2015, 20(9): 3115-3129. doi: 10.3934/dcdsb.2015.20.3115

Quantitative analysis of time delays of glucose - insulin dynamics using artificial pancreas

1. 

Department of Applied Mathematics, Delhi Technological University, Delhi-110042, India, India

Received  October 2014 Revised  June 2015 Published  September 2015

Time delay in insulin secretion, its absorption and action is a point of consideration in artificial pancreas as it may prove fatal in the extreme situation. The present mathematical model deals with two time delays out of which one occur in insulin secretion and second in its absorption and action. The model assess the change in glucose - insulin dynamics after the induction of different values of these time delays in their respective range. Also, simulation is performed over the model to quantify the amount of two time delays to avoid diabetic comma, which has not been explored much.
Citation: Saloni Rathee, Nilam. Quantitative analysis of time delays of glucose - insulin dynamics using artificial pancreas. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3115-3129. doi: 10.3934/dcdsb.2015.20.3115
References:
[1]

S. Bennett, A History of Control Engineering,, IEE Control Engrg. Ser., 47 (1993), 1930. doi: 10.1049/PBCE047E.

[2]

Brandt, Starvation and Diabetes Mellitus,, Chapter 6, ().

[3]

C. Cobelli, E. Renard and B. Kovatchev, Artificial pancreas: Past, present, future,, Diabetes, 60 (2011), 2672. doi: 10.2337/db11-0654.

[4]

A. Haidar, L. Legault, M. Dallaire, A. Alkhateeb, A. Coriati, V. Messier, P. Cheng, M. Millette, B. Boulet and R. Rabasa-Lhoret, Glucose-responsive insulin and glucagon delivery (dual-hormone artiicial pancreas) in adults with type 1 diabetes: a randomized crossover controlled trial,, CMAJ, 5 (2013).

[5]

M. Huang, J. Li, X. Song and H. Guo, Modeling impulsive injections of insuin: Towards artificial pancreas,, Siam J. App. math. Society for Industrial and Applied Mathematics, 72 (2012), 1524. doi: 10.1137/110860306.

[6]

L. Jiaxu and K. Yang, Analysis of a model of the glucose-insulin regulatory system with two delays,, SIAM J. APP. Math., 67 (2007), 757. doi: 10.1137/050634001.

[7]

D. B. Keenan, J. J. Mastrototaro, G. Voskanyan and G. M. Steil, Delays in minimally invasive continuous glucose monitoring devices: A review of current technology,, Journal of Diabetes Science and Technology, 3 (2009), 1207. doi: 10.1177/193229680900300528.

[8]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Math. Sci. Eng., (1993).

[9]

J. Li, Y. Kuang and C. Mason, Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two time delays,, J. Theoret. Biol., 242 (2006), 722. doi: 10.1016/j.jtbi.2006.04.002.

[10]

K. Rebrin and G. M. Steil, Can interstitial glucose assessment replace blood glucose measurements?, Diabetes Technol Ther, 2 (2000), 461. doi: 10.1089/15209150050194332.

[11]

L. F. Sampine and S. Thompson, Solving DDEs in matlab,, Appl. Numer. Math., 37 (2001), 441. doi: 10.1016/S0168-9274(00)00055-6.

[12]

C. Simon and G. Brandenberger, Ultradian oscillations of insulin secretion in humans,, Diabetes, 51 (2002). doi: 10.2337/diabetes.51.2007.S258.

[13]

J. Sturis, K. S. Polonsky, E. Mosekilde and E. V. Cauter, Computer model for mechanisms underlying ultradian oscillations of insulin and glucose,, Amer. J. Physiol., 260 (1991), 801.

[14]

I. M. Tolic, E. Mosekilde and J. Sturis, Modeling the insulin-glucose feedback system: The significance of pulsatile insulin secretion,, J. Theoret. Biol., 207 (2000), 361.

