NOTES AND DISCUSSIONSThe downloaded PDF for any Note in this section contains all the Notes in this section.

Using physics to investigate blood flow in arteries: A case studyfor premed students

Michael A. Waxmana�

Department of Natural Sciences, University of Wisconsin–Superior, Superior, Wisconsin 54880

�Received 24 December 2009; accepted 14 March 2010�

Concepts from magnetism, hydrodynamics, and electrical circuits are used to investigate bloodcirculation and motivate premed students in their physics studies. © 2010 American Association of PhysicsTeachers.

�DOI: 10.1119/1.3379292�

Most premed students don’t like physics. Almost all of theauthor’s physicians over the years, having learned what sub-jects he teaches, have confided how challenging they foundtheir undergraduate courses in physics and physical chemis-try and how irrelevant those subjects seemed at the time.This attitude is not helpful, both in regard to the students’short-term goals �physics is an important component of theMCAT� and in regard to their later professional success be-cause physics is important for understanding the functioningof living organisms.

The challenges and importance of effectively teachingphysics to premed students have long been recognized �see,for example, Refs. 1–4�. I will describe a case study thatincreases the motivation of these students in introductoryphysics. I have also designed a series of activities wherestudents need to integrate their knowledge from differentbranches of physics and beyond and apply their knowledgein an unfamiliar context. These skills are crucial for theirsuccess on the MCAT.3 It will be shown that in the study ofblood circulation, concepts from magnetism, hydrodynamics,and electrical circuits are necessary and interconnected.

My starting point is a textbook problem5 on using a mag-netic field to determine the velocity of blood flow in an ar-tery. Before I assign the problem, I ask my students if theythink that our blood contains any particles that can bestrongly affected by a magnetic field. After some thought, afew students �particularly those who have already taken bio-chemistry� recall that there are ions in blood. We briefly dis-cuss their nature, particularly the fact that proteins arecharged particles, and that the buffer that stabilizes the pH ofblood at nearly 7.35 contains bicarbonate ions, HCO3

− .Monoatomic ions, such as Na+ and K+, are contained inblood as well.

After this discussion, problem 1 is assigned: “A heart sur-geon examines the flow rate of blood through an artery usingan electromagnetic flow meter �see Fig. 1�. Electrodes A andB make contact with the outer surface of the blood vessel,which has interior diameter 3.00 mm. For a magnetic fieldmagnitude of 0.0400 T, an emf of 160 �V appears betweenthe electrodes. Calculate the speed of the blood.”5 To solve

the problem, it is necessary to understand that the resulting

970 Am. J. Phys. 78 �9�, September 2010 http://aapt.org/ajp

emf is essentially the motional emf. As usual, the balancecondition between the magnetic and electric forces yields

� = �V/Bd , �1�

giving �=1.33 m /s, where � is the speed of the blood flow,�V is the emf, B is the magnetic field, and d is the innerdiameter of the artery. This problem is simple enough forsome of students to proudly produce the solution.

An interesting question to ask at this point is whether theeffect would be diminished by the presence in blood of bothnegative and positive charges. The answer is negative be-cause the magnetic field will deflect the positive and negativeions in opposite directions �positive ions upward, towardelectrode A, and negative ions downward, toward B�, withthe corresponding electric fields adding up.

It might seem to the students that the described device ishighly invasive because the electrodes in Fig. 1 appear topenetrate the artery. Actually, the electromagnetic flowmeters are implemented as catheters with the external diam-eter as small as 0.5 mm.6–9 Interestingly, the described in-strument is used to measure the velocity of fluxes not only inblood but also in various other conductive liquids, such aswaste water and slurry.10

True to my usual routine,11 the next step is to assign afollow-up problem for extra credit in a test: “A heart surgeonexamines the condition of an artery using two devices con-sidered in your recent homework problem. One device, at apoint X along the artery, has magnetic field magnitude of0.040 T, and an emf of 250 �V appeared between its elec-trodes. Another device, at another point Y along the sameartery, has magnetic field magnitude of 0.060 T, and an emfof 475 �V appeared between its electrodes. What quantita-tive conclusions, if any, can be reached about the conditionof the artery at points X and Y, if there is no blood vesselbranching between points X and Y?”

A basic understanding of fluid flow has been cited as theknowledge most lacking in medical students,2 and the solu-tions given by students to this bonus problem usually con-firm their lack of knowledge convincingly. Some students,for example, suggest that there is an unequal quantity of

blood flowing through points X and Y. In actuality for blood

970© 2010 American Association of Physics Teachers

treated as an incompressible fluid, the volume flow rates �must be equal due to the lack of branching between points Xand Y, which leads to the equality �see Ref. 5, p. 477�

� � �X��dX2 /4� = �Y��dY

2 /4� . �2�

Equation �2� yields

��XdX�dX = ��YdY�dY , �3�

and thus

�XdX

�YdY=

dY

dX. �4�

According to Eq. �1�, at each point, X and Y,

�idi = �Vi/Bi, �5�

where i=X or Y. We substitute Eq. �5� into Eq. �4� and find

dY

dX=

�VX

�VY

BY

BX. �6�

According to Eq. �6�, dY /dX=0.79, that is, the artery is nar-rowed by about 21% at point Y.

