pressure transmission in pulmonary arteries related to frequency

Pressure Transmission in Pulmonary ArteriesRelated to Frequency and Geometry

By Ernst O. Attinger, M.D.

• The difficulties in analyzing pulsatile pres-sure and flow in the circulatory system canhardly be overestimated. Although the so-called Navier-Stokes equation expresses thegeneral relation between pressure gradientand the resulting motion of a viscous New-tonian liquid, its exact solution depends uponthe choice of boundary conditions. In pulsa-tile flow fluid motion A'aries from point topoint and from moment to moment and musttherefore be described by a time-dependentthree-dimensional vector field. The presentstate of knowledge in the physical as well asbiological sciences is inadequate for an exactsolution since we are dealing with a non-New-tonion fluid, in a viseo-elastic system ofbranched, tapered tubes with multiple changesin geometry.

One of the characteristics of the behaviourof the vascular system is the dispersion of thepulse wave during its travel. A number ofattempts have been made to analyze this fea-ture Avith the help of analogues, an approachintroduced into hemodynamics by B. H.Weber.1 The progress of the engineering sci-ences led to a rapid extension of these meth-ods, with models ranging from rubber tubesto transmission lines and wave guides. Basedon a number of simplifying assumptions,Womersley2 has developed a coherent physi-cal theory of pulsatile flow and his analysishas been applied by McDonald,3'4 Taylor,5"7

Hardung,8- 9 Fry10 and others to the evalua-tion of experimental results obtained frommodels and from studies of the peripheral cir-culation .

From the Research Department of the PresbyterianHospital, Philadelphia, Pennsylvania.

Supported by U. S. Public Health Service GrantsH-5024 and H-6836 and by a V. S. Public HealthService Fellowship GSP-14037 from the Institute ofGeneral Medical Sciences.

Beceived for publication January 8, 1963.

Circulation Research, Volume XII, June 196S

Since little is known about the pulsatilebehaviour of the pulmonary circulation, thisstudy of the transmission of the pressurepulse in the pulmonary artery of the livingdog was undertaken. We will first present ashort outline of the basic theory and then ex-amine the experimental results against thisbackground.

THEORY

The theoretical analysis is based on a sim-plified model of a uniform vascular segment,the behaviour of which can be described bylinear differential equations. The justificationof basing such a model on a number of as-sumptions and approximations lies in the tre-mendous simplification of the analyticaltreatment. Once the basic usefulness of amodel has been shown, it can always be modi-fied and improved as new experimental databecome available.

The behaviour of the vascular system isdetermined by its physical parameters: 'mass,elasticity and viscosity. Neglecting secondaryflows, the fluid moves both in radial and axialdirections. The response of such a S3?stemto a driving force, such as that generatedby the heart, will be different at various fre-quencies, and this frequency dependence isdetermined not only by its individual compo-nents but also by the way in which they areinterconnected. The response of a linear sys-tem to any driving force can be predictedonce its frequency behaviour is known. Toanalyze the response of the system at various,frequencies, one can lump its parameters atone point, which means that the response toan excitation is at every instant identicalthroughout the system, i.e., one uses simpledifferential equations to describe the system.The Windkessel theory is a classical exampleof this approach. If the error introduced bythis approximation is not tolerable, one has

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to describe the model in terms of partial dif-ferential equations, where one thinks of thevarious parameters as distributed throughoutthe system. As an example, if we take a pul-monary artery of 10 em length, a heart rateof 120/min and a pulse wave velocity of 200cm/sec, we observe a phase difference betweenthe two end points of 36° for the fundamentalfrequency and 180°, or half a cycle length forthe fifth harmonic. Since such phase differ-ences are far from negligible for our pur-poses, we have to resort to an analysis usingdistributed parameters, i.e., all the variablesare a function of distance as well as of time.

The complicated form of pressure and flowpulse are not amenable to an easy analyticalapproach. However, any periodic functionwhich has at most a finite number of dis-continuities, minima and maxima (Dirichletconditions), can be represented as a linearsuperposition of sine waves, which are gener-ally more manageable in a mathematicaltreatment. Such a Fourier analysis is a per-fectly valid method of representing a singlepulse wave. However, the calculations basedon such an analysis require that the systembe linear. The vasculature is certainly non-linear in most of its properties; however, theerror introduced by assuming linearity isprobably less than the measuring error insuch experiments.2- *• ° The simple harmoniccomponents of the pulse wave are propagatedat a definite velocity and one has to considerthe relation between the shortest essentialwave length and the dimensions of the systemin deciding upon the method of analytical ap-proach. If the shortest wave length of thedisturbance is very large compared with themaximal dimension of the system acted upon,the maximal phase difference in the compo-nent wave between any two points must bevery small, and the wave can be consideredto have the same instantaneous value at allpoints in the system, i.e., it can be regardedas lumped at a point.

In order to get a clearer picture of wavemechanics, consider a water wave of simpleharmonic shape and travelling at a constant

velocity. A cork floating on the wave andrestrained from moving horizontally will beobserved to bob up and down with simpleharmonic motion. It is apparent that the ve-locity of propagation of the harmonic wave,the wave length (i.e., the distance from crestto crest) and the frequency with which thecork bobs up and down are not independent.In fact, the frequency with which the corkbobs is equal to the ratio of wave velocity overwave length. Hence, the wave, which is prop-agated at a definite velocity, is not only asimple harmonic function of distance, but ata fixed point represents a simple harmonicfunction of time as well. Such a behaviourcan be described mathematically by an equa-tion of the form :*

V 4 - l ^ = o (1)V- ot-

Equation 1 is a three-dimensional wave

equation, where P represents the pressurevector and V the wave velocity. The operatorV2 stands for the second partial derivativeswith respect to the coordinates of the quantityfollowing it. Separating the pressure vectorinto its three components in the direction ofthe three spatial coordinates, equation 1 canbe expanded into three scalar equations.

Since we are dealing with a cylindrical ves-sel, we choose the components Pr, P e and Pz

of the pressure vector in the direction of ra-dius, circumference and longitudinal axis ofthe vessel. If we now assume that the pres-sure is uniform throughout any given crosssection of the vessel, the radial component Pr

of the vector P is a function only of the axialcoordinate z and the time, t. This assumption,which is implicit in all present methods formeasuring blood pressure, is not strictly true,but is a good approximation for our purpose,since the radial dimension of the vessel is lessthan 2.5% of the wave length at the highestfrequencies of interest. Under these condi-

*Por details of the derivation the reader is referredto any textbook of electromagnetic field theory.11"13

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PRESSURE TRANSMISSION IN PULMONARY AETERIES 625

tions the solution of the wave equation be-comes :

P(z, t) = fi(z - vt) + f2(z + vt) (2)where P(z, t) is the amplitude of the pres-sure wave and f2 and f2 are arbitrary func-tions.

