Masticatory muscle anatomy and feeding efficiency of the American beaver, Castor canadensis (Rodentia, Castoridae)

Philip G. Cox & Hester Baverstock

Online Resource 1: Calculation of biomechanical metrics

The following biomechanical calculations follow as closely as possible Appendix 1 of Druzinsky (2010) with minor adjustments for operating in three dimensions instead of two. Four assumptions were made:

  1. The system is in static equilibrium
  2. Only vertical forces are resisted by the TMJ
  3. The entire horizontal component of muscle force is resisted at the bite point
  4. The muscles are equally activated on both sides of the head

The mass (M) and fibre lengths (FL) of the masticatory muscles of the beaver were measured directly from the dissected muscles as outlined in the Materials and Methods. 3D landmark co-ordinates of muscle origin and insertion points were recorded from the virtual reconstruction of the skull and mandible created in Avizo 8.0 (FEI, Hillsboro, OR, USA). Muscle density (D) was estimated to be 1.0564 g/cm3(Murphy & Beardsley, 1974), and an intrinsic muscle stress value (k) of 0.3 N/mm2 (van Spronsen et al, 1989) was used.

The physiological cross-sectional area (PCSA) of each muscle is calculated as follows:

The force (F) produced by each muscle is then estimated as:

In theory, PCSA and intrinsic muscle stress should also be multiplied by the cosine of the pinnation angle of the muscle fibres to give an accurate estimate of muscle force. However, as the pinnation angles of most masticatory muscles are small, their cosines are close to one and can be ignored (Druzinsky, 2010). Pinnation angles are larger in the temporalis and medial pterygoid, so muscle forces may be overestimated for these two muscles.

In muscles in which the fibre direction varied substantially, two or three lines of action were calculated (anteriormost, posteriormost, and, in the case of three lines, a midline), and the force was divided equally between them. Each muscle part was then effectively treated as a separate muscle.

The force produced by each muscle can be decomposed into a vertical component that rotates the mandible around the jaw joint, and a horizontal component that translates the lower jaw in the antero-posterior axis, but results in no rotation owing to the lack of articular eminence in rodents. The lateral components to muscle force were ignored as they would be cancelled out by bilateral activation of the muscles.To determine the vertical and horizontal components of each muscle force, it was necessary to define occlusal and coronal planes. The occlusal plane was defined as the plane containing the landmarks representing the left and right TMJs plus the cranial end of the wear facet on the upper right incisor. The coronal plane was defined as the plane perpendicular to the occlusal plane that contained the landmarks representing both TMJs.

The angle (φ) of the muscle force vector(running from origin to insertion)(a) to either plane is calculated using a normal (n) to that plane:

The vertical (FV) component and horizontal (FH) components of each muscle forceare calculated thus:

The total muscle force in each direction (MF) is then determined by summing the individual muscle forces:

The moment arm (MA) of a muscle is defined as the perpendicular distance from the muscle force vector to the fulcrum. In the case of the masticatory muscles, this is calculated by first determining the angle (θ) between the muscle force vector (a) and the line running from the temporo-mandibular joint (TMJ) and the muscle insertion (b) via the dot product:

and then calculating MA as follows:

The moment (MOM) of a muscle is given by:

And the total moment produced by the masticatory muscles (MUSCMOM) is:

Given static equilibrium, the moment generated by the joint reaction force (TMJMOM) is equal and opposite to the total moment produced by the masticatory muscles. Thus, by the formula for calculating moments above:

Where TMJFis the joint reaction force and TMJMA is the distance from the jaw joint to the bite point.

The vertical component of muscle force (MFV) is resisted by both joint reaction force and the vertical component of bite force (BFV):

So,

The TMJ only resists vertical forces so the horizontal component of muscle force (MFH) is entirely resisted by the horizontal component of bite force (BFH):

So,

The overall bite force (BF) is then calculated:

Using the virtual reconstruction of the mandible, the angle (θI) between the occlusal plane and the long axis of the lower incisor (a line passing through the straightest part of the wear facet on the lingual incisor surface) was measured. The orientation of the incisor at a gape of X° (θI-X) is calculated by:

The orientation of the bite force to the occlusal plane (θBF) is calculated:

So, the proportion of bite force directed along the long axis of the lower incisor (IBF) is:

To recalculate the length of the moment arms, the bite force and the proportion of bite force directed along the long axis of the lower incisor at 30° gape, the landmarks representing the muscle insertion points on the mandible were rotated 30° around an axis running between the two landmarks representing the left and right TMJ.The point (x,y,z) rotated about the line (a,b,c) with unit direction vector (u,v,w) (where u2 + v2 + w2 = 1) by the angle θ will have the following co-ordinates:

References

Druzinsky RE (2010) Functional anatomy of incisal biting in Aplodontiarufa and sciuromorph rodents – Part 2: Sciuromorphy is efficacious for production of force at the incisors. Cells Tissues Organs 192: 50-63

Murphy RA, Beardsley AC (1974)Mechanical properties of the cat soleus muscle in situ. Am J Physiol 227: 1008-1013

van Spronsen PH, Weijs WA, Valk J, Prahl-Andersen B, van Ginkel FC (1989) Comparison of jaw-muscle bite-force cross-sections obtained by means of magnetic resonance imaging and high-resolution CT scanning. J Dent Res 68: 1765-1770

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