When discussing the test problems during the next classmeeting and after presenting the solution, I ask students ifthey believe that having plaque blocking the artery at somepoint so that its diameter is decreased by 21% seems verysignificant. Many students answer negatively, which makesthem quite surprised when I announce that the ability of theartery to support the blood flow is decreased by 61%. Indeed,in reality the velocity of blood across the cross-section of theartery is not constant because the blood speed is attenuatedby the walls. Because of the cylindrical symmetry of theartery, the resulting flow consists of a series of concentric,telescoping layers, with the central portion flowing most rap-idly and with the outermost layer stationary at the artery’swall. To determine the exact shape of this flow, let usconsider12 a cylinder of blood with a length L and a radius rcentered on the artery’s axis �see Fig. 2�. The pressure dif-ference, PA− PB �caused by the pumping action of the heart�,drives the blood, whose motion is opposed by the drag force,−��2�rLd� /dr, which acts on the surface of the cylinder ofradius r in the center of the artery �slowing it down through

Fig. 1. Geometry of Problem 1 from Ref. 5. Reproduced with permission,�www.cengage.com/permissions.

friction from the adjacent layer of blood which is closer to

971 Am. J. Phys., Vol. 78, No. 9, September 2010

the artery wall and therefore slower�. Here � is theviscosity,13 and � is the velocity of the blood flow at thedistance r from the center of the artery. We use Newton’sfirst law to equate these two forces for the stationary flow ofblood and obtain

�r2�PA − PB� = − 2�rL�d�

dr. �7�

We integrate from r to R �where R=d /2, the artery innerradius�, use the no-slip boundary condition at the arterywalls, ��R�=0, and find

PA − PB

2�L�

r

R

rdr = − ��

0

d� . �8�

Equation �8� yields the following expression for the velocityprofile across the artery’s cross-section:

��r� = −1

4�

�P

�z�R2 − r2� , �9�

where �P /�z= �PB− PA� /L is the blood pressure gradientalong the artery. The negative sign in Eq. �9� indicates thatthe direction of the flow is opposite to that of the bloodpressure gradient. Students can integrate the result of Eq. �9�for all the circular layers of thickness dr to find the resultingvolume flow rate13 of the blood,

� = �0

R

2�r���r��dr

=�

2�

��P��z

�0

R

�rR2 − r3�dr

=�

8�

��P��z

R4, �10�

which is the famous relation derived experimentally by theFrench physician Jean Léonard Marie Poiseuille.12–14

Equation �10� has several aspects worthy of discussion.First, the blood flow rate depends very strongly on the innerradius of the artery,15 ��R4, which yields a drastic decreasein the artery’s ability to pass blood even for a relatively smalldecrease in its diameter �61% decrease in the blood flow ratefor 21% reduction in the artery’s diameter in our example�.To pump the same amount of blood through the artery, our

Fig. 2. Flow of blood in artery. The drag forces shown act on the cylinderwith surface area 2�rL.

body would need to, according to Eq. �10�, increase the

971Notes and Discussions

blood pressure gradient �P /�z along the artery, which sug-gests a connection between the deposits of plaque in arteries�atherosclerosis� and hypertension.

A series of questions can be discussed at this point. Howmuch leeway is there for the pressure differential, and thushow much constriction can be compensated? What does thepressure differential �P along the artery have to do with theblood pressure measured in the doctor’s office? To fully an-swer the latter question, students need a simple introductionto the physiology and physics of circulation �see, for ex-ample, Refs. 13, 14, and 16�. Because the heart pumps bloodin pulses, the pressure in arteries varies between the maximaland minimal values �see Fig. 3�. An interesting aspect ofblood pressure measurements is the use of turbulence.13,14

Fig. 3. Blood pressure in various blood vessels of the circulation system�Ref. 16�. Reproduced with permission.

