SNAPSHOTS

OF

HEMODYNAMICS

BASIC SCIENCE FOR THE CARDIOLOGIST

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2.

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B. Swynghedauw (ed.): Molecular Cardiology for the Cardiologist. SecondEdition. 1998 ISBN 0-7923-8323-0B. Levy, A. Tedgui (eds.): Biology of the Arterial Wall. 1999

ISBN 0-7923-8458-XM.R. Sanders, J.B. Kostis (eds.): Molecular Cardiology in ClinicalPractice. 1999 ISBN 0-7923-8602-7B.Ostadal, F. Kolar (eds.): Cardiac Ischemia: From Injury to Protection. 1999

ISBN 0-7923-8642-6H. Schunkert, G.A.J. Riegger (eds.): Apoptosis in Cardiac Biology. 1999

ISBN 0-7923-8648-5A. Malliani, (ed.): Principles of Cardiovascular Neural Regulation in Healthand Disease. 2000 ISBN 0-7923-7775-3P. Benlian: Genetics of Dyslipidemia. 2001 ISBN 0-7923-7362-6D. Young: Role of Potassium in Preventive Cardiovascular Medicine. 2001

ISBN 0-7923-7376-6E. Carmeliet, J. Vereecke: Cardiac Cellular Electrophysiology. 2002

ISBN 0-7923-7544-0C. Holubarsch: Mechanics and Energetics of the Myocardium. 2002

ISBN 0-7923-7570-XJ.S. Ingwall: ATP and the Heart. 2002 ISBN 1-4020-7093-4W.C. De Mello, M.J. Janse: Heart Cell Coupling and Impuse Propagation inHealth and Disease. 2002 ISBN 1-4020-7182-5P.P.-Dimitrow: Coronary Flow Reserve – Measurement and Application: Focuson transthoracic Doppler echocardiography. 2002 ISBN 1-4020-7213-9G.A. Danieli: Genetics and Genomics for the Cardiologist. 2002

ISBN 1-4020-7309-7F.A. Schneider, I.R. Siska, J.A. Avram: Clinical Physiology of the Venous System.2003. ISBN 1-4020-7411-5Can Ince: Physiological Genomics of the Critically Ill Mouse. 2004

ISBN 1-4020-7641-XWolfgang Schaper, Jutta Schaper: Arteriogenesis. 2004

ISBN 1-4020-8125-1eISBN 1-4020-8126-X

Nico Westerhof, Nikos Stergiopulos, Mark I.M. Noble: Snapshots of Hemodynamics:An aid for clinical research and graduate education. 2005

ISBN 0-387-23345-8eISBN 0-387-23346-6

KLUWER ACADEMIC PUBLISHERS – DORDRECHT/BOSTON/LONDON

Snapshots of Hemodynamics

An aid for clinical research andgraduate education

Nico WesterhofLaboratory for Physiology,

VU University medical centerAmsterdam, the Netherlands

Nikos StergiopulosLaboratory of Hemodynamics and Cardiovascular Technology,

Swiss Federal Institute of TechnologyLausanne, Switzerland

Mark I.M. NobleCardiovascular Medicine,

Aberdeen UniversityAberdeen, Scotland

Springer

eBook ISBN: 0-387-23346-6Print ISBN: 0-387-23345-8

Print ©2005 Springer Science + Business Media, Inc.

All rights reserved

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

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©2005 Springer Science + Business Media, Inc.

Visit Springer's eBookstore at: http://ebooks.kluweronline.comand the Springer Global Website Online at: http://www.springeronline.com

CONTENTS

PREFACEACKNOWLEDGEMENT

PART A BASICS OF HEMODYNAMICSCHAPTER 1CHAPTER 2CHAPTER 3CHAPTER 4CHAPTER 5CHAPTER 6CHAPTER 7CHAPTER 8CHAPTER 9CHAPTER 10CHAPTER 11

