CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



6ll 



an observed curve in which recirculation is not promi- 

 nent, it does not follow that the same distributive 

 law holds in those cases in which recirculation is 

 prominent (and in which goodness of fit cannot be 

 tested), nor that it holds for other vascular beds. 



WASHOUT FROM A MIXING CHAMBER 



When Hamilton et al. (17) discovered that the 

 downlimb of the sudden-injection indicator-dilution 

 curve from the dog heart-lung preparation could be 

 fitted by a simple exponential they recognized the 

 model which the fit described. Whenever a quantity 

 decays exponentially, the equivalent statement is 

 that the rate of decay is proportional to the amount 

 remaining. 



Consider a container of volume, f, filled with 

 fluid and constantly well stirred. Introduce a quantity, 

 9„ , of indicator into the container at zero time. Its 

 concentration is qo/V. Now let fluid begin flowing 

 through the container at constant rate, F, in such a 

 way as to mix it immediately and thoroughly with 

 the contents of the container. Indicator remaining 

 in the container at time / is q{t) and its concentration 

 is q{t)/V. The rate at which indicator leaves the 

 container is F q{t)/V, but this is also the rate at 

 which the quantity of indicator remaining in the 

 container decreases, or 



dq{t) 

 dt 



. q{t) = F C{t) 



On integration this yields 



C{t) -- C^e y 



(64) 



(65) 



where Co = qo/V. 



Newman and his colleagues (21) suggested that 

 equation 65 could be used to measure the volume 

 of the segment of the cardiovascular system through 

 which indicator flowed. The slope, k, of the ex- 

 ponential downlimb is determined graphically, 

 equated to F/V. F is determined in the conventional 

 manner from the relation F = qjo C{t) dt, and V 

 is obtained. 



It is obvious, however, that equation 65 is de- 

 cidedly not the usual indicator-dilution curve. It 

 begins at time zero, at which it is maximum, and 

 falls monotonically. If appearance time, a, is included 

 in the model, equation 65 becomes a lag exponential 

 of the form 



C(t) = 



o, t < a 



I > a 



(66) 



where Co is concentration for t — a and is not, as 

 we shall see, q„/V. 



When a = o, k is indeed i/l, not only because 

 / = V/F, and so the relation follows from the de- 

 velopment of equation 65, but also because t of an 

 exponential is i/k, as can be shown by multiplying 

 both sides of equation 65 by /, integrating between 

 t = o and < = 00 , and dividing by q/F. 



This relation is not true for a lag exponential. 

 When I is determined from equation 66, k = i/{t — a). 

 If k is now equated falsely with F/V, the estimate of 

 V will be too small. For example, I is usually approxi- 

 mately equal to 2 a for the central circulation in 

 dog and in man. For t = 2 a, k in equation 66 equals 

 2 F/V, that is, the volume falsely estimated from the 

 downslope k is only half the real volume. 



Furthermore, as we have hinted, Ca in equation 

 66 is not the initial quantity of indicator divided by 

 the volume. The proof follows. 



Recall that, because all the indicator must come out 

 eventually, q = Ff^ C(t) dt = Ca Fja e-'^c-"' dt. 

 Equating the integral on the right to q/F €„ and 

 integrating, Ca = qk/F.Butk = i /(/ — a). Therefore, 



Ca = ql{V - aF) 



That is, the initial concentration, C^ , of an indicator- 

 dilution curve which is a lag exponential, is greater 

 than q/V. 



Let us examine the physical meaning of this more 

 closely. If we have two exponential curves which 

 are superimposable, e.xcept that one is maximum at 

 zero time and one is displaced in time so that its 

 maximum (and first non-zero) value is at time a, 

 then Co = Ca , where Co is the value of the concentra- 

 tion at zero time for the first curve and Ca is the value 

 at time a for the lag exponential. Since the slope, k, 

 is the same for the two curves, k = 1/^1= 1/(^2 ~ a), 

 where subscript i refers to the first curve (beginning 

 at / = o) and subscript 2 refers to the second curve 

 (beginning at / = a). Therefore, ~ti = ti — a. If 

 the flow through the two systems is the same, then 

 the volumes must be unequal. Then, tx = V\/F, 

 h = V./F = /"i + a. Whence V. = Fi h/ih - a), 

 that is. To is larger than I'l . If the initial quantity of 

 indicator is dissolved in V^ so that its concentration 

 Ca = Co = q/Vi , it cannot, therefore, be distributed 

 evenly through Vo . Indeed this is the physical mean- 

 ing of the lag in the lag exponential. During the 

 lag period, a, a volume of fluid equal to a F, flows 

 out of the system. The indicator is therefore dis- 

 tributed initially only in a quiescent volume equiva- 

 lent to the difference between the total volume and 

 the volume a F which escapes the system. 



