Biomedical Engineering Reference
In-Depth Information
terms of cellular parameters, the second Piola-Kirchhoff stress tensor is given by:
A
1
+
f
active
(I
4
)
a
0
⊗
σ
ff
σ
ff
ten
I
4
A
2
c
trop
A
2
A
max
2
pass
I
4
S
muscle
S
iso
f
pass
(I
4
)
=
+
+
a
0
,
(8.2)
where
c
trop
is the amount of troponin present within a half-sarcomere and
A
ma
2
is the
maximal possible concentration of attached myosin heads in the post-powerstroke
state. Further, the expression
A
2
/A
ma
2
in Eq. (
8.2
) is comparable to the
α
-term and
presents a cellular-based activation variable that ranges between 0 and 1. The ad-
ditional term containing
(A
1
+
A
2
)/c
trop
can be interpreted as an additional pas-
sive stiffness that is induced by a specific contraction. The justification for this
term is based on the assumption that each attached myosin head, either in the pre-
or post-powerstroke state, adds additional passive stiffness to the system. Further-
more, the cellular parameters in the second Piola-Kirchhoff stress tensor (
A
1
and
A
2
) are still multiplied by a rather macroscopic force-length relationship
f
active
(I
4
)
and
f
pass
(I
4
)
, which are derived from experiments on whole muscles. The justifica-
tion for doing so is that the model of Shorten et al. (
2007
) in its current form is only
valid for isometric contractions. Therefore the pre- and post-powerstroke concentra-
tions (
A
1
and
A
2
) have been multiplied by the normalized force-length relationship
for active contractions, i.e.
f
active
(I
4
)
, to account for the probability by how much
the actin and myosin filaments overlap.
The cellular parameters
A
1
and
A
2
change from grid point to grid point along the
1D muscle fibers, and from 1D fiber to 1D fiber. A relative small grid spacing has to
be used for the propagation of the action potential along the skeletal muscle fibers
within the 1D grids, due to the sharp gradients occurring. Furthermore, a relative
large number of fibers within a skeletal muscle have to be considered to represent
physiological conditions. Due to the computational work involved, the second Piola-
Kirchhoff stress tensor as given in Eq. (
8.1
), cannot be evaluated at each grid point
of the 1D meshes. A FE discretization of the mechanical problem, in which each
1D grid point would coincide with a Gauss point of the 3D mesh, would lead to an
overly large mechanical problem. Therefore, the cellular variables are homogenized.
For all grid points of the 1D meshes, the closest Gauss point of the 3D mesh is
sought, where the respective cellular values are homogenized by averaging. A grid
convergence study justifying the proposed homogenization is presented in Röhrle et
al. (
2008
).
An output of the above described electromechanical framework is given in
Fig.
8.2
. There, the stimulation times of every second motor unit are shown by ver-
tical strikes on the horizontal lines (each line represents one MU). The simulation
shows that, like in reality, the larger motor units (higher MU numbers) are activated
later than the smaller motor units (lower MU numbers) and that the average recruit-
ment frequency of all MUs increases, if a linear excitatory drive function as input
to the Fuglevand et al. (
1993
) motor unit recruitment model is chosen. The result-
ing force output behaves sigmoidal with a slow average change in curvature at the
beginning and end of the simulation and a relatively linear section in the middle.
The results in Fig.
8.2
essentially present the solution to a forward dynamics
problem. The recruitment of particular motor units serves as input while the model's
