1154 
MONITORING 
tope may be permitted to enter D by recircula- 
tion; instead, they must be sequestered or 
eliminated. 
The radiotracer stimuli are injected at rates 
a (t) and v (t) at the arterial and venous sites, 
respectively. The injection experiments are de- 
scribed mathematically as follows : 
Injection 
experiment 
A 
Injection 
experiment 
V 
(a(t) = ijt) u(t), 
'v(t) =0, 
ia(t) = 0, 
(1) 
(2) 
/v(t) = i,(t) u(t). 
Here, u (t) is the unit positive step function de- 
fined by 
u(t) = 
1, t > 0, 
0, t < 0. 
We thus identify t = 0 as the start of injection 
for each experiment. The injection rate func- 
tions ia(t) and iv(t) may have any forms pro- 
vided only that the integrals 
00 
f^= f t°ia (t) dt 
0 
and 
00 
/ f>iv(t)dt,n = 0,1,2, 
0 
(3) 
defining their nth moments exist. In particular, 
Ho and vo must represent, respectively, the total 
amounts of tracer qna and Qov administered in 
the two injections. The requirements for the lo- 
cation of the two injection sites are: (a) that 
the entire upstream or arterial bolus, on first 
transit of the detector field D after injection, 
pass through h(t) ; and (b) that the entire 
downstream or venous bolus, on first transit of 
D after injection, follow the same route (s) as 
departing tracer from the first-transit arterial 
bolus. Stated differently, between the two injec- 
tion sites, only the system of interest may inter- 
vene. A further equivalent interpretation is 
that the present method yields the transit-time 
distribution of the composite system lying be- 
tween any two injection sites satisfying condi- 
tions (a) and (b) above. Thus, for real vascu- 
lar systems in which injections of tracer must 
be made into blood vessels at locations some- 
what removed from the vascular bed of interest, 
the measured transit-time distributions will re- 
flect additional delay and dispersion of the bol- 
uses due to their passage through these vessels. 
In view of the foregoing discussion, sites 1 and 
2 in Figure 2 are seen to be appropriate loca- 
tions for the arterial and venous injections, 
while sites 3 and 4 are not. 
The responses corresponding to the stimuli 
described by equations (1) and (2) are denoted 
by qa(t) and qT(t), respectively. These func- 
tions represent the amount of radiotracer pres- 
ent within the detector field at any time after 
the beginning of injection experiments A and 
V, respectively, in excess of any tracer already 
present in D from previous injections. The de- 
tector responses observed will be proportional 
to qa(t) and q^it) provided the effective detec- 
tion coefficient remains time-invariant. Some 
response curves similar in form to those which 
might be obtained from the two injection exper- 
iments indicated in Figure 2 are shown in Fig- 
ures 3 and 4, where the normalized responses 
and 
qa(t) = qa(t)/qoa (4a) 
qv(t) = qv(t)/qov (4b) 
are plotted. The hypothetical response curves in 
^{t) 
Figure 3.— Hypothetical Arterial (a) and Venous (v) 
Detector Responses for Model of Figure 2 without 
Elimination of Tracer. 
