the physics of pulsatile blood flow with particular reference to small vessels a1

The physics of pulsatile blood flow withparticular reference to small vessels

E. O. Attinger

Theoretical aspects of the analysis of pulsatile blood flow are discussed in the first part of thepaper. In the second part the physical characteristics of the vascular system are described, andin the last part the behavior of the system is analyzed in terms of the outlined theory.

FJLoi

ollowing the work of Frank1 and Witzig2

early in this century, an increasing num-ber of attempts have been made during thelast decade to increase our understandingof pulsatile blood flow by applying analyti-cal rather than purely descriptive methodsfor its investigation.3"6 While for a numberof biological systems appropriate theorieswere already available from the physicalsciences, the situation in the cardiovascularfield was much more complicated. Thetheoretical hydrodynamicists concernedthemselves primarily with ideal fluids andwere never particularly interested in pul-satile flow of real liquids through pipes.The general theoretical development intheir field was limited primarily by the factthat nonlinearities and instabilities are soubiquitously present in fluid flows. As a

From the Research Institute, Presbyterian Hos-pital, The School of Veterinary Medicine, andthe Moore School of Electrical Engineering,University of Pennsylvania, Philadelphia, Pa.

The original work reported in this paper was doneduring the tenure of a special fellowship fromthe National Institutes of Health 5-F3-GM-14037 and supported by Research Grants H-6836 and FR 00148 from the United StatesPublic Health Service.

consequence, the practical applications ofhydrodynamics are based primarily onempirical relationships rather than on theo-retical predictions. The situation in therapidly developing field of rheology issimilar. Because of its practical impor-tance, large efforts have gone into thestudy of stress-strain and pressure-flow re-lations of various materials such as plasticsand wood, and of suspensions. The similar-ity of the problems in rheology, hydro-dynamics, and cardiovascular physiologyhas resulted in a rather extensive exchangeof ideas between these different disci-plines, beginning with Poiseuille, a physi-cian interested in blood flow through cap-illaries, who first described laminar flowthrough tubes and stimulated much of thelater theoretical work.

The analysis of a system aims at a quan-titative description which permits the pre-diction of its behavior under a variety ofcircumstances. This behavior is considera-bly more complex in biological than inman-made systems. In order to make ananalytical approach to a biosystem at allpossible it becomes necessary to introducesimplifying assumptions, the judicious se-lection of which represents one of the mostcritical points of departure for any analy-sis and requires considerable insight into

973

974 Attinger Inoestigative OphthalmologyDecember 1965

biological phenomena. No matter how so-phisticated the theoretical approach, itsvalidity cannot be established by argumentalone; experimental evidence is necessaryto support it, and here we find ourselvesfaced with a major problem. The primarydata required for such an analysis are pres-sure, flow, and volume. These variableschange in pulsatile flow both with time andwith location within the system. At manysites they are not measurable without seri-ously disturbing the system and even if thebest available equipment and utmost careare used, measurement errors in the orderof 5 per cent are unavoidable. Since theeffects of some of the parameters whichhave been introduced in various analyticaltreatments are of the same order of mag-nitude, they cannot be assessed with anyreal confidence.

The first part of this paper deals withthe theoretical aspects of the analysis ofpulsatile blood flow. In the second part thephysical characteristics of the vascular sys-tem are described and in the last part thebehavior of the system is analyzed in termsof the outlined theory. For the sake ofbrevity, only the more significant aspectsof the various problems are discussed; formore complete and detailed discussions thereader is referred to two recent publica-tions.7' s

Theoretical aspects

We begin with Newton's second equa-tion:

Force = mass x acceleration,which we apply to a unit volume of anincompressible Newtonian liquid in a cy-lindrical, elastic vessel segment under con-ditions of laminar flow (Fig. 1). To get apicture of laminar flow, imagine that theflowing liquid is composed of an infinitenumber of concentric layers, each forminga sleeve around the layers inside it. Theoutermost layer is in contact with the ves-sel wall and moves the same way as thewall does. Let this velocity be zero. Theadjacent layer has a finite velocity, the nextmoves even faster, and so on. In the center

of the tube, the velocity is largest. Thisvelocity difference between adjacent layersproduces shear forces, and the fluid ele-ments within are subjected to shear strains.The ratio between this shearing stress (ve-locity gradient) and the rate of strain iscalled the coefficient of viscosity (/A). Fora Newtonian liquid, the relation betweenstress and rate of strain must satisfy twoconditions; The coefficient of viscosity isconstant; and if stress is zero, the rate ofstrain is also zero. In other words, a plot ofthese two variables must yield a straightline going through the origin.

