Med. & bioL Engng. Vol. 6, pp. 627-636. Pergamon Press, 1968. Pr inte d in G r e a t Britain

THE EFFECT OF ARTERIAL WALL THICKNESS AND CONDUCTIVITY ON ELECTROMAGNETIC FLOWMETER READINGS* ROBERT H. EDGERTONt:~ Assistant Professor of Engineering, Dartmouth College, Hanover, New Hampshire, U.S.A. Abstract--The performance of electromagnetic flowmeters is reviewed in the light of data concerning the tensor conductivity of arterial walls. The muscular structure and its effect on external contttct flowmeters is discussed. The situations where the wall effects must be considered are outlined and experimentally demonstrated. Maxwell's equations for the external contact electromagnetic flowmeter are solved considering the difference in conductivity of the wall radially and angularly. The importance of the blood conductivity to the wall conductivity ratio is also demonstrated. It is shown that for situations where the ratio of inside diameter to outside diameter of an artery changes during a flowmeter test, the sensitivity of the flowraeter changes. This is shown to be an important consideration in research studies where vasodilation and vase-constructors are used. The applicability of in vitro flowmeter calibrations to in viva measurements is also considered. 1 2 3

LIST O F SYMBOLS a b B E j K

insideradius of the artery outsideradius of the artery magnetic flux density electricalfield density current density electrical conductivity of the wall k~ n2 -kl r radial coordinate Uw S 2bBVm U electrical potential AU potential difference between electrodes V fluid velocity Z axial coordinate permeability electrical conductivity of the fluid 0 angle relative to the B field contact resistance between wall and fluid second order tensor - - vector Subscripts w wall f fluid m mean

radial tangential axial 1. I N T R O D U C T I O N

THE ELECTROMAGNETIC flowmeter designed to measure blood flow in arteries has been utilized in many studies both clinically and in experimental research. In the use of this apparatus the calibration is usually performed on an excised artery using either blood or saline and often in a saline bath. This procedure will not be satisfactory in situations where the arterial wall properties and thickness change. These situations may in particular occur when drugs are introduced producing vasodilation and vasoconstriction in an active vascular bed. KoLnq (1960) points out that the thickness of the wall of the artery does not effect the flowmeter reading as the conductivity of the wall is nearly the same as that of the blood. This may

* Received 16 March 1968; in revised form 22 May 1968. t Present address: School of Engineering, Oakland University, Rochester, Michigan 48063, U.S.A. ,+ This work was supported in part by Research Grant HE~8771 from the National Institutes of Health, Bethesda, Maryland. 627 D

628

ROBERT H. E D G E R T O N

be erroneous if the arterial wall has the properties noted by BURGERand VAN DONGBN (1960) who have shown that the electrical conductivity along and across muscle tissue may be considerably different. They show that the conductivity along the fibers may be roughly ten times that across the muscle fibers. If this is true, (measurements are now in progress to determine more accurately this situation), a correction must be made in flowmeter readings. Similarly, if the calibration is made in a situation where saline in a bath is allowed to permeate the muscle wall the conductivity will be different than in the in vim situation and an error in the calibration may be observed. Difficulties in accurate flow measurement and adequate calibration will be encountered primarily in studies where the thickness of the arterial wall is changed considerably. These situations occur primarily in the smaller arteries which are more muscularly active. These effects become very important in flow measurements on a strongly reacting vascular system such as the gastric bed. In these situations several effects can occur which may alter flowmeter readings; (1) a pressure rise in the circulation can cause a thinning of the wall of the artery, (2) a vaso constriction of the artery itself may thicken the wail and, (3) chemical addition can alter the conductivity of the walls relative to the blood. The effects of wall property changes on the sensitivity of electromagnetic flowmeters have been considered experimentally by both KOLIN (1960) and SPENCER and DENISON (1960). In these experiments, however, chemical and thickness effects were not effectively separated. The data in the experiments noted does not represent the range of conditions which could be encountered in research on small vessels nor those in which the wall thickness is altered appreciably. SPENCER and DENISON note that placing a vein within an artery and thus doubling the wall thickness does not change the sensitivity. They state that balancing effects are present which cause this to be true. The data available in their paper is not sufficient, however, for

analyzing what the balancing effects which occur might be. It might also be noted that most researchers recommend a calibration in vivo at the completion of an experiment, This is not an acceptable procedure for several reasons. First, the wall properties and thicknesses may change, or be altered, during an experiment and, second, the distention and muscular state of the wall will, in general, not be the same if pressures are not also simulated. In the following analysis the tensor conductivity properties of arterial walls is considered and the sensitivity of flowmeters to wall thickness and property variations examined. 2. ANALYSIS

The usual electromagnetic flowmeter used in medical research is of the transverse field type (Fig, 1).

