Abstract
An islet population model is proposed for pancreatic insulin secretion. Without detailing the chain of biochemical events giving rise to the delivery of insulin packets, the effect of the islets’ bursting response to varying glucose concentration is described by a simple second order nonlinear model, of the same functional form for all islets, but with a random distribution of parameter values over the one million islets considered. The islet equations are coupled to a traditional model of the glucose/insulin dynamics to complete a description of the feed-back control of the glucose/insulin system. The model is thus based upon the completely random cooperation of a large number of independent controllers, all reacting to the same prevailing plasma glucose concentrations, but with distributed reaction characteristics. It is shown that the proposed model is able to replicate in silico different observed phenomena such as low frequency glycemia–insulinemia oscillations (ultradian oscillations, with a period between 50 and 150 min, amplified by constant glucose administration and entrained by an oscillating exogenous glucose infusion), as well as concordant induction of high-frequency insulin oscillations by a rapid periodic pulsatile glucose infusion. In order to reproduce by simulation all of the above observed phenomena, a single set of (hyper-)parameters has been used throughout, showing that it is indeed possible that a single model may explain the results of several published experimental protocols.
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References
Anderson GE, Kologlu Y, Papadopoulus C (1967) Fluctuations in postabsorptive blood glucose in relation to insulin release. Metabolism 16: 586–596
Ashcroft FM, Rorsman P (1989) Electrophysiology of the pancreatic β-cell. Prog Biophys Mol Biol 54: 87–143
Bennet DL, Gourley SA (2004) Asymptotic properties of a delay differential equation model for the interaction of glucose with plasma and interstitial insulin. Appl Math Comput 151: 189–207
Bergman RN, Ider YZ, Bowden CR, Cobelli C (1979) Quantitative estimation of insulin sensitivity. Am J Physiol 236: 667–677
Bertram R, Satin L, Zhang M, Smolen P, Sherman A (2004) Calcium and glycolisis mediate multiple bursting modes in pancreatic islets. Biophysics 87: 3074–3087
Bertram R, Sherman A (2004) A calcium-based phantom bursting model for pancreatic islets. Bull Math Biol 66: 1313–1344
Bertuzzi A, Salinari S, Mingrone G (2007) Insulin granule trafficking in β-cells: mathematical model of glucose-induced insulin secretion. Am J Physiol Endocrinol Metab 293: E396–E409
Blackman RB, Tukey JV (1958) The measurement of power spectra. Dover Publications, New York
Bowden CR, Bergman RN, Marsh DJ (1980) Cause of glucose oscillations during glucose infusion: periodic variation in glucose uptake. Am J Physiol 238: E395–E407
Chou HF, Ipp E (1990) Pulsatile insulin secretion in isolated rat islets. Diabetes 39: 112–117
Clausen JO, Borch-Johnsen K, Bergman RN, Hougaard P, Winther K, Pedersen O (1996) Insulin sensitivity index, acute insulin response, and glucose effectiveness in a population-based sample of 380 young healthy caucasians. J Clin Invest 98: 1195–1209
Cook D.L. (1983) Isolated islets of Langerhans have slow oscillations of electrical activity. Metabolism 32: 681–685
De Gaetano A, Arino O (2000) Mathematical modelling of the intravenous glucose tolerance test. J Math Biol 40: 136–168
Eddlestone GT, Goncalves A, Bangham JA (1984) Electrical coupling between cells in islets of Langerhans from Mouse. J Membr Biol 77: 1–14
Engelborghs K, Lemaire V, Bélair J, Roose D (2001) Numerical bifurcation analysis of delay differential equations arising from physiological modeling. J Math Biol 42: 361–385
Frankel BJ, Atwater I, Grodsky GM (1981) Calcium affects insulin release and membrane potential in islet beta-cells. Am J Physiol Cell Physiol 240: C64–C72
Gammaitoni L, Hänggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 1: 223–287
