Estimation of Ventricular Volumes by a Constant Infusion Indicator

Estimation of Ventricular Volumes by aConstant Infusion Indicator DilutionTechniqueBy Aaron B. Shaffer, M.D.

• The sudden injection indicator dilutionmethod for the estimation of ventricular vol-umes J has been applied both in the experi-mental animal 2~7 and in man.8-12 This methodpresupposes effective uniform distribution ofindicator in each end diastolic volume(EDV), a circumstance which may not oc-cur.18 Further, it has not yet been practical toevaluate in vivo the possible effects of "non-mixing" on the accuracy of the method. A con-stant infusion indicator dilution method ofmeasuring ventricular volume offers, at leasttheoretically, an approach to these problems.It may shed new light on an aspect which isbasic to the behavior of injected indicator, i.e.,the nature of exchange among the three com-partments of a ventricle. These are the di-astolic venous return, the end systolic volume(£SV), and the stroke volume (SV). Thepurposes of this report are to give the back-ground for such an approach and to showthat this hitherto undescribed method is feas-ible and has promise.

MethodsIf indicator is infused at a constant rate into a

ventricle it is to be expected that indicator con-centration measured at the root of the efferentvessel will rise in a stepwise fashion with eachbeat to reach a plateau at a maximum concen-tration (Cmax) which, in accordance with the

From the Cardiovascular Institute, Michael ReeseHospital and Medical Center, Chicago, Illinois.

Supported by Grant N-63-1 from the ChicagoHeart Association and in part by Grant HE-06375from the National Heart Institute, U. S. Public HealthService.

A preliminary report was presented at the 36thScientific Sessions, American Heart Association, LosAngeles, California, in October 1963. {Circulation28:802, 1963.)

Received for publication January 31, 1964.

168

principles of flow measurement by constant in-fusion,14' 16 should approximate the ratio of therate of indicator infusion to the rate of blood flow.The latter is equivalent to the ratio of the amountof dye infused per cardiac cycle to the strokevolume. Until Cmam is approached more indica-tor enters the ventricle with each beat than leavesit. Near Cmaa there is an equilibrium as a uni-form distribution of indicator between points ofinjection and sampling is approached and as theamount of indicator leaving approaches equalitywith the amount being infused. The rate of riseof concentration of indicator with each beattowards Cmaa depends at least in part on theratio of SV to EDV. If indicator were uniformlydistributed over EDV so that effectively theamount present divided by EDV would equal theconcentration of indicator in the next stroke vol-ume, then the rate of rise of concentration, as-suming of course steady state conditions, woulddepend entirely on this ratio and

EDV =SV ( 1 ) #

where Cn — any reference concentration at out-flow as concentration rises with each beat andCmaw = maximal concentration.

A second approach to the determination ofvolume is permitted by constant infusion, i.e., toestimate the amount of indicator present in theventricle at equilibrium, because the amount ofindicator present in end systole divided by CmM

equals ESV. A larger or smaller ESV, for thesame conditions of flow, will simply hold more orless indicator at the same Cmai. Since the calcu-lation can be based simply on the amount of indi-cator entering the ventricle during each cycle,less the amount leaving the ventricle in eachsystole from the onset of the infusion to Cmam

we have:

(2)

* Derived in Appendix.

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VENTRICULAR VOLUME BY CONSTANT INFUSION 169

where the expression in brackets is represented bythe shaded area in figure 1. The nature of mixingwithin the entire ventricle is not critical in thiscase. The only assumptions, apart from constancyof flow and volumes during the inscription of thecurve, are that the untagged blood in ESV prior toonset of infusion is completely replaced by taggedblood at Cmax, and that each concentration of in-dicator sampled at the outlet is representative ofthe average concentration of indicator ejectedin that systole. This approach is analogous to thatused for measuring, by constant infusion, thecirculating blood volume between two points.16

Since prevailing evidence suggests that homo-geneous mixing of indicator is unlikely in theEDV of the same cycle in which it enters theventricle, a simple mathematical model was de-vised in order to outline the possible effects ofnonmixing of indicator on the beat-concentrationcurve. Indicator was regarded as being continu-ously introduced into the ventricle and as beingdistributed over a volume smaller than EDV,but including the diastolic venous return, priorto each systole. Working assumptions were thatejected indicator was uniformly mixed in thestroke volume and (for convenience) that resid-ual indicator was effectively distributed overESV. This was programmed on a digital com-puter to yield the successive indicator concentra-tions (expressed as fractions of CmM) whichwould be sampled at the outflow for variousratios of SV to EDV and, within these, for various

