# Mathematical Modeling of the Glucose-Insulin System During Peritoneal Dialysis With Glucose-Based Fluids

The purpose of this study was to analyze the effect of peritoneal dialysis with glucose-based solution on plasma glucose and insulin responses in patients on continuous ambulatory peritoneal dialysis (CAPD), describe the glucose-insulin system using a mathematical model, and identify abnormalities in this system. Six-hour dwell studies—using glucose 3.86% solution with a volume marker—were performed in 13 stable, fasting, nondiabetic CAPD patients. We used a mathematical model based on the previous works of Stolwijk and Hardy (1974) and Tolic et al (2000) to estimate the parameters of glucose-insulin system, insulin sensitivity index (S_{l}), and glucose effectiveness at basal (S_{G}) and zero (GEZI) insulin. The individual peaks in plasma glucose and insulin concentrations occurred after 30–60 minutes of the dwell, with the average increase of 52% and 168% over the initial values, respectively. Increased insulin resistance was found in most of these patients. Both clinical and simulation results demonstrated a high interpatient variability in glucose and insulin kinetics and glucose-insulin system parameters in the patients. We demonstrated a successful control of increasing plasma glucose by insulin, despite an increased insulin resistance, during CAPD.

From the *Institute of Biocybernetics and Biomedical Engineering, Warsaw, Poland; and †Divisions of Baxter Novum and Renal Medicine, Department of Clinical Science, Intervention and Technology, Karolinska Institutet, Stockholm, Sweden.

Submitted for consideration November 2009; accepted for publication in revised form September 2010.

Reprint Requests: Magda Galach, PhD, IBBE, 4 Trojdena Str, Warsaw 02-109, Poland. Email: magda@ibib.waw.pl.

Glucose homeostasis in the body is maintained by complex mechanisms involving many different molecules, cell types, and organs, resulting in a tight regulation of circulating glucose levels. In this study, the kinetics of two key regulatory factors, glucose and insulin, in plasma during a peritoneal dwell with hypertonic glucose solution is investigated in detail. Despite the importance of glucose-insulin regulation during peritoneal dialysis (PD), this topic was fully studied only in a few papers.^{1–4}

The purpose of this study was to 1) analyze glucose and insulin transport and plasma profiles during a peritoneal dwell with glucose 3.86% solution in nondiabetic continuous ambulatory PD (CAPD) patients, 2) describe the glucose-insulin control system using a mathematical model, 3) test the model using clinical data, and 4) identify potential abnormalities in the glucose-insulin control system in these patients.

Metabolic problems due to insulin resistance and glucose intolerance are common in end-stage renal disease (ESRD) patients; however, there is a lack of practical and sensitive enough methods to characterize the degree of insulin resistance. For PD patients, a peritoneal dwell with glucose 3.86% solution could represent an analog of an oral glucose tolerance test and may perhaps be used for the assessment of insulin resistance and the glucose-insulin control system in PD patients.

## Materials and Methods

### Clinical Data

Six-hour dwell studies using glucose 3.86% solution were performed in 13 stable, fasting, nondiabetic CAPD patients. The dwell studies were performed in the morning after an overnight fast and after the drainage of the overnight dialysis fluid (always with 1.36% glucose). Detailed data on solute and fluid transport were obtained as described previously with frequent sampling of dialysis fluid at 0, 3, 15, 30, 60, 90, 120, 180, 240, 360 minutes and plasma at 0, 30, 60, 120, 180, and 360 minutes (as regards glucose and insulin) or at 0, 180, and 360 minutes (for other solutes).^{5} Glucose and urea concentrations were measured with an IL 919 system (Instrumentation Laboratory, Milan, Italy), sodium with flame photometry, and insulin by an immunoreactive method at the Department of Clinical Chemistry, Karolinska Hospital. Radioiodinated human serum albumin (RISA) was used as a volume marker.^{5} Dialysate samples were analyzed for RISA activity on an Intertechnique CG Gamma Counter (Intertechnique, Plaisir, France). The calculations of dialysis fluid volumes were performed with peritoneal fluid absorption and sample volumes taken into account.^{5}

To quantify insulin resistance and β-cell function (insulin secretion) the homeostatic model assessment (with parameters HOMA-IR and HOMA-B, respectively) was used^{6,7}:

where *C* _{Glucose} is the fasting plasma glucose concentration in mmol/L, and *C* _{Insulin} is the fasting plasma insulin concentration in mU/L.

