Fayetteville State University DigitalCommons@Fayetteville State University Math and Computer Science Working Papers College of Arts and Sciences 10-2015 Mathematical Modeling and Simulations of the Pathophysiology of Type-2 Diabetes Mellitus Frank K. Nani Fayetteville State University, [email protected] Mingxian Jin Fayetteville State University, [email protected] Follow this and additional works at: http://digitalcommons.uncfsu.edu/macsc_wp Part of the Medical Biomathematics and Biometrics Commons Recommended Citation Nani, Frank K. and Jin, Mingxian, "Mathematical Modeling and Simulations of the Pathophysiology of Type-2 Diabetes Mellitus" (2015). Math and Computer Science Working Papers. Paper 25. http://digitalcommons.uncfsu.edu/macsc_wp/25 This Conference Proceeding is brought to you for free and open access by the College of Arts and Sciences at DigitalCommons@Fayetteville State University. It has been accepted for inclusion in Math and Computer Science Working Papers by an authorized administrator of DigitalCommons@Fayetteville State University. For more information, please contact [email protected]. 2015 8th International Conference on BioMedical Engineering and Informatics (BMEI 2015) Mathematical Modeling and Simulations of the Pathophysiology of Type-2 Diabetes Mellitus Frank Nani and Mingxian Jin Department of Mathematics and Computer Science Fayetteville State University Fayetteville, NC 28301, USA Abstract --The pathophysiology of Type 2 Diabetes Mellitus and cardio-metabolic factors. Recent emerging risk factors (T2DM) is modelled using a coupled system of non-linear include sleep deprivation, drug-induced metabolic changes, deterministic differential equations. An attempt is made to environmental pollutants, low birth weight and fetal construct to a clinically plausible mathematical model that malnutrition [3, 4, 5, 6, 7]. incorporates the homeostasis associated with endocrinological regulation of glucose and glycogen levels in the human body, by Many mathematical models in the literature were devoted the hormones, insulin and glucagon. The model variables include mainly to the dynamics of glucose in relation to the hormone, the concentrations of glucose in the venous blood plasma, the insulin. The basic model was proposed by Bolie [9], who used concentration of glycogen in the liver/tissues, the concentration of the following system of ordinary differential equations the hormone glucagon, and the concentration of insulin in the venous blood plasma. The physiological interactions between the dG = − a G − a I + P model parameters are depicted by clinically measurable rate dt 1 2 (1.1) constants and biophysically quantifiable stoichiometric dI coefficients. The processes of gluconeogenesis, glycogenolysis, and = − a G − a I pulsatile insulin secretion during type 2 diabetes are modelled dt 3 4 using plausible auxiliary functions. Investigative computer simulations are performed to elucidate various hypothetical where G = G(t) represents glucose concentration, I = I(t) scenarios of glycemia, patho-physiology of T2DM and insulinoma represents insulin concentration, and P, a1, a2, a3, a4 are model associated hypoglycemia which results from excessive insulin parameters. production probably due to a tumor. This study has demonstrated the necessity of simultaneous monitoring of plasma A clinically relevant minimal model of glucose-insulin glucose, glucagon, insulin, and glycogen levels in the proper dynamics was proposed by Bergman, Bowden, and Cobelli in assessment of the pathophysiology of type 2 diabetes and during [10]. De Gaetano Arino [11] constructed a dynamical model of determination of the therapeutic efficacy of anti-diabetic drugs. insulin-glucose dynamics using an aggregated integro- differential delay equations. Several other models were Keywords--Mathematical modeling; diabetes mellitus; dynamics constructed by the following authors: Li et. al. [12], Cobelli of glucose; glycogen; glucagon; insulin and Tomaseth [13], Lam et. al. [14], Mari [8]. The main objective of the research presented in this I. INTRODUCTION paper is to construct a clinically plausible model that depicts Type 2 diabetes mellitus (T2DM) is characterized by the homeostasis of glucose-insulin regulatory dynamics and insulin resistance, insulin hypo-secretion to β cell impairment the pathophysiology of T2DM. and/or over production of glucose. A person is diagnosed as diabetic if the fasting venous plasma glucose concentration is II. DEFINITIONS OF MODEL PARAMETERS greater than 126 mg/dL and/or the 2-hour post-glucose load of 75g of anhydrous glucose yield a venous plasma glucose The explicit non-linear deterministic mathematical concentration of 200 mg/dL [1]. Alternative novel tests for equations describing the patho-physiology of T2DM are presented in this section. diagnosis include the use of glycated hemoglobin (HbA1C) with suggested levels of about 6.5% [2]. x 1 (t ) : the concentration of glucose in the blood plasma of The pre-disposing historical high risk factors for T2DM the patient at any time t during T2DM. include increasing age (old age), central obesity, dietary x (t ) : the concentration of glycogen in the blood plasma of polyphagia of animal fats, carbonated drinks, lack of physical 2 the patient at any time t during T2DM. exercises, familial genetic factors, history of gestational : the concentration of glucagon in the blood plasma of diabetes, polycystic ovary syndrome, and severe mental x 3 ( t ) illness, presence of hypertension, hyperlipidemia, ethnicity the patient at any time t during T2DM. 978-1-5090-0022-7/15/$31.00 ©2015 IEEE 296 x 4 (t ) : the concentration of insulin in the blood plasma of A. The Glucose Dynamics Equation the patient at any time t during T2DM. The rate of change of glucose in the body is equal to the the rate constant depicting exogenous input into the post prandial input ( G1 f1 (t) ), plus the rate of glucose G i (xi ) : ith compartment by either post-prandial uptake or production by gluconeogenesis (S1(x3, x4)) which is breakdown intravenous (i.v.) infusion or by subcutaneous injection of pyruvate into glucose mediated by insulin and glucagon, [i = 1,2,3,4] . plus the rate of glucose production by glycogenolysis σ involving action of glucagon on glycogen ( 1 x 2 x 3 ), minus S ( x , x ) : the rate of hepatic glucose synthesis by 1 3 4 the rate of loss of glucose during glycogenesis due to action of gluconeogenesis insulin on glucose ( a x x ), minus the rate of catabolism of the rate of glycogen bio-synthesis. 14 1 4 S 2 ( x 3 , x 4 ) : glucose during glycolysis ( k1 x1 ), minus the linear rate of S 3 ( x1 , x 2 ) : the rate of glucagon bio-synthesis in alpha cells degradation of glucose by utilization during exercise and of the pancreas. physiologic activity in the brain, skeletal muscles and cardiac S ( x , x ) : the rate of insulin bio-synthesis in beta cells of muscles ( k01 ). The glucose dynamics equation is shown as 4 1 4 the pancreas. below: In particular, the following homeostasis functions are dx used: 1 = G f (t) + S (x , x ) + σ x x − a x x − k x − k dt 1 1 1 3 4 1 2 3 14 1 4 1 1 01 c x S ( x , x ) = 1 3 (3.1) 1 3 4 1 + μ x 1 4 c x B. The Glycogen Dynamics Equation S ( x , x ) = 2 4 2 3 4 1 + μ x Glycogen dynamical equation depends on the post- 2 3 c x prandial rate of glycogen input G f (t ) ; the bio-synthesis of = 3 2 2 2 S 3 ( x 1 , x 2 ) 1 + μ x glycogen term S2(x3, x4) which is mediated by glucagon and 3 1 (2.1) insulin and stored in body tissues including muscle; the rate of c x S ( x , x ) = 4 1 de novo synthesis of glycogen due to action of insulin on 4 1 2 1 + μ x 4 2 glucose (σ x x ) and mediated by glycogen synthase; the k : the rate constant depicting catabolic degradation 2 1 4 i rate of catabolism of glycogen by glycogenolysis in liver ( [i = 1,2,3,4] pertaining to exponential decay kinetics. − a 23 x 2 x 3 ); rate of loss of glycogen by enzymatic/exponential degradation mediated by glycogen k0i : the rate constant depicting catabolic degradation phosphorylase (–k x ); and rate of linear degradation of [i = 1,2,3,4] pertaining to linear decay kinetics. 2 2 glycogen (–k02). The catabolic breakdown of glycogen is (a14 , a 41 ) : stoichiometric kinetic constant involved in the mediated primarily by glucagon but there are other hormones action of insulin (x ) on glucose ( x ) . that are involved. These include cortisol, somatotropin, and 4 1 catecholamines. The glycogen dynamical equation thus takes σ stoichiometric kinetic constant involved in the 2 : the form: production of glycogen. dx (a , a ) : stoichiometric kinetic constant involved in the 2 = G f (t) + S (x , x ) + σ x x − a x x − k x − k 23 32 dt 2 2 2 3 4 2 1 4 23 2 3 2 2 02 action of glucagon ( x ) on glycogen (x ) . 3 2 (3.2) σ 1 : stoichiometric kinetic constant involved in the production C. The Glucagon Dynamics Equation of glucose. The rate of change of glucagon in the body equals to the fi (t) = sin nt represents pulsatile input function for each i ⎡⎤ rate of exogenous input (G3f3(t)) by intravenous or intra- [i = 1,2,3,4] . muscular injection, plus rate of biosynthesis of glucagon and glycogen (S3(x1, x2)) mediated by the levels of glucose and glycogen, minus the rate of loss of glucagon during III. MODEL EQUATIONS FOR TYPE-2 DIABETES MELLITUS glycogenolysis (a32 x2 x3) in the liver, minus degradation of In this section, an elaborate physiological description and glucagon enzymatically (k3x3), and minus linear catabolism of mathematical formulation of the model equations are glucagon (k03). The glucagon dynamical equation thus takes presented. the following form: 297 dx 3 = G f (t) + S (x , x ) − a x x − k x − k dt 3 3 3 1 2 32 2 3 3 3 03 (3.3) x1-Glucose x2-Glycogen D.