American Institute of Mathematical Sciences

2016, 13(1): 83-99. doi: 10.3934/mbe.2016.13.83

Mathematical analysis of a model for glucose regulation

 1 Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, United States, United States 2 College of Public Health, The Ohio State University, Columbus, OH 43210, United States 3 Department of Medicine, The Ohio State University, Columbus, OH 43210, United States, United States 4 Mathematical Biosciences Institute and College of Public Health, The Ohio State University, Columbus, OH 43210, United States

Received  April 2015 Revised  July 2015 Published  October 2015

Diabetes affects millions of Americans, and the correct identification of individuals afflicted with this disease, especially of those in early stages or in progression towards diabetes, remains an active area of research. The minimal model is a simplified mathematical construct for understanding glucose-insulin interactions. Developed by Bergman, Cobelli, and colleagues over three decades ago [7,8], this system of coupled ordinary differential equations prevails as an important tool for interpreting data collected during an intravenous glucose tolerance test (IVGTT). In this study we present an explicit solution to the minimal model which allows for separating the glucose and insulin dynamics of the minimal model and for identifying patient-specific parameters of glucose trajectories from IVGTT. As illustrated with patient data, our approach seems to have an edge over more complicated methods currently used. Additionally, we also present an application of our method to prediction of the time to baseline recovery and calculation of insulin sensitivity and glucose effectiveness, two quantities regarded as significant in diabetes diagnostics.
Citation: Kimberly Fessel, Jeffrey B. Gaither, Julie K. Bower, Trudy Gaillard, Kwame Osei, Grzegorz A. Rempała. Mathematical analysis of a model for glucose regulation. Mathematical Biosciences & Engineering, 2016, 13 (1) : 83-99. doi: 10.3934/mbe.2016.13.83
References:
 [1] I. Ajmera, M. Swat, C. Laibe, N. Le Novère and V. Chelliah, The impact of mathematical modeling on the understanding of diabetes and related complications, CPT: Pharmacometrics & Systems Pharmacology, 2 (2013), 1-14. doi: 10.1038/psp.2013.30. [2] American Diabetes Association, Standards of medical care in diabetes-2014, Diabetes Care, 37 (2014), S14-S80. [3] E. Bartoli, G. P. Fra and G. P. Carnevale Schianca, The oral glucose tolerance test (OGTT) revisited, Eur J Intern Med, 22 (2011), 8-12. doi: 10.1016/j.ejim.2010.07.008. [4] R. N. Bergman, Lilly lecture 1989. Toward physiological understanding of glucose tolerance. Minimal-model approach, Diabetes, 38 (1989), 1512-1527. doi: 10.2337/diabetes.38.12.1512. [5] R. N. Bergman, The minimal model of glucose regulation: A biography, in Mathematical Modeling in Nutrition and the Health Sciences (eds. J. A. Novotny, M. H. Green and R. C. Boston), Advances in Experimental Medicine and Biology, Kluwer Academic/Plenum, New York, 537 (2003), 1-19. doi: 10.1007/978-1-4419-9019-8_1. [6] R. N. Bergman, Minimal model: Perspective from 2005, Horm Res, 64 (2005), 8-15. doi: 10.1159/000089312. [7] R. N. Bergman, Y. Z. Ider, C. R. Bowden and C. Cobelli, Quantitative estimation of insulin sensitivity, Am J Physiol, 236 (1979), E667-E677. [8] R. N. Bergman, L. S. Phillips and C. Cobelli, Physiologic evaluation of factors controlling glucose tolerance in man: measurement of insulin sensitivity and $\beta$-cell glucose sensititivy from the response to intravenous glucose, J Clin Invest, 68 (1981), 1456-1467. doi: 10.1172/JCI110398. [9] P. J. Bingley, P. Colman, G. S. Eisenbarth, R. A. Jackson, D. K. McCulloch, W. J. Riley and E. A. Gale, Standardization