Abstract
We have separated the effect of insulin on glucose distribution/transport, glucose disposal, and endogenous production (EGP) during an intravenous glucose tolerance test (IVGTT) by use of a dual-tracer dilution methodology. Six healthy lean male subjects (age 33 ± 3 yr, body mass index 22.7 ± 0.6 kg/m^{2}) underwent a 4-h IVGTT (0.3 g/kg glucose enriched with 3–6% d-[U-^{13}C]glucose and 5–10% 3-O-methyl-d-glucose) preceded by a 2-h investigation under basal conditions (5 mg/kg ofd-[U-^{13}C]glucose and 8 mg/kg of 3-O-methyl-d-glucose). A new model described the kinetics of the two glucose tracers and native glucose with the use of a two-compartment structure for glucose and a one-compartment structure for insulin effects. Insulin sensitivities of distribution/transport, disposal, and EGP were similar (11.5 ± 3.8 vs. 10.4 ± 3.9 vs. 11.1 ± 2.7 × 10^{−2}ml · kg^{−1} · min^{−1} per mU/l;P = nonsignificant, ANOVA). When expressed in terms of ability to lower glucose concentration, stimulation of disposal and stimulation of distribution/transport accounted each independently for 25 and 30%, respectively, of the overall effect. Suppression of EGP was more effective (P < 0.01, ANOVA) and accounted for 50% of the overall effect. EGP was suppressed by 70% (52–82%) (95% confidence interval relative to basal) within 60 min of the IVGTT; glucose distribution/transport was least responsive to insulin and was maximally activated by 62% (34–96%) above basal at 80 min compared with maximum 279% (116–565%) activation of glucose disposal at 20 min. The deactivation of glucose distribution/transport was slower than that of glucose disposal and EGP (P < 0.02) with half-times of 207 (84–510), 12 (7–22), and 29 (16–54) min, respectively. The minimal-model insulin sensitivity was tightly correlated with and linearly related to sensitivity of EGP (r = 0.96, P < 0.005) and correlated positively but nonsignificantly with distribution/transport sensitivity (r = 0.73, P = 0.10) and disposal sensitivity (r = 0.55, P = 0.26). We conclude that, in healthy subjects during an IVGTT, the two peripheral insulin effects account jointly for approximately one-half of the overall insulin-stimulated glucose lowering, each effect contributing equally. Suppression of EGP matches the effect in the periphery.
- glucose kinetics
- compartment modeling
- d-[U-^{13}C]glucose
- 3-O-methyl-d-glucose
- insulin action
- glucose transport
- glucose disposal
- endogenous glucose production
- intravenous glucose tolerance test
Glossary
New Model
- EGP_{0}
- Endogenous glucose production extrapolated to zero insulin concentration (mmol/min)
- EGP_{b}
- EGP at basal insulin concentration (mmol/min)
- F_{01}
- Total non-insulin-dependent glucose flux (mmol/min)
- g_{1}(t), g_{3}(t)
- Concentrations of d-[U-^{13}C]glucose and 3-O-methyl-d-glucose in the accessible compartment (mmol/l)
- G(t)
- Total glucose concentration in the accessible compartment (mmol/l)
- I(t), I_{b}
- Plasma insulin and basal (preexperimental) plasma insulin (mU/l)
- k_{03}
- Transfer rate constant of 3-O-methyl-d-glucose excretion (min^{−1})
- k_{12}
- Transfer rate constant from nonaccessible to accessible compartment (min^{−1})
- k_{a1}, k_{a2},k_{a3}
- Deactivation rate constants (min^{−1})
- k_{b1}, k_{b2},k_{b3}
- Activation rate constants (min^{−2} per mU/l)
- q_{1}(t),q_{2}(t)
- Masses of d-[U-^{13}C]glucose in the two compartments (mmol)
- q_{3}(t),q_{4}(t)
- Masses of 3-O-methyl-d-glucose in the two compartments (mmol)
- Q_{1}(t),Q_{2}(t)
- Masses of native glucose in the two compartments (mmol)
- Q_{10}
- Initial mass of native glucose in the accessible compartment (mmol)
- S_{IT}, S_{ID}, S_{IE}
- Insulin sensitivity of glucose distribution/transport, glucose intracellular disposal, and EGP (ml · min^{−1} · kg^{−1} per mU/l)
- u_{1}(t), u_{3}(t)
- Bolus doses of d-[U-^{13}C]glucose and 3-O-methyl-d-glucose administered at 0 and 120 min (mmol/min)
- U(t)
- Bolus dose of the unlabeled glucose administered at 120 min (mmol/min)
- V
- Distribution volume of the accessible compartment (liters)
- x_{1}(t),x_{2}(t),x_{3}(t)
- Remote effect of insulin on glucose distribution/transport, glucose disposal, and EGP, respectively (min^{−1})
Two-Compartment Minimal Model
- D
- Administered dose of d-[U-^{13}C]glucose at 120 min (mmol)
- F_{01}
- Constant component of glucose uptake (fixed at 1 mg · kg^{−1} · min^{−1})
- g(t)
- plasma concentration of d-[U-^{13}C]glucose (mmol/l)
- I_{b}
- Basal (postexperimental) insulin concentration (mU/l)
- k_{21}, k_{12},k_{02}
- fractional rate parameters (min^{−1})
- k_{a}
- Deactivation rate constant (min^{−1})
- k_{b}
- Activation rate constant (min^{−2} per mU/l)
- k_{p}
- Proportional term of glucose disposal (min^{−1})
- MCR
- Basal metabolic clearance rate of glucose (ml · kg^{−1} · min^{−1})
- q_{1}(t),q_{2}(t)
- Masses of d-[U-^{13}C]glucose in the two compartments (mmol)
- S
- Insulin sensitivity (ml · kg^{−1} · min^{−1} per mU/l)
- V
- Volume of the accessible compartment (liters)
- x(t)
- Remote insulin (min^{−1})
One-Compartment Minimal Model
- D
- (Total) glucose dose (mmol)
- G(t)
- Plasma concentration of total (labeled and unlabeled) glucose (mmol/l)
- G_{b}
- Basal (postexperimental) glucose concentration (mmol/l)
- I_{b}
- Basal (postexperimental) insulin concentration (mU/l)
- p_{1} = S_{G}
- Glucose effectiveness (min^{−1})
- p_{2}
- Deactivation rate constant (min^{−1})
- p_{3}
- Activation rate constant (min^{−2} per mU/l)
- S_{I}
- Insulin sensitivity (min^{−1} per mU/l)
- V
- Distribution volume (liters)
- x(t)
- Remote insulin (min^{−1})
insulin is a potent anabolic hormone, which activates metabolic pathways to regulate glucose metabolism and maintain homeostasis. Insulin stimulates glucose transmembrane transport and intracellular glucose disposal while also suppressing endogenous glucose production (EGP).