FIG. 1. Electromagnetic flowmeter schematic.

The magnetic field is perpendicular to the flow axis of the tube and an e.m.f. V X Jg, is induced by the flow. Electrodes on the outside of the tube perpendicular to the B field are connected to a voltmeter circuit and the change in voltage with flow is recorded. Ohm's law may be written for this situation as

.~ = ~- (.~ + ~ x .~),

(1)

where ~ is the tensor conductivity of the medium and j-is the current density. E -k ~" X B is the electric field relative to the moving fluid, and is the field due to charges in space.

ARTERIAL WALL THICKNESS ON ELECTROMAGNETIC FLOWMETER READINGS Most commercial flowmeters use a periodic magnetic field which contributes an induced voltage which is in quadrature with the flow signal. Since this is a source of error many electronic techniques have been utilized to minimize it. In any event the signal is conditioned electronically so the flowmeter is to read the voltage V" X B as if it were a steady field situation. Let us then consider Maxwell's equations as those which apply in the steady magnetic field situation Curl ~ = 0,

(2)

Curl B = ~t).

(3)

and The electric field E may then be expressed as the gradient of a potential = -- grad U.

(4)

From (3) one can show that d i v ] = 0,

(5)

or that there is no accumulation of charge in the media. From (5) and (1) the equation governing the behavior of the electromagnetic flowmeter is div [~(E + ~" X B)] -= div [~ grad U + ~(iz X B)I = 0.

(6)

The following assumptions are now made concerning the conductivity tensor ~. (1) The flowing fluid is homogeneous so that 7~ = const = e for the blood. (This is not strictly correct and the effect of this will be examined in a later paper.) (2) The conductivity of the wall may be considered as a second order tensor with principle conductivities: kl (radially), k2 (tangentially), ks (axially). This assumption is made from an examination of the structure of the walls of arteries. (The magnitude of these conductivities is being experimentally determined at present. Typical expected values are given later, assuming muscle conductivities.) The potential problem to be solved then reduces to the usual equation for the potential

629

in the fluid (SHERCLIFF,1962). (The potential is divided into a potential Uw in the wall and Ut in the fluid.) W UI

=

B

0V ~-r sin 0.

(7)

The potential problem for the wall becomes:

k, 0( r Jr

OVw r Jr/

k, O2Vw +r ~

00 ~

0.

(8)

The magnetic field B does not appear in equation (8) because there is no axial flow of fluid in the walls. This is written in cylindrical coordinates (r, 0) and the assumption is made that there is no variation in Uw in the axial, Z direction. (This is not true in general and end effects will provide a further correction.) The Boundary conditions to be applied will be assumed as follows: (1) At the outside radius b there is no radial flow of currerit. i.e. at r = b, \--~r/,=b = 0.

(9)

(2) At the interface between the fluid and the wall the current flow is equal in the radial direction:

(OU$)

(OUw~

(10)

* \ Jr ] , = , -----/21 \ Jr ] r = , " (3) At the interface between the fluid and the wall there may be a contact resistance -r so that the potential at a may be different in the wall and the fluid. The potential drop at the interface may be considered as a 'jr loss.' Since

oue

j = -- e ~

and "r is the contact resistance.

ous Uy-- Uw = --'re Jr "

(11)

A solution of equation (8) in polar coordinates may be found for the wall potential by assuming

Vw = R(r)O(0).

(12)

630

ROBERT H. EDGERTON From the boundary condition (10)

The variables may then be separated to give R'

r ~-

R" R

+ r2

ke | kl |

.

( nC) kl n A a "-1 -- an+x = ~f' (a).

(13)

k2 If ~ is defined as equal to n e then the potential

(19)

From the boundary condition (11) C A a" q- ~ = f (a) q- ~xf' (a).

(20)

in the wall can be expressed as Uw=

rn §

sin0.

