679

Cerebral Blood Flow Determination by Rapid-Sequence ComputedTomography

A Theoretical Analysis1

Leon Axel, M.D., Ph.D.

Dynamic computed tomography (CT) using rapid-sequence scanning can be used to deter-mine cerebral blood flow nonlnvaslvely following an intravenous bolus Injection of contrastmaterial. Since the contrast material remains intravascular in the normal brain, principlesof IndIcator dilution analysis for nondiffuslble Indicators are applIed to the time course of thechanges in contrast concentration. While a delay is Introduced by the relatively prolongedIntravenous injection, this can be corrected for if the arteries are seen on the scan; the cor-rected mean transit time of the first passage of the bolus through the vessels gives the bloodflow per unit vascular volume. To find the blood flow per unit of total tissue volume requiresmeasuring the concentration of contrast material In the blood, which cannot always be donedirectly from brain scans with current CT equIpment; however, a relative value for total tis-sue flow can be found by using the area under the curve of contrast concentration as a func-tion of time, as this area is proportional to the fractional vascular volume of the tissue.

INDEX TERMS: Blood, flow dynamics #{149}Brain, blood supply (Skull and contents, low flow state, 1[0].760) #{149}Comput-ed tomography, contrast enhancement #{149}Head, computed tomography, 1 [0J.121 1

Radiology 137:679-686, December 1980

Ne�

A CONVENIENT, relatively noninvasive means of as-sessing blood flow within the brain and other organs

is constantly being sought; some current methods werereviewed by HiIaI (4). Radiopaque contrast media havebeen used to study blood volume and flow, with theirconcentration being measured by densitometry (5, 6),videodensitometry (1 2), and fluorescence excitation (8,16). Computed tomography (CT) has been used to estimatecerebral blood volume, based on the degree of contrastenhancement (10), but there have been problems with thismethod (17). The recent capability of rapid-sequencescanning suggests that blood flow could be assessed basedon changes in CT number following a bolus injection ofiodinated contrast material (15).

In order to extract useful information from rapid-sequence scans, an understanding of indicator dilutionprinciples is required. Excellent reviews of both diffusibleand nondiffusible indicators have been presented by Zierler(21) and Antman (1); however, both laboratory and clinical

analysis of cerebral blood flow has been limited primarilyto diffusible indicators (2, 7), as analysis of rapid-sequencescans with a nondiffusible indicator, such as iodinatedcontrast material in the normal brain, is not intuitively ob-vious. I wish to review some of the basic concepts of in-dicator dilution analysis and relate them to rapid-sequenceCT scanning with a nondiffusible indicator, particularly forthe study of cerebral blood flow.

One difficulty in extending the use of indicator dilutiontechniques to calculation of tissue perfusion is that con-ventional iodinated contrast media diffuse rapidly into theextravascular space in tissues other than the intact centralnervous system (9, 1 1 , 14). If this takes a good deal longerthan the circulation time, it may be possible to separatecontrast enhancement into two phases, vascularand ex-travascular, by compartment analysis, since a simplepartition coefficient will probably be inadequate to describethis nonequilibnium situation. In the analysis presentedhere, I am assuming that extravascular diffusion is negli-gibie (as in the intact brain) or capable of being correctedfor, i.e., contrast material is a nondiftusible indicator.Future contrast agents may behave more like ideal non-diffusible indicators.

THEORETICAL BACKGROUND

The CT number of any tissue changes following the in-jection of an iodinated contrast material. Since this changeis proportional to the concentration of that material, it canbe used as an indicator. When a known amount of an in-dicator is added to a fluid system, its concentration allowscertain deductions about the volume and flow of the fluid.This basic concept has had many applications in physi-ology.

An example of a standard indicator study is shown in

1 From the Department of Radiology, University of California Medical Center, San Francisco, Calif. Revised version received March 7,

1980 and accepted July 17.This study was supported by USPHS grant GN-01272-14 and the Radiology Research and Education Foundation, University of California.

San Francisco, Calif. sjh

F

Fig. 2. Schematic diagram of a system with a central mixingvolume and a fluid flow F. � symbolic mixer.

