Badeer, "Hemodynamics for Medical Students"

25:44-52, 2001.Advan in Physiol EduHenry S. BadeerHEMODYNAMICS FOR MEDICAL STUDENTS

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HEMODYNAMICS FOR MEDICAL STUDENTS

Henry S. Badeer

Department of Biomedical Sciences, Creighton University School of Medicine,Omaha, Nebraska 68178

The flow of blood through the cardiovascular system depends on basic prin-

ciples of liquid flow in tubes elucidated by Bernoulli and Poiseuille. The

elementary equations are described involving pressures related to velocity,

acceleration/deceleration, gravity, and viscous resistance to flow (Bernoulli-Poiseuille

equation). The roles of vascular diameter and number of branches are emphasized. In

the closed vascular system, the importance of gravity is deemphasized, and the

occurrence of turbulence in large vessels is pointed out.

ADV PHYSIOL EDUC 25: 4452, 2001.

Key words: streamline flow; volume flow; velocity; viscous flow pressure; kinetic

pressure; gravitational pressure and potential; vascular diameter; number of branches;

turbulence

Hemodynamics is the study of the relationship amongpressure, viscous resistance to flow, and the volumeflow rate (Q) in the cardiovascular system. Confusionand erroneous thinking have prevailed because of thefailure to distinguish the various sources of pressurethat may exist and their roles in driving the blood inthe closed cardiovascular system. Of particular im-portance is the influence of gravitational pressure incausing or hindering flow. Gravity can induce flow inan open system (e.g., from an open container filledwith liquid), but it does not cause liquid to flow in aclosed system such as the circulatory system (3).

The basic elementary equations related to the flow offluids are associated with two names, Daniel Bernoulli(Fig. 1) and J. L. M. Poiseuille (Fig. 2). Bernoulliconsidered, for simplicity, fluids that are nonviscousor frictionless (so-called ideal) in which pressuremay originate from two possible sources: pressuredue to the action of gravity on the fluid column(weight of fluid) and/or the pressure related tochanges in the velocity of the fluid (inertial forces).The former is commonly called hydrostatic pres-sure, which is often used indiscriminately to describe

other sources of pressure as well, such as pressurerelated to viscous resistance to flow. We will definethe pressure due specifically to gravity as gravita-tional pressure, in contrast to other sources of pres-sure. The pressure to increase velocity is described asaccelerative pressure, and the pressure resultingfrom decrease in velocity as decelerative pressure.

The Bernoulli equation is often written as follows:

I N N O V A T I O N S A N D I D E A S

1043 - 4046 / 01 $5.00 COPYRIGHT 2001 THE AMERICAN PHYSIOLOGICAL SOCIETY

VOLUME 25 : NUMBER 1 ADVANCES IN PHYSIOLOGY EDUCATION MARCH 2001

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The sum of the three components of the equationremains constant in a given nonviscous flow system.P is the accelerative or decelerative pressure of fluidif there is a change in velocity and/or is due to gravityif there is an elevation from a given reference plane. ris the density of the fluid, g is the gravitational accel-eration (9.8 m/s2), h is the height of the fluid from agiven reference plane, v# is the mean velocity of thefluid in a flowtube system (v# 5 volume Q/cross-sectional area of tube).

The second item (rgh) refers to the gravitationalpotential related to the position or elevation of thefluid from the given reference. The same is true for asolid mass above or below a reference plane.

The third term (rv#2/2) is the pressure due to themotion of the fluid. It is called kinetic or dynamicpressure. It cannot be recorded unless the fluid is

stopped or decelerated (hence it is also called im-pact pressure). This pressure or force is most obvi-ous in solids. For instance, to move a stopped car, agreat force is needed or stopping a fast moving carwill damage the car and the object involved in stop-ping by converting kinetic energy to force (or pres-sure). In the Bernoulli equation, all three items areinterconvertible. Acceleration of fluid converts P (lat-eral pressure) to rv#2/2; hence P falls. This is thewell-known Bernoulli principle in which increasingvelocity of fluid at a region drops the pressure at thatregion. It finds many applications in everyday life,e.g., lift of an airplane, garden sprays, flowmeters, etc.Conversely, deceleration converts rv#2/2 to lateralpressure. Figure 3 illustrates these conversions. Itshould be remembered that Bernoullis equation ap-plies to flow that is steady (not pulsatile) and laminar(or streamline) in which the incompressible fluidmoves as a series of layers. When flow is fully devel-oped, the molecules immediately adjacent to the walldo not move at all and those in the center movefastest. The velocity profile is parabolic. The average

FIG. 2.J. L. M. Poiseuille (17991869; reproduced, with per-mission, from Fishman AP and Richards DW. Circula-tion of Blood: Men and Ideas. New York: Oxford Uni-versity Press, 1964).