Now consider the motion of fluid in thisvessel segment. The pertinent dimensionsof the vessel are: Ro, outside radius; Rj,inside radius; h, wall thickness; and 1, thelength along the z-axis. The fluid elementmoves with a velocity, V, which has twocomponents, one parallel to the z-axis (Vz)and one parallel to the r-axis (Vr). Simi-larly, the pressure gradient V P, which pro-vides the net driving force for the element,also has two components (SP/8z andSP/8r). (Since the vessel has a circularcross section, the third components in thetangential direction VQ and SP/89 areeliminated on the basis of symmetry.) Ac-cording to Newton's equation the totalpressure gradient is equal to the total ac-celeration of the fluid element minus theforce necessary to overcome viscous resis-tance, as shown in the tipper equation ofFig. 1. In steady flow the accelerationterm disappears, and the only force neces-sary to maintain the motion of fluid is thatrequired to balance the viscous resistanceof the liquid. Under these conditions thereis no motion of the elastic vessel wall andthe relation between flow and pressurereduces to the familiar equation of Poi-seuille:

(1) 8 AIn pulsatile flow the vessel wall moves

during each cardiac cycle: during systolethe radius increases, during diastole it de-creases. We therefore have to write not

Volume 4Number 6

Physics of pulsatile blood flow 975

only two equations for the fluid motion(one in the direction parallel to the vesselaxis and one in the radial direction) butwe also have to consider the forces in-volved in the movement of the vessel wallitself (Panel C in Fig. 1). The force whichcauses this motion is the pressure gradientbetween the inside and the outside of thevessel (transmural pressure). The ratio be-tween transmural pressure and the result-ing deformation of the wall is called theelastic modulus E.

(2) E =stress

strain Ax/x,,

where Ax/x0 is the relative change in onedimension, say length. Since the vessel wallis three dimensional, there are three elasticmoduli, one relating pressure to change inradius (tangential modulus), one relatingpressure to change in length (longitudinalmodulus), and one relating pressure tothe change in wall thickness (radial modu-lus). These three moduli are related by sixPoisson ratios (o^), where Poisson ratiois defined as

(3) strain in a direction i at right anglesto the stress

<r,j z= •strain in the direction j of the stress

SCHEMA OF FLUID AND WALL MOTION IN DISTENSIBLE ELASTIC TUBE

PRESSURE GRADIENT = DENSITY X TOTAL ACCELERATIONMINUS (ELASTIC AND VISCOUS LOSSES)

Fig. 1. Diagram to illustrate the equations of motion for a fluid and a wall element in anelastic tube.

976 Attinger Investigative OphthalmologyDecember 1965

For example, the Poisson ratio for astretched rubber band would be

Aw/w(>

where w0 = the initial width= initial length= change in width= change in length

For isotropic materials (which vesselwall is not!) the elastic moduli are thesame in all directions, and if the volumeof the material does not change understress, there is only one Poisson ratio witha value of one half.

According to Newton's third law theremust be a force which counterbalances theexcess pressure inside the vessel. Thisforce is the tension developed by the elas-tic, collagenous, and muscular elements ofthe vessel wall. The relation between pres-sure (P) and wall tension (T) was firstgiven by Laplace:

(4) T = Pr/liwhere h = wall thickness.

As it turns out, the physical propertiesof the vessel wall (elasticity, viscosity,and inertia) are frequency-dependent,and the static elastic modulus is some-what lower than the modulus at the fre-quency of the heartbeat. Except for theaddition of a term which characterizes theelastic forces of the vessel wall, the formof the equations for wall motion is identi-cal with those for the motion of the liquid.However, the equation of wall motion isexpressed in terms of wall displacement,since we are primarily interested in therelation between stress and strain. Theequation for fluid motion, on the otherhand, is written in terms of velocity be-cause here the relations between stressand rate of strain are most pertinent. Theequations for the motion of fluid and themotion of the wall are inseparably linked.At the boundary between the blood andthe vessel wall they must hold simulta-neously. Furthermore, the peripheral vas-cular resistance, or better, impedance,

which varies inversely with the fourthpower of the radius, is highly sensitiveto a change in transmural pressure.

Womersley and others3"5 have solvedthe Navier-Stokes equations for pulsatileflow in a uniform, elastic tube, and thissolution is the basis of much of the recentwork on arterial hemodynamics. (For ex-cellent discussions of Womersley's theoryand the assumption upon which his solu-tion is based, see McDonald7 and Fry.9)For an assumed pressure gradient AP =M cos (wt — 0) the solution for volumeflow is

(5) Q =

where a- = r2 —

M sin (wt - 0 + e)

w = radian (angular), frequencyv = kinematic viscosity (viscosity/densi-

ty) of the blood.

M' and e are basically the ratio and thephase angle difference of two Bessel func-tions of complex argument, and depend onwall thickness, Poisson ratio, and longi-tudinal tethering of the vessel.

As (o -» 0, M'/or -> ys and e -> 0, i.e.,the solution is that of Poiseuille.

Streeter5 has solved the equation of mo-tion for pulsatile flow using the method ofcharacteristics. In order to fit his experi-mental to his theoretical data he used dif-ferent coefficients of viscosity for forwardand backward flow because he assumeddifferent conditions in the boundary layerand flow patterns under these two con-ditions. However, further experimental dataare necessary to evaluate the relative meritsof this approach.

In considering these equations and theirsolutions, we have to remember that thedifferent variables are functions of spaceand time. The parameters of the systemcannot, therefore, be treated as lumped(concentrated at one point), but must beconsidered distributed throughout the sys-tem.10 This is most easily demonstratedif one looks at the deformation which oc-

Volume 4Number 6


curs both in the pressure and in the flowpulse as they travel from the heart towardthe periphery. The vascular system exhibitsa highly complex geometry and the rel-ative proportion of its structural compo-nents varies markedly throughout the vas-cular bed. As a consequence, wave re-flections as well as selective damping areintroduced, leading to the striking dis-persion of pressure and flow pulses. It isapparent that the theoretical analysis can-not be applied to the cardiovascular sys-tem without reducing arbitrarily the com-plexity of the biological system.