TIME (seconds)Fig. 1 . Curve representing arterial concentration of an indicator

(blue dye T-1824)after a single brief intravenous injection; appearanceis delayed due to passage through the lungs. The break in the smoothcurve marks the appearance of recirculating dye; the dashed line Is anextrapolation representing the expected curve in the absence of re-circulation. [Redrawn from Meier and Zierler (13) and reproducedby permission of the author and publisher]

F

680 LEON AXEL December 1980

One such method is bolus injection. If a mass of contrast

0,

20

a::I-2Iii0z00

Figure 1 . Following intravenous bolus injection, the arterialconcentration is measured. After a delay while the indicatortraverses the pulmonary circulation, it rises to a peak andthen drops off until a second, smaller peak caused by re-circulation appears.

Use of indicator dilution curves to determine blood flow,as well as the more complex curves resulting from’infusionof the indicator into the capillary bed, will be consideredlater in this paper. First, however, I wish to consider somesimpler situations, beginning with the use of an indicatorto find the (unknown) volume of a container, of static fluid.If a mass m of the indicator is distributed in a container offluid of volume V, the average concentration is definedas

mc=- (1)

V

Since the mass is known, the concentration can be mea-sured and used to calculate the volume of the fluid.

The dynamic case of a flowing fluid is more complex.An an initial model, consider an idealized flow systemconsisting of a central volume with single inflow and out-flow orifices and no recirculation. Contrast material is in-jected into the fluid at the site of inflow, and its concen-tration is sampled at the site of outflow (Fig. 2): for ex-

ample, the central volume could represent the pulmonarycirculation and the flow the cardiac output. The followingassumptions are made; (a)the contrast material mixes withthe fluid before sampling, so that its motions are repre-sentative of the fluid; (b) the amount of contrast materialis small enough not to disturb the flow of the fluid; and (C)

all of the injected contrast material eventually leaves thesystem. If a known amount of contrast material is injectedat one point and the resulting concentration at another pointis measured, generalizations of Equation 1 allow us tocalculate the volume and flow of fluid between these twopoints, with the method used depending on the type of in-jection made (some of which are not clinically prac-tical).

Conceptually, flow is easiest to determine if contrastmaterial is injected at a constant rate. For a constant in-

jection at rate I (mg/sec.) into a steady fluid flow F (mI/sec.), the downstream concentration c (mg/mi) at equi-librium will be given by

c=L (2)

and the calculated flow would simply be inversely pro-portional to the measured concentration. In the clinicalsituation, however, equilibrium would not be reached dueto recirculation, so that this method is not practical.

Flow can also be calculated from the transient response

of a flow system to a step injection, i.e., abrupt initiationor cessation of a constant injection of contrast material.The rate with which the system approaches a new equl-libnium will be determined by the amount of flow throughthe system (tissue) and the volume distribution of thecontrast material within the tissue. This method is useful

for radionuclide flow studies with diffusible indicators,wherein the resulting concentration curves are used tocompute flow per unit tissue volume (assuming the parti-tion coefficient of the indicator is known); and the samemethod can be applied to CT measixement of the washout

of stable xenon from the brain (2). However, It cannot beused with conventional iodinated contrast media, due onceagain to recirculation, whtch makes it difficult to achieveequilibrium with a constant injection. Therefore, we mustconsider methods which do not require that equilibrium bereached.

Vol. 137 CEREBRAL BLOOD FLOW DETERMINATION BY RAPID-SEQUENCE CT

material is injected as an instantaneous bolus at the inflow 1 , it is the time value most directly related to flow in thispoint, the concentration at the outflow point as a function system. To calculate the mean transit time, we set t 0

of time, c(t), will rise to a peak and return to zero (Fig. 1). when the contrast material enters the central volume andOver any short interval L�t, the amount of contrast material measure c(t) either at the outflow orifice or within theleaving the system is given by Fc(t)LIt. Since we are as- central volume (assuming concentration is the same atsuming that all of the injected material leaves the system, both points). For a central volume V,as noted earlier, the sum (or, in the limit as the intervals�tapproach zero, the integral) of these amounts of con- (6)trast material must equal the mass originally injected (inthe absence of recirculation): thus as shown by Meier and Zierler (13)(Appendix I). Thus the

