Mathematical Analysis of Hemodynamic Pulse Wave in Human Fluid-structure Interaction.
ABSTRACT
Mathematical study of human pulse wave was studied with the view to gaining an insight into physiological situations. Fluid –Structure interaction (FSI) in blood flow is associated with pressure pulse wave arising from ventricular ejection. Solution of the coupled system of nonlinear 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.
TABLE OF CONTENTS
Title page
Certificate of Approval ——————————————-i
Acknowledgement ——————————————-ii
Dedication ——————————————-v
Abstract ——————————————-vi
Chapter One
Overview of Fluid-Structure Interaction Problem 1
1.0 Introduction ——————————————-1
1.1 Body vessels ——————————————-1
1.2 Arteries and their structure ——————————————-4
1.3 Blood ——————————————-6
1.4 Blood Pressure (BP) ——————————————-7
1.5Arterial Pulse ——————————————-8
1.6 Pulse Pressure ——————————————-9
1.7 Fluid-Structure Interaction ——————————————-10
1.8 Pulse Wave ——————————————-11
1.9 Resonance ——————————————-12
1.10 Aim and objectives of the study ——————————————-12
1.11 Scope and limitations of the study ——————————————-13
1.12 Methodology ——————————————-13
1.13 Significance of the study ——————————————-14
Chapter Two
Literature Review ——————————————-15
Chapter Three
Fluid-Wall interaction and non-linear pulse wave models in blood flow ——————————————-21
3.1.0 Generalized equation of motion of viscous fluid ——————————————-21
3.1.1 Action of fluid on the wall ——————————————-26
3.1.2 Fluid –Structure interaction model: Problem presentation ——————————————-29
3.1.3 Fluid –Structure coupling ——————————————-36
3.2.0 Model of non-linear arterial pulse: Problem presentation ——————————————-39
3.2.1 Linear superposition of forward and backward ABP waves ——————————————-41
Chapter Four
Solutions to model problems ——————————————-44
4.1.0 Solution of fluid-wall interaction problem ——————————————-44
4.1.1 Weak Formulation and Variational Form ——————————————-45
4.1.2 Rescaled Problem and asymptotic expansion ——————————————-50
4.1.3 Weak Formulation ——————————————-51
4.1.4 Energy estimates after rescaling ——————————————-52
4.1.5 Asymptotic Expansions ——————————————-53
4.1.6 Justification for asymptotic expansions ——————————————-54
4.1.7 Reduced problem using Expansion I ——————————————-55
4.1.8. Reduced problem using Expansion II ——————————————-58
4.2.0. Nonlinear arterial pulse model ——————————————-62
4.2.1 Methods of Solution of non-Linear Wave model problem ——————————————-67
4.2.2 The Tanh (hyperbolic tangent) Method of Solution ——————————————-68
4.3.0 Bilinear Method ——————————————-73
4.3.1 Solitons by bilinear method ——————————————-75
4.4 Systolic and Diastolic PW Representation ——————————————-80
Chapter Five
Results and discussions ——————————————-83
5.1.0 Features of: ——————————————-84
5.1.1 Tanh method ——————————————-84
5.1.2 Bilinear Method ——————————————-84
5.2.0 Physiological Analysis using solitary waveform ——————————————-85
5.2.1Distance effect ——————————————-85
5.2.2 Short and tall statures ——————————————-86
5.2.3 Time effects ——————————————-91
5.2.4 Harmonic Components of Arterial Pulse Waves ——————————————-94
5.2.5 Heart-Organ Resonance ——————————————-95
5.2.6 Hypertension and vaso-active Agents ——————————————-96
5.2.6 Dying Process ——————————————-98
5.3.0 Summary and Conclusion ——————————————-99
5.4.0 Recommendation(s) for further studies ——————————————-101
References ——————————————-102
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.
REFERENCES
Abbaoui K and Cherruault. New ideas for proving convergence of decomposition
methods. Comput. Maths. Appl., 29(7):103-108,1995Almanasreh H. Blood Flow Modelling in Compliant Arteries; Deriving effective
equations using asymptotic analysis methods. M.Sc Thesis,Chalmers University of
Technology (2007).Atabek HB, Wave propagation through a viscous fluid contained in a tethered
initially stressed,orthotropic elastic tube, Biophys.J.8:626-649,1968.Avolio A P. ,Van Bortel, L M., Boutouyrie P, Cockcroft J R., McEniery C M., Protogerou
A D. Roman M J., Safar ME., Segers P.,SmulyanH, Role of Pulse Pressure Amplification
in Arterial Hypertension: Seventh International Workshop on Structure and Function of the
Vascular System, Hypertension.Vol. 54, doi: 10.1161/ HYPERTENSIONAHA.109.134379
(2009).Babin , A. and Figotin A.,Linear superposition in nonlinear wave dynamics, Rev.
Math. Phys. 18, 971 (2006), DOI: 10.1142/S0129055X06002851Barbu V,Grujic Z, Lasiecka N, and Tuffaha A, Smoothness of weak solutions to a nonlinear fluid-structure interaction model. Indiana Univ.Math.,J. 57:1173-1207 ,2008
Bakirtas I and Demiray H.Amplitude modulation of nonlinear waves in a fluid-filled
tapered elastic tube.Appl.Math and Comput.154; 747-767,2004.Becker, K.L., ed. (2001). Growth and Development in the Normal Infant and Child,
Table 7.1. Principles and Practice of Endocrinology and Metabolism (3 ed.). Philadelphia,
Pa.: Lippincott, Williams & Wilkins. p. 69.Berger DS, Li JKJ., Laskey WK and,Noordergraaf .A, Repeated reflection of waves in
the systemic arterial system. Am. Physiol.Soc., 269-281,1993.Berger DS, Li JK, Noordergraaf A. Differential effects of wave reflections and
peripheral resistance on aortic blood pressure: a model-based study. Am J Physiol Heart
Circ Physiol 266 H1626–H1642, (1994).
StudentsandScholarship Team.