## Abstract

The fasting proinsulin-to-insulin ratio is a currently used marker of β-cell dysfunction. This ratio is calculated at the basal condition, but its behavior in dynamic conditions, i.e., during glucose stimulation, could be more informative. Given the different kinetics of the peptides, a mathematical model was necessary to analyze the oral glucose tolerance test (OGTT) data of insulin, C-peptide, and proinsulin in 55 healthy (NGT), 30 impaired glucose-tolerant (IGT), and 31 type 2 diabetic (T2DM) subjects. The model provided for secretion and disappearance of the peptides and an index of β-cell function under dynamic conditions. Total proinsulin secretion during the OGTT was not different (*P* > 0.053) among NGT (0.17 ± 0.01 mmol/l in 3 h), IGT (0.22 ± 0.02), and T2DM (0.21 ± 0.02) subjects. The proinsulin-to-insulin molar ratio measured from basal samples was higher (*P* < 0.0001) in T2DM (0.39 ± 0.05) than in NGT (0.14 ± 0.01) and IGT (0.13 ± 0.02) subjects, and similar results (*P* < 0.003) were found by the dynamic index (0.27 ± 0.04, 0.14 ± 0.01, 0.15 ± 0.01 in T2DM, NGT, IGT subjects, respectively). The basal ratio significantly correlated with the dynamic index, and the regression line slope was lower than 1 (0.43 ± 0.08, 0.61 ± 0.10, and 0.56 ± 0.03 in NGT, IGT, and T2DM subjects, respectively, *P* < 0.0001). Impaired β-cell function in T2DM could then be indicated by proinsulin-to-insulin indexes at both basal and dynamic phases.

- proinsulin secretion
- proinsulin clearance
- impaired glucose tolerance
- type 2 diabetes
- mathematical modeling

proinsulin is a stable precursor of insulin, the hormone modulating blood glucose. Inside the pancreatic β-cell, proinsulin is synthesized and then transformed into mature insulin, ready to be released into the blood-stream, mainly under glucose stimulation. However, not all of the proinsulin becomes insulin, and a low percentage of the secretion product still contains proinsulin-like molecules (7). In terms of blood concentration, proinsulin accounts for 10–20% of normal fasting insulin, but it may reach values as high as 50% in type 2 diabetes, leading to the hypothesis that conversion from proinsulin to insulin is impaired in this disease (23). Thus, to detect possible β-cell dysfunction, the fasting proinsulin-to-insulin molar ratio is commonly computed. However, this index may be misleading, because elevated levels of proinsulin could also be due to altered kinetics (for instance, reduced clearance). A recent study (15) showed that, under maximal stimulation of insulin secretion induced by secretagogues such as glucagon-like peptide-1 (GLP-1), the proinsulin-to-insulin molar ratio decreases in impaired glucose-tolerant subjects, eventually reaching values not different from those of normal glucose-tolerant subjects. This suggests that caution should be used with the basal proinsulin-to-insulin ratio as a marker of β-cell defect. Therefore, studying the behavior of this ratio in dynamic conditions seems to be paramount.

The present study had three aims, the first being to develop a method for the investigation of proinsulin secretion and kinetics in individual subjects during an oral glucose tolerance test (OGTT), the simple test for diagnosing metabolic disorders. For this purpose, we developed a mathematical model that, incorporating known physiological evidence, describes the peripherally measured plasma proinsulin concentration, yielding measurements of its dynamic secretion and clearance. The second aim was to compute indexes of β-cell function under hyperglycemic stimulation of the β-cell and to compare them with the commonly used fasting proinsulin-to-insulin molar ratio. The third aim was to examine the proinsulin kinetic behavior and the proinsulin-based indexes of β-cell function in different groups of subjects (control, former gestational diabetic, and type 2 diabetic) to evaluate the differences in these indexes in the progression of the metabolic derangement (22, 23).