� reciprocal of the mean transit time for an instantaneous

som Fc(t)dt (3) bolus gives the flow per unit volume of the system. Thisequation could be used clinically to estimate the centralIf m is known and c(t) is measured, we can solve Equation volume, with flow being determined from Equation 4 and3 for F (assuming constant flow), which gives t from Equation 5, and will be used later in this paper to

determine flow in tissue based on the CT measurement ofF m (4) vascular volume and the mean transit time t.S � c(t)dt One problem with calculating � in this way is that when

0 mixing is not perfect, the concentration of contrast material

The integral in the denominator of Equation 4 is the area in the veins will not exactly equal that in the tissue, andunder the curve representing concentration plotted as a therefore meas�iring the tissue level will not give an ac-function of time. If flow is pulsatile rather than constant, curate value of t. Although we cannot measure the actualthis equation yields the mean value of F. Calculation of flow concentration in the draining veins, we can use the tissueusing Equation 4 is the basis of the clinical determination concentration in Equation 5 to calculate the approximateof cardiac output by dye or thermal dilution (1 8), in which mean transit time. However, this will tend to result in un-the sites of injection and sampling are on opposite sides derestimation of mean transit time and overestimation ofof the heart. This equation, often referred to as the flow per unit volume unless the tissue behaves like aStewart-Hamilton equation, could also be used to deter- simple well-stirred compartment, with a monoexponentialmine cardiac output with dynamic CT. washout (Appendix II). Fortunately, when calculating rel-

Note that the use of Equation 4 does not require a bolus ative rather than absolute values of blood flow, this wouldinfusion; however, for practical reasons (such as recir- not pose a significant problem, because the relative errorculation), the injection is usually kept as brief as possible. would be similar in different regions as discussedAlso, note that the injection need not be made directly into below.the inflow orifice, provided that the contrast material mixes AI’IOth& tfl�5ns of approximating the mean transit timewith the total flow. Similarly, the sampling site need not is determining the equivalent width of the curve showing

be the outflow orifice but may be a more peripheral artery, change in indicator concentration as a function of time,as long as it reflects the time course of the change in d&iiied as the area of the curve divided by its height, whichconcentration: there may be a brief delay, but the area gives the exact value when an instantaneous bolus isunder the curve should remain the same. Finally, the employed (20). The problem arises from the practicalintegral of the change in CT number of an artery on difficulty of delivering such a bolus: in particular, an in-sequential scans after contrast injection will reflect only fravanOti5 injection would be delayed and prolonged beforethe cardiac output, not the flow through that artery. reaching the arteries. However, ft is possible to correct for

We have seen how the area under the outflow con- this in the fqllowing manner. Since the injection has its owncentration curve resulting from a bolus injection can be mea�’� time tinj, defined in a manner analogous to Equation

used to calculate a central flow such as the cardiac output. �, it � be ShOWfl that the ObSOfVOd fl1�fl tranSit time t�j�

To extract even more information, we can use the first equals the sum of the true mean transit time t (as foundmoment of the curve to define the mean transit time t with an instantaneous bolus) (approximated by the firstas moment, as defined by Equation 5) plus the mean injection

time, according to the equationf tc(Odt (7)�= Ocx, (5)

c(t)d t as shown by Zienler (21) (Appendix Ill). Thus even though,5 0 in practice, contrast injections are not instantaneous, if wecan find the mean injection time, we can, in principle,

The first moment is equivalent to the center of gravity of correct the observed concentration curve to find the meanthe shape defined by the time-concentration curve; and transit time that would have been achieved with a trulywhile it is distinct from both the peak time and other values instantaneous injection, and then use Equation 4 to find theused to characterize curves such as that shown in Figure flow per unit volume.

- Vt=

iobs tlnj +t

68 1 Netroradiology

(13)

682 LEON AXEL December 1980

There is another way of assessing the response of aflow system to a bolus injection besides using t. The de-scending portion of the concentration curve after a bolusinjection is commonly represented by an exponential

function (i.e., a straight line on a semilogarithmic plot), andthis is frequently used to advantage in extrapolating thecurve beyond the time recirculation is first observed. Forthe particular case of washout from a single compartmentin the central mixing volume, the time course of the changein concentration would indeed be an exponential functionof the form

c(t) COO�T (8)

where Co is the concentration at time t 0 and the timeconstant ris given by

V

F

Thus if the volume is known and washout is exponential,the observed rate of washout is another means of as-sessing flow.

