Mathematical Model for Glucose-Insulin Regulatory System of Diabetes Mellitus

Transcription

1 Advances in Applied Matheatical Biosciences. ISSN Volue, Nuber (0), pp. 9- International Research Publication House Matheatical Model for Glucose-Insulin Regulatory Syste of Diabetes Mellitus Sandhya and Deepak Kuar Departent of Matheatics, Faculty of Engineering & Technology, MRIU, Faridabad, Haryana, India Abstract This paper works a new approach to regulate the blood glucose level of diabetes. We proposed a new atheatical odel for the study of diabetes ellitus. The odel takes into account all plasa glucose concentration, generalized insulin and plasa insulin concentration. The nuerical solution presents the coplex situation of diabetic patients. Coputer siulations are used to evaluate the effectiveness of the proposed work. Keywords: diabetes ellitus, glucose-insulin regulatory syste, atheatical odel. Introduction It is now coonly aditted that diabetes is sweeping the globe as a silent epideic largely contributing to the growing burden of non-counicable diseases and ainly encouraged by decreasing levels of activity and increasing prevalence of obesity. The recent reports released by the World Health Organization [] and the International Diabetes Federation [] Diabetes ellitus is a disease in glucose-insulin endocrine etabolic syste characterized by hyperglyceia resulting fro defects in insulin secretion, insulin action or both. The two ost coon fors of diabetes are due to either a diinished production of insulin (Type diabetes), or diinished response by the body to insulin (Type diabetes). Both lead to hyperglyceia, which largely causes the acute signs of diabetes: excessive urine production, resulting copensatory thirst and increased fluid intake, blurred vision, unexplained weight loss, lethargy and changes in energy etabolis. In 980 expert coittee of WHO (World Health Organization) proposed classification of Diabetes Mellitus and naed it as IDDM (Insulin Dependent Diabetes Mellitus) or Type and NIDDM (Non Insulin Dependent Diabetes Mellitus) or Type.But in 98 Type and Type naes were oitted and only IDDM & NIDDM were known as the types of Diabetes Mellitus []. Huan

2 0 Sandhya and Deepak Kuar bodies need to aintain glucose concentration level in a narrow range 70-0 g/dl. If one s glucose concentration level is significantly out of the noral range, this person is considered to have the plasa glucose proble. With or without diabetes, onitoring of blood sugar levels is crucial to a person s health. Diabetes is a etabolic illness that can affect other organs of the body. Acquiring diabetes ay aggravate a siple disease too. Internal organs like heart, kidney, lungs, liver, pancreas, or even the libs are the favorite target of this etabolic disease. Senses ay also be affected. Norally, it s the sense of sight that is targeted. Occurrence of diabetes ust not create panic if onitored carefully. Diabetes will reain as elevated blood sugar levels if treated properly. But if it starts to reach other organs or other processes in the body, then it s tie to be ore cautious. The basic thing that a patient or a noral person ust know is blood sugar level. Having stored knowledge will ake it easy for a person to react properly regarding diabetes at a given tie. Norally, blood glucose is at.8 to 0. g/dl. The ean noral blood glucose level is 7 g/dl. But it is iportant to note that blood glucose level is lowest in the orning before the first eal is taken. Two to three hours after a eal, blood glucose will elevate depending on what kind of food was taken. For a noral person without diabetes, blood sugar levels reach as high as 7 g/dl after a eal is taken. This level will return to noral rate after soe tie. The tie consued for the blood sugar level to return to noral rate is faster in noral healthy person than in person with diabetes. Diabetics ust aintain a blood sugar level of 80 g/dl after eating. In diabetics, the easureent is done two hours after eal. Higher levels than 80 g/dl indicates that the person has taken too uch carbohydrates or fatty food. Classic syptos of diabetes ellitus such as frequent urination, excessive thirst and fatigue accopany these laboratory results in aking the diagnosis. Diabetes is associated with a large nuber of abnoralities in insulin etabolis, ranging fro an absolute deficiency to a cobination of deficiency and resistance, causing an inability to dispose glucose fro the blood strea. Three factors: Insulin sensitivity, Glucose effectiveness, and pancreatic responsiveness, referred to in Pacini and Bergan [], play an iportant role for glucose disposal. Failure in any of these ay lead to ipaired glucose tolerance, or, if severe, diabetes. The literature dealing with odeling for diabetes is ainly concerned with glucose and insulin dynaics. Matheatical Model Matheatical odels have provided one ean of understanding Diabetes dynaics. There are various odels based on glucose and insulin distributions and those odels have been used to explain glucose/insulin interaction. All these odels are valid under certain conditions and assuption. Although these odels ay be useful in research setting, they all have liitations in predicting blood glucose in real-tie clinical situation because of the inherent requireent of frequently updated inforation about the odels variable like glucose loads and insulin availability []. Consider a atheatical odel coprised of glucose level G, glucose uptake activity

