A critical review of mathematical models and data used in diabetology

The literature dealing with mathematical modelling for diabetes is abundant. During the last decades, a variety of models have been devoted to different aspects of diabetes, including glucose and insulin dynamics, management and complications prevention, cost and cost-effectiveness of strategies and epidemiology of diabetes in general. Several reviews are published regularly on mathematical models used for specific aspects of diabetes. In the present paper we propose a global overview of mathematical models dealing with many aspects of diabetes and using various tools. The review includes, side by side, models which are simple and/or comprehensive; deterministic and/or stochastic; continuous and/or discrete; using ordinary differential equations, partial differential equations, optimal control theory, integral equations, matrix analysis and computer algorithms.

[1]  Abdesslam Boutayeb,et al.  The double burden of communicable and non-communicable diseases in developing countries. , 2006, Transactions of the Royal Society of Tropical Medicine and Hygiene.

[2]  R. Anderson,et al.  AUTOMATION IN ANALYTICAL CHEMISTRY. , 1966 .

[3]  David M Eddy,et al.  Archimedes: a trial-validated model of diabetes. , 2003, Diabetes care.

[4]  Y. Hamamoto,et al.  Guidelines for computer modeling of diabetes and its complications. , 2004, Diabetes care.

[5]  S. Genuth,et al.  The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin-dependent diabetes mellitus. , 1993, The New England journal of medicine.

[6]  L. Heinemann,et al.  Sensors for glucose monitoring: technical and clinical aspects , 2001, Diabetes/metabolism research and reviews.

[7]  M. Fantone,et al.  Report of the Expert Committee on the Diagnosis and Classification of Diabetes Mellitus , 1997, Diabetes Care.

[8]  W. Herman.  Diabetes modeling. , 2003, Diabetes care.

[9]  Claudio Cobelli,et al.  The minimal model of glucose disappearance: optimal input studies , 1987 .

[10]  D. Spodick.  Multiple risk factor intervention trial. , 1985, The American journal of cardiology.

[11]  R. Kahn.  Dealing with complexity in clinical diabetes: the value of archimedes. , 2003, Diabetes care.

[12]  R S Parker,et al.  The intravenous route to blood glucose control. , 2001, IEEE engineering in medicine and biology magazine : the quarterly magazine of the Engineering in Medicine & Biology Society.

[13]  G. Reaven,et al.  Comparison of Insulin Secretion and In Vivo Insulin Action in Nonobese and Moderately Obese Individuals with Non-insulin-dependent Diabetes Mellitus , 1982, Diabetes.

[14]  Richard N Bergman,et al.  The minimal model of glucose regulation: a biography. , 2003, Advances in experimental medicine and biology.

[15]  G. Segre,et al.  Modeling Blood Glucose and Insulin Kinetics in Normal, Diabetic and Obese Subjects , 1973, Diabetes.

[16]  R. Bain,et al.  The effect of angiotensin-converting-enzyme inhibition on diabetic nephropathy. The Collaborative Study Group. , 1993, The New England journal of medicine.

[17]  J. Levy,et al.  Correct Homeostasis Model Assessment (HOMA) Evaluation Uses the Computer Program , 1998, Diabetes Care.

[18]  D L DeMets,et al.  The Wisconsin Epidemiologic Study of Diabetic Retinopathy. IX. Four-year incidence and progression of diabetic retinopathy when age at diagnosis is less than 30 years. , 1989, Archives of ophthalmology.

[19]  A. Boutayeb,et al.  The effect of physical exercise on the dynamics of glucose and insulin. , 2002, Journal of biomechanics.

[20]  R. Andres,et al.  Manual feedback technique for the control of blood glucose concentration , 1966 .

[21]  R. Bellazzi,et al.  The subcutaneous route to insulin-dependent diabetes therapy. , 2001, IEEE engineering in medicine and biology magazine : the quarterly magazine of the Engineering in Medicine & Biology Society.