(14)

Where A and C are constants to be found from the boundary conditions. The solution to the potential in the fluid may be shown in a similar manner to be UI = f(r) sin 0,

(15)

wheref(r) is the radial variation in UI satisfying the equation 0 OV Or (re f ' -- rf) = r e B 0-7"

(16)

(21) If the denominator is defined as M the potential in the wall may be written as = ~

(17)

where I'm is the average velocity in the robe. Now the boundary conditions (9), (10), and (11) with the expression for Uw andf(r) may be rewritten to find A and C. From the boundary condition (9) atr=b,

ving (17), (18), (19), and (20) it may be shown that B a" I'm C= a n+1 1 kin (a" + ~r) (an-1 \-~-ff b'---; + a "-1

L)

If the velocity is axisymmetric so that V = V(r) only, then the expression for f may be solved using standard techniques. In this situation the form of the potential distribution is not required so that this equation need not be solved explicitly. Integrating equation (16) from r = 0 to r = a gives ae f ' (a) = a f(a) = --Ba e I'm

The set of equations (17), (18), (19), (20) are four equations involving four unknowns A, D, f(a) and f'(a) which may be solved simultaneously. Since Uw at the electrodes is the potential difference of interest, finding A and C will be sufficient for determining the performance characteristics. C From (18) A = ~-~ and from simultaneous sol-

-t-

(Vw)~,~-- AUw

=

b" M

(23)

The usual way to present the flow meter characteristics is to define a sensitivity S representing the observed voltage at the electrodes divided by the ideal voltage generated, AU,o S = B--'----~m 2b "

therefore

S=

(22)

The potential difference between the two wall electrodes is then 4 B a ~ I'm (Uw)~,~ -

~Uw O'r = 0

C A = --. ben

sin 0.

(18)

(24)

If this is expressed in full using (23) and the definition of M it becomes:

2a ~

(25)

ARTERIAL WALL THICKNESS ON ELECTROMAGNETIC FLOWMETER READINGS

I f the contact resistance 9 is zero this reduced to S=

2

(i)'-'+ (!)'+'

(!).+,] (26)

Equations (25) and (26) may be shown to reduce to the equations 2a 2

S =

a S q - b ~q-k-!~

(~

ductivity of 0"23 and 0.65 were taken as typical values from Ref. 2. Since the structure of arteries suggests that the tangential conductivity is greater than the radial conductivity, typical values of the ratio of tangential to radial conductivity were taken as n = = 1, 4 and 16. The situation in which the conductivity of the walls is considered as scalar is shown in Fig. 3. This

(27)

1 q-~

(b ' - a

9

....._~

/:~

= 0.23

....

when n = 1 (where kl is the scalar wall conductivity), and

,r

0.6

//~/

S

2a , t-

S =

631

(28)

/// / / //~///

0.4

a S q-b ~ q- ~--! (b~ - - a S)

// 0.2

whenz=0andn= 1. These are the expressions for the case of a wall with uniform conductivity k~ (SH~RCUFF, 1962).

0

0.2

0.4

O.G

0.~

].O

a/b

3. DISCUSSION The performance of a flowmeter as described by the parameter S when the contact resistance is zero, is shown in Fig. 2. For illustration the ratios of radial wall conductivity to fluid con1.0 - K~ .... -~=i

0.8

___

"~'1 = 0.23 = 0.65

0.6

$

k2 n2 =~'1

n=l

0.4

0.2

/ i

0.2

0.4

0.6

0.8

1.0

~v'b

Flo, 2. Theoretical sensitivity of an electromagnetic flowmeter to the tensor conductivity of the wall as a function of the inside to outside diameter of the artery. (From equation 28).

FIG. 3. Theoretical sensitivity of an electromagnetic flowmeter to differences in scalar conductivities between the wall Kx and the fluid as a function of the ratio of inside diameter to outside diameter of the artery. (From equation 26). demonstrates that only a very small change in a flowmeter calibration can be produced by a scalar change. The actual behavior of the flowmeter thus cannot be described by a scalar variation, and tensor effects must be considered. While the behavior of the flowmeter is determined from equations (24) and (26) as depicted in Fig. 2, the explanation of the experimental behavior requires a more careful examination of these equations. In order to discuss the significance of these factors a better measure than the sensitivity S is required. In experimental measurements the variables recorded are the total flow rate Q and AU~. the voltage AUw. ~ is thus a better measure of the e.m.f, performance. Using equation (24) and the expression for the total flow rate Q =

632

ROBERT H. E D G E R T O N

Vrar:a~, the slope of the Uw vs. Q relation can be expressed as A Uw =

S

(2bB 1 \-~a~/.

