Biomedical Engineering Reference
In-Depth Information
to movement of a musculoskeletal system. The first methodology focuses in par-
ticular on modeling principles of skeletal muscle motor-unit recruitment to form a
basis to further investigate the use of EMG signals in forward-dynamics simulations.
The second one describes a new methodology to achieve forward-dynamics simu-
lations, when the involved skeletal muscles are represented as three-dimensional
continuum-mechanical models.
8.2 Constitutive Modeling of Skeletal Muscles
The continuum-mechanical models considered in this work are based on solving the
governing equations of finite-elasticity theory using the FE method. Solving for the
mechanical deformation due to skeletal muscle activity or due to a change in the
muscle's attachment location, i.e. movement of a connected bone, requires the eval-
uation of a stress tensor, e.g., the second Piola-Kirchhoff stress tensor. In continuum
mechanics, a relation between the applied strain and the corresponding stress has
to be provided, which characterizes the behavior of the underlying material. Such
a relation is commonly known as constitutive equation. In general, skeletal muscle
tissue is modeled as a transversely isotropic and incompressible hyperelastic ma-
terial. For hyperelastic materials, the constitutive equation can be derived from a
Helmholtz free-energy function.
Given the ability of skeletal muscles to contract upon an externally induced stim-
ulus, e.g., a nerve signal, the overall mechanical behavior of a skeletal muscle is
typically split into two parts: a passive part describing the mechanical behavior of
the so-called ground matrix of a skeletal muscle,
S
matrix
, and an active part describ-
ing the contractile behavior of the muscle,
S
active
.
The second Piola-Kirchhoff stress tensor used within the theory of finite elasticity
is described herein by
4
∂Ψ
muscle
(I
1
,I
2
,I
3
,I
4
,α)
∂I
i
∂I
i
∂
C
S
muscle
=
2
i
=
1
σ
ff
I
4
f
passive
(I
4
)
(
a
0
⊗
p
I
3
C
−
1
pass
=
c
1
I
+
c
2
(I
1
I
−
C
)
−
+
a
0
)
=:
S
iso
=:
S
aniso
α
σ
ff
I
4
f
active
(I
4
)
(
a
0
⊗
ten
+
a
0
)
,
(8.1)
=:
S
active
where
Ψ
muscle
is the Helmholtz free-energy,
I
1
−
I
4
are the standard invariants,
a
0
denotes the local direction of the skeletal muscle fibers,
C
is the right Cauchy-Green
tensor,
I
is the identity tensor,
σ
ff
σ
ff
=
=
0
.
03 MPa are the maximal passive
pass
ten
