1152 
MONITORING 
be recovered by means of a deconvolution proce- 
dure. 
Because of certain relative advantages of- 
fered in some cases by the use of residue detec- 
tion, it appears useful to extend its application, 
via Stephenson's method, to measurements in 
which early recirculation interferes with the 
primary response. The purpose of this paper, 
therefore, is to present theoretical results which 
allow measurement of blood flow by residue de- 
tection when recirculation cannot be ignored. In 
it, we discuss the mathematical analysis by 
means of which mean transit time and higher 
moments of the transit-time distribution of 
blood flow in a vascular bed can be deduced 
from externally-monitored radioactive tracer 
being transported through the bed by first and 
subsequent circulations. We show that in cer- 
tain favorable cases, our method is capable of 
accounting for interference due to recirculating 
radiotracer carried into perfused regions 
within the field of view of the detector and ad- 
jacent to the vascular bed of interest. No appeal 
to any transport model is made in our analysis. 
The results are therefore of general validity in 
the sense that they are independent of any spe- 
cial assumptions as to the transport mecha- 
nisms involved. In particular, we do not invoke 
the requirement of fully dispersed flow basic to 
compartmental analysis, nor do we make any 
assumptions as to the degree of equilibration of 
tracer between vascular and extravascular 
spaces. Because it does not depend on transport 
models, our method implies nothing about the 
shapes of experimental response curves. Conse- 
quently, the computational procedures for de- 
ducing the mean transit-time and higher mo- 
ments of the transit-time distribution from ex- 
perimental data can be exclusively numerical if 
so desired. 
MATHEMATICAL MODEL 
For the present application, the circulation is 
properly modeled as a closed-loop flow system. 
Figure 1 is a schematic representation of the 
cardiovascular system in its essentials. The my- 
ocardial circulation is considered as encom- 
passed within the system labeled systemic cir- 
culation, while the components labeled right 
heart and left heart are considered to represent 
flow systems consisting of the blood within the 
heart chambers. All blood vessels are regarded 
as being components of either the systemic or 
the pulmonary circulations, so that the arrows 
leading between components may be considered 
merely as delayless and dispersionless transfer 
lines. 
In the residue-detection method of measuring 
blood flow,- a bolus of radioactively-labeled 
tracer is injected into the inflow of the system 
of interest, and the subsequent content of tracer 
in the system is monitored as a function of time 
by means of a suitable radiation detector coUi- 
mated over a region containing the system. In 
many cases, recirculating radioisotope inter- 
feres with the primary clearance record which 
would be obtained in the absence of recircula- 
tion. This interference occurs in two ways, viz., 
(a) by contributing added radioactivity to the 
system of interest, and (b) by contributing 
added radioactivity to other systems adjacent to 
the system of interest which are also within the 
field of view of the detector. We have found that 
the problem of obtaining the transit-time distri- 
bution and its moments for a particular system 
by means of residue detection in these circum- 
stances can be solved through application of 
Stephenson's principle. For this purpose, we 
employ, in addition to an arterial injection up- 
stream of the system of interest, a second injec- 
SYSTEMIC 
CIRCULATION 
LEFT 
HEART 
RIGHT 
HEART 
PULMONARY 
CIRCULATION 
Figure 1. — Schematic Representation of the Mam- 
malian Circulation. 