APPLICATION TO RAPID-SEQUENCE CT

The most practical means of administering iodlnatedcontrast material is a brief intravenous injection. if thematerial satisfies the assumptions given earlier for an in-

dicator, the above theory can be used to calculate cardiacoutput and pulmonary blood volume by means of serialscans of the pulmonary artery and thoracic aorta after anintravenous bolus injection. (It is assumed here and in thediscussion which follows, first, that initially there is negli-gible fluid or contrast exchange between vascular andextravascular compartments, and, second, that recircu-Iation can be corrected for by extrapolation after it hasbegun.) However, this theory must be extended to applyto calculation of tissue perfusion: in so doing, I offer a newapproach to the analysis of blood flow with nondiffusibleindicators as applied to rapid-sequence CT scanning.

To correct for recirculation, the duration of the bolus inthe tissue must be shorter than the recirculation time. Inorder to find absolute rather than relative values for thefractional vascular volume, we must know either (a) theactual time course of the vascular concentration or (b) thevascular concentration itself at the time of the delayedscan (assuming the blood/brain barrier is intact). However,relative values may suffice for many clinical situations.

We cannot determine the actual mass of contrast ma-terial delivered to the tissue by its arterial supply followinginjection. Therefore, the Stewart-Hamilton equation(Equation 4) cannot be used to calculate tissue perfusion.Essentially, there are two principal methods of assessingtissue perfusion based on CT measrement of the vascularphase of enhancement following an intravenous bolusinjection. One method is based on determination of themean transit time of the contrast material, which requiresknowledge of the time course of arterial concentration inorder to correct for delay and prolongation of the bolus; the

other is based on the rate of washout of contrast material(assuming it is exponential). However, for a nondiffusibleindicator, both of these methods define only flow per unitvascular volume; thus in order to compute flow per unittissue volume, one must know what fraction of the tissueis vascular volume. This fractional vascular volume canbe determined by considering the area under the contrastconcentration curve as follows. Given a volume V of thetissue being scanned, a fraction f of this tissue will beoccupied by vascular volume according to the equation

f= � =!!.�!2

Vvasc + Viflt�tftj� + VoslIs V(10)

where V�8�0, � and V�11� are the volumes oc-cupied by the vascular space, interstitium, and cells, re-spectively. In the brain, the vascular volume is primarily

(9) composed of the capillary bed and veins (4). Thus if a re-gion of interest in the scan is chosen so as to avoid themajor blood vessels, the measured change in CT numberwill reflect the tissue blood pool. The average concen-tration of contrast material in the tissue, c�, which is whatwould actually be measured by a CT scanner, is smallerthan the intravascular concentration c�, by the fraction f,as shown by

c;=fcv (11)

If the tissue is perfused by blood flow F, the total amountof contrast material delivered to the tissue via the arteriesis the integral of the product of Ftimes the arterial con-centration c8(O. According to our earlier assumption, thistotal amount of contrast material must be equal to theamount leaving the tissue, which is given by the integralof the product of the flow times the vascular space con-cenfration (= the concentration in the draining veins).Thus

IFCaU�)dt= I Fc�AOdt� 1Fc�(t)dt (12)

%/o JO fjo

This relationship holds even if the capillary and venousconcentrations are not equal (Appendix IV). if the flowremains steady, it will cancel out, and we find that

s:s:

Thus if we know Ca(t) (e.g., from an arterial samplingcatheter), or If a vessel large enough to minimize volumeaveraging (such as the aorta) appears in the scan plane(outside the region of interest), we can use Equation 13 tofind ffrom c�(t); and conversely if a large enough venousstructure (such as the superior sagittal sinus) can be seenon the scan, its contrast concentration can be used inEquation 13 rather than Ca(t). Note that any arterial site canbe sampled, even with a time lag, since it is the integralwhich is important. Also, the integral of tissue contrastconcentration over time tells us only the fractional vascularvolume, not the flow, unlike the analysis of central flow

Vol. 137

with the Stewart-Hamilton equation. This is because fora given value of f, although increased flow will bring morecontrast material to the tissue, it will also wash it out morequickly. Finally, it must be kept in mind that the descendingportion of the concentration curve must be extrapolatedin order to compute the remaining portion of the integralafter recirculation begins.

