The present analysis tests the hypothesis that quantifiable disruption of the glucose-stimulated insulin-secretion dose-response pathway mediates impaired fasting glycemia (IFG) and type 2 diabetes mellitus (DM). To this end, adults with normal and impaired fasting glycemia (NFG, n = 30), IFG (n = 32), and DM (n = 14) were given a mixed meal containing 75 g glucose. C-peptide and glucose were measured over 4 h, 13 times in NFG and IFG and 16 times in DM (age range 50–57 yr, body mass index 28–32 kg/m2). Wavelet-based deconvolution analysis was used to estimate time-varying C-peptide secretion rates. Logistic dose-response functions were constructed analytically of the sensitivity, potency, and efficacy (in the pharmacological sense of slope, one-half maximal stimulation, and maximal effect) of glucose's stimulation of prehepatic insulin (C-peptide) secretion. A hysteresis changepoint time, demarcating unequal glucose potencies for onset and recovery pathways, was estimated simultaneously. According to this methodology, NFG subjects exhibited distinct onset and recovery potencies of glucose in stimulating C-peptide secretion (6.5 and 8.5 mM), thereby defining in vivo hysteresis (potency shift −2.0 mM). IFG patients manifested reduced glucose onset potency (8.6 mM), and diminished C-peptide hysteretic shift (−0.80 mM). DM patients had markedly decreased glucose potency (18.8 mM), reversal of C-peptide's hysteretic shift (+4.5 mM), and 30% lower C-peptide sensitivity to glucose stimulation. From these data, we conclude that a dynamic dose-response model of glucose-dependent control of C-peptide secretion can identify disruption of in vivo hysteresis in patients with IFG and DM. Pathway-defined analytic models of this kind may aid in the search for prediabetes biomarkers.
laboratory studies indicate that soluble nutrients, gut peptides, systemic hormones, neural inputs, and intraislet factors, such as diffusible transmitters and metabolites, control the secretion of insulin, glucagon, and somatostatin under physiological conditions (14, 32, 37). Because of the complexity of homeostatic networks, mathematical models are often required to quantify the loss or erosion of reciprocal linkages among metabolic signals. Indeed, pathophysiological studies suggest that multiple regulatory defects may exist in type 2 diabetes mellitus (DM) and possibly in impaired fasting glycemia (IFG) (8, 18, 20, 25, 30). Although computer-assisted simulations have shed light on plausible homeostatic control mechanisms (7, 23, 27, 28, 34), analytic estimation methods are needed to quantify the type and degree of impairment in pathophysiology. The present work introduces an analytical model of the interactive regulation of C-peptide secretion (stimulated) by glucose. The dynamic model was validated in normoglycemic adults (normal fasting glycemia, NFG) and then applied to multiple groups of subjects spanning a full spectrum of glycemic derangements from IFG to type 2 DM. The goal was to identify structural changes in the regulatory relationships that could potentially form a framework for establishing a biomarker for prediabetes classification and detection.
The experimental basis for the model is the hypothesis that insulin/C-peptide secretion is stimulated by glucose and that the dose-response relationship changes under impaired glycemia (9–11, 13, 15, 17, 33). Whereas our previous analyses were based on linear modeling (22, 26), physiological dose-response relationships are characteristically nonlinear, saturable, and time delayed, as exemplified by glucose-insulin coupling (6, 19, 32). Accurate quantification of glucose/C-peptide dynamics must therefore embody such characteristics. Moreover, if de facto dynamics also include dose-response potentiation or downregulation, the analytical construct must allow for such adaptations. The present model quantifies allowable dose-response shifts or hystereses via a new nonlinear analytical formalism. The formalism assesses C-peptide dynamics under changing glucose concentrations while evaluating possible hysteresis. Hysteresis is here defined as a shift in glucose-stimulatory potency at an estimable time (T) after food ingestion. This approach unveils prominent glucose-C-peptide hysteresis in subjects with NFG, and markedly impaired glucose-hormone interactions in patients with IFG or DM. The pharmacological meaning of dose-response sensitivity, potency and efficacy are given in the appendix.