VISCOSITYLAW OF POISEUILLEBERNOULLI’S EQUATIONTURBULENCE

ARTERIAL STENOSISRESISTANCEINERTANCEOSCILLATORY FLOW THEORYLAW OF LAPLACEELASTICITYCOMPLIANCE

PART B CARDIAC HEMODYNAMICSCHAPTER 12CHAPTER 13CHAPTER 14CHAPTER 15CHAPTER 16CHAPTER 17CHAPTER 18CHAPTER 19

CARDIAC MUSCLE MECHANICSPRESSURE-VOLUME RELATION

PUMP FUNCTION GRAPHWORK, ENERGY AND POWEROXYGEN CONSUMPTION & HEMODYNAMICSPOWER AND EFFICIENCYCORONARY CIRCULATIONASSESSING VENTRICULAR FUNCTION

PART C ARTERIAL HEMODYNAMICSCHAPTER 20CHAPTER 21CHAPTER 22CHAPTER 23CHAPTER 24CHAPTER 25CHAPTER 26CHAPTER 27CHAPTER 28

WAVE TRAVEL AND VELOCITYWAVE TRAVEL AND REFLECTION

WAVEFORM ANALYSISARTERIAL INPUT IMPEDANCEARTERIAL WINDKESSELDISTRIBUTED MODELSTRANSFER OF PRESSUREVASCULAR REMODELINGBLOOD FLOW AND ARTERIAL DISEASE

PART D INTEGRATIONCHAPTER 29CHAPTER 30

DETERMINANTS OF PRESSURE & FLOWCOMPARATIVE PHYSIOLOGY

PART E APPENDICESAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5APPENDIX 6APPENDIX 7

TIMES & SINES: FOURIER ANALYSISBASIC HEMODYNAMIC ELEMENTSVESSEL SEGMENTBASIC ASPECTSBOOKS FOR REFERENCESYMBOLSUNITS AND CONVERSION FACTORS

INDEX

viixi

137

111517212529313541

495157636971758191

9799

105109113121127131137143

147149155

161163167169173177179181

183

PREFACE

This book is written to help clinical and basic researchers, as well as graduatestudents, in the understanding of hemodynamics. Recent developments ingenetics and molecular biology on the one hand, and new non-invasivemeasurement techniques on the other hand, make it possible to measure andunderstand the hemodynamics of heart and vessels better than ever before.Hemodynamics makes it possible to characterize, in a quantitative way, thefunction of the heart and the arterial system, thereby producing informationabout what genetic and molecular processes are of importance forcardiovascular function.

We have made the layout of the book such that it gives a short overviewof individual topics, in short chapters only giving the essentials, so that it iseasy to use it as a quick reference guide. It is not necessary to read the bookfrom cover to cover. Each chapter is written in such a way that one is able tograsp the basic and applied principles of the hemodynamic topic. If moredetails or broader perspectives are desired one can go to the other, related,chapters to which the text refers, or the textbooks listed in the ‘Referencebooks’ section (Appendix 5).

To bring the essentials even more directly across, every chapter starts witha ‘box’ containing a figure and caption, which give the basic aspects of thesubject. It is often sufficient to study the contents of this box alone to obtainthis basic information. Chapters end with a section ‘Physiological andClinical relevance’ to place the information into perspective. The part inbetween, called ‘Description’, can be used to get more detailed informationand find some references to more detailed work.

More comprehensive information on the subjects discussed can be foundin textbooks on physiology and cardiology, as well as special books onhemodynamics. A number of these books are listed in the section ‘Referencebooks’ (Appendix 5). More literature also can easily be found on the internet.In the chapters the number of references therefore is limited to a few only.

NW, NS, and MIMN

How to use the

Snapshots of Hemodynamics

Chapter 2 Law of Poiseuille Chapter and title

DescriptionWith laminar flow through a uniform tube of radiusthe velocity profile over the cross-section is aparabola.

Physiological and clinical relevanceThe more general form of Poiseuille’s law given

above, i.e., allows us to derive resistance,

R, from mean pressure and mean flow measurement.

The box contains a figure witha short text that illustrates themain message of the chapter.

The ‘Description” sectiongives the essential backgroundand discusses the differentaspects of the subject.

The ‘Physiological andclinical relevance’ sectionplaces the subject in a broaderphysiological context andshows clinical applications.

A limited number of refe-rences is given. Major refe-rence books are given inAppendix 5.

References1.

2.