2 ^ 3 _ , 4_ ,

FIGURE 1

Graphic representation of indicator dilution curve asobtained by continuous infusion of indicator into aventricle with sampling at the root of the efferentvessel. Concentration of indicator on ordinate; heartcycles from onset of infusion (indicated by arrow)along abscissa. Concentration rises hy diminishing in-crements with each beat to a plateau (Cmax). Heightof shaded area between each step and Cmal, whenmultiplied by stroke volume, is the difference betweeninput and output of indicator for that cycle. The sumof these quantities is the total accumulation. Discussedin text and appendix.

volumes of immediate distribution of indicator.Another consideration was whether SV derivedentirely, partially, or not at all from the volumeof immediate distribution of indicator (fig. 2).

Twenty-one satisfactory determinations of leftventricular volume were obtained in 12 dogs, ofwhich five were anesthetized with pentobarbital(25 mgAg iv) and seven with chloralose (120ing/kg iv), the latter being on assisted respira-tion. Indocyanine green dye was the indicatorand a Colson densitometer was used to measurethe concentration of dye in blood. A no. 6Cournand catheter for infusion of indicator wasadvanced in retrograde fashion from the femoralartery to the left ventricle and sometimes on intothe left atrium near the mitral valve. A 35-cmlength of yellow "Kifa" catheter was passed viathe right carotid artery to the sampling site atthe root of the aorta. Catheter tip locations wereconfirmed by pressure measurements and fluoros-copy. Pressure pulses and indicator dilutioncurves were recorded by a Sanbom multichannelbiological recorder. Indicator was infused bymeans of a Harvard pump at a rate of 24.25mg/min. To avoid lag in infusion rate at theonset, the catheter was initially full of indicatorand the pump was started with the infusiondirected into a side connection. The volume ofthe catheter-densitometer sampling system was0.95 mL Blood was drawn through the samplingsystem by means of a Harvard pump arrangementat a rate of 2.6 ml/sec giving a mean transittime through the sampling system of 0.37 sec.In practice, the duration of withdrawal throughthe sampling system for a given curve was about20 sec, and the duration of infusion about 10sec. Calibration of the curves was carried outwith the same sampling system; known amountsof the specific dye used were added to aliquotsof blood of the dog under study. Flow wasderived using the formula for cardiac output byconstant infusion, i.e., cardiac output (liter/man)

Z/minCmajT(mg/liter)

where / = mg of indicator in-

fused, and this was divided by heart rate to givestroke volume (average). Cardiac output wasdetermined independendy in some animals bymeans of "slug" injection of indicator into theright atrium with sampling at the root of theaorta.

ResultsThe form of the mathematically derived

beat-concentration curve rising, as it does,by decreasing increments towards Cmax is simi-lar to that of the charge on a condenser as the

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170 SHAFFER

END DIASTOLE IEDV

EJECTION IESV SV

END DIASTOLE 2EDV

EJECTION 2ESV SV

"COMPLETEM I X I N G "

INCOMPLETEM I X I N G °

: : : : b * i " •. • •. •. . . .

: : :• • .

FIGURE 2

Diagram to show that if, during continuous infusion into a ventricle, indicator entering duringa given cycle is uniformly distributed over EDV ("complete mixing"), the actual site of originof SV from EDV obviously has no effect on outflow concentration. If immediately enteringindicator does not come to be uniformly distributed over EDV ("incomplete mixing"), thenfor the same ratio of SV to EDV and for the same relative volume of immediate distribution ofindicator, it is to be noted that the concentration of indicator at the outflow during the firstand subsequent ejections can vary markedly according to whether SV comes a) entirely, b)partially, or c) not at all, from the volume of immediate distribution. In this figure, each dotrepresents a unit of indicator, and indicator is distributed according to the conventions used inthe mathematical model.

latter is being charged through a resistor.*These curves become straight lines when

Q(1 — p;—) is plotted beat for beat on a semi-log graph. Under assumed conditions of im-mediate uniform distribution of constantlyinfused indicator in EDV ("complete diastolic"mixing), the derived indicator dilution curvesplotted as above formed straight lines whichpassed through the origin of the graph. Thenumber of steps required to approach Cmax

* < ? , • = l-ae-bt, where Qt is the charge at a given

time, t, while the charge is changing from itsoriginal level (£>0) to the new maximum (Qmllx),

a = —_ and b is the reciprocal of the timeYmax

constant (RC) of the circuit

varied in an inverse manner with the ratio ofSV to EDV (S/V). Concentration at outflowobviously reflected closely indicator concen-tration in the previous EDV. EDV and ESV,as recalculated using equations 1 and 2 re-spectively, accurately reflected the originalvolumes.

