Modelling of delay in glucose-insulin dynamics

• Autorzy: Baranowski, J.
• Języki publikacji: EN

Abstrakty

EN

In this paper a model of glucose-insulin dynamics is analysed. Since the model is in the form of a delay differential equation, a finite dimensional approximation is desirable. Two methods of such approximation are discussed. The first method is based on a discretisation and the second is based on Galerkin projections. Both methods are thoroughly described. A comparison of methods is executed for wide range of approximation orders and illustrated with graphs.

Słowa kluczowe

EN

modelling; delay; glucose; insulin

• Czasopismo: Bio-Algorithms and Med-Systems
• Rocznik: 2010
• Tom: Vol. 6, no. 12
• Strony: 123--130
• Opis fizyczny: Bibliogr. 39 poz., rys., tab.

Twórcy

autor

Baranowski, J.

• Akademia Górniczo-Hutnicza, Katedra Automatyki, Kraków

Bibliografia

• 1. Banks H., Kappel F. (1979), Spline approximations for functional differential equations. J. Differential Equations 34(3), 496-522.
• 2. Bayly P. V., Halley J. E., Mann B. R, Davies M. A. (2003), Stability of Interrupted Cutting by Temporal Finite Element Analysis. ASME Journal of Manufacturing Science and Engineering 125, 220-225.
• 3. Bennet D., Gourley S. (2004), Global stability in a model of the glucose-insulin interaction with time delay. Euro. Jnl of Applied Mathematics 15,203-221.
• 4. Bronsztejn I., Siemiendajew K., Musiol G., Muhlig H. (2004), Nowoczesne kompendium matematyki. PWN, Warszawa.
• 5. Chee F., Fernando T. (2007), Closed-loop Control of Blood Glucose. Lecture Notes in Control and Information Sciences. Springer, Berlin - Heidelberg.
• 6. Deng K„ Xiong Z., Huang Y. (2007), The Galerkin continuous finite element method for delay-differential equation with a variable term. Applied Mathematics and Computation 186,1488-1496.
• 7. Drozdov A., Khanina H. (1995), A model for ultradian oscillations of insulin and glucose. Mathematical and Computer Modelling 22(2), 23-38.
• 8. Elsgolts L. E., Norkin S. B. (1973), Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Mathematics in Science and Engineering 105. Academic Press, New York. Translated from Russian by J. L. Casti.
• 9. Geraghty E. (2008), Delay differential equations in modeling insulin therapies for diabetes. Final year project submitted to the Dept. of Mathematics of the University of Portsmouth. Supervisor: Dr. Athena Makroglou.
• 10. Ghosh D., Saha R, ChowdhuryA. R.(2007), On synchronization of a forced delay dynamical system via the Galerkin approximation. Communications in Nonlinear Science and Numerical Simulation 12, 928-941.
• 11. Gibson J. S. (1983), Linear quadratic optimal control of heraditary differential systems: Infinite dimensional Riccati equations and numerical approximation. SIAM J. Contr. Optimiz. 21(1), 95-139.
• 12. Gu G., Khargonekar P. R, Lee E. B. (1989), Approximation of infinite-dimensional systems. IEEE Transactions on Automatic Control 34(6), 610-618.
• 13. S. H. A.-A., F. M. A.-S. (2000), Approximation of time-delay systems. In: Proceedings of the American Control Conference. Chicago, Illinois, 2491-2495.
• 14. Horng I.-R., Chou J.-H. (1986), Analysis and parameter identification of time delay systems via shifted Jacobi poly-nomials. International Journal of Control 44(4), 935-942.
• 15. Hwang C, Chen M.-y. (1986), Analysis of time-delay sys­tems using the Galerkin method. International Journal of Control 44(3), 847-866.
• 16. Insperger T., Stepan G. (2002), Semi-discretization Method for Delayed Systems. International Journal for Numerical Methods in Engineering 55(5), 503-518.