A function such as fi(z — vt) represents adisturbance travelling along the z-axis witha velocity v, while f2(z + vt) corresponds toa wave propagating in the negative z— direc-tion. The mathematical form of the functionsfi and f2 depends upon the type of disturb-ance creating the wave. For sine waves, theywill be sine or cosine functions of (z ± vt).

In the presence of friction losses we obtaina damping term in the wave equation :

3z2 V2" { 3t2 3t j (3)For a Newtonian fluid and laminar flow in

a linear system the damping term is directlyproportional to the viscosity of the conduct-ing medium. The friction losses result in anexponential decay of the wave as it propa-gates away from the source. The solution fora sinusoidal disturbance Po sin &>t, in the di-rection of the positive z-axis, becomes then

P(z,t) = Poe-«z sin(<ut - /3z) (4)This is more conveniently written in ex-

ponential form:P(z, t) =Poe-i"e)ut (5)

where y = a + j/8 = the propagation-constanta = damping constant/? = phase constant

For the case of the superposition of a down-stream and an upstream travelling wave, thepressure at any point is given by adding thesolutions for each wave. This results in anexpression of the form:

P(z, t) = Pie)ut— ̂ + P2e'"' + ^ (6)

where Pie1"' and P2eJajt represent the disturb-

ance creating the downstream and upstreamtravelling wave respectively.

The propagation constant is, of course, afunction of the physical properties of theconducting medium and is always a complexnumber. Its real part, a, the damping con-stant, describes the amplitude variation withdistance and its imaginary part, /?, the phase

constant, indicates the way in which the phaseof the wave changes, i.e., the velocity at whichthe transient disturbance is propagated.

The velocity at which a simple harmonicdisturbance travels is the phase velocity,mathematically described by:

where Vp = phase velocityAl = distance over which the phase

velocity is measuredA© = difference in phase angle be-

tween the two measurementsA. = wave lengthf = frequency

The phase constant fi is inversely propor-tional to the wave length, and is experimen-tally evaluated by measuring the phase dif-ference of the wave over a given distance.The damping constant a is roughly propor-tional to frequency.14

In a simple elastic system, the phase ve-locity is related to the physical properties ofthe system by the Moens-Korteweg equation:

V2 - — f«• )

where B = modulus of elasticity of the ves-sel wall

p = mass of blood per unit volumeR = vessel radiusd = thickness of vessel wall

Theoretically the determination of phasevelocity offers an easy experimental approachto the evaluation of the elastic properties ofa blood vessel in vivo. As Hardung14 hasshown, the values calculated for phase veloc-ity, in rubber tubes and arteries fit, in gen-eral, quite well with those predicted fromtheory, provided the stress and strain distri-bution is considered in evaluating the modu-lus of elasticity.

In contrast to the phase velocity, the group-velocity expresses the speed at which thecompound wave and hence the energy travels.Skilling13 gives the following analogy for thetwo velocities: the ripples along the back ofa caterpillar travel at the phase velocity,whereas the bulk of the beast moves at thegroup velocity. In general, the group veloc-

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626 ATTINGER

SUPERPOSITION OF INCIDENT AND REFLECTED WAVES ( TIME FIXED)

FIGURE 1

Superposition of incident and reflected pressurereaves proximal to a branch point of a vessel.full circles = incident waveopen circles = sum of incident and reflected wavestars = reflected tvavedotted lines = transmitted wavef is a damped sine function of (z ± vt)

ity would be expected to be close to the fa-miliar foot-to-foot velocity of the pressurepulse. Because the pressure pulse changes itsshape considerably along the arterial bed,points of equal pressure on the rising andfalling parts of the pressure curve will havedifferent transmission times and therefore thefoot-to-foot velocity is only an approximationfor the group velocity.

In addition to the propagation constant,the behaviour of a uniform segment of thevascular tree is determined by a second pa-rameter, the characteristic impedance Zo. Zo

relates pressure to flow at the entrance of aninfinitely long uniform vessel. In contrast tothe vascular resistance which, in the hemo-dynaniic literature, is defined as the ratio ofmean pressure to mean flow (and representsa real number), impedance includes the ef-fects of the visco-elastic and inertial proper-ties of the vascular system and its contents.Similar to the propagation constant, it ischaracterized both by a magnitude and anangle.

The relation between pressure and flow atthe end of the segment under discussion iscalled the load impedance. Similar to the

characteristic impedance, it contains visco-elastic, resistive and inertial components andis therefore a complex variable. The furtherout one goes in the arterial bed, the less be-comes the influence of the imaginary compo-nents visco-elasticity and inertia, and thecloser the load impedance approaches a pureresistance. Since we are dealing with a dis-tributed parameter network, each segment tobe considered is very small and its load im-pedance represents the input impedance ofthe following segment. Two consecutive seg-ments with different properties, in otherwords, mismatch between characteristic andload impedance, give rise to reflected waves,and the wave observed in the steady statewill be the super-position of the incident wavetravelling downstream and the reflected wave,travelling upstream.

Figure 1 shows a diagram of a blood vesselending in two branches. A steady state sinu-soidal pressure generator is assumed at thevessel entrance. The pressure wave drawnwith full dots at the bottom of the figure,travels for two wave lengths along the vesselto the right and is partly transmitted andpartly reflected at the branch point. Thedotted line represents the transmitted, thestarred line the reflected wave. The line char-acterized by open circles is the actual pres-sure wave upstream of the branching point,i. e., the measured pressure, which is the vec-tor sum of the incident and reflected wave.The amount and the phase of reflectionare determined by the relation between char-acteristic and load impedances. Since fric-tiona] losses occur, the amplitude of the in-cident wave decreases along the vessel towardthe branch point and the amplitude of thereflected wave decreases toward the origin ofthe vessel. Because of the frictional losses, notrue standing waves can be produced.

In figure 2 the same waves are shown againin form of a polar plot over the wave lengthjust prior to the branch point. Each waveis represented as a vector, i.e., by amplitudeand angle. Distance is expressed in terms ofangles, one wave length corresponds to 360°

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PRESSURE TRANSMISSION IN PULMONARY ARTERIES 627

VARIATION OF PRESSURE VECTOR ALONG AUNIFORM VESSEL ONE WAVE LENGTH LONG

1/2

A - LOCUS OF INCIDENT VECTORB - LOCUS OF REFLECTED VECTORg— LOCUS OF RESULTANT VECTOR

FIGURE 2

Polar plot of incident, reflected and resultant waveover one wave length prior to a branch point.Note the ellipsoidal form of the resultant loave andits non-uniform rotation rate as indicated by thecross bars at each eighth of a> wave length.P*z = pressure vector of incident wave at entranceof vesselp- = pressure vector of reflected wave at branchpointpo = pressure vector of resultant wave at vesselentrancePl = pressure vector of resultant wave at branchpoint

or a full rotation of the pressure vector. Thevector A representing the downstream travel-ling pressure wave starts on the real positiveaxis (zero phase angle) with a magnitude P+.It rotates uniformly clockwise over 360° dur-ing one wave length, decaying exponentiallyduring this travel because of friction-losses.The rate of decay and rotation are determinedby the attenuation and phase constant, andat the end of one rotation (z = I) its magni-tude is Y+e~al. The vector B represents thepart of the incident wave which is reflectedat the branch point z = I. In this diagramits magnitude P~ is 2/3 of the incident waveand its augle 50°. During its travel from the

branch point upstream B also decays expo-nentially and rotates uniformly clockwise.After travelling one wave length its magni-tude is P~ea!. Since the reflection coefficient

