CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



605 



to fit some more specific expression, and there is 

 the further advantage that comprehension of the 

 distribution function might reveal something about 

 the nature of laws governing distribution of blood 

 through vascular nets. There are two ways to ap- 

 proach the problem. One is to examine the con- 

 centration-time curves obtained experimentally and, 

 by curve-fitting, derive an empirical expression for 

 the concentration as a function of time or some 

 other useful relation. The other is to assume that 

 certain given laws govern the distribution, predict 

 the distribution function from these laws, and test 

 the observed concentration-time curve for goodness 

 of fit. 



Empirical Expressions 



Perhaps the first empirical expression is attributable 

 to Allen (i), the hydrologist to whom reference was 

 made earlier, who elected to treat his observed 

 concentration-time curves (following sudden-injec- 

 tion) as triangles. The area is, of course, then simply 

 l^^Cp (T — a), where Cp is the peak concentration, a 

 is the appearance time, and T is the time at which 

 concentration returns to zero; t is simply J-^ (T -\- p + 

 a), where p is the time at which peak concentration 

 occurs. Allen's penstock flows must have been 

 turbulent, since a was very long compared to (7" — a) 

 and it did not make much diff'erence what time was 

 used in the interval a to T. These approximations, 

 though perhaps adequate for their original purpose, 

 are apt to be unsatisfactory for biological application 

 owing to the large error and to the fact that recircula- 

 tion obscures the curve so that T may not be esti- 

 mated with confidence. 



Indeed, most attempts at empirical definition of 

 the indicator concentration-time curve have been 

 inspired by the practical necessity of extracting the 

 primary curve from the observed curve obscured by 

 recirculating indicator. 



Hamilton's observation, cited previously, that the 

 downlimb of the sudden-injection curve was ex- 

 ponential, or reasonably close to exponential, was 

 made on the heart-lung preparation in the dog. 

 The exponential fit has been verified repeatedly, 

 although not invariably, for the sudden-injection 

 concentration-time curve obtained when indicator 

 was distributed through the heart and pulmonary 

 vascular bed in the intact dog and in man. However, 

 when recirculation begins shortly after peak con- 

 centration is attained it is impossible to estimate 

 accurately the slope of the exponential downlimb, 

 and extrapolation by this method cannot be used. 



For this reason, several students of the problem asked 

 whether or not indicator-dilution curves committed 

 themselves, so to speak, as soon as they had written 

 the rising limb and reached peak concentration. 

 To this end, Dow (5) examined a large number of 

 indicator dilution curves obtained by sudden-injec- 

 tion into the cardiopulmonary circuit in dogs and 

 in man. After testing various combinations of a 

 number of measurable factors, Dow reported that 

 the following formula correlated best with the area 

 under the concentration-time curve, corrected for 

 recirculation: /)Cp/[3 — (0.9 /;/«)], whence 



^ = qls - {o.g p/a)]/pCp 



(47) 



where p is the time (after injection) at which peak 

 concentration, Cp , occurs and a is appearance time. 



Keys et al. (16) found that in their data on man, 

 Dow's formula systematically underestimated flow, 

 calculated from F = q/jg c(t) dt, where c{t) is cor- 

 rected for recirculation. 



Use of />, Cp, and a in formulas designed to estimate 

 area arises naturally from the fact that the indicator 

 concentration-time curve approximates a linear 

 rising phase to a peak followed by exponential decay. 

 Concentration as a function of time in this approxi- 

 mation is 



o, t < a 



c{t) = {cp(t - a)/(p - a),a ^ t ^ p 



Cp (>"'<'~f> , p ^ t 



(48) 



The area under the curve is Cp[(p — a) / -2 -\- i/A'] 

 which is to be compared with Dow's empirical 

 formula. 



A closely related approach was taken by Hetzel 

 et al. (15), who compared the area of what they 

 called the forward triangle with the total area of the 

 curve corrected for recirculation. The forward triangle 

 is defined by the values which c{t) assumes in equa- 

 tion 48 for a ^ < ^ /) and its area is Cp (p — a)/2. 

 The ratio of the forward triangle to the total area 

 was determined in a number of experiments in man 

 in which indicator traversed the heart and pul- 

 monary vascular bed. The ratio varied somewhat 

 with the site of injection but in general was sur- 

 prisingly stable at about 0.35. If equation 48 is a 

 reasonable approximation of c(t), the ratio is 

 k{p — a)/2 + k{p — a). The fact that the observed 

 ratio was about I-3 implies that k ;^ i/(p — a), and 

 this is an anticipated result because, as we shall see 

 later, k is related closely to ; — a, and J is sufficiently 

 near p to make it plausible that some empirical 

 correlation might be found. 