3 Matheatical Model for Glucose-Insulin Regulatory Syste X and insulin level I. Many paraeters have been taken and on the basis of these paraeters values a atheatical odel is fored. This odel includes the basal values also i.e. G b and I b. The odal is defined as: dg = G+ I+ G dt dx dt b = X+ I I + I b b di I G I I dt = b All the variables and paraeters values used in atheatical odels are described as: G (t) The plasa glucose concentration at tie t (g/dl) X (t) The generalized insulin variable for the reote copartent (in-) I (t) The plasa insulin concentration at tie t (μu/l) Gb This is the basal preinjection value of plasa glucose (g/dl) Ib This is the basal preinjection value of plasa insulin (μu/l) Insulin independent rate constant of glucose rate uptake in uscles, liver and adipose tissue ( in ). The rate of decrease in tissue glucose uptake ability ( in ). The insulin independent increase in glucose uptake ability in tissue per unit of insulin concentration Ib ( in (μu/l)). The rate of the pancreatic β-cells release of insulin after the glucose injection and with glucose concentration above h [(μu/l) in- (g/dl)-] The threshold value of glucose above which the pancreatic β-cells release insulin. The first order decay rate for insulin in plasa (in-) pancreatic β-cells release insulin Nuerical Solution The analysis is done on the noral person as well as on different types of diabetic patient i.e patient, patient, patient.basically there are patients who are suffering fro Diabetes ellitus but the results fro each patient is different and it is explained with the help of graphs and paraeters values. Glucose is given to the patients then we studied the plasa glucose concentration, plasa insulin concentration and generalized insulin variable in the body of patients. The graph for these types of patients is obtained. There are soe variables and sybols used in the graphs: G(t) The plasa glucose concentration at tie t (g/dl) *

4 Sandhya and Deepak Kuar X(t) The generalized insulin variable for the reote copartent (in-). I(t) The plasa insulin concentration at tie t (μu/l) o Data for Noral Person The study of Glucose, insulin, and plasa concentration is done on the noral persons. The study of this case is shown by the graph( fig.). Fig. shows the study for 0 hours. It show that when glucose is given to the noral persons the glucose concentration level becoe very high and as tie passes the level becoe stable.the sae can be seen in case of plasa insulin concentration but when we see the graph for generalized insulin variable there is no change even after soe tie it will reain sae. The values of paraeters for noral person are given in Table:. Tie Figure : Glucose-insulin regulatory syste (a)(*) Plasa glucose Concentration (b) Plasa insulin Concentration (o) (c) generalized insulin variable (.). Table Paraeter Values x G b 80 I b 7

