Biomedical Engineering Reference
In-Depth Information
The principal components are obtained by computing the spectral decomposition
Σ
PMP
T
.
M
is the diagonal matrix of the eigenvalues
σ
m
, or variances, sorted
in decreasing order and
P
is the orthonormal matrix (in the
L
2
-norm sense) of the
eigenvectors
p
m
.The
m
th
loading
l
m
of the PCA decomposition is given by the
formula:
=
l
m
=
p
m
[
β
(
i
)
.
i
]
(5.5)
i
In this equation,
p
m
[
i
]
is the
i
th
element of the
m
th
eigenvector of
Σ
,
β
(
i
)
=
(
β
(
i
)
1
− β
1
,...β
(
i
n
− β
n
)
T
is the
n ×
3
matrix that gathers the
n
centered moment
of patient
i
. As a result, the initial velocity field of the
m
th
mode is
v
0
l
m
(
x
)=
k
K
W
(
x, x
k
)
l
m
. The variability captured by this
m
th
deformation mode between
[
−ησ
m
;+
ησ
m
]
is visualized by deforming the atlas
T
with the deformations
φ
−m
and
φ
+
m
parameterized by the moments
β−ηl
m
and
β
+
ηl
m
respectively. Selecting
the first
p
modes only among the
N −
1
possible modes (where
N
is the number of
patients) allows to explain a percentage
p
m
=1
σ
m
/
trace
of the total variance.
The orthogonal projection of each patient's initial velocity field onto the selected
PCA subspace gives a unique shape vector. This simply corresponds to the
coordinates of the projection in the basis constituted by the chosen eigenmodes:
(
Σ
)
<v
(
i
)
0
β
(
i
)
j
− β
s
i,m
=
l
k
−
¯
v
0
,v
0
l
m
>
W
=
[
]
K
W
(
x
j
,x
k
)
.
(5.6)
j,k
Using PCA we have reduced the amount of data needed to represent the shape of a
patient by two or more orders of magnitude. The precision of the representation is
controlled by the number of components of the PCA subspace. However, it has to
be observed that modes with low variances may still be relevant to external clinical
parameters. For instance, a mode that captures a local bulging is probably more
related to the pathology than a global scaling of the shape although this bulging is
not very visible in the population and could be considered as noise in the model.
Consequently, we are fairly conservative in the selection of the PCA subspace and
select the modes based on their relationship with the clinical parameters of interest
and not their variance, as described in the following sections.
5.3.2.2
Identifying Factors Between Shape and Clinical Features
The
s
i,m
's quantify the amount of variability along the
m
th
mode present in patient
deformation. We can thus investigate the heart shape by relating these shape vectors
to clinical parameters that quantify the pathology. Ordinal clinical parameters are
investigated using non-parametric rank-based statistics. Kruskal-Wallis analysis of
variance is applied to find effects between the investigated parameters and shape
[
36
]. If an effect is found, post-hoc two-sample Wilcoxon test is used to determine
which levels differ [
36
]. Continuous clinical parameters are investigated using linear
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