These pathways are measurable in particular tissues/organs by use of imaging methods such as nuclear magnetic resonance spectroscopy (18, 42) or multiple-tracer dilution techniques (11,39). Techniques such as the glucose clamp or the minimal-model analysis of an intravenous glucose tolerance test (IVGTT) provide whole body aggregated measures of insulin action (5) without the ability to separate the three effects, with the exception of estimating EGP and peripheral glucose uptake during a labeled IVGTT (15,27) or clamp (34, 35).
The sensitivity of glucose transport to insulin and the temporal pattern of its activation are currently unknown at the whole body level. Similarly, it is currently unknown to what extent the three pathways contribute to glucose lowering during dynamic conditions such as an IVGTT. Recent studies with mice indicate that a muscle-specific insulin receptor knockout does not alter glucose tolerance (12), raising questions about the relative importance of insulin-activated pathways.
The present study was designed to estimate simultaneously the effect of insulin on glucose distribution/transport, glucose disposal, and EGP during an IVGTT. The aim was to compare the three pathways both in terms of their sensitivities to insulin and in their abilities to lower plasma glucose concentration. We employed a dual-tracer technique with the administration ofd-[U-^{13}C]glucose, a stable-label tracer indistinguishable from native glucose, and 3-O-methyl-d-glucose, a marker of glucose transport. The data were analyzed using a novel model of glucose kinetics. Validation of an EGP estimate was facilitated by administering and reconstructing a variable infusion of another stable-label tracer,d-[6,6-^{2}H_{2}]glucose.
METHODS
Subjects and Experimental Protocol
Six healthy lean male subjects (age 33 ± 3 yr, body mass index 22.7 ± 0.6 kg/m^{2}; means ± SE) participated in the study, which was approved by the Ethics Committee, Guy's and St. Thomas' National Health Service Trust. Subjects provided written informed consent.
The subjects underwent a 4-h IVGTT preceded by a 2-h investigation of glucose kinetics under basal conditions. The study was carried out after an overnight fast. No medication was given, and the subjects followed their standard diet regimen before the study.
At ∼0830, a Venflon cannula was inserted into a vein in each antecubital fossa. One cannula was used for sampling and the other for the administration of glucose infusates, which were given via a Harvard pump (Harvard Instruments, Millis, MA) and an IVAC 560 pump (IVAC, San Diego, CA). The experiment commenced after a rest period of ∼30 min. The subjects remained supine during the studies and were allowed to sip water but otherwise had no oral intake.
At 0 min, an intravenous bolus was given ofd-[U-^{13}C]glucose (5 mg/kg body wt; Cambridge Isotopes Laboratories, Promochem, Herts, UK) and 3-O-methyl-d-glucose (8 mg/kg; Sigma-Aldrich, Gillingham, Dorset, UK).
At 120 min, an intravenous glucose bolus was administered (0.3 g/kg, 50% aqueous dextrose solution over 1 min) enriched withd-[U-^{13}C]glucose (subject 1: 20 mg/kg; subjects 2–6: 10 mg/kg; the dose was reduced, because satisfactory resolution of measurements was achieved with the lower dose and to limit costs) and 3-O-methyl-d-glucose (subject 1: 32 mg/kg, subjects 2–6: 16 mg/kg).
A variable, discontinuous (piecewise constant) intravenous infusion ofd-[6,6-^{2}H_{2}]glucose (range: 0–0.4, average: 0.2 mg · kg^{−1} · min^{−1}; MassTrace, Woburn, MA) with a time step of 5–30 min started at 90 min and continued until the end of the study. The purpose of the infusion was to validate calculations of EGP.
Samples were taken at −10, −5, 0, 4, 6, 8, 10, 12, 14, 16, 18, 20, 25, 30, 40, 50, 60, 80, 100, 105, 110, 115, 119, 122, 123, 124, 125, 128, 130, 132, 134, 136, 138, 140, 144, 148, 152, 160, 165, 170, 180, 190, 200, 210, 220, 230, 240, 260, 280, 300, 330, and 360 min. Samples were analyzed for insulin, C-peptide, glucose,d-[U-^{13}C]glucose, 3-O-methyl-d-glucose, andd-[6,6-^{2}H_{2}]glucose.
Assays
All samples were immediately centrifuged, separated, and stored at −20°C until assayed.
Mass spectrometry analysis.
Details of analysis have been published previously (41). In brief, plasma samples were derivatized to obtain volatile esters of penta-O-trimethylsilyl-d-glucose-O-methoxime by use of a modification of a method described by Laine and Sweeley (33). Gas chromatography was performed on a Hewlett-Packard model 5890 Series II with a Hewlett-Packard model 7673 autosampler (Hewlett-Packard, Woking, UK). Mass spectrometry analysis was performed with a Hewlett-Packard model 5971A mass-selective detector. In the selected-ion monitoring mode, the following ions were measured: mass-to-charge ratio (m/z) 261 (M + 0) and 264 (M + 3) for 3-O-methyl-d-glucose and the internal standard 3-O-methyl-d-[^{2}H_{3}]glucose (MassTrace); m/z 319 (M + 0), 321 (M + 2), 322 (M + 3) and 323 (M + 4) for the determination ofd-[6,6-^{2}H_{2}]glucose andd-[U-^{13}C]glucose.
Insulin, C-peptide, and glucose assay.
Sample tubes for glucose contained fluoride oxalate and those for C-peptide K-EDTA and Trasylol. Plasma glucose was measured using an enzymatic method on a Clandon glucose analyzer (Yellow Springs Instrument, Yellow Springs, OH) with a 1.5% within-assay coefficient of variation (CV). Plasma immunoreactive insulin and plasma C-peptide were measured by double-antibody radioimmunoassay techniques. The within-assay CVs were 6 and 5%, respectively.
Data Analysis
Tracer-to-tracee ratio and tracer concentrations.
The tracer-to-tracee ratio (TTR) was calculated on the basis of the work by Cobelli et al. (21), with further elaboration by Rosenblatt et al. (38). TTR represents a ratio of exogenously originating glucose to endogenously originating glucose in the sample. In the case of 3-O-methyl-d-glucose, the endogenous component is replaced by the internal standard.