## MATERIALS AND METHODS

*Subjects.* A total of 116 volunteers participated in this study. Control subjects were 55 young females with normal glucose tolerance [NGT, age 30 ± 2 yr, body mass index (BMI) 24 ± 1 kg/m^{2}]. Thirty females were former gestational diabetic subjects, studied 4–6 mo after delivery (age 31 ± 2 yr, BMI 28 ± 1 kg/m^{2}), all with impaired glucose tolerance (IGT) according to criteria of the Expert Committee on the Diagnosis and Classification of Diabetes Mellitus (16). The third group included 31 subjects with early type 2 diabetes (T2DM; age, 55 ± 3 yr; M/F, 21/10; BMI, 28 ± 1 kg/m^{2}; fasting glucose, 166 ± 14 mg/dl; glycated hemoglobin, 8.0 ± 0.5%). These patients were on diet therapy and without any diabetic complications or atherosclerotic lesions and with normal kidney, thyroid, and liver function. No subject showed any clinical sign of inflammatory processes or any pathological laboratory test pointing to a subclinical infection at the time of the study. The subjects were recruited from the outpatient department of the University Clinic of Vienna. Some of the subjects have been part of already published studies (10, 11).

*Test and assays.* All subjects underwent an OGTT. After an overnight fast and a collection of basal samples, the subject drank water in which 75 g of glucose had been diluted. Blood samples for the determination of plasma concentrations of glucose, insulin, C-peptide, and proinsulin were taken at 10, 20, 30, 60, 90, 120, 150, and 180 min. Blood was rapidly centrifuged, and plasma was properly stored for the following assays. Insulin (Serono Diagnostics, Freiburg, Germany), C-peptide (CIS Bio International, Gif-sur-Yvette, France), and proinsulin (Linco Research, St. Charles, MO) were determined in duplicate by commercially available radioimmunoassay kits with an interassay coefficient of variation (CV) of <5% for insulin and C-peptide and <8% for proinsulin. Cross-reactivity for C-peptide and insulin assays was <12% for human proinsulin (HPI); cross-reactivity of the proinsulin assay was 95% for HPI Des 31,32, or 100% for HPI, and <0.1% for HPI Des 64,65, for insulin and for C-peptide. Glycated hemoglobin (upper limit of normal range 5.8%) was quantified by on-line high-pressure liquid chromatography (C-R4A Chromatopac; Shimadzu, Kyoto, Japan) from capillary blood.

*Mathematical model.* Concentration data of insulin, C-peptide, and proinsulin were analyzed by means of a mathematical model that yields parameters related to insulin secretion, clearance, and hepatic extraction, and proinsulin kinetics.

Proinsulin and insulin are not released equimolarly, and the plasma concentration of proinsulin is usually only a small proportion of that of insulin (7), with clearance of proinsulin being lower than that of insulin (24). In the present study, proinsulin secretion was assumed to be proportional to that of C-peptide and insulin (which are released equimolarly). The kinetics of the three peptides are described by linear time-invariant first-order equations 1 2 3 where CP(*t*), I(*t*), and PI(*t*) are measured plasma concentrations (pmol/l) of C-peptide, insulin, and proinsulin, respectively. *Equations 1* and *2* assume that insulin and C-peptide secretion rate per unit volume can be represented by a smooth function CPS(*t*) (pmol·l^{-}^{1}·min^{-}^{1}), expressed by continuous piecewise polynomials (splines). Liver degradation of insulin is accounted for by *F* (nondimensional), representing the fraction of the secreted hormone that reaches the peripheral circulation, where it is measured. Parameters *k*_{CP} and *k*_{I} are the fractional clearance rates (min^{-}^{1}) of C-peptide and insulin, respectively. This model of insulin secretion and kinetics was introduced in a previous study (18) and validated against direct measurements obtained with hepatic arteriovenous differences (20).