The underlying premise was that erosion or loss of regulatory control in prediabetes and diabetes results in quantifiable disruption of dynamic (time-dependent) dose-response relationships between glucose and C-peptide. The goal was to estimate and compare dynamic regulation in NFG, IFG, and DM and to explore quantifiable distinctions among the dynamics of NFG/NGT, NFG/IGT, IFG/NGT, IFG/IGT, IFG/DM, and DM.
After approval from the Mayo Institutional Review Board, 30 subjects with NFG, 32 with IFG, and 14 with DM gave written informed consent to participate in the studies. Ten other individuals with NFG and normal glucose tolerance (NGT) were each studied twice at least 1 wk apart to evaluate test-retest reliability of the model. Moreover, the sampling rate for these 10 individuals was quite rapid (5 min for hours 0–2, 10 min for hours 2–4). Detailed subject characteristics have previously been published (1, 3). This new study utilizes a subset of the previously published data never analyzed in this fashion. In brief, all subjects were Caucasian, in good health, at a stable weight, and did not engage in regular vigorous physical exercise. The absolute range of ages was 50–57 yr, and of body mass index 28–32 kg/m2. Oral hypoglycemic medications were discontinued at least 3 wk prior to study in DM subjects. At the time of study, subjects were on no medications other than stable doses of thyroid hormone, low-dose aspirin, HMG-CoA reductase inhibitors, SSRI antidepressants, or antihypertensives that are metabolically neutral (low-dose thiazide, calcium channel blockers, or losartan).
Blood sampling and meal/glucose-tolerance test.
Volunteers were instructed to follow a weight maintenance diet containing 55% carbohydrate, 30% fat, and 15% protein for at least 3 days before the study date. Subjects were admitted to the Mayo Clinical Research Unit the evening before the study and ate a standard 10 kcal/kg meal (55% carbohydrate, 30% fat, and 15% protein) between 1830 and 1900. After fasting overnight, subjects ingested a standard mixed meal consisting of three scrambled eggs, 55 g of Canadian bacon (or 47 g of steak), and Jell-O containing 75 g glucose enriched (to ∼4%) with [1-13C]glucose, as previously described (2). For NFG and IFG subjects, glucose and C-peptide were sampled every 10 min from time 0–30 min, every 15 min from time 30–90 min, and every 30 min from time 90–240 min (13 time points altogether). For DM subjects, glucose and C-peptide were measured every 5 min from 0–20 min, every 10 min from 20–60 min, every 15 min from 60–90 min, and every 30 min from 90–240 min (16 time points).
NFG and NGT were determined by a glucose-tolerance test (GTT) comprising 75 g of glucose given orally in the morning after a 12-h fast. Normal values were a fasting plasma glucose (FPG) concentration of ≤5.1 mM and a 2-h glucose concentration ≤7.7 mM. IFG was defined by FPG of 6.1–6.9 mM, DM by FPG of ≥7.0 mM or 2-h postprandial glucose of ≥11.1 mM. IGT was defined by a 2-h postprandial glucose concentration of ≥7.8 mM and <11.1 mM. No subjects were studied with FPG values of 5.2 to 6.0 mM inclusive, in order to limit heterogeneity. Using these GTT criteria, exploratory subanalysis could be performed in the following six categories: NFG/NGT (18 subjects), NFG/IGT (12 subjects), IFG/NGT (7 subjects), IFG/IGT (16 subjects), IFG/DM (9 subjects), and DM (14 subjects). Figure 1, A and B, illustrates data obtained in one subject from each of the six categories, i.e., C-peptide concentration and secretion along with matching glucose concentration time profiles.
There are also clinical experimental results in the literature where intravenous (iv) glucose was administered rather than a meal. There are strengths and weaknesses in using either a standard mixed meal or iv glucose. The standard meal has the advantage that it consists of carbohydrates, fat, and protein and requires absorption and first-pass liver extraction. A disadvantage is that there are also effects due to gut hormones, to a lesser degree, on insulin secretion due to gut hormones. A variety of important secretagogs: GLP-1 and GIP, as well as ghrelin (from the stomach in addition to the pancreas) and others, are induced by absorption by the gut. An advantage of iv glucose is that the gut hormones are not released. A disadvantage, one could argue, is that metabolism is being observed in a nonnormal setting. It was decided in the present experiments that the mixed meal would be used. In the discussion, the analysis of intravenous glucose tolerance test (IVGTT) data for 62 young normal subjects is briefly described; the same hysteresis shift is observed as for the mixed-meal NFG/NGT subjects.
