Mathematical analysis of hemodynamic pulse wave in human fluid-structure interaction

Project Abstract

<p> </p><p>Mathematical study of human pulse wave was studied with the view to gaining an insight into<br>physiological situations. Fluid –Structure interaction (FSI) in blood flow is associated with<br>pressure pulse wave arising from ventricular ejection. Solution of the coupled system of nonlinear<br>PDEs that arose from the FSI was sought in order to determine pressure. Further study on<br>pressure pulse waves showed that the Korteweg-de Vries (KdV) equations hold well for the<br>propagation of nonlinear arterial pulse wave. Solutions of the KdV equation by means of the<br>hyperbolic tangent (tanh) method and the bilinear method each yielded solitons. The solitons<br>describe the peaking and steepening characteristics of solitary wave phenomena.<br>The morphologies of the waves were studied in relation to the length occupied by the waves<br>(which corresponds to length of arterial segment and stature) and the left ventricular ejection<br>time (LVET). The study showed that both stature and LVET are independent descriptors of<br>cardio-vascular state.</p><p>&nbsp;</p> <br><p></p>

Project Overview

<p> OVERVIEW OF FLUID-STRUCTURE INTERACTION<br>1.0 Introduction:<br>In this work we analyzed hemodynamic pulse waves (PW) in human fluid-structure<br>interaction problems. The work engaged mathematical models to show, among other things,<br>that arterial pressure which has systolic and diastolic components generates PW which are<br>enough to determine the physiological state of each of the internal organs, especially of the<br>heart. The understanding of some of the terms used in this work may be necessary. In<br>subsection 1.1 below some of such terms are explained.<br>1.1 Body vessels<br>In anatomy, a vessel is a tubular structure that conducts body fluid: a duct that carries fluid,<br>especially blood or lymph to parts of the body. Thus, blood vessels are blood-carrying ducts.<br>Blood vessels are in three varieties: arteries, veins and capillaries.<br>Arteries<br>The main arteries are:<br>Pulmonary arteries: Carry deoxygenated blood from the body to the lungs where it is<br>oxygenated and freed of carbon dioxide.<br>Systemic arteries: They deliver blood to the arterioles, then to the capillaries where gases<br>and nutrients are exchanged.<br>2<br>Aorta: This artery is supplied with blood from the left ventricle of the heart<br>via the aortic valve. It is the root systemic artery, and it branches to daughte arteries. It carries<br>blood away from the heart.<br>Arterioles: These are the smallest of the true arteries. They regulate blood pressure and<br>deliver blood to the capillaries.<br>Carotid, subclavian, mesenteric, renal, iliac arteries and the celiac trunk are branches of<br>the aorta<br>Venules are the small blood vessels that transfer blood from the capillaries to the veins.<br>Veins<br>– They are large collecting vessels, such as the subclavian, jugular, renal, and<br>iliac veins. They carry blood at low pressures.<br>– Venae cavae are the largest veins, which they carry blood into the heart.<br>Capillaries<br>These are the smallest blood vessels (about 5-10μm in diameter).They form part of<br>microcirculation. Arteries divide into arterioles and continue to narrow, and as they reach the<br>muscles they become capillaries. Capillaries do not transport blood. They are specially<br>designed for the passage of substances, mainly oxygen and carbondioxide. They are thinwalled<br>and are composed only of endotheliac cells, which allow easy passage of substances.<br>A notable feature of capillary beds is their control of blood flow through auto regulation. This<br>helps an organ to maintain constant flow despite changes in central blood pressure. New<br>capillaries can be formed by pre-existing capillaries in a process called angiogenesis. Fig.1.1<br>shows the location of various arteries in the body.<br>3<br>Fig1.1 Human arterial system. (Available at <a target="_blank" rel="nofollow" href="http://www.chakras.org.uk/chakra_yoga_health_holistic_arterial.gif">www.chakras.org.uk/chakra_yoga_health_holistic_arterial.gif</a>)<br>4<br># Artery Name Function<br>0 Arterial system Canals that carry blood from the heart to the organs<br>1 Posterior Vessel that carries blood to the head.