A MATHEMATICAL MODEL FOR 
MEASURING BLOOD FLOW BY RESIDUE 
DETECTION WHEN RADIOTRACER 
RECIRCULATION INTERFERES 
K. B. Larson and D. L. Snyder* 
We have developed a mathematical model for meas- 
uring blood flow in a vascular system by external moni- 
toring of radiotracers when tracer recirculation is not a 
late event and must be taken into consideration. Addi- 
tionally, the model accounts for interfering recircula- 
tion of tracer to other perfused tissues also within the 
detector field of view. Central to the model is the use of 
two injections of tracers — an intra-arterial and an intra- 
venous — upstream and downstream of the particular 
organ or region of interest. Thus, two residue curves 
are obtained. We have developed equations indicating 
how to employ the two residue curves in order to deter- 
mine the mean transit time of tracer through the vascu- 
lar system of interest, as well as higher moments of the 
transit-time distribution if these are desired. The requi- 
site numerical calculations need not rely on curve-fitting 
of the data. Although our method can yield compart- 
mental parameters if these are appropriate and desired, 
our equations do not depend for their validity on any 
model of tracer transport within the system of interest. 
We describe some experiments designed on the basis of 
our dual-injection method to account for recirculating 
tracer in blood-flow measurements of various organs of 
anesthetized dogs, and give some indication of the 
agreement between blood-flow values predicted on our 
theory with those determined independently. 
INTRODUCTION** 
Zierler^ was the first to show, without invok- 
ing special assumptions as to transport mecha- 
nisms in a vascular region of interest, that the 
ratio of blood flow to volume of distribution of a 
radioactively-labeled indicator could be meas- 
ured by external monitoring of the radioactiv- 
ity following rapid arterial injection of the in- 
dicator. He has called this technique residue 
* School of Medicine, Washington University, Saint Louis, Mis- 
souri. 
** The theoi'etical studies leported herein were supported by Na- 
1 tional Institutes of Health Research Grant RR-00396 from the Divi- 
sion of Research Resources. 
detection, contrasting it with outflow detec- 
tion.2 The latter is synonomous with the 
term "indicator dilution," in which the flow- 
weighted concentration history of tracer in the 
venous blood is continuously monitored by use 
of a catheter or by other means. 
The mathematical analyses central to the in- 
terpretation of kinetic tracer data in most of 
the circulation studies reported in the literature 
have been based on models which, in the inter- 
est of simplicity, have ignored recirculation. 
Such models have been widely employed both in 
residue and in outflow detection. An important 
limitation of these models is that they can be 
used successfully only in those situations in 
which the major portion of the response curves 
are obtainable prior to onset of the first recircu- 
lation of tracer. A further complication which 
limits the usefulness of the models used in resi- 
due detection is due to interference from radio- 
tracer recirculating into perfused regions ad- 
jacent to the system of interest and within the 
field of view of the detector. 
When recirculation is a late event, an arbi- 
trary extrapolation of the primary response 
curve does not result in large error.^ However, 
when recirculation appears relatively early 
after injection of tracer, the primary response 
curve cannot be recovered confidently in this 
simple fashion. In such a circumstance, a prin- 
ciple first suggested by Stephenson^ can be 
applied. According to this principle, injec- 
tions of tracer and concentration measurements 
are required at multiple sites in the cardiovas- 
cular system. This permits specific account of 
recirculation to be taken ab initio in the mathe- 
matical model, and allows the transit-time dis- 
tribution for the vascular system of interest to 
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