Mathematical Model for Diabetes Mellitus with Impulsive Injections of Glucose-insulin

Impulsive injections of glucose and insulin analogues are very important strategies for the control of diabetes mellitus. We mainly imitate diabetes patients take insulin before eating, and eating approximately as a pulse blood glucose injection, as a result, a new mathematical model with impulsive injections of both glucose and insulin at different fixed times is formulated in this paper. Using the discrete dynamical system determined by the stroboscopic map, we show that the existence and uniqueness of a positive globally asymptotically stable periodic solution for type I diabetes. By impulsive comparison theorem, we obtain the glucose concentration level of the system is uniformly bounded above and below for type Ⅱ diabetes. Numerical analysis verifies our theoretical results.

Glucose-stimulated insulin secretion in isolated pancreatic islets: Multiphysics FEM model calculations compared to results of perifusion experiments with human islets

Because insulin released by the β-cells of pancreatic islets is the main regulator of glucose levels, the quantitative modeling of their glucose-stimulated insulin secretion is of obvious interest not only to improve our understanding of the processes involved, but also to allow better assessment of β -cell function in diabetic patients or islet transplant recipients as well as the development of improved artificial or bioartificial pancreas devices. We have recently developed a general, local concentrations-based multiphysics computational model of insulin secretion in avascular pancreatic islets that can be used to calculate insulin secretion for arbitrary geometries of cultured, perifused, transplanted, or encapsulated islets in response to various glucose profiles. Here, experimental results obtained from two different dynamic glucose-stimulated insulin release (GSIR) perifusion studies performed by us following standard procedures are compared to those calculated by the model. Such perifusion studies allow the quantitative assessment of insulin release kinetics under fully controllable experimental conditions of varying external concentrations of glucose, oxygen, or other compounds of interest, and can provide an informative assessment of islet quality and function. The time-profile of the insulin secretion calculated by the model was in good agree- ment with the experimental results obtained with isolated human islets. Detailed spatial distributions of glucose, oxygen, and insulin were calculated and are presented to provide a quantitative visualization of various important aspects of the insulin secretion dynamics in perifused islets.

Design and Analysis of a Sliding Table Controller for Diabetes

A discrete insulin infusion based on long-time interval measurement is the classic technique for diabetes treatment. Nevertheless, in this research, a closed-loop control system was proposed for continuous drug infusion to overcome the drawbacks of these typical discrete methods and develop more practical diabetes therapy systems. A blood glucose-insulin system was implemented relying on continuous insulin injection model. Based on this model, two controllers were designed to deal with the control dilemma of the resulting highly nonlinear plant. The controllers designed in this paper are: proportional integral derivative (PID), and sliding table controllers. Simulation results have shown that the sliding table controller can outperform the PID controller even with severe circumstances of disturbance in glucose, such as exercise, delay or noise in glucose sensor and nutrition mixed meal absorption at meal times.

Modeling and qualitative analysis of diabetes therapies with state feedback control

For the therapies of diabetes mellitus, a uovel mathematical model with two state impulses： impulsive injection of insulin and impulsive injection of glucagon, is proposed. To avoid hypoglycemia and hyperglycemia, the injections of insulin and glucagon are determined by closely monitoring the plasma glucose level of the patients. By using differential equation geometry theory, the existence of periodic solution and the attrac- tion region of the system have been obtained, which ensures that injections in such an automated way can keep the blood glucose concentration under control. The simula- tion results verify that the better insulin injection strategy in closed-loop control is a larger dose but longer interval rather than a smaller dose but shorter interval. Besides, our numerical analysis reveals that medicine studies and practice that slow down the insulin degradation are helpful for the plasma glucose control. Our findings can provide significant guidance in both design of artificial pancreas and clinical treatment.