of IVGTT to predict IDDM, Diabetes Care, 15 (1992), 1313-1316. doi: 10.2337/diacare.15.10.1313. [10] V. Biourge, R. W. Nelson, E. Feldman, N. H. Willits, J. G. Morris and Q. R. Roger, Effect of weight gain and subsequent weight loss on glucose tolerance and insulin response in healthy cats, J. Vet Intern Med., 11 (1997), 86-91. doi: 10.1111/j.1939-1676.1997.tb00078.x. [11] Z. T. Bloomgarden, Approaches to treatment of type 2 diabetes, Diabetes Care, 31 (2008), 1697-1703. doi: 10.2337/dc08-zb08. [12] E. Bonora and J. Tuomilehto, The pros and cons of diagnosing diabetes with A1C, Diabetes Care, 34 (2011), S184-S190. doi: 10.2337/dc11-s216. [13] R. Boston, D. Stefanovski, P. Moate, O. Linares and P. Greif, Cornerstones to shape modeling for the 21st Century: Introducing the AKA-Glucose project, in Mathematical Modeling in Nutrition and the Health Sciences (eds. J. A. Novotny, M. H. Green and R. C. Boston), Advances in Experimental Medicine and Biology, Kluwer Academic/Plenum, New York, 2003, 21-42. doi: 10.1007/978-1-4419-9019-8. [14] R. C. Boston, D. Stefanovski, P. J. Moate, A. E. Sumner, R. M. Watanabe and R. N. Bergman, MINMOD Millennium: A computer program to calculate glucose effectiveness and insulin sensitivity from the frequently sampled intravenous glucose tolerance test, Diabetes technology & therapeutics, 5 (2003), 1003-1015. [15] A. Boutayeb and A. Chetouani, A critical review of mathematical models and data used in diabetology, BioMedical Engineering OnLine, 5 (2006), p43. doi: 10.1186/1475-925X-5-43. [16] A. Caumo, R. N. Bergman and C. Cobelli, Insulin sensitivity from meal tolerance tests in normal subjects: A minimal model index, J Clin Endocrinol Metab, 85 (2000), 4396-4402. doi: 10.1210/jcem.85.11.6982. [17] Centers for Disease Control and Prevention, National diabetes fact sheet: National estimates and general, information in diabetes and prediabetes in the United States, 2011. [18] H. P. Chase, D. D. Cuthbertson, L. M. Dolan, F. Kaufman, J. P. Krischer, D. A. Schatz, N. H. White, D. M. Wilson and J. Wolfsdorf, First-phase insulin release during the intravenous glucose tolerance test as a risk factor for type 1 diabetes, J Pediatr, 138 (2001), 244-249. doi: 10.1067/mpd.2001.111274. [19] Y. J. Cheng, E. W. Gregg, L. S. Geiss, G. Imperatore, D. E. Williams, X. Zhang, A. L. Albright, C. C. Cowie, R. Klein and J. B. Saaddine, Association of A1C and fasting plasma glucose levels with diabetic retinopathy prevalence in the U.S. population: Implications for diabetes diagnostic thresholds, Diabetes Care, 32 (2009), 2027-2032. doi: 10.2337/dc09-0440. [20] S. Colagiuri, C. M. Lee, T. Y. Wong, B. Balkau, J. E. Shaw, K. Borch-Johnsen and D.-C. W. Group, Glycemic thresholds for diabetes-specific retinopathy: Implications for diagnostic criteria for diabetes, Diabetes Care, 34 (2011), 145-150. doi: 10.2337/dc10-1206. [21] A. De Gaetano and O. Arino, Mathematical modeling of the intravenous glucose tolerance test, J Math Bio, 40 (2000), 136-168. doi: 10.1007/s002850050007. [22] W. S. Eldin, M. Emara and A. Shoker, Prediabetes: A must to recognise disease state, Int J Clin Pract, 62 (2008), 642-648. doi: 10.1111/j.1742-1241.2008.01705.x. [23] A. Festa, K. Williams, A. J. Hanley and S. M. Haffner, Beta-cell dysfunction in subjects with impaired glucose tolerance and early type 2 diabetes: comparison of surrogate markers with first-phase insulin secretion from an intravenous glucose tolerance test, Diabetes, 57 (2008), 1638-1644. [24] R. G. Hahn, S. Ljunggren, F. Larsen and T. Nyström, A simple intravenous glucose tolerance test for assessment of insulin sensitivity, Theor Biol Med Model, 8 (2011), p12. doi: 10.1186/1742-4682-8-12. [25] J. Li, Y. Kuang and B. Li, Analysis of IVGTT