The choice of these simplifying assump-tions depends very much upon the physi-cal properties of the system, which arebriefly outlined in the next section.

Physical properties of the vascular system

A. Geometry. Fig. 2 shows a simplifieddiagram of the system. It can roughly bedivided into four parts:

PULMONARY

CORONARY 1 0 %

8RAIN 1 5 %

MUSCLE 1 5 %

SKIN 1 0 %

HEPATIC

KIDNEY 2 5 %

, SPLENIC

MESENTERIC

Fig. 2. Schema of the circulation. The numbersindicate the approximate percentage of the cardiacoutput fed into each of the six parallel beds. Notethat there are two capillary systems in the kidneyand three in the splanchnic circulation. LA, leftatrium, LV, left ventricle, RA, right atrium, RV,right ventricle.

1. Two pumps in series, the left andright heart.

2. A distributing system, the arteries,leading from each ventricle into the pe-riphery.

3. An exchange system, the capillaries,where metabolites diffuse across the capil-lary membrane both from and into thetissue.

4. A collecting system, the veins, whichtransport the blood back to the pump.

Note that the peripheral circulation isarranged in a number of parallel beds,each perfusing an organ system and fedfrom a common pump. It is obvious thatthe distribution of flow between these var-ious beds will depend on their respectiveimpedances. At rest about 10 per cent ofthe total cardiac output flows through theheart, 15 per cent through the brain, 25per cent through the splanchnic bed, 25per cent through the kidneys, 15 per centthrough the muscles, and 10 per centthrough the skin. Similarly, there are anumber of parallel pathways through thelungs. Here, however, we deal with a low-pressure system, and hence the distribu-tion of blood flow through the variouslobes will depend primarily on the hydro-static pressure difference, and thereforeupon the position of the animal.11 Duringexercise, after meals, or in hot or cold en-vironment these proportions may changeconsiderably. For instance, the fraction ofthe cardiac output going to the eye or theheart of the submerged duck increasesfourfold as compared to that in the non-submerged state, while the flow to the pan-creas or the kidneys falls to one-tenth ofits previous value.12 Hence, it is well toremember that the blood supply to an or-gan depends not only on the cardiac out-put but also on the state of the localvasculature.

As discussed in the previous section, thegeometrical basis for an analytical treat-ment of pulsatile blood flow is a single,circular tube. Fig. 3 shows how large anapproximation this represents. The dataare based upon the values calculated by


flafNAME

No. OF BRANCHESRADIUS cm.LENGTH cm.VOLUME cm.3

R E S I S T A N C E 1 " ^

6a1.5403064

E:::i:

5a40.1520

603.9-103

4a

600.051050

1.6-105

3a

1800.03125

1.2-105

2a

40-106

.001.225

2.1010

1

12-108

.0004.160

3.9-1011

2b

80-106

.0015.2110

4-109

3b

1800.075

130

3.2-103

4b

600.1210

270.5-104

iiiiii

i

5b40.320220250

6b

1.625405026

Fig. 3. Geometry of peripheral vascular tree of the dog. Diagram illustrating the massivechanges in total cross section along the peripheral vascular bed. Blocks are numbered as fol-lows: 1, capillaries. 2a, arterioles. 2b, veinules. 3a, terminal arterial branches. 3b, terminalveins. 4a, main arterial branches. 4b, main venous branches. 5a, large arteries. 5b, large veins.6a, aorta. 6b, vena cava. Resistance values pertain to the total effect of one segment.

Green13 for a 13 kilogram dog. A singletube, the aorta, with a cross section ofabout 0.8 cm.2 branches progressively into12 x 10s capillaries, each of which has across section of about 5 x 10~7 cm.2 Thetotal cross section of the capillaries repre-sents an area of about 600 cm.2 Hence,for a 13 kilogram dog with a cardiac out-put of 1.8 L. per minute or 30 cm.3 per sec-ond, the average linear velocity of the flow-ing blood decreases from 37.5 cm. per sec-ond in the aorta to 0.05 cm. per second ina capillary. Since blood is a non-New-tonian liquid, the viscosity of which fallsas the rate of strain (velocity) increases,particularly at high hematocrits,4 the pres-sure-flow relations in the various segmentscan be expected to be different on thisbasis alone. Note, however, that at anyone time only a fraction of the total capil-lary bed (60 to 75 per cent) is perfused.

Although the total cross section increases,the cross section of an individual arterydecreases during its course toward the pe-riphery. As an example, the mean radiusof the aortas of ten large dogs was foundto vary from 1.2 cm. in the ascending aortato 0.75 cm. in the middle of the descend-ing aorta, to 0.5 cm. in the abdominal aorta.The corresponding radius in the externaliliac artery was 0.27 cm.15 Over the same

distance the elastic modulus of the arterialwall increases from 3 x 10° dyn. cm."2 to12 x 10G dyn. cm,"2' 7 i.e., the wall materialof the arteries is about four times stifferin the periphery as compared to the centralpart of the aortic tree.

On the venous side the system decreasesfrom the total cross section of the capillarybed to that of the vena cava. Here, dataon the physical characteristics of the in-dividual segments are even scarcer thanon the arterial side. It is, however, reason-able to assume that this system also isnonuniform along its length. Furthermore,because it is a low pressure system, its di-mensions vary considerably more withpressure changes than those of the highpressure system.

