where C Glucose is the fasting plasma glucose concentration in mmol/L, and C Insulin is the fasting plasma insulin concentration in mU/L.
The model proposed by Stolwijk and Hardy8,9 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 as8,9:
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 concentrations8,9:
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
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 function10:
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 plasma8,9:
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; PSs,p is a permeability surface area product for solute “s” and pore “p”; J Vp is a fluid flow rate through the pores “p”; σs,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; αpore is the part of L pS accounted for the specific type of pore; and ΔP 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 depended15:
where f(t) = 1 + 0.6875e−t/50,
is the time-depended permeability surface area for solute “s” and pore “p” used in the equations in three-pore model, PSs,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.
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:
are the initial values of the glucose and insulin plasma concentrations, respectively.
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).
According to the homeostasis model assessment, 12 patients were insulin resistant (using as a cut off point for HOMA-IR of 2.6, according to6) 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.
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 minutes4 or 120 minutes3 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.
1.Heaton A, Johnston DG, Haigh JW, et al
: Twenty-four hour hormonal and metabolic profiles in uraemic patients before and during treatment with continuous ambulatory peritoneal dialysis. Clin Sci (Lond)
69: 449–457, 1985.
2.Wideroe TE, Smeby LC, Myking OL: Plasma concentrations and transperitoneal transport of native insulin and C-peptide in patients on continuous ambulatory peritoneal dialysis. Kidney Int
25: 82–87, 1984.
3.Lindholm B, Karlander SG: Glucose tolerance in patients undergoing continuous ambulatory peritoneal dialysis. Acta Med Scand
220: 477–483, 1986.
4.Delarue J, Maingourd C, Couet C, et al
: Effects of oral glucose on intermediary metabolism in continuous ambulatory peritoneal dialysis patients versus healthy subjects. Perit Dial Int
18: 505–511, 1998.
5.Waniewski J, Heimburger O, Werynski A, et al
: Diffusive and convective solute transport in peritoneal dialysis with glucose as an osmotic agent. Artif Organs
19: 295–306, 1995.
6.Ascaso JF, Pardo S, Real JT, et al
: Diagnosing insulin resistance by simple quantitative methods in subjects with normal glucose metabolism. Diabetes Care
26: 3320–3325, 2003.
7.Alssema M, Dekker JM, Nijpels G, et al
: Proinsulin concentration is an independent predictor of all-cause and cardiovascular mortality: An 11-year follow-up of the Hoorn Study. Diabetes Care
28: 860–865, 2005.
8.Stolwijk JE, Hardy JD: Regulation and control in physiology, in Mountcastle VB (ed), Medical Physiology
. St. Louis, CV Mosby, 1974, pp. 1343–1358.
9.Khoo M: Physiological Control Systems; Analysis, Simulation, and Estimation. New York, IEEE Press, 2000.
10.Tolic IM, Mosekilde E, Sturis J: Modeling the insulin-glucose feedback system: The significance of pulsatile insulin secretion. J Theor Biol
207: 361–375, 2000.
11.Kahn SE, Prigeon RL, McCulloch DK, et al
: The contribution of insulin-dependent and insulin-independent glucose uptake to intravenous glucose tolerance in healthy human subjects. Diabetes
43: 587–592, 1994.
12.Galach M, Werynski A, Waniewski J, et al
: Kinetic analysis of peritoneal fluid and solute transport with combination of glucose and icodextrin as osmotic agents. Perit Dial Int
29: 72–80, 2009.
13.Waniewski J, Debowska M, Lindholm B: How accurate is the description of transport kinetics in peritoneal dialysis according to different versions of the three-pore model? Perit Dial Int
28: 53–60, 2008.
14.Rippe B, Levin L: Computer simulations of ultrafiltration profiles for an icodextrin-based peritoneal fluid in CAPD. Kidney Int
57: 2546–2556, 2000.
15.Waniewski J: A mathematical model of local stimulation of perfusion by vasoactive agent diffusing from tissue surface. Int J Cardiovasc Eng
4: 115–123, 2004.
16.Getty L, Hamilton-Wessler M, Ader M, et al
: Biphasic insulin secretion during intravenous glucose tolerance test promotes optimal interstitial insulin profile. Diabetes
47: 1941–1947, 1998.
Copyright © 2011 by the American Society for Artificial Internal Organs
17.Bergman RN: Lilly lecture 1989. Toward physiological understanding of glucose tolerance. Minimal-model approach. Diabetes
38: 1512–1527, 1989.