When indicator which enters the ventricleduring the preceding cycle is distributed overa volume less than EDV before systole, therate of rise of indicator concentration at theoutflow will depend on more than S/V, i.e., itwill also depend to a varying extent on thesize, relative to EDV, of the volume of im-mediate distribution of indicator, but evenmore on the site in EDV from which SV isderived, in relation to this volume. In this re-gard, analysis of the theoretical data revealedthe following:

1) When SV is drawn entirely from theCircmUtion Research, Vol. XV, Arngnjl 1964


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1 2 3 4 5 6 7 8 9

0.9FIGURE 3

Slopes of a series of six theoretical indicator dilutioncurves as sampled at the outflow, resulting from con-tinuous ventricular infusion of indicator starting atend systole of heart cycle 0, calculated for a ventricleof EDV = 100 and SV = 30. Steps of curves 1 and 6are shown. In curves 1 to 5 inclusive, the volume ofimmediate distribution of indicator prior to systole ineach cycle is 50. Curve 1 results when SV derives en-tirely from the part of EDV which is not the volumeof immediate distribution. Curve 2 results when SV isderived three-quarters from the part of EDV whichis not the volume of immediate distribution ofindicator, and one-quarter from the volume of im-mediate distribution. Curve 3 results when SV isdrawn from these two volumes in proportion to theirrelative magnitudes, i.e., one-half from each volume.This curve is the same as would result from idealcomplete mixing in EDV. Curve 4 results when SVis derived three-quarters from the volume of immedi-ate distribution and one-quarter from the other. Curve5 results when SV is drawn entirely from the volumeof immediate distribution, as is also true of curve 6except that for the latter the volume of immediatedistribution was reduced to 40. Note that the concen-tration at heat 1 of curve 1 is zero and that, for eachcurve, the higher the concentration at beat 1, themore gradual is the slope. ESV as recalculated fromeach curve was essentially the correct ESV, the cal-

CircuUtion Reietrch, Vol. XV, August 1964

volume of immediate distribution of indica-tor, concentration at outflow reflects concen-tration in that volume only and not the averageconcentration of dye in the ventricle. Concen-tration at the outflow after the first cycle of in-fusion will of course be higher than the ratioof amount infused per beat to EDV, but willthen rise more slowly with each beat thanwould be the case for complete mixing at thesame S/V, and will take more steps to ap-proach Cmax. The curves, plotted as above, donot pass through the origin of the graph. EDVas calculated from the ratio of successive con-centrations (equation 1) overestimates actualEDV and the degree of overestimation ofEDV increases as the volume of immediatedistribution is made smaller. Theoretically, inthese circumstances, calculated EDV will ap-proach infinity as the volume of immediatemixing prior to systole approaches SV in mag-nitude.

2) When SV is drawn only from that partof EDV into which immediately entering dyeis not distributed before systole, no indicatorat all will appear at the outflow following thefirst cycle of infusion and the concentrationfollowing the next cycle will equal, in thismodel, the ratio of the amount of indicatorinfused in a cycle to ESV. Concentration willthen rise rapidly with each beat (more sothan is the case with complete mixing at thesame S/V). The curves plotted as above donot pass through the origin. EDV as calculatedfrom equation 1 will be maximally underes-timated, and the figure yielded will be in factESV. In this case the actual volume of imme-diate distribution of indicator will not affectvolume determination at all.

3) When SV is drawn from the volume ofimmediate distribution of indicator and fromthe volume of residual indicator, in proportionto the relative magnitudes of these latter twovolumes, then the concentration at outflowwith each beat, while representing no partic-

culated value underestimating the actual (70) by nomore than 3%. EDV as recalculated from each curvewas as follows: curve 1, 70 (which is actually ESV);curve 2, 82; curve 3, 100; curve 4, 127; curve 5, 175;curve 6, 280.


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172 SHAFFER

ular concentration within the ventricle, willrepresent average concentration of indicatorin EDV and the resulting curve will be thesame as if complete mixing in EDV had oc-curred. If this condition of "complete systolic"mixing is met, actual size of volume of initialdistribution of indicator does not matter.

