References
- Ahrén , B. and Pacini , G. 2004 . Importance of quantifying insulin secretion in relation to insulin sensitivity to accurately assess beta cell function in clinical studies . Eur. J. Endocrinol. , 150 : 97 – 104 .
- Ashwell , C. G. , Amiel , S. A. , Bilous , R. W. , Dashora , U. , Heller , S. R. , Hepburn , D. A. , Shutler , S. D. , Stephens , J. W. and Home , P. D. 2006 . Improved glycaemic control with insulin glargine plus insulin lispro: a multicentre, randomized, cross-over trial in people with Type 1 diabetes . Diabet. Med. , 23 : 285292
- Barbalat , I. 1959 . Systems d'equations differential d'oscillations nonlinearies . Rev. Roumaine Math. Pure. Appl. , 4 : 267 – 270 .
- Bennett , D. L. and Gourley , S. A. 2004 . Periodic oscillations in a model of the glucose-insulin interaction with delay and periodic forcing . Dyn. Syst. , 19 : 109 – 125 .
- Bennett , D. L. and Gourley , S. A. 2004 . Global stability in a model of the glucose-insulin interaction with time delay . Eur. J. Appl. Math. , 15 : 203 – 221 .
- Bergman , R. N. 2002 . Pathogenesis and prediction of diabetes mellitus: lessons from integrative physiology, in Irving L. Schwartz Lecture . Mount Sinai, J. Med. , 60 : 280 – 290 .
- Bergman , R. N. , Finegood , D. T. and Ader , M. 1985 . The assessment of insulin sensitivity in vivo . Endocr. Rev. , 6 ( 1 ) : 45 – 86 .
- Bergman , R. N. , Finegood , D. T. and Kahn , S. E. 2002 . The evolution of beta-cell dysfunction and insulin resistance in type 2 diabetes . Eur. J. Clin. Invest. , 32 ( Suppl. 3 ) : 35 – 45 .
- Bolie , V. W. 1961 . Coeffcients of normal blook glucose regulation . J. Appl. Physiol. , 16 : 783 – 788 .
- Dalla Man , C. , Caumo , A. and Cobelli , C. 2002 . The oral glucose minimal model: estimation of insulin sensitivity from a meal test . IEEE Trans. Biomed. Eng. , 49 : 419 – 429 .
- Daugherty , K. K. 2004 . Review of Insulin Therapy . J. Pharm. Pract. , 17 : 10 – 19 .
- De Gaetano , A. and Arino , O. 2000 . Mathematical modeling of the intravenous glucose tolerance test . J. Math. Biol. , 40 : 136 – 168 .
- Doran , C. V. , Chase , J. G. , Shaw , G. M. , Moorhead , K. T. and Hudson , N. H. 2005 . Derivative weighted active insulin control algorithms and intensive care unit trials . Control Eng. Pract. , 13 : 1129 – 1137 .
- Engelborghs , K. , Lemaire , V. , Belair , J. and Roose , D. 2001 . Numerical bifurcation analysis of delay differential equations arising from physiological modeling . J. Math. Biol. , 42 : 361 – 385 .
- Fernqvist , E. , Linde , B. , Ostman , J. and Gunnarsson , R. 1986 . Effects of physical exercise on insulin absorption in insulin-dependent diabetics. A comparison between human and porcine insulin . Clin. Physiol. , 6 : 487 – 497 .
- Goodyear , L. and Kahn , B. 1998 . Exercise, glucose transport, and insulin sensitivity . Ann. Rev. Med. , 49 : 235 – 261 .
- Gottesman , I. , Mandarino , L. and Gerich , J. 1983 . Estimation and kinetic analysis of insulin-independent glucose uptake in human subjects . Am. J. Physiol. Endocrinol. Metab. , 244 : E632 – E635 .
- Hildebrandt , P. 1991 . Skinfold thickness, local subcutaneous blood flow and insulin absorption in diabetic patients . Acta Physiol. Scand. Suppl. , : 41 – 45 . 733 – 753 .
- Hirsch , I. B. 1999 . Type 1 Diabetes Mellitus and the use of flexible insulin regimens . Am. Fam. Physician , 60 : 2343 – 2352 . 2355 – 2356 .
- Hirsch , W. M. , Hanisch , H. and Gabriel , J. P. 1985 . Differential equation models of some parasitic infections:methods for the study of asymptotic behavior . Comm. Pure Appl. Math. , 38 : 733 – 753 .
- Keener , J. P. 2001 . Diffusion induced oscillatory insulin secretion . Bull. Math. Biol. , 63 : 625 – 641 .
- Keener , J. and Sneyd , J. 1998 . Mathematical Physiology , New York : Springer-Verlag .
- M. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964
- Kuang , Y. 1993 . “ Delay differential equations with applications in population dynamics ” . In The Series of Mathematics in Science and Engineering , Vol. 191 , Boston : Academic Press .
- Li , J. and Kuang , Y. 2007 . Analysis of a model of the glucose-insulin regulatory system with two delays . SIAM J. Appl. Math. , 67 : 757 – 776 .
- J. Li and Y. Kuang, Pharmacokinetical models of subcutaneous injection of insulin analogues for type 1 diabetes, in review
- Li , J. , Kuang , Y. and Li , B. 2001 . Analysis of IVGTT glucose-insulin interaction models with time delay . Discrete Contin. Dyn. Syst. Ser. B , 1 : 103 – 124 .