The formulas to calculate TTR and concentrations ofd-[6,6-^{2}H_{2}]glucose,d-[U-^{13}C]glucose, and recycled glucose are given in appendix .
Modeling glucose kinetics.
The new model describes the basal period and the IVGTT. We used a two-compartment structure, which described adequately and simultaneously the kinetics ofd-[U-^{13}C]glucose, 3-O-methyl-d-glucose, and native glucose and reflects current physiological knowledge (1, 16). The accessible glucose compartment (where measurements are made) represents plasma and tissues that equilibrate quickly with plasma. It contains plasma distribution space and a portion of the interstitial distribution space (30). The nonaccessible compartment represents the slowly equilibrating pool and contains the remaining interstitial space and the intracellular distribution space.
It has been shown that insulin stimulates the transfer from the accessible to the nonaccessible compartment (inward transfer) but fails to stimulate the reverse (outward) transfer at insulin concentrations comparable to those observed during an IVGTT (20). This observation of the selective stimulation of inward transfer is consistent with the greater stimulation of inward transport across the cell membrane in human skeletal muscle (11), reflecting recruitment of glucose transporters. Insulin has also been shown to enhance vasodilatation of skeletal muscle vasculature (3) and to increase muscle blood flow and its dispersion (45). The stimulation of the inward transfer might therefore represent insulin action on glucose transport and glucose distribution, although studies with l-[^{14}C]glucose in rat (47) and dog (43) have shown that glucose distribution is not affected by insulin. Farther on in the text, we refer to the effect on inward intercompartmental transfer as the effect on distribution/transport. The inward intercompartmental transfer represents the transport across the endothelium, transport into tissues, potential recruitment of distribution space, and transport into cells by glucose transporters.
3-O-methyl-d-glucose is transported by the same specific transporters as native glucose, i.e., it has identical inward and outward transmembrane fractional rates but, crucially, does not undergo further metabolism intracellularly (10, 13). It is renally excreted (25).
The model includes non-insulin-dependent glucose utilization, described as a constant outflow (i.e., independent of glucose concentration) from the accessible glucose compartment. Others have used a combination of a constant and proportional outflow (15), but our preliminary work on the model excluded the proportional component, because it tended to converge to zero when estimated from the data. Normally, the ability of glucose to promote its own disposal (glucose effectiveness) is modeled via the non-insulin-dependent pathway. In the present model, it is included in the insulin-dependent pathway (removal from the nonaccessible compartment), being consistent with recent observations (17) and the fact that the non-insulin-dependent utilization, which corresponds to the utilization by the central nervous system, red blood cells, kidneys, and liver, is saturated at euglycemia. The insulin-dependent utilization represents the insulin-stimulated intracellular glucose disposal (glucose phosphorylation) in muscle and adipose tissues and is represented by an outflow from the nonaccessible compartment.
The model includes the effect of insulin on EGP suppression in a formulation independent of glucose concentration. The model structure is shown in Fig. 1.
Formally, the model consists of two compartments representing the kinetics of d-[U-^{13}C]glucose Equation 1 Equation 2 Equation 3two compartments representing the kinetics of 3-O-methyl-d-glucose Equation 4 Equation 5 Equation 6and two compartments representing kinetics of native glucose Equation 7 Equation 8 Equation 9Insulin action is modeled by postulating three effect compartments (representing so-called remote insulin), affecting, in turn, glucose distribution/transport, disposal, and production Equation 10 Equation 11 Equation 12The meanings of the symbols are as follows:q _{1}(t) andq _{2}(t) represent the masses ofd-[U-^{13}C]glucose in the accessible and the nonaccessible compartments, respectively (mmol);q _{3}(t) andq _{4}(t) represent the masses of 3-O-methyl-d-glucose in the two compartments, respectively (mmol); Q _{1}(t) andQ _{2}(t) represent the masses of native glucose in the two compartments, respectively (mmol);Q _{10} represents the initial mass of native glucose in the accessible compartment (mmol);x _{1}(t),x _{2}(t), andx _{3}(t) represent the (remote) effect of insulin on glucose distribution/transport, glucose disposal, and EGP, respectively (min^{−1}); k _{12}represents the transfer rate constant from the nonaccessible to the accessible compartment (min^{−1}); F _{01}is the total non-insulin-dependent glucose flux (mmol/min); u_{1}(t) and u_{3}(t) represent bolus doses of d-[U-^{13}C]glucose and 3-O-methyl-d-glucose, respectively, administered at 0 and 120 min (mmol/min); U(t) represents the bolus dose of the unlabeled glucose administered at 120 min (mmol/min);k _{03} represents the transfer rate constant of 3-O-methyl-d-glucose excretion (min^{−1}); k _{a1},k _{a2}, and k _{a3} represent deactivation rate constants (min^{−1});k _{b1}, k _{b2}, andk _{b3} represent activation rate constants (min^{−2} per mU/l); I(t) and I_{b}represent plasma insulin and basal (preexperimental) plasma insulin, respectively (mU/l); EGP_{0} represents endogenous glucose production extrapolated to the zero insulin concentration (mmol/min); V represents the distribution volume of the accessible compartment (liters); g_{1}(t) and g_{3}(t) represent concentrations of d-[U-^{13}C]glucose and 3-O- methyl-d-glucose, respectively (mmol/l); and G(t) is the total glucose concentration (mmol/l).
Insulin sensitivity of glucose distribution/transport (S_{IT}) and glucose disposal (S_{ID}) describe the effect of insulin on the metabolic clearance rate (MCR) of glucose. Insulin sensitivity of EGP (S_{IE}) represents a reciprocal concept to the change in the metabolic clearance (a change in EGP expressed as a glucose volume per unit time), and overall sensitivity [S_{I(T+D+E)}; all ml · min^{−1} · kg^{−1} per mU/l] describes the combined effect of insulin.
S_{IT} represents the change in the glucose clearance rate due to elevated glucose distribution/transport while annulling the other two effects. Similarly, S_{ID} and S_{IE} represent the independent effect of insulin due to stimulated glucose disposal and suppressed EGP, respectively (see appendix for a formal definition of the sensitivities).