### Mathematical Model

The model proposed by Stolwijk and Hardy^{8,9} with modifications according to Tolic *et al.* ^{10} was used in this study, but to adjust the model to the clinical data obtained during PD in fasting, nondiabetic patients, we introduced some modifications (Figure 1).

In brief, basic equations of the model describe regulation of glucose concentration in extracellular fluid by insulin.^{8,9} The distribution compartment for glucose and insulin was the extracellular fluid with the assumed constant volume *V* of 15 L.^{8,9} It was also assumed that concentrations of small solutes in extracellular fluid and plasma were equilibrated. Glucagon is not taken into account because in this specific case (dialysis with glucose-based fluid) supplementary load of glucose causes an increase of glucose concentration, and glucagon secretion to plasma is not activated.

Constant hepatic glucose production (*G* _{g,ex}), and transport from dialysis fluid in the peritoneal cavity (*Q* _{g,D}) are included in the model. Two types of glucose utilization processes are taken into account:

- Insulin-independent glucose utilization whereby glucose leaving the blood enters cells through facilitated diffusion that depends only on the extracellular-to-intracellular glucose concentration gradient. In most circumstances, the intracellular glucose concentration is low and can therefore be ignored, and the rate
*U*_{ind}of this process may be described as^{8,9}:

where λ is the rate parameter for insulin independent cellular uptake of glucose.

2. Insulin-dependent glucose utilization whereby glucose enters cells (muscle and adipose tissue) by insulin-facilitated diffusion at the rate *U* _{d} proportional to glucose (*C* _{g,ex}) and to insulin (*C* _{i,ex}) extracellular concentrations^{8,9}:

where γ is the rate parameter for insulin-dependent cellular uptake of glucose. The parameter *S* _{l}=γ*/V* is called the insulin sensitivity index.^{11} The parameter

, where

is the initial (fasting) insulin extracellular concentration, is called the glucose effectiveness at basal insulin, and the parameter GEZI=λ*/V* is called the glucose effectiveness at zero insulin.^{11}

Although glucose can also be excreted by the kidneys, this pathway is of quantitatively lower importance especially in ESRD patients and was not taken into account.

The glucose mass balance in the extracellular compartment can be described by the following equation:

where *Q* _{g,D} is the flow rate at which glucose is transported between dialysis fluid and blood. *Q* _{g,D} is described using the three-pore model for peritoneal transport.^{12,13}

A similar mass balance equation can be formulated for extracellular insulin. The pancreatic insulin production rate, *G* _{i,ex}, controlled by glucose concentration is specified by the following function^{10}:

where *R* _{m} is the maximum rate of insulin production, *a* is the sensitivity parameter of insulin production to glucose concentration, and *C* _{1} is the threshold value of glucose concentration.^{10}

Insulin is destroyed through a reaction involving insulinase at a rate *D* _{i} proportional to insulin concentration in plasma^{8,9}:

where α is the rate parameter.

Thus, insulin mass balance equation has the following form:

where *Q* _{i,D} is the flow rate at which insulin is transported between dialysis fluid and blood.

The model can also be extended for additional external sources of glucose or insulin to describe, for example, glucose load during the meal or injection of insulin during the dwell.