Gilon P, Ravier MA, Jonas JC, Henquin JC (2002) Control mechanisms of the oscillations of insulin secretion in vitro and in vivo. Diabetes 51: S144–S151
Goodner CJ, Koerker DJ, Stagner JI, Samols E (1991) In vitro pancreatic hormonal pulses are less regular and more frequent than in vivo. Am J Physiol Endocrinol Metab 260: E422–E429
Goodner CJ, Walike BC, Koerker DJ, Ensinck JW, Brown AC, Chideckel EW, Palmer J, Kalnasy L (1977) Insulin, glucagon and glucose exhibit synchronous sustained oscillations in fasting monkeys. Science 195: 177–179
Henquin JC, Meissner HP, Schmeer W (1982) Cyclic variations of glucose-induced electrical activity in pancreatic B cells. Pflugers Arch 393: 322–327
Jenkins GM, Watts DG (1968) Spectral analysis and its applications. Holden-Day, San Francisco
Keener J, Sneyd J (1998) Mathematical physiology. Springer, New York
Lang DA, Matthews DR, Peto J, Turner RC (1979) Cyclic oscillations of basal plasma glucose and insulin concentrations in human beings. N Engl J Med 301: 1023–1027
Li J, Kuang Y, Mason CC (2006) Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. J Theor Biol 242: 722–735
Liptser RS, Shiryaev AN (1974) Statistics of random processes. Springer, Berlin
Longo EA, Tornheim K, Deeney JT, Varnum BA, Tillotson D, Prentki M, Corkey BE (1991) Oscillations in cytosolic free Ca2+, oxygen consumption, and insulin secretion in glucose-stimulated Rat pancreatic islets. J Biol Chem 266: 9314–9319
Mari A, Camastra S, Toschi E, Giancaterini A, Gastaldelli A, Mingrone G, Ferranini E (2001) A model for glucose control of insulin secretion during 24 hours of free living. Diabetes 50(suppl 1): S164–S168
Mari A, Tura A, Gastaldelli A, Ferranini E (2002) Assessing insulin secretion by modeling in multiple-meal tests. Role of potentiation. Diabetes 51(suppl 1): S221–S226
Matthews DR, Hermansen K, Connolly AA, Gray D, Schmitz O, Clark A, Orskov H, Turner RC (1987) Greater in vivo than in vitro pulsatility of insulin secretion with synchronized insulin and somatostatin secretory pulses. Endocrinology 120: 2272–2278
Meissner HP (1976) Electrophysiological evidence for coupling between beta-cells of pancreatic islets. Nature 262: 502
Millsaps K, Pohlhausen K (1975) A mathematical model for glucose insulin interaction. Math Biosci 23: 237–251
Nyengaard JR, Bendtsen TF (1992) Glomerular number and size in relation to age, kidney weight, and body surface in normal man. Anat Rec 232: 194–201
Ookhtens M, Marsh DJ, Smith SW, Bergman RN, Yates FE (1974) Fluctuations of plasma glucose and insulin in conscious dogs receiving glucose infusions. Am J Physiol 226: 910–919
Palumbo P, Panunzi S, De Gaetano A (2007) Qualitative behavior of a family of of delay-differential equations models of the glucose-insulin regulatory system. Discrete Contin Dyn Syst Ser B 7(2): 399–424
Panunzi S, Palumbo P, De Gaetano A (2007) A discrete single delay model for the intra-venous glucose tolerance test. Theor Biol Med Model 4: 35
Pedersen MG, Bertram R, Sherman A (2005) Intra- and inter-islet synchronization of metabolically driven insulin secretion. Biophys J 89: 107–119
Pincus SM (1991) Approximate entropy as a measure of system complexity. Proc Natl Acad Sci USA 88: 2297–2301
Polonski KS, Given BD, Van Cauter E (1988) Twenty four hour profiles and pulsatile patterns of insulin secretion in healthy and obese subjects. J Clin Invest 81: 442–448
Pørksen N (2002) The in vivo regulation of pulsatile insulin secretion. Diabetologia 45: 3–20
Pørksen N, Juhl C, Hollingdal M, Pincus SM, Sturis J, Veldhuis JD, Schmitz O (2000) Concordant induction of rapid in vivo pulsatile insulin secretion by recurrent punctuated glucose infusions. Am J Physiol Endocrinol Metab 278: E162–E170
Pørksen N, Munn S, Steers J, Vore S, Veldhuis J, Butler P (1995) Pulsatile insulin secretion accounts for 70% of total insulin secretion during fasting. Am J Physiol 269: E478–E488