<br>3 External carotid Neck vessel that carries blood to the face<br>4 Internal carotid Neck vessel that carries blood to the brain<br>5 Common carotid (left) Carries blood to the left side of the neck<br>6 Brachio- cephalic Main vessel of the arm<br>7 Left subclavian Carries blood beneath the left clavicle<br>8 Right coronary artery Feeds the tissues of the right side of the heart with blood<br>9 Thoracic aorta Main artery of the thorax<br>10 Celiac trunk Carries blood to the thoracic cavity<br>11 Renal Carries blood to the kidneys<br>12 Superior mesenteric Carries blood to the upper part of the abdomen<br>13 Abdominal aorta Main artery in the abdominal area<br>14 Inferior mesenteric Carries blood to the lower part of the abdomen.<br>15 Common illiac Principal artery of the human lower limb<br>16 Internal iliac Internal branch of the iliac artery<br>17 External iliac External branch of the iliac artery<br>18 Profunda femoris Carries blood towards the inside of the thigh<br>19 Peroneal Carries blood to the lower leg<br>20 Lateral planter Carries blood to the side of the sole of the foot<br>21 Dorsalis pedis Carries blood to the dorsal part of the foot<br>22 Plantar arc Carries blood to the instep area<br>23 Medial plantar Carries blood to the median of the sole of the foot<br>24 Anterior tibial Carries blood to the front part of the lower leg<br>25 Posterior tibial Carries blood to the back part of the lower leg<br>26 Popliliteal Carries blood to the back of the foot<br>27 Femoral Carries blood to the thigh<br>28 Superficial palmar arch Situated beneath the skin of the palmar arc of the hand<br>29 Ulnar Situated in the area of the ulnar<br>30 Common interosseous Situated between the two bones of the forearm<br>31 Gonadal (Genital) Carries blood to the genital organs<br>32 Radial Situated in the area of the radius<br>33 Brachial Carries blood to the arm<br>34 Profunda brachial Carries blood towards the interior of the arm<br>35 Axillary Carries blood to the armpit<br>36 Right subclavian Carries blood beneath the right clavicle<br>37 Right vertebral Situated on the right, carries blood to the vertebrae<br>38 Common carotid (right) Carries blood to the right of the neck<br>39 Superior thyroid Carries blood to the thyroid<br>40 Lingual Carries blood to the tongue<br>41 Facial Carries blood to the face<br>42 Maxillary Carries blood to the maxillae<br>43 Superficial temporal Carries blood to the surface of the skin, in the area of the temples<br>Table 1.1 The 43 human main arteries and related function (Almanasreh (2007))<br>1.2 Arteries and their structure<br>All relatively large arteries have similar basic structure. The artery consists of the outermost<br>layer known as the tunica adventitia. This layer is composed of connective tissue. The inner<br>5<br>layer is known as the tunica media, and is made of smooth muscle cells and elastic tissues.<br>The innermost layer is known as the tunica intima. This layer is in direct contact with the<br>flowing blood. The lumen is the hollow internal cavity in which the blood flows (as seen in<br>Fig. 1.2).<br>Fig 1.2 Photomicrograph of the cross section of an artery showing the tunica intima, tunica media, and tunica externa.<br>Available at <a target="_blank" rel="nofollow" href="http://www.mhprofessional.com/product.php?isbn=0071472177">http://www.mhprofessional.com/product.php?isbn=0071472177</a><br>The endothelium of the intima is surrounded by sub-endothelial connected tissue. Around this<br>there exists a layer of vascular smooth muscle, which is developed in arteries. There is a<br>further layer of connective tissue known as the adventitia, which contains nerves that supply<br>blood to the muscular layer as well as nutrient to the capillaries in the larger blood vessels.<br>6<br>Blood vessels do connect and form anastamosis (a region of diffuse vascular supply). In<br>event of blockages anastamoses provide critical alternative route for the flow of blood.<br>In course of blood circulation arteries mainly carry blood away from the heart. The<br>capillaries link the arteries to the veins, and the veins carry the blood back to the heart.<br>Besides blood circulation, arteries (and blood vessels as a whole) help to measure vital health<br>statistics such as pulse and blood pressure. We can measure heart rate, or pulse, by touching<br>an artery. The rhythmic contraction of the artery as the heart beats keeps pace with the pulse.<br>The proximity of the artery to the surface of the skin enhances the accurate measurement of<br>the heart’s pulse by touching the artery. The heart itself is deeply protected.<br>1.3 Blood<br>Talking about whole blood, we think of the formed elements that are suspended in plasma.