The Murgo JP, Westerhof N, Giolma JP,Altobelli SA. Aortic input impedance in normalman: relationship to pressure wave forms.Circulation 1980;62:101-116.Murgo JP, Westerhof N, Giolma JP, Altobelli

ACKNOWLEDGEMENT

The authors wish to thankDr. Ben Delemarre, Prof. Dr. Walter Paulus and Prof. Dr. P Segers forreading the manuscript and giving many excellent suggestions.We thank Jan Paul Barends for all his help in aspects of layout andpresentation.

xi

Part ABasics of Hemodynamics

Chapter 1 VISCOSITY

Description

Consider the experiment shown in the figure in the box. The top plate ismoved with constant velocity, v, by theaction of a shear force F, while the bottomplate is kept in place (velocity is zero). Theresult is that the different layers of bloodmove with different velocities. The differencein velocity in the different blood layerscauses a shearing action between them.

The rate of shear, is the relativedisplacement of one fluid layer with respectto the next. In general, the shear rate is theslope of the velocity profile, as shown in thefigure on the right. In our particular example the velocity profile is linear,going from zero at the bottom to v at the top plate. Therefore, the slope of thevelocity profile, and thus the rate of shear, is equal to v/h, h being thedistance between the plates. The units of shear rate are 1/s. The force neededto obtain a certain velocity, is proportional to the contact area, A, betweenfluid and plates. It is therefore convenient, instead of force, to use the termshear stress, defined as the force per area with units Pa or

We may think of the following experiment: we pull the top plate atdifferent velocities v and we measure the shear force F. Then we plot theshear stress, against the shear rate, The resulting relation is given in thefigure in the box and the slope is the viscosity:

The units of viscosity are or Poise with1 Pa·s =10 Poise. Fluids with a straight relationship between shear stress andshear rate are called Newtonian fluids, i.e., viscosity does not depend onshear stress or shear rate. Viscosity is sometimes called dynamic viscosity in

VELOCITY and shear rate for ageneral profile.

contrast to the kinematic viscosity, which is defined as viscosity divided bydensity thus

Viscosity of blood

Blood consists of plasma and particles, with99% of the particle volume taken by the redblood cells, RBC’s, or erythrocytes. Thus thered blood cells mainly determine thedifference between plasma and bloodviscosity. The viscosity of blood thereforedepends on the viscosity of the plasma, incombination with the hematocrit (volume %of red blood cells, Ht) and red celldeformability. Higher hematocrit and lessdeformable cells imply higher viscosity. The relation between hematocrit andviscosity is complex and many formulas exist. One of the simplest is the oneby Einstein:

Einstein’s relation for the viscosity offluids containing particles applies onlyto very low particle concentrations.Nevertheless, it gives some indication.The viscosity of plasma is about 0.015Poise (1.5 centipoise, cP) and theviscosity of whole blood at aphysiological hematocrit of 40 - 45% isabout 3.2 cP, or Pa·s.

Blood viscosity depends not only onplasma viscosity and hematocrit, but

also on the size, shape and flexibility of the red blood cells. For instance, thehematocrit of camel blood is about half of that of human blood, but thecamel’s red blood cells are more rigid, and the overall effect is a similar bloodviscosity.

Anomalous viscosity or non-Newtonian behavior of blood

The viscosity of blood depends on itsvelocity. More exactly formulated, whenshear rate increases viscosity decreases.At high shear rates the doughnut-shapedRBC’s orient themselves in the directionof flow and viscosity is lower. Forextremely low shear rates formation ofRBC aggregates may occur, therebyincreasing viscosity to very high values. Ithas even been suggested that a certainminimum shear stress is required beforethe blood will start to flow, the so-calledyield stress. In large and medium sizearteries shear rates are higher than so viscosity is practically constant.

4 Basics of Hemodynamics

VISCOSITY of plasma and blood.

VISCOSITY as function of shear rate forhematocrit of 48.

VISCOSITY as function of hematocrit.

Viscosity 5The physiological range of wall shear stress is 10 to 20 or 1 to 2Pa, with 1Pa = 0.0075 mmHg. Several equations exist that relate shear stressand shear rate of blood, e.g., Casson fluid, and Herschel-Bulkley fluid [1,2].

