Image Processing Reference
In-Depth Information
nondiffusible tracer leaving the VOI at a time t is given by
t
∫
qt
out
()
=
q h d
0
ττ
( )
(19.1)
0
where h(t) is the
, i.e., the probability density function of the
tracer transit time through the VOI. The transport function, h(t), is a characteristic
of the system and has the dimensions of 1/time. In particular, h(t) is dependent
on the flow and vascular structure of the VOI. When integrated over the interval
0 to infinity, the area is unitary and dimensionless. Therefore,
transport function
∞
∫
ht dt
()
=
1
(19.2)
0
19.2.2
R
F
ESIDUE
UNCTION
Following the definition of h(t), the amount of tracer remaining in the VOI is
given by q
minus the amount that left the VOI:
0
t
t
∫
∫
qt q q
()
=−
h d
()
ττ
= −
q
1
h d
()
ττ
(19.3)
in
0
0
0
0
0
The function
t
∫
Rt
()
=−
1
0
ττ
h
( )
d
(19.4)
is called
and describes the fraction of tracer still present in the
VOI after a time t following an ideal bolus injection. R(t) is a dimensionless,
positive, decreasing function of time for which
residue function
R
(01
=
(19.5)
Once h(t) and, consequently, R(t) are known for a given VOI, the concentra-
tion curve of the tracer at the exit and of that retained in the VOI can be predicted
for any known input function to the VOI.
19.2.3
C
B
V
EREBRAL
LOOD
OLUME
In case of an intact BBB, the amount of blood in a given VOI measures the central
blood volume (CBV). From dynamic images acquired during bolus injection of
a contrast agent, CBV can be determined from the ratio of the areas under the
concentration time curve of the tracer within a given VOI, C
(t), and the
concentration time curve of the tracer in the feeding vessel to the VOI, C
VOI
(t)
AIF
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