B. Pressure-volume relations. The stress-strain relations with respect to linear di-mensions have been discussed in the pre-vious section. Their over-all effect deter-mines the instantaneous volume of a par-ticular vascular segment. The latter canalso be described by the vessel disten-sibility, C, which in terms of vessel di-mensions and elastic modulus becomes:

(6)dV

E(2a + 1)

Volume 4Number 6


where dV/dP is the change in volume perunit pressure change dP, and a is the ratiobetween radius and wall thickness. In thearterial system, the mean distending pres-sure is in the order of 130 x 103 dyn. cm."2

(100 mm. Hg) upon which is superim-posed the oscillating pulse pressure. Sincethe arteries are quite stiff (E = 3 to 12dyn. cm."2 x 10°) this pressure change re-sults in a change in radius of only 2 to5 per cent. This rather simplified analysis isbased upon the assumption that the ves-sel length does not change as the trans-mural pressure changes. In reality, thedimensions of the vessel change not onlyin the circumferential direction, but alsoalong the axis and in wall thickness. Hencethere would be 3 moduli of elasticity toconsider which are related for an aniso-tropic, nonisovolumetric material by sixPoisson ratios. Usually isovolumetry anda single Poisson ratio of 0.5 are assumed,but, although this assumption has beenproved to be incorrect,10 the effects uponvascular dynamics are not yet measurablein the in vivo system. While the availableevidence indicates17 that the elastic proper-ties of the arterial tree are linear over thenormal operating range (100 to 200 dyn.cm.~2 x 103) nonlinear effects are consid-erably more important in the venous sys-tem.

As the volume of a completely collapsedvein is increased, the vessel cross sectionchanges from a very flat ellipse to a circlewithout initially producing a wall tension.Once wall tension (and hence a trans-mural pressure) begins to develop, therise in pressure per unit volume changeis at first rather small but increases rapidlythereafter. For instance, Green's data13

indicate that for the inferior vena cava ofa dog the distending pressures are as fol-lows: for a volume of 2 cm.3, P = 0, forV = 8 cm.3, P = 2 cm. H2O, for V = 12cm.3, P = 12 cm. H2O, and for V = 14cm.3, P = 24 cm. H2O). Hence, the disten-sibility varies from practically infinity at avolume of 0 to 2 cm.3 to 0.166 x 10"3 cm.5

dyn.~T at a volume of 14 cm.3

Of the total blood volume, about 22 percent resides in the pulmonary circulation,14 per cent in the heart, and 64 per centin the peripheral circulation. The distri-bution among the vessels of the latter is asfollows: about 17 per cent is found in ar-teries with a diameter greater than 0.3 mm.,about 17.8 per cent in the microcirculation(radius less than 0.05 mm.), and 64 per centin the venous bed. Based on geometricalconsiderations, as well as on pressure-flowand pressure-volume relations, we have es-timated the volume distribution in the dif-ferent parallel beds of the peripheral cir-culation as follows10: splanchnic bed 38 percent, muscle 21 per cent, brain 17 per cent,skin 14 per cent, kidney 7.1 per cent, cor-onary circulation 3 per cent. The splanchnicbed can absorb or release roughly 40 percent of its normal volume; similar relationshold for the vascular bed of the skin andmuscle.ls

C. Pressures and flows in the vascularbed. The total change in mean pressureacross the peripheral vascular bed is inthe order of 100 mm. Hg (130 x 103 dyn.cm."2). The main pressure drop (65 percent) occurs through the arterioles (meandiameter 0.02 cm., mean length 0.2 cm.).In contrast, the fall in mean pressure alongthe large arteries is very small, althoughthe pulse pressure increases. The pressuredrop across the capillaries will dependgreatly on the state of the arterioles. Ifthey are dilated, their contribution to thetotal pressure loss will be less and thatof the capillaries will increase. It is goodto remember that the control of the capil-lary pressure, and of the gradient acrossthe capillary bed is a major factor for thecontrol of fluid exchange across the capil-lary membrane. The fall in pressure alongthe venous system represents only about 10per cent of the total pressure loss. Theserelations are similar for both the peripheraland the pulmonary circulation. One cancalculate that a reduction of the cross sec-tion of the capillary bed by one half in-creases the total resistance across the pe-ripheral vascular bed by about 30 per cent;


if half the arterioles are blocked, the resis-tance increases by 60 per cent, and so on.During its travel the pressure wavechanges considerably in shape (Fig. 4).The amplitude of the pressure pulse in-creases, the front of the wave becomessteeper, and the sharp inflection at the in-cisura is smoothed out and then disap-pears entirely by the time the wave reachesthe lower abdominal aorta. The time re-quired for the pressure pulse to travel fromthe heart to the femoral artery is between50 and 100 msec. Between these larger

arteries and the small arteries with aninternal diameter of 200 /u, the meanblood pressure falls by 10 to 15 per centand the pulse pressure decreases. Thepressure is still markedly pulsatile and,although it is dampened somewhat, theavailable evidence indicates that the mainpressure drop occurs in the arteries of lessthan 200 /x internal diameter.19

Flow in the large arteries is markedlypulsatile. In the ascending aorta of thedog, the average volume flow is in theorder of 30 to 40 cm.3 per second. The peak

E. c. G.

I. CAROTID

2. ARCH

3. THORACIC AORTA10 CM

4. THORACIC AORTA20 CM

5. ABD. AORTA25 CM

6. ABD. AORTA30 CM

7. ILIAC

8. FEMORAL

0 100 200 m secFig. 4. Change in tlie shape of a pressure wave traveling from the heart toward tlie periphery.The site of measurement and its distance from the aortic arch are marked beside each pres-sure pulse. Note the time delays between the different measuring sites.

