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October  2015, 11(4): 1149-1164. doi: 10.3934/jimo.2015.11.1149

Modelling and optimal control of blood glucose levels in the human body

Zahra Al Helal 1, , Volker Rehbock 1, and  Ryan Loxton 1,

1. 

Department of Mathematics and Statistics, Curtin University, GPO Box U1987 Perth, Western Australia 6845, Australia, Australia, Australia

Received  June 2014 Revised  September 2014 Published  March 2015

Regulating the blood glucose level is a challenging control problem for the human body. Abnormal blood glucose levels can cause serious health problems over time, including diabetes. Although several mathematical models have been proposed to describe the dynamics of glucose-insulin interaction, none of them have been universally adopted by the research community. In this paper, we consider a dynamic model of the blood glucose regulatory system originally proposed by Liu and Tang in 2008. This model consists of eight state variables naturally divided into three subsystems: the glucagon and insulin transition subsystem, the receptor binding subsystem and the glucose subsystem. The model contains 36 model parameters, many of which are unknown and difficult to determine accurately. We formulate an optimal parameter selection problem in which optimal values for the model parameters must be selected so that the resulting model best fits given experimental data. We demonstrate that this optimal parameter selection problem can be solved readily using the optimal control software MISER 3.3. Using this approach, significant improvements can be made in matching the model to the experimental data. We also investigate the sensitivity of the resulting optimized model with respect to the insulin release rate. Finally, we use MISER 3.3 to determine optimal open loop controls for the optimized model.
Keywords: dynamic model, optimal control, optimal parameter selection problem., Blood glucose.
Mathematics Subject Classification: Primary: 49M37; Secondary: 65P99, 92C45, 92C5.
Citation: Zahra Al Helal, Volker Rehbock, Ryan Loxton. Modelling and optimal control of blood glucose levels in the human body. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1149-1164. doi: 10.3934/jimo.2015.11.1149
References:
[1]

F. Chee and T. Fernando, Closed Loop Control of Blood Glucose,, Springer, (2007).   Google Scholar

[2]

R. Hovorka, V. Canonico, L. J. Chassin, U. Haueter, M. Massi-Benedetti, M. O. Federici, T. R. Pieber, H. C. Schaller, L. Schaupp, T. Vering and M. E. Wilinska, Nonlinear model predictive control of glucose concentration in subjects with type 1 diabetes,, Physiological Measurement, 25 (2004), 905.  doi: 10.1088/0967-3334/25/4/010.  Google Scholar

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IDF Diabetes Atlas, 5th Edition, International Diabetes Federation,, Brussels, (2011).   Google Scholar

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L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER3 Optimal Control Software: Theory and User Manual Version 3,, University of Western Australia, (2004).   Google Scholar

[5]

M. Korach-André, H. Roth, D. Barnoud, M. Péan, F. Péronnent and X. Leverve, Glucose appearance in the peripheral circulation and liver glucose output in men after a large C starch meal,, The American Journal of Clinical Nutrition, 80 (2004), 881.   Google Scholar

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L. Kovács, B. Kulcsár, A. György and Z. Benyó, Robust servo control of a novel type 1 diabetic model,, Optimal Control Applications and Methods, 32 (2011), 215.  doi: 10.1002/oca.963.  Google Scholar

[7]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey,, Journal of Industrial and Management Optimization, 10 (2014), 275.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[8]

W. Liu and F. Tang, Modelling a simplified regulatory system of blood glucose at molecular levels,, Journal of Theoretical Biology, 252 (2008), 608.  doi: 10.1016/j.jtbi.2008.02.021.  Google Scholar

[9]

R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints,, Automatica, 44 (2008), 2923.  doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[10]

G. Marchetti, M. Barolo, L. Jovanovic and H. Zisser, An improved PID switching control strategy for type 1 diabetes,, In Proceeding of the 28th IEEE EMBS Annual International Conference, (2006), 5041.  doi: 10.1109/IEMBS.2006.259541.  Google Scholar

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R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy,, World Scientific, (1993).  doi: 10.1142/9789812832542.  Google Scholar

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K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems,, Longman Scientific and Technical, (1991).   Google Scholar

show all references

References:
[1]

F. Chee and T. Fernando, Closed Loop Control of Blood Glucose,, Springer, (2007).   Google Scholar

[2]

R. Hovorka, V. Canonico, L. J. Chassin, U. Haueter, M. Massi-Benedetti, M. O. Federici, T. R. Pieber, H. C. Schaller, L. Schaupp, T. Vering and M. E. Wilinska, Nonlinear model predictive control of glucose concentration in subjects with type 1 diabetes,, Physiological Measurement, 25 (2004), 905.  doi: 10.1088/0967-3334/25/4/010.  Google Scholar

[3]

IDF Diabetes Atlas, 5th Edition, International Diabetes Federation,, Brussels, (2011).   Google Scholar

[4]

L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER3 Optimal Control Software: Theory and User Manual Version 3,, University of Western Australia, (2004).   Google Scholar

[5]

M. Korach-André, H. Roth, D. Barnoud, M. Péan, F. Péronnent and X. Leverve, Glucose appearance in the peripheral circulation and liver glucose output in men after a large C starch meal,, The American Journal of Clinical Nutrition, 80 (2004), 881.   Google Scholar

[6]

L. Kovács, B. Kulcsár, A. György and Z. Benyó, Robust servo control of a novel type 1 diabetic model,, Optimal Control Applications and Methods, 32 (2011), 215.  doi: 10.1002/oca.963.  Google Scholar

[7]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey,, Journal of Industrial and Management Optimization, 10 (2014), 275.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[8]

W. Liu and F. Tang, Modelling a simplified regulatory system of blood glucose at molecular levels,, Journal of Theoretical Biology, 252 (2008), 608.  doi: 10.1016/j.jtbi.2008.02.021.  Google Scholar

[9]

R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints,, Automatica, 44 (2008), 2923.  doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[10]

G. Marchetti, M. Barolo, L. Jovanovic and H. Zisser, An improved PID switching control strategy for type 1 diabetes,, In Proceeding of the 28th IEEE EMBS Annual International Conference, (2006), 5041.  doi: 10.1109/IEMBS.2006.259541.  Google Scholar

[11]

R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy,, World Scientific, (1993).  doi: 10.1142/9789812832542.  Google Scholar

[12]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems,, Longman Scientific and Technical, (1991).   Google Scholar

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