ACTUAL PHA5E OF PRESSURE

APPARENT PHASE VELOCITY

PHASE VELOCITY OF INCIDENT

REFLECTION SITE

DISTANCE FROM REFLECTION SITE

FIGURE 3

Change vti amplitude (top), phase (center) andphase velocity (bottom) of the incident and actualcompound wave along a uniform blood vesselat varying distances from a reflection site. With-out reflection the phase velocity is independentof distance "I". In the presence of reflections theapparent phfise velocity oscillates around the truephase velocity. The numbered circles refer to therelations expected between apparent and truephase velocity in the pulmonary vusculature onthe basis of a simple model (see text).

is different from unity and also has an angle,the incident and reflected wave differ both inamplitude and phase. The net pressure (Vec-tor C) at any point in the vessel is the vectorsura of the incident and reflected wave at thatpoint and its locus describes an elliptic spiral,which varies both in magnitude and in phaseas a function of distance. At z = 0 its valueis Po = P + + P~ coi, at the branch point itsmagnitude is Pi = P + e ~ a l + P~. Note thatalthough the incident and the reflected vectorrotate uniformly, the resultant vector does

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628 ATTINGER

not. This is indicated by the difference in arclength between the cross bars drawn at eacheighth of a wave length. Hence its phasechange over a wave length is not equal to360°. This means, of course, that the wavelength of the resulting wave is not exactlythe same as the wave length of the incidentwave. (This same fact is also apparent fromfigure 1.) As the phase angle increases alongthe vessel, the two waves will be alternativelyin phase and in phase opposition, and thusthe magnitude of pressure will ripple as afunction of distance. Since at any point thepressure is a sine function of time, the mag-nitude of the wave goes through maxima andminima, a quarter-wave length apart. Underthese conditions, the group velocity also var-ies with frequency and bears no longer a sim-ple relation to the actual movement of energy.

The relations between amplitude, phase andphase velocity of the incident and the actualwave composed of incident and reflected wavealong a uniform vessel are summarized in fig-ure 3. In the absence of reflections, the be-haviour is characterized by equation 5. At agiven moment the amplitude of the pressurewave decays exponentially and its phase lin-early along the vessel. As a result, the phasevelocity is constant throughout the course ofthe vessel. In the presence of a reflected wavethe behaviour is characterized by -equation 6,and amplitude, phase and phase velocity os-cillate as a function of distance around theirtrue value, namely the respective value forthe incident wave. I t is this true phase ve-locity and its related parameters, such as wavelength, which are of primary interest in he-modynamics. One has to arrive at some esti-mate of these values from the experimentallyobserved phase velocity which, over a giveninterval, depends upon the location of theobservation points and the distance betweenthem. If the measurements are made over asmall interval the apparent phase velocitywill be very sensitive to changes in the re-flection coefficient. If the apparent phasevelocity is determined at a distance of lessthan 1/4 wave length from the reflection point

it is always in excess of the true phase veloc-ity, regardless of the type of termination(open or closed). If the measurement is madeseveral wave lengths away from the reflectionpoint, the apparent phase velocity fluctuatesless and less about its true value. Hence wecan fix two limits for the value of the truephase velocity in the pulmonary vasculature.At low frequencies, where the length of thebed is considerably greater than a quarter-wave length, the measured phase velocity willbe in excess of the true phase velocity. Withincreasing frequency the observed phase ve-locity will approach the true phase velocityas an asymptote.

The phase velocity is not only an interest-ing parameter per se, but enters explicitlyinto the relations between pressure gradientand oscillatory motion of the blood, as wellas those between vessel-expansion and flowwithin. A better knowledge of its behaviouris essential for the understanding of the be-haviour of pulsatile flow. The physiologicalsignificance of pulsatile flow is often under-estimated, although a recent review summar-izes the evidence for its importance forcapillary circulation, kidney function andcellular metabolism.15

EXPERIMENTAL

MethodsThe experiments were carried out in ten mongrel

dog's, weighing between 15 and 20 kg. The ani-mals were anesthetized with jSTembutal, intubatedand artificially ventilated through a Harvardrespiration pump. After thoracotomy, two or threecatheters, made from Nylon tubing* (20 cm long,2.5 mm 0. D., 0.4 mm wall thickness), wereintroduced through small branches of the upperand lower lobe arteries, advanced to the desiredmeasuring points in the main, right or left pul-monary artery, and connected to Statham straingauges P23D. The results obtained with this systemwere compared in two experiments with thoseobserved simultaneously from two catheter tipmanometers (Dallonst catheters), placed at thesame measuring points as the Nylon catheters.

Measurements were made with the lungs either

*Polypenco Nylon tubing, DuPont Form 101, Mfg.by Polymer Corp., Reading, Pennsylvania.

tManufactured by Dallons Laboratories, El Se-gundo, California.

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inflated to a transpulmonary pressure of 25cm H20 or deflated (transpulmonary pressure 5cm H20). If necessary, suecinylcholine was usedto eliminate any respiratory movement during therecording periods, which lasted from 20 to 30seconds. Transpulmonary pressure was maintainedat a fixed level by applying the desired pressurefrom an oxygen tank through an overflow valveto the airways. The valve consisted of a glasstube, connected to a sideann of the oxygen lineand submersed either 5 or 25 cm below thewater level in a suction bottle.

In five animals the chest was closed and sealedafter obtaining the open chest measurements. ANylon tube, connected to a Statham P23D gaugewas left in the pleural cavity through whichintrapleural pressure was monitored. Care wastaken to expand the lungs fully before sealingthe chest wall in order to minimize the effectsof pneumothorax. In these cases the overflowvalve was adjusted for the value of intrapleuralpressure in order to obtain the desired transpul-monary pressure during the recording periods.

In several experiments, lobar arteries wereclamped as close to the origin as possible for10 to 20 seconds. In four experiments additionalcatheters were introduced through a small lobarvein branch and wedged into the venous sideof the capillary system. Through these cathetersan estimate of the pressure on the venous sideof the capillary system was obtained. This is sobecause only a small outflow segment is obstructedby the catheter, so that the latter transmits thedynamic pressure at the next upstream branchpoint for an alternate flow path. Weibel1" hasshown that such pathways exist on the venousside through the peribronchial plexus venosus.Although these experiments lasted several hours,no infarction was observed in the eannulatedsegments. Figure 3 in Lloyd's paper17 also indi-cates that appropriate drainage exists underthese experimental conditions.

