MATHEMATICAL ANALYSIS OF HEMODYNAMIC PULSE WAVE IN HUMAN FLUID-STRUCTURE INTERACTION

Table of Contents

Title page

Certificate of Approval

Acknowledgement

Dedication

Abstract

Chapter One

Overview of Fluid-Structure Interaction Problem

1.0 Introduction

1.1       Body vessels

1.2       Arteries and their structure

1.3       Blood

1.4       Blood Pressure (BP)

1.5       Arterial Pulse

1.6       Pulse Pressure

1.7       Fluid-Structure Interaction

1.8       Pulse Wave

1.9       Resonance

1.10     Aim and objectives of the study

1.11     Scope and limitations of the study

1.12     Methodology

1.13     Significance of the study

Chapter Two

Literature Review

Chapter Three

Fluid-Wall interaction and non-linear pulse wave models in blood flow

3.1.0 Generalized equation of motion of viscous fluid

3.1.1 Action of fluid on the wall

3.1.2 Fluid –Structure interaction model: Problem p resentation

3.1.3 Fluid –Structure coupling

3.2.0 Model of non-linear arterial pulse: Problem presentation

3.2.1 Linear superposition of forward and backward ABP waves

Chapter Four

Solutions to model problems

4.1.0 Solution of fluid-wall interaction problem

4.1.1 Weak Formulation and Variational Form

4.1.2 Rescaled Problem and asymptotic expansion

4.1.3 Weak Formulation

4.1.4 Energy estimates after rescaling

4.1.5 Asymptotic Expansions

4.1.6 Justification for asymptotic expansions

4.1.7 Reduced problem using Expansion I

4.1.8. Reduced problem using Expansion II

4.2.0. Nonlinear arterial pulse model

4.2.1 Methods of Solution of non-Linear Wave model problem

4.2.2 The Tanh (hyperbolic tangent) Method of Solution

4.3.0   Bilinear Method

4.3.1 Solitons by bilinear method

4.4 Systolic and Diastolic PW Representation

Chapter Five

Results and discussions

5.1.0 Features of:

5.1.1 Tanh method

5.1.2 Bilinear Method

5.2.0 Physiological Analysis using solitary waveform

5.2.1Distance effect

5.2.2 Short and tall statures

5.2.3 Time effects

5.2.4 Harmonic Components of Arterial Pulse Waves

5.2.5 Heart-Organ Resonance

5.2.6 Hypertension and vaso-active Agents

5.2.6 Dying Process

5.3.0 Summary and Conclusion

5.4.0 Recommendation(s) for further studies

References     

Appendices

Abstract

Mathematical study of human pulse wave was studied with the view to gaining an insight into physiological situations. Fluid –Structure interact ion (FSI) in blood flow is associated with pressure pulse wave arising from ventricular ejection. Solution of the coupled system of non-linear PDEs that arose from the FSI was sought in order to determine pressure. Further study on pressure pulse waves showed that the Korteweg-de Vries (KdV) equations hold well for the propagation of nonlinear arterial pulse wave. Solutions of the KdV equation by means of the hyperbolic tangent (tanh) method and the bilinear method each yielded solitons. The solitons describe the peaking and steepening characteristics of solitary wave phenomena.

The morphologies of the waves were studied in relation to the length occupied by the waves (which corresponds to length of arterial segment and stature) and the left ventricular ejection time (LVET). The study showed that both stature and LVET are independent descriptors of cardio-vascular state.

CHAPTER ONE

OVERVIEW OF FLUID-STRUCTURE INTERACTION

1.0 Introduction:

In this work we analyzed hemodynamic pulse waves (PW) in human fluid-structure interaction problems. The work engaged mathematical models to show, among other things, that arterial pressure which has systolic and diastolic components generates PW which are enough to determine the physiological state of each of the internal organs, especially of the heart. The understanding of some of the terms used in this work may be necessary. In subsection 1.1 below some of such terms are explained.

1.1 Body vessels

In anatomy, a vessel is a tubular structure that conducts body fluid: a duct that carries fluid, especially blood or lymph to parts of the body. Thus, blood vessels are blood-carrying ducts. Blood vessels are in three varieties: arteries, veins andcapillaries.

Arteries

The main arteries are:

Pulmonary arteries: Carry deoxygenated blood from the body to the lungs where it is oxygenated and freed of carbon dioxide.....

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