
Mathematical Biosciences and Engineering, 2019, 16(2): 759781. doi: 10.3934/mbe.2019036.
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A mathematical model for the robust blood glucose tracking
Department of Mathematics, University of Central Arkansas, 201 Donaghey Avenue, Conway, AR 72035, USA
Received: , Accepted: , Published:
Keywords: diabetes; robust blood glucose tracking; feedback and feedforward control; parameter uncertainty; internal model
Citation: Weijiu Liu. A mathematical model for the robust blood glucose tracking. Mathematical Biosciences and Engineering, 2019, 16(2): 759781. doi: 10.3934/mbe.2019036
References:
 1. E. Ackerman, J.W. Rosevear andW. F. McGuckin, A mathematical model of the glucosetolerance test, Phys. Med. Biol., 9 (1964), 203–213.
 2. E. Ackerman, L. C. Gatewood, J. W. Rosevear and G. D. Molnar, Model studies of blood glucose regulation, Bull. Math. Biophys., 27 (1965), 21–37.
 3. B. Ashley and W. Liu, Asymptotic tracking and disturbance rejection of the blood glucose regulation system, Math. Biosci., 289 (2017), 78–88.
 4. R. N. Bergman, Y. Z. Ider, C. R. Bowden and C. Cobelli, Quantitative estimation of insulin sensitivity, Am. J. Physiol. Endocrinol. Metab., 236 (1979), E667–E677.
 5. R. N. Bergman, L. S. Phillips and C. Cobelli, Measurement of insulin sensitivity and βcell glucose sensitivity from the response to intraveous glucose, J. Clin. Invest., 68 (1981), 1456–1467.
 6. R. N. Bergman, D. T. Finegood and M. Ader, Assessment of insulin sensitivity in vivo, Endocrine Reviews, 6 (1985), 45–86.
 7. R. N. Bergman, Toward physiological understanding of glucose tolerance, Minimalmodel approach, Diabetes, 38 (1989), 1512–1527.
 8. A. Bertoldo, R. R. Pencek, K. Azuma, J. C. Price, C. Kelley, C. Cobelli and D. E. Kelley, Interactions between delivery, transport, and phosphorylation of glucose in governing uptake into human skeletal muscle, Diabetes, 55 (2006), 3028–3037.
 9. J. Carr, Applications of Center Manifold Theory, Applied Mathematical Sciences 35, Springer, New York, 1981.
 10. C. Cobelli, G. Federspil, G. Pacini, A. Salvan and C. Scandellari, An integrated mathematical model of the dynamics of blood glucose and its hormonal control, Math. Biosci., 58 (1982), 27– 60.
 11. K. Fessel, J. B. Gaither, J. K. Bower, G. Gaillard and K. Osei, Mathematical analysis of a model for glucose regulation, Mathematical Biociences and Engineering, 13(2016), 83–90.
 12. L. B. Freidovich and H. K. Khalil, Performance recovery of feedbacklinearizationbased designs, IEEE Trans. Automat. Contr., 53 (2008), 2324–2334.
 13. W. T. Garvey, L. Maianu, J. H. Zhu, G. BrechtelHook, P. Wallace and A. D. Baron, Evidence for defects in the tra cking and translocation of GLUT4 glucose transporters in skeltal muscle as a cause of human insulin resistance, J. Clin. Invest., 101 (1998), 2377–2386.
 14. C. J. Goodner, B. C. Walike, D. J. Koerker, J. W. Ensinck, A. C. Brown, E. W. Chideckel, J. Palmer and L. Kalnasy, Insulin, glucagon, and glucose exhibit synchronous, sustained oscillations in fasting monkeys, Science, 195 (1977), 177–179.
 15. O. I. Hagren and A. Tengholm, Glucose and insulin synergistically activate phosphatidylinositol 3 kinase to trigger oscillations of phosphatidylinositol 3,4,5trisphosphate in βcells, J. Biol. Chem., 281 (2006), 39121–39127.
 16. B. C. Hansen, K. C. Jen, S. B. Pek and R. A.Wolfe, Rapid oscillations in plasma insulin, glucagon, and glucose in obese and normal weight humans. J. Clin. Endocr. Metab., 54 (1982), 785–792.
 17. A. Klip and M. Vranic, Muscle, liver, and pancreas: Three Musketeers fighting to control glycemia, Am. J. Physiol. Endocrinol. Metab., 291 (2006), E1141–E1143.
 18. R. Hovorka, Continuous glucose monitoring and closedloop systems, Diabetic Med., 23 (2006), 1–12.
 19. J. Huang, Nonlinear output regulation, theory and applications, Society for Industrial and Applied Mathematics, Philadelphia, 2004.
 20. H. Kang, K. Han and M. Choi, Mathematical model for glucose regulation in the wholebody system, Islets, 4 (2012), 84–93.
 21. D. A. Lang, D. R. Matthews, J. Peto, and R. C. Turner, Cyclic oscillations of basal plasma glucose and insulin concentrations in human beings, New Engl. J. Med., 301 (1979), 1023–1027.
 22. J. Lee, R. Mukherjee and H.K. Khalil, Output feedback performance recovery in the presence of uncertainties, Syst. Control Lett., 90 (2016), 31–37.
 23. J. Li, Y. Kuang and C. C. Mason, Modeling the glucoseinsulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays, J. Theor. Biol., 242 (2006), 722–735.
 24. W. Liu and F. Tang, Modeling a simplified regulatory system of blood glucose at molecular levels, J. Theor. Biol., 252 (2008), 608–620.