The distances between catheter openings weremeasured at autopsy by threading a markedcatheter through the various vessels at a transpul-monary pressiire of 15 cm H2O. Since the vascularbed is somewhat longer in the inflated as eom-pai'ed to the deflated lung, our results overestimatethe phase velocity somewhat for the deflatedand underestimate it for the inflated lung. Thiserror is probably less than 5% because the changein vessel length is proportional to the cube rootof the change in lung volume.

All pressures were recorded simultaneously to-gether with an ECG, intratraeheal or transpul-monary and peripheral arterial pressure on an8-ehannel San born oscillograph at a paper speedof 100 mm/see. The overall frequency response

Circulation Reich, Volume XII, June 1963

of the sen sing-recording system was flat up to 30cycle/sec and without phase shift between individ-ual channels, as evaluated by dynamic calibrationwith a sine pump similar to the one described byTaylor.5 Since the distances involved and hencephase angles and transit times are much smallerthan in the peripheral vasculature, these frequencyrequirements are quite critical. Static calibrationwas carried out with a water manometer. Theobtained tracings were manually sampled at inter-vals of 0.01 second and the digitalized data sub-jected to Fourier analysis, using first an IBM650 and later an IBM 1620 digital computer.

The Fourier representation of the pressure pulsecan he written :

CO

p(t) = a0 + % (an c o s n t + K s i n n t) (9)n = 1

whieh can be rearranged to:

(10)

where: an = — J ' p (t) cos nt dt

bn = —. f p(t) sin n19t dt

\In addition to the sine and cosine coefficients thecomputer printed out the magnitude Cn and thephase angle 0 n . Initially this was done for thefirst 10 harmonics, establishing the frequency be-haviour between 0 and 30 cycle/sec, since the lattervalue represents the frequency of the tenth har-monic for a heart rate of 180. However, it wasfound that the harmonies above the sixth werewithin the noise level of the measuring system andthe analysis was consequently reduced to the firstsix harmonics.

Even the amplitudes of the fifth and sixthharmonies were often below 2 cm H2O. Sincewe were only able to measure pressures accuratelyto 0.25 cm H2O with our system, the error asso-ciated with the calculation of phase angles andphase velocities for the fifth and sixth harmonicwas between 10% and 20%. In contrast, themeasurement error for the first four harmonicsdid not exceed 5%. The revised computer programcontained also the calculation of phase angledifferences, phase-velocity and attenuation accord-ing to equations 6 and 7. At least three con-secutive pressure pulses were analyzed for eachrecording period. Two or three sets of measure-


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6 3 0 ATTINGEK

FOURIER ANALYSIS OF A PRESSURE CURVEIN THE PULMONARY ARTERY

FIGURE 4Actual tracing of a pressure pulse in the pul-monary artery and its reconstruction from fiveharmonics by Fourier analysis.

mcnts were obtained for each lung volume. Carewas taken to ventilate the dog properly betweenthe various recording periods. Running time ofthe computer was approximately three minutes perpressure pulse and more than 1000 pulses wereanalyzed in these ten animals.

In order to evaluate the geometry of the pul-monary arterial tree, vascular casts of 12 doglungs were obtained in situ both in the supineand prone position with the lungs either inflatedat a transpulmonary pressure of 25 cm H2Oor deflated at a transpulmonary pressure of: 5cm HoO. Batson's corrosion solution No. 17,* anicthylmethacrylate monomer which is polymerizedto the solid polymer at room temperature, wasinjected under a pressure of 25 cm H 2 0 . In theinfla ted lung, the solution usually did not passthrough the capillary bed, while the easts of thedeflated lung extended into the pulmonary venoussystem and'i!he left heart. The solution hardenedwithin 30 minutes and maceration of the prepa-ration was obtained by immersion in a solutionof potassium hydroxide (500 g KOH to 1000 gof H2O) for 24 to 48 hours. The smallest vesselsreproduced by the cast had an internal diameterof % mm. Shrinkage of the cast as tested in aseries of glass tubes was found to be 15% in vol-ume and 5.5% in diameter. All measurementswere made with a micrometer and are reporteduncorreeted for shrinkage.

•Manufactured \>y H. D. Justi and Son, Inc.,Philadelphia, Pennsylvania.

ResultsFigure 4 shows the fit between an actual

tracing and its reconstruction from the firstfive harmonics. An analysis of the figure re-veals that 97% of the harmonic content isassociated with the first two harmonics andmore than 90% with the first five harmonics.The mean square error of the fit was less than2% for the first four harmonics and decreasedonly 2/10% when two additional harmonicswere added. In other words, the first fourharmonics of the Fourier series represent anexcellent approximation of the pressure pulseand no significant improvement in the ap-proximation can be expected by the additionof a few more terms in the Fourier expansion.This most fortunate circumstance could havebeen predicted from the low amplitudes ofthe higher harmonics. In table 1 are listedthe results obtained from the analysis of 30pressure pulses in one animal. In this experi-ment three Nylon and two Dallons catheterswere placed into the pulmonary arterial bed.One of the Nylon and one of the Dallons cath-eters were introduced through a right upperlobe artery into the main pulmonary artery,the other Dallons and a second Nylon cathe-ter through a lower lobe artery into the rightpulmonary a r t e ^ 7.3 cm apart from the firstmeasuring point. The third Nylon catheterwas placed through a middle lobe artery intothe right pulmonary artery at a distance of5.6 em from the proximal and 1.7 em fromthe distal measuring point. Except for thesecond harmonic, the results in columns 3aand 3b are in excellent agreement. The bet-ter frequency response of the Dallons cathe-ters as tested by a sine pump was, in ourexperience, offset by baseline drifts and clot-ting problems, which did not exist in the othermeasuring system. The table shows that theapparent phase velocity varied as a functionof frequency and lung volume and was sig-nificantly larger in the periphery (column 2)as compared to the more proximal part of thepulmonary artery (column 1), both in theinflated and deflated lung. The relativelylarge standard error in column 2 for all fre-

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PHASE VELOCITY IN PULMONARY ARTERY (5D56I)

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FIGURE 5

Apparent phase velocity in the pulmonary arteryas a function of frequency. Measurements fromright pulmonary artery over a distance of 5.3 cm.

quencies and for the sixth harmonic in theother columns is due to the limitations of themeasuring systems with respect to sensitivityand measuring distance as discussed under' ' Methods.''

Figure 5 represents the data obtained inanother animal, with the catheters located inthe proximal part of the right pulmonaryartery, 5.3 em apart. The apparent phasevelocity is plotted as a function of frequency,both for closed and open chest and inflatedand deflated lungs. In all four conditionsthere is a marked frequency dependence ofthe apparent phase velocity, particularly atlow frequencies. At higher frequencies thevalues appear to approach an asymptoticvalue around 200 cm/sec. In general, theapparent phase velocity Mras larger for thedeflated lung as compared to the inflated lung.In contrast, a comparison between the dataof closed and open chest preparations showedno consistent trend.