Viscosity also depends on the size ofblood vessel. In small blood vessels and athigh velocities, blood viscosity apparentlydecreases with decreasing vessel size. Thisis known as the Fahraeus-Lindqvist effect,and it begins to play a role in vesselssmaller than 1 mm in diameter. Red bloodcells show axial accumulation, while theconcentration of platelets appears highest atthe wall. The non-Newtonian character ofblood only plays a role in the microcircula-tion.

Viscosity depends on temperature. Adecrease of 1°C in temperature yields a 2% increase in viscosity. Thus in acold foot blood viscosity is much higher than in the brain.

How to measure viscosity

Blood viscosity is measured using viscometers. Viscometers consistessentially of two rotating surfaces, as a model of the two plates shown in thebox figure. Blood is usually prevented from air contact and temperature iscontrolled. When comparing data on viscosity one should always keep inmind the measurement technique, as results are often device dependent.

Physiological and clinical relevance

The anomalous character of blood viscosity results from the red blood cells,and the effects are mainly found in the microcirculation at low shear andsmall diameters. The effects are of little importance for the hemodynamics oflarge arteries. Thus, in hemodynamics, it may be assumed that viscosity isindependent of vessel size and shear rate.

Determination of blood viscosity in vivo is almost impossible. Inprinciple, the pressure drop over a blood vessel and the flow through it,together with vessel size, can be used to derive viscosity on the basis ofPoiseuille’s law. However, the vessel diameter in Poiseuille’s law (Chapter 2)appears as the fourth power, so that a small error in the vessel diameter leadsto a considerable error in the calculated viscosity. Also, the mean pressuredrop over a segment of artery is typically a fraction of 1 mmHg. Moreover,hematocrit is not the same in all vessels due to plasma skimming effects. Andfinally, Poiseuille’s law may only be applied when there are no effects of inletlength (see Chapter 2).

The main purpose of the circulation is to supply tissues with oxygen.Oxygen supply is the product of flow and oxygen content. The hematocritdetermines the (maximum) oxygen carrying capacity of blood and itsviscosity, and therefore the resistance to blood flow. These counteractingeffects on oxygen transport result in an optimal hematocrit of about 45 in thehuman at sea level, with a small difference between males and females. Itappears that in mammals, blood viscosity is similar, but the hematocrit is not

VISCOSITY as function of vesselsize.

6 Basics of Hemodynamicsbecause of the different size, shape and flexibility of the red blood cells asmentioned above.

Low hematocrit, as in anemia, decreases oxygen content and viscosity ofblood. The former lowers oxygen supply and the latter increases blood flowthus increasing supply. Inversely, polycythemia increases oxygen content butlowers blood flow. At high altitude, where oxygen tension is lower and thusoxygen saturation in the blood is lower, a larger hematocrit is advantageous.In endurance sports higher hematocrit is more efficacious during increasedoxygen demand. This is the reason EPO is sometimes used by the athletes.

References

1.2.

Merrill EW. Rheology of blood. Physiol Rev 1969;49:863-888.Scott Blair GW, Spanner DC. An introduction to biorheology. 1974,Amsterdam, Elsevier Sc Publ.

Chapter 2 LAW OF POISEUILLE

Description

With laminar and steady flow through a uniform tube of radius the velocityprofile over the cross-section is a parabola. The formula that describes thevelocity (v) as a function of the radius, r is:

is the pressure drop over the tube of length (l), and is blood viscosity.At the axis (r = 0), velocity is maximal, while at the wall thevelocity is zero. Mean velocity is:

and is found atBlood flow (Q) is mean velocity times the cross-sectional area of the tube,

giving:

This is Poiseuille’s law relating the pressure difference, and the steadyflow, Q, through a uniform (constant radius) and stiff blood vessel. Hagen, in1860, theoretically derived the law and therefore it is sometimes called thelaw of Hagen-Poiseuille. The law can be derived from very basic physics(Newton’s law) or the general Navier-Stokes equations.

The major assumptions for Poiseuille’s law to hold are:

The tube is stiff, straight, and uniform

8 Basics of HemodynamicsBlood is Newtonian, i.e., viscosity is constant

The flow is laminar and steady, not pulsatile, and the velocity at the wallis zero (no slip at the wall).

INLET LENGTH. Flow entering a sidebranch results in skewed profile. Ittakes a certain inlet length before thevelocity develops into a parabolicprofile again.