glucose-insulin interaction models with time delay, Discrete and Continuous Dynamical Systems - Series B, 1 (2001), 103-124. doi: 10.3934/dcdsb.2001.1.103. [26] M. A. Marini, E. Succurro, S. Frontoni, S. Mastroianni, F. Arturi, A. Sciacqua, R. Lauro, M. L. Hribal, F. Perticone and G. Sesti, Insulin sensitivity, beta-cell function, and incretin effect in individuals with elevated 1-hour postload plasma glucose levels, Diabetes Care, 35 (2012), 868-872. [27] R. Muniyappa, S. Lee, H. Chen and M. J. Quon, Current approaches for assessing insulin sensitivity and resistance in vivo: Advantages, limitations, and appropriate usage, Am J Physiol Endocrinol Metab, 294 (2008), E15-E26. doi: 10.1152/ajpendo.00645.2007. [28] D. M. Nathan, M. B. Davidson, R. A. DeFronzo, R. J. Heine, R. R. Henry, R. Pratley, B. Zinman and American Diabetes Association, Impaired fasting glucose and impaired glucose tolerance: Implications for care, Diabetes Care, 30 (2007), 753-759. doi: 10.2337/dc07-9920. [29] A. Nittala, S. Ghosh, D. Stefanovski, R. Bergman and X. Wang, Dimensional analysis of MINMOD leads to definition of the disposition index of glucose regulation and improved simulation algorithm, BioMedical Engineering OnLine, 5 (2006), 44-57. [30] T. Nozaki, H. Tamai, S. Matsubayashi, G. Komaki, N. Kobayashi and T. Nakagawa, Insulin response to intravenous glucose in patients with anorexia nervosa showing low insulin response to oral glucose, J Clin Endocrinol Metab, 79 (1994), 217-222. [31] G. Pacini and R. N. Bergman, MINMOD: A computer program to calculate insulin sensitivity and pancreatic responsivity from the frequently sampled intravenous glucose tolerance test, Comput Meth Prog Bio, 23 (1986), 113-122. doi: 10.1016/0169-2607(86)90106-9. [32] S. Panunzi and A. DeGaetano, Pitfalls in model identification: Examples from glucose-insulin modelling, in Data-driven Modeling for Diabetes (eds. V. Marmarelis and G. Mitsis), Lecture Notes in Bioengineering, Springer Berlin Heidelberg, 2014, 117-129. doi: 10.1007/978-3-642-54464-4_5. [33] M. Stumvoll, B. J. Goldstein and T. W. van Haeften, Type 2 diabetes: Principles of pathogenesis and therapy, Lancet, 365 (2005), 1333-1346. doi: 10.1016/S0140-6736(05)61032-X. [34] N. van Riel, Eindhoven University of Technology, Department of Biomedical Engineering, Department of Electrical Engineering, BIOMIM & Control Systems, 1-21. [35] N. van Riel, GLUC_MM_MLE2012 Maximum Likelihood Estimation of minimal model of glucose kinetics, http://bmi.bmt.tue.nl/sysbio/parameter_estimation/gluc_mm_mle2012.m, 2012, Accessed: 2015-02-24. [36] L. Zhang, G. Krzentowski, A. Albert and P. J. Lefebvre, Risk of developing retinopathy in Diabetes Control and Complications Trial type 1 diabetic patients with good or poor metabolic control, Diabetes Care, 24 (2001), 1275-1279. doi: 10.2337/diacare.24.7.1275.

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References:
 [1] I. Ajmera, M. Swat, C. Laibe, N. Le Novère and V. Chelliah, The impact of mathematical modeling on the understanding of diabetes and related complications, CPT: Pharmacometrics & Systems Pharmacology, 2 (2013), 1-14. doi: 10.1038/psp.2013.30. [2] American Diabetes Association, Standards of medical care in diabetes-2014, Diabetes Care, 37 (2014), S14-S80. [3] E. Bartoli, G. P. Fra and G. P. Carnevale Schianca, The oral glucose tolerance test (OGTT) revisited, Eur J Intern Med, 22 (2011), 8-12. doi: 10.1016/j.ejim.2010.07.008. [4] R. N. Bergman, Lilly lecture 1989. Toward physiological understanding of glucose tolerance. Minimal-model approach, Diabetes, 38 (1989), 1512-1527. doi: 10.2337/diabetes.38.12.1512. [5] R. N. Bergman, The minimal model of glucose regulation: A biography, in Mathematical Modeling in Nutrition and the Health