Figure 6 shows the change in apparent

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CHANGE IN PHASE VELOCITY WITH

(50561, RT. PULM. ARTo>o= 3. 03 c ,LUNGS DEFLATED,FIRST 6 HARMONICS)

FIGURE &

Change in apparent phase velocity with distance. Each group of 6 bars representsthe phase velocities of the first 6 harmonics measured over the distance indicated onthe bottom diagram.

phase velocity with distance in the deflatedJung of the same animal. In this instance, thesecond catheter was withdrawn in steps tothe points marked on the diagram of theright pulmonary artery at the bottom of thefigure. Each set of six bars represents theapparent phase velocity for the first six har-monics over the indicated distance. The wid-est variations in phase velocity occurred overthe distances involving the major branchingpoints, A, C and D. For the inflated lung,the results were similar, except that the ap-parent phase velocity was lower and thevariation over the measuring distance C hadshifted to B. These marked oscillations ofapparent phase velocity with frequency canbe reduced by proper clamping of majorbranches. Figure 7 depicts the results fromone such experiment. With all the majorbranches of the left pulmonary artery except

two lower lobe arteries clamped, the phasevelocity varied only between 320 and 500 cm/sec as a function of frequency over the rangestudied. Unclamping of the various branchesdid not significantly alter the phase A'elocityof the first two harmonics, but introduced apeak velocity of 1050 cm/sec for the fourthharmonic.

The overall results obtained in all ten ani-mals are presented in table 2. The apparentphase velocity for the fundamental frequencyis in the order of 250 cm/sec for the inflatedand 325 cm/sec for the deflated lung. I t var-ies as a function of frequency with peak A'e-locities at the frequencies of the third andfifth harmonics.

The pulsations of the pressure pulse aretransmitted across the capillary bed. Figure8 shows the amplitude of the various harmon-ics of the pressure pulse in the pulmonary

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lu,

IPLITUDE OT PRE5SURE PUL5E

• ALL BRANCHES CLAMPED¥ LLL JNCLAMPED* LML, LLL UNCLAMPED

NCHEST. LUNGS DEFLATED

FIGURE 7Change in apparent phase velocity with frequencywith major branches clamped and undamped.

artery of the right middle lobe and from acatheter placed in a pulmonary venous wedgeposition in the same lobe. The first two har-monics are perfectly well transmitted into thevenous side of the capillary bed, particularlyin the deflated lung. For the inflated lung theamplitude is lower than in the deflated lung.This difference is most apparent for the pul-monary venous wedge pressure in the closedchest. Since the distance between the twomeasuring sites was not measurable in thisinstance, the apparent phase velocity couldnot be calculated. However, from the dataobtained, one can determine the attenuationor inverse gain of the pressure pulse as afunction of frequency. The results for theexperiment of figure 8 are represented infigure 9, where amplitude and angle of thetransfer function are plotted as a function offrequency in the complex attenuation plane.

The coordinates - of such a locus plot rep-resent the real and the imaginary part of thetransfer-function respectively. The frequen-cies investigated in this particular experimentrange from zero to 12 cycle/sec, the heart ratebeing 2 cycle/sec (fig. 8). Connecting the at-tenuation for the various frequencies in nu-merical order by a smooth curve results in alogarithmic spiral. If the pulse were trans-mitted without attenuation such a plot wouldrepresent a unit circle. For the nonpulsatile

Circulation Research. Volume XII, June 19CS

CLOSED CHEST

— PULM. ARTER1 *INFLATEOPULM. VENOUS WEDGE • £

FIGURE S

Change in amplitude of pulse pressure with fre-quency. Note the transmission of the first twoharmonics into the venous side of the capillarybed, particularly ivith the lungs deflated.

term (f = 0) the attenuation, is close to unityon the real axis. As the frequency increasesthe attenuation becomes progressively larger.Note that for the inflated lung the plot couldonly be extended to three harmonics becauseof the large amount of damping present. Onemust conclude that pulmonary capillary flowis less pulsatile during positive pressure infla-tion as compared to deflation. In contrast tothe pulmonary capillary bed, a plot of atten-uation against frequency in the pulmonaryartery forms an ellipsoid. Figure 10 repre-sents such an example recalculated from thedata shown in figure 5. The amplitude of thepressure pulse increases over the measureddistance for all frequencies studied except forthe fifth and sixth harmonies in the closedchest preparation, since all these points lie

TABLE 2

Average Phase Velocities (cm/see) in PulmonaryArtery. (10 dogs, open chest)

Harmonic

123456

Irl

255292445289411263

lflated

± 44± 44± 126± 107± 201± 54

Deflated

324 ±330 ±426 ±321 ±468 ±224 ±

7579

12185

20944


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634 ATTINGEE

FIGURE 9

Lotus plot of attenuation across the pulmonarycapillaries. Note the marked attenuation particu-larly at the higher harmonics. The coordinate axesrepresent the real and the imaginary axis of the

attenuation function J . The frequencies rangeG

from zero to 12 cycle/sec counter-clockwise (seetext).

within the \mit circle. For comparison, a lo-cus plot for data obtained simultaneously inthe thoracic aorta over a distance of 12 cm isshown at the bottom of the figure. In this casethere is a significant attenuation of all har-monic components except the first. The exactshape of such a plot depends not only on thevessel parameters but also on the amount ofreflections present, as Konnigeris pointed out.

The geometry of the larger branches of thepulmonary arterial tree is shown in figures11 and 12, and the data obtained from 12vascular casts are summarized in tables 3 to5. Note that the main pulmonary artery andits two major branches form essentially threeadjoining edges of a cube. The left pulmo-nary artery slopes caudally in a gentle curvewhile the right continues on a straight coursefor approximately 4 cm, over which the upper

i+ 1

THORACIC AORTA

FIGURE 10

Locus plots of inverse gain (attenuation) for rightpulmonary artery and thoracic aorta. Data ob-tained simultaneously with those of figure 5. Fre-quency range is 0 to 18 cycle/sec, counter-cloclc-ivise. Note thai in the pulmonary artery the pres-sure pulse increases over the measured distance forall frequencies (except 15 and 18 cycle/sec in theclosed chest). In contrast; the pressure pulse de-creases in the thoracic aorta over a> distance of12 cm, except for the first harmonic (see text).

lobe artery arises. It curves then suddenlycaudally with an angle of close to 45°. Sincewe were impressed by the ellipsoidal shape ofthe casts of the pulmonary artery, we meas-ured the longest diameter and, at a rightangle to it, a second (minor) diameter at thepoints marked on figure 12. The ratio of thesediameters was never unity, not even betweenbranching points within the investigated re-gion of the vascular tree. The cross sectionis therefore not circular and hence we usedthe approximation formula for the area ofan ellipse (-ab) for the calculation of thecross section, where a and b are the majorand minor semiaxes of the ellipse. This, andthe higher perfusion pressures employed byPatel et al.,19 may account for the fact thatour values are considerably lower than those