- Li , J. , Kuang , Y. and Mason , C. 2006 . Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two time delays . J. Theor. Biol. , 242 : 722 – 735 .
- Mahaffy , J. M. and Edelstein-Keshet , L. 2007 . Modeling cyclic waves of circulating T cells in autoimmune diabetes . SIAM J. Appl. Math. , 67 ( 4 ) : 951 – 937 .
- Makroglou , A. , Li , J. and Kuang , Y. 2006 . Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview . Appl. Numerical Math. , 56 : 559 – 573 .
- Mayfield , J. A. and White , R. D. 2004 . Insulin therapy for type 2 diabetes: Rescue, augmentation, and replacement of β -cell function . Am. Fam. Physician , 70 : 489 – 500 .
- Mooradian , A. D. and Thurman , J. E. 1999 . Drug therapy of postprandial hyperglycaemia . Drugs , 57 : 19 – 29 .
- Mukhopadhyay , A. , DeGaetano , A. and Arino , O. 2004 . Modeling the intra-venous glucose tolerance test: a global study for a single-distributed-delay model . Discrete Contin. Dyn. Syst. Ser. B , 4 : 407 – 417 .
- Murphy , N. P. , Keane , S. M. , Ong , K. K. , Ford-Adams , M. , Edge , J. A. , Acerini , C. L. and Dunger , D. B. 2003 . Randomized cross-over trial of insulin glargine plus lispro or NPH insulin plus regular human insulin in adolescents with Type 1 Diabetes on intensive insulin regimens, . Diabetes Care , 26 : 799 – 804 .
- Palumbo , P. , Panunzi , S. and De Gaetano , A. 2007 . Qualitative behavior of a family of delay-differential models of the glucose-insulin system . Discrete Contin. Dyn. Syst. Ser. B , 7 : 399 – 424 .
- Panunzi , S. , Palumbo , P. and De Gaetano , A. 2007 . A discrete single-delay model for the Intra-Venous Glucose Tolerance Test . Theor. Biol. Med. Model , 4
- Pöksen , N. , Hollingdal , M. , Juhl , C. , Butler , P. , Veldhuis , J. D. and Schmitz , O. 2002 . Pulsatile insulin secretion: detection, regulation, and role in diabetes . Diabetes , 51 : S245 – S254 .
- Rassam , A. G. , Zeise , T. M. , Burge , M. R. and Schade , D. S. 1999 . Optimal administration of lispro insulin in hyperglycemic type 1 diabetes, . Diabetes Care , 22 : 133 – 136 .
- Recasensa , M. , Aguileraa , E. , Mornigoa , R. , Casamitjanab , R. , Nicolettic , F. , Gomisa , R. and Conget , I. 2003 . Insulin lispro is as effective as regular insulin in optimising metabolic control and preserving – cell function at onset of type 1 diabetes mellitus . Diabetes Res. Clin. Pract. , 60 : 153 – 159 .
- Simon , C. and Brandenberger , G. 2002 . Ultradian oscillations of insulin secretion in humans . Diabetes , 51 : S258 – S261 .
- Sturis , J. , Polonsky , K. S. , Mosekilde , E. and Van Cauter , E. 1991 . Computer model for mechanisms underlying ultradian oscillations of insulin and glucose . Am. J. Physiol. , 260 : E801 – E809 .
- Sturis , J. , Scheen , A. , Leproult , R. , Polonsky , K. and Van Cauter , E. 1995 . 24-hour glucose profles during continuous or oscillatory insulin infusion . J. Clin. Invest. , 95 : 1464 – 1471 .
- Thompson , C. L. , Dunn , K. C. , Menon , M. C. , Kearns , L. E. and Braithwaite , S. S. 2005 . Hyperglycemia in the hospital . Diab. Spectr. , 18 ( 1 ) : 20 – 27 .
- Tolic , I. M. , Mosekilde , E. and Sturis , J. 2000 . Modeling the insulin-glucose feedback system: the signifcance of pulsatile insulin secretion . J. Theor. Biol. , 207 : 361 – 375 .
- Topp , B. , Promislow , K. , De Vries , G. , Miura , R. M. and Finegood , D. T. 2000 . A Model of β -cell mass, insulin, and glucose kinetics: pathways to diabetes . J. Theor. Biol. , 206 : 605 – 619 .
- Wanant , S. and Quon , A. J. 2000 . Insulin receptor binding kinetics: modeling and simulation studies . J. theor. Biol. , 205 : 355 – 364 .
- Wang , H. 2004 . Positive periodic solutions of functional differential equations . J. Diff. Equ. , 202 : 354 – 366 .
- Wang , H. , Li , J. and Kuang , Y. 2007 . Mathematical modeling and qualitative analysis of insulin therapies, . Math. Biosci. , 210 : 17 – 33 .
- Wilinska , M. E. , Chassin , L. J. , Schaller , H. C. , Schaupp , L. , Pieber , T. R. and Hovorka , R. 2005 . Insulin kinetics in type-1 diabetes: continuous and bolus delivery of rapid acting insulin . IEEE Trans. Biomed. Eng. , 52 : 3 – 12 .