Wavelet-based deconvolution analysis, under regularization constraints, was used to estimate time-varying C-peptide secretion rates (see the appendix). Logistic dose-response functions were constructed analytically to quantify the sensitivity, potency, and efficacy of glucose's stimulation of prehepatic insulin (C-peptide) secretion. These dose-response terms are used in the classical pharmacological senses of slope, one-half-maximally stimulatory concentration, and maximal effects of glucose on insulin secretion. A hysteresis changepoint time, demarcating unequal glucose potencies for onset and recovery pathways, was calculated simultaneously.
A technical challenge was the sparseness of clinical sampling. In particular, 13 time-distributed measurements of C-peptide and glucose were obtained in NFG and IFG. Both markers were assayed at 16 time points in DM subjects and at 37 time points in healthy individuals evaluated for model reliability. Meal ingestion evokes a strong increase in blood glucose lasting for 90–120 min along with sustained responses in insulin/C-peptide. Hence, changes in mean responses over time are much greater than associated measurement errors. As a result of this, we show in the appendix that interpolation of the sparse data to a 5-min equally spaced version for analysis is justified.
The a priori hypothesis was that there is detectable impairment of glucose's stimulation of C-peptide secretion in IFG and DM compared with NFG. The hypothesis corresponds to a decrease in potency, sensitivity, and/or efficacy. For a small number of hypotheses, such as comparing NFG and DM, one may apply two-sample t-tests (e.g., with Welch-calculated degrees of freedom) (29). For multiple-hypotheses testing, one-way ANOVA is preferred, so long as group variances and sample sizes are the same, followed by Tukey's post hoc tests for a maximal contrast. However, the variances of parameters in the DM group were often larger than those in the other groups, and the six sample sizes (glycemic cohorts) ranged from 7 to 18 subjects. Thus, for the small number of a priori-chosen specific comparisons, we present specific two-sample t-results in the text and multicomparison ANOVA in the tables.
To test the appropriateness of 5-min interpolation, GTT data were obtained from 10 normal subjects, each studied twice. In the 20 data series, primary measurements were made every 5 min from 0 to 120 min (where 0 denotes glucose ingestion), and then every 10 min from 120 to 240 min (37 time points all together). The last 2-h 10-min data were first interpolated to a 5-min scale (resulting in 49 time points). These 5-min data were then used in two ways: first, deconvolution and dose-response analyses were performed on the 5-min data (49 values); second, the 5-min data were subsampled to match the exact times obtained in the NFG and IFG patient series (13 time points) and then reinterpolated back to the 5-min level. The appendix presents the validity of the interpolation to the sparsely sampled patient series.
The primary hypothesis was that estimates of glucose's concentration-dependent stimulation of C-peptide secretion would differ in subjects with NFG (n = 30), IFG (n = 32), and DM (n = 14). As a first step, we represented the time-varying relationships as two-dimensional curves: Fig. 2 (data are from the glucose concentration and C-peptide secretion profiles in Fig. 1B). Time evolution of the interlinked signals is represented by three successive colors (red, green, blue) and corresponding symbols (circles, triangles, squares) to denote the first 1 h, the next 1 h, and the last 2 h after meal/glucose ingestion. NFG/NGT curves evolved from values of low glucose concentration and low C-peptide secretion rate to values of high glucose concentration and high C-peptide secretion rate and then returned toward starting values via a different pathway. Nonidentical onset and recovery pathways in response to glucose ingestion suggest downregulation of the C-peptide secretion rate, here termed hysteresis.
