Biomedical Engineering Reference
In-Depth Information
5.
LAGRANGIAN AND EULERIAN STRAIN MEASUREMENTS FROM
AN NURBS MODEL
Lagrangian strain maps are produced from our model by describing the de-
formation of the left ventricle in the Lagrangian reference frame. If the spatial
coordinates are represented by
X
at time
t
=0and by
x
at time
t>
0 then,
in the Lagrangian reference frame, the mapping
χ
(
X
) warps
X
into
x
, that is,
x
=
χ
(
X
)=
V
(
X
)+
X
. The deformation gradient tensor can be written as
F
=
∇
χ
(
X
)=
∇
V
(
X
)+
∇
X
. The elements of the deformation gradient tensor
are
∂x
∂X
∂x
∂Y
∂x
∂Z
,
∂y
∂X
∂y
∂Y
∂y
F
=
∂Z
,
,
(35)
∂z
∂X
∂z
∂Y
∂z
∂Z
where the displacement field
χ
(
µ, ν, ω
) relates the coordinates of a material point
in the reference configuration
X
(
X, Y, Z
) with its coordinates at a later time point:
x
(
x, y, z
):
µ
=
x
−
X,
ν
=
y
−
Y,
ω
=
z
−
Z.
(36)
In order to solve for each of the three partials of the components of the dis-
placement field necessary for calculation of the deformation gradient tensor,
F
,
we calculate the following vector quantities:
∂
∂
∂X
∂u
+
∂
∂Y
∂u
+
∂
∂Z
∂u
,
∂u
=
χ
∂X
χ
∂Y
χ
∂Z
(37)
∂
∂
∂X
∂v
+
∂
∂Y
∂v
+
∂
∂Z
∂v
,
∂v
=
χ
∂X
χ
∂Y
χ
∂Z
(38)
∂
χ
∂w
=
χ
∂X
∂
∂X
∂w
+
χ
∂Y
∂
∂Y
∂w
+
∂
χ
∂Z
∂Z
∂w
,
(39)
where (
u, v, w
) are the parametric directions of the NURBS model. This yields
the following linear system:
µ
X
µ
Y
µ
Z
X
u
X
v
X
w
∂
(
µ, ν, ω
)
∂
(
u, v, w
)
=
.
ν
X
ν
Y
ν
Z
Y
u
Y
v
Y
w
(40)
ω
X
ω
Y
ω
Z
Z
u
Z
v
Z
w
Considering the relationships in (36), the following linear system is derived:
−
1
1+
µ
X
µ
Y
µ
Z
x
u
x
v
x
w
X
u
X
v
X
w
=
F
=
ν
X
1+
ν
Y
ν
Z
y
u
y
v
y
w
Y
u
Y
v
Y
w
.
ω
X
ω
Y
1+
ω
Z
z
u
z
v
z
w
Z
u
ω
v
ω
w
(41)