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TABLE 3Cross Section and Diameter Ratios of 12 Canine Pulmonary Arteries at MeasuringSites Listed in Figure 11 (mean and SE^

Site1

23

4

0

C789

101112

45.148.942.346.131.028.014.012.410.1

8.720.4

R

110.0± 5.4it 6.4i t 3.2± 3.3it 2,5± 2.6± 1.3± 1.5± .9± .7± 1.9

Area mmL

± 11.941,1 i52,4 ±40.0 it39.5 ±30.3 i26.6 i12,7 i10.4 i t

10.9 i19.2 ±

4.3

g.a4.04.03.89.22.1

.7

1.01.7

1.721.911.611.301.251.301.131.211.241.071.20

Diameter rK

1.246 itit .094± .16it .12± .08it .06i t .07it .03± .003± .03± .006it .05

atio f

L.0681.38 ±1.25 it1.30 i t1.38 ±1.22 i1.40 ±1.14 ±.1.17 ±.

1.14 ±1.28 it

.062

.05

.06

.08

.06

.07

.03.04

.05.08

TABLE 4

Cross Section Ratios at Branch Points and Lengths of Various Pulmonary ArteryB SBranches

Branch poir

(mean

it

and SE)

RD istance-cm

LCro

Koss section ratio

L

1

4 • 83

6 + B5

7 + 106

11 • 127

UL art.ML art.CL urt.LI™ art.XITJLI art .

4.0

6.6

6.9

8.5

14.115.415.518.315.7

± .2

it .2

it .2

it .3

± .7± .5± .6it .57± .7

2.4 ± .2

4.6 it .2

6.1 ± .2

12.8 it .514.7 ± .6

15.4 ± .613.9 ± .7

0.78

4.6 it.2

.95 1.03

1.22

1.04 1.13

*Branch points are designated as in figure 11. Distances are measured from pulmonaryartery valves either to the various branch points (first five rows) or to the end of thelongest lobar branch in each of the 12 oasts.

U L , ML, CL, LL refer to upper, middle, cardiac and lower lobes, respectively. 21 and 2rstand for left and right pulmonary artery, respectively.

reported by these investigators, although thecross section ratios of the sum of the twodaughter vessels to the mother vessel are sim-ilar for the first branch point, which was theonly one studied by them. Calculation of thecross section on the basis of a circular area,the radius of which was obtained by measur-ing the circumference, showed an average

Circulation Research, Volume XII, June 19GS

ratio of 0.82 between elliptic to circular crosssection, the largest difference being in theright main pulmonary artery (0.72). The spa-tial orientation of the major and minor axisvaried from segment to segment and is prob-ably influenced by the shape of: the adjacentstructures. Except for the right pulmonaryartery no significant difference in cross section


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636 ATTINGEE

RIGHT

FIGURE 11

Photograph of a> east of the pulmonary arterialtree.

or diameter ratio due to body position or lungvolumes was observed in our limited sample(table 5). Because of changes in flow andtransmural pressure such differences may ex-ist in the intact animal under dynamic con-ditions. Although the diameter ratio of theright pulmonary artery was significantlylarger in the supine as compared to the proneposition, the calculated cross section appearedto be independent of the bod3r position. Themarked flattening of the right pulmonaryartery in the supine position is apparentlydue to the weight of the niediastinal organs.The ratio of the combined cross section ofright and left pulmonary artery to the crosssection of the main pulmonary artery was lessthan unity, a finding already reported byPatel.19 The cross section ratios of the othermajor branch points (table 4) were ratherclose to unity and significantly less than the

LEFT

FIGURE 12

Schematic diagram of the pulmonary arterial treewith the measuring sites referred to in tables 3-5.

ratio of 1.26 postulated by Fleisch.20 Thevolume of the arterial tree was obtained fromthe cast weight and the specific weight of theeast material. The average volume was 24.3± 1.8 ml, which is in excellent agreement withthe values reported by Peisal et al.21 in theliving dog.

DiscussionWe can now attempt to compare the ex-

perimental results with those predicted fromtheory. The superposition of incident and re-flected waves results in a compound wave,with nodes and antinodes at intervals of aquarter wave length for each frequency. Ourresults show that the apparent phase velocityvaries both with frequency and distance, asthe theory predicts (figs. 5 and 6). However,the agreement between theory and experimentis only qualitative, as the following argumentshows. From the average heart rate of 120per minute and an average true phase velocityof 250 mm/see, as estimated from the extra-polation of the results in table 2 to an asymp-totic value, one obtains a wave length of 125cm for the fundamental and of 21 cm for the

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8

S3

J

1

+1 +1

+1 +1

+1 +1

1 I •%

Circulatum Research,

ai c5 ft ftVolume XII. June 196S

sixth harmonic. The arterial bed with anaverage length of 16 cm (table 4) represents,therefore, 1/8 of a wave length for the fun-damental and 3/4 of a wave length for thesixth harmonic. Under these conditions theapparent phase velocity in the investigatedsegments of the pulmonary vasculature shouldbe higher than the true phase velocity for thefirst three harmonies and lower for the fourth,fifth and sixth harmonics, as indicated by thenumbered circles in figure 3. This is clearlynot the case for our experimental results.

On the other hand, one can attempt to ex-plain the dispersion of the pressure pulse bydifferent propagation velocities and selectivedamping of the various harmonics, associatedwith mass and friction in the central part ofthe arterial bed.22 However, with this assump-tion the correspondence between theory andexperimental results is even worse. In con-trast to the pulmonary capillary bed, whereall harmonies are markedly attenuated (figs.8 and 9), there is no damping of the pressurepulse in the pulmonary artery (fig. 10). Asa matter of fact, all the harmonics increasein amplitude over the distance measured.Physically, there are only two possibilities toaccount for such an increase, namely the ad-dition of energy, which would require anactive system, or the superposition of incidentand reflected waves. In contrast to the pul-monary artery, the plot of figure 10 showssome damping for the higher frequencies inthe thoracic aorta. Similar data have beenreported by McDonald and Taylor3; however,these authors have also shown that the am-plitude of the higher harmonies begins to in-crease as the pulse wave approaches the pe-ripheral arterial bed.

