Biomedical Engineering Reference
In-Depth Information
Fig. 5.10
Analysis of the variance and dependency of the shape and covariates using PLS.
(
a
) Cumulative variance of PLS modes with respect to shape and BSA. (
b
) CCA correlation
coefficients between BSA and PLS modes with exponential fitted curve in
black
1
N−
1
If we define the covariance matrices
V
UZ
=
U
T
Z
, then the matrix
V
−
1
/
2
XX
V
XY
V
−
1
/
2
Γ
can be seen as a mult
i-variate g
eneralization of the uni-
dimensional correlation coefficient
r
=
σ
XY
/
√
σ
XX
σ
YY
. The sought correlations
are obtained by SVD decomposition of
Γ
:
YY
=
ASB
T
,
Γ
=
(5.11)
where S is the diagonal matrix of the correlation coefficients between correlation
vectors and A and B are rotation matrices of correlation vectors, i.e.,
A
T
A
=
B
T
B
Id
. In our application, Y is a one-column matrix. Hence, S has only
one non-null coefficient R, which is the overall correlation between the PLS shape
vectors X and BSA. B is a scalar equal to
=
±
1
that determines the direction of
BSA correlation. The elements of the first correlation vector of A, denoted by
ρ
,
relate to the amplitude and direction of correlations of each predictor, namely each
PLS mode, when Y varies along the direction defined by the sign of B. In other
words, when BSA varies by 1, the
k
th
predictor varies by
BRρ
[
k
]
. We can therefore
compute a generative average model of heart growth by artificially increasing BSA
and deforming the atlas T with the growth deformation
Φ
parametrized by the
moments
μ
=
B.R.Σ
k
ρ
[
k
]
p
k
,where
p
k
is the
k
th
PLS loading.
5.3.3.3
Interpretation
The growth model computed on the rToF data-set is shown in Fig.
5.11
. This model
shows an expected overall growth of both ventricles as body surface area increases.
We can also see the caving of the septum into the right ventricle as time passes
and the elongation of the right ventricular outflow tract which is observed in these
patients over time.
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