Transport of solutes between dialysate and extracellular fluid is described using the three-pore model, which is based on the assumption that peritoneal membrane is heteroporous with three types of pores.^{12–14} Water and solutes are transported through two types of pores: small and large, whereas the third type—ultrasmall pore that is impermeable for solutes—plays an important role in fluid transport. Therefore, the solute transport rate, *Q* _{s,D}, is the sum of the solute flows through the pores (*J* _{Ssolute,pore}):

where “Small” and “Large” stands for type of pores; *L* is a peritoneal absorption rate; *C* _{s,D} is concentration of solute “s” in dialysate; PS_{s,p} is a permeability surface area product for solute “s” and pore “p”; *J* _{Vp} is a fluid flow rate through the pores “p”; σ_{s,p} is the solute “s” osmotic reflection coefficient for pore “p” and depends on solute “s” and pore “p” radiuses; is the mean membrane solute concentration; *LpS* is the membrane ultrafiltration coefficient; α_{pore} is the part of *L* _{p}S accounted for the specific type of pore; and Δ*P* and ΔΠ are the hydrostatic and osmotic pressure differences between the blood and dialysate, respectively.^{12–14}

Additionally, the effect of vasodilation is taken into account, and therefore, some parameters are assumed to be time depended^{15}:

where *f(t) = 1 + 0.6875*e*−t/50*,

is the time-depended permeability surface area for solute “s” and pore “p” used in the equations in three-pore model, PS_{s,p} is a constant value without vasodilation, *L* _{p} *St* is the time-depended membrane ultrafiltration coefficient used in the equations in three-pore model, and *L* _{p} *S* is a constant value without vasodilation.

### Computer Simulations

The model was solved using ode45 solver of Matlab v. R2007b software (MathWorks Inc., USA) based on an explicit 4th and 5th order Runge-Kutta formula. The data of each patient separately were used as target values for estimation of the model parameters done using Matlab function fminsearch (Nelder-Mead type simplex search method) with the aim to minimize the function that described the sum of absolute differences between theoretical predictions and clinical data. Twelve parameters were selected for the fitting procedure carried out in four stages, see Tables 1 and 2 for details. Computer simulations of a single 6-hour dwell with the 3.86% glucose solution using a three-pore model and the clinical data from the whole dwell period (solute plasma concentrations were interpolated by linear spline functions) were performed in stage 1.^{12,13} During stages 2–4 of the parameter estimation, only data from the initial 2 hours of the dwell study were used to minimize the influence of possible subsequent external glucose load (as some patients may have had a small breakfast after 2 hours) on the simulation results and to make the evaluations similar to that for oral glucose tolerance test.^{3,4} It was also assumed that each patient before the dwell study was in the metabolic steady state, and therefore, the following algebraic equations were applied during the estimation of parameters:

where

and

are the initial values of the glucose and insulin plasma concentrations, respectively.

### Statistical Methods

Correlations between parameters of the model and clinical data were assessed by the Pearson's correlation coefficient. Clinical results were evaluated using the paired two-tailed t test. All values were expressed as mean ± standard deviation (SD).

## Results

According to the homeostasis model assessment, 12 patients were insulin resistant (using as a cut off point for HOMA-IR of 2.6, according to^{6}) with mean HOMA-IR of 3.8 ± 1.4 and mean HOMA-B of 220.3 ± 131.8.

Plasma glucose concentration increased significantly during the first hour of the dwell from 5.2 ± 0.6 mmol/L (mean ± SD) to 7.9 ± 1.2 mmol/L (*p* < 0.00001) and then slowly decreased to 6.7 ± 1.1 mmol/L (*p* < 0.0005) after 2 hours of the dwell (Figure 2). Therefore, the individual plasma glucose peaks were observed after 30–60 minutes of the dwell with an average increase of 52% over the initial glucose value. Only in one case, the plasma glucose concentration exceeded 10 mmol/L during the first hour. After 2 hours, glucose concentration was <9 mmol/L in all patients, and in 7 of 13 patients, it was not higher than 7 mmol/L. At the end of the dwell (after 6 hours), mean plasma glucose concentration was 6.2 ± 1.3 mmol/L, and only in three patients, it exceeded 7 mmol/L.

