Biomedical Engineering Reference
In-Depth Information
and Delise (
1973
), Gestrelius and Borgström (
1986
), Lee and Schmid-Schönbein
(
1996a
,
b
), Miftakhov and Abdusheva (
1996
), Rachev and Hayashi (
1999
), Yang et
al. (
2003a
,
b
), Zulliger et al. (
2004
), Herrera et al. (
2005
), Bates and Lauzon (
2007
),
Bursztyn et al. (
2007
), Stålhand et al. (
2008
) and Murtada et al. (
2010
). All these
models have the main restriction that they are implemented in form of so-called
stand-alone programmes. Hence, only limited estimations are possible as the chemo-
mechanical behaviors of smooth muscles significantly depend on their geometry
undergoing large deformations it is essential to take a three-dimensional modeling
approach into account. However, to the authors knowledge, there only exists one
three-dimensional coupled chemomechanical modeling approach presenting three-
dimensional boundary-value problems, see Schmitz and Böl (
2011
). Herein,
steady
state
characteristics of the calcium concentration are presented, only.
The present contribution concentrated on the development of a three-dimensional
chemomechanical SM model including
dynamic
behavior of the calcium concen-
tration. Section
5.2
introduces the governing equations of a coupled boundary-
value problem for SM chemomechanics. Before the manuscript is concluded with
Sect.
5.4
, Sect.
5.3
shows illustrative numerical examples.
5.2 Field Equations of Smooth Muscle Chemomechanics
5.2.1 Kinematics
As this work focuses on the modeling of vascular smooth muscle tissue, an aniso-
tropic material with outstanding directions for collagen bundles and SMC layers has
to be considered. Collagen bundles as well as SMC layers are aligned tangentially
with the wall of the vessel (Herlihy and Murphy,
1973
; Walmsley and Murphy,
1987
; Dahl et al.,
2007
) accomplished by the angle
Φ
, see Fig. 1 in Schmitz and
Böl (
2011
). The in-wall dispersions
Θ
c
/
s
(c
=
=
SM layer) lead-
ing to arbitrary direction vectors which are able to describe the collagen and SMC
orientations in the reference configuration using so-called unit direction vectors
collagen and s
⎛
⎝
⎞
⎠
.
cos
Θ
c
/
s
cos
Φ
cos
Θ
c
/
s
sin
Φ
sin
Θ
c
/
s
M
c
/
s
=
(5.1)
Consequently, the structural tensors
Z
c
/
s
=
M
c
/
s
⊗
M
c
/
s
(5.2)
can be constructed by means of the dyadic product including the directional infor-
mation of a certain SMC layer
M
s
or collagen fiber bundle
M
c
. This allows the
computation of corresponding stretches
λ
c
/
s
=
I
4
,
c
/
s
=
C
:
Z
c
/
s
,
(5.3)