<br>The red blood cells (RBCs) constitute major part of the formed elements. The ratio of this<br>part to the other constituents of whole blood is known as hermatocrit.<br>Fig1.3 Formed elements of blood, Dugdale (2010)<br>7<br>It is the preponderance of the RBCs in the whole blood composition that make them very<br>important in determining the flow characteristics of blood. RBCs aggregate at low shear rates<br>(values &lt; 100dyn s-1) and form rouleaux. This has the effect of increasing the viscosity of<br>blood. Rouleaux disaggregation occurs as shear rate increases, resulting in the shear-thinning<br>characteristics that cause the non-Newtonian behavior of blood. In effect, blood viscosity<br>decreases. We consider blood as Newtonian when its shear-thinning characteristic disappears<br>as a result of increase in shear rate beyond the low shear rate region (Ku (1999), Kang<br>(2002)).<br>1.4 Blood Pressure (BP)<br>Blood pressure is the force being exerted on the walls of the arteries in the event of blood<br>being transported to parts of the body. It is customary to use the blood flowing from the<br>arteries to measure blood pressure because it is transported at a higher pressure than the blood<br>in the veins. BP is measured using two numbers (see Blood pressure chart in Appendix A).<br>The first number, which is usually higher, is taken when the heart beats during systole (the<br>contraction of the heart during which blood is pumped into the arteries), as the heart rests<br>between cycles.The systolic pressure is the peak pressure during heart contraction while the<br>second number is taken when the heart relaxes during diastole (rhythmic expansion of the<br>heart’s chambers at each heartbeat during which they fill with blood). The diastolic pressure<br>is the minimum pressure between contractions. Each of the numbers is recorded in<br>millimeters column of mercury (mmHg). It is normal for BP to increase in course of exercise<br>and to decrease when asleep. If BP stays too high or too low, there may be the risk of heart<br>disease.<br>8<br>The heart pumps blood out through a main artery known as the dorsal aorta. This main aorta<br>divides and branches out into several smaller arteries so that each region of the body has a<br>system of arteries that supply it with fresh oxygenated blood.<br>When the heart beats (during systole) the artery is filled with blood and it expands. When the<br>heart relaxes (during diastole) the artery contracts and exerts force that would push the blood<br>along. The integrity of blood flow and efficient circulation is the synergy between the heart<br>and the artery.<br>1.5 Arterial pulse<br>Pulse may be explained in terms of regular beat of blood flow as the regular expansion of an<br>artery, caused by the heart pumping blood through the body, or in terms of single beat of<br>blood flow as a single expansion and contraction of an artery, caused by a beat of the heart.<br>Usually, in medicine, pulse is the tactile arterial palpation by trained fingertips. Such<br>palpation may be in any place where an artery can be compressed against a bone. Such places<br>include neck (carotid artery), the wrist (radial artery), behind the knee (popliteal artery), on<br>the inside of the elbow (brachial artery), and near the ankle joint (posterior tibial artery), as<br>shown in Fig.1.4.<br>Pulse may be used in expediency as a tactile method of determination of systolic blood<br>pressure to a trained observer, but this cannot be said about diastolic blood pressure. Below<br>are the physiological pulse rates at rest (the resting heart rate (HRrest) is a person’s heart rate<br>when they are at rest, that is lying down but awake, and not having recently exerted<br>themselves <a target="_blank" rel="nofollow" href="http://en.wikipedia.org/wiki/Heart_rate#At_rest)">http://en.wikipedia.org/wiki/Heart_rate#At_rest)</a>.