The frequency dependence of phase velocityand input impedance has been demonstrateby a number of investigators for Hag periph-eral3-8' 23"26 as w e l l a s for t h e pulmonary ar-terial bed.27'2S It is difficult to see how theseresults could be interpreted Otherwise thanin terms of reflected waves. It is, however, notalways realized that the similarity betweena model of a stogie viscoelastic tube and the


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638 ATTINGEK

living vascular system with its multiplebranch points may be quite remote. Any ofthe models proposed describes only a shortand uniform vascular segment and cannot ac-count for the complex geometry of the vas-cular bed (fig. 11). As pointed out in thesection on theory, the input impedance ofeach such segment represents also the load ofthe previous segment and a mismatch at ajunction gives rise to reflections. Since thesesegments are quite short, reflected waves mustarise in very close succession and superimposethemselves upon the downstream travellingwave. Only if the number of reflection sitesor the reflection coefficients are small will aseparation of the various components be pos-sible. It is apparent that under these condi-tions the outlined theory is not refined enoughto allow a quantitative theoretical evaluationof experimental results in animals. It is,nevertheless, useful for the understanding ofwave propagation and reflection, as well asfor a qualitative interpretation of our results.The existence of multiple reflection sites doesnot invalidate the concepts of apparent andtrue phase-velocity, as applied to pressurepulses in the circulation, although it makestheir interrelation much more complex.

Some of the recent experimental results inthe peripheral circulation have been inter-preted as indicating that the terminations ofthe large vessels supplying pelvis and thighare the dominant reflection sites in relation tothe aorta.3'9i 24 "Woinersley2 calculated theamount of reflection to be 14% at the bifur-cation of the aorta and about 7% at the mul-tiple branching of the upper aorta. Taylor0

and Ronnigerls estimated the magnitude ofreflection at the termination of the femoralvascular bed at approximately 50% and con-eluded that the total reflection from the vas-cular bed is much greater than that from thesites of branching of major arteries. Our ex-perimental design was based on these calcu-lations. It is well known that if the iutra-alveolar pressure exceeds the pulmonarycapillary pressure, the capillaries collapse.Hence, inflating the lungs -#ith a positive

pressure higher than the mean pulmonaryartery pressure will transform the pulmonaryarterial tree to a state resembling a closedpipe while during deflation of the lungs thesystem should exhibit a more open-ended typeof reflection. Although there was a markeddifference with respect to transmission ofpulse pressure and attenuation through thecapillary bed under these two experimentalconditions (figs. 8 and 9). the effects upon thefrequency and distance dependence of theapparent phase velocity were considerablysmaller than those of clamping the majorbranches (figs. 5 and 7). It is impossible todetermine reflection coefficients on the basisof pressure measurements alone, but the data(figs. 6 and 7) indicate that the primary re-flection sites in the pulmonary vasculatureare represented by the major branch points.Our values for phase velocity are in goodagreement with the 250 cm/see which Caroand McDonald27 calculated from P a t e l 'data. They investigated the frequency be-haviour of the pulmonary arterial bed in therabbit and found the average phase velocityto be aboi;t 85 cm/sec at 6-11 cycle/sec, andconsiderably higher at lower frequencies. Therelatively stable value of their apparent phasevelocity at higher frequencies may be relatedto the longer interval (in terms of wavelength) over which the measurements weremade. Due to an averaging effect, the discrep-ancies between measured and true phase ve-locity tend to be reduced if the determinationis made over a long distance (fig. 3).

There are several reasons why Womersley'se

estimates for reflections in the peripheral cir-culation are not valid for the pulmonary vas-cular tree. He calculated the amount of re-flection as functions of cross section ratio andthe non-dimensional parameter a. The latteris uniquely determined by vessel radius, an-gular velocity and kinematic viscosity of theliquid. His plots show a minimum of reflec-tion for a cross section ratio between 1.1 and1.3. This minimum, which represents theoptimum for the energy transfer across thejunction, is associated with the higher of these

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values if the disparity in phase velocity be-tween the original artery and its branches islarge and vice versa. According to our data,the branch point of the main pulmonary ar-tery has a cross section ratio of only 0.78. Arough calculation following Womersley 's de-velopment, but neglecting the throttling effectdue to viscosity, yields a reflection coefficientof 55%. Womersley's calculation assumes acircular tube and equal mass loading of allbranches. Neither of these assumptions isvalid for the pulmonary vasculature (tables3 and 4), because of the absence of circularsymmetry. Since the total cross section of alllobar branches is only some 8% larger thanthe cross section of the main pulmonary ar-tery (table 3), the velocity of blood flow mustremain nearly constant over a large part ofthe pulmonary arterial bed. The kinetic en-ergy of the blood entering the small vesselsis therefore considerably higher than in theperipheral circulation and the energy lossassociated with multiple reflections could com-pensate for this to .some extent. For a cardiacoutput, of 2 liters/min the linear velocity ofsteady flow is aboiit 30 cm/sec in the majorpulmonary arteries, a value which has to bededucted from the measured phase velocity,since the latter represents the sum of truephase velocity and mean flow velocity.3

Furthermore, the presence of right anglesand changes in the spatial orientation of themajor axis at the branch point may alter thereflection pattern. Although at present nophysical theory is available for the wavepropagation through a medium of such com-plex geometry, it is not surprising that thelargest variations in phase velocity are en-countered in the initial segment of the pul-monary artery (fig. 6). The elliptic shape ofthe cross section is not due to gravitationalforces, as Rodenbeck29 seems to imply, sincethe spatial orientation of the major and minorserniaxes varies considerably between thevarious branches. The presence of an ellipticcross section has further implications. Ac-cording to the Laplace relation, wall tensionis proportional to pressure and wall thickness

Circulation Research, Volume XII. June 19 09

and inversely proportional to the vessel ra-dius for a circular cross section. For the el-lipse this problem is much more complicated,since the tangential stress is not only a func-tion of the radius, but also of the angle. Thisimplies either an unequal distribution of thestresses along the circumference of the vesselor structural non-uniformity. The details ofthe physical relations under these conditionsare to be published subsequently.

Barthel30 has estimated the Poiseuille re-sistance for an elliptic as compared to a cir-cular cross section. From his calculations theratio of the resistances under these conditionsis as follows:

b2

= 2>h3

(a2 + b2)where ^ = coefficient of viscosity

I = lengthr = radius of circular cross section

a + b = the major and minor semiaxes ofthe elliptic cross section

Hence, for a given pressure gradient, onewould expect a smaller flow rate in the ellipticvessel because of its larger flow resistance ascompared to the circular vessel. Only forsmall eccentricities of the ellipse is this dif-ference negligible.

Fry10 has reviewed the application ofhydrodjmamics to the living cardiovascularsystem and discussed the simplifying assump-tions which would make it possible to deriveinstantaneous velocity from the axial pres-sure gradient. Our results indicate that someof these assumptions may not be valid in thepulmonary artery. Consider first the inletlength.31 At the entrance of the vessel thevelocity distribution is uniform, although afunction of time. As the flow proceeds furtherdown the tube, this uniform velocity profilechanges its form gradually. Because of viscousdrag, the fluid particles close to the wall slowdown and in order to maintain the same rateof flow, the particles near the axis of the tubeaccelerate. These changes in the velocity pro-file continue over a certain distance, the inletlength, until finally the profile becomes inde-pendent of entrance-effects and we have de-


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640 ATTINGER

velofiing clthe pulthe flowFurther:at the firstbends in tin

Because of the multiple branch-heart, the straight part of

irtery is not long enough forJfoecome free of entrance-effects,

cross section ratio of 0.78ichpoint as well as the sharp

'oximal parts of the system willintroduce vortex systems and convective ac-celeration pressure gradients which can not beneglected. Changes in the spatial relations ofmajor and minor axes of the different seg-ments and their branches will introduce addi-tional complications in the relation betweenpressure gradient and flow velocity as pre-dicted by the Navier-Stokes equation. On thebasis of the presently existing theories themagnitude of these secondary effects cannotbe predicted, and most of these relations willhave to be worked out on more realistic modelsbefore a comprehensive physical theory canbe established.

