Skip to main navigation menu Skip to main content Skip to site footer

Vol. 13 No. 2 (2022)
Vol. 13 Núm. 2 (2022): Vol. 13 Núm. 2 (2022): Vol 13, Núm.2 (2022): Julio-Diciembre

Artículos de investigación / Research papers

Optimal Blood Glucose Control Through Continuous Insulin Infusion

https://doi.org/10.19053/01217488.v13.n2.2022.14173

Published 2022-07-12

  • Edwar Fabián Panqueba Moreno
    • ,
  • Jorge Mauricio Ruiz Vera

Edwar Fabián Panqueba Moreno
Universidad Nacional de Colombia

Jorge Mauricio Ruiz Vera


Abstract

In this paper, we formulate the problem of the insulin supply regimen in a diabetic patient as an optimal control problem, so that there is no overdose or insufficiency of the hormonal drug under different feeding styles. The interaction between glucose and insulin is modeled as a nonlinear system of ordinary differential equations. We prove the existence and global uniqueness of the solution of the system, as well as for the optimal control. The optimal control problem is solved directly by using the quadratic sequential programming method. The numerical results suggest setting, according to the patient’s dietary style, the prescribed glucose concentration level to be maintained during the day. The analytical and numerical study of this proposal is expected to be helpful in future insulin pump developments.

Keywords

Optimal control, Nonlinear programming, lucose-insulin dynamic, diabetes, mathematical biology

PDF (Español)