(29)

u,,)

-C-/,

1.1 1.0

n=l

0.8

n--2

0.6

f o r n = 1 and--kl = 1

a

U@)

T,q

0.4.

n=4

then

(av ] = Q It

2__B_B. r~b

I

This is the slope of the line of the experimental data for an ideal situation. Let us discuss the ratio

Q ]A

0.2

(30)

0

,

I ,

.4

.2

I

.6

1

!

.8

1.0

R=kl 0"

b-'io.4. E.m.f. voltage to flow ratio compared to the ideal as a function of the ratio of radial wall conductivity to fluid conductivity for several values of n =

S

(31)

is the ratio of tangential to radial wall conductivity), more important than the decrease in radial

where

\--Q-].4

is the actual slope of the

AUw vs.

Q curve. The effects of the change in conductivity o f the walls and the effect of a changing wall thickness on flowmeter performance m a y now be examined in terms of this new parameter.

-0--In

conductivity.

Similarly, if ~ > 1 and n = 1,

wall shorting is important and the signal drops (curve 2). (2) F r o m the curve o f n = 2, (i.e. k= kl) the signal drops, because of wall shorting for the kl case - - = 1 (curve 3).

Figure (4) shows the ratio

y1,

1.2 1.0

as a function of the ratio --klfor different values of n for a fixed a/b = 0.9. This illustrates the chemical effects possible if the wall conductivity is altered. The effects on the actual voltage A Uw produced by a given flow rate Q is shown in Fig. 5. The effects m a y be summarized as follows: (1) F r o m the curves for n --- 1 one can see that if the conductivity of the wall is made less than that of the blood, the signal increases for a given flow rate (curve 1). This is because the angular conductivity of the wall also decreases so that there is less wall shorting. This effect is

n:l

"Uw)

0.8

-

0.6

WI^

-'

kl ~"

"~'Uw)

T/)

0.4. 0.2 O

Fro. 5.

~ i 0.2

~__4. 11=4. 0.4

0.6 a./b

0,8

1,0

E.m.f. voltage to flow ratio compared to the ideal as a function of the inside radius of the tube.

A R T E R I A L WALL THICKNESS O N E L E C T R O M A G N E T I C F L O W M E T E R R E A D I N G S

(3) Compensation may occur in the case where the radial wall conductivity drops and the tangential conductivity decreases also, but less than the radial conductivity. For example if

[A Uw~ the ratios ~ = 0.3 and n = 2 then \ Q ]~=1. (r [AUw~

This occurs because the conductivity effect of the angular shorting is most important, and if this ks decreases enough it will offset the increasing sigrial due to kl being decreased~ Curve (4) or curve (41) are possible depending upon the magnitude of kl/o and n (see Fig. 4 also). This may be the compensating effect observed by SPENCEg and DENISON, (1960). As a typical example of compensating effects, consider the drying of an artery which should cause the ratio --klto decrease.

2 and/c~ = 0.65, S goes from 0.7 to 0.82.

-r

0.878 originally to 0.911 finally. An increased signal results because of an increase in pressure. (Fig. 6) This effect has been observed in a physiological situation in which vasoconstrictor drugs have been used to increase the pressure. An increase in the flowmeter reading of 8 per cent was observed with an increase in pressure when the bed was perfused with a constant volume flow pump. The explanation is that as the wall becomes thinner the total tangential resistance is higher and the total radial resistance is less. On the other hand if n = 1 then for the same situation as far as a and K are concerned

///~/o7kl 4~1

///

n,1

,deal

~U w

If the artery is

homogeneous then the signal will increase if the flow meter is a no current flow device, since there will be less current flow in the artery and less voltage due to current loops. If on drying, however, the tangential conductivity does not decrease as rapidly as the radial conductivity the tangential shorting will become important and the signal will diminish. These effects will compensate each other so that the net change in the signal may be very small. Another consideration of importance is the changing of the lumen diameter with the outside diameter fixed. This might be the result of increased pressure, or chemically produced constriction or dilation. As an example take the a case where ~ changes from 0.9 to 0.95. For n =