Sciences (eds. J. A. Novotny, M. H. Green and R. C. Boston), Advances in Experimental Medicine and Biology, Kluwer Academic/Plenum, New York, 537 (2003), 1-19. doi: 10.1007/978-1-4419-9019-8_1. [6] R. N. Bergman, Minimal model: Perspective from 2005, Horm Res, 64 (2005), 8-15. doi: 10.1159/000089312. [7] R. N. Bergman, Y. Z. Ider, C. R. Bowden and C. Cobelli, Quantitative estimation of insulin sensitivity, Am J Physiol, 236 (1979), E667-E677. [8] R. N. Bergman, L. S. Phillips and C. Cobelli, Physiologic evaluation of factors controlling glucose tolerance in man: measurement of insulin sensitivity and $\beta$-cell glucose sensititivy from the response to intravenous glucose, J Clin Invest, 68 (1981), 1456-1467. doi: 10.1172/JCI110398. [9] P. J. Bingley, P. Colman, G. S. Eisenbarth, R. A. Jackson, D. K. McCulloch, W. J. Riley and E. A. Gale, Standardization of IVGTT to predict IDDM, Diabetes Care, 15 (1992), 1313-1316. doi: 10.2337/diacare.15.10.1313. [10] V. Biourge, R. W. Nelson, E. Feldman, N. H. Willits, J. G. Morris and Q. R. Roger, Effect of weight gain and subsequent weight loss on glucose tolerance and insulin response in healthy cats, J. Vet Intern Med., 11 (1997), 86-91. doi: 10.1111/j.1939-1676.1997.tb00078.x. [11] Z. T. Bloomgarden, Approaches to treatment of type 2 diabetes, Diabetes Care, 31 (2008), 1697-1703. doi: 10.2337/dc08-zb08. [12] E. Bonora and J. Tuomilehto, The pros and cons of diagnosing diabetes with A1C, Diabetes Care, 34 (2011), S184-S190. doi: 10.2337/dc11-s216. [13] R. Boston, D. Stefanovski, P. Moate, O. Linares and P. Greif, Cornerstones to shape modeling for the 21st Century: Introducing the AKA-Glucose project, in Mathematical Modeling in Nutrition and the Health Sciences (eds. J. A. Novotny, M. H. Green and R. C. Boston), Advances in Experimental Medicine and Biology, Kluwer Academic/Plenum, New York, 2003, 21-42. doi: 10.1007/978-1-4419-9019-8. [14] R. C. Boston, D. Stefanovski, P. J. Moate, A. E. Sumner, R. M. Watanabe and R. N. Bergman, MINMOD Millennium: A computer program to calculate glucose effectiveness and insulin sensitivity from the frequently sampled intravenous glucose tolerance test, Diabetes technology & therapeutics, 5 (2003), 1003-1015. [15] A. Boutayeb and A. Chetouani, A critical review of mathematical models and data used in diabetology, BioMedical Engineering OnLine, 5 (2006), p43. doi: 10.1186/1475-925X-5-43. [16] A. Caumo, R. N. Bergman and C. Cobelli, Insulin sensitivity from meal tolerance tests in normal subjects: A minimal model index, J Clin Endocrinol Metab, 85 (2000), 4396-4402. doi: 10.1210/jcem.85.11.6982. [17] Centers for Disease Control and Prevention, National diabetes fact sheet: National estimates and general, information in diabetes and prediabetes in the United States, 2011. [18] H. P. Chase, D. D. Cuthbertson, L. M. Dolan, F. Kaufman, J. P. Krischer, D. A. Schatz, N. H. White, D. M. Wilson and J. Wolfsdorf, First-phase insulin release during the intravenous glucose tolerance test as a risk factor for type 1 diabetes, J Pediatr, 138 (2001), 244-249. doi: 10.1067/mpd.2001.111274. [19] Y. J. Cheng, E. W. Gregg, L. S. Geiss, G. Imperatore, D. E. Williams, X. Zhang, A. L. Albright, C. C. Cowie, R. Klein and J. B. Saaddine, Association of A1C and fasting plasma glucose levels with diabetic retinopathy prevalence in the U.S. population: Implications for diabetes diagnostic thresholds, Diabetes Care, 32 (2009), 2027-2032. doi: 10.2337/dc09-0440. [20] S. Colagiuri, C. M. Lee, T. Y. Wong, B. Balkau, J. E. Shaw, K. Borch-Johnsen and D.