Biomedical Engineering Reference
In-Depth Information
representations for the undeformed brain shapes of subjects
i
=1
, ..., n
s
. Simi-
larly, the vectors
U
c
i
+
U
d
i,j
are representations of the deformed brain shapes of
subjects
i
=1
, ..., n
s
for values of the tumor model parameters Θ
j
,
j
=1
, ..., n
m
.
Assuming that
U
c
i
,
i
=1
, ..., n
s
, are independent realizations of a Gaussian
random vector, principal component analysis (PCA) is applied to these vectors to
yield the mean
µ
c
and the 3
n
p
×
m
c
matrix
V
c
whose columns are the first
m
c
principal components (
m
c
≤
n
s
−
1). Next, we compute the component of
U
d
i,j
in the subspace orthogonal to the columns of
V
c
as
U
d
i,j
=
U
d
i,j
−
V
c
V
c
U
d
i,j
U
d
i,j
,
j
=1
, ..., n
s
, are independent re-
alizations of a Gaussian random vector, PCA is performed on these vectors to
yield the mean
Further, assuming that for each
j
,
V
d
j
whose columns are the
first
m
d
j
principal components associated with eigenvalues
λ
d
j
,l
,
l
=1
, ..., m
d
j
(
m
d
j
and the 3
n
p
×
m
d
j
matrices
µ
d
j
n
s
−
1). It is now possible to approximate the discrete displacement map
V
f
between the atlas and a subject with a simulated tumor with parameters Θ
j
,
j
=1
, ..., n
m
, as follows:
≤
U
f
≈
µ
c
+
V
c
a
+
µ
d
j
+
V
d
j
b
j
.
b
j,
1
, ..., b
j,m
d
j
T
Each of the vectors
a
and
b
j
=
follows a Gaussian distri-
bution with decorrelated components, with that of
b
j
explicitly stated here as
m
d
j
b
j,l
λ
d
j
,l
1
−
0
.
5
f
j
(
b
j
)=
exp
m
d
j
2
πλ
d
j
,l
l
=1
l
=1
for
j
=1
, ..., n
m
.
4.4.2. Statistical estimation
Given an approximate deformation map
ϕ
f
(between a real tumor patient's
image and the atlas image) obtained by the direct use of deformable image regis-
tration, the goal of the methods presented here is to obtain an estimate Θ of the
tumor model parameters. The displacement map
u
f
defined in a similar manner
to Eq. (15) is also discretized over all the atlas voxels in
B
A
\
M
A
and represented
by a vector
U
f
. Owing to the orthogonality of
V
c
for all
j
, the component
of this displacement that is caused by the tumor can be found by
V
d,j
to
U
d
=
U
f
−
µ
c
V
c
a
,