Biomedical Engineering Reference
In-Depth Information
Therefore, for calculation of the deformation gradient tensor, the partial derivatives
of the model with respect to its parameterization in the Cartesian coordinate system
need to be known. For NURBS models using Cartesian coordinates, this is derived
analytically directly from the model. For cylindrical coordinates, this is found by
differentiating the relationships in Eq. (5):
∂x
∂
(
u, v, w
)
=
∂r
∂
(
u, v, w
)
cos
φ
∂φ
∂
(
u, v, w
)
−
r
sin
φ
,
(42)
∂y
∂
(
u, v, w
)
=
∂r
∂
(
u, v, w
)
sin
φ
+
r
cos
φ
∂φ
∂
(
u, v, w
)
,
(43)
∂z
∂
(
u, v, w
)
∂z
∂
(
u, v, w
)
=
.
(44)
Similar results are obtained for the case of prolate spheroidal coordinates (Eq. (8)):
∂x
∂
(
u, v, w
)
=
δ
cosh
λ
sin
η
cos
φ
∂λ
∂
(
u, v, w
)
∂η
∂
(
u, v, w
)
+
δ
sinh
λ
cos
η
cos
φ
(45)
∂φ
∂
(
u, v, w
)
−
δ
sinh
λ
sin
η
sin
φ
,
∂y
∂
(
u, v, w
)
=
δ
cosh
λ
sin
η
sin
φ
∂λ
∂
(
u, v, w
)
∂η
∂
(
u, v, w
)
+
δ
sinh
λ
cos
η
sin
φ
(46)
∂φ
∂
(
u, v, w
)
−
δ
sinh
λ
sin
η
cos
φ
,
∂z
∂
(
u, v, w
)
=
δ
sinh
λ
cos
η
∂λ
∂
(
u, v, w
)
∂η
∂
(
u, v, w
)
−
δ
cosh
λ
sin
η
.
(47)
1
2
(
F
T
F
−
1
), where
The Lagrangian strain tensor,
E
, is then given by
E
=
1
is the identity matrix. Similarly, we can use the Eulerian displacements found
in step 4 of Table 1 to calculate the deformation gradient tensor,
F
, which is used
2
(
1
−
(
FF
T
)
−
1
). From the strain
tensors we can calculate the various normal, shear, and principal strain values that
describe the local deformation of the myocardium.
1
to calculate the Eulerian strain tensor,
G
=