4) If SV is drawn mainly from the volumeof immediate distribution of indicator, the ac-tual EDV will be overestimated but the added

"systolic" mixing will make the volume ofimmediate distribution effectively larger andthe error in calculation of EDV will be less.Similarly, if SV is drawn mainly from that partof EDV into which immediately entering in-dicator is not distributed before systole, EDVwill be underestimated but not to the maximalextent.

5) ESV as calculated using equation 2closely mirrors the actual ESV regardless of

TABLE 1

Ventricular Volume Data in the Dog

Dogno. Weight

Infilling catheterlocation Heart rite

Strokevolume SV/EDV E D V EDV

Pentobarbital aneathesia

14.5

15.5

LV inflowLV apexLV outflow

beats/mln

222214207

ml

13.011.014.0

0.160.200.28

LV apexLV apexLA at mitralLV at mitral

11112596

102

13.09.0

11.011.0

0.310.220.210.20

LV = left ventricle.LA = left atrium.SV = stroke volume.

EDV = end diastolic volume.

ml

815550

42415255

ml/kg

5.63.83.4

2

3

4

5

Average

6

7

18.6

24.0

19.0

16.4

kg

13.6

23.0

LV midLV at mitralLA at mitral

LV apex

LA at mitral

LV apex

LV apex

LV inflow

167176172

150

154

150

168

26.022.025.0

22.0

9.5

12.0

Chloralose anesthesia

beau/min

123

136

ml

9.0

31.0

0.250.220.22

0.23

0.35

0.40

0.28

0.44

0.34

104100114

96

27

30

ml

20

91

5.65.46.1

4.0

1.4

1.8

3.4

ml/kg

1.5

4.0

2.72.63.43.5

9

10

11

12

Average

15.0

16.5

20.5

17.2

LV inflowLV inflowLV outflow

LV mid

LV apex

LV apex

167167150

118

65

100

116

20.019.019.0

18.0

36.0

24.0

0.210.300.32

0.25

0.46

0.38

0.34

956359

72

78

63

6.3A3,3.9

4.4

3.8

3.7

3.6

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the volume of immediate distribution of in-dicator or origin of SV.

Thus, for a given S/V there may be an in-finite number of possible curves which mayaccurately reflect EDV, or may underestimateor overestimate it. These concepts are illus-trated in figure 3.

Left ventricular volumes in the dog asdetermined by the constant infusion indicatordilution method are shown in table 1. Repre-sentative curves are shown in figures 4 and 5.The volumes as tabulated were estimated us-ing equation 1 only, in this initial study. Inorder to use equation 2 with confidence, trig-gering of the onset of infusion at an appro-priate time in the cardiac cycle (end diastolefor atrial infusion, end systole for ventricularinfusion) would be necessary. The biologicalcurves of concentration rose discontinuouslyby decreasing increments towards CmM andtended to become linear (fig. 4) when plot-ted as were the mathematically derived curves.The calculated volumes were within the rangeof those obtained by the instantaneousmethod.1'4'ia When several determinationswere made in one animal there was consid-erable scatter of results, probably because theconditions of this study make a steady state

unlikely; also the position of the infusingcatheter was changed. It is noteworthy thatthe curves obtained by infusion into the leftatrium near the mitral valve did not differmaterially in appearance from those obtainedby ventricular infusion. The volumes calcu-lated from atrial infusion curves did not differfrom those of ventricular infusion any morethan the latter differed among themselves.This was the case despite entry of indicatorinto the ventricle throughout the cardiac cycleduring ventricular infusion and only in diastoleduring atrial infusion.

There is no theoretical reason to doubt thevalidity of using Cmam for the calculation ofcardiac output, as was done in this study, thetechnical limitations of indicator dilutionmethods in general being recognized.13'18'17

On the occasions in which sudden injectionindicator dilution curves, obtained by right-sided injection of indicator and aortic rootsampling, were interspersed among constantinfusion ventricular volume curves, the out-puts calculated from each type of curveshowed good agreement.

Discussion

A well-defined relationship exists between

ECC

u

5

10

, iiliji

ill: ::

- ::i

i l*. "5

•'•}'

•I:

h

r

m

A

P

Q\-

¥I

•

sy

r-

\A

! •

:' r

::3

\ f"

• iC

3.*:

|

ri

„ _

I

r:

ill!