FIG. 1.(Reproduced from HYDRODYNAMICS by Daniel Ber-noulli and HYDRAULICS by Johann Bernoulli. Trans-lated by T. Carmody & H. Kobus from Latin. New York:Dover, 1968).



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velocity (v#) is equal to one-half the maximum velocityat the center of the circular tube.

The main shortcoming of the Bernoulli equation isthat it overlooks the viscosity of fluids. This wasstudied by a French physician named Poiseuille and aGerman named Hagen. With the use of glass capillarytubes (rigid) of uniform diameter and a homogeneousliquid (water) with streamline flow, Poiseuille foundthat the pressure gradient between two points (P1 2P2) varies.

1. Directly with the distance between P1 and P2 (L).Therefore,P1 2 P2 ~ L (constant Q, r, and h)where r is internal radius and h is viscosity ofliquid (Fig. 4A).

2. Directly with the flow rate, Q (Fig. 4B).P1 2 P2 ~ Q (constant L, r, h)

3. Directly with the h (Fig. 4C)P1 2 P2 ~ h (constant L, Q, r)

4. Inversely with the 4th power of radius (Fig. 4D)

P1 2 P2 ~1

r4(constant L, Q, h)

Therefore, P1 2 P2 ~ LhQ

r45 k Lh

Q

r4

k was found to be8

p

So

P1 2 P2 58

p

LhQ

r4or Q 5

~P1 2 P2!pr4

8Lh(2)

This relationship is known as Poiseuilles equation orthe Hagen-Poiseuille equation. It can be derived alsomathematically. It should be emphasized that P1 2 P2in the Poiseuille equation is strictly limited to theviscous loss of pressure in a flowtube, excluding thepressures shown in the Bernoulli equation. In otherwords, it excludes the difference in the gravitationalor accelerative-decelerative pressures that may existbetween the two points in a tube system. To empha-size its importance, we refer to this pressure as the

FIG. 3.Model showing a rigid tube with a narrow segment in the middle toillustrate the conversion of accelerative-to-kinetic and kinetic-to-decel-erative pressure (P) in a steadily flowing nonviscous liquid (no fric-tional heat). However, the principle applies equally well to viscousfluids. Thus, when velocity increases, lateral (side) P drops and viceversa (Bernoullis principle). v# , Mean velocity.



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viscous flow pressure (1). Viscous flow pressuredrops continuously along a flowtube because of fric-tional heat between the particles of the fluid. Thisheat cannot be reconverted into pressure (consideredlost).

In a single rigid tube of uniform diameter with thesteady flow of a homogeneous fluid, one can calculatethe Q if P1 2 P2, L, r, and h are known.

Viscous resistance to flow. It is customary to referto 8Lh/pr4 as the resistance (R) to flow between twopoints in a single flowtube. Hence, the Poiseuilleequation may be generalized as P1 2 P2 5 RQ or R 5P1 2 P2/Q. If Q is constant, P1 2 P2 may be taken toindicate R. This equation is analogous to Ohms law inelectrical circuits, E 5 IR (volts 5 amperes 3 ohms)or I 5 E/R. The general equation is applicable alsowhen tubes divide and subdivide and then reunite, as

FIG. 4.A: pumping a homogeneous viscous liquid at a steady rate through a rigid tube of uniform diameter. Thelateral Ps drop along the tube, indicating the viscous flow P gradients. The perfusion Ps between thedifferent points vary directly with the distance between the points (L). Drop of P is due to friction (heatloss). B: keeping L, viscosity (h) and radius (r) constant, and doubling the flow rate (Q) doubles theperfusion P (P1 2 P2). C: keeping Q, L, and r as in A, and doubling the h doubles (P1 2 P2). D: effect ofdiameter on viscous flow P gradient. Reducing the diameter to one-half increases the pressure gradient16 times, other things being as in A.



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it is in the circulatory system. The branches are tubesthat are placed in parallel, and the total R (RT) ofsuch a flow system is similar to the equation in elec-trical circuits

1

RT5

1

R11

1

R21

1

R3(3)

and so on.

If flow is kept constant, the more the branches, theless is the RT or the pressure drop (Fig. 5). Anotherconsequence of parallel tubes is that the RT is lessthan any of the individual R. For example, if theresistances of the four parallel elements in Fig. 5Cwere equal, then R1 5 R2 5 R3 5 R4. Therefore,1/RT 5 4/R1 and RT 5 R1/4.

The influence of number of vessels is an importantprinciple to remember in the circulatory system. Agood example is the very small pressure drop in therenal glomerular capillary network that consists of2040 capillaries in parallel. Figure 6 represents thepressure drop in the systemic vascular circuit. It isseen that the arterioles are the chief resistant vessels,

FIG. 5.Other things being equal (Q, diameter, L, and h), viscousflow pressure gradients vary inversely with the numberof parallel tubes (reproduced, with permission, fromH. S. Badeer and D. H. Petzel. Am J Nephrol 15: 95, 1995).

