Mathematical analysis of a model for glucose regulation

Kimberly  Fessel; Jeffrey B.  Gaither; Julie K.  Bower; Trudy  Gaillard; Kwame  Osei; Grzegorz A.  Rempała

doi:10.3934/mbe.2016.13.83

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

Mathematical analysis of a model for glucose regulation

Kimberly Fessel ^1, , Jeffrey B. Gaither ^1, , Julie K. Bower ^2, , Trudy Gaillard ^3, , Kwame Osei ^3, and Grzegorz A. Rempała ^4,

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

Abstract
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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.

Keywords: parameter estimation, glucose tolerance test, insulin sensitivity., Diabetes, minimal model.

Mathematics Subject Classification: Primary: 92B; Secondary: 92C6.

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. Google Scholar
[2]	American Diabetes Association, Standards of medical care in diabetes-2014, Diabetes Care, 37 (2014), S14-S80. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	R. N. Bergman, Minimal model: Perspective from 2005, Horm Res, 64 (2005), 8-15. doi: 10.1159/000089312. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	Z. T. Bloomgarden, Approaches to treatment of type 2 diabetes, Diabetes Care, 31 (2008), 1697-1703. doi: 10.2337/dc08-zb08. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	N. van Riel, Eindhoven University of Technology, Department of Biomedical Engineering, Department of Electrical Engineering,, BIOMIM & Control Systems, (): 1. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

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. Google Scholar
[2]	American Diabetes Association, Standards of medical care in diabetes-2014, Diabetes Care, 37 (2014), S14-S80. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	R. N. Bergman, Minimal model: Perspective from 2005, Horm Res, 64 (2005), 8-15. doi: 10.1159/000089312. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	Z. T. Bloomgarden, Approaches to treatment of type 2 diabetes, Diabetes Care, 31 (2008), 1697-1703. doi: 10.2337/dc08-zb08. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	N. van Riel, Eindhoven University of Technology, Department of Biomedical Engineering, Department of Electrical Engineering,, BIOMIM & Control Systems, (): 1. Google Scholar
[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. Google Scholar
[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. Google Scholar

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