Glucose absorption (calculated as a glucose mass absorbed from dialysis fluid) at the 1st, 2nd, and 6th hour of the dwell was 125.9 ± 40.6 mmol, 188.7 ± 51 mmol, and 313.4 ± 52 mmol, respectively. There were strong, positive, statistically significant correlations between maximum glucose concentration increase during 2 hours of the dwell and both glucose absorption at the 2nd and 6th hours (*r* = 0.83 and *r* = 0.82 with *p* < 0.001, respectively). Glucose absorption at the second hour was also correlated with maximum glucose concentration during 2 hours (*r* = 0.6, *p* < 0.05).

The differences among patients in plasma insulin levels were much higher than for glucose. The maximum insulin concentration was achieved after the first hour of the dwell in 10 patients; circulating insulin increased from 17.1 ± 5.5 mU/L at the beginning of the dwell to 42.2 ± 21.5 mU/L (*p* < 0.005) at the first hour and then showed tendency to decrease (to 36.2 ± 14.4 mU/L at the second hour), but the difference in the insulin concentration between the first and second dwell hour was not statistically significant (*p* > 0.1), Figure 2. For the remaining three patients, the maximum insulin concentration was achieved already after the first 30 minutes of the dwell; circulating insulin increased from 14.4 ± 5.8 mU/L at the beginning of the dwell to 34.7 ± 13.2 mU/L at 30 minutes and then showed tendency to decrease (to 17.0 ± 2.6 mU/L at the second hour). Thus, the individual plasma insulin peaks were observed after 30–60 minutes of the dwell with an average increase of 168% over the initial insulin value. The plasma insulin concentration was <50 mU/L throughout the whole dwell for 8 of 13 patients.

The mathematical model for the glucose-insulin system with the fitted parameter values (Table 3) was able to describe well the clinical data concerning plasma glucose concentration (Figure 2). The percentage errors (PEs) were <10% for all patients, and in 9 of 13 patients they were <5% (Table 4). The agreement of the simulation results with clinical data was slightly worse for plasma insulin than for glucose (Table 4). For 6 of 13 patients, the PE of simulation results for insulin exceeded 10% (but for 11 of 13 patients, this error was <15%, Table 4).

The mean parameter values (for 10 patients for whom insulin concentration peak was achieved after the first hour of the dwell) and their reference values from previous studies are presented in Table 3. The highest discrepancies between the estimated parameters and the reference values were found for the rate of insulin-independent cellular uptake of glucose λ and the insulin utilization rate α (over 730% and 480%, respectively). For the three patients with fast insulin generation (peak at 30 minutes) the fitted α, γ, and *R* _{m} values were much higher than for the other 10 patients (α: 8.65 ± 6.36 *vs.* 0.74 ± 0.44 L/min; γ: 0.0104 ± 0.0118 *vs.* 0.0032 ± 0.0046 (L/min)/(mU/L); *R* _{m}: 1528 ± 1548 *vs*. 235 ± 205 mU/min).

Insulin sensitivity index, *S* _{I}, was lower than in the young healthy adults, as may be expected for older, uremic patients, whereas glucose effectiveness parameters, *S* _{G} and GEZI, were within normal, albeit higher than the average value, range in this control population (Table 3).

There were correlations between γ (and *S* _{I}) and plasma glucose concentration at 120 minutes (*r* = −0.80, *p* < 0.01), and between λ (and GEZI) and plasma glucose (*r* = 0.62, *p* < 0.05) and plasma insulin (*r* = 0.57, *p* < 0.05) concentration decreases between 60 and 120 minutes. The maximal insulin generation rate *R* _{m} correlated with plasma insulin concentration peak (*r* = 0.87, *p* < 0.005), and the sensitivity of insulin production to glucose concentration *a* correlated positively with the HOMA-IR and negatively with the inverse of HOMA-IR (*r* = 0.67, *r* = −0.67, both with *p* < 0.05). Additionally, the ratio *R* _{m}/*C* _{1} correlated with both the HOMA-IR and the inverse of HOMA-IR (*r* = 0.70, *r* = −0.68, both with *p* < 0.05). No statistically significant correlations between parameters of the model and HOMA-B were found.