Quantification of glucose's time- and concentration-dependent effects required estimation of an onset segment, a hysteretic inflection time T, and a recovery segment. Hysteresis due to a rightward shift in the recovery vis-à-vis onset glucose dose-response curve would define reduced glucose potency, whereas a leftward shift would define increased glucose potency. To assess the importance of incorporating hysteresis into dose-response modeling, one-potency (nonhysteresis) and two-potency (hysteresis) model fits were compared. Figure 3A displays maximum likelihood-based estimates of dose-response curves for the one-potency (dashed) and two-potency (solid) models plotted over time in three colors, as before. Statistical comparisons of model fits utilized a generalized likelihood ratio test, which was applied to the 76 (30 + 32 + 14) subjects' data individually. Despite the statistical penalty for two additional parameters in the hysteresis model (i.e., recovery potency and hysteresis delay time, T), 27 of 30 (NFG), 29 of 32 (IFG), and 9 of 14 (DM) model comparisons rejected the one-potency in favor of the two-potency model at P ≤ 0.01.
Figure 3B illustrates two-potency (hysteresis) model fits for two additional subjects in each of the NFG, IFG, and DM groups. Table 1 summarizes, for the three primary groups, all parameter estimates for logistic regressions of C-peptide secretion rates on glucose concentrations. The most prominent group distinction was erosion and reversal of the EC50 shift (hysteresis). Mean EC50 shifts (mM glucose) were −2.0, −0.8, and +4.6 for NFG, IFG, and DM, respectively (all 3 comparisons differ at P ≤ 0.005). This was due primarily to significant differences in EC50 onset values. Specifically, glucose/C-peptide dose-response curves exhibited 1) strong rightward (desensitizing) hysteresis (onset EC50 6.5 < recovery EC50 8.5 mM) in NFG; 2) slight rightward hysteresis (onset EC50 8.6 < onset EC50 9.4 mM) in IFG; and 3) completely reversed (leftward potentiating) hysteresis (onset EC50 18.8 > recovery EC50 14.2 mM) in DM (P ≤ 0.002 for all 3 onset EC50 comparisons). In contrast, only the recovery EC50 of NFG differed from that of DM at the P ≤ 0.05 level (P = NS for IFG vs. NFG or DM). There were no detectable differences in C-peptide secretory sensitivity to glucose concentrations (absolute slope) or in the efficacy (maximum) of glucose's stimulation of C-peptide secretion. Basal (nonstimulated) C-peptide secretion was higher in each step from NFG to IFG to DM (P < 0.01). The time from meal onset to the inflection point in the hysteretic curve (T) was longer in DM than in NFG or IFG (P < 0.01 for both). Thus, the three primary groups were distinguishable with respect to EC50 onset, hysteresis shift, and basal C-peptide secretion.
The secondary hypothesis was that glucose/C-peptide and dose-response functions discriminate among the six categories of NFG/NGT, NFG/IGT, IFG/NGT, IFG/IGT, IFG/DM, and DM. Table 2 gives estimated parameter means (±SE); Fig. 4 presents mean plots; and Fig. 5 displays EC50 shifts (hysteresis) of the six glucose/C-peptide functions. Post hoc comparisons revealed group contrasts in C-peptide dynamics in terms of 1) onset EC50, wherein NFG/NGT differed from IFG/IGT, IFG/DM, and DM (at P < 0.01), NFG/IGT, IFG/NGT, and IFG/IGT differed from IFG/DM and DM (P < 0.01), and IFG/DM differed from DM (P < 0.005); 2) EC50 shift, wherein NFG/NGT differed from IFG/NGT, IFG/IGT, IFG/DM, and DM at P < 0.05; 3) basal C-peptide secretion, wherein the DM value (P < 0.001) exceeded that of all others; and 4) sensitivity of C-peptide secretion to glucose stimulation, which was lower in IFG/DM than any of the first four subgroups (P < 0.05), and lower in DM than IFG/IGT (P < 0.05). In contrast, maximal (efficacy of) glucose-induced C-peptide secretion was similar. A systematic trend existed for glucose EC50, such that values were higher in the recovery than in the onset phases in each glycemic stratum except IFG/DM (no difference) and DM (significantly lower recovery than onset EC50). The result was reversal of potency-shift direction (algebraic difference between rising and falling glucose EC50 values) from a rightward shift in NFG/NGT (−2.25 mM glucose) to a leftward shift in IGT/DM (+0.37 mM glucose) and DM (+4.54 mM glucose) (both < 0.002 vs. NFG/NGT) (Fig. 4). Compared with DM, there were 1) lower basal C-peptide secretion rates in all other categories (P ≤ 0.005), 2) greater sensitivity to glucose in IGF/IGT (P ≤ 0.05) but not in others, 3) lower onset (but not recovery) glucose EC50 values in all other categories (P ≤ 0.003), and 4) lesser rightward shifts in glucose EC50 in all non-DM categories except IFG/DM (P < 0.05) (Table 2). The hysteresis onset time for DM was significantly delayed compared with the other five subgroups except IFG/IGT (P < 0.01).