[22]  Norman Fleischer,et al.  The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin-dependent diabetes mellitus. The Diabetes Control and Complications Trial Research Group. , 1993 .

[23]  David M Eddy,et al.  Validation of the archimedes diabetes model. , 2003, Diabetes care.

[24]  R. Ratner.  Type 2 diabetes mellitus: the grand overview , 1998, Diabetic medicine : a journal of the British Diabetic Association.

[25]  L C Gatewood,et al.  Model studies of blood-glucose regulation. , 1965, The Bulletin of mathematical biophysics.

[26]  H. Kern,et al.  Steroid Diabetes in the Guinea Pig: Studies on Islet-cell Ultrastructure and Regeneration , 1970, Diabetes.

[27]  K. M. West.  Epidemiology of diabetes and its macrovascular complications. , 1979, Diabetes care.

[28]  Bruce H. R. Wolffenbuttel,et al.  Effects of ramipril on cardiovascular and microvascular outcomes in people with diabetes mellitus: results of the HOPE study and MICRO-HOPE substudy , 2000, The Lancet.

[29]  Yang Kuang,et al.  Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview , 2006 .

[30]  A. Bagust,et al.  An economic model of the long-term health care burden of Type II diabetes , 2001, Diabetologia.

[31]  M I Mackness,et al.  Design of the Collaborative AtoRvastatin Diabetes Study (CARDS) in patients with Type 2 diabetes , 2002, Diabetic medicine : a journal of the British Diabetic Association.

[32]  R. Holman,et al.  A model to estimate the lifetime health outcomes of patients with Type 2 diabetes: the United Kingdom Prospective Diabetes Study (UKPDS) Outcomes Model (UKPDS no. 68) , 2004, Diabetologia.

[33]  E. Lewis,et al.  Renoprotective effect of the angiotensin-receptor antagonist irbesartan in patients with nephropathy due to type 2 diabetes. , 2001, The New England journal of medicine.

[34]  A. Zbrozek,et al.  Model of Complications of NIDDM: II. Analysis of the health benefits and cost-effectiveness of treating NIDDM with the goal of normoglycemia , 1997, Diabetes Care.

[35]  M. Brandeau.  Modeling Complex Medical Decision Problems with the Archimedes Model , 2005, Annals of Internal Medicine.

[36]  R. Stevens,et al.  UKPDS 60: Risk of Stroke in Type 2 Diabetes Estimated by the UK Prospective Diabetes Study Risk Engine , 2002, Stroke.

[37]  M. Farnier.  Heart Protection Study (HPS) , 2002 .

[38]  K. M. West.  Standardization of definition, classification, and reporting in diabetes-related epidemiologic studies , 1979, Diabetes Care.

[39]  Andrea Mari,et al.  Mathematical modeling in glucose metabolism and insulin secretion , 2002, Current opinion in clinical nutrition and metabolic care.

[40]  C Cobelli,et al.  Quantitative Estimation of Beta Cell Sensitivity to Glucose in the Intact Organism: A Minimal Model of Insulin Kinetics in the Dog , 1980, Diabetes.

[41]  A statistical approach to the determination of stability for dynamical systems modelling physiological processes , 2000 .

[42]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[43]  C. Aris Chatzidimitriou Dreismann.  New frontiers in theoretical biology , 1996 .

[44]  Hsin-i Wu,et al.  A case study of type 2 diabetes self-management , 2005, Biomedical engineering online.

[45]  R. Holman,et al.  Erratum: The UKPDS risk engine: A model for the risk of coronary heart disease in type II diabetes (UKPDS 56) (Clinical Science (2001) 101 (671-679)) , 2002 .

[46]  Andrea De Gaetano,et al.  Modeling the intra-venous glucose tolerance test: A global study for a single-distributed-delay model , 2004 .

[47]  Claudio Cobelli,et al.  Models of subcutaneous insulin kinetics. A critical review , 2000, Comput. Methods Programs Biomed..