<br>9<br>newborn<br>(0-3 months<br>old)<br>infants<br>(3 — 6<br>months)<br>infants<br>(6 — 12<br>months)<br>children<br>(1 — 10<br>years)<br>children over 10<br>years<br>&amp; adults, including<br>seniors<br>welltrained<br>adult<br>athletes<br>100-150 90–120 80-120 70–130 60–100 40–60<br>Table 1.2 Normal pulse rates at rest, in beats per minute (BPM):US(20<br>Fig1.4 Arterial pulse points (at: <a target="_blank" rel="nofollow" href="http://www.nakedscience.org/mrg/AnatomyLectureNotesUnit7CirculatorySystem-">http://www.nakedscience.org/mrg/AnatomyLectureNotesUnit7CirculatorySystem-</a><br>TheBloodVessels_files/image006.gif)<br>1.6 Pulse Pressure (PP)<br>This is the amount of pressure that is required to create the feeling of a pulse.The variation of<br>pressure within the artery produces pulse which is transmitted through the artery; hence the<br>name arterial pressure pulse .When the heart’s left ventricle contracts, systemic arterial<br>pressures are generated. PP is most easily defined as being the amount of pressure required to<br>create the feeling of a pulse. The amount of pulse is created is measured (in mmHg) by the<br>Systolic versus Diastolic difference of blood pressures. It is mainly related to the amount of<br>blood ejected by each heart beat, stroke volume and the elasticity of the major arteries. If you<br>10<br>have a resting blood pressure is (systolic/diastolic) 120/80 mmHg, then your PP is 40, which<br>is a healthy PP.A consistent high resting PP is very harmful, and is most likely to quicken the<br>normal ageing of the heart, brain and kidney. In recent times researches have been carried out<br>to determine the relationship between pulse pressure and hypertension (Martin et al (1995)).<br>There is the possibility that resistance vessel structural adaptation in hypertension may be<br>closely related to pulse pressure than to other blood pressure parameters. Studies show that<br>the widening of PP is seen as a consequence of a loss of compliance in the large conduit<br>arteries and increased wave reflections from the periphery, and less to increased resistance in<br>peripheral arteries (Safar (1993)).It is most likely to be evident in, and prognostic of, cardiovascular<br>abnormally.<br>1.7Fluid-Structure Interaction (FSI)<br>The notion of fluids-structure interaction with regard to human dynamical structure is the<br>interplay of blood flow and the arterial conduit. The flow of blood via the arterial conduit<br>induces concomitant motion of the arterial wall, and thus there exists some interfacial region<br>within which fluid-structure interaction (FSI) is felt. Flow usually occurs in radial, angular<br>and axial directions. The arterial wall is expected to be compliant in order to sustain the<br>integrity of the flow. In compliant arteries flow in radial direction induces radial dilatation,<br>whilst axial flow induces longitudinal (axial) stretch.<br>In the main, the circulatory system of animated life owes its integrity to FSI. Within<br>reasonable expectation this interaction would present some mixed blessing to human<br>physiological state. Within physiological range of flow and pressure, the body would be held<br>at normal cardio-vascular state. On the unpleasant version, there could be incipient pathology<br>in the physiological state when the interaction fails to yield the desired goal. Beside various<br>hemodynamic para- meters we take exceptions at arterial pressure pulse wave. We shall give<br>11<br>attention to details on the nonlinear waves generated by the pulse. The injection of blood<br>from the left ventricle into the aorta generates far more than traveling waves in the arterial<br>system, otherwise forced stationary oscillations of the entire arterial system would be<br>untenable; thus pressure pulse would not likely reveal details about the resonance conditions<br>of the whole body in a defined manner. It is against this background that we are attracted to<br>the mathematical analysis of pulse waves and the resonance conditions there from, in a bid to<br>furnish medics with some contribution to wave-related issues in physiology. As we attest to<br>the solitary nature of the pulse waves, Wang et al (2010) reminds us in more physiological<br>terms that the resonance conditions of the whole body, sequel to the wave train must be a<br>benchmark for normal conditions of organs. This reminder is worthwhile in treating the<br>waveforms alongside their harmonic components. With this treatment, salient information<br>could be supplied regarding organ patho-physiology and hypertension. In this regard medics<br>could be availed of the benefits derivable from this work.<br>1.8 Pulse Wave (PW)<br>Pulse wave is a very complex physiological phenomenon observable, and detectable in blood<br>circulation. Any segment of blood ejected and transported through the artery in event of heart<br>systole is transformed between kinetic and potential energy. There are three observable<br>coherent phenomena on each artery or venous segment affected by pulse wave: blood flow<br>(flow pulse), the change in BP (pressure pulse) and the extension in transverse profile<br>(volume pulse). PPW is generated from the combination of the incident wave (i.e. the<br>pressure wave generated by the left ventricle in systole) and waves reflected back from the<br>periphery.