*Equation 3* describes the proinsulin kinetics. A block diagram of the model is depicted in Fig. 1. Proinsulin passes through the liver, enters the peripheral circulation, and is finally cleared by the kidneys. The same compartmental model structure found valid for the other two peptides was adopted also for proinsulin. Zilker et al. (24) described with a triexponential function the decrease of bolus-injected biosynthetic human proinsulin that would yield a three-compartment model. However, in that study, proinsulin reached supraphysiological levels, whereas a monocompartmental model is deemed acceptable when the low levels and the slow dynamics proper of an OGTT are considered, as already discussed for insulin, C-peptide, and amylin dynamics (5, 18, 19). *Equation 3* incorporates two terms, one related to its secretion assumed to be proportional to that of C-peptide, *f*_{PI}·CPS(*t*), with *f*_{PI} being the proinsulin-insulin cosecretion factor (nondimensional) that also accounts for possible liver degradation, and the other term describing proinsulin disappearance, *k*_{PI}·PI(*t*), with *k*_{PI} the fractional clearance rate (min^{-}^{1}). The initial conditions of the three differential equations are given by the basal level of the three peptides measured immediately before the administration of the oral glucose load.

*Equation 3* should be considered as an approximation of dynamic proinsulin secretion and elimination during the OGTT, rather than a general constraint to be valid also during the basal state. In fact, the molar ratio of proinsulin to insulin secretion is progressively reduced during long-lasting stimulated conditions (15). The cosecretion factor *f*_{PI} quantifies, therefore, an average molar ratio of proinsulin to insulin secretion during the dynamic phase of the OGTT, and it is not constrained by basal concentrations.

To solve model *Eqs. 1–3* for any single subject, parameter *k*_{CP} was fixed to a constant value, equal to 0.0615 min^{-}^{1}, as derived from a large number of previously studied subjects (e.g., Refs. 2 and 9). From *Eqs. 1* and *2*, CPS(*t*) and the values of *F* and *k*_{I} were estimated such that they yield the best fit between measured and model-reconstructed C-peptide and insulin plasma concentrations. CPS(*t*) was used as forcing input in *Eq. 3* to estimate proinsulin kinetic parameters *f*_{PI} and *k*_{PI.} The total amount of secreted insulin (TIS; pmol/l in 3 h), was computed by integrating CPS(*t*) over the 180-min interval of the OGTT. The total amount of secreted proinsulin (TPS; pmol/l in 3 h), was then computed as the product between TIS and *f*_{PI}. The distribution volumes of the peptides were not taken into account, because only fractional parameters were considered.

The implementation of the model equations was performed using the software tool PANSYM (17); parameter estimations were performed using MATLAB (The Mathworks).

*Model-based metabolic indexes. Equations 2* and *3* can be used to find an expression of proinsulin-to-insulin molar ratio under glucose stimulation of the β-cell. We introduce the hypothesis that a constant glucose stimulation eventually would lead to a new steady state, different from the basal. *Equations 2* and *3* then yield 4 5 where I_{ss} and PI_{ss} would be the new steady-state insulin and proinsulin plasma concentrations, respectively; CPS_{ss} is C-peptide secretion rate at the new steady state, and the model parameters *F, k*_{I}*, f*_{PI}*, k*_{PI} are those estimated from the dynamic data measured during the glucose stimulation.

From *Eqs. 4* and *5*, by simplifying for CPS_{ss}, we obtain 6

We used the model parameters *f*_{PI}*, k*_{PI}*, F,* and *k*_{I}, estimated during the OGTT, to compute the index of proinsulin-to-insulin molar ratio under glucose stimulation according to *Eq. 6*. Thus, despite the fact that this index can be influenced by the basal proinsulin-to-insulin molar ratio, PI_{b}/I_{b}, it is computed under glucose stimulation, and hence it can be different from the basal ratio. Actually, the OGTT does not strictly lead to new steady state; however, because of its slow dynamics, we assume that the model parameters estimated from the OGTT yield an acceptable approximation. Furthermore, an actual new steady-state condition is difficult to obtain in vivo, unless a long and complex experiment is performed.

The proinsulin-to-insulin molar ratio is commonly used to evaluate possible defects in the β-cell function. However, the information that it provides is influenced by the liver extraction of insulin. Therefore, to have a more direct index of β-cell function, we introduce another index, based on proinsulin and C-peptide, which is not extracted by the liver. From *Eqs. 1* and *3*, on the basis of similar considerations that yield *Eq. 6*, a new index can be computed as 7

To evaluate possible differences in basal or stimulated conditions, the model-derived indexes PI_{ss}/I_{ss} and PI_{ss}/CP_{ss} were compared with the corresponding basal ratios, PI_{b}/I_{b} and PI_{b}/CP_{b}, respectively.