The model has twelve parameters: k _{12},k _{03}, F _{01},k _{a1}, k _{b1},k _{a2}, k _{b2},k _{a3}, k _{b3}, EGP_{0}, Q _{10}, and V, with an alternative parameterization: S =k _{b1}/k _{a1}, S =k _{b2}/k _{a2}, and S =k _{b3}/k _{a3}. The model is theoretically identifiable (proof not shown). The glucose concentration of the two tracers and the total glucose was zero, weighted at 122 and 123 min during the parameter estimation process to allow glucose distribution to be completed within the accessible compartment.
Stable-label two-compartment model of glucose kinetics during IVGTT.
For comparison, we estimated parameters of the stable-label two-compartment model of glucose kinetics during an IVGTT described by a set of differential equations (46) where q _{1}(t) andq _{2}(t) are masses of the tracer glucose in the two compartments (mmol); V is the volume of the accessible compartment (liters); k _{p} is the proportional term of glucose disposal (min^{−1});k _{21}, k _{12}, andk _{02} are fractional rate parameters (all min^{−1}); x(t) represents the remote insulin (min^{−1}); k _{b}(min^{−2} per mU/l) and k _{a}(min^{−1}) have similar meaning as p _{3}and p _{2} of the one-compartment minimal model;F _{01} is the constant component of glucose uptake [fixed at 1 mg · kg^{−1} · min^{−1}(8)]; g(t) is plasma concentration ofd-[U-^{13}C]glucose (mmol/l); and D is the administered dose of d-[U-^{13}C]glucose at 120 min (mmol). The proportional term of glucose disposalk _{p} is constrained to produce insulin-independent utilization three times higher than the insulin-dependent utilization at the basal glucose concentration (G_{b}) and the basal insulin concentration (I_{b}) guaranteeing theoretical identifiability of the model. The model has six parameters k _{21},k _{12}, k _{02}, V,k _{a}, and k _{b}. All measurements were included in parameter estimation.
The basal MCR of glucose (ml · kg^{−1} · min^{−1}) and the insulin sensitivity index S (ml · kg^{−1} · min^{−1} per mU/l) are calculated as
Minimal model of glucose kinetics during IVGTT.
We also estimated parameters of the minimal model to enable comparison with the newly developed model. The minimal model of glucose kinetics after an IVGTT is described by two differential equations (6) where G(t) is the plasma concentration of total (labeled and unlabeled) glucose (mmol/l); x(t) represents the remote insulin (min^{−1}); G_{b} is the basal glucose concentration (mmol/l); D is the (total) glucose dose (mmol); and p _{1} (min^{−1}),p _{2} (min^{−1}),p _{3} (min^{−2} per mU/l), and V (liters; the distribution volume) are model parameters. Insulin sensitivity (S_{I}; min^{−1} per mU/l) is defined as the ratio S_{I} =p _{3}/p _{2} and glucose effectiveness as S_{G} = p _{1}.
The glucose concentration was zero weighted from 2 to 5 min after the administration of the unlabeled glucose bolus.
Free-format reconstruction of EGP.
EGP was also calculated without imposing a relationship to insulin by using a variation of deconvolution methodology. Details are given inappendix . In brief,d-[U-^{13}C]glucose and 3-O-methyl-d-glucose (i.e., excluding native glucose) were employed to estimate the time-variant unit impulse response of the glucose system. An advanced numerical approach (regularized deconvolution with nonnegative constraint) calculated free-format EGP.
These free-format calculations of EGP were validated by reconstructing the infusion of the validation tracer (d- [6,6-^{2}H_{2}]glucose) (seeappendix ). The difference between the reconstructed rates and the actual rates of the validation tracer indicates the accuracy of the free-format estimate of EGP.
Parameter estimation.
Model parameters were estimated by employing a nonlinear, weighted, least squares algorithm. The weight was defined as the reciprocal of the square of the measurement error.
The measurement errors associated withd-[U-^{13}C]glucose,d-[6,6-^{2}H_{2}]glucose, and 3-O-methyl-d-glucose were determined experimentally from duplicate measurements. Below a threshold concentration of 0.153 mmol/l, the standard deviation (SD) of the measurement error associated withd-[U-^{13}C]glucose was constant at 0.00133 mmol/l; above the threshold, the coefficient of variation (CV) of the measurement error was constant at 0.87%. Ford-[6,6-^{2}H_{2}]glucose, the threshold concentration was 0.130 mmol/l, SD was 0.00593 mmol/l, and the CV was 4.57%. For 3-O-methyl-d-glucose, the threshold concentration was 0.07 mmol/l, SD was 0.00188 mmol/l, and the CV was 2.68%.
The accuracy of parameter estimates was obtained from the inverse of the Fisher information matrix (14). The SAAM II v1.1.1 package (SAAM Institute, Seattle, WA) was employed to carry out the calculations.
Statistical Analysis
Correlations were evaluated employing the Pearson correlation coefficient. Analysis of variance (ANOVA) with Tukey's post hoc analysis was employed to assess the relative contributions of the three insulin effects and their combinations on glucose lowering. ANOVA was also employed to compare the three insulin sensitivities. Values are represented as means ± SE or as mean (95% confidence interval) (log transformed to assure normality) unless stated otherwise.
RESULTS
Plasma Glucose, Insulin, and Glucose Tracers
The profiles of plasma glucose, plasma insulin,d-[U-^{13}C]glucose, and 3-O-methyl-d-glucose during an IVGTT are shown in Fig. 2. Plasma glucose concentration was raised from the basal level of 5.4 ± 0.2 mmol/l to a maximum of 14.9 ± 1.2 mmol/l at 122 min. Plasma insulin and plasma C-peptide concentrations increased from 8.0 ± 0.2 mU/l and 0.43 ± 0.07 nmol/l to a maximum of 96.5 ± 56.9 mU/l and 1.67 ± 0.75 nmol/l at 125 and 128 min, respectively. We observed a high interindividual variability in the insulin response to the unlabeled glucose bolus (Fig. 2).
The insulin profiles and the C-peptide profiles (not shown) demonstrated that the bolus of glucose tracers at 0 min did not stimulate insulin secretion.
After the two boluses at 0 and 120 min, bothd-[U-^{13}C]glucose and 3-O-methyl-d-glucose presented a double-peak profile with the peaks well defined. The profiles were smooth and confirmed the low level of measurement error. The average concentration of recycled glucose was 0.007 mmol/l, with a maximum concentration of 0.015 ± 0.004 nmol/l at 125 min and a decrease to 0.006 ± 0.003 mmol/l at the end of the study (profile not shown). These values were negligible in comparison withd-[U-^{13}C]glucose andd-[6,6-^{2}H_{2}]glucose concentrations.