' - • : •

• • ' *

'i1

V

BEATS

0.1 -

FIGURE 4

Left panel. Constant infusion indicator dilution curve in the dog, obtained by left ventricularinfusion and aortic root sampling. Concentration of indicator on ordinate. Arrow indicatesonset of infusion corrected for lag in sampling system. Concentration rises stepwise with eachbeat by decreasing increment* towards a maximum. Right panel. Several successive concentra-

tions show a straight-line relationship when plotted as 1 — -on a semi-log graph. Stroke

volume was 19 ml, SV/EDV as calculated from the curve was 0.30, and EDV was 63 ml (dogno. 9, second curve, table 1).

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174 SHAFFER

ECG*?

o

10 L

FIGURE 5

Constant infusion indicator dilution curve in the dog, obtained by left ventricular infusion andaortic root sampling. ECG indicates ventricular bigeniiny. Both the curve itself and the pressurepulses obtained during the study suggest that only every other beat was effective. Data fromthis curve not included in table 1. Labelled as in figure 4.

sudden injection and constant infusioncurves,14'lr> the latter type being in essencethe integral of the former. In the case ofventricular volume curves, under assumedconditions of complete mixing of indicator inEDV, the ascending concentration curve ofconstant infusion is the "integral" of the sud-den injection curve which would result if in-dicator were infused in similar fashion but forone heart cycle only. Thus, under ideal con-ditions, the one curve can be derived from theother. Furthermore, the form of either curvewould be independent of the position of theinfusing catheter within the ventricle. As longas all the indicator infused over a given cyclewould be in the ventricle during the corres-ponding ejection, the infusing catheter couldeven be in the atrium. It is also true in thesecircumstances that the expressions for calcu-lating EDV and ESV from a constant infu-sion curve have a common derivation (seeAppendix).

Some evidence indicates, however, that evenunder ideal practical conditions the circula-tory system introduces deviations from theory

in the sense that indicator distribution, assampled at a given site for purpose of meas-uring cardiac output, does not appear repre-sentative of overall indicator distribution.10

The requirements for accurate measurementof any volumes by indicator dilution methodsare somewhat more stringent than those formeasuring cardiac output.15 The proximity ofpoint of injection to point of sampling, whichis necessary when measuring ventricular vol-umes, can reasonably be expected to makeconditions deviate from theory even more.In vivo observations relating to this point1S

give no support to the assumption that in-jected indicator is immediately and uniformlydistributed over EDV; the practical effectsof nonmixing on the accuracy of ventricularvolume determination by indicator dilutionmethods are, however, unknown.

While it is not to be expected that the im-mediate distribution of indicator produced byconstant infusion will differ materially fromthat produced by sudden injection, the formerapproach appears to offer one immediate ad-vantage: ventricular volume can be estimated

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in two ways, one of which does not dependupon the assumption of immediate distributionof infused indicator over EDV. Calculation ofvolume on the basis of the amount of indicatorheld by the ventricle at Cmax, however, stilldepends on two assumptions:

1) The concentration of indicator sampledat the outflow with each beat is representa-tive of the average concentration in that strokevolume. This is obviously implicit in the as-sumption of complete mixing in EDV andwhether mixing in EDV is complete or not,the concentration sampled at outflow is theonly index of the amount of indicator ejected.Since cardiac output as calculated from Cmax

of the ventricular volume curve in the pres-ent study compared well with cardiac outputas determined by standard sudden injectionindicator techniques, and since this has alsobeen shown more formally to hold for suddeninjection ventricular volume curves,17 this as-sumption appears valid for practical purposes.

2) Indicator is uniformly distributed overESV at Cmax. If indicator could be infusedinto the atrium so that each cycle of infusionof indicator enters the ventricle in the corres-ponding diastole, it would be a reasonablecorollary that untagged blood in the ventriclewill be replaced by tagged blood, barring thepresence of stagnant pools, and that the dis-tribution will approach homogeneity in ESVat Cmax since indicator is entering at a constantrate with the diastolic venous return and resid-ual amounts of each cycle of infusion will bein the ESV. In the case of infusion directlyinto the ventricle, any immediately enteringindicator, presumably poorly distributed,which is not ejected immediately, will thenhave as long to distribute as if it had beeninfused into the atrium.

The mathematical model used in this study,as a first approximation of what might happenif indicator entering the ventricle in a givencycle were not uniformly distributed overEDV, results in indicator being distributedover the ventricle in two volumes which con-tain differing concentrations during the earlypart of infusion. Concentrations in the twovolumes approach equality at CmaJ as concen-

CircmUtion Rtsirrcb, Vol. XV. August 1964

tration at outflow approaches Cmai. In thesecircumstances, calculation of volumes based onthe ratio of successive concentrations variesaccording to the conditions illustrated in figure3, while volumes calculated from the samecurve on the basis of amount of indicator pres-ent in the ventricle at Cmax are unaffected.The basic reason for this would appear to bethat equation 1 depends on uniform distribu-tion of indicator in each EDV while equation2 depends only on uniform distribution of in-dicator in ESV at Cmax. From the practicalpoint of view, any valvular regurgitation thatmay be created by the infusing catheter willbe included in ESV as calculated by equation2.