In curved vessels and distal tobranching points the velocity profile is notparabolic and the blood flow profile needssome length of straight tube to develop,this length is called inlet length. The inletlength depends on the Reynolds number(Re, see Chapter 4) as:

with D vessel diameter. For the aorta meanblood flow is about 6 1/min, and thediameter 3 cm, so that the mean velocity is~ 15 cm/s. The Reynolds number istherefore ~ 1350. This means that is~ 80, and the inlet length ~240 cm, whichis much longer than the length of the entireaorta. In the common iliac artery the

Reynolds number is about 500 and diameter 0.6 cm giving an inlet length of~18 cm. In other, more peripheral arteries the inlet length is much shorter buttheir length is shorter as well. Clearly, a parabolic flow profile is not evenapproximated in the arterial system. Nevertheless, the law of Poiseuille canbe used as a concept relating pressure drop to flow.

A less detailed and thus more general form of Poiseuille’s law iswith resistance R being:

This law is used in analogy to Ohm’s law of electricity, where resistanceequals voltage drop/current. The analogy is that voltage difference iscompared to pressure drop and current to volume flow. In hemodynamics wecall also it Ohm’s law. Thus:

This means that resistance can be calculated from pressure and flowmeasurements.

Calculation of wall shear stress

THE SHEAR RATE at the wallof a blood vessel can becalculated from the ‘rate ofchange of velocity’ at the wall,as indicated by angle

The wall shear rate can be calculated from theslope of the velocity profile near the wall (angle

in the figure above), which relates to thevelocity gradient, tan near the wall (seeChapter 1). The derivative of the velocity profilegives the shear rate Shear stressis shear rate times viscosity Theshear rate at the vessel axis, r = 0, is zero, and atthe wall, it is so the blood cells encounter a range ofshear stresses and shear rates over the vessel’s cross-section.

Law of Poiseuille 9

SHEAR STRESS at the wall can alsobe calculated directly by the balanceof pressure and frictional forces.

The shear stress at the wall can also becalculated from basic principles. For anarterial segment of length l, the forceresulting from the pressure difference

times the cross-sectional area,should equal the opposing force

generated by friction. This frictional forceon the wall equals the shear stress, timesthe lateral surface, Equating these

forces gives and

This formulation shows that with constant perfusion pressure an increase inviscosity does not affect wall shear stress.

The wall shear stress may also be expressed as a function of volume flowusing Poiseuille’s law

this is a more useful formula for estimating shear stress because flow andradius can be measured noninvasively using ultrasound or MRI, whereaspressure gradient cannot.

Example of the use of Poiseuille’s law to obtain viscosity

A relatively simple way to obtainviscosity is to use a reservoir thatempties through a capillary. Knowing thedimensions of the capillary and usingPoiseuille’s law viscosity can becalculated. Even simpler is thedetermination of viscosity relative to thatof water. In that case only a beaker andstopwatch are required. The amounts ofblood and water obtained for a chosentime are inversely proportional to theirviscosities. The practical design based onthis principle is the Ostwald viscometer.

Murray’s law

Murray’ law (1926) was originallyproposed by Hess in 1913 and assumesthat the energy required for blood flow and the energy needed to maintain thevasculature is assumed minimal [1]. The first term equals pressure times flowand, using Poiseuille’s law, this is The second term isproportional to vessel volume and thus equals with b a proportionalityconstant. The total energy, is:

A WIDE BORE RESERVOIR main-taining constant pressure, provides theblood flow through a capillary. Theapplication of Poiseuille’s law, orcomparison with water, gives absoluteor relative viscosity, respectively.

The minimal value is found for and this leads to:

10 Basics of HemodynamicsFor a bifurcation it holds that

and thus

with two equal daughters it holds that:

and we find that

The area of both daughters together is the area of the mothervessel. This area ratio is close to the area ratio predicted by Womersley onthe basis of the oscillatory flow theory, to obtain minimal reflection of wavesat a bifurcation, namely between 1.15 and 1.33 [2]. Thus Murray’s lawsuggests a minimal size of blood vessels and an optimum bifurcation [1].

Physiological and clinical relevance

The more general form of Poiseuille’s law given above, i.e., allowsus to derive resistance, R, from mean pressure and mean flow measurement.

