Biomedical Engineering Reference
In-Depth Information
4.4. Statistical Estimation of Mass-Effect Model Parameters
Now, with the availability of the biomechanical model of the brain tumor
mass-effect described in Section 3, it is possible to apply such a model to the atlas
image in order to produce a version of the atlas image with a tumor. This atlas
image will hopefully look more similar to that of the patient, which will make the
deformable registration between the two images easier. However, in order to use
the biomechanical model, values of this model's parameters Θ that correspond to
the patient must be estimated. This is achieved through a statistical estimator as
follows.
4.4.1. Statistical model training
The goal of this step is to create a statistical model for the deformation
ϕ
f
,
which will aid in estimation of Θ for a particular tumor image. First, the defor-
mation maps
ϕ
c
i
,
i
=1
, ..., n
s
, between the atlas and MR images of
n
s
normal
subjects are obtained using HAMMER, a deformable image registration approach
designed for normal-to-normal brain image registration [52]. Simulations of the
mass-effect of tumor growth are then conducted for each subject
i
for values Θ
j
,
j
=1
, ..., n
m
, covering a range of the model parameters to produce the deforma-
tions
ϕ
d
i,j
,
i
=1
, ..., n
s
,
j
=1
, ..., n
m
.
A problem preventing the collection of statistics on
ϕ
d
i,j
directly is that the
domains of these maps are different for different values of
i
and
j
. This precludes
the point-to-point comparison of these deformation maps. To overcome this prob-
lem, for all tumor model simulations, regions
T
A
j
and
D
A
j
are defined in the atlas
space based on Θ
j
and mapped to each subject's space via
ϕ
c
i
,
i
=1
, ..., n
s
. Next,
for
X
A
∈
A
\
T
A
j
,
i
=1
, ..., n
s
,
j
=1
, ..., n
m
, define the displacement maps
u
d
i,j
(
X
A
)
≡
ϕ
f
i,j
(
X
A
)
−
ϕ
c
i
(
X
A
)
(12)
≡
ϕ
d
i,j
(
ϕ
c
i
(
X
A
))
−
ϕ
c
i
(
X
A
)
,
(13)
u
c
i,j
(
X
A
)
≡
ϕ
c
i
(
X
A
)
−
X
A
,
(14)
u
f
i,j
(
X
A
)
≡
ϕ
f
c
i
(
X
A
)
−
X
A
,
(15)
which implies
u
f
i,j
(
X
A
)=
u
c
i
(
X
A
)+
u
d
i,j
(
XA
)
.
(16)
For different
i
=1
, ..., n
s
but the same
j
=1
, ..., n
m
, the domains of
u
d
i,j
are the
same. An example of a tumor model simulation and the involved displacement
maps is shown in Figure 20.
Discrete versions of the displacement maps
u
d
i,j
are constructed by
sampling their Cartesian components for all voxels in the atlas with coordinates
X
A
k
u
c
i
and
∈
B
A
\
M
A
,
k
=1
, ..., n
p
, to yield the 3
n
p
×
1 vectors
U
c
i
and
U
d
i,j
,
respectively. It is possible to see that the vectors
U
c
i
act as displacement-based
