Image Processing Reference
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is overestimated, which invariably occurs, for instance, in slices that are close to
the apex. Buller and coworkers [103,47] introduced an improvement on this method
by estimating at each location the angle between the wall and the imaging plane.
Later, Bolson and Sheehan [104,105] introduced the Center Surface method (a true
3-D extension of the centerline method), which makes use of a reference medial
surface to compute the chords and subsequently wall thickness.
9.3.2.3
Strain Analysis
Strain analysis (SA) is a method to describe the internal deformation of a con-
tinuum body. It is a promising tool to study and quantify myocardial deformation.
Here, we shall briefly introduce some of the concepts related to strain analysis.
A comprehensive exposition of this theory can be found in Fung [106].
To describe the deformation of a body, the position of any point in the body
needs to be known with respect to an initial configuration; this is called the
reference
state
. Moreover, to describe the position a reference frame is needed. In the fol-
lowing description, a Cartesian reference frame will be assumed. It is also common
to use curvilinear coordinates because some of the expressions simplify.
A myocardial point,
M
r
, has coordinates {
y
i
} and a neighboring point,
M
′
r
has coordinates {
y
i
+
dy
i
}. Let
M
r
be moved to the coordinates {
x
i
} and its
neighbor to {
x
i
dx
i
}. The deformation of the body is known completely if we
know the relationship
+
xxyyy
i
=
(, , )
123
i =1,2,3
(9.6)
i
or its inverse,
yyxxx
i
=
(, , )
123
i =1,2,3
(9.7)
i
For every point in the body, we can write
xyu
i
=+
i = 1,2,3
(9.8)
i
i
where
u
i
is called the
displacement
of the particle
M
r
. In order to characterize
the deformation of a neighborhood, the first partial derivatives of Equation 6-
Equation 8 are computed. These derivatives can be arranged in matrix form to
define the
deformation gradient tensor
:
F
= [
y
i
], (
i
,
j
= 1, 2, 3). The defor-
mation gradient tensor enables estimation of the change in length between the
neighboring points{
y
i
} and {
y
i
+
∂
x
i
/
∂
dy
i
}, when they are deformed into {
x
i
} and
{
x
i
+
dx
i
}. Let
d
r
and
d
be these lengths before and after deformation. Then
3
3
∑
∑
dl
2
−=
=
dl
2
2
E dy dy
(9.9)
r
ij
i
j
i
=
1
j
1
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