Since--

633

This ratio changes from

q FiG. 6. Summary of e.m.f, performance as a function of the wall conductivity variations.

the AUw decreases from 1.05 to 1.008. In general the signal voltage for a given flow rate may either increase or decrease as a result of changes in wall properties or for changes in wall thickness. A decrease in voltage for a given flow rate will occur if increased wall shorting is the net result of the changes. The signal observed can be made greater than the ideal if the wall is poorly conducting, as shown in Fig. 6. This occurs because the wall shorting is reduced. For example, the voltage produced on the inside of the tube is the same as that outside if no current flows and there is no shorting around the sides. The difficulty of using

634

ROBERT H. EDGERTON For the in vivo part the tensor conductivity will become important as kl/~ < 1 and n > 1. The spleenic artery should provide a situation kl where - - < 1 and n > 1. From preliminary

this method to improve the signal is that the impedence matching problem between the voltmeter and the source impedence becomes important and then the signal will drop as the impedence of the source is increased, This is not of concern in medical electromagnetic flowmeter design since the impedence of the vessel is small compared to the flowmeter voltmeter impedence for most biological applications.

G

measurements the muscular structure appears to be such that the ratio n does not change in saline, even though k, and k, both increase in the in vitro situation. The walls of the spleenic artery are thinner than the femoral artery b u t they contain a larger fraction of muscular material and are very active. In these arteries the ratio of wall conductivities is expected to be much larger and hence the sensitivity of the flow meters for them reduced as shown. The difficulty with the spleenic artery is illustrated by the pressure adjusted curves. This data is disconcerting in that one would expect that if one wished to improve the calibration one would try to simulate the in vivo pressure. The studies to date have shown that this not the case and the artery reaction must be examined further. Figure 7 illustrates the experimental situations which can be encountered with arteries which have active muscular walls. The range of variation shown, however, can be explained theoretically if the tensor conductivity of walls is considered.

4. DISCUSSION OF PRELIMINARY EXPERIMENTS Preliminary experiments have been performed using a Medicon 2 mm diameter e m.f. probe on spleenic and femoral arteries of dogs. The results of several experiments are shown in Fig. 7. (AUw is the flowmeter output voltage not the voltage produced at the electrodes.) The ideal performance shown is taken as that in which saline is circulated through the probe with no artery, since, in that case, the conductivity is uniform and constant. The in vitro femoral artery curve shown is closest to this ideal because the wall has been permeated with saline by soaking overnight. The femoral wall has very little muscular structure so that its conductivity in vitro is nearly isotropic and corresponds closely to the case k,/a = 1 for the in vitro part (a saline fluid in a saline artery). 0.5 /

0.4 au w (volts 0.3

0.2

~

/Invitro femoral I , / S a l i n e only (ideal)

/ / /

,n v ,. . o . o ~ . .

,,

/ // InvJ~ro splenic 1, higher '// / pressure adjusted V /~.lnvivo femoral I1 / /// Invivo splenic I and II /// i n v i t r o splenic I and il

/ / ~ / /

Invivo femoral i

///~//~71nvitr~ splenic II, higher // / pressure

O.1

20

,

,

40

60

i 80

Q m i / min.

PIG. 7. Experimental e.m.f, voltage vs. flowrate for splenic and femoral dog arteries in vivo and in vitro using a Medicon 2mm dia. e.m.f, probe.

ARTERIAL WALL THICKNESS ON ELECTROMAGNETIC FLOWMETER READINGS 5. CONCLUSIONS The purpose of this paper has been to illustrate that the external electrode electromagnetic flowmeter can give erroneous results and that corrections can be made if a knowledge of the chemistry, structure and geometry of the wall is known. In many studies these corrections are not needed because the changes may compensate each other and an in vivo calibration may be possible. In those cases where the structure of the artery is modified during an experiment in vivo calibration is not satisfactory and a correction for these effects is required. The theoretical analysis shows that difficulties can be encountered, particularly with muscularly active arteries such as the spleenic artery, where the wall thickness and conductivity ratio may be significantly altered in physiological situations. With small arteries more careful calibration procedures must be employed than with large arteries since the ratio of lumen diameter to wall thickness is not as large nor as stable. The chemical state of these small arteries may also be altered more easily. It has been shown that the errors in flowmeter readings cannot be accounted for by considering the wall as a homogeneous structure. By considering the wall as inhomogeneous with different conductivities in the radial and tangential direc-