In practice, the arteries and often even the large veinsin the head are too small to allow an accurate determina-tion of contrast concentration directly from the scan, due

to the fact that their volume is averaged together with thatof the surrounding parenchyma on current CT scanners.Therefore, we cannot solve Equation 13 directly to find thefractional vascular volume of the brain from the scan dataalone. However, it is reasonable to assume that the inte�’al

of the arterial concentration will have the same value forall tissues in the brain. For example, in the case of uniiat-erai carotid arterial stenosis without major collateral flow,there is less flow on the side downstream from the ste-nosis. However, the arterial concentration of contrastmaterial in the vessels supplying the affected side will besimilar to that on the opposite side except for a time lag;and the integral of the arterial concentrations will be ex-actly the same by the argument used above to obtainEquations 12 and 13, replacing Ca and c�, by the concen-trations upstream and downstream from the stenosis, re-spectively. Thus the denominator of the fraction in Equation13 will be similar for all parts of the brain within the scan,and we can compare their relative blood volumes bycomparing the integrals of their vascular enhancement

CEREBRAL BLOOD FLOW DETERMINATION BY RAPID-SEQUENCE CT

able to determine t�nJ from the scans alone by employinga small enough region of interest around the image of amajor artery, since, again, it is the shape rather than theheight of the Ca(t) curve that determines t1�1; also, thechoice of t 0 is not crucial, since t can be set at anyvalue as long as the same one is used to calculate both tInJ

and t�. The mean transit time depends on both fend V/F,i.e., we must know the fractional blood volume in order tocompute flow per unit tissue volume from the mean transittime t. We can use Equation 14 to find the relative valueof ffor the region scanned, but this still gives only relativeflow per unit volume of tissue. To find the absolute valueof F, we first need to find the absolute value of f, whichmeans either (a) measuring c8(f) or c� or (b) performing

an “equilibrium” delayed scan with a simultaneous venous

blood sample to determine contrast concentration (as-suming the blood/brain barrier is intact). However, evenknowledge of the relative flow may be clinically useful,e.g., relative flows for opposite sides of the brain or regionssupplied by different vessels can be compared.

The other means of assessing flow with CT, in principle,is to use the time constant r of the washout phase in thetissue (assuming it behaves exponentially and the expo-nential phase is long enough to find r accurately) asdefined in Equation 8. This scan can be used to find theeffective flow per unit volume as in Equation 9, usingthe equation

(16)

In principle, we could instead determine f by rescanningafter waiting for the contrast material to approach equi-librium and taking a simultaneous venous blood samplefor determination of contrast concentration (10). However,enhancement may then be too low for accurate mea-surements with reasonable doses of contrast material (17),and there is a greater chance of extravascular contrastaccumulation in abnormal areas of the brain.

Let us now consider in more detail the mean transit time

method of analyzing tissue blood flow. We can find themean transit time t by Equation 5 using either c�, or c�,since t depends only on the shape of the curve and not onits height. For an instantaneous injection, mean transit timeis related to blood flow by an extension of Equation 6,

since only the vascular volume determines the transit timeof nondiffusible indicators (such as contrast material) inthe brain. However, for a prolonged arterial infusion, suchas that produced by an intravenous bolus injection, wemustuse Equation 7 to correct the observed mean transittime t� for the mean injection time t� in order to calculatethe mean transit time that would have been found with ahypothetical instantaneous bolus t. Note that we may be

Again, to find flow per unit volume of the entire tissue from(1 4) the rate of washout, we see that we need to know f; and

to find the absolute rather than relative flow, we wouldagain need to know c8(O or c� or do a delayed scan anda simultaneous venous blood sample for the contrastconcentration (once again assuming the blood/brain barrier

is intact). However, there are three additional difficultieswith this method: (a) the tissue may not behave like asimple single-compartment mixing chamber; (b) washoutcan be calculated only after the contrast infusion, whichmay not be practical with an intravenous injection; and (C)even granting an exponential washout phase before re-circulation, it may not be long enough to allow accuratedetermination of the time constant. With both of thesemethods, before flow can be determined from the scans,

the duration of the scan must be equal to or less than thetime needed for the contrast material to flow through thetissues. For a normal brain, the cerebral circulation time(artery to vein) is on the order of 3.4 seconds (3); however,angiographic densitometry indicates washout half-timeson the order of 1 second for the cerebral cortex and basalganglia and 2 seconds for the white matter (6), so thatideally one should use a scanner capable of subsecondscans.

