The Modeling And Computation Of Pulse Wave Propagation In Human Systemic Arterial Network

The temporal profile of the blood pressure is believed to be an important indicator of the state of human body.This thesis focuses on mathematical models to study formation mechanisms of the normal and abnormal blood pressures in human body.In detail,the thesis includes the following three parts.The first part,including the first four chapters,is to derive an one-dimensional model of blood flow based on the conservations of mass and momentum.It also includes the boundary conditions of the model based on the properties of blood flow in the large ar-terial system.In particular,there are two types of outflow boundary conditions.One is analogous to the electric circuit,including the constant resistance model and the Wind-kessel model,the other type is based on the hydrodynamics of blood flow in small arteries,including the structured tree model and the random tree model.In Chapter 4,we provide a systematic procedure to extract the parameters of the three-element Windkessel model from the impedances of an arterial tree or from experimental measurements of the blood flow rate and pressure.The theoretical analysis and numerical simulations show that a small truncation radius and the complex capacitance in the improved Windkessel model can reduce modeling errors.The second part studies the wave propagation and reflection in the arterial system.According to the one-dimensional model,we establish new formulations representing two waves propagating in a single vessel and reformulated the boundary conditions.We then use the new system to study the wave propagation in a single artery and wave reflections at junctions.Our results show that the heart initiates a wave into the arterial tree;there are two wave propagating along each artery nearly linearly and independently;the wave reflection at each junction is largely determined by the radii at the junction;at each outlet,the reflected wave mainly contains low-frequency modes because of the compressibility property of the outflow boundary condition.The traditional numerical methods,such as the Lax-Wendroff method,for the one-dimensional model are severely constrained by the CFL condition due to the large wave speed,giving a very tiny time step size.This makes the computational cost rather high.In Chapter 5,we develope a new numerical algorithm by exchanging the spatial and tempo-ral coordinates with the new formulation mentioned before.The Lax-Wendroff method is applied in the exchanged coordinate system.This method allows a much larger time step size than the traditional method.In this method,the boundary conditions are satisfied by an iterative method according to the reformulated boundary conditions.Our numerical studies show that the new algorithm is stable and it can achieve over 15 times of accel-eration compared to the traditional methods.This numerical method provides us with a highly efficient tool in studying the blood flow in different arterial systems.Based on the wave propagation and reflection in an arterial tree,we have identified the origins of different components of the normal radial blood pressure,which is mainly composed of the main wave,the pre-dicrotic wave,dicrotic notch,and the dicrotic wave.According to our results,the main wave is essentially determined by the wave initiated by the heart;the pre-dicrotic wave mainly comes from a large positive reflection at the junction at the end of the abdominal aorta;the dicrotic notch is mainly caused by a strong negative reflection at the hands;and the dicrotic wave is mainly determined by the total reflection of the whole arterial system.Moreover,the disappearance of the pre-dicrotic wave and the dicrotic notch in the wiry pulse in hypertension is caused by the increased wave speed,which decreases the time lag between the dicrotic notch and the main wave;the disappearance of the pre-dicrotic wave in the slippery pulse may be due to the weak-ening of the positive reflection at the end of abdominal aorta in pregnancy.The structure of a junction plays an essential role in determining the wave reflection at the junction.In the last part,we study effects of the wall shear stress on the junctional structures in an arterial tree.In Chapter 7,we take into account the pulsation of blood flow and propose that the L₂norm of wall shear stress is a uniform constant in all the arteries,which is derived from the minimization of energy cost in a network with fluctuations in flow.The numerical results qualitatively agree with experimental scaling results and show the effect of wall shear stress on the chronic adaptation of the arterial structure with an unsteady blood flow.In all,the thesis summaries our study of the pulse wave propagation in the large arterial system using mathematical modeling.The theoretic analysis and the numerical results provide us new insight into the relation between the blood pressure and the variation of the arterial system.