635

tions the observed variation of flowmeter readings can be explained, but a determination of what property variations occur is required to make a correction. To make corrections to measurements of blood flow using an electromagnetic flowmeter the following work is required. (1) Identify those arteries where difficulty is expected. (2) Develop an experimental technique which will determine the correction to be made to the flowmeter reading. For example: (a) A calibration electrical signal could be applied to the arterial wall when the magnetic field is off to determine if the wall conductivity has changed during an experiment. (b) An actual measurement of the wall conductivity in vivo may also be possible. REFERENCES BURGER, H. and VAN DONOEN, R. (1960) Specific resistance of body tissues. Phys. Med. Biol. 5, 431. KOLIN, A. (1960) Circulatory system: methods, blood flow determination by electromagnetic method. Meal. Phys. 3, 141. SH~ReUFF, J. A. (1962) The Theory of Electromagnetic Flow Measurement, pp. 23-26 Cambridge University Press. SPENCER, M. P. and DENISON,A. B. Jr. (1960) Squarewave electromagnetic flowraeter for surgical and experimental application. Meth. Med. Res. 8, 321.

636

ROBERT H. EDGERTON

EFFET DE L'EPAISSEUR DE LA PAROI ARTERIELLE ET DE LA CONDUCTIVITE SUR LA LECTURE DES DEBITMETRES ELECTROMAGNETIQUES Sommalre--L'article pr~sente les performances des d6bitm~tres 61ectromagn6tiques h la lumibre des dorm6es concemant la conductivit6 des parois art~rielles. On y examine l'effet de la structure musculaire sur le d6bitm6tre ~t contact exteme. Les cas o/~ les effets de paroi doivent ~tre pris en consideration sont expos6s et d6montr6s exp6rimentalement. Les ~quations de Maxwell pour le d6bitm~tre 61ectromagn6tique h contact externe sont r6solues en consid~rant les diff6rences de conductivit6 radiales et angulaires de la paroi. On y pr6sente l'importance du rapport de la eonductivit6 sanguine ~tla conductivit6 de la paroi. On montre aussi que dans les cas off le rapport du diam6tre int6rieur au diam~tre ext6rieur d'une art~re varie au cours d'une mesure, la sensibilit6 du d6bitm6tre varie 6galement: cette demi6re consid6ration est importante dans les &udes oh la vasodilatation et les vasoconstricteurs sont utilis6s. On y 6tudie la validit6 des calibrations de d6bitm~tres in vitro relatives anx mesures in vivo.

EINFLUSS VON DICKE UND LEITF~HIGKEIT DER ARTERIENWAND AUF DIE ANZEIGE BEI ELEKTROMAGNETISCHEN FLOWMETERN Ztt~ammenfassung--Ober die Leistung elektromagnetischer Flowmeter wird im Lichte von Ergebnissen fiber die Leitf~ihigkeit von Arterienwiinden berichtet. Die Muskelstruktur und ihr EinfluB auf Flowmeter mit AuBenkontakt wird diskutiert. Es wird beschrieben und experimenteli gezeigt, warm die Wandeffekte beriichsichtigt werdcn miissen. Fiir die elektromagnetischen Flowmeter mit AuBenkontakt werden die Maxwell-Gleichungen gel0st unter Beriicksichtigung der radi/iren und angul/iren Leitfiihigkeitsunterschiecle tier Wand. Die Bedeutung des Verhiiltnisses der Blut- und der Wandleitfiihigkeit wird ebenfalls demonstriert. Es wird gezeigt, daB sich bei .~mderung des Verhaltnisses von Innen- lind AuBendurchmesser einer Arterie wiihrend des Flowmeterbetriebs die Empfmdlichkeit des Flowmeters ver~ndert. Bei Untersuchtmgen unter Verwendung von Vasodilatatoren und Vasokonstriktoren shad diese 13bedegtmgen wichtig. Die Anwendbarkeit yon in vitro-Eichung des Flowmeters auf die in vivo-Messungen wird ebenfalls untersucht.