-C. W. Group, Glycemic thresholds for diabetes-specific retinopathy: Implications for diagnostic criteria for diabetes, Diabetes Care, 34 (2011), 145-150. doi: 10.2337/dc10-1206. [21] A. De Gaetano and O. Arino, Mathematical modeling of the intravenous glucose tolerance test, J Math Bio, 40 (2000), 136-168. doi: 10.1007/s002850050007. [22] W. S. Eldin, M. Emara and A. Shoker, Prediabetes: A must to recognise disease state, Int J Clin Pract, 62 (2008), 642-648. doi: 10.1111/j.1742-1241.2008.01705.x. [23] A. Festa, K. Williams, A. J. Hanley and S. M. Haffner, Beta-cell dysfunction in subjects with impaired glucose tolerance and early type 2 diabetes: comparison of surrogate markers with first-phase insulin secretion from an intravenous glucose tolerance test, Diabetes, 57 (2008), 1638-1644. [24] R. G. Hahn, S. Ljunggren, F. Larsen and T. Nyström, A simple intravenous glucose tolerance test for assessment of insulin sensitivity, Theor Biol Med Model, 8 (2011), p12. doi: 10.1186/1742-4682-8-12. [25] J. Li, Y. Kuang and B. Li, Analysis of IVGTT glucose-insulin interaction models with time delay, Discrete and Continuous Dynamical Systems - Series B, 1 (2001), 103-124. doi: 10.3934/dcdsb.2001.1.103. [26] M. A. Marini, E. Succurro, S. Frontoni, S. Mastroianni, F. Arturi, A. Sciacqua, R. Lauro, M. L. Hribal, F. Perticone and G. Sesti, Insulin sensitivity, beta-cell function, and incretin effect in individuals with elevated 1-hour postload plasma glucose levels, Diabetes Care, 35 (2012), 868-872. [27] R. Muniyappa, S. Lee, H. Chen and M. J. Quon, Current approaches for assessing insulin sensitivity and resistance in vivo: Advantages, limitations, and appropriate usage, Am J Physiol Endocrinol Metab, 294 (2008), E15-E26. doi: 10.1152/ajpendo.00645.2007. [28] D. M. Nathan, M. B. Davidson, R. A. DeFronzo, R. J. Heine, R. R. Henry, R. Pratley, B. Zinman and American Diabetes Association, Impaired fasting glucose and impaired glucose tolerance: Implications for care, Diabetes Care, 30 (2007), 753-759. doi: 10.2337/dc07-9920. [29] A. Nittala, S. Ghosh, D. Stefanovski, R. Bergman and X. Wang, Dimensional analysis of MINMOD leads to definition of the disposition index of glucose regulation and improved simulation algorithm, BioMedical Engineering OnLine, 5 (2006), 44-57. [30] T. Nozaki, H. Tamai, S. Matsubayashi, G. Komaki, N. Kobayashi and T. Nakagawa, Insulin response to intravenous glucose in patients with anorexia nervosa showing low insulin response to oral glucose, J Clin Endocrinol Metab, 79 (1994), 217-222. [31] G. Pacini and R. N. Bergman, MINMOD: A computer program to calculate insulin sensitivity and pancreatic responsivity from the frequently sampled intravenous glucose tolerance test, Comput Meth Prog Bio, 23 (1986), 113-122. doi: 10.1016/0169-2607(86)90106-9. [32] S. Panunzi and A. DeGaetano, Pitfalls in model identification: Examples from glucose-insulin modelling, in Data-driven Modeling for Diabetes (eds. V. Marmarelis and G. Mitsis), Lecture Notes in Bioengineering, Springer Berlin Heidelberg, 2014, 117-129. doi: 10.1007/978-3-642-54464-4_5. [33] M. Stumvoll, B. J. Goldstein and T. W. van Haeften, Type 2 diabetes: Principles of pathogenesis and therapy, Lancet, 365 (2005), 1333-1346. doi: 10.1016/S0140-6736(05)61032-X. [34] N. van Riel, Eindhoven University of Technology, Department of Biomedical Engineering, Department of Electrical Engineering, BIOMIM & Control Systems, 1-21. [35] N. van Riel, GLUC_MM_MLE2012 Maximum Likelihood Estimation of minimal model of glucose kinetics, http://bmi.bmt.tue.nl/sysbio/parameter_estimation/gluc_mm_mle2012.m, 2012, Accessed: 2015-02-24. [36] L. Zhang, G. Krzentowski, A. Albert and P. J. Lefebvre, Risk of developing retinopathy in Diabetes Control and Complications Trial type 1 diabetic patients with good or poor metabolic control, Diabetes Care, 24 (2001), 1275-1279. doi: 10.2337/diacare.24.7.1275.
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