Volume 4Number 6


velocity during systole may reach 100 to200 cm. per second. The ejection periodis followed by a short interval duringwhich blood actually flows backward. Overthe remainder of the cardiac cycle the ve-locity is zero. Toward the periphery thishighly peaked flow pattern is progressivelydampened out, backflow disappears, andforward flow is present over the wholecycle. Both the pattern and the distributionof flow depend markedly on the characteris-tics of the particular vascular bed. In ar-teries of 1 mm. diameter the pattern isstill quite oscillatory and available evidencesuggests that the pulsatile flow charactermay be preserved even in pulmonary capil-laries. 20 On the venous side the flow inthe inferior and superior vena cava isstrongly pulsatile. Backflow may be presentin the latter where the magnitude of theoscillatory components is of the same orderas that of the mean flow.21 No reliable dataare yet available on the dynamic flow pat-terns in the other parts of the venous vas-culature.

As far as the blood flow through the eyeis concerned, the average flow rate per 100Gm. of retina is in the order of 500 cm.3

per minute. This represents a total flowof about 0.5 cm.3 per minute for the cateye.22 About 20 per cent flows through theretinal vessels, the remaining 80 per centthrough the choriocapillaries. The lattersupply the nutrient flow for a tissue withan extremely high metabolism (Qo2 = 31).Direct observation of the microcirculationof the retina indicates that blood flow inthe retinal vessels is relatively steady and

Table I. Change in blood viscosity withpeak flow rate in pulsatile flow

Hematocrit(%)958568.547

Apparent

a

10060187

viscosity

b7025126

(centipoise)*

c

6425

41.7

"Column a, 0.25 cm.3 per second; column b, 1 cm.3 persecond; column c, 4 cm.3 per second.

that all the capillaries appear to be per-fused.23

D. The physical characteristics of blood.Since blood represents a suspension of cellsin a colloidal solution its behavior is notthat of a Newtonian liquid. The valuesgiven for the viscosity of plasma are gen-erally about 1.39 centipoise and those forwhole blood between 1.7 and 3.7 cp. at37° C.7 The viscosity at low rates of shearis considerably higher. Typical values ob-tained by Kunz and Coulter14 for oscil-latory flow are listed in Table I.

Their evidence also indicates that theorientation of the red cells within the ve-locity field changes during the cardiaccycle. Close to the vessel wall there is arelatively cell-poor layer, the width ofwhich fluctuates from instant to instant, de-creases as the hematocrit increases andwhose effects become relatively less im-portant in larger vessels, where the flowapproaches that of a homogeneous fluid.The higher velocities in the center in com-bination with a peripheral layer lead tocore concentrations which are lower thanif the velocity were uniform over the crosssection. Such a dynamic effect is but onedemonstration of the complex interactionof velocity and concentration distributions.At bifurcations the concentration of sus-pended particles is usually lower in theside branch and is affected primarily bythe rate of discharge in the two branches,the branch size, and the concentration up-stream.24 As the red blood cells pass fromthe arterioles into the capillaries their shapechanges from that of an elliptic disc tothat of a thimble.25

The analysis of pulsatile flow

We are now in a position to interpretthe behavior of the cardiovascular systemwithin the background provided by thetheory. The first problem in any analyticalprocedure is to express the measured var-iable as a number. A convenient way ofdoing this is the use of mean values, i.e.,integration with respect to time or space,for example: cardiac output, minute ven-


tilation, peripheral resistance, diffusion ca-pacity, etc. However, for certain purposesthis approach neglects significant charac-teristics. There are a number of mathemat-ical techniques which make it possible toexpress any arbitrary function of time asa series of terms. For periodic and quasi-periodic phenomena, such as encounteredin pulsatile flow, the so-called Fourier se-ries are particularly suited. With this tech-nique flow and pressure values are expressedas the sum of a number of sine waves,which permits a mathematical treatmentof the data in terms of meaningful physicalconcepts. In such an analysis (Fig. 5) onefinds that the amplitude of the variousharmonics (frequency components) of thepressure pulse increases from the arch ofthe aorta both toward the fore and thehindpart of the body.20 (This, of course,is in accordance with the over-all increasein pulse pressure illustrated in Fig. 4.) Inthis figure the distances from the aorticarch in centimeters are plotted on the ab-scissa and the amplitude of the first 4 har-monics (from top to bottom) on the ordi-

nate. It will be seen that the increase inamplitude for the first 3 harmonics is rathersteep until the inguinal ligament and thatthereafter the amplitude falls off. Towardthe head the increase occurs only in thefirst harmonic and is much smaller. Thefact that the amplitude changes of thevarious harmonics are not parallel indi-cates that this behavior is not only fre-quency dependent. However, it clearlyshows that the vessels become stiffer to-ward the periphery. Fig. 6 shows how thepulse wave velocity (foot-to-foot velocity)changes in the various arteries. Again dis-tance in centimeters from the heart isplotted on the abscissa. At a mean pres-sure of 200 cm. H2O the wave velocity inthe aortic arch is approximately 4 metersper second. It increases to 7 m per secondin the carotid, to 8 m per second at the bi-furcation of the aorta and to 14 m per sec-ond in the arteries of the foot.27 Since ac-cording to the Moens-Korteweg equation:

(7)Eh

2'rp[symbols as defined earlier),

cm

151

14

13

12-

I I -

10-

9-

I

7"

6-

5-

4-

3-

2-

1-

NjO CAR. BIF.