*Data analysis.* The areas under the curves were computed by integration with the trapezoidal rule. All data and results are presented as means ± SE unless otherwise designated. The unpaired *t*-test was used for statistical comparisons among the different groups, whereas the paired *t*-test was used within the same group. Agreement between each basal ratio and corresponding dynamic index was investigated according to the Bland-Altman method (3). Regression between each basal ratio and corresponding index in dynamic, stimulated conditions was studied by using one-way analysis of covariance (1) with separate slopes: a linear model was used to fit the response (the model-based index), and two dummy variables were introduced to take into account the group nominal effect with its three levels (NGT, IGT, and T2DM). This yielded a different slope of the regression line for each of the three groups of subjects. Age and BMI of the subjects were also included as added covariate regressor terms in the model. However, because they did not reach statistical significance, these two covariates were removed in the final analysis. The regression line intercept was assumed equal for the three groups.

## RESULTS

*OGTT.* In Table 1, the basal values of insulin, C-peptide, and proinsulin, as well as the corresponding area under the concentration curves (AUC) during the OGTT, are reported for the three groups of subjects. Basal insulin and C-peptide were lower in NGT than in IGT and T2DM (*P* < 0.01). Basal proinsulin was higher in T2DM than in IGT and NGT (*P* < 0.0001), and in IGT slightly higher than in NGT (*P* = 0.04).

The AUC of insulin was lower in T2DM than in IGT (*P* < 0.005), but not lower than in NGT (*P* = 0.08). There was no significant difference between NGT and IGT (*P* = 0.4). The AUC of C-peptide was lower in T2DM than in both IGT and NGT (*P* < 0.02), and in NGT lower than in IGT (*P* < 0.007). The AUC of proinsulin was not significantly different between T2DM and IGT (*P* = 0.4), but in both was higher than in NGT (*P* < 0.007).

The instant-by-instant molar ratio of plasma concentration of proinsulin and insulin during the OGTT resulted in a U-shaped curve, which shows that the molar ratio is not constant during dynamic conditions (Fig. 2).

*Model parameters.* The model successfully reproduced the experimental plasma concentration data for any single individual of the various groups. Figure 3 shows the mean model-reconstructed patterns of insulin, C-peptide, and proinsulin concentration curves during the OGTT for each group of subjects.

The values of the model-estimated parameters are reported in Table 2. The coefficients of variation (CV) of the estimates, which were derived from the parameter covariance matrix, are shown in Table 3, which shows that parameter estimation was acceptable in terms of precision.

Neither the hepatic insulin fractional extraction, nor the insulin and proinsulin fractional clearances, were significantly different in the three groups (Table 2) (*P* = 0.08, *P* = 0.26, and *P* = 0.12, respectively). On the contrary, the proinsulin-insulin cosecretion factor was found to be higher in T2DM subjects with respect to NGT and IGT subjects (*P* < 0.04). On the other hand, the total amount of insulin secretion was found to be higher in IGT subjects than in other groups (*P* < 0.01). When the cosecretion factor is multiplied by the total amount of insulin secretion to obtain total proinsulin secretion, the results yield a nonsignificant difference for any group (*P* = 0.053).

The patterns of insulin and proinsulin secretion rate during the OGTT for the three groups are reported in Fig. 4.

*Basal ratios and dynamic indexes.* Comparisons between the basal proinsulin-to-insulin molar ratio from experimental data and the index in dynamic condition due to OGTT stimulation (*Eq. 6*), and between the basal proinsulin-to-C-peptide molar ratio and the index from *Eq. 7* are reported in Table 4. PI_{b}/I_{b} was found to be higher in T2DM than in the other groups (*P* < 0.0001). A similar result was found when the index *f*_{PI}·*k*_{I}*/F*·*k*_{PI} (*P* < 0.003) was considered. Also, PI_{b}/CP_{b} was higher in T2DM (*P* < 0.001), as well as the index *f*_{PI}·*k*_{CP}*/k*_{PI} (*P* < 0.0007).