[48]  R. Turner,et al.  Homeostasis model assessment: insulin resistance and β-cell function from fasting plasma glucose and insulin concentrations in man , 1985, Diabetologia.

[49]  Ovide Arino,et al.  Mathematical modelling of the intravenous glucose tolerance test , 2000, Journal of mathematical biology.

[50]  M. Shichiri,et al.  Development of a Miniaturized Glucose Monitoring System by Combining a Needle-Type Glucose Sensor With Microdialysis Sampling Method: Long-term subcutaneous tissue glucose monitoring in ambulatory diabetic patients , 1994, Diabetes Care.

[51]  M. D. Corte,et al.  On a Mathematical Model for the Analysis of the Glucose Tolerance Curve , 1970, Diabetes.

[52]  Irene M Stratton,et al.  Risk factors for myocardial infarction case fatality and stroke case fatality in type 2 diabetes: UKPDS 66. , 2004, Diabetes care.

[53]  D. Goldstein,et al.  Defining the relationship between plasma glucose and HbA(1c): analysis of glucose profiles and HbA(1c) in the Diabetes Control and Complications Trial. , 2002, Diabetes care.

[54]  R. Sridhar,et al.  A mathematical model for the control mechanism of free fatty acid-glucose metabolism in normal humans. , 1970, Computers and biomedical research, an international journal.

[55]  P. McEwan,et al.  Guidelines for computer modeling of diabetes and its complications: response to American Diabetes Association Consensus Panel. , 2005, Diabetes care.

[56]  Yang Kuang,et al.  Analysis of IVGTT glucose-insulin interaction models with time delay , 2001 .

[57]  Juhani Knuuti,et al.  Insulin and myocardial blood flow. , 2003, Cardiovascular research.

[58]  Edward H. Twizell,et al.  An age structured model for complications of diabetes mellitus in Morocco , 2004, Simul. Model. Pract. Theory.

[59]  L. Svenson,et al.  A model for non-communicable disease surveillance in Canada: the prairie pilot diabetes surveillance system. , 2004, Chronic diseases in Canada.

[60]  David M. Eddy,et al.  Archimedes: a new model for simulating health care systems--the mathematical formulation , 2002, J. Biomed. Informatics.

[61]  Claudio Cobelli,et al.  Optimal input design for identification of compartmental models: theory and application to a model of glucose kinetics , 1985 .

[62]  C. Cobelli,et al.  A physiological simulation model of the glucose-insulin system , 1999, Proceedings of the First Joint BMES/EMBS Conference. 1999 IEEE Engineering in Medicine and Biology 21st Annual Conference and the 1999 Annual Fall Meeting of the Biomedical Engineering Society (Cat. N.

[63]  M Aickin,et al.  The global diabetes model: user friendly version 3.0. , 2000, Diabetes research and clinical practice.

[64]  Abdesslam Boutayeb,et al.  A non-linear population model of diabetes mellitus , 2006 .

[65]  E D Lehmann,et al.  A physiological model of glucose-insulin interaction in type 1 diabetes mellitus. , 1992, Journal of biomedical engineering.

[66]  S. Yusuf,et al.  Effects of an angiotensin-converting-enzyme inhibitor, ramipril, on cardiovascular events in high-risk patients. The Heart Outcomes Prevention Evaluation Study Investigators. , 2000 .

[67]  T. Deutsch,et al.  Application of computers in diabetes care--a review. II. Computers for decision support and education. , 1995, Medical informatics = Medecine et informatique.

[68]  R. Bergman.  Pathogenesis and prediction of diabetes mellitus: lessons from integrative physiology. , 2002, The Mount Sinai journal of medicine, New York.

[69]  R. B. Kerr,et al.  Insulin-sensitive and insulin-insensitive types of diabetes mellitus. , 1939 .

[70]  Paterson Kr.  Diabetes epidemiology: present and future. , 1999 .