EXT. CAROTID

ARCH DIAPH. BIF. ING. LIG.

COMMON CAROTID I BR. | THORACIC AORTA | ABD. AORTA [ ILIAC I HIND LIMBCEPH.

60 cm

Fig. 5. Changes in the amplitude of the first (top), second, third, and fourth (bottom) har-monies of the pressure pulse, as the pressure pulse travels from the heart both cephalad andcaudad into the periphery. Pressure measurements were obtained in intervals of 2 to 5 cm.;the distances are indicated on the abscissa. Note the marked increase in the first three har-monies as the wave approaches the inguinal ligament and the decrease thereafter. The in-crease toward the head is less striking.

Volume 4Number 6


the wave velocity c increases proportion-ally to the square root of the elasticmodulus, the expected increase in the lat-ter would be respectively 1.3 fold in thecarotid, 1.4 fold at the bifurcation, and1.9 fold in the arteries of the foot, if theratio wall thickness/radius remains con-stant. The change in the shape of thepressure wave shown in Fig. 4 and Fig. 5can be explained by a combination of threemechanisms originating from the geometryand nonuniformity of the vascular wall:wave reflections, viscous losses, and elastictapering.

A. Sudden changes in impedance areassociated with wave reflections. At suchpoints only a part of the pulse wave travel-ing peripherally is transmitted, the re-mainder is reflected back toward the heart.

Since frictional losses occur, the amplitudeof the incident wave decreases along thevessel toward the periphery and the am-plitude of the reflected wave decreases to-ward the origin of the vessel. Because ofthese frictional losses no true standingwaves can be produced, and there areprobably no secondary reflections. At pres-ent the theory of wave reflections appearsto give the best explanation for the peaksand troughs in the amplitude of the var-ious harmonics of pressure and flowpulses.26 Although these patterns can beanalyzed for descriptive purposes as thesuperposition of two waves, one has tothink of the superposition of a number ofwaves, each reflection originating at a dif-ferent site.

B. Losses are introduced because of the

m/sec PULSE WAVE VELOCITY (FOOT) IN VARIOUS ARTERIES

10 -

5-

ARCH DIAPH. BIF

! ! iCAROTID M E A N PRESSURE 200 CMHIO (UPPER) KNEE FOOT

MEAN PRESSURE 5 0 - 6 0 CMH<0 (UOWCH )

20 10 10 20 30 40 50 70 80FROM HEART

Fig. 6. Pulse wave velocity (foot-to-foot) measured throughout the arterial tree of a 50 pounddog. Distances are measured from the heart, both cephalad and caudad. The two waveswere obtained at mean pressures of 200 and 50 to 60 cm. H«O, respectively. Note the markedincrease in pulse wave velocity, as the wave travels away from the heart. (From data pre-sented by McDonald, D. A., Navarro, A., Martin, R., and Attinger, E. O.: Federation Proceed-ings 24: 334, 1965.)


viscous properties of the liquid and thewall. The viscosity of the blood is, ofcourse, one of the important factors forthe pressure drop across the vascular bed(Equations 1 and 5). It has been proposedthat a frequency dependence of wall vis-cosity may account for the change in theshape of the pressure wave. However,available data from in vitro experimentsdo not support this suggestion.28'29 Ourpresent measuring techniques are not ac-curate enough to evaluate the effects ofwall viscosity on the transmission of thepressure wave in the living animal.

C. Elastic tapering, i.e., the progressivestiffening of the vessel wall toward theperiphery has the following effects uponthe over-all function of the cardiovascularsystem30:

It reduces the oscillatory component ofthe work required to maintain the cardiacoutput.

It reduces the over-all distensibility ofthe arterial system.

It amplifies the pressure pulse as ittravels toward the periphery.

Since both flow and pressure are pul-satile, the concept of peripheral resistancehas to be modified in order to account forthe frequency-dependent behavior of thesystem. Instead of using only the ratio ofmean pressure and mean flow, the pressure-flow relation has to be evaluated for allthe pertinent frequencies. The frequency-dependent parameter obtained in this wayis called the impedance, and is, of course,identical with peripheral resistance at zerofrequency. Fig. 7 shows the input impe-dance of a 60 pound dog measured simul-taneously in the ascending aorta, the de-scending aorta (mid-thoracic), the abdom-inal aorta (below the renal arteries), andthe carotis communis. The oscillatoryterms represent only a fraction of the DCimpedance term. The impedance valuesare normalized with respect to the DCvalue. Note that the magnitude of the lat-ter as well as that of the oscillatory com-ponents increases progressively toward theperiphery.

