Mathematical Model And Its Algorithm Studying For The Human Pulmonary Valve

The human heart delivers more than 7,000 liters of blood to other parts of the body every day,which is caused by regular contraction and relaxation of the heart valve.However,in real life,due to functional abnormalities or structural abnormalities in a single or multiple valves,the heart valve may be described as a spectrum of disease,which will cause a certain degree of problems in delivery.The pulmonary valve is one of the most important valve tissues in human heart.It is necessary for us to do more researches on human pulmonary valve from the perspective of applied mathematics,which will contribute to diagnose and treat the diseases associated with pulmonary valves in clinical medicine,and provide an important theoretical for the medical operations such as pulmonary valve injury repair,synthetic or valve replacement.Currently,many scholars have conducted a large number of researches both at home and abroad,and have obtained corresponding research results on the pulmonary valve.But most scholars focus on the comparison of two valves based on the fact that the pulmonary valve and the aortic valve are similar.Only a few scholars conduct pure research about pulmonary valve.So far,the mathematical model has not been established under the consideration of the mechanical properties of heart valves.Therefore,the complete characterization of the pulmonary valve is the study of this subject in a cardiac cycle.The research content of this thesis mainly consists of the following aspects:(1)We have analyzed the basic structure of the pulmonary valve and determined the basic shape of the pulmonary valve in the existing research;Based on the structural and physiological characteristics of the pulmonary valve,the valve is described as a cylindrical shell in the elastic shell;We have fitted the parametric equation of the valve combining the size of the valve,and verified the fitting results by the other data.Thus,the geometric reproduction of the human heart pulmonary valve was determined.(2)A dynamic model of the human pulmonary valve has been established based on the theory of the elastic shell.Starting from the Koiter's static model and introducing the time variable,the Koiter model of the heart valve elastic dynamics has been set up as the fourth-order development partial differential equation.For the proposed model,we use the finite element method and the Newmark method to full-discretize the time and space,and analyze the existence,uniqueness and convergence of the fully discrete solution.Calculating the differential geometry on the surface of the pulmonary valve includes the covariant and contravariant components of the metric tensor,the covariant and mixed component of the curvature tensor,and the Christoffel symbols,determing the geometric parameters based on the geometric models and the material parameters based on the relevant literature,we analyse the deformation of the pulmonary valve during a cardiac cycle.Finally,we solve the displacement utilization model at each point on the neutral surface of the pulmonary valve under different steps,and visualize the calculation results.Thus,a complete functional simulation of the pulmonary valve during a heartbeat cycle has finished,which is the key of this thesis.(3)Based on the proposed dynamic Koiter model,the elliptical membrane shell model is studied under the assumption of the special geometry and the function space.Similarly,the full space-time discretization scheme is obtained by the discretizing on the space and time.The existence,uniqueness and convergence of the solution in the fully discrete scheme are provided.In addition,we have analyzed the stability and error estimate for the fully discretized scheme.Finally,the numerical experiments are carried out for the ellipsoidal shells and the spherical shells,We have simulated the deformation process of the membrane shell after a certain period of time,and visualized the displacement change results.In addition,the convergences of errors with respect to space size and time size for ellipsoidal shell are obtained to verify the stability and convergence of the numerical scheme.