FIG. 6.Viscous flow pressures in the systemic vascular circuit. Note, the greatest dropof P (P1 2 P2) is in the arterioles, which represent the chief resistance vessels.Next largest drop is in the capillaries. There is very little viscous resistance inthe large arteries and veins. [Reproduced, with permission, from A. C. Guyton.Textbook of Medical Physiology (7th ed.). Philadelphia PA: Saunders, 1986).



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which is explained by the smaller number and greatereffective viscosity of blood in arterioles comparedwith that of capillaries. Whereas in a single tube withhomogeneous liquid flow, it is possible to calculateresistance by applying the equation 8Lh/pr4, in acomplex vascular network, it is impossible to use thisequation. One has to resort to the general equation,R 5 P1 2 P2/Q. If Q is constant, P1 2 P2 indicates theR (in mmHg z ml21 z min21 or peripheral resistanceunits) of the system.

If resistances are placed in series, the RT is equal tothe sum of the individual Rs. Some authors prefer touse the reciprocal of R, which is designated as con-ductance. Conductance 5 1/R.

Bernoulli-Poiseuille equation. It was pointed out ear-lier that Bernoullis equation is incomplete because itoverlooks the viscosity of fluids (h). Likewise, Poi-seuilles equation is incomplete because it neglects grav-itational pressure and gravitational potential as well asaccelerative-decelerative pressures. The more realistic

equation is a combination of the two, which has beendesignated as the Bernoulli-Poiseuille equation (1).

Figure 7 illustrates all of the components of this equa-tion in a graphic way. Flow of liquids is due to thegradients in total pressure or energy (PDV; excludingthermal) that takes into account the various sourcesof pressure or energy.

In the circulatory system, the influence of kinetic,compared with viscous flow pressure, is relatively

FIG. 7.Model illustrating all the components of the Bernoulli-Poiseuille equation.(Reproduced, with permission, from H. S. Badeer. The Physics Teacher 32:426, 1994).



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small because of its magnitude. It is significant in afew areas of the circulatory system. For instance, itplays a role in driving blood from the left ventricleinto the aorta during the latter part of the period ofejection (see cardiac cycle). Also, it plays a role in thepulmonary circuit and the filling of the ventricles withblood from the atria (2). In other parts of the circu-lation, the contribution of velocity of flow (hencekinetic pressure) in driving blood is negligible.

Although normally in the circulatory system, the ac-celerative-decelerative and kinetic components ofpressure are not very significant, gravitational pres-sure and gravitational potential of blood become sig-nificant in the upright position of man and animals. Ifa flowtube with an open aperture is directed upwardor downward (open system), gravity hinders upward

flow or facilitates downward flow. If the liquid ispumped upward and is discharged to a higher gravi-tational potential, the pump develops pressure toovercome gravitational pressure (rgh) and to over-come the viscous resistance of the tube (QR). It liftsthe liquid to a higher gravitational potential (rgh; Fig.8A).

If now the same tube is bent down as shown in Fig.8B, the liquid is returned to its original level of grav-itational potential. This design eliminates the work ofthe pump to overcome gravitational effects (pressureand potential), because the gravitational effects in thedownward limb counterbalance the gravitational ef-fects in the upward limb of the circuit (counterbal-ancing principle; Fig. 9). The pump now developspressure to overcome the viscous resistance of the

FIG. 8.Model showing the reduction of work of pump in an inverted U tubesystem. In position (a), the pump develops P to overcome the gravita-tional P (rgh) in addition to viscous P and kinetic P. In position (b), thegravitational effects in the upward limb are counterbalanced by thegravitational effects in the downward limb of the loop, leaving only theviscous P gradient (QR) and the kinetic pressure [r(v#2/2)].



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tube and impart velocity. This may be considered tobe a closed system.

A similar situation exists in the circulatory system,where blood is pumped out of the ventricles and isreturned to the ventricles by way of the veins andatria in a U or inverted U design below and above theheart (closed system). In other words, gravity neitherfavors nor hinders blood flow in the upright posi-tion. This important principle in hemodynamics isoften overlooked or misunderstood. To illustrate, letus say that in the upright position of a person, themean pressure in the right atrium is 2 mmHg and themean pressure in the dorsal vein of the foot is 90mmHg. Most of the 90 mmHg is due to the pressure ofthe column of blood from the foot to the right atrium(rgh). If this gravitational pressure is ;85 mmHg,then the driving or perfusion pressure from the footto the right atrium is not 902 mmHg, but is (9085)-2 5 3 mmHg. In other words, perfusion pressure