References

  1. Ackerman, E., Gatewood, L., Rosevear, J. & Molnar, G. Model studies of blood-glucose regulation. Bulletin Of Mathematical Biology. vol. 27, pp. 21-37, 1965. DOI: https://doi.org/10.1007/BF02477259
  2. Ajmera, I., Swat, M., Laibe, C., Novère, N. & Chelliah, V. The impact of mathematical modeling on the understanding of diabetes and related complications. CPT: Pharmacometrics & Systems Pharmacology, vol. 2, pp. 1-14, 2013. DOI: https://doi.org/10.1038/psp.2013.30
  3. Alonso Cisneros, I. Técnicas de control en modelos matemáticos para la diabetes. Universidad de Cantrabria, 2015.
  4. Andrikov, D. & Kurbanov, S. Development of a Linear Model of Diabetes Mellitus with a PID Controller and a Method for Automatic Adjustment of the PID Controller. Intelligent Sustainable Systems, pp. 43-49, 2022. DOI: https://doi.org/10.1007/978-981-16-6309-3_5
  5. Apablaza,P.,Soto,N.,Román,R.&Codner,E. Nuevas Tecnologías en diabets. Revista Médica Clínica Las Condes, vol. 27, pp. 213-226, 2016. DOI: https://doi.org/10.1016/j.rmclc.2016.04.011
  6. Barbu, V. Mathematical methods in optimiza- tion of differential systems. Springer Science & Business Media, 2012.
  7. Barragán, F. Técnicas de control robusto H2/H para sistemas controlados a través de redes de comunicación. Universidad de Sevilla, 2012.
  8. Beato, P. Sistemas de páncreas artificial disponibles en el mercado español: ¿qué aportan al paciente?. Revista Diabetes. vol. 58, pp. 28-32, 2019.
  9. Bergman, R., Ider, Y., Bowden, C. & Cobelli, C. Quantitative estimation of insulin sensitivity. American Journal Of Physiology- Endocrinology And Metabolism, 236, E667, 1979. DOI: https://doi.org/10.1152/ajpendo.1979.236.6.E667
  10. Bolie, V. Coefficients of normal blood glucose regulation. Journal Of Applied Physiology, vol. 16, pp. 783-788, 1961. DOI: https://doi.org/10.1152/jappl.1961.16.5.783
  11. Bulirsch, R. & Stoer, J. Introduction to numerical analysis. Springer, 1996.
  12. Fatsecret Chile Calculadora conteo de calorías y carbohidratos por porción de alimento, 2021. https://www.fatsecret.cl/calorias-nutricion/
  13. Fisher, M. & Teo, K. Optimal insulin infusion resulting from a mathematical model of blood glucose dynamics. IEEE Transactions On Biomedical Engineering, vol. 36, pp. 479-486, 1989. DOI: https://doi.org/10.1109/10.18755
  14. Staten, M., Stern, M., Miller, W., Steffes, M., Campbell, S. & Workgroup, I. Insulin assay standardization: leading to measures of insulin sensitivity and secretion for practical clinical care. Diabetes Care, vol. 33, pp. 205-206, 2010. DOI: https://doi.org/10.2337/dc09-1206
  15. Huang, M., Li, J., Song, X. & Guo, H. Modeling impulsive injections of insulin: towards artificial pancreas. SIAM Journal On Applied Mathematics, vol. 72, pp. 1524-1548, 2012. DOI: https://doi.org/10.1137/110860306
  16. Kapur, J. Mathematical modelling. New Age International, 1988.
  17. Knopp, J., Holder-Pearson, L. & Chase, J. Insulin Units and Conversion Factors: A Story of Truth, Boots, and Faster Half-Truths. Journal Of Diabetes Science And Technology, vol. 13, pp. 597-600, 2019. DOI: https://doi.org/10.1177/1932296818805074
  18. König Quantifying the Contribution of the Liver to Glucose Homeostasis: A Detailed Kinetic Model of Human Hepatic Glucose Metabolism. PLOS Computational Biology, vol. 8, pp. 1-17, 2012. DOI: https://doi.org/10.1371/journal.pcbi.1002577
  19. Leon, B., Alanis, A., Sanchez, E., Ornelas- Tellez, F. & Ruiz-Velazquez, E. Neural inverse optimal control applied to type 1 diabetes mellitus patients. Analog Integrated Circuits And Signal Processing, vol. 76, pp. 343-352, 2013. DOI: https://doi.org/10.1007/s10470-013-0109-8
  20. ̈Medtronic Minmed Inc Información básica sobre el tratamiento con bomba de insulina. (2015),
  21. Mirzaee, A., Dehghani, M. & Mohammadi, M. Optimal impulsive blood glucose control through multiple injections. Chaos, Solitons & Fractals, vol. 152, pp. 111326, 2021. DOI: https://doi.org/10.1016/j.chaos.2021.111326
  22. Myfitnesspal Calculadora conteo de calorías y carbohidratos por porción de alimento, 2021. https://www.myfitnesspal.com/es/food/search
  23. Nocedal, J. & Wright, S. Numerical opti- mization. Springer Science & Business Media, 2006.
  24. Pan, D., Wang, K., Zhou, Z., Liu, X. & Shen, J. FAHP-based Mathematical Model for Exercise Rehabilitation Management
  25. of Diabetes Mellitus. arXiv, 2022. https://arxiv.org/abs/2201.07884
  26. Parker, R., Doyle, F. & Peppas, N. A model- based algorithm for blood glucose control in
  27. Type I diabetic patients. IEEE Transactions On Biomedical Engineering, vol. 46, pp. 148-157, 1999. DOI: https://doi.org/10.1109/10.740877
  28. Pedregal, P. Introduction to optimization. Springer New York, 2006.
  29. Rathee, S. & Nilam ODE models for the man- agement of diabetes: A review. International Journal Of Diabetes In Developing Countries, vol. 37, pp. 4-15, 2017. DOI: https://doi.org/10.1007/s13410-016-0475-8
  30. Song, X., Huang, M. & Li, J. Modeling Impulsive Insulin Delivery in Insulin Pump with Time Delays. SIAM Journal On Applied Mathematics, vol. 74, pp. 1763-1785, 2014. DOI: https://doi.org/10.1137/130933137
  31. Swan, G. An optimal control model of diabetes mellitus. Bulletin Of Mathematical Biology, vol. 44, pp. 793-808, 1982. DOI: https://doi.org/10.1016/S0092-8240(82)80043-8
  32. Uvaliyeva, I. & Ismukhamedova, A. Mathe- matical Model for Early Diagnosis of Diabetes Mellitus 2021 International Young Engineers Forum (YEF-ECE), pp. 104-108, 2021. DOI: https://doi.org/10.1109/YEF-ECE52297.2021.9505105
  33. Velásquez, S., Velásquez, R., Leyton, M., Borjas, J. & Custodio, A. Modelado del control de la regulación de Glucosa. Universidad, Ciencia Y Tecnología, vol. 17, 2013.
  34. Wang, L. PID Control System Design and Automatic Tuning using MATLAB/Simulink. Wiley-IEEE Press, 2020. DOI: https://doi.org/10.1002/9781119469414

Downloads

Download data is not yet available.

Similar Articles

1 2 3 > >> 

You may also start an advanced similarity search for this article.