• 17. Insperger I, Mann B. R, Stepan G., Bayly P. V. (2003), Stability of Up-milling and Down-milling, Part 1: Alternative Analytical Methods. International Journal of Machine Tools and Manufacture 43(1), 25-34.
• 18. Ito K., Teglas R. (1986), Legendre-tau approximations for functional differential equations. SIAM Journal on Control and Optimization 24(4), 737-759.
• 19. Ito K., Tran H., Manitius A. (1991), A Fully-discrete Spectral Method for Delay-differential Equations. SIAM Journal on Numerical Analysis 28(4), 1121-1140.
• 20. Kappel F., Kunisch K. (1981), Spline approximations for neutral functional differential equations. SIAM Journal on Numerical Analysis 18(6), 1058-1080.
• 21. Li D., Zhang C. (2010), Nonlinear stability of discontinuous Galerkin methods for delay differential equations. Applied Mathematics Letters 23,457-461.
• 22. Li J., Kuang Y. (2007), Analysis of a model of the glucose-insulin regulatory system with two delays. SIAM J. Appl. Math. 67(3), 757-776.
• 23. Li J., Kuang Y., Mason C. C. (2006), Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. Journal of Theoretical Biology 242, 722-735.
• 24. Liu W., Tang F. (2008), Modeling a simplified regulatory system of blood glucose at molecular levels. Journal of Theoretical Biology 252, 608-620.
• 25. Makila P. M., Partington J. R. (1999), Laguerre and Kautz shift approximations of delay systems. International Journal of Control72(10), 932-946.
• 26. Makroglou A., Li J., Kuang Y. (2006), Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview. Applied Numerical Math­ematics 56, 559-573.
• 27. MitkowskiW. (1991), Stabilizacja systemów dynamicznych. WNT, Warszawa.
• 28. O'Meara N. M, Sturis J, Cauter E. V., Polonsky K. S. (1993), Lack of control by glucose of ultradian insulin secretory oscillations in impaired glucose tolerance and in non-insulin-dependent diabetes mellitus. Journal of Clinical lnvestigation 92, 262-271.
• 29. Saupe D. (1983), Global bifurcation of periodic solutions to some autonomous differential delay equations. Applied Mathematics and Computing 13,185-211.
• 30. Scheen A., Sturis J, Polonsky K. S., Cauter E. V. (1996), Alterations in the ultradian oscillations of insulin secretion and plasma glucose in aging. Diabetologia 39,564-572.
• 31. Shampine L. F., Gladwell I., Thompson S. (2003), Solving ODEs with MATLAB. Cambridge University Press, New York.
• 32. Simon C, Brandenberger G. (2002), Ultradian oscillations of insulin secretion in humans. Diabetes 51(1), 258-261.
• 33. Sturis J., Cauter E. V., Blackman J. D., Polonsky K. S. (1991), Entrainment of pulsatile insulin secretion by oscillatory glucose infusion. Journal of Clinical Investigation 87, 439-445.
• 34. Sturis J., Polonsky K. S., Shapiro E. T, Blackman J. D, O'Meara N., Cauter E. V. (1992), Abnormalities in the ultradian oscillations of insulin secretion and glucose levels in Type 2 (non-insulin-dependent) diabetic patients. Diabetologia 35, 681-689.
• 35. Tatoń J., Czech A., Bernas M. (2008), Diabetologia klinicz­na. Wydawnictwo Lekarskie PZWL, Warszawa.
• 36. Tolić I. M, Mosekilde E., Sturis J. (2000), Modeling the insulin-glucose feedback system: The significance of pulsatile insulin secretion. Journal of Theoretical Biology 207, 361-375.
• 37. Wahi R, Chatterjee A. (2005), Galerkin projections for delay differential equations. Transactions of the ASME. Journal of Dynamic Systems, Measurement, and Control 127, 80-87.
• 38. Wang H., Li J., Kuangb Y (2007), Mathematical modeling and qualitative analysis of insulin therapies. Mathematical Biosciences 210,17-33.
• 39. Yoon M. G., Lee B. H. (1997), A new approximation method for time-delay systems. IEEE Transactions on Automatic Control 42(7), 1008-1012.