Mathematical Analysis of Hemodynamic Pulse Wave in Human Fluid-structure Interaction

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Mathematical Analysis of Hemodynamic Pulse Wave in Human Fluid-structure Interaction


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.


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
Appendices 112



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 and capillaries.


The main arteries are:

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

Systemic arteries: They deliver blood to the arterioles, then to the capillaries where gases and nutrients are exchanged.

Aorta: This artery is supplied with blood from the left ventricle of the heart via the aortic valve. It is the root systemic artery, and it branches to daughte arteries. It carries blood away from the heart.

Arterioles: These are the smallest of the true arteries. They regulate blood pressure and deliver blood to the capillaries.

Carotid, subclavian, mesenteric, renal, iliac arteries and the celiac trunk are branches of the aorta Venules are the small blood vessels that transfer blood from the capillaries to the veins.


– They are large collecting vessels, such as the subclavian, jugular, renal, and iliac veins. They carry blood at low pressures.

– Venae cavae are the largest veins, which they carry blood into the heart.


These are the smallest blood vessels (about 5-10μm in diameter).They form part of microcirculation. Arteries divide into arterioles and continue to narrow, and as they reach the muscles they become capillaries. Capillaries do not transport blood. They are specially designed for the passage of substances, mainly oxygen and carbondioxide. They are thinwalled and are composed only of endotheliac cells, which allow easy passage of substances.

A notable feature of capillary beds is their control of blood flow through auto regulation. This helps an organ to maintain constant flow despite changes in central blood pressure. New capillaries can be formed by pre-existing capillaries in a process called angiogenesis. Fig.1.1 shows the location of various arteries in the body.

Fig1.1 Human arterial system. (Available at

# Artery Name Function

0 Arterial system Canals that carry blood from the heart to the organs

1 Posterior Vessel that carries blood to the head.

3 External carotid Neck vessel that carries blood to the face

4 Internal carotid Neck vessel that carries blood to the brain

5 Common carotid (left) Carries blood to the left side of the neck

6 Brachio- cephalic Main vessel of the arm

7 Left subclavian Carries blood beneath the left clavicle

8 Right coronary artery Feeds the tissues of the right side of the heart with blood

9 Thoracic aorta Main artery of the thorax

10 Celiac trunk Carries blood to the thoracic cavity

11 Renal Carries blood to the kidneys

12 Superior mesenteric Carries blood to the upper part of the abdomen

13 Abdominal aorta Main artery in the abdominal area

14 Inferior mesenteric Carries blood to the lower part of the abdomen.

15 Common illiac Principal artery of the human lower limb

16 Internal iliac Internal branch of the iliac artery

17 External iliac External branch of the iliac artery

18 Profunda femoris Carries blood towards the inside of the thigh

19 Peroneal Carries blood to the lower leg

20 Lateral planter Carries blood to the side of the sole of the foot

21 Dorsalis pedis Carries blood to the dorsal part of the foot

22 Plantar arc Carries blood to the instep area

23 Medial plantar Carries blood to the median of the sole of the foot

24 Anterior tibial Carries blood to the front part of the lower leg

25 Posterior tibial Carries blood to the back part of the lower leg

26 Popliliteal Carries blood to the back of the foot

27 Femoral Carries blood to the thigh

28 Superficial palmar arch Situated beneath the skin of the palmar arc of the hand

29 Ulnar Situated in the area of the ulnar

30 Common interosseous Situated between the two bones of the forearm

31 Gonadal (Genital) Carries blood to the genital organs

32 Radial Situated in the area of the radius

33 Brachial Carries blood to the arm

34 Profunda brachial Carries blood towards the interior of the arm

35 Axillary Carries blood to the armpit

36 Right subclavian Carries blood beneath the right clavicle

37 Right vertebral Situated on the right, carries blood to the vertebrae

38 Common carotid (right) Carries blood to the right of the neck

39 Superior thyroid Carries blood to the thyroid

40 Lingual Carries blood to the tongue

41 Facial Carries blood to the face

42 Maxillary Carries blood to the maxillae

43 Superficial temporal Carries blood to the surface of the skin, in the area of the temples

Table 1.1 The 43 human main arteries and related function (Almanasreh (2007))

1.2 Arteries and their structure

All relatively large arteries have similar basic structure. The artery consists of the outermost layer known as the tunica adventitia. This layer is composed of connective tissue. The inner layer is known as the tunica media, and is made of smooth muscle cells and elastic tissues.