While it is certainly true that a numberof additional phenomena, such as the ef-fects due to entrance length, non-New-tonian behavior of flow, and nonlaminarflow patterns will have to be included ina final analysis of the behavior of the largevessel, they appear to be too small to bereadily detected with the measuring equip-ment presently available. Hence, it seemsjustifiable to consider the flow behaviorin large vessels in terms of continuummechanics, i.e., local variations at the mi-croscopic level are replaced by statisticalaverages. This is not true for the micro-circulation.30 Here the detailed flow be-havior of blood cannot be described interms of bulk rheological parameters ofblood. If we were to count the moleculesof the various plasma components and theportions of red cells contained in a unitvolume a large number of times, wecould not only arrive at an average densityof each component, but could also define alocal average velocity. In larger vessels wecan extend this time average to a spaceaverage taken at a single fixed time; how-ever, in the microcirculation there is novolume element in the flow possessing thisaverage property. Direct observation of themicrocirculation indicates that the flow ofplasma and erythrocytes is completely ran-dom both with respect to their relative pro-portions and with respect to direction alongthe vessel axis. I know of no theory whichcould adequately describe blood flowthrough the microcirculation in any detail.

Conrad and Green32 have evaluated he-modynamic changes in peripheral vasculardisease. The vasculature of normal indi-viduals is characterized by a vasodilatorresponse to alcohol, resulting in a markedincrease in the flow rate, with respect toboth the mean and the oscillatory com-ponents. The pressure drop through largearteries and the pulse wave velocity didnot change but the resistance of thesmall vessels fell to one third of the controlvalue.

In patients with vasospastic disease, thevalues for blood flow were considerably

Volume 4Number 6


IZ I

1.0 -

.9 -

. 7 •

.6 -

. 5 -

.4 -

.3 -

.2 -

.1 -

NORMALIZED IMPEDANCES IN 60 LB. DOG

T ASC. AORTA Zo=4. 4« I03

• DESC. AORTA Zo=6. 2-ICP

• ABDOM. AORTA Zorl2.9«IOs

O CAROTIDS Zo= 48* I03

dyn cm5sec

2.75 5.5 8.25 11.0 13.75 16.5 c/sec

Fig. 7. Magnitude of the input impedance in various vascular beds of a 60 pound dog as afunction of frequency. Pressures and flows were measured simultaneously in the ascending,descending, lower abdominal aorta and in the common carotid. The impedance values arenormalized. The value for the DC impedance is given in the inset.

reduced. There was a significant increasein the pressure drop through the large ar-teries and the resistance through smallvessels was markedly elevated. Pulse wavevelocity remained unchanged. Although,following the administration of alcohol,marked improvement in vascular perfor-mance was observed, the values of theparameters measured remained in generalbelow normal values of the controls. Inarterial occlusion, the observed pressure

drop through the large arteries (brachialto digital artery) was increased 80 foldin the average as compared to the nor-mal. As a consequence, blood flow (bothaverage value and oscillatory components)was greatly reduced. A rise in small vesselresistance (3 fold as compared to normal)and a decrease in pulse wave velocity wasalso found in these cases. The response ofthese parameters to vasodilation by alcoholwas rather small.


Conclusion

It is apparent that the analysis of thepressure-flow relations of pulsatile bloodflow in a complex vascular bed is not asimple matter. However, it is of consider-able importance, both in terms of cardiacwork and in terms of organ perfusion. Al-though the evidence is not clear-cut, it hasoften been suggested that an organ func-tions better if it is perfused by means ofpulsatile rather than steady flow. Further-more, pulsatile flow is not restricted to thearterial bed alone. In both the inferior andthe superior vena cava blood flow is oscil-latory. These oscillations are probably dueprimarily to back flow associated withatrial contraction. However, the study ofthe dynamic behavior of blood flow in veinshas hardly begun.

The frequency-dependence of vascularimpedance suggests that there is an opti-mal rate at which cardiac work is minimalfor a given cardiac output. Calculationsfor the pulmonary bed of dogs show thatthe total power dissipated falls from 300mwatts at a heart rate of 30 per minuteto 100 mwatts at 180 per minute.33 Thepower losses due to the pulsatile compo-nents, therefore, can represent a consider-able fraction of the total power dissipation.Furthermore, the distribution of flow andvolume between the various vascular bedsis a frequency-dependent phenomenon. Itwill be of great interest to carry this analysisinto the smaller vessels, once adequate dataabout their dimensions and pressure-flowpatterns become available. The ophthal-mologist is in the enviable position of be-ing able to provide such data for the smallvessels in the fundus of the eye. By care-ful microphotographic techniques he canevaluate not only the flow pattern, but thephysical properties of the vessel wall aswell, using the pulse wave velocity as anindex for the latter.

I am grateful to Dr. D. A. McDonald of theResearch Institute of the Presbyterian Hospitalin Philadelphia for allowing me to use some ofhis unpublished material (Figs. 4 to 6).

REFERENCES1. Frank, O.: Grundform des arteriellen Pulses,

Ztschr. Biol. 37: 483, 1899.2. Witzig, K.: Cber erzwungene Wellenbeweg-

ungen iiher incompressibler Fliissigkeiten inelastischen Rohren, Doctoral thesis, Univer-sity of Bern, 1914.