## Discussion

The clinical data, as well as simulation results, demonstrated a successful control of the increasing plasma glucose by insulin in these patients. The interpatient variability in insulin profiles was higher than for glucose profiles. Typically, the insulin concentration at 1 hour of the dwell was higher than from that measured at 30 minutes, but in three patients, maximum of insulin concentration was obtained within 30 minutes—in these patients, the insulin release was faster than that for all other patients, and the applied model reproduced this effect with less accuracy. We speculate that the problem might result from simplifications in the model concerning description of insulin generation. Many studies showed that insulin secretion in response to sudden glucose load is biphasic with transient first phase followed by a sustained second phase.^{16} In the model, it is assumed that there is only one phase, and in most cases, such assumption seems to be appropriate—the first phase has only a limited impact on insulin generation and may be difficult to observe when glucose is continuously (for hours) transported to blood. However, in the three patients, the first phase might be more pronounced than in the other patients.

The average deviations and percentage differences between the clinical data and simulated dialysate volume and solute concentrations (Table 1) showed, in most cases, a good agreement between the measured and simulated values. The only exception was insulin dialysate concentration (Table 1). A possible reason for this difference between simulated and clinical data is that dialysate insulin concentration was low throughout the whole dwell, close to the limit of the accuracy of the measurement method, and therefore, the errors of measurement was high.

It should also be noted that the parameters of the three-pore model had to be modified to provide a good agreement between the theoretical curves and clinical data (Table 3).

Clinical data indicate that PD with glucose 3.86%, despite the amount of glucose absorbed during the whole dwell being comparable with that absorbed during an oral glucose tolerance test (OGTT, 56 ± 8.9 g *vs*. 50 g in Ref. 4 or 66 ± 9.5 g in Ref. 3), evokes weaker response to glucose load than OGTT. The absorption of glucose during PD dwell is much slower than during OGTT, and therefore, both glucose and insulin peak plasma concentrations are lower during PD than during OGTT.^{3,4} Nevertheless, the time to maximum concentration is similar for peritoneal dwell and OGTT. During OGTT, the plasma glucose concentration peak is observed after 60 minutes^{4} or 120 minutes^{3} of the test *vs*. 60 minutes during the dwell with glucose 3.86% (Figure 2). Plasma insulin concentration peak is observed in 60 minutes of OGTT and the peritoneal dwell (Figure 2).^{3,4}

A high interpatient variability of the glucose-insulin system parameter values was observed. Additionally, the values of the parameters differed significantly from the previously published ones, which, in general, described a healthy subject.^{11} This indicates that in the patients on CAPD, major alterations in glucose regulation were present, especially in the insulin utilization rate and insulin-independent cellular uptake of glucose.^{10} The estimated model parameters have close mathematical relationships with the parameters *S* _{I}, *S* _{G}, and GEZI estimated in many previous studies on glucose-insulin system by the minimal model.^{11,17} Insulin sensitivity index, *S* _{I}, was found to be lower in uremic patients on PD than in healthy young men, Table 3, as expected. All three parameters had higher variability in the patients than in the control population (Table 3). However, even in healthy people, the values of these parameters may be much different between different subpopulations.^{17} In our patients, one may expect high variability in the insulin resistance and glucose tolerance because of different degree of impairment of the function of the glucose-insulin system.

The applied model involves some simplifications, but an extended version may also be considered in the future if the measurements are done more frequently. This extended model should take into account the reduction of hepatic glucose generation when glucose level is increased and the restricted transport of insulin between plasma and interstitial fluid.

Our results indicate that a PD dwell with glucose 3.86% solution can be used to study insulin resistance and glucose intolerance in PD patients using the proposed model. The model allows for the precise assessment of glucose absorption from the peritoneal dialysate to the blood, and the estimation of the basic parameters that are used for the description of the insulin sensitivity and insulin-dependent and glucose-independent glucose uptake to cells and the insulin response to increased glucose level. The status of the glucose-insulin control system exhibits large interpatient variations among nondiabetic PD patients, which can be due to differences in peritoneal glucose absorption, cellular glucose uptake, and insulin resistance.

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