<br>The shape of PW changes as moves to the distal arteries. This change could be largely due to<br>the pulse wave reflection and arterial tapering.<br>12<br>Pulse waveform varies in different vessels in the same individual. It is depends on the (viscoelastic)<br>properties of the artery (inducing wave amplification as it travels from more elastic<br>central to stiffer peripheral arteries), the viscosity of the blood, wave reflection and wave<br>dispersion.<br>1.9 Resonance<br>The term resonance as used here is in association with mechanical systems. In this regard it is<br>used to describe large oscillation at natural frequency. In the event of physiological arterial<br>flow, there is increased amplitude of oscillation of the organs (the mechanical system), when<br>they are subjected to vibration arising from the PW.The wave transmits through the artery<br>(source) proximal to the organ in question. Each of the organs (specifically the heart, liver,<br>lungs, kidney and spleen) has its own natural frequency. Each of the organs is influenced by<br>both the input pulsatile blood and by the harmonic driving force, both of which are<br>proportional to the PP and the connecting site of the local main artery. PP is good enough to<br>reveal information about the resonance conditions of the whole body. Wang et al (1990),<br>therefore was of the view that periodic injection of blood from the left ventricle would not<br>generate only traveling waves, but would also induce forced stationary oscillation of the<br>arterial system. This work buys this view, and would use the organ specific harmonics to<br>analyze the patho-physiological state of the body.<br>1.10 Aim and objectives of the study<br>The aim of this work is to determine whether stature (height) and LVET each can be<br>implicated in cardiac-vascular events.<br>The objectives of this work are:<br>13<br>— To carry out a study of hemodynamic pressure pulse wave due to left<br>ventricular ejection.<br>— To use pulse waveforms to analyze the human physiological state.<br>— To extend the frontiers of existing knowledge of the subject under consideration.<br>1.11 Scope and limitations of the study<br>This work studied hemodynamic pulse waves arising from the heart’s left ventricular ejection<br>and the FSI. It studied the effect of stature on pulse waveforms and analyzed the<br>physiological implications of the waveforms.<br>The study was non-invasive (as applicable in most medical practice). However, the work<br>relied on some experimental findings where necessary. We must agree, without prejudice,<br>that it takes maturity to know that models should be used, not believed (Henry (2012)).<br>1.12 Methodology<br>A set of nonlinear partial differential equations was used to describe the pressure-induced<br>viscous flow of the fluid. Another set of nonlinear partial differential equations was used to<br>describe the motion of the arterial wall. We expressed the FSI as a coupled system and the<br>equation of the radial contact force was obtained. The model of arterial pulse was built on the<br>system of equations governing the FSI. Analytic method was used to get the solutions of the FSI<br>problem which yielded the desired equation of arterial pressure. In a similar way, the solution of<br>the arterial pulse was sought. We used a combination of the hyperbolic tangent (tanh) and the<br>bilinear methods of solution to get the solutions of the arterial pulse problem. We obtained<br>solitary wave (soliton-) solutions. We therefore described pulse waves as solitons. Matlab<br>Software was used for the processing and plotting of graphs of our solutions and our analysis.<br>We went further to analyze the physiology underlying the pulse waves. Stature and the LV<br>14<br>ejection time were the parameters used for clinical purposes. In each of the cases the values of<br>the parameters were varied in order to study the (patho-) physiology underlying arterial pulse<br>waveforms.<br>1.13 Significance of the study<br>This work is important in many respects:<br>(i) It would underscore the importance of arterial pulse waveforms in determining pathophysiological<br>state of humans.<br>(ii) It would help medics to decide the course of their treatment to related pathologies.<br>15 <br></p>