74 Chapter 7: Upper Limb Model
muscle belly with two separate proximal tendons. The gap between the two tendon heads is filled with isotropic soft tissue. The fibre direction in the tendon tissue zone is assumed to behave fusiform-like.
The cross-section at the ends of the muscles is much smaller compared to the ends of the muscle tissue with rigid tendons. Furthermore, the tendons have to withstand relatively small deformations, hence, at the distal surfaces, all nodes were fixed in all spatial directions.
Bipennate muscles have a slim aponeurotic tissue layer which deeply penetrates into the muscle belly, see Figure 2.12c. The aponeurosis is a very fine and slim tissue layer which has a strong impact on the mechanical behaviour of the overall muscle-tendon complex. To represent the triceps more realistically, a central line of elements is defined as aponeurotic tissue layer, which connects the proximal and distal ends of the muscle. Compared to the thickness of the aponeurosis, the size of the element is large. Therefore, the mechanical behaviour of the aponeurotic tissue layer is homogenised using the material variable γM which is chosen to be γM = 0.99. As the tendon is approximately 1000 times stronger than muscle tissue, a linear interpolation is not constructive. Imagine a tissue composition of 10% tendon and 90% muscle where muscle has the stiffness of 1 and tendon of 1000.
The interpolated bulk has a stiffness of 100.9 which is still 100 times stiffer than muscle.
A tissue consisting of 1% tendon and 99% muscle is still eleven times stiffer than pure muscle tissue. Hence, the chosen material parameter γM is chosen to be γM = 0.99.
Table 7.2: Skeletal muscle material parameter set, as introduced in Section4.4, for triceps and biceps brachii when the muscles are considered as a muscle-tendon complex.
tissue contribution parameter triceps biceps source
muscle
isometric c1M
c2M
3.56·10−2 MPa 3.86·10−3 MPa
3.56·10−2MPa 3.86·10−3MPa
Hawkins and Bey (1994) passive c3M
c4M
4.02·10−7 MPa 38.5 [-]
3.57·10−8MPa 42.6 [-]
Zheng and Mak (1999)
active
∆Wasc 0.30 [-] 0.25 [-]
adapted from G¨unther et al.
(2007)
∆Wdesc 0.10 [-] 0.15 [-]
νasc 4.00 [-] 3.00 [-]
νdesc 4.00 [-] 4.00 [-]
λoptf 1.3 [-] 1.35 [-]
-σmax 0.30 MPa 0.30 MPa
-tendon
isotropic c1T
c2T
2.31 MPa 1.15·10−6 MPa
2.31 MPa 1.15·10−6MPa
Weiss and Gar-diner (2001) passive c3T
c4T
7.99 MPa 16.6 [-]
7.99 MPa 16.6 [-]
Weiss and Gar-diner (2001)
7.3 Equivalent Static System 75
It consists of the humerus, the forearm, the antagonistic muscle pair, and an external force F to increase the DoFs of the system by one. The angle between the humerus and the forearm is defined as the elbow flexion angleθ and defines the position of the forearm.
The humerus is supported in all spatial directions and the forearm rotates around the fulcrum of the elbow joint. The forearm is represented including a rigid, bend-resistant corner at the fulcrum to represent the location of the olecranon. The muscle attachment points coincide with the volumetric model. The forearm’s rotation matrix is defined by an axis and an angle. The rotation angle is defined to be the elbow flexion angle and the rotation axis is defined by a vector product of a vector along the humerus and the orientation of a vector from the fulcrum and to the distal end of the forearm.
As both muscles insert at the forearm, the position of the forearm defines the muscles’
lengths and lever arms, i.e. the muscles’ kinematics only depend on the elbow angle.
