Biomedical Engineering Reference
In-Depth Information
membership function at each pixel for a specified number of classes and minimizing
the objective function
J
FCM
.
X
X
C
J
FCM
¼
u
jk
jjI
j
v
k
jj
:
(5)
j2
O
k¼
1
X
C
0
u
jk
1
u
jk
¼
1
(6)
;
;
k¼
1
where
is the image domain and
u
jk
is the fuzzy membership at location
j
for class
k
. The membership functions (which form a partition of unity), represent the degree
of similarity between the intensity at that location and the centroid of its class. Thus,
a high membership value means that the data value at that location is close to the
centroid for that particular class. Our meshless model contains three classes which
are healthy tissue (as the same properties for white and brain matter is used without
introducing noticeable errors in computed displacements in [
3
]) ventricles and
tumor, therefore C
O
3.
Statistical material properties
MP
j
at each location j are interpolated based on
these membership functions from FCM as followings
¼
X
C
MP
j
¼
u
jk
MP
k
:
(7)
k¼
1
We assign the material properties (Young's modulus and Poisson's Ratio)
directly to the integration points in the domain.
3
Implementation and Verification
To obtain a quantitative assessment of the accuracy of the proposed approach, we
modeled a brain MRI slice with clearly visible ventricles, tumor, and craniotomy
using both statistical meshless model and finite element model. The patient-specific
data was obtained from a set of pre-operative MRIs of a patient undergoing brain
tumor surgery at the Department of Surgery, Brigham and Women's Hospital
(Harvard Medical School, Boston, Massachusetts, USA). We compared simulation
results from the two models with the same constitutive model and boundary
conditions.
For both the finite element model and the statistical meshless model, we used
skull striping to remove the nonbrain parts so that boundary conditions could be
applied. We used a deformable-model-based brain extraction method [
11
] for its
robustness, fully automated execution and high accuracy. Generation of the
meshless grid is instantaneous. Based on the extracted brain surface, meshless