(15)

CONCLUSION

Dynamic CT scanning may offer a useful means of

fa 5 c�(t)dt

- Vt= f-

F

VT =

F

683 Netroradiology

and hence

_t= I � (A1.5)#{149}J0

684 LEON AXEL December 1980

V=R (A1.6)

measuring blood flow noninvasively. Although many sim-plistic assumptions were made in the model presentedhere, the method derived should give clinically useful re-suIts. Promising areas of clinical application include (a)evaluation of extracranial carotid disease and follow-upof superficial temporal-artery bypass, (b) determinationof vascular spasm following subarachnoid hemorrhage,and (C) assessment of tumor blood flow as an aid to diag-nosis and treatment planning. It may also be useful instudying the basic pathophysiology of processes such ascerebral edema and infarction. Compared to regional an-giodensitometry and conventional radionuclide methods

of assessing blood flow, rapid-sequence CT has the sig-nificant advantage of being able to separate overlyingstructures; moreover, a simple intravenous injection ofiodinated contrast material facilitates calculation ofphysiological data that would otherwise require moreinvasive or difficult procedures.

Department of RadiologyRoom M-396University of California Medical CenterSan Francisco, Calif. 94143

REFERENCES

1 . Antman S: Foundations of indicator-dilution theory. [In]Bloomfield DA, ad: Dye Curves: The Theory and Practice ofIndicator Dilution. Baltimore, University Park Press, 1974,pp 2 1-40

2. Drayer BP. Wolfson SI< Jr. Reinmuth CM, et al: Xenon enhancedCT for analysis of cerebral integrity, perfusion, and blood flow.Stroke 9:123-130, Mar-Apr 1978

3. �eltz T: Normal cerebral circulation time as determined bycarotid angiography with sodium and methylglucamine diatrizoate(Urografin). Acts Radiol [Diagn] 7:331-336, Jul 1968

4. Hilal SK: Cerebral hemodynamics assessed by anglography.[In] Newton TH, Potts DC: Radiology of the Skull and Brain. An-giography. St. Louis, Mo., Mosby, 1974, Vol 2, Book I, Chapt54, pp 1049-1085

5. Hllal SK: Densftometric study of the regional cerebral circulation

in intracranial tumors and arteriovenous malformations. [IniTaveras JM, Flschgold H, Dilenge D. ed: Recent Advances in theStudy of Cerebral Circulation. Springfield, III., Thomas, 1970,Chapt 14, pp 136-155

6. Hilal SK, Aesch JA, Amplatz K: Determination of the carotidand regional cerebral blood flow by a radiographic technique.Acta Radiol [Diagn] 5:232-240, 1966

7. Ingvar OH, Cronqvist S. Ekberg A, et al: Normal values of regionalcerebral blood flow in man, Including flow and weight estimatesof gray and white matter. A preliminary summary. Acta NeurolScand 41 (Suppl 14):72-78, 1965

8. Kaufman L, Shames DM, Greenspan RH, et aI: Cardiac outputdetermination by fluorescence excitation in the dog. Invest Rarliol7:365-368, Sep-Oct 1972

9. Kormano M, Dean PB: Extravascular contrast material: the majorcomponent of contrast enhancement. Radiology 121:379-382,Nov 1976

10. Ladurner 0, Zilkha E, Sager WD, et ai: Measurement of regionalcerebral blood volume using the EMI 1010 scanner. Br J Radiol52:371-374, May 1979

1 1 . Lagemann K: Pharmakokinetik anglographischer Kontrastmlttelunter besonderer BerUckslchtigung des extravasalen Raumes.Eine experimentelle �undlagenstudie an Hunden zur Charak-terisierung der Angiographica. II. Mitteilung: Pharmakokinetikeines angiographlschen Kontrastmittels unter den Bedingungeneiner selectiven-’angiographischen’-Appllkation. FortschrGob Roentgenstr Nuklearmed 123:515-521, Dec 1975