Pørksen N, Nyholm B, Veldhuis JD, Butler PC, Schmitz O (1997) In humans at least 75% of insulin secretion arises from punctuated insulin secretory bursts. Am J Physiol Endocrinol Metab 273: E908–E914
Ravier MA, Sehlin AJ, Henquin JC (2002) Disorganization of cytoplasmic Ca2+ oscillations and pulsatile insulin secretion in islets from ob/ob mice. Diabetologia 45: 1154–1163
Santos RM, Rosario LM, Nadal A, Garcia-Sancho J, Soria B, Valdeolmillos M (1991) Widespread synchronous [Ca 2+] i oscillations due to bursting electrical activity in single pancreatic islets. Pflugers Arch 418: 417–422
Sha L, Westerlund J, Szurszewski JH, Bergsten P (2001) Amplitude modulation of pulsatile insulin secretion by pancreatic ganglion neurons. Diabetes 50: 51–55
Shapiro ET, Tillil H, Polonsky KS, Fang VS, Rubenstein AH, Van Cauter E (1988) Oscillations in insulin secretion during constant glucose infusion in normal man: relationship to changes in plasma glucose. J Clin Endocrinol Metab 67: 307–314
Simon C, Brandenberger G, Follenius M (1987) Ultradian oscillations of plasma glucose, insulin, and C-peptide in man during continuous enteral nutrition. J Clin Endocrinol Metab 64: 669–674
Simon C, Follenius M, Brandenberger G (1987) Postprandial oscillations of plasma glucose, insulin and C-peptide in man. Diabetologia 30: 769–773
Sirek A, Vaitkus P, Norwich KH, Sirek OV, Hunger RH, Harris V (1985) Secretory patterns of glucoregulatory hormones in prehepatic circulation of dogs. Am J Physiol 249: E34–E42
Smolen P (1995) A model for glycolytic oscillations based on skeletal muscle phosphofructokinase kinetics. J Theor Biol 174: 137–148
Stagner JI, Samols E, Weir GC (1980) Sustained oscillations of insulin, glucagon and somatostatin from the isolated canine pancreas during exposure to a constant glucose concentration. J Clin Invest 65: 939–942
Sturis J, Blackman JD, Van Cauter E, Polonski KS (1991) Entrainment of pulsatile insulin secretion by oscillatory glucose infusion. J Clin Invest 87: 439–445
Sturis J, Polonsky KS, Mosekilde E, Van Cauter E (1991) Computer model for mechanisms underlying ultradian oscillations of insulin and glucose. Am J Physiol 260: E801–E809
Sturis J, Pugh WL, Tang J, Ostrega DM, Polonsky JS, Polonsky KS (1994) Alterations in pulsatile insulin secretion in the Zucker diabetic fatty rat. Am J Physiol 267: E250–E259
Tolić IM, Mosekilde E, Sturis J (2000) Modeling the insulin-glucose feedback system: the significance of pulsatile insulin secretion. J Theor Biol 207: 361–375
Tornheim K (1997) Are metabolic oscillations responsible for normal oscillatory insulin secretion?. Diabetes 46: 1375–1380
Toschi E, Camastra S, Sironi AM, Masoni A, Gastaldelli A, Mari A, Ferranini E, Natali A (2002) Effect of acute hyperglycemia on insulin secretion in humans. Diabetes 51(Suppl.1): S130–S133
Tsaneva-Atanasova K, Zimliki CL, Bertram R, Sherman A (2006) Diffusion of calcium and metabolites in pancreatic islets: killing oscillations with a pitchfork. Biophys J 90: 3434–3446
Valdeolmillos M, Gomis A, Sanchez-Andres JV (1996) In vivo synchronous membrane potential oscillations in Mouse pancreatic β-cells: lack of co-ordination between islets. J Physiol 493: 9–18
Van Cauter E, Desir D, Decoster C, Féry F, Balasse EO (1989) Nocturnal decrease of glucose tolerance during constant glucose infusion. J Clin Endocrinol Metab 69: 604–611
Verdonk CA, Rizza RA, Gerich JE (1981) Effects of plasma glucose concentration on glucose utilization and glucose clearance in normal man. Diabetes 30: 535–537
Zhang M, Goforth P, Bertram R, Sherman A, Satin L (2003) The Ca2+ dynamics of isolated Mouse β-cells and islets: implications for mathematical models. Biophys J 84: 2852–2870
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Palumbo, P., De Gaetano, A. An islet population model of the endocrine pancreas. J. Math. Biol. 61, 171–205 (2010). https://doi.org/10.1007/s00285-009-0297-0
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DOI: https://doi.org/10.1007/s00285-009-0297-0