[71]  Abdelaziz Chetouani,et al.  Dynamics of a disabled population in Morocco , 2003, Biomedical engineering online.

[72]  R N Bergman,et al.  Assessment of insulin sensitivity in vivo. , 1985, Endocrine reviews.

[73]  A. Zbrozek,et al.  Model of Complications of NIDDM: I. Model construction and assumptions , 1997, Diabetes Care.

[74]  Y. Z. Ider,et al.  Quantitative estimation of insulin sensitivity. , 1979, The American journal of physiology.

[75]  Claudio Cobelli,et al.  Optimal equidose inputs and role of measurement error for estimating the parameters of a compartmental model of glucose kinetics from continuous- and discrete-time optimal samples , 1988 .

[76]  C. Cobelli,et al.  On optimality of the impulse input for linear system identification , 1988 .

[77]  C. Cobelli,et al.  Validation of mathematical models of complex endocrine-metabolic systems. A case study on a model of glucose regulation , 1983, Medical and Biological Engineering and Computing.

[78]  Abdesslam Boutayeb,et al.  International Journal for Equity in Health the Burden of Non Communicable Diseases in Developing Countries , 2005 .

[79]  E R Carson,et al.  Validation of a metabolic prototype to assist in the treatment of insulin-dependent diabetes mellitus. , 1993, Medical informatics = Medecine et informatique.

[80]  H. Hethcote.  A Thousand and One Epidemic Models , 1994 .

[81]  Z-H Lam,et al.  Active insulin infusion using optimal and derivative-weighted control. , 2002, Medical engineering & physics.

[82]  Abdesslam Boutayeb,et al.  Age structured models for diabetes in East Morocco , 2002, Math. Comput. Simul..

[83]  S. Fowler,et al.  Reduction in the incidence of type 2 diabetes with lifestyle intervention or metformin. , 2002 .

[84]  M. Charles,et al.  Devices for Insulin Administration , 1990, Diabetes Care.

[85]  M. Shichiri,et al.  Closed-loop subcutaneous insulin infusion algorithm with a short-acting insulin analog for long-term clinical application of a wearable artificial endocrine pancreas. , 1997, Frontiers of medical and biological engineering : the international journal of the Japan Society of Medical Electronics and Biological Engineering.

[86]  Ping Zhang,et al.  Guidelines for computer modeling of diabetes and its complications , 2004 .

[87]  P. Raskin,et al.  Report of the expert committee on the diagnosis and classification of diabetes mellitus. , 1999, Diabetes care.

[88]  E. Gómez,et al.  Multi-Agent Architecture for the Provision of Intelligent Telemedicine Services in Diabetes Management , 2003 .

[89]  A. Boutayeb,et al.  A mathematical model for the burden of diabetes and its complications , 2004, Biomedical engineering online.

[90]  M. Kalergis,et al.  The Role of the Glycemic Index in the Prevention and Management of Diabetes: A Review and Discussion , 2005 .

[91]  Sélam Jl,et al.  Devices for insulin administration. , 1990 .

[92]  H. Parving,et al.  The effect of irbesartan on the development of diabetic nephropathy in patients with type 2 diabetes. , 2001, The New England journal of medicine.

[93]  A. Bagust,et al.  Deteriorating beta-cell function in type 2 diabetes: a long-term model. , 2003, QJM : monthly journal of the Association of Physicians.

[94]  F. Jiwa.  Diabetes in the 1990s--an overview. , 1997, Statistical bulletin.

[95]  R. Holman,et al.  UKPDS58--modeling glucose exposure as a risk factor for photocoagulation in type 2 diabetes. , 2002, Journal of diabetes and its complications.

[96]  V. Bolie,et al.  Coefficients of normal blood glucose regulation. , 1961, Journal of applied physiology.

[97]  R. Holman,et al.  The UKPDS risk engine: a model for the risk of coronary heart disease in Type II diabetes (UKPDS 56). , 2001, Clinical science.