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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:
𝑃𝐶𝑆𝐴 =
𝑀
𝐹𝐿 × 𝐷
The force (F) produced by each muscle is then estimated as:
𝐹 = 𝑃𝐶𝑆𝐴 × 𝑘
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,
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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:
sin 𝜑 =
𝑎. 𝑛
|𝑎| × |𝑛|
The vertical (FV) component and horizontal (FH) components of each muscle force are
calculated thus:
𝐹𝑉 = 𝐹 × sin 𝜑 𝑂𝐶𝐶𝐿𝑈𝑆𝐴𝐿
𝐹𝐻 = 𝐹 × sin 𝜑𝐶𝑂𝑅𝑂𝑁𝐴𝐿
The total muscle force in each direction (MF) is then determined by summing the individual
muscle forces:
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𝑀𝐹𝑉 = ∑ 𝐹𝑉
𝑀𝐹𝐻 = ∑ 𝐹𝐻
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:
𝑎. 𝑏
|𝑎| × |𝑏|
cos 𝜃 =
and then calculating MA as follows:
𝑀𝐴 = 𝑏 × sin 𝜃
The moment (MOM) of a muscle is given by:
𝑀𝑂𝑀 = 𝐹 × 𝑀𝐴
And the total moment produced by the masticatory muscles (MUSCMOM) is:
𝑀𝑈𝑆𝐶𝑀𝑂𝑀 = ∑ 𝑀𝑂𝑀
Given static equilibrium, the moment generated by the joint reaction force (TMJ MOM) is equal
and opposite to the total moment produced by the masticatory muscles. Thus, by the formula
for calculating moments above:
𝑀𝑈𝑆𝐶𝑀𝑂𝑀 = −𝑇𝑀𝐽𝑀𝑂𝑀 = −𝑇𝑀𝐽𝐹 × 𝑇𝑀𝐽𝑀𝐴
𝑇𝑀𝐽𝐹 = −
𝑀𝑈𝑆𝐶𝑀𝑂𝑀
𝑇𝑀𝐽𝑀𝐴
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Where TMJF is 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):
𝑀𝐹𝑉 + 𝑇𝑀𝐽𝐹 + 𝐵𝐹𝑉 = 0
So,
𝐵𝐹𝑉 = −(𝑀𝐹𝑉 + 𝑇𝑀𝐽𝐹 )
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):
𝑀𝐹𝐻 + 𝐵𝐹𝐻 = 0
So,
𝐵𝐹𝐻 = −𝑀𝐹𝐻
The overall bite force (BF) is then calculated:
𝐵𝐹 = √(𝐵𝐹𝑉 )2 + (𝐵𝐹𝐻 )2
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:
𝜃