The mean basal proinsulin-to-insulin molar ratio was not different from the corresponding dynamic index in NGT (*P* = 0.6) and IGT (*P* = 0.09), whereas T2DM PI_{b}/I_{b} was higher (*P* < 0.0001). When the proinsulin-to-C-peptide ratio was considered, statistical difference was found in NGT (*P* < 0.0001) and IGT (*P* < 0.0001) but not in T2DM (*P* = 0.3). The Bland-Altman plot is shown in Fig. 5. The limits of agreement between the basal proinsulin-to-insulin ratio and corresponding index are not negligible with respect to the mean values of the compared variables (see Table 4). As a consequence, basal ratios and corresponding indexes cannot be considered equivalent. Similar results were found for the proinsulin-to-C-peptide ratio (not shown). Lack of equivalence was confirmed by regression analysis. However, although the slopes of regression line were significantly different from 1, a significant relationship was found between each basal ratio and the corresponding index (Fig. 6). The slope for the proinsulin-to-insulin ratio was 0.43 ± 0.08 (*P* < 0.0001) for NGT, 0.61 ± 0.10 (*P* < 0.0001) for IGT, and 0.56 ± 0.03 (*P* < 0.0001) for T2DM; that for the proinsulin-to-C-peptide ratio was 0.52 ± 0.07 (*P* < 0.0001), 0.76 ± 0.10 (*P* < 0.0001), and 0.65 ± 0.03 (*P* < 0.0001), respectively.

## DISCUSSION

Proinsulin is used to evaluate β-cell function, because its elevated plasma levels are considered a sign of the worsening of the pancreatic activity (11, 23). In particular, the fasting proinsulin-to-insulin molar ratio is used as a marker of β-cell impairment (22). However, this ratio is based on the concentration of insulin and proinsulin in the periphery; thus it cannot directly indicate a possible defect at the β-cell level, because it can be affected by several other factors of insulin and proinsulin kinetics that can vary significantly in different conditions (20). To provide an index of β-cell function based on estimation of both secretion and kinetics of the peptides, mathematical modeling is necessary (6, 13, 18, 21).

To our knowledge, this study is the first to investigate endogenous proinsulin behavior during a dynamic test by exploiting mathematical modeling. The model well reconstructed the measured patterns of insulin and proinsulin peripheral concentration (Fig. 3), as assessed by the value of residuals (not shown). The parameter estimates were evaluated in terms of precision (4), as shown in Table 3. The CV of the estimates of insulin secretion, proinsulin-insulin cosecretion factor, and proinsulin clearance were satisfactory, although they slightly worsen in T2DM. As already observed in other studies (20), estimates of insulin hepatic extraction and insulin clearance were higher, but still acceptable.

Previous pharmacokinetic investigations performed by Galloway et al. (8) showed that proinsulin clearance ranged between 2.7 and 3.7 ml·kg^{-}^{1}·min^{-}^{1}. With the assumption of a proinsulin distribution volume of 99 ml/kg (8), proinsulin fractional clearance ranged between 0.027 and 0.037 min^{-}^{1}, in good agreement with that estimated by our model (*k*_{PI}, Table 2). Another study (24) found proinsulin clearance of 120 ml/min and proinsulin distribution volume of 9,300 ml, thus yielding proinsulin fractional clearance of 0.013 min^{-}^{1}. The slight discrepancy with our *k*_{PI} can be due to the administration of a pharmacological dose of proinsulin that yielded after 3-h serum proinsulin levels ∼400 pmol/l, which are 40 times higher than endogenous values. To the best of our knowledge, no studies in humans have reported direct assessments of the proinsulin-insulin dynamic cosecretion factor and the proinsulin secretion rate into the portal vein.

Our experimental findings confirm the increased fasting proinsulin-to-insulin molar ratio observed in T2DM (23). Knowing that insulin secretion at the β-cell level is better described by C-peptide, we have also computed the fasting proinsulin-to-C-peptide molar ratio, which was higher in T2DM as well. The difference between the two ratios is the influence of hepatic extraction. However, the two were found to behave similarly.