excludes the difference in gravitational pressure be-tween two points and refers only to the viscous flowpressure gradient. Although the model in Fig. 8 refersto a rigid tube system, in a collapsible tube systemsuch as the circulatory, the same principle appliesexcept that in the upright position, gravitational pres-sure alters the diameter of blood vessels above andbelow the heart. Gravitational pressure drops abovethe heart and increases below the heart. Accordingly,vessels above the heart tend to collapse, and vesselsbelow the heart distend. The veins being more com-pliant are affected most (compliance is the ratio ofchange of volume to change of pressure). Venousreturn from the lower parts of the body tends to bereduced and causes a drop of cardiac output andarterial pressure. There are important protectivemechanisms to maintain arterial pressure and bloodflow to the head in the upright position (see carotidsinus reflex).

Students who are interested in experimenting mayuse the model described by Smith (5).

Distinction between volume Q and velocity offlow. It is important to distinguish between volumeQ and velocity of flow. Q is volume per unit time,whereas velocity is distance per unit time. By defini-tion, v# 5 Q/cross-sectional area, or in a circular tube

v# 5Q

pr2(5)

If we substitute for Q from the Poiseuille equation weobtain

v# 5 ~P1 2 P2!pr4

8Lh3

1

pr25 ~P1 2 P2!

r2

8Lh(6)

Consequently, if P1 2 P2, L, and h are kept constant,increasing the radius increases the velocity of flow.This may appear paradoxical because increasing rincreases the cross-sectional area, and this might beexpected to decrease the velocity. That would be trueif Q were to remain unchanged; but we see from thePoiseuille equation that when r increases and P1 2 P2is kept constant, Q would increase very markedly (byr4). Because flow increases by r4 and the cross-sec-

FIG. 9.Model showing the counterbalancing of the rgh in theneck arteries by the equal and opposite rgh in theneck veins. Arrow with a plus (1) sign, the directionof increasing P; arrow with a minus (2) sign, thedirection of decreasing pressure. (Reproduced, withpermission, from H. S. Badeer. Comp Biochem Physiol118A: 575, 1997).



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tional area increases by r2, velocity, which is flowdivided by cross-sectional area, increases by r2(r4/r2).

This situation occurs in the body when blood vesselsdilate locally without causing any change in meanarterial blood pressure. Both blood flow and velocityincrease during local vasodilation in a tissue and viceversa.

Turbulent flow. In contrast to streamline flow, tur-bulent flow occurs when fluid particles move ran-domly in all directions in a flowtube or vessel. Suchflow requires more energy or pressure than stream-line flow. Osborne Reynolds (18421916) studied thisin a model and established that turbulent flow occurswhen a number called Reynolds number exceeds2,000. It is calculated as follows: Reynolds number 5dv# (r/h) (dimensionless), where d is the internaldiameter of tube (cm) and r is density.

From this equation, it is seen that large tubes withhigh velocity of flow tend to develop turbulence.Aortic and pulmonary artery flow during ejectiontend to develop turbulence that contributes to thefirst heart sound.

In disease, turbulence occurs when blood flowsthrough narrow cardiac valves at high velocity orwhen arteries are narrowed locally (e.g., carotid ar-tery). Vibrations are produced that may be heard with

a stethoscope. Turbulence also occurs when bloodflows in opposite directions (regurgitation through acardiac value). These abnormal sounds over the heartare known as murmurs (4).

Turbulent flow occurs when one is taking arterialpressure with the cuff around the arm and stetho-scope at the elbow. When cuff pressure is reducedslowly, the jet of blood passing under the cuff witheach heart beat flows through the stationary bloodbelow the cuff and causes vibrations of the artery thatcan be heard with a stethoscope (Korotkoff sounds)(4).

Address for reprint requests and other correspondence: H. S.Badeer, Dept. of Biomedical Sciences, Creighton Univ. School ofMedicine, Omaha, NE 681780405.

Received 3 May 2000; accepted in final form 7 December 2000

REFERENCES

1. Badeer HS and Synolakis CE. The Bernoulli-Poiseuille equa-tion. The Physics Teach 27: 598601, 1989.

2. Burton AC. Physiology and Biophysics of the Circulation. (2nded.) Chicago: Year Book Medical, 1972, p. 104106.

3. Hicks JW and Badeer HS. Gravity and the circulation: openvs. closed systems. Am J Physiol Regulatory IntegrativeComp Physiol 262: R725R732, 1992.

4. Rushmer RF. Cardiovascular Dynamics (4th ed.) Philadel-phia: Saunders, 1976, p. 184 and 433.

5. Smith AM. A model circulatory system for use in undergraduatephysiology laboratories. Am J Physiol Adv Physiol Educ 22:S92S99, 1999.



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Badeer, "Hemodynamics for Medical Students"