relaxation time; and Hct is the hematocrit value. AIF
characterizes the rate of change of contrast agent con-
centration in the blood plasma.
3.Excluding Background Voxels
In our study, the background voxels could be
excluded using a flexible threshold. This threshold
was defined as the average maximum contrast agent
concentration C
max,average
plus p times the standard
deviation of C
max,average
. Because the contrast agent
concentration in background was much lower than in
arterial vessels, the final AIF detection results could
achieve satisfactory performance in the mentioned
DCE-MRI datasets although the factors p were prede-
termined experientially. In particular, the factors
were, respectively, set to 1.5 and 1.2 for rat and
human datasets in our experiments.
4.Calculating Areas under Concentration Curves
According to the small area of arterial regions, only a
small fraction of contrast agent concentration curves
corresponds to arterial regions on the DCE-MRI data-
sets. Thus, we computed the area under concentra-
tion curve for each voxel and excluded a percentage
(P
AUC-S
) of voxels with the smallest areas. A percent-
age (P
AUC-L
) of remaining voxels with the largest areas
were also excluded.
5.Calculating Peak Values of Concentration Curves
In manually detected AIF curves, the concentration
curve has the following characteristics, such as early
tracer arrival, high peak height, and quick wash-out
(9). In our study, a percentage (P
CTC
) of voxels with
smallest peak values were excluded in each slice.
6.Application of Fast-AP Clustering
The concentration curves of the remaining voxels were
regarded as the input data points in the Fast-AP clus-
tering. The final AIF was determined by averaging the
voxels in the most appropriate cluster, and the arte-
rial regions were extracted accordingly. The K-means
and AP clustering were also implemented using the
same automatic methodology for extracting potential
arterial voxels. The detailed description of the applica-
tions of AP and Fast-AP clustering in AIF detection
were presented in the next subsections.
Application of AP Clustering in AIF Detection
Recently, Frey and Dueck (17) originally proposed a
completely data-driven clustering algorithm in Sci-
ence, called affinity propagation (AP). In this cluster-
ing algorithm, all data points can be equally
considered as exemplars. The aim of AP is to find the
optimal set of exemplars by maximizing the sum of
similarities between the data points and their corre-
sponding exemplars.
Let X = fX
1
,X
2
,...,X
M
g be the set of voxels obtained
from the rat or human DCE-MRI datasets, and s(i,k)
denote the similarity between two data points x
i
and
x
k
. In conventional clustering methods, negative
Euclidean distance is a common selection for the
choice of similarity
sði; kÞ¼-
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðC
i
-C
k
ÞðC
i
-C
k
Þ
T
q
[2]
where C
i
represents the concentration–time curve of
the ith voxel. The preference of point k, called as
s(k,k), is a priori assumption of data point x
k
to sever
as an exemplar. The data point x
k
with higher value of
preference s(k,k) has more opportunities to be chosen
as an exemplar. Meanwhile, the number of exemplars
(clusters) is influenced by the predefined preference.
Low value of preference will lead to small number of
exemplars (clusters), while high value tends to pro-
duce many exemplars (clusters) (17).
In our study, all responsibilities r (i,k) and availabil-
ities a(i,k) are set to 0 initially. Their values are itera-
tively adjusted by setting (18,19):
rði; kÞ sði; kÞ- max
k
0
;s:t:k
0
6¼k
faði; k’Þþsði; k’Þg [4]
rðk; kÞ sðk; kÞ- max
k
0
;s:t:k
0
6¼k
faðk; k’Þþsðk; k’Þg [5]
aðk; kÞ
X
i’;s:t:i’6¼k
maxf0; rði’; kÞg [6]
aði; kÞ minf0; rðk; kÞþ
X
i’;s:t:i
0
=2fi;kg
maxf0; rði’; kÞgg
[7]
The index of exemplar c
i
associated to x
i
is finally
defined as:
c
i
argmax
1kM
frði; kÞþaði; kÞg [8]
According to the above-mentioned AP clustering pro-
cedure, the messages exchanged between data points
are updated until an appropriately set of exemplars
and clusters emerges. However, the simple updating
procedures of the responsibilities r(i,k) and availabil-
ities a(i,k) often lead to oscillations. T o overcome this
disadvantage, the responsibilities and availabilities
can be updated as follows (18,20)
R
tþ1
¼ gR
t-1
þð1-gÞR
t
A
tþ1
¼ gA
t-1
þð1-gÞA
t
(
[9]
where R and A, respectively, denote the responsibility
and availability vectors; g僆[0.5,1] represents the
damping factor; and t is the number of iterations. The
damping factor and maximum number of iterations
are respectively set to g = 0.9 and T = 2000 in our
study.
Application of Fast-AP Clustering in AIF Detection
In Frey and Dueck (17), the authors have demon-
strated the superior performance of the original AP
clustering on some different examples. In AP cluster-
ing, the number of clusters need not be prespecified
as in k-center clustering. If the number of clusters
should be prespecified in advance, this clustering
method would unfortunately lead to low clustering
efficiency. To overcome this limitation, Wang and
Detection of AIF in DCE-MRI Using AP Clustering 1329