To overcome the limits of the fasting proinsulin-to-insulin molar ratio, we investigated the proinsulin-to-insulin molar ratio under a dynamic condition due to OGTT stimulation. However, this ratio changes significantly during the OGTT (Fig. 2) because of the different kinetics of the two substances. Because an index continuously changing in time cannot be of practical usefulness, the problem arises of finding a method to compute a compact, unique index, also during the OGTT. To this aim, we derived model-based indexes of proinsulin-to-insulin, as well as proinsulin-to-C-peptide, relationships.

Despite different kinetics of the peptides, the dynamic indexes and fasting molar ratios yielded the same information regarding the different groups of subjects: in particular, both indexes were found to be higher in T2DM, as expected (see Fig. 6 and Table 4). However, basal and dynamic indexes were not equivalent. Although significant difference in the mean values of basal and corresponding indexes was not always found, the lack of equivalence was shown by both the Bland-Altman method and regression analysis. The dynamic indexes remain lower than basal ratios, but regression lines were found statistically significant and different from zero. Our results, showing indexes of proinsulin-to-insulin molar ratio under stimulated conditions lower than the basal ratios, confirm the findings of several studies. Nagamatsu et al. (12) observed that prior exposure of isolated islets to a sufficiently high stimulus of glucose does not affect the proinsulin synthesis but increases the conversion rate of proinsulin to insulin, thus yielding a decreased proinsulin-to-insulin molar ratio. Rhodes and Halban (14) found no increase in intracellular proinsulin-to-insulin conversion under glucose stimulation of isolated islets but did observe an increased release from the islets in the medium of both proinsulin and insulin. The rate of increase of insulin was higher than that of proinsulin, thus again yielding a decreased molar ratio (14). A similar decrease during β-cell stimulation was obtained in vivo by Stumvoll et al. (15) during hyperglycemic clamp with additional administration of GLP-1 and arginine (15).

It is worth noting that mathematical modeling allows segregation of the main metabolic processes (secretion and disappearance) in determining a given value of proinsulin-to-insulin or proinsulin-to-C-peptide ratios. For instance, higher values of the proinsulin-to-insulin molar ratio in T2DM appear to be due mainly to a higher proinsulin-insulin cosecretion factor. Although a 30% difference in the cosecretion factor, and no difference in the other parameters involved in the index computation, results in an index more than doubled, there is no contradiction, because parameters are combined nonlinearly.

Insulin secretion (TIS) in IGT was higher than in NGT and T2DM. This suggests that the β-cell function is not impaired and that the augmented secretion is presumably due to the still-effective mechanism that compensates for the increased insulin resistance. This compensatory mechanism, instead, appears to fail in T2DM.

In conclusion, we have introduced a mathematical model that allows the investigation of proinsulin secretion and clearance in a single subject during an OGTT. Dynamic indexes are correlated with basal proinsulin-to-insulin molar ratios, which provide similar information. Thus the basal proinsulin-to-insulin ratios can still be useful in characterizing the β-cell status, even when compared with more sophisticated indexes.

## Acknowledgments

We acknowledge useful discussions with A. Brazzale and A. Mari of the Institute of Biomedical Engineering (ISIB) of the Italian National Research Council (CNR).

This work is part of an ongoing cooperative project between ISIB-CNR (formerly LADSEB-CNR) of Padua and the Third Medical Clinic of the University of Vienna, Austria. It was funded initially by a Progetto Bilaterale CNR to K. Thomaseth. R. Prager is currently affiliated with the Krankenhaus der Stadt Wien in Lainz. Part of this study was funded by the Project FWF-Austria, P14515, to A. Kautzky-Willer. A short description of the model was introduced in the Health Sciences track at the Conference of the Society for Computer Simulation, San Diego, CA, January 2000; preliminary results have been presented at the Scientific Sessions of the 62nd Annual Meeting of the American Diabetes Association, San Francisco, CA, June 2002.

## Footnotes

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- Copyright © 2003 by American Physiological Society