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
,
 
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