12. Lantz B: Relative flow measured by roentgen vldeodensftometryin hydrodynamic model. Acts Radiol [Diagn] 16:503-5 19,Sep 1975

13. Meler P. Zierler KI: On the theory of the indicator-dilution methodfor measurement of blood flow and volume. J Appl Physiol 6:731-744, Jun 1954

14. Newhouse JH: Fluid compartment distribution of intravenousiothalamate In the dog. Invest Radiol 12:364-367, Jul-Aug1977

15. Norman D, Berninger W, Boyd D, et al: Dynamic computedtomography. Presented at the XI Symposium Nettoradlologi-cum, Wiesbaden, Germany, June 4-10, 1978

16. Phelps ME, Grubb RL Jr, Ter-Pogossian MM: In vivo regionalcerebral blood volume by x-ray fluorescence: validation ofmethod. J AppI Physlol 35:741-747, Nov 1973

17. Phelps ME, Kuhi DE: Pitfalls In the measurement of cerebralblood volume with computed tomography. Radiology 121:375-377, Nov 1976

18. Sorensen MB, Bille-Brahe NE, Engell HC: Cardiac output mea-strement by thermal dilution: reproducibility and comparison withthe dye-dilution technique. Ann Surg 183:67-72, Jan 1976

19. Zierler KL: Circulation times and the theory of indicator-dilutionmethods for determining blood flow and volume. [ml HamiltonWF, ed: Circulation. Handbook of Physiology, Sect 2, Vol 1.Washington DC, American Physiological Society; Baltimore,Williams & Wilkins, 1962, Chapt 18, pp 585-615

20. Zierler KL: Equations for measuring blood flow by externalmonitoring of radioisotopes. Clrc Res 16:309-321, Apr 1965

21 . Zierler KL: Theoretical basis of indicator-dilution methods formeasuring flow and volume. Circ Res 10:393-407, Mar 1962

APPENDIX I

To demonstrate that the mean transit time is the ratio of volume toflow, I follow the general argument of Meier and Zierler (13). For a bolusinjection of a mass m of the indicator (contrast material) at time t 0,there is a range of times at which particles of the indicator leave thesystem (tissue). The distribution function Ii(t) is defined as

h(o=�;!! (A1.1)

where t�O is the fraction of indicator leaving the system per unit of time,F is the fluid flow, and �l) is the concentration of the exiting fluid.Accordin9 to Equation 3, this is equal to

1*0= c(fl (A1.2)

s:c�t)dtSince we assume the fluid (blood) particles move in the same manneras the indicator particles, h(t�also gives the distribution of the timedurations for the fluid particles to traverse the tissue, i.e., the fractionof the fluid particles remaining in the system for total time t . . . t + dtis f4tjdt. Since fluid enters (and leaves) this system at a rate F, thevolume occupied by fluid particles requiring total times between tandt + dtto traverse the system Is

dVFtl�t)dt (A1.3)

Thus the total volume occupied by fluid flowing through the systemis

From Equation 5,

VF ffh(�t (A1.4)#{149}Jo

1 _ss: � �

1i�S t�o�it+�5#{176}’ t�t�it- ...j

x[i_s5t�Odt+�j5t2�fldt_...] (A3.8)

Vol. 137 CEREBRAL BLOOD FLOW DETERMINATION BY RAPID-SEQUENCE CT 685 Nettoradk�ogy

or

- V

t F

APPENDIX II

(A1.7)

The first moment of the draining venous concentration of an indicatorafter a bolus arterial input (Equation 5) gives the mean transit time ofthe system (Appendix I) and thus the flow per unit volume accessibleto the indicator. While the effect of a prolonged input can be readilycorrected for (Appendix III), with CT it is actually the residual indicator(contrast material) in any tissue (primarily the capillary bed and extra-vascular space) rather than the concentration in the draining veins thatis measured; for a bolus injection, the exit concentration of the draining

veins will be proportional to the time derivative of the residual tissueconcentration (20). With perfect mixing, the residual indicator con-centration is an exponential function (Appendix IV); thus the exit con-centration will be an exponential function of the same shape, and thefirst moments of the tissue and venous concentrations will be equal.If mixing is poor, there will be a delay before contrast material appearsin the draining veins; for a bolus injection, the residual tissue concen-tration will show an initial plateau and then fall off while the initial exitconcentration will be low, rising to a peak when the residual concen-tration is falling most rapidly. The worst case would be transit of the