Modeling Glucose Kinetics
The sample fit of the model tod-[U-^{13}C]glucose, 3-O-methyl-d-glucose, and total glucose is shown in Fig. 3. Table1 lists the parameters of the model and includes EGP at the basal insulin concentration. Before the injection of unlabeled glucose, the fractional transfer rate from the accessible to the nonaccessible compartment (inward rate) and insulin-mediated glucose disposal were 0.0266 ± 0.0044 and 0.0042 ± 0.0012 min^{−1}. After the bolus administration at 120 min, the suppression of EGP followed the profile presented by the activated glucose disposal more closely than that of glucose distribution/transport (Fig. 4). EGP was suppressed by 70% (52–82%) (relative to basal) within 60 min of the bolus administration. Glucose distribution/transport was maximally activated by 62% (34–96%) above basal at 80 min compared with maximum 279% (116–565%) activation of glucose disposal at 20 min. The deactivation of glucose distribution/transport was slower than that of glucose disposal and EGP (P < 0.02) with half-times of 207 (84–510), 12 (7–22), and 29 (16–54) min, respectively.
Insulin sensitivities of distribution/transport, disposal, and EGP were similar [see Table 2; P= nonsignificant (NS), ANOVA]. Insulin sensitivity of distribution/transport was positively correlated with that of disposal (S_{IT} vs. S_{ID}, r = 0.82,P < 0.05) but not with that of EGP (S_{IT}vs. S_{IE}, r = 0.58, P = 0.23). EGP and disposal sensitivities were not significantly correlated (S_{IE} vs. S_{ID}, r = 0.50,P = 0.32).
The plot of weighted residuals is shown in Fig.5. The average SD of weighted residuals during the basal period tended to be smaller than that after the unlabeled glucose bolus (d-[U-^{13}C]glucose: 1.6 vs. 2.4; 3-O-methyl-d-glucose: 1.4 vs. 1.6; total glucose: 1.5 vs. 2.0), indicating an average misfit slightly above the measurement error for all three substrates and a slightly better fit during the basal period than during the IVGTT.
Simulation runs with the model facilitated the separation of the three effects of insulin on glucose lowering during IVGTT. The model was run in eight configurations: 1) the three effects [i.e., the remote insulin compartments x _{1}(t),x _{2}(t), andx _{3}(t)] following their nominal (stimulated) levels during IVGTT; 2) the three effects fixed at their basal (i.e., 120-min) levels; 3–5) one effect following its nominal level and the other two effects fixed at their basal levels; and 6–8) two effects following their nominal levels and one effect fixed at its basal level.
Figure 6 shows the results ofconfigurations 1 and 3–5 relative to the baseline configuration (2) (the three effects fixed at their basal levels). It is demonstrated that suppression of EGP has the greatest and longest impact on glucose lowering and accounts, at its maximum, for ∼3 mmol/l out of the lowering magnitude of 6 mmol/l. The effects of stimulated glucose distribution/transport and stimulated glucose disposal are smaller but similar.
Glucose-lowering activities were quantified by area under the curve (AUC) of glucose differential profiles shown in Fig. 6 (see Fig.7). Stimulation of disposal and stimulation of distribution/transport account each independently for ∼25 and 30% of the overall glucose-lowering AUC. Suppression of EGP is more influential (P < 0.01, ANOVA) and accounts for ∼50% of the overall glucose-lowering AUC. The combination of stimulated glucose disposal and stimulated glucose distribution/transport is less potent than the two combinations associated with EGP suppression (P < 0.05, ANOVA).
EGP
Individual profiles of EGP obtained by 1) free-format calculations and 2) model-based calculations are shown in Fig. 8, documenting a similar pattern obtained by the two methods. When the two methods were compared, AUCs associated with the EGP profiles were identical during the basal period (1,269 ± 70 vs. 1,265 ± 71 μmol · kg^{−1} · min^{−1} for 120 min; P = NS, paired t-test), but during the IVGTT the free-format method gave 16% lower AUC (1,455 ± 180 vs. 1,737 ± 156 μmol · kg^{−1} · min^{−1} for 240 min; P < 0.005, paired t-test), which was highly correlated with that obtained by the model-based calculations (r = 0.95, P < 0.005).
The calculations are validated by the reconstructed discontinuous infusion of the validation tracer. The actual infusion rates and the reconstructed infusion rates are shown in Fig.9, documenting the ability of the free-format method to calculate even discontinuous appearance rates throughout the experiment.
Quantitative analysis shows that 96 ± 5% of the total validation infusion was recovered (statistically not different from 100%) with a mean square error of 0.24 ± 0.02 μmol · kg^{−1} · min^{−1}, which suggests good accuracy of free-format EGP calculations.
Stable-Label Two-Compartment Model of Glucose Kinetics during IVGTT
The results of the stable-label two-compartment model are given in Table 3. The two-compartment model insulin sensitivity was tightly correlated with distribution/transport sensitivity (S vs. S_{IT}:r = 0.92, P < 0.01). It also correlated positively, but nonsignificantly, with disposal sensitivity and sensitivity of EGP (S vs. S_{ID}:r = 0.73, P = 0.10; S vs. S_{IE}: r = 0.81,P = 0.05). S was about one-half the sum of S_{IT} and S_{ID}. The mean deactivation rate of insulin action was similar to that of the deactivation of glucose disposal (k _{b} vs. k _{b2}, 0.0953 ± 0.0143 vs. 0.0683 ± 0.0207 min^{−1}).
Minimal Model of Glucose Kinetics during IVGTT
The results of the minimal model of glucose kinetics are given in Table 4. The minimal model insulin sensitivity index was tightly correlated with and linearly related to sensitivity of EGP (S_{I} vs. S_{IE}: r = 0.96, P < 0.005) (see Fig.10). It also positively, but nonsignificantly, correlated with disposal and distribution/transport sensitivities (S_{I} vs. S_{ID}: r = 0.55, P = 0.26; S_{I} vs. S_{IT}:r = 0.73, P = 0.10). This indicates that the minimal model measures a mixture of the three indexes but primarily reflects the insulin sensitivity of EGP.
It was of interest to investigate the relationship between glucose effectiveness S_{G} and parameters of the newly developed model. However, S_{G} was not correlated with the non-insulin-dependent utilization, the inward/outward fractional transfer rates at basal conditions, or the basal EGP. This is possibly due to a narrow range of S_{G} values.
