Biomedical Engineering Reference
In-Depth Information
for positive constants a;b 2R 0 . Under the natural assumption that at time
t = 0 there is no radioactivity in the compartment (i.e., at the beginning of
the scanning process the tracer has not yet been injected into the human or
animal body and we have h(0) = f(0) = 0), Equation (9.1) can be expressed
as a Laplace convolution, i.e.,
Z t
f(t) = (a h(
·
) exp(b
·
)) (t) = a
h() exp (b (t)) d .
(9.2)
0
The compartment model can easily be extended to N compartments, as pre-
sented in Figure 9.1(b). The tracer concentration in the blood pool interacts
no longer with just a single region of tissue but with N different regions of
tissue, and the concentration f(t) is simply modeled as the sum of these in-
teractions f n (t), i.e.,
X
f(t) =
f n (t) ,
n=1
for
d
dt f n (t) = a n h(t) b n f n (t) .
Equation (9.2) therefore changes to
Z t
X
f(t) =
h()
a n exp (b n (t)) d .
(9.3)
0
n=1
Compartment modeling can also be used to derive more advanced models
incorporating more complex physical relations among different parameters.
A very simple non-linear model to quantify myocardial perfusion has been
proposed in [4]. The radioactive tracer distribution in myocardial tissue, f(t),
(a) One Compartment Model
(b) N Compartment Model
FIGURE 9.1: The one compartment model and its generalization to N com-
partments.
 
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