CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



6.3 



form, but its slope would not, in general, be related 

 to the volume through which the distribution func- 

 tion is exponential. 



For example, if /;(/) is the distribution function 

 through a parabolic flow system, J/2 /', then for an 

 instantaneous mixing chamber in series with a 

 parabolic flow system, 



Cd) = 



qk 



f 



-k(l- 



2 f2 



where h is over-all appearance time (= a -{- }-2 t-i) 

 and ^2 is mean transit time through the parabolic 

 flow system. Integration yields C(t) as an infinite 

 series : 



CU) = 



+ ke- 



-k ((-a) 



In t/h 



I ! 



2 ! 



X-3 (fl 



+ 



3 ■ 3 



A«) 



4- 



For t = b, C(b) = o, and the limit of C(t) as < — > 00 

 is also zero. A clearer notion of the function follows 

 from inspection of its derivative which simplifies to 



dC(t) 



q k li 



1 F P 



k C{t) 



(68) 



When t = h, fl C{t)/dt = q k i, e'' "/a F h- > o, that 

 is, C(t) increases from zero. It reaches a maximum 

 value at C(p) = q <■> c*" "/2 F p'-, from which it will 

 fall more slowly than a simple exponential, owing 

 to the positive term in the right-hand member of 

 equation 68. 



MODELS CONCERNED ONLY WITH THE HEART 



ejected forward and there is no regurgitation. At 

 the end of systole there remains in the ventricle a 

 volume r,, = r — Ve . During diastole, a volume 

 equal to ]'e , containing no indicator, flows from the 

 auricle into the ventricle. The concentration of 

 indicator at time zero, at the end of diastole, is q/V. 

 A quantity q Ve/]' is ejected with the first systole. 

 Therefore a quantity q — q (Ve/V) = 9 [i — (Vb/V)] 

 remains in the ventricle. By the end of the first 

 diastole this is distributed in a volume V at con- 

 centration [i — (Ve/V)] q/V. During the second 

 systole, a quantity (q Ve/V)[i — (Ve/V)] is ejected, 

 and the quantity remaining is 9 [i — (Ve/V)] — 

 q Ve/V [i - (I'^/F)] = q[i - (Ve/V)^. During 

 the i'-th diastole the quantity of indicator remaining 

 in the ventricle is 9 [i — (Ve/V)]' and its concentra- 

 tion during the i-th. systole is q/V[i — (Ve/V)Y = 

 1/^ (Vr/V)'. If the heart rate is constant at, say, 

 k beats per unit time, then the concentration of 

 indicator at ventricular outflow during systolic 

 ejection is C(t) = q/V {Vr/VY", where for n k -^ 

 t < (n -\- i) k, and n is an integer, / assumes the 

 value of n. 



As the situation is permitted to become more compli- 

 cated, by introducing regurgitant flow, for example, 

 concentration will appear as the sum of various 

 combinations of volume, each term, in general, con- 

 taining t as a factor in the exponent. (Actually, 

 the exponent is dimensionless because k has dimension 

 I /time. In fact, because flow equals stroke volume 

 and beats per unit time, F = k Ve, and A:, therefore, 

 is \/t.) 



A neat solution for a two-chambered system, in- 

 cluding regurgitant flow, is given by McClure and 

 his colleagues (18). 



If indicator distribution is limited to the chambers 

 of the heart, then Newman's model becomes more 

 plausible, and may do very well as a first approxi- 

 mation despite evidence that mixing of blood is 

 indeed not complete and immediate in the right 

 ventricle. Interest in this application has been stimu- 

 lated by clinical concern for a method with which to 

 quantify valvular incompetence and the amount of 

 regurgitant flow from one chamber back to another. 



For the simplest case, which will suffice to demon- 

 strate the nature of the general argument, consider 

 that an amount of indicator, q, is introduced into a 

 ventricle during diastole. The end diastolic volume, 

 with which indicator is assumed to be mixed com- 

 pletely, is V. During systole a quantity of blood, Ve, is 



SUMMARY 



When an indicator is introduced into a flow 

 system in such a way as to be distributed at once 

 throughout the inflow (for sudden-injection, concen- 

 tration at inflow is q/F dt; for constant-injection, 

 concentration at inflow is I/F), then, providing it 

 does not recirculate before the concentration at 

 outflow has returned to zero, the curve of its con- 

 centration as a function of time is exactly the shape of 

 the distribution function of transit times through 

 the system. The area under the sudden-injection 

 indicator-concentration-time curve, in the absence 

 of recirculation, is q/F, and so provides a measure of 

 flow. The maximum, and plateau, concentration of 