DISCUSSION
By combining a dual-tracer methodology with a new model of glucose kinetics, we have separated three effects of insulin during an IVGTT. Insulin sensitivities of glucose distribution/transport, disposal, and production were quantified and their contributions to glucose lowering were assessed.
We were able to partition whole body insulin action into effects on glucose distribution/transport and glucose disposal by simultaneously administering 3-O-methyl-d-glucose andd-[U-^{13}C]glucose, whereas glucose production was determined by additionally considering the kinetics of native glucose. The new model describes the kinetics of the two glucose tracers and native glucose by using a two-compartment structure for glucose and a one-compartment structure for each insulin effect. The model describes both steady-state (pre-IVGTT) and dynamic (IVGTT) conditions.
The results show that the liver plays an important role in the restoration of glucose homeostasis after an IVGTT. Suppression of EGP accounts for approximately one-half of the overall glucose-lowering effect. The peripheral effect accounts for the other half and is divided into two approximately similar components, which are attributed to the stimulation of distribution/transport and disposal.
EGP suppression results from both direct insulin effect (e.g., at the site of the liver) and indirect effect [e.g., FFA mediated (7)]. Thus our measure of insulin sensitivity of EGP suppression is a combined index of direct and indirect effects.
The conventional assessment of the insulin effect during IVGTT employs the (one-compartment) minimal model (6) with the administration and analysis of native glucose. As a result, the minimal model S_{I} is a mixture of the three insulin effects. Our analysis shows that, unexpectedly, the suppression of EGP dominates the mixed measure of S_{I}, as indicated by its highest correlation with S_{IE} (r = 0.96,P < 0.005). The other two effects are also represented (0.55 ≤ r ≤ 0.73). S_{I} is often associated with the insulin effect in the periphery, but our study shows that S_{I} reflects primarily liver sensitivity in healthy subjects.
Inward and outward transmembrane glucose transports have been estimated by Bonadonna and colleagues (9, 11) in skeletal muscle by use of a dual-tracer methodology. The rates were calculated by analyzing washout curves of l-glucose and labeled 3-O-methyl-d-glucose at the steady state. During fasting conditions in healthy subjects, values of inward transport were higher (0.066 ± 0.004 vs. 0.027 ± 0.004 min^{−1}) and values of outward transport smaller (0.038 ± 0.003 vs. 0.065 ± 0.011 min^{−1}), but still compatible with the present study.
Another estimate on the whole body level was obtained from a two-compartment model of glucose kinetics during basal and insulin-stimulated (hyperinsulinemic glucose clamp) conditions (20, 24). During basal conditions, similar values of the outward transport were obtained. The inward transport rate was slightly higher (0.043 ± 0.005 vs. 0.027 ± 0.004 min^{−1}), but it should be noted that glucose disposal was partitioned in a 3:1 ratio between the accessible and nonaccessible compartments to overcome an identifiability problem. During the insulin-stimulated conditions (∼100 mU/l), the inward transport increased approximately twofold (which compares with values obtained in the present study), whereas the outward transport was slightly reduced.
At basal state, insulin exerts a relatively small control over glucose disposal. Data shown in Table 1 indicate that a model-based estimate of insulin-dependent glucose uptake is ∼1.4 μmol · kg^{−1} · min^{−1}(EGP_{b} − F _{01}) or ∼13% of the total glucose turnover. At basal conditions, non-insulin-dependent glucose uptake dominates. This is in agreement with observations made by others (8) and is also compatible with studies showing that acute suppression of basal insulin levels has only a limited effect on whole body glucose utilization [<20% (22,23)].
The relative “unimportance” of insulin-dependent disposal at basal insulin also explains that insulin is “less” effective in promoting glucose uptake than at physiological hyperinsulinemia. Raising basal insulin by 50% results, in our model, in a 50% increment in insulin-dependent glucose uptake, but this increases whole body glucose disposal by only ∼7%. However, when insulin-dependent disposal dominates, such as during hyperinsulinemic clamps (e.g., insulin infusion at 0.5 mU · kg^{−1} · min^{−1}), insulin has a nearly proportional effect on glucose disposal (subject to reaching saturation of its action).
The model gives temporal patterns of insulin actions. Glucose disposal is rapidly activated fourfold above basal and quickly deactivated, with a half-time of 12 min. Glucose distribution/transport is activated much more slowly, 1.6-fold above basal [compared with 1.6- to 2.0-fold increase in GLUT4 content in skeletal muscle plasma membranes in healthy subjects at physiological levels of insulin (48)]; it is deactivated again very slowly, with a half-time of 200 min, and remains elevated at the end of the study. EGP is suppressed in a similar temporal pattern as glucose disposal is activated. The half-time of deactivation of EGP suppression (30 min) compares with that of glucose disposal.
Together, these data show that insulin activates and deactivates intracellular disposal and EGP quickly, whereas glucose distribution/transport is activated/deactivated slowly, explaining the “memory” effect of insulin, i.e., the observation that glucose clearance is elevated well beyond the time when insulin returns to its basal level.
The effect of insulin and its partitioning can be expressed at steady-state conditions or during dynamic conditions. Insulin sensitivities and clearance rates such as the minimal model S_{I} or the newly defined S_{IT} (and the end-stage glucose infusion rate during glucose clamps) are measures of insulin action at an incremental insulin concentration extrapolated to steady-state conditions. On the other hand, AUCs of glucose-lowering profiles, such as those evaluated in the present study, are measures of insulin effects during specific dynamic conditions. The benefit of the former is that they are, in principle, independent of the test and characterize the metabolic system. However, they may provide an inaccurate impression about the amount of glucose removed via the insulin-dependent pathway. This is due to the nonlinearity of the glucose system and the delayed onset of insulin actions. The present study therefore evaluates both modalities (i.e., insulin sensitivities and glucose-lowering AUCs), because they provide complementary information.
Insulin temporal action can be represented in three ways. First, there are the effect (remote) compartments, which represent the direct stimulation of physiological phenomena such as the recruitment of glucose transporters or the stimulation of glucose phosphorylation (see Fig. 4, top). Second, glucose fluxes quantify the flow rates between/to/out of compartments and provide information about the absolute movement of the glucose mass (see Fig. 4, bottom). Finally, glucose-lowering profiles reflect changes in plasma glucose due to a negative net balance of glucose fluxes (see Fig. 6 for an example).