[15]

H. Wang, J. Li and Y. Kuang, Mathematical modeling and qualitative analysis of insulin therapies,, Math. Biosci., 210 (2007), 17. doi: 10.1016/j.mbs.2007.05.008.

[16]

W. K. Ward, R. G. Massoud and C. J. Szybala, In vitro and in vivo evaluation of native glucagon and glucagon analog (MAR-D28) during aging: lack of cytotoxicity and preservation of hyperglycemic effect,, J Diabetes Sci Technol, 4 (2010), 1311. doi: 10.1177/193229681000400604.

show all references

References:
[1]

S. Bennett, A History of Control Engineering,, IEE Control Engrg. Ser., 47 (1993), 1930. doi: 10.1049/PBCE047E.

[2]

Brandt, Starvation and Diabetes Mellitus,, Chapter 6, ().

[3]

C. Cobelli, E. Renard and B. Kovatchev, Artificial pancreas: Past, present, future,, Diabetes, 60 (2011), 2672. doi: 10.2337/db11-0654.

[4]

A. Haidar, L. Legault, M. Dallaire, A. Alkhateeb, A. Coriati, V. Messier, P. Cheng, M. Millette, B. Boulet and R. Rabasa-Lhoret, Glucose-responsive insulin and glucagon delivery (dual-hormone artiicial pancreas) in adults with type 1 diabetes: a randomized crossover controlled trial,, CMAJ, 5 (2013).

[5]

M. Huang, J. Li, X. Song and H. Guo, Modeling impulsive injections of insuin: Towards artificial pancreas,, Siam J. App. math. Society for Industrial and Applied Mathematics, 72 (2012), 1524. doi: 10.1137/110860306.

[6]

L. Jiaxu and K. Yang, Analysis of a model of the glucose-insulin regulatory system with two delays,, SIAM J. APP. Math., 67 (2007), 757. doi: 10.1137/050634001.

[7]

D. B. Keenan, J. J. Mastrototaro, G. Voskanyan and G. M. Steil, Delays in minimally invasive continuous glucose monitoring devices: A review of current technology,, Journal of Diabetes Science and Technology, 3 (2009), 1207. doi: 10.1177/193229680900300528.

[8]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Math. Sci. Eng., (1993).

[9]

J. Li, Y. Kuang and C. Mason, Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two time delays,, J. Theoret. Biol., 242 (2006), 722. doi: 10.1016/j.jtbi.2006.04.002.

[10]

K. Rebrin and G. M. Steil, Can interstitial glucose assessment replace blood glucose measurements?, Diabetes Technol Ther, 2 (2000), 461. doi: 10.1089/15209150050194332.

[11]

L. F. Sampine and S. Thompson, Solving DDEs in matlab,, Appl. Numer. Math., 37 (2001), 441. doi: 10.1016/S0168-9274(00)00055-6.

[12]

C. Simon and G. Brandenberger, Ultradian oscillations of insulin secretion in humans,, Diabetes, 51 (2002). doi: 10.2337/diabetes.51.2007.S258.

[13]

J. Sturis, K. S. Polonsky, E. Mosekilde and E. V. Cauter, Computer model for mechanisms underlying ultradian oscillations of insulin and glucose,, Amer. J. Physiol., 260 (1991), 801.

[14]

I. M. Tolic, E. Mosekilde and J. Sturis, Modeling the insulin-glucose feedback system: The significance of pulsatile insulin secretion,, J. Theoret. Biol., 207 (2000), 361.

[15]

H. Wang, J. Li and Y. Kuang, Mathematical modeling and qualitative analysis of insulin therapies,, Math. Biosci., 210 (2007), 17. doi: 10.1016/j.mbs.2007.05.008.

[16]

W. K. Ward, R. G. Massoud and C. J. Szybala, In vitro and in vivo evaluation of native glucagon and glucagon analog (MAR-D28) during aging: lack of cytotoxicity and preservation of hyperglycemic effect,, J Diabetes Sci Technol, 4 (2010), 1311. doi: 10.1177/193229681000400604.

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