The innermost layer is known as the tunica intima. This layer is in direct contact with the flowing blood. The lumen is the hollow internal cavity in which the blood flows (as seen in Fig. 1.2).

Fig 1.2 Photomicrograph of the cross section of an artery showing the tunica intima, tunica media, and tunica externa.

The endothelium of the intima is surrounded by sub-endothelial connected tissue. Around this there exists a layer of vascular smooth muscle, which is developed in arteries. There is a further layer of connective tissue known as the adventitia, which contains nerves that supply blood to the muscular layer as well as nutrient to the capillaries in the larger blood vessels.

Blood vessels do connect and form anastomosis (a region of diffuse vascular supply). In event of blockages anastomoses provide critical alternative route for the flow of blood.

In course of blood circulation arteries mainly carry blood away from the heart. The capillaries link the arteries to the veins, and the veins carry the blood back to the heart.

Besides blood circulation, arteries (and blood vessels as a whole) help to measure vital health statistics such as pulse and blood pressure. We can measure heart rate, or pulse, by touching an artery. The rhythmic contraction of the artery as the heart beats keeps pace with the pulse.

The proximity of the artery to the surface of the skin enhances the accurate measurement of the heart’s pulse by touching the artery. The heart itself is deeply protected.

1.3 Blood

Talking about whole blood, we think of the formed elements that are suspended in plasma.

The red blood cells (RBCs) constitute major part of the formed elements. The ratio of this part to the other constituents of whole blood is known as hermatocrit.

Fig1.3 Formed elements of blood, Dugdale (2010)

It is the preponderance of the RBCs in the whole blood composition that make them very important in determining the flow characteristics of blood. RBCs aggregate at low shear rates (values 1.4 Blood Pressure (BP)

Blood pressure is the force being exerted on the walls of the arteries in the event of blood being transported to parts of the body. It is customary to use the blood flowing from the arteries to measure blood pressure because it is transported at a higher pressure than the blood in the veins. BP is measured using two numbers (see Blood pressure chart in Appendix A).

The first number, which is usually higher, is taken when the heart beats during systole (the contraction of the heart during which blood is pumped into the arteries), as the heart rests between cycles.The systolic pressure is the peak pressure during heart contraction while the second number is taken when the heart relaxes during diastole (rhythmic expansion of the heart’s chambers at each heartbeat during which they fill with blood). The diastolic pressure is the minimum pressure between contractions. Each of the numbers is recorded in millimeters column of mercury (mmHg). It is normal for BP to increase in course of exercise and to decrease when asleep. If BP stays too high or too low, there may be the risk of heart disease.

The heart pumps blood out through a main artery known as the dorsal aorta. This main aorta divides and branches out into several smaller arteries so that each region of the body has a system of arteries that supply it with fresh oxygenated blood.

When the heart beats (during systole) the artery is filled with blood and it expands. When the heart relaxes (during diastole) the artery contracts and exerts force that would push the blood along. The integrity of blood flow and efficient circulation is the synergy between the heart and the artery.

1.5 Arterial pulse

Pulse may be explained in terms of regular beat of blood flow as the regular expansion of an artery, caused by the heart pumping blood through the body, or in terms of single beat of blood flow as a single expansion and contraction of an artery, caused by a beat of the heart.

Usually, in medicine, pulse is the tactile arterial palpation by trained fingertips. Such palpation may be in any place where an artery can be compressed against a bone. Such places include neck (carotid artery), the wrist (radial artery), behind the knee (popliteal artery), on the inside of the elbow (brachial artery), and near the ankle joint (posterior tibial artery), as shown in Fig.1.4.

Pulse may be used in expediency as a tactile method of determination of systolic blood pressure to a trained observer, but this cannot be said about diastolic blood pressure. Below are the physiological pulse rates at rest (the resting heart rate (HRrest) is a person’s heart rate when they are at rest, that is lying down but awake, and not having recently exerted themselves


(0-3 months old)


(3 — 6 months)


(6 — 12 months)


(1 — 10 years)

children over 10 years & adults, including seniors
100-150 90–120 80-120 70–130 60–100 40–60

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