 25. W. Liu, C. Hsin, and F. Tang, A molecular mathematical model of glucose mobilization and uptake, Math. Biosciences, 221 (2009), 121–129.
 26. W. Liu, Elementary Feedback Stabilization of the Linear Reaction Diffusion Equation and the Wave Equation, Mathematiques et Applications, Vol. 66, Springer, 2010.
 27. W. Liu, Introduction to Modeling Biological Cellular Control Systems, Modeling, Simulation and Applications, Vol. 6, Springer, 2012.
 28. K. Ma, H. K. Khalil and Y. Yao, Guidance law implementation with performance recovery using an extended highgain observer, Aerosp. Sci. Technol., 24 (2013), 177–186.
 29. C. D. Man, A. Caumo, R. Basu, R. A. Rizza, G. Toffolo and C. Cobelli, Minimal model estimation of glucose absorption and insulin sensitivity from oral test: validation with a tracer method, Am. J. Physiol. Endocrinol. Metab., 287 (2004), E637–E643.
 30. C. D. Man, M. Campioni, K. S. Polonsky, R. Basu, R. A. Rizza, G. Toffolo and C. Cobelli, Twohour sevensample oral glucose tolerance test and meal protocol: minimal model assessment of βcell responsivity and insulin sensitivity in nondiabetic individuals, Diabetes, 54 (2005), 3265– 3273.
 31. C. D. Man, R. A. Rizza and C. Cobelli, Meal simulation model of the glucoseinsulin system, IEEE Trans. Biomed. Eng., 54 (2007), 1740–1749.
 32. H. Nishimura, F. Pallardo, G. A. Seidner, S. Vannucci, I. A. Simpson and M. J. Birnbaum, Kinetics of GLUT1 and GLUT4 glucos transporters expressed in Xenopus oocytes, J. Biol. Chem., 268 (1993), 8514–8520.
 33. A. E. Panteleon, M. Loutseiko, G. M. Steil and K. Rebrin, Evaluation of the effect of gain on the meal response of an automated closedloop insulin delivery system, Diabetes, 55 (2006), 1995– 2000.
 34. A. R. Sedaghat, A. Sherman and M. J. Quon, A mathematical model of metabolic insulin signaling pathways, Am. J. Physiol. Endocrinol. Metab., 283 (2002), E1084–E1101.
 35. E. T. Shapiro, H. Tillil, K. S. Polonsky, V. S. Fang, A. H. Rubenstein and E. V. Cauter, Oscillations in insulin secretion during constant glucose infusion in normal man: relationship to changes in plasma glucose, J. Clin. Endocr. Metab., 67 (1988), 307–314.
 36. C. Simon, G. Brandenberger and M. Follenius, Ultradian oscillations of plasma glucose, insulin, and Cpeptide in man during continuous enteral nutrition, J. Clin. Endocr. Metab., 64 (1987), 669–674.
 37. J. T. Sorensen, A Physiological Model of Glucose Metabolism in Man and its Use to Design and Assess Improved Insulin Therapies for Diabetes, PhD Thesis, Massachusetts Institute of Technology, 1985.
 38. G. M. Steil, K. Rebrin, C. Darwin, F. Hariri and M. F. Saad, Feasibility of automating insulin delivery for the treatment of type 1 diabetes, Diabetes, 55 (2006), 3344–3350.
 39. J. Sturis, E. V. Cauter, J. D. Blackman and K. S. Polonsky, Entrainment of pulsatile insulin secretion by oscillatory glucose infusion, J. Clin. Invest., 87 (1991), 439445.
 40. J. Sturis, K. S. Polonsky, E. Mosekilde and E. V. Cauter, Computer model for mechanisms underlying ultradian oscillations of insulin and glucose, Am. J. Physiol. Endocrinol. Metab., 260 (1991), E801–E809.
 41. G. Toffolo and C. Cobelli, The hot IVGTT twocompartment minimal model: an improved version, Am. J. Physiol. Endocrinol. Metab., 284 (2003), E317–E321.
 42. G. Toffolo, M. Campioni, R. Basu, R. A. Rizza and C. Cobelli, A minimal model of insulin secretion and kinetics to assess hepatic insulin extraction, Am. J. Physiol. Endocrinol. Metab. 290 (2006), E169–E176.
 43. I. M. Tolic, E. Mosekilde and J. Sturis, Modeling the insulinglucose feedback system: the significance of pulsatile insulin secretion, J. Theor. Biol., 207 (2000), 361–375.
 44. R. C. Turner, R. R. Holman, D. Matthews, T. D. Hockaday and J. Peto, Insulin deficiency and insulin resistance interaction in diabetes: estimation of their relative contribution by feedback analysis from basal plasma insulin and glucose concentrations, Metabolism, 28 (1979), 1086– 1096.
 45. O. Vahidi, K. E. Kwok, R. B. Gopaluni and L. Sun, Developing a physiological model for type II diabetes mellitus, Biochem. Eng. J., 55 (2011), 716.
 46. O. Vahidi1, K. E. Kwok, R. B. Gopaluni and F. K. Knop, A comprehensive compartmental model of blood glucose regulation for healthy and type 2 diabetic subjects, Med. Biol. Eng. Comput., 54 (2016), 13831398.
 47. R. R. Wolfe, J. R. Allsop and J. F. Burke, Glucose metabolism in man: Responses to intravenous glucose infusion, Metabolism, 28 (1979), 210–220.
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