The three temporal profiles differ due to structural properties of the glucose system. For example, glucose disposal (i.e., the insulin-dependent uptake) is activated fast, but due to its structural distance from the accessible (plasma) compartment, glucose lowering will first be initiated in the nonaccessible (intracellular) compartment, where it will depend on glucose availability, and then glucose removal will be propagated into the accessible compartment via an increased concentration gradient, creating a lag between the temporal patterns.
Glucose distribution/transport is activated most slowly and reaches a smaller fraction of its asymptotic (i.e., achievable at the steady state) value during an IVGTT compared with, say, the suppression of EGP. Together with the structural effect on the temporal patterns as discussed above, this means that, although the three pathways have identical effects on glucose clearance at steady-state conditions as exemplified by identical sensitivities, during dynamic conditions the glucose-lowering potency differs. It therefore follows that the study of the glucose system under dynamic conditions provides important and complementary information about the actual contribution of metabolic pathways to glucose lowering compared with the projected contributions obtained from studies at steady-state conditions such as glucose clamps.
The model includes a specification of EGP and its suppression that is independent of glucose concentration. In a separate analysis, we examined this assertion and postulated that EGP(t) = EGP_{0}[1 −x _{3}(t) −k _{L}G(t)], wherek _{L} represents the effect of glucose on EGP suppression (1). However, such a formulation was not appropriate, as we observed no improvement in the model fit, poor parameter accuracy, and no improvement in the Akaike information criterion (data not shown).
It is generally perceived that both glucose and insulin contribute to EGP suppression during an IVGTT (4). However, a study in subjects with type 1 diabetes clearly demonstrated that glucose on its own does not suppress EGP during an IVGTT (37). The study involved two IVGTTs in each subject, one with “normal” insulin response and the second with basal insulin. Endogenous glucose concentration was calculated; this concentration corresponds to glucose originating from endogenous sources, and the calculations are model independent. During the IVGTT with basal insulin, endogenous glucose concentration remained unchanged for ∼90 min after a glucose bolus, with a tendency of endogenous glucose to increase toward the end of the IVGTT. This shows that EGP is not suppressed by glucose during an IVGTT when insulin is maintained at basal level.
When insulin is elevated during an IVGTT, as in the present case, there is no direct evidence of an exacerbated effect of glucose on EGP suppression. On the contrary, with the use of data obtained from glucose clamp studies, an increase in glucose has little or no further effect when glucose production is already markedly suppressed by elevated insulin concentrations (2, 36).
It therefore follows that, during glucose clamps, glucose substantially suppresses its production at basal insulin, but this effect is negligible at elevated insulin. During shorter exposure, such as during an IVGTT, the glucose effect is not present even at basal insulin, supporting our formulation of EGP suppression.
We further assessed the correctness of the specification of EGP. We estimated a subset of model parameters related tod-[U-^{13}C]glucose and 3-O-methyl-d-glucose but omitting native glucose (i.e., using Eqs. 1-6, 10, and 11 ) fromd-[U-^{13}C]glucose and 3-O-methyl-d-glucose measurements (results not shown). The parameter estimates were virtually identical to those calculated from the complete model (with the exception of parametersk _{b1} and S in subject 5, which were, respectively, six times higher and two times smaller), demonstrating model consistency and coherence.
The validity of model-derived EGP is supported by a good comparison with free-format estimation of EGP (see Fig. 8). The free-format estimation is, in turn, validated by the infusion and accurate recovery of the validation tracer (see Fig. 9).
The two-compartment minimal model by Vicini et al. (46) can be seen as the immediate precursor of our new model. The insulin sensitivities S_{IT} and S_{ID} dissect the two-compartment minimal model index S . It appears that S reflects primarily the sensitivity of the distribution/transport pathway, but its deactivation reflects that of the disposal pathway. Thus S mixes properties of both pathways. However, the minimal model underestimates the sensitivity of the periphery (S is about one-half of the sum of S_{IT} and S_{ID}).
In our study, we used nonarterialized venous blood compared with the traditional use of arterialized venous blood. It is known that, during an IVGTT, a significant arteriovenous difference exists during 1–7 min in the case of glucose and 1–2 min in the case of insulin concentration (26). This difference has little effect on estimates of insulin sensitivity using the minimal model (26). In our model, only two glucose measurements at 4 and 6 min are expected to be underestimated, potentially affecting parameters estimated from the early stage of IVGTT, i.e., the volume of distribution, non-insulin-dependent glucose disposal (>80% of total disposal), and the inward transport (modulated by S_{IT}). However, the effect on S_{IT} is likely to be small, as above-basal, insulin-stimulated distribution/kinetics are elevated in later stages of IVGTT, providing information for their accurate estimation.
The new model does not have insulin actionsx _{i}(t) centered on the basal insulin (i.e., insulin action is not zero at basal insulin) as do the two minimal models. The new model represents insulin action above zero insulin concentration.
The traditional specification ofx _{i}(t) to represent above-basal action has theoretically attractive properties: it is not assumed that at zero insulin the action is zero (as our model does). However, its practical use suffers from several drawbacks. First, basal insulin is measured with error, and it is well recognized that this measurement error propagates into error in the estimate of insulin sensitivity. Furthermore, postexperimental values are used for basal insulin, because these are normally lower than preexperimental values and there is need to avoid negative insulin action at the end of IVGTT. Such a choice is difficult to justify on purely physiological grounds. Second, we wanted a model that could be used under other experimental scenarios such as glucose clamp (40) to make direct comparison with IVGTT-derived indexes without the need to “correct” for different basal insulin values.
The partitioning of the peripheral effect into the distribution/transport effect and the disposal effect rests on using an adequate model structure and obtaining a good fit to 3-O-methyl-d-glucose andd-[U-^{13}C]glucose profiles. Physiological considerations guided the construction of the model (see data analysis). The evaluation of the residuals demonstrates the ability of the model to fit the data throughout the experiment (see Fig. 5).
Normalized residuals were at times outside the −2 to +2 range. This applies mainly to 3-O-methyl-d-glucose andd-[U-^{13}C]glucose after administration of boluses and to unlabeled glucose toward the end of the IVGTT. The accuracy of analytical techniques associated with 3-O-methyl-d-glucose andd-[U-^{13}C]glucose measurements is very high, resulting in very large weights when calculating weighted residuals. Sampling time error during frequent sampling stages (after administration of boluses) therefore exerts an exacerbated effect on the lack of fit and explains part of the observations. EGP is known to decrease during prolonged fasting, and this may explain the lack of fit toward the end of the experiment in relation to the unlabeled glucose.
