Modeling and Simulation of Distensible Blood Vessels and Glucose-Insulin System - Application to Therapeutic Management of Stroke and Diabetes.
A Thesis Submitted to the School of Postgraduate Studies, University of Lagos.
A mathematical model of blood flow in distensible vessels and glucose insulin kinetics during simple clinical tests such as insulin and glucose tolerance test were developed. A number of problems were identified with the approach and methods of solution, and as a result a better numerical technique was used to solve the model. The issues most often addressed consider the application of faster and more accurate numerical methods. One-dimensional fluid dynamical model for blood flow and pressure in larger systemic arteries was developed using the Saint-Venant equation. Furthermore a new approach was developed to solve the one-dimensional non-linear equations of blood flow using the Riemann based methods constructed within the finite volume framework. Such methods are noted for their good conservation and shock capturing capabilities and have a number of desirable properties, most noticeably the ability to predict continuities in the solution. The developed equations of blood flow were transformed into Riemann problem and solved. The developed models would be useful in predicting anomalies in the arterial system and free of the pitfalls in the previous models. Physiological values were used to simulate different arterial models with mild stenoses (33.3% and 66.7% area reduction in radius) in one third, half and two third of the vessel length at a steady state flow situation. The mathematical model equations describing the glucose insulin kinetics during simple clinical tests such as insulin and glucose tolerance test were developed using an infinite number of continuous stirred tanks in series. The aim is to quantify in individual patients, macroscopic physiological process that may be associated with the development of diabetic disease, and to explore the possibility of applying the model in clinical studies with numerically significant group of subjects. The gut-blood system was modeled as an infinite number of continuous stirred tanks in series, which could be used as simulators of the entire blood glucose-insulin system in addition to the Bergman’s minimal model. Laplace transformation technique was used to solve the equations analytically with a view to determine model parameters and glucose disappearance rate constant in an individual subjected to short insulin tolerance test (SITT). To verify the model, the rate constants derived from the model for the subjects studied were compared with those obtained from measurement. The comparison was done using statistical analysis, student t-test and analysis of variance (ANOVA). Both the two analysis showed that there was no significant difference between the model-based and measured values. The exit age distribution function was determined. Furthermore, a software package was developed to assist clinicians in early screening of patients at risk and treatment procedures for stroke and diabetic patients in any crisis situations.