SummaryThe propagation of the pulse wave through

the canine pulmonary arterial tree has beenevaluated by means of harmonic analysis. Theapparent phase velocity varied both with fre-quency and distance as one would expect inthe presence of reflected waves. The phasevelocity was about 230 cm/sec for the inflatedand 325 cm/see for the deflated lung. Theo-retical considerations as well as experimentalresults indicate that the major branches rep-resent the primary reflection site for thepulse wave. The pulse pressure is transmittedacross the pulmonary capillary bed, althoughat a high attenuation, and the transmissionvaries markedly with lung volume.

The dimensions of the pulmonary arterialtree were evaluated from casts. Data for thelength and cross sections of the variousbranches are given. The volume of the arterialtree as represented by the casts was 24 ml.The cross section of the various branches rep-resents an ellipse rather than a circle, and thecross section ratios of the major branch pointsis considerably less than those postulated foroptimal energy transfer. The implications ofthe geometrical findings for the relation be-

tween pressure gradient and flow are brieflydiscussed.

AcknowledgmentThe author is deeply indebted to Mr. J. Maginnia

for making the facilities of the Computer Centerat the Drexel Institute of Technology available forthis study. He also wishes to acknowledge theinvaluable assistance of Drs. J. Dickson, III, andS. Onodera during the experiments and of B. Pennock,M. S., M. Labance, E. E., and Dr. H. Sugawara inthe analysis of the results.

References1. WEBER, E. H.: cit. by Lambossy, P.: Aper§u

historique et critique sur le probleme desondes dans un liquide compressible enfermedans un tube elastique. Helv. Physiol. Phar-macol. Acta 8: 209, .1950.

2. WOMEESLEY, J. E.: An elastic tube theory ofpulse transmission and oscillatory flow in mam-malian arteries. WADC Tech. Eep. TE-56-614,

1957.3. MCDONALD, D. A., AND TAYLOR, M. G.: Hydro-

dynamics of the arterial circulation. Proc.Biophys. Phys. Chem. 9: 107, 1959.

4. MCDONALD, D. A.: Blood Plow in Arteries. Lon-don, Edw. Arnold Publ., 1960.

5. TAYLOR, M. G.: An approach to an analysis ofthe arterial pulse wave II. Phys. Med. Biol.1: 321, 1957.

6. TAYLOH, M. G.: An approach to the analysis ofthe arterial pulse wave I. Phys. Med. Biol.1: 258, 1957.

7. TAYLOR, M. G.: Experimental determination ofthe propagation of fluid oscillations in a tubewith a visco-elastic wall; together with ananalysis of the characteristics required in anelectrical analogue. Phys. Med. Biol. 4: 63,1959.

8. HARDTJNG, V.: Wellenwiderstand und Impedanzender geraden Sehlauchleitung. Arch. Kreislauf-forsch. 29: 77, 1958.

9. HARDDNG, V.: Physikalische Analyse der in derAorta des Hundes und ihren Hauptasten regis-trierten Druekwellen. Arch. Kreislaufforseh.37: 87, 1962.

10. FRY, D. L.: Certain aspects of hydrodynamicsas applied to the living cardiovascular system.IEE Trans, on Med. El.: ME 6: 252, 1959.

11. STRATTON, J. A.: Electromagnetic Theory. NewYork, McGraw-Hill Book Co., Inc., 1941.

12. PLONSEY, E., AND COLLIN, E. E.: Principles andApplications of Electromagnetic Fields. NewYork, McGraw-Hill Book Co., Inc., 1961.

13. SKILLETQ, H. H.: Fundamentals of ElectricWaves. New York, 2nd ed. John Wiley andSons, 1948.



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14. HABDUNG, V.: Propagation of pulse waves inviscoelastic tubing. Handbook of Physiology,Section I I , Vol. I Circulation, ed. by W. F.Hamilton and P. Dow. Baltimore, Williamsand Wilkins Co. 1962, p. 107.

15. WILKENS, H., REGELSON, W., AND HOFFMEISTER,

F. S.: Physiologic importance of pulsatile bloodflow. New Engl. J. Med. 267: 443, 1962.

16. WEIBEL, E.: Blutgefassanastomosen in der men-

schlichen Lunge. Z. Zellforschg. Mikroskop.Anat., Abt. Histochem. 50: 653, 1950.

17. LLOYD, C. T., AND WRIGHT, G. W.: Pulmonary

vascular resistance and vascular transmuralgradient. J. Appl. Physiol. 15: 241, 1960.

IS. RONNIGER, R.: Tiber eine Methode zur iiber-sichtlichen Darstellung hamodynamischer Zu-sammenhange. Arch Kreislaufforsch. 21: 127,1954.

19. PATEL, D. J., SCHILDER, D. P., AND MALLOS,A. J . : Mechanical properties and dimensionsof the major pulmonary arteries. J. Appl.Physiol. 15: 92, 1960.

20. FLEISCH, A.: Beziehung zwischen Stamm- undAstquersehnitt im Arterien system. Z. Anat.Entwicklungsgesehichte 64: 543J 1922.

21. FEISAL, K. A., SONI, J., AND DUBOIS, A. B.:

Pulmonary arterial blood volume in the dog.Federation Proc. 20: 106, 1961.

22. PETERSON, L. H.: Peripheral circulation. Ann.Rev. Physiol. 19: 255, 1957.

23. PORJE, I. G.: Studies of the arterial pulse wave,particularly in the aorta. Acta Physiol. Scand.13, Suppl. 42, 1946.

24. FARROW, R. L., AND STACY, R. W.: Harmonic

analysis of aortic pressure pulses in the dog.Circulation Res. 9: 395, 1961.

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der Pulswellengesehwindigkeit durch Reflex-ionen. Z. Biol. I l l : 367, 1960.

26. MALINDZAK, G. S.: Digital computer analysisof some transmission line characteristics ofthe mammalian arterial system. Dissertation.Ohio State Univ., 1961.

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of pulsatile pressure and flow in the pulmonaryvascular bed. J. Physiol. 157: 426, 1961.

28. CARO, C. G., AND HARRISON, G. K.: Observations

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ERNST O. ATTINGERPressure Transmission in Pulmonary Arteries Related to Frequency and Geometry

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