K. B. LARSON AND D. L. SNYDER 
1155 
q(t) 
1 - 
0 
Figure 4. — Hypothetical Arterial (a) and Venous (v) 
Detector Responses for Model of Figure 2 with Elimi- 
nation of Tracer. 
each of the figures are shown exhibiting multi- 
ple recirculation peaks, and each curve is shown 
approaching, with increasing time, a common 
normalized steady-state value, defined by 
= qa(oo) = qv(oo). (5) 
Here qa(oo) and qv(co) are the normalized 
steady-state values of the amount of tracer 
within D for the two injection experiments. 
These can be nonzero only when there is no loss 
of tracer from the system during the course of 
the measurements. In this case, the tracer elimi- 
nation or sequestration rate is zero. When this 
rate is positive, the final steady-state values 
q„ ( 00 ) and q,- ( oo ) will be zero. This is the situ- 
ation depicted in Figure 4. 
MEAN TRANSIT TIME 
I A detailed analysis of the model described 
! above appears elsewhere.^ We provide here the 
I principal results which we have deduced from 
the model by applying the principle of conserva- 
tion of tracer mass together with elementary 
linear systems theory. 
The first moment 
I J th(t)dt (6) 
0 
of the transit-time distribution, or mean trans- 
it time,^ is given on our model by 
00 
t= (l-q...)-i J [q„(t) -q,(t)]dt. (7) 
0 
(Our model is also capable of yielding higher 
moments of h(t). In Reference 7 we give ex- 
plicit expressions for these. We show further 
that by using the higher moments it is possible 
to compute the parameters of a model conjec- 
tured to represent the transport processes for 
the region of interest. The particular represen- 
tation selected in Reference 7 to demonstrate 
such a determination is the multicompartmental 
model.) 
The above result allows the mean transit time 
to be obtained from the experimental responses 
by purely numerical methods, without appeal to 
any special assumptions regarding the shapes 
of the response curves qa(t) and qv(t). Equa- 
tion (7) for the mean transit time t holds for 
injections with general time courses, provided 
the normalized arterial and venous injection 
functions ia(t) and iv(t) are identical and that 
their first moments exist. When ia (t) iv (t) , 
the more general equations of Reference 7 must 
be used in place of Equation (7) . 
Equation (7) for the mean transit time t may 
be given a simple geometric interpretation : the 
mean transit time is numerically equal to the 
area between the normalized arterial and ven- 
ous response curves, qa(t) and qv(t), divided 
by the quantity 1 — q . In this interpretation, 
the area between the two curves is taken as pos- 
itive when qa(t) > qT(t) and negative when 
qa(t) < qv(t), so that t is equal to the al- 
gebraic sum of the areas, divided by 1 — q,^. 
These areas are shown as the shaded regions in 
Figures 3 and 4, labelled with the appropriate 
algebraic sign. 
The above results, which account for total or 
partial recirculation of tracer, also remain valid 
when there is no tracer recirculation. Thus, the 
results hold equally well for residue-detection 
models which, for simplicity, ignore recircula- 
tion effects altogether. In the present model, the 
steady-state content q ^. of tracer within the de- 
tector field D will be zero in the absence of tra- 