When mixing in EDV is incomplete, the rateof rise of a constant infusion ventricular vol-ume curve, and the calculation of EDV fromsuch a curve using equation 1, will vary withthe position of the infusing catheter insofar aschanges in position will alter the extent towhich stroke volume is derived from thevolume of immediate distribution of indicator.Relative insensitivity of the sudden injectionventricular volume curve to the position of theinjecting catheter has been noted and takenas an indication that mixing in EDV is com-plete.0 An alternate explanation is that, re-gardless of where in the ventricle the indica-tor is injected, the residual indicator is al-ways based in the same part of the ventricle,i.e., the ESV.

Thus, once the mixing of immediately en-tering indicator in EDV is incomplete, thecurve obtained by sudden injection would becomparable only to the constant infusioncurve obtained by infusion of indicator intoESV. Further, derivation of a constant infu-sion curve from a given sudden injectioncurve, for purposes of estimating the amountof dye which would be present in ESV at Cmax

under conditions of constant infusion, wouldbe difficult because the early part of the sud-den injection curve is usually unsuitable foranalysis on account of the mode of injec-tion.4' Is On the other hand, one advantageof the constant infusion method is that thedeflection in the curve of concentration re-


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176 SHAFFER

suiting from the first cycle of infusion of indi-cator is not subject to the artifact frequentlynoted in the early part of a sudden injectioncurve.

The shape of the curves obtained in the dogin the present study indicate that the constantinfusion method for assessing ventricular vol-umes is feasible in vivo. The similarity of thecurves obtained by infusion on either side ofthe mitral valve suggests the feasibility ofusing atrial injection to outline ventricularvolume. The amount of indicator solution in-fused with each beat near the mitral valve,being only a few per cent of the venous returnper beat, should insure that no given aliquotof indicator remains longer than one heart cy-cle in the atrium. Such an atrial infusion wouldmake it possible to "tag" the ventricular in-flow. This in turn points to a possible means ofstudying how diastolic inflow to the ventricleis partitioned between that amount which ex-changes with ESV and the amount which con-tributes to the next stroke volume. If diastolicvenous return should mix completely withESV prior to systole, then it is likely that in-dicator will behave in the same way regard-less of where or how it is infused into theventricle and the same indicator dilution curveshould be obtained regardless of the part ofEDV from which SV is derived. Such a blend-ing of diastolic venous return and ESV wouldin fact probably be fundamental to "completemixing" of indicator in EDV. If diastolic ve-nous return and ESV do not blend prior to thenext systole it would seem an obvious conse-quence that infused or injected indicator can-not be instantaneously distributed over EDV.In the latter circumstance it is reasonable tosuppose that the form of an indicator dilutioncurve sampled at the outlet of a ventricle willdepend in part on the behavior of the volumeinto which the indicator is immediately intro-duced. In theory, as outlined above, EDVcould either be overestimated or underesti-mated, depending on the contribution of thisvolume to SV, while direct calculation of ESVappears relatively insensitive. Studies to date,using the sudden injection method, suggestthat immediate complete mixing is unlikely,13

even when indicator is forcefully injectedthrough a multiple holed catheter18* andthat immediate diastolic inflow contributesheavily to the next stroke volume.13

An objection which might be raised to theconstant infusion method is that the resultingcurve may be more sensitive to variations inheart rate during its inscription than wouldbe a sudden injection curve. Beat-to-beat vari-ations in actual heart volume in these circum-stances may be relatively small while theamount of dye infused per beat varies directlywith the duration of the cardiac cycle.