bolus injection through the capillaries like a piston with simultaneousentry into the veins, so that the tissue concentration would be constantuntil time T and then drop to zero while the exit concentration wouldbe a brief bolus” at time T. The resulting first moments would differby a factor of 2, with the first moment of the tissue concentration onlyhalf the actual mean transit time. In general, tissue concentration curveswould be somewhere in between these two extremes, so that the actualmean transit time of the system should be between 1 and 2 times thecalculated first moment of the tissue concentration response to a bolusinjection. The shape of the tissue response curve is consistent, so thatthere may be a simple correction factor in the range of 1 to 2 that could

be used to multiply the calculated first moment of the tissue concen-tration so as to estimate the actual mean transit time.

and

�o=

s:(A3.2)

and the convolution f*g is defined as above. The mean transit time(first moment) Is defined as

The Laplace transform

�f= s:��g= s:�

-tf*g= ftf*�t

L(t�=#{149}.10

(A3.3)

(A3.4)

of the convolution of two functions is equal to the product of the twotransforms

or

L(f� g) LU�L(g) (A3.5)

s: e�stt* �1t= s:e�st��tS� e�t�t�it

We can expand these exponentials as power series:

=5(i_st+��_...)��t

or, separating the integrals,

1” ( �2�2 �)�t)dtx 1-st+---..

�/0 � 2!

APPENDIX Ill

If instead of an instantaneous contrast injection at time t 0, thereis a more general input function, we can consider it to be composedof a continuous series of instantaneous injections, with the tissue re-sponse to each one being superimposed. The resulting tissue con-centrations as a function oftime are given by the convolution ofthe inputfunction I�O and the response for an instantaneous bolus � wherethe convolution is defined as

f*g1�t)*g(t) c I�s)�t-s)ds (A3.1)Jo

Given the input function and corresponding tissue response, it is pos-

sible to compute the tissue response to an instantaneous bolus. How-ever, as we only need the mean transit time of the bolus response (thefirst moment as defined in Equation 5), things are much simpler; themean transit times of the observed tissue response and input function

can simply be subtracted without requiring a full deconvolution. Althoughthe additive property of the first moments of convolved functions is oftencited as a general property of distributions (19, 21), a direct demon-stration is presented here. For two functions, F(t� and � the nor-malized equivalents are defined as

itt) F(t�

s:

We recognize the mean transit time in the second terms:

= 1 - S(�f + t9) + . .. (A3.9)

Equating coefficients of like powers of s in the two power series, wefind for the second term on each side that

tfsg 4 + t� (A3.10)

i.e., the mean transit times of two convolved functions are additive.

APPENDIX IV

In order to demonstrate that the area under the curve representingindicator (contrast material) concentration as a function of time is thesame for the capillaries as it is for the arteries, consider a brief arterialbolus of duration L�t, with constant concentration c� during this time.

V 5c��(Odt_2!2= (A4.4)F Fc,�,4t

Vcap

or

S: c�(fldt = c�4t

A4 2 The right-hand side of the last equation is merely the area under the( . ) arterial Indicator concentration curve. Since we can consider any ar-

bitrary arterial Input as composed of a series of such brief injections,if the resulting capillary indicator concentrations can be superimposed,

the areas should still be equal, i.e.:

s: c�t= s: c�,dt

This same argument can be extended in a straightforward manner to

(A4.3) demonstrate that the area under the capillary Indicator curve is alsoequal to the area under the venous indicator concentration curve.

686 LEON AXEL December 1980

For a steady flow F through the tissue, this is equivalent to a bolus

injection of quantity q� of contrast material with

q�Fc�4t (A4.1)

which will be diluted into the capillaries. Although mixing may not becomplete in the capillary blood volume V�, the effect of the finiteresolution of CT scanners is such that the effective initial concentrationCo of contrast material in the capillary bed Is

Fc�4tco= Vc

The mean transit time tttWOUgh the capillary system is equal to the ratioof volume to flow and also to the area under the curve representingcapillary contrast concentration c� divided by c� (20)(correcting forrecirculation): thus

t=!r= 5c�(Odt