The wall shear stress, i.e., the shear force on the endothelial cells plays animportant role in short term, second to minutes, and long term, weeks,months or years, effects. Short-term effects are vasomotor tone and flowmediated dilatation (FMD). Long-term effects are vascular remodeling,endothelial damage, changes in barrier function, and atherosclerosis.

It is still not possible to directly measure wall shear stress or shear rate invivo. Shear rate is therefore derived from the velocity profile. Velocityprofiles can be measured with MRI and Ultrasound Doppler. From thevelocity profile the velocity gradient is often calculated. However, thecalculations to obtain shear rate require extrapolation, because very near thewall velocity measurements are not possible. To calculate wall shear stressthe blood viscosity near the wall has to be known as well, but viscosity closeto the wall is not known because of plasma skimming. Plasma skimmingrefers to the relative absence of erythrocytes in the region near the wall. Alsothe diameter variation over the heartbeat is almost impossible to account for.

Wall shear stresses are which is about 10,000 timesless than the hoop stress (Chapter 9). Despite this enormous difference inmagnitude, both stresses are equally important in the functional wall behaviorin physiological and pathological conditions (see Chapters 27 and 28).

References

1.2.

Weibel E. Symmorphosis. 2000, Cambridge MA, Harvard Univ Press.Womersley JR. The mathematical analysis of the arterial circulation in a stateof oscillatory motion. 1957, Wright Air Dev. Center, Tech Report WADC-TR-56-614.

Chapter 3 BERNOULLI’S EQUATION

Description

The Bernoulli equation can be viewed as an energy law. It relates bloodpressure (P) to flow velocity (v). Bernoulli’s law says that if we follow ablood particle along its path (dashed line in left figure in the box) thefollowing quantity remains constant:

where is blood density, g the acceleration of gravity and z the elevationwith respect to a reference horizontal surface (i.e., ground level or heartlevel). The equation of Bernoulli says that as a fluid particle flows, the sumof the hydrostatic pressure, P, potential energy, and the dynamicpressure or kinetic energy, remains constant. One can easily deriveBernoulli’s equation from Newton’s law: Pressure forces+ gravitational forces = mass × acceleration. Strictlyspeaking, the Bernoulli equation is applicable only ifthere are no viscous losses and blood flow is steady.

Physiological and clinical relevance

Bernoulli’s law tells us that when a fluid particledecelerates pressure increases. Conversely, when a fluidparticle accelerates, such as when going though a severestenosis, pressure drops.

Because of the direct relationship between pressureand velocity, the Bernoulli equation has found severalinteresting clinical applications, such as the Gorlin [2]

PRESSURES, andand velocities,

and in ventricularlumen and valvularstenosis.

12 Basics of Hemodynamicsequation for estimating the severity of an aortic or mitral valve stenosis. Letus consider flow through a stenosed valve according to the figure on theprevious page.

Applying Bernoulli’s law

and

The flow Q is the same at both locations, thus where andare the cross-sectional areas of ventricle and valve, respectively.

Substituting this into the Bernoulli’s equation we obtain:

Since the cross-sectional area of the stenosed valve is much smaller thanthe cross-sectional area of the ventricle the equation can besimplified to:

When velocity in the stenosis, is expressed in m/s the pressure drop (P, inmmHg) is approximately

Earlier this approach was used to approximate to estimate effective area[2], of the valvular stenosis by measuring flow and pressure gradient (e.g.,using a pressure wire).

When the pressure is in mmHg and flow in ml/s, this gives an effective area:If pressure recovery downstream of the vena

contracta is included then:

Calculation of aortic valvular area

Doppler velocimetry applied to both the valvularannulus and the aorta allows for the directcalculation of valve area. Since volume flow is thesame, the product of velocity and area is also thesame at both locations. Thus AORTIC VALVE AREA

can be derived fromDoppler velocity mea-surements, in aorta andvalve, and aortic area.

Jets

Jets and vena contracta are formed when blood flow emerges from anopening such as a valve, and play a role in valvular stenosis and regurgitation(see figure on the enxt page). The contraction coefficient, i.e., the area ratioof the jet and the valve depends on the shape of the valve. The coanda effectis the phenomenon that a jet along the, atrial or ventricular, wall appears