stabilisation period following caloric restriction (Table 1, the complete parameter sets are provided in S2 Table). The rate parameters for both glycerol and NEFA concentration gradient based diffusion from the plasma to the adipose space (P_GLY and P_NEFA) increase significantly following caloric restriction, indicating an increase in the rate of glycerol and NEFA transport in and out of the adipose tissue. A large, although not significant, decrease can be seen in the insulin delay parameter for LPL lipolysis of circulating triglyceride, τ_LPL, (157 mins at baseline to 113 mins following weight stabilisation) which is not accompanied by a strong change in the value for K_ad, the rate parameter for that term. In addition, there is a four minute decrease in the adipose tissue insulin delay parameter τ_AT. The reduction in the insulin delay parameters indicates a less damped response to insulin signalling following caloric restriction, indicative of improved insulin sensitivity.

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Table 1. Parameter values estimated for data collected at baseline and following caloric restriction.

https://doi.org/10.1371/journal.pcbi.1007400.t001

Decomposition of glycerol and NEFA model predictions.

The estimated values for P_GLY and P_NEFA increased significantly following caloric restriction, but it remained unclear if the observed differences in the glycerol and NEFA fluxes following caloric restriction (Fig 4) could be entirely attributed to these increases in the rates of uptake and release of both metabolites. In Fig 6, our refined model’s predictions are used to decompose the calculated adipose glycerol and NEFA fluxes over the duration of the mixed meal challenge test into their constituent reactions. This allows for visualisation of how the rates of specific, unmeasured reactions change following caloric restriction. The model predicts an increase in the rate of ATL lipolysis within the adipose tissue following caloric restriction (Fig 6B and 6E, red lines), with a predicted 56% increase in the rate of lipolysis of stored triglyceride (0.229 μmol/100 ml tissue/min at baseline to 0.359 μmol/100 ml tissue/min following caloric restriction), resulting in the observed increase in efflux of both NEFA and glycerol from the adipose tissue in the fasting state (Fig 6A and 6D, purple lines). An increase in ATL lipolysis would be expected following such a period of caloric restriction as the triglyceride stores in the adipose tissue are hydrolysed to supply NEFA for use in other tissues. In the later postprandial phase, the lower concentrations of circulating triglyceride follow caloric restriction (total AUC 642.2 μmol a baseline to just 463 μmol following the diet intervention), results in the reduced efflux of glycerol and NEFA from LPL lipolysis after 120 minutes (Fig 6A and 6D, yellow lines).

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Fig 6. Decomposition of model glycerol and NEFA flux predictions into constituent reactions.

Comparison of model predictions of rates of reactions contributing to the (A) glycerol flux in the plasma compartment, (B) rates of glycerol rate of reactions in adipose compartment, (C) plasma and adipose compartment glycerol concentrations, (D) NEFA flux in the plasma compartment, (E) rates of NEFA reaction in adipose compartment, and (F) plasma and adipose compartment NEFA concentrations. Solid lines represent baseline estimates and dashed lines represents estimates following weight stabilisation.

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Interestingly, the peak in the rate of LPL occurs approximately 40 mins earlier (230 mins at baseline versus 190 mins following caloric restriction) reflecting the estimated decrease in τ_LPL (Fig 6A and 6E, yellow lines). Note, there is no decrease in the peak time of measured arterial insulin (S4E Fig), however there is a decrease in both the fasting and postprandial plasma insulin concentrations (total AUC 6821.5 μU of insulin at baseline to 5291.8 μU following caloric restriction). Despite the decrease in circulating concentrations of insulin the postprandial inhibition of ATL lipolysis and stimulation of re-esterification by insulin occur at faster rates. The magnitude of slope describing the rate of postprandial inhibition of ATL lipolysis increases from 0.019 at baseline to 0.0023 following caloric restriction. Concurrently, the rate of stimulation of re-esterification increases from 0.0055 at baseline to 0.0078 following the diet intervention. These increases are indicative of improved responsiveness to circulating insulin, as reflected with the decrease in the estimated values for both insulin delay parameters.