As one only considers an equivalent static system and assumes rigid bones, it is sufficient to identify, for each insertion or origin, one node on each mesh of the ulna and radius as the muscles’ insertion points. The muscles’ length are defined by the linear distance between the origin and insertion points. The origin is fixed whereas the location of the insertion point varies depending on the elbow flexion angle. The chosen nodes on the bone are tracked for the physiologically reasonable range of motion of the forearm, i.e. 10◦ < θ <150◦. The muscle length is defined as change of the distance between origin and insertion relative to its initial position at θ0 = 60◦. Then, for each of the muscle lengths relative to the initial position, a separate third-order polynomial has been obtained using MATLAB (2014)’s least-squares fitting functionality. The respective relative muscle lengths are depicted in Figure 7.6a and Figure 7.6b.
Compared to the triceps brachii, the biceps brachii has a much larger change of length.
The reason is that the location of the attachments’ site is further away from the elbow’s fulcrum.
0 50 100 150
−40
−20 0 20 40 60 80
elbow angle θ [°]
muscle displacement [mm]
triceps
x
triceps
y
triceps
z
triceps
total
(a) triceps brachii
0 50 100 150
−40
−20 0 20 40 60 80
elbow angle θ [°]
muscle displacement [mm]
biceps
x
biceps
y
biceps
z
biceps
total
(b) biceps brachii
Figure 7.6: Relative change of muscle length is determined by subtracting the muscle length at θ0 = 60◦ from the current muscle length. As the z-direction is close to parallel to the longitudinal axis of the muscle, it is dominating the mechanical behaviour of the muscle.
The muscle lever arms are determined using two different methods. The first method, introduced by An et al. (1984), is widely used and its related errors are considered to
76 Chapter 7: Upper Limb Model
be small. It is also known as the tendon-displacement method and is derived using the principle of virtual work and basic geometry and defines the moment arm as the derivative of the muscle length with respect to the joint angle. The idea basically is that the change of length of a string fixed at two ends which is wrapped around a pulley is caused only by that part of the string which wraps around the pulley with a diameter l. Hence, the lever arms can be defined by:
lm = dLm
dθ as ∆Lm = 2π lm ∆θ
2π , ∀ m ={T, B}, (7.1) whereLm is the muscle’s length andlm is the distance from the joint centre, or the pulley to the tendon or string. The second part of (7.1) exactly describes the part of the string which is wrapped around the pulley.
Within the first method, the lever arm is a scalar-valued parameter lacking all infor-mation of the actual location and orientation of the acting members. Unlike rigid-body simulations, where the muscle force direction is an assumed parameter derivable from the kinematics, the volumetric model enables to introduce a second method which determines the lever arm by considering the force direction depending on the muscle activation, the muscle geometry, the fibre orientation, and the restricting neighbours.
As the muscle force orientation can change dramatically within the range of a possible acting situation and a new point of action can be derived in order to respect the mechanical situation of the muscle (see Chapter 7.5), a more detailed description of the lever arm is required to advantageously use this results.
The lever arm resulting from the vector-resulting description is determined by the shortest distance between the muscle reaction force, Fm, and the location of the elbow’s fulcrum, xf and can to be determined by basic geometry. To determine the vector-resulting lever arm, firstly, a plane is defined using a support point and a vector that is normal to the plane. The fulcrum of the elbow is chosen to be the support point while the normal direction of the plane is defined by the muscle reaction force. Secondly, a line is defined by a point and a vector defining its orientation. The line’s support point is the point of action of the muscle’s insertion face while the orientation of the line is also defined by the muscle reaction force. As the plane and the line are perpendicular to each other, they always have one intersection which can be derived by:
lm = (xf −xm)×Fm
kFmk , ∀ m={T, B}. (7.2)
Herein, lm is the vector representing the shortest distance between the line and the elbow fulcrum, xf.
The resulting scalar-valued moment can be determined by the norm of the vector prod-uct or by the prodprod-uct of the norm of the lever arm and the force, as the two vectors are, by definition, perpendicular to each other, i.e.
Mm =klm×Fmk= klmk · kFmk=lm·Fm, ∀ m ={T, B}. (7.3) Herein, the scalar-valued lever arm and muscle reaction force are defined by lm = klmk and Fm =kFmk, respectively.
To investigate the static equilibrium of the upper limb, the equivalent static system is introduced (Figure 7.7a). A free body diagram of the forearm is defined by the black