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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

In this paper, we study the problem of the robust blood glucose tracking. Tracking here means that the error between a state variable of a system under control and its desired time-varying reference converges to zero over time. Robustness here means that a controller designed for a system can tolerate a small variation of the system parameters. Since the parameters in the blood glucose regulation system differ in people, such a robust controller is useful in the insulin pump technology: an insulin pump equipped with such a robust controller could be used in a group of people. Thus, in our study, parameter uncertainties are introduced into a mathematical model of the blood glucose regulation system. Using an actual blood glucose level as feedback and an exogenous glucose input and a desired glucose reference as feedforward, we design a robust feedback and feedforward controller, which drives the blood glucose to track the desired time-varying glucose reference for any small uncertainties. Numerical simulations with published experimental blood glucose data are conducted to further confirm our theoretical results.
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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): 759-781. doi: 10.3934/mbe.2019036


  • 1. E. Ackerman, J.W. Rosevear andW. F. McGuckin, A mathematical model of the glucose-tolerance 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 β-cell 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, Minimal-model 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 feedback-linearization-based designs, IEEE Trans. Automat. Contr., 53 (2008), 2324–2334.
  • 13. W. T. Garvey, L. Maianu, J. H. Zhu, G. Brechtel-Hook, 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 3,4,5-trisphosphate in β-cells, 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 closed-loop 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 whole-body 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 glucose-insulin 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 high-gain 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 seven-sample oral glucose tolerance test and meal protocol: minimal model assessment of β-cell 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 glucose-insulin 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 closed-loop 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 C-peptide 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), 439-445.
  • 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 two-compartment 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 insulin-glucose 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), 7-16.
  • 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), 1383-1398.
  • 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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