Salient outcomes of the present model-based analyses are that 1) estimated C-peptide secretion rates can be related nonlinearly to sequential glucose concentrations via onset and recovery (hysteretic) dose-response functions; 2) glucose-dependent C-peptide hysteresis declines significantly across NFG, IFG, and DM, resulting in complete reversal of the direction of hysteretic shifts in DM (respectively, Δvalues of −2.0, −0.80, and 4.54 mM glucose); and 3) basal secretion (per min) of C-peptide (0.028 nM) is markedly higher in DM than in IFG (0.012 nM) and NFG (0.007 nM), delineating impaired islet regulation. Accordingly, we propose that a suitably framed analytical formalism can allow 1) estimation of distinguishable in vivo dose-response potencies during the onset and recovery phases of glucose/meal-stimulated C-peptide secretion, and 2) quantification of strong differences in pathway dynamics (hysteretic shifts) among NFG, IFG, and DM, using sparse data typical of clinical glucose tolerance testing. Moreover, the dose-response model exhibited good test-retest reliability in 10 healthy adults each studied twice.
DM compared with NFG/NGT was characterized by an elevation of basal C-peptide secretion, decreases in the onset, recovery, and hysteretic difference (shift) of glucose's potency in stimulating C-peptide secretion, a delay in the time of the C-peptide secretory maximum (hysteresis inflection point), and paradoxical potentiation rather than inhibition of glucose hysteresis. Exploratory subgroup analyses of NFG/NGT, NFG/IGT, IFG/NGT, IFG/IGT, IFG/DM, and DM permitted finer pathophysiological distinctions. In particular, DM was further distinguishable from all other subgroups by reduced potency of glucose's initial stimulation of C-peptide secretion, e.g., DM compared with IFG/DM yielded a higher EC50 of 18.5 ± 2.7 vs. 11.2 ± 1.3 mM glucose.
The analytical framework developed here required several innovations. First, because sampling intervals were on the order of the C-peptide slow half-life, a procedure was needed to capture sufficiently the secretion and elimination that proceed de facto between sampled times. Because of the specific nature of the experimental setting, the concentrations were interpolated to the 5-min scale, allowing more accurate estimation of secretion rates. Interpolation was justified by comparison of interpolated and actual 5-min sampling in 10 subjects. Second, a wavelet-based regularization method was utilized to estimate C-peptide secretion rates using the interpolated hormone concentrations and populational estimates of peptide half-lives (see appendix). An assumption was that C-peptide secretion is a suitable surrogate of portal-venous insulin secretion, given the former's relative robust kinetics and low hepatic extraction (16). Third, a logistic dose-response construct was developed to accommodate nonlinearity and saturability of putative effector-response linkages (21, 31). Maximum-likelihood estimation was implemented to estimate all dose-response parameters and the time of putative dose-response reversal (hysteresis inflection) simultaneously. The combined strategies allowed calculation of endogenous potencies of effector-regulated peptide secretion before and after a hysteresis changepoint, an estimated changepoint time, peptide secretory sensitivity to changing effector concentrations, and the efficacy of an effector in stimulating peptide secretion. Thus, these methods may have utility in other investigative contexts.