3. Womersley, J. R.: The mathematical analysisof the arterial circulation in a state of oscil-latory motion, Wright Air Development Cen-ter, Tech. Report WADC-TR 56-614, 1958.

4. Lambossy, P.: Oscillations forcees d'un liq-uide incompressible et visqueux dans un tuberigide, Helvet. physiol. acta 25: 371, 1952.

5. Streeter, V. L., Keitzer, W. F., and Bohr,D. F.: Energy dissipation in pulsatile flowthrough distensible tapered vessels, in At-tinger, E. O., editor: Pulsatile blood flow,New York, 1964, McGraw-Hill Book Co., Inc.

6. Evans, R. L.: A unifying approach to bloodflow theory, J. Theor. Biol. 3: 392, 1962.

7. McDonald, D. A.: Blood flow in arteries,London, 1960, Edward Arnold, Ltd.

8. Attinger, E. O., editor: Pulsatile blood flow,New York, 1964, McGraw-Hill Book Co., Inc.

9. Fry, D. L., and Greenfield, J. C : Mathe-matical approach to hemodynamics, in At-tinger, E. O., editor: Pulsatile blood flow,New York, 1964', McGraw-Hill Book Co., Inc.

10. Attinger, E. O., and Anne, A.: Simulation ofthe cardiovascular system, Ann. New YorkAcad. Sc. In press.

11. West, J. B., Dollery, C. T., and Naimark, A.:Distribution of blood flow in isolated lung,J. Appl. Physiol. 19: 713, 1964.

12. Johansen, J.: Regional distribution of circu-lating blood during submersion asphyxia in theduck, Acta physiol. scandinav. 62: 1, 1964.

13. Green, H. D.: Physics of the circulation, inMedical physics, vol. 1, Chicago, 1944, Year-book Medical Publishers Inc.

14. Kunz, A. L., and Coulter, N. A.: Pressure-flow properties of blood and glycerol duringdynamic oscillation in a rigid tube, Fed.Proc. 24: 334, 1965.

15. Fry, D. L., Griggs, D. M., and Greenfield,J. C : In vivo studies of pulsatile blood flow,in Attinger, E. O., editor: Pulsatile bloodflow, New York, 1964, McGraw-Hill BookCo., Inc.

16. Attinger, F. M. L.: Some problems concern-ing the application of the theory of elasticityto the blood vessel wall, Doctoral thesis, Uni-versity of Pennsylvania, 1964.

17. Attinger, E. O., Anne, A., and McDonald, D.A.: Use of Fourier series for the analysis ofbiological systems. Submitted for publication.

18. Mayerson, H. S.: Blood volume and its regu-lation, Am. Rev. Physiol. 27: 307, 1965.

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Physics of pulsatile blood floiu 987

19. Sugiura, T., and Freis, E. D.: Pressure pulsein small arteries, Circulation Res. 11: 838,1962.

20. Attinger, E. O.: Pressure transmission in pul-monary arteries related to frequency andgeometry, Circulation Res. 12: 623, 1963.

21. Pinkerson, A. L., Luria, M. H., and Freis, E.D.: Vena cava flows in normal and abnor-mal heart rhythms, Fed. Proc. 24: 278, 1965.

22. Friedman, E., Kopald, H. H., and Smith,T. R.: Retinal and choroidal blood flow,INVEST. OPHTH. 3: 539, 1964.

23. Friedman, E., Smith, T. R., and Kuwabara,T.: Retinal microcirculation in vivo, INVEST.OPHTH. 3: 217, 1964.

24. Bugliarello, C , Kapur, C, and Hsiao, G.:Profile viscosity and other characteristics ofblood flow in a nonuniform shear field, Tr.Fourth Internat. Rheology Congress, 1965,Interscience Publishers.

25. Cuest, M. M., Bond, T. P., Cooper, R. G.,and Derrick, J. R.: Red blood cells changein shape in capillaries, Science 142: 1319,1963.

26. McDonald, D. A., and Attinger, E. O.: Thecharacteristics of arterial pressure wave prop-agation in the dog. Submitted for publication.

27. McDonald, D. A., Navarro, A., Martin, R.,and Attinger, E. O.: Regional pulse wave ve-locities in the arterial tree, Fed. Proc. 24:334, 1965.

28. Bergel, D. H.: Arterial viscoelasticity, inAttinger, E. O., editor: Pulsatile blood flow,New York, 1964, McGraw-Hill Book Co.,Inc.

29. Hardung, V.: Propagation of pulse waves inviscoelastic tubings, in Handbook of Physi-ology, Circulation, vol. 1, Washington, D. C,1962, American Physiological Society.

30. Taylor, M. G.: Wave travel in arteries, inAttinger, E. O., editor: Pulsatile blood flow,New York, 1964, McGraw-Hill Book Co., Inc.

31. Wayland, H.: A rheologist looks at the micro-circulation, Third European Conference onMicrocirculation, 1964.

32. Conrad, M. C, and Green, H. D.: Hemo-dynamics of large and small vessels in pe-ripheral vascular disease, Circulation Res. 29:847, 1964.

33. Bargainer, J. D., and Milnor, W. R.: Influ-ence of heart rate on hydraulic power re-quired for pulmonary blood flow, Fed. Proc.24: 706, 1965.

the physics of pulsatile blood flow with particular reference to small vessels a1

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