K. B. LARSON AND D. L. SNYDER 
1153 
tion of radioactive tracer at the venous outflow. 
We show our method to be valid when there is 
recirculation of tracer to the system of interest 
whether or not adjacent tissues are also within 
the detector field. 
Figure 2 represents the flow model in more 
detail than Figure 1. Let the interior, D, of the 
circle in this figure represent the field of view 
of a radiation detector used in a residue-detec- 
tion measurement. An arbitrary portion of ei- 
ther the systemic or the pulmonary circulations 
is contained within D. Flow elements, which 
may be blood vessels, vascular labyrinths, vas- 
culatures of entire organs, or the entire sys- 
temic or pulmonary circulations, are character- 
ized by their system functions h(t), G(t), and 
F(t). These functions are the transit-time 
frequencies,*'^ which characterize the various 
flow elements. Let h(t) represent the system 
whose flow parameters we wish to determine, 
and let an arbitrary number of systems G(t) be 
also present within D. Let h (t) and the G (t) be 
interconnected to form a flow system of arbi- 
trary complexity within D. Let the composite 
system consisting of all systems outside D have 
DETECTOR 
FIELD, D 
Figure 2. — General Model for Residue Detection. 
a resultant transit-time distribution F(t). The 
component subsystems in F(t) can be arbitrary 
and can intercommunicate in an arbitrary fash- 
ion. Recirculating tracer can enter and leave the 
region within D and the system F(t) by means 
of an arbitrary number of inlets and outlets 
through which the fluxes x(t) and y(t) can be 
convective or diffusive. Let the rates at which 
tracer is metabolized, sequestered or otherwise 
eliminated from the circulation be given by 
z (t) . Directions of tracer flux between elements 
in Figure 2 are indicated by arrows. The inter- 
connecting lines can be thought of as delayless 
and dispersionless transfer paths, with the dou- 
ble lines representing multiple routes. The sin- 
gle lines associated with h(t) refer to the fact 
that we are concerned here with describing a 
method of measuring flow only in systems hav- 
ing a single convective inlet and a single convec- 
tive outlet. Diffusion of tracer across the bound- 
aries of the system of interest is inadmissible if 
the mean transit time obtained from the meas- 
urements is to be used for determining blood 
flow. (Systems with multiple convective inlets 
and outlets may be treated by the present 
method through the use of more than two injec- 
tions; however, we shall not pursue this ques- 
tion here.) 
The usual requirements for the validity of 
tracer stimulus-response methods concerning 
linearity and stationarity are assumed to hold 
for the systems we consider here. Thus, the ele- 
mentary and composite systems depicted in Fig- 
ure 2 are assumed to be linear only with respect 
to tracer stimuli and responses, a condition eas- 
ily met.*' The systems are not required to be lin- 
ear with respect to changes in concentration 
nor with respect to perturbations in flow rates 
of traced substance. Instead, the system is re- 
garded as being in a steady state, that is, con- 
centrations and flow rates of traced substance 
are assumed to be time-invariant. It is further 
assumed that h (t) for the tracer is the same as 
that for the substance being traced. This im- 
plies that no changes in chemical properties of 
the label which would affect its kinetic behavior 
within the detector field are permitted. Thus, if 
the label undergoes metabolic changes any- 
where outside the detector field D, none of the 
products of metabolism incorporating radioiso- 