Objective estimation of the time interval from meal onset to hysteretic recovery of dose-response functions disclosed a significantly greater delay for glucose-driven C-peptide secretion in DM compared with NFG/NGT, NFG/IFG, and IFG/NGT (i.e., 116 vs. 88, 78, and 84 min, respectively). The exact basis for selectively prolonged glucose/C-peptide dose-response delays is not yet known.
The present in vivo estimate of the potency of glucose in driving C-peptide secretion in NFG/NGT (EC50 = 6.0 mM) is similar to the Km reported for glucokinase (6.0 mM) in human islets (36). A strong rightward shift in the recovery glucose/C-peptide dose-response curve (denoting a 2.25 ± 0.37 mM reduction in glucose potency) was observed ∼90 min after meal ingestion. A rightward hysteretic shift also occurs in the rat in vivo after prolonged fasting (5), and in human pancreata perfused ex vivo after glucose exposure (27). DM was associated with a complete reversal of normal glucose/C-peptide hysteresis, namely a major leftward shift reflecting a 4.54 mM enhancement of glucose potency. Exploratory analysis of other dysglycemic states revealed intermediate hystereses during oral-glucose exposure.
Caveats include the relatively small cohorts studied (NFG, n = 30; IFG, n = 32; and DM, n = 14), response heterogeneity among subjects, irregular hysteretic profiles in some patients with DM, and limitations of the glucose-containing mixed-meal paradigm itself (4). Consequently, outcomes should be confirmed in larger groups of subjects and as well after glucose administration in the absence of a meal.
Finally, to compare the mixed-meal results with those from the application of the IVGTT, IVGTT data were analyzed by the model. There were 62 young, normal subjects (31 males, 31 females), with C-peptide and glucose concentrations measured over 4 h at 25 time points (0, 2, 4, 6, 8, 10, 15, 20, 22, 25, 26, 28, 31, 35, 75, 90, 120, 180, and 240 min). As was done for the mixed-meal data, the concentrations were interpolated to equally spaced 5-min values. The C-peptide secretion rates were estimated by previously published methods that have been widely applied (35). In Fig. 6, the results for four (of the 62) subjects are displayed. In rows 1 and 2 are the C-peptide concentrations and secretion rates, respectively. The glucose concentrations are plotted in row 3. In row 4, the dashed black curve represents the C-peptide secretion (y-axis) and glucose concentration (x-axis) values evolving over time. The two-potency dose-response model is displayed as solid red curves, with green asterisks (*) representing the time point at which the hysteresis occurs (i.e., the switch from one curve to the other). Estimated dose-response parameters for the 62 subjects are given in Table 3. For both the male and female groups, there were statistically significant positive EC50 shifts (hysteresis), as there were for the NFG/NGT subjects in the mixed meal. There was no detectable difference in the EC50 shifts between sexes.
Because of the nature of the IVGTT test, the recovered C-peptide secretion rates appear to primary reflect first-phase C-peptide/insulin release. The concentrations of C-peptide and glucose were larger than those observed in the mixed meal, as were the EC50 onset, recovery, and difference values. The mean onset time of hysteresis for the 62 subjects is ∼20 min (21.0 min in males, 19.6 min in females) compared with 84 min for the mixed-meal data for NFG subjects. This difference may reflect IVGTT's rapid first-phase effect and mixed meal's emphasis on the second phase of secretion, as well as possible influences by gut hormones after meal ingestion.
In conclusion, analytical estimation of pairwise glucose and C-peptide dose-response relationships identifies respective loss and reversal of glucose-induced C-peptide hysteresis in IFG and DM compared with NFG, delayed hysteresis and elevated basal C-peptide secretion in DM, and attenuation of maximal C-peptide secretion in DM compared with IFG. More rapid onset of hysteresis occurs after IVGTT. Thus, quantification of hysteretic (onset and recovery) dose-response linkages for glucose-C-peptide stimulation may offer a framework for methods to enhance discrimination among prediabetic states.
This work was supported in part via the Center for Translational Science Activities (CTSA) Grant no. 1 UL 1 RR-024150 to the Mayo Clinic and Foundation from the National Center for Research Resources (Rockville, MD), R01 National Institute of Diabetes and Digestive and Kidney Diseases DK-073148 and DK50456 and R21 National Institute on Aging AG-029215 from the National Institutes of Health (Bethesda, MD), and DK-29953 from the US Public Health Service.