Summary

A constant infusion indicator dilutionmethod for estimating ventricular volumes isintroduced and its feasibility demonstrated.Using this method, volumes can be calculatedby two different approaches, one of whichdoes not assume effective distribution of in-dicator throughout end diastolic volume(EDV). This offers a means of assessing theeffects of nonmixing on the form of the step-wise rise in concentration of the constant in-fusion curve, which has relevance to the basi-cally related stepwise decline in concentrationfollowing sudden injection of indicator. It issuggested that the degree of mixing of dias-tolic venous return in the previous end systolicvolume (ESV) is basic to the distribution ofindicator introduced into the ventricle. If mix-ing is incomplete, the site of origin of strokevolume (SV) from EDV is an importantfactor in determining the form of an indicatordilution curve and the EDV as calculated fromratios of successive concentrations. The pos-sible range of effects of nonmixing on the formof the constant infusion curve for ventricularvolume, as derived from a mathematical mod-el, is outlined. End diastolic volumes, as esti-mated from ratios of successive concentrationsin a series of constant infusion indicator dilu-

* Even if the force or amount of injection couldbring about uniform distribution of indicator in theEDV of the first cycle following injection, distributionof indicator in EDV of subsequent cycles would nolonger be related to mode of injection.

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tion curves obtained experimentally, are tabu-lated, and as a group do not differ materiallyfrom those obtained by the sudden injectionmethod.

AcknowledgmentThe author is grateful to Dr. R. Estin, Department

of Physics, Illinois Institute of Technology, for hishelp in developing the mathematical concepts andfor preparing the computer data; to Dr. L. N. Katzfor his advice in the preparation of the manuscript;and to Mr. A. Ellis for help with the animal experi-ments.

References1. HOLT, J. P.: Estimation of the residual volume

of the ventricle of the dog's heart by two in-dicator dilution technics. Circulation Res. 4:187, 1956.

2. HOLT, J. P.: Effect of plethora and hemorrhageon left ventricular volume and pressure. Cir-culation Res. 5: 273, 1957.

3. HOLT, J. P.: Regulation of the degree of empty-ing of the left ventricle by the force of ven-tricular contraction. Circulation Res. 5: 281,1957.

4. RAPAPORT, E. , WrEGAND, B. D . , AND BRISTOW,

J. D.: Estimation of left ventricular residualvolume in the dog by a thermodilution method.Circulation Res. 11: 803, 1962.

5. FERGUSON, R., BRISTOW, D., MINTZ, F., AND

RAPAPORT, E.: The effects of pericardial tam-ponade on left ventricular volumes and func-tion as calculated from aortic thermodilutioncurves. Clin. Res. 11: 100, 1963.

6. BRISTOW, J. D., FERGUSON, R. E., MINTZ, F.,

AND RAPAPORT, E.: The influence of heartrate on left ventricular volumes in dogs. J. Clin.Invest. 42: 649, 1963.

7. BRISTOW, J. D., FERGUSON, R. E., MINTZ, F.,

AND RAPAPORT, E.: Thermodilution studies ofventricular volume changes due to isoproter-enol and bleeding. J. Appl. Physiol. 18: 129,1963.

8. FREIS, E. D., RIVARA, C. L., AND CILMORE, B. L.:

Estimation of residual and end-diastolic vol-umes of the right ventricle of men withoutheart disease, using the dye-dilution method.Am. Heart J. 60: 898, 1960.

9. FOLSE, R., AND BRAUNWALD, E.: Determination

of fraction of left ventricular volume ejectedper beat and of ventricular end-diastolic andresidual volumes. Experimental and clinicalobservations with a precordial dilution technic.Circulation 25: 674, 1962.

10. RAPAPORT, E., WONG, M., FERCUSON, R. E.,

CircuUtio* Rtitarcb, Vol. XV, Amfust 1964

BERNSTEIN, P., AND WIEGAND, B. D.: Right

ventricular volumes in patients with and with-out heart failure. Federation Proc. 22: 402,1963.

11. PALEY, H. W., WEBSLER, A. M., AND SCHOEN-

FELD, C. D.: The effect of upright posture onleft ventricular volume in man. Clin. Res. 12:105, 1964.

12. BRISTOW, J. D., FARRFHT, C, LEWIS, R. P., AND

GRISWALD, H. E.: Left ventricular volumestudies in man by thermodilution. Clin. Res.12: 76, 1964.

13. SWAN, H. J. C , AND BECK, W.: Ventricular non-

mixing as a source of error in the estimationof ventricular volume by the indicator-dilutiontechnic. Circulation Res. 8: 989, 1960.

14. HAMILTON, W. F., AND REMINGTON, J. W.: Com-

parison of the time concentration curves inarterial blood of diffusible and non-diffusiblesubstances when injected at a constant rateand when injected instantaneously. Am. J.Physiol. 148: 35, 1947.

