Biomedical Engineering Reference
In-Depth Information
and active stiffnesses along the fiber direction,
p
is the hydrostatic pressure, and
α
∈[
0
,
1
]
is an internal variable that describes the level of activation. Further,
λ
=
√
a
0
·
Ca
0
is the fiber stretch, and
f
passive
and
f
active
are the normalized force-length
relationships describing, for the active part, the overlap of actin and myosin, and
hence the ability to generate tension through crossbridge dynamics. For the passive
part,
f
passive
describes the nonlinear behavior of the skeletal muscle tissue due to
stretch. The normalized force-length relationships are a common tool to incorporate
the physiological behavior of the fiber stretch in the muscle's constitutive equation
(e.g., Blemker et al.,
2005
; Röhrle and Pullan,
2007
; Böl and Reese,
2007
).
The second Piola-Kirchhoff stress tensor, as defined in Eq. (
8.1
), forms the basis
for all continuum-mechanical investigations of skeletal muscle mechanics discussed
in this work. In case of modeling the skeletal muscle as an electromechanical-active
tissue, the activation parameter
α
is replaced by cellular variables (cf. Sect.
8.4
).
8.3 The Electromechanical Skeletal Muscle Model
The cellular parameters within the continuum-mechanical model are based on func-
tional and structural characteristics of skeletal muscles. Each skeletal muscle fiber
is connected to an
α
-motoneuron at the neuromuscular junction at the middle of
the fiber's length. Each
α
-motoneuron connects to a number of fibers that are dis-
tributed over some part of a muscle and transfers electrical signals from the central
nervous system to these muscle fibers. Such an electrical signal induces a change in
the membrane potential of the skeletal muscle fibers at the neuromuscular junction
(stimulation). The potential change, referred to as action potential, spreads along the
length of the fiber towards its ends, and induces a number of biophysical processes
in the sarcomeres (basic units of a muscle fiber) eventually leading to a contraction.
Hereby it is important to note that individual muscle fibers are electrically isolated
from each other, i.e. a propagating action potential along one fiber does not induce
an electrical signal in neighboring fibers, but mechanically coupled. To allow such
a setting, in which electrical signals can independently propagate along the length
of specific muscle fibers, one-dimensional structures representing the muscle fibers
(the micro-structure) need to be embedded within the three-dimensional geometrical
representation of the entire skeletal muscle.
For this purpose, a three-dimensional FE model of the tibialis anterior muscle
(TA) has been generated based on the male Visible Human data set (Spitzer and
Whitlock,
1998
). Based on a manual digitization process, a tri-quadratic Lagrange
FE representation of the superficial part and the deep part of the TA has been created
using a least-squares fitting process similar to the one described by Bradley et al.
(
1997
). Particular care was taken to align the
ξ
1
-local coordinate direction of the
FE mesh with the anatomical muscle fiber direction of the TA's muscle fibers. This
choice of aligning the muscle fibers with a local FE coordinate direction eased the
embedding of the 1D fiber meshes and, later on, the numerical solution of action-
potential propagation along the muscle fibers. The muscle-fiber distribution is based
on published data obtained from diffusion-tensor MRI (Lansdown et al.,
2007
).