No conflicts of interest, financial or otherwise, are declared by the author(s).
Author contributions: D.M.K. and J.D.V. analyzed data; D.M.K. and J.D.V. interpreted results of experiments; D.M.K. and J.D.V. edited and revised manuscript; D.M.K., R.B., Y.L., A.B., G.B., and J.D.V. approved final version of manuscript; R.B., Y.L., A.B., G.B., and J.D.V. conception and design of research; R.B., Y.L., A.B., and G.B. performed experiments; J.D.V. and D.M.K. drafted manuscript.
We thank Donna Scott and Jill Smith for support of manuscript preparation, Ashley Bryant for data analysis and graphics, the Mayo Immunochemical Laboratory for assay assistance, and the Mayo research nursing staff for implementing the protocol. We thank Dr. Robert Rizza (Mayo) for enlightening discussions.
Estimation of the dependency of C-peptide secretion rates on glucose concentrations, only concentrations of the two having been observed, required a novel approach. From experience with the modeling of testosterone secretion as a function of LH as well as cortisol secretion as a function of ACTH, estimation methods were formulated to recover secretion rates first and then model interdependency secondarily (35). Various methods have been developed to recover a secretion rate from the observed concentrations. Let X(t) and Z(t) denote the concentration and secretion at time t. Suppose that the fast and slow rates of elimination are (α(1), α(2)), with their fractions being, respectively, a and (1 − a). The observations are made at discrete times. Suppose for the moment that they are equally spaced: t1, t2, …, tn, Δt = ti − ti−1, and let ã(1) = exp(−α(1) Δt, ã(2) = exp(−α(2) Δt. An exact solution of the corresponding differential equation model is the integral equation solution:(1)
Most endocrine modeling to date has been directed toward pulsatile hormones, for which finite-parameter models are well established. The latter approach makes the deconvolution of the secretion rate Z well behaved. In the present sampling context, however, there are no generally accepted finite-parameter models for the secretion rates of C-peptide. As an alternative, to make the deconvolution well behaved, one can implement a regularization procedure. The basic idea of regularization is to construct, as a function of the amount of observed data (n), a parameter space whose dimension grows with, but at a slower rate than, n and which has asymptotic properties like smoothness. One simple parameter space of allowable secretion rates would be a Fourier representation (using a sum of sine and cosine functions). The difficulty with a Fourier approach is that it often yields an inordinate number of both high and low frequencies. Another consideration, implemented here, is a regularization procedure using wavelet basis functions to represent the unknown secretion rates (Z) in Eq. 1. This allows for both high and low frequencies with fewer terms. Haar functions were used in the wavelet formulation (24). Moreover, the requirement of nonnegativity of the estimated secretion rates (Z) is itself a regularizing property (12). In the case of Haar functions, which are piece-wise constant, Eq. 1 reduces to(2)
and, hence, no numerical integration is required.
The present objectives were to obtain secretion rates (Z) of C-peptide and then model their dependency on glucose concentrations. To this end, we assume literature-estimated population fast (α(1)) and slow (α(2)) rates of elimination and corresponding partitioning coefficients (a(1) and a(2)) for C-peptide, i.e., rapid half-life of 4.95 min, fractional slow/total amplitudes of 0.24, and slow half-life of 32 min (16). What one observes are C-peptide concentrations at the sampled time points plus random error:(3)
where the ε's are IID Gaussian values, with mean zero and variance σ2Cpep. The resulting likelihood functions (L) for C-peptide secretion is Gaussian. To regularize the estimation, we append a penalty term to the (minus) log-likelihood function; in the present case, the penalty function (Ω, below) involves the integral of the squared second derivative of Z (the secretion rate being estimated). The minus sign corresponds to the fact that most maximizations are actually implemented as minimization of the negative. The second-derivative constraint, which forces regularity onto the solution and makes the estimation well posed, was one of the first formulations of regularization. The function minimized is(4)
Z(t) ≥ 0, for all t, and the ψj's are the Haar wavelet bases.