5 Matheatical Model for Glucose-Insulin Regulatory Syste Data gor Patient The study of Glucose, insulin, and plasa concentration is done on the Diabetic patients. The study of this case is shown by the graph (fig.). Fig. shows the study for 0 hours. It show that show that initially the glucose level is very high but after giving the glucose to the patient there is still no ajor fall in glucose level. After 0 hours fro 0 g/dl it falls to only about 7 g/dl. But when we see the graph for generalized insulin variable as well as for plasa insulin concentration even after soe tie it is sae and no change in its concentration level. The values of paraeters for patient is given in Table: Tie Figure : Glucose-insulin regulatory syste (a)(*) Plasa glucose Concentration (b) Plasa insulin Concentration (o) (c) generalized insulin variable (.) Table Paraeter Values x Gb 80 Ib 7 Data for Patient The study of Glucose, insulin, and plasa concentration is done on the Diabetic patients of another type. The study of this case is shown by the graph (fig.). Fig.

6 Sandhya and Deepak Kuar shows the study for 0 hours. It show that show that initially the glucose level is very high but after giving the glucose to the patient there is still no ajor fall in glucose level. In this graph plasa insulin concentration reains the sae but when we see the graph for generalized insulin variable there is the inor fall after soe tie. The values of paraeters for patient are given in Table :. Tie Figure : Glucose-insulin regulatory syste (a)(*) Plasa glucose Concentration (b) Plasa insulin concentration(o) (c) generalized insulin variable(.). Table Paraeter Values x Gb 80 Ib 7 Data for Patient The study of Glucose, insulin, and plasa concentration is done on the Diabetic patient s type. The study of this case is shown by the graph (fig.). Fig. shows the study for 0 hours. Again the level of glucose is sae as in the case of other patients. In this graph plasa insulin concentration & generalized insulin variable have a inor fall after soe tie. The values of paraeters for patient are given in Table:.

7 Matheatical Model for Glucose-Insulin Regulatory Syste Tie Figure : Glucose-insulin regulatory syste (a)(*) Plasa glucose Concentration (b) Plasa insulin Concentration (o) (c) generalized insulin variable (.). Table Paraeter Values x Gb 80 Ib 7 Discussion In this paper a atheatical odel is developed for different kind of diabetic patients. Different graphs are obtained for noral as well as diabetic a patient who shows the variability of Glucose, insulin and plasa glucose concentration. The odel showed clearly the results given according to different scenarios. In conclusion, the present work presents and justifies a atheatical odel of long-ter diabetes progression. Diabetes anageent is one of iportant issues in the field of huan regulatory systes. It shows the difference of glucose-insulin regulatory syste, between a noral person and diabetic person. The glucose concentration of diabetic patient does not coe down after a certain tie which shows the evidence that the person suffer fro diabetes. The current effort refines the previous odel by Yasini et. al. (009) and akes the resulting odel directly useful for clinical purposes through a careful assessent of the relevant paraeters.

8 Sandhya and Deepak Kuar Acknowledgeent The authors wish to thank Ms. Prerna Pandit, Departent of Pharacy, and College of Pharacy, IILM Acadey of Higher Learning, Greater Noida, U.P. for helpful and insightful discussion concerning this work. References [] The world health report: Today's challenges. [ website Geneva, World Health Organization [] International Diabetes Federation: IFD report. [ website 00. [] Sarah Wild, Gojka Roglic, Anders Green, Richard Sicree, Hilary King(00): Global Prevalence of Diabetes, Estiates for the year 000 and projections for 00. Diabetes Care 7:07 0. [] Pacini, G. and Bergan, R.N. (98) MINMOD: A coputer progra to calculate insulin sensitivity and pancreatic responsively fro the frequently sapled intravenous glucose tolerance test. Coputer Methods and Progras in Bioedicine,, -. [] Sh. Yasini, M.B. Naghibi-Sistani, A Karipour (009): Agent-based Siulation for Blood Glucose Control in Diabetic Patients. International journal of Applied science, Engineering and Technology : [] Raed abu zitar (00): Towards neural network odel for insulin / glucose in Diabetes. International journal of coputing and inforation sciences, Vol, No..