Fig. 4. Electronic circuit used to model blood circulation in

972 Am. J. Phys., Vol. 78, No. 9, September 2010

Constriction introduced by the inflatable cuff makes theblood flow become turbulent �full of eddies, swirls, andripples� and thus audible through a stethoscope �unlike thelaminar, smooth, and quiet flow,13 which is quantitativelydescribed by Eq. �10��. When the applied pressure from thecuff exceeds the heart’s output pressure, the artery collapses,and no noise of the flowing blood can be heard through thestethoscope. The corresponding pressure is called systolicpressure. Then the pressure in the cuff is slowly lowereduntil the sound of the turbulent blood flow becomes audibleagain �as the flow resumes� and then disappears �as the flowbecomes laminar� to yield the diastolic pressure.13 The aver-age of the systolic and diastolic pressures, the mean arterialpressure,16 is the pressure that propels the blood to the tis-sues �see Fig. 3�. We can see from Fig. 3 that in accord withthese arguments, the most rapid decline in blood pressure inthe human body occurs in the arterioles, not the aorta orarteries, because the arterioles’ smaller diameters increasethe resistance to blood flow.

We now discuss the final part of our case study, namely,electrical circuits. When my premed students learn �particu-larly, from the introduction to the textbook5� that this topic isone of the most important items for the physics section of theMCAT, they become upset. In their minds, knowledge ofelectrical circuits is important only for people who design orrepair instrumentation. In reality, many important aspects ofblood circulation �such as the functioning of pacemaker tis-sue in the heart�17 and the entire human circulationsystem18,19 can be successfully modeled using electrical cir-cuits. In these models, the current, voltage, charge, resis-tance, and capacitance of the electrical circuit correspond tothe flow rate, blood pressure, volume, resistance, and capaci-

human body �Ref. 19�. Reproduced with permission.

972Notes and Discussions

tance of the cardiovascular system19 �see Fig. 4�. The resis-tance of the cardiovascular system is caused by the bloodviscosity, elasticity of the blood-vessel walls produces ca-pacitance, and, in some models, inertia of the blood flowyields the inductance �see Ref. 18, pp. 41–77�. The param-eters of any element of the electrical circuit can be deter-mined, and this analogy allows modeling of pathologicalconditions such as left ventricular failure, aortic stenosis, orhypertension by adjusting the parameters of the circuitcomponents.19 Although a full-scale description of the entiremodel is not appropriate here, we can offer students a simpleproblem conveying the spirit of this model.

Problem 3. A portion of a certain artery in the human bodycan be normally represented in an electrical-circuit model asa 4.5 resistor. As a result of atherosclerosis, the innerdiameter of this artery has decreased from 6.0 to 4.0 mm,and a by-pass has been made, which is 5.0 mm in diameter.Assuming that the blood flow remains laminar, what shouldthe resistance of the corresponding element in the electrical-circuit model be changed to?

It should be clear to students that the resistance of a bloodvessel is inversely proportional to R4 �or d4�.18 Therefore, theresistance of the analog to the original artery increased to4.5 �6.0 mm /4.0 mm�4=23 , resistance of the by-passshould be 4.5 �6.0 mm /5.0 mm�4=9.3 , and the com-bined resistance of those two is 1 / ��1 /23 �+ �1 /9.3 ��=6.6 . The fact that the combined resistance of the dam-aged artery plus by-pass �6.6 � is still significantly higherthan the original resistance of the healthy artery �4.5 �,even though there are now two vessels working in parallel�instead of the single one originally�, with the diameter ofeach of the two in the new system �5 and 4 mm� almost thesame as that of the original healthy artery �6 mm�, againillustrates the strong dependence of the rate of blood flow onthe diameter of the vessel, ��d4.

I emphasize that my goal was not to introduce a rigorousdescription of blood flow in a magnetic field �or a full-scalemodel using electrical circuits�. Such a description requiressophistication much higher than can be expected from anaverage premed student �for a more sophisticated discussion,see Refs. 17–20�. Rather, my goal was to motivate thesestudents by demonstrating the relevance of the principles ofphysics to understanding how the human body works and toshow how important it is to integrate knowledge from differ-ent areas of physics to probe a single function of an organ-

ism.

973 Am. J. Phys., Vol. 78, No. 9, September 2010

ACKNOWLEDGMENTS

The author is grateful to Jim Lane and Myron Schneider-went, as well as to the reviewers, for comments that wereuseful in improving the clarity and the overall quality of themanuscript.

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7Some noninvasive tools for measuring the velocity of blood flow havebeen developed as well. Those include nuclear magnetic resonance flowmeters �Ref. 8� and ultrasonic instruments �Ref. 9�.

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15Qualitatively, the fourth power of R in the Poiseuille law occurs due tothe combination of two factors. First, in the vessels of larger diameter, thewalls are far from most of the blood. Thus the blood is free to flow morerapidly, yielding a greater average speed �av. Second, due to the greatercross-section of the larger vessels and because the total flow rate ����av�cross-section�, the difference between the flow rates for large andsmall vessels becomes even greater.

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field,” Phys. Fluids 17, 077103 �2005�.

973Notes and Discussions