It is important to stress that the validity of our model is limited to the conditions observed in the present study; i.e., the model describes kinetic events related to glucose disposal and glucose production during basal conditions and during an IVGTT. Extrapolations to other conditions, such as sustained hyperinsulinemia observed during glucose clamps, need model modifications such as the inclusion of saturable inward transport (40).
There are considerable interindividual variations in the partitioning of insulin sensitivities and the relative and absolute contribution of the three pathways to glucose lowering. Further studies investigating the effect of various pathological conditions and diseases, particularly type 2 and type 1 diabetes, are warranted to link molecular, cellular, and whole body defects (9, 12, 18,48).
In conclusion, we have partitioned the effect of insulin on glucose kinetics during an IVGTT into three components, which represent stimulated glucose distribution/transport, stimulated glucose disposal, and suppressed endogenous glucose production. In healthy subjects, the two peripheral effects account jointly for approximately one-half of the overall insulin-stimulated glucose lowering, each effect contributing equally. Suppression of endogenous glucose production matches the effect in the periphery.
Acknowledgments
This work was in part supported by the Special Trustees of St. Thomas', Diabetes UK, and the European Community Information Society Technologies Programme (the Adicol Project IST-1999–14027). Susan C. Wood provided secretarial support.
CALCULATING TRACER-TO-TRACEE RATIO
The formulas to calculate TTR ofd-[6,6-^{2}H_{2}]glucose (z _{D}), d-[U-^{13}C]glucose (z _{U}), and recycled glucose (z _{R}) in the mixture of tracee and tracers (sample) are where r_{D}, r_{U}, and r_{R} are the raw isotope (area) ratios M + 2/M + 0,M + 4/M + 0, and M + 3/M + 0, respectively, in the sample; r_{DN}, r_{UN}, and r_{RN} are the raw isotope ratiosM + 2/M + 0, M + 4/M + 0, and M + 3/M + 0, respectively, in the preexperiment sample (tracee only); and the common denominator D_{c} is calculated as The calculations assume thatd-[U-^{13}C]glucose molecules can be recycled into glucose molecules with m/z of M + 2 andM + 3, but not M + 4, and that the same amount of d-[U-^{13}C]glucose molecules is recycled in molecules with m/z of M + 2 andM + 3 (31, 32).
The formula to calculate the TTR of 3-O-methyl-d-glucose (z_{O}) in the mixture of the tracer and the internal standard is where r_{O} is the raw isotope ratio M+ 0/M + 3 in the mixture, and r_{ON} is the raw isotope ratio M + 0/M + 3 in the internal standard.
The concentrations of d-[U-^{13}C]glucose (g_{U}),d-[6,6-^{2}H_{2}]glucose (g_{D}), and recycled glucose (g_{R}) are obtained as where C is the (total) glucose concentration in the sample. The concentration of 3-O-methyl-d-glucose (g_{O}) is obtained as where g_{S} is the concentration of the internal standard 3-O-methyl-d-[^{2}H_{3}]glucose.
The derivation of the formulas follows principles described elsewhere (29).
DEFINING INSULIN SENSITIVITIES
Insulin sensitivity of glucose distribution/transport (S_{IT}), glucose disposal (S_{ID}), EGP (S_{IE}), and overall sensitivity [S_{I(T+D+E)}; all ml · min^{−1} · kg^{−1} per mU/l] are defined as The basal values x _{1b},x _{2b}, and x _{3b} correspond to pre-IVGTT quantities (i.e., values at 120 min). Similarly, G(t) and x _{2}(t) represent pre-IVGTT values for the purpose of evaluating insulin sensitivities.
FREE-FORMAT CALCULATIONS OF EGP AND RECONSTRUCTION OF VALIDATION TRACER INFUSION
The endogenous glucose concentration G_{E}(t) due to EGP after an IVGTT was calculated using a model-independent approach (19). Only the assumption of the isotopic indistinguishability was made.
The total glucose concentration G(t) includes the endogenous component G_{E}(t), the unlabeled exogenous glucose component G_{X}(t), and the tracer component g_{U}(t) Equation 13The components g_{U}(t) and G_{X}(t) are fixed at a ratio E, E = g_{U}(t) = G_{X}(t), which is given by the (d-[U-^{13}C]glucose) enrichment of the glucose bolus administered at 120 min. Solving Eq. 13 for G_{E}(t), we obtain Equation 14The general form of the relationship between glucose appearance a(t) and the resulting glucose concentration c(t) is described by an integral equation Equation 15where h(t,τ) is the (time-variant) unit impulse response of the glucose system. The impulse response h(t,τ) was defined as a sum of two exponentials (a function of τ) for t between two sampling points where values A_{1,i}, A_{2,i}, λ_{i,1}, and λ_{i,2} were determined from the two-compartment model of d-[U-^{13}C]glucose defined by Eqs. 1, 2, 3, 10, and 11 .
A separate parameter estimation was carried out to calculate the time-variant unit impulse response. The concentrations ofd-[U-^{13}C]glucose and 3-O-methyl-d-glucose excluding the concentration of native glucose were employed to estimate a subset of model parameters (omitting EGP-related parameters, i.e., 8 parameters:k _{12}, k _{03},F _{01}, S ,k _{b1}, S ,k _{b2}, and V) for the purposes of free-format reconstruction of EGP. The submodel is defined by Eqs. 1-6,10, and 11 . The integral equation Eq. 15 is ill conditioned and was therefore solved using a regularization method (28, 44). The regularization component consisted of the norm of second differences to provide a piecewise constant appearance rate a(t). The regularization coefficient, which defines the extent of smoothing, was chosen individually by assessing the fit to the data and the distribution of residuals.
EGP(t) was calculated by substituting the endogenous glucose concentration G_{E}(t) (see Eq.14 ) for c(t) in Eq. 15 . The infusion ofd-[6,6-^{2}H_{2}]glucose (the validation tracer) was reconstructed by substitutingd-[6,6-^{2}H_{2}]glucose concentration for c(t) in Eq. 15 .
Footnotes
Address for reprint requests and other correspondence: R. Hovorka, Metabolic Modelling Group, Centre for Measurement and Information in Medicine, City University, Northampton Square, London EC1V 0HB, UK (E-mail:r.hovorka{at}city.ac.uk).
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First published January 2, 2002;10.1152/ajpendo.00304.2001
- Copyright © 2002 the American Physiological Society