One issue in regularization is how to choose the value λ. A theory of such, cross-validation, was developed in the 1970s to answer this, but it is quite difficult to apply except in the simplest of linear models. In practice, λ is chosen by trial and error, which actually works quite well. In the present setting, λ = 20 was used for C-peptide (again, chosen by trial and error).
Denote the solutions as (ZCpep,i, i = 1, …, n). Thereby, we have estimated the secretion rates independently of their relationship to glucose feedforward (on C-peptide). We now model this dependency. The corresponding glucose concentrations are denoted by YGlc,i. To quantify pairwise coupling of putative effectors and responses, the second stage of analysis employed a four-parameter logistic dose-response model. The model comprises exponential potency terms A1 or A2 for the onset and recovery, respectively, of stimulation; B (sensitivity, a measure of maximal absolute slope); C (efficacy or asymptotic response); and D (baseline). Physiological potency is calculated from these coefficients as A1/B and A2/B. These ratios denote the magnitude of the dose-response input, glucose concentration, exerting 50% of the stimulatory effect, EC50. For regression of C-peptide on glucose, B is positive (i.e., stimulatory), since the dose-response curve slopes upward. A unique feature is inclusion of a simultaneously estimated time, T, when the EC50 or IC50 changes from A1 to A2. These relationships are given by(5)
The no-hysteresis (one-potency) models are given by6
To test the appropriateness of 5-min interpolation, GTT data obtained from 10 normal subjects each studied twice after 75 g of glucose were analyzed. In the 20 data series, primary measurements were made every 5 min from 0 to 120 min (where 0 denotes glucose ingestion) and then every 10 min from 120 to 240 min (37 time points all together). The last 2-h 10-min data were first interpolated to a 5-min scale (resulting in 49 time points). The 5-min data were then used in two ways: first, deconvolution and dose-response analyses were performed on the 5-min data (49 values); second, the 5-min data were subsampled to match the exact times obtained in the NFG and IFG patient series (13 time points) and then reinterpolated back to the 5-min level. Figure A2A presents primary 5-min concentrations (solid lines) and secondarily interpolated concentrations (dashed lines) in two healthy subjects. Figure A2B shows, along with primary 5-min data (solid), reinterpolated 5-min data (dashed), the estimated secretion rates for C-peptide, the corresponding glucose concentrations, and the two-potency (hysteresis) dose-response functions. Comparing parameter estimates for the two (visit and revisit) sessions using t-values showed that none was significant at the 0.01 level, and two were significant at the 0.05 level. In Fig. A3, for six of the ten individuals (distinct from the previous two), the two-potency dose-response functions are shown for C-peptide secretion on glucose concentration. Based on a Tukey-Studentized test value, none of the 28 was significant at 0.05. Moreover, Spearman's nonparametric rank correlation between matching dose-response parameter estimates revealed good test-retest reliability (Table A1). Figure A4 displays the regressions of the estimated parameters from the visit one two-potency models vs. that parameter in the visit two two-potency model, with the estimated correlations denoted. The strongest test-retest correlations were observed for the efficacy (r = 0.879) and sensitivity (r = 0.697) of glucose in driving C-peptide.
The extent of abnormal dynamics in various subgroups can be explored by creating a symmetric interval estimate (estimated mean ± 2× estimated SD) of a 95% probability interval for the populational distribution of each parameter for NFG/NGT rather than a confidence interval for the mean. Because individual parameter distributions have substantial overlap between subgroups, multiple parameters may be required to reduce the probabilities of misclassification. To illustrate, Fig. A5 plots C-peptide logistic dose-response parameter estimates for three parameters: EC50 shift (hysteresis), x-axis; basal secretion rate, y-axis; and EC50 onset, z-axis. The subpanels are: top left, NFG/NGT, NFG/IGT, and IFG/NGT; top right, NFG/NGT and IFG/IGT; bottom left, NFG/NGT, IFG/IGT, and IFG/DM; and bottom right, NFG/NGT and DM. Such plots suggest the potential to develop prediabetes classifiers built upon multiple parameters.
- Copyright © 2012 the American Physiological Society