15. ZIERLER, K. L.: Circulation times and the theoryof indicator-dilution methods for determiningblood flow and volume. In Handbook of Phys-iology, sect. II, Circulation, vol. I, ed. byW. F. Hamilton. Washington, D. C , Am.Physiol. Soc., 1962.

16. SLEEPER, J. C , THOMPSON, H. K., JR., MCINTOSH,

H. D., AND ELSTON, R. C : Reproducibility ofresults obtained with indicator-dilution tech-nique for estimating cardiac output in man.Circulation Res. 11: 712, 1962.

17. RHODE, E. A., KINES, H., AND HOLT, J. P.:

Studies on the accuracy of ventricular volumemeasurements by the indicator dilution tech-nique. Physiologist 6: 263, 1963.

18. IRISAWA, H., WILSON, M. F., AND RUSHMER,

R. F.: Left ventricle as a mixing chamber.Circulation Res. 8: 183: 1960.

AppendixDERIVATION OF EXPRESSIONS FOR END DIASTOLICAND END SYSTOLIC VOLUMES

Expressions for estimating ventricular end diastolicand end systolic volumes from a constant infusionindicator dilution curve can be derived in varioussvays. The most rigorous and direct treatment is out-lined below.*

Inject indicator into a ventricle at a constant rateof / mg/min. Let ventricular contraction rate be pbeats/min. Then I/p is constant rate of injection perbeat Let q(n) be the quantity of indicator in the

* We are indebted to a member of the EditorialBoard of Circulation Research for this treatmentwhich is reproduced with his permission.


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178 SHAFFER

ventricle at the end of diastole at the nth beat. Letthis quantity be distributed uniformly over end dias-tolic volume, V. The quantity of indicator ejected

cduring the next systole is -^-q(n), where S is stroke

volume. q(n) is the difference between the total inputup to the nth cycle, which is nl/p and the total output

up to the nth cycle, which is — <?(');

that is

q(n) =nl 5 -p V

(3)

Similarly for the quantity present at end diastole onthe ( n + l)th cycle,

vhence

Cm M = I/PS (7)

Since pS is cardiac output, CO, equation 7 is thefamiliar equation,

CMM = 1/CO

Substituting equation 7 into equation 6,

(7a)

A C = ^ - [ C M M - C ( n ) ] (8)

which when solved for V is equivalent to equation 1.To estimate ESV in terms of the amount of dye in

the ventricle at Cmax, one starts again with equation3. Since it has been shown that I/p = SCnax,since q(i)/V has been defined as C(<), then

and

(3a)

(4)

Subtracting equation 3 from 4

-q(n) = ±q= y ~ Y q(n) (5)

which is the quantity of indicator present at end di-astole. Sum the output due to the next ejection to getthe quantity present at end systole, which is

(3b)

Let -^-= C be concentration in ventricle at end di-

astole. Then from equation 5,

C( n + 1 ) — u( n ) = aC = —-rr 77- O l n ) ( o )

pv VEquations 5 and 6 are difference equations, akin to

differential equations. Since the first term on the rightis constant and since the second term, or output term,can never exceed the first term, C(n) grows to amaximum, at which output equals input and AC = 0.This is proved by differentiating equation 6, i.e., bytaking the difference, AC, between a pair of successive

cdifferences to get A'C = — ^ AC, which indicates

that AC decreases with n as C(n) increases. There-

I S „fore, set AC = 0 to find

When C(n)=Cmax, q,(n) will have reached itsmaximum, qmwx for ESV, because output =: input andA<7 = 0. Then

= ESV- [ " C ^ - (2)

pV

The quantity in brackets is then graphically repre-sented by the shaded area in figure 1, and this multi-plied by stroke volume is the total accumulation ofindicator in ESV when concentration at outflowreaches CmM. The important point is that, with theinput of indicator per beat being known, the onlyrequirements for estimation of ESV are that the con-centration of indicator ejected with each beat, asmeasured at the outflow, is representative of theamount ejected and that there is effective uniformdistribution of indicator over ESV at Cmal. Immediatedistribution of indicator in any EDV is not specificallyrequired.

CircmUiio, Rtsurcb, Vol. XV, Axgvt 1964


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AARON B. SHAFFEREstimation of Ventricular Volumes by a Constant Infusion Indicator Dilution Technique

Print ISSN: 0009-7330. Online ISSN: 1524-4571 Copyright © 1964 American Heart Association, Inc. All rights reserved.is published by the American Heart Association, 7272 Greenville Avenue, Dallas, TX 75231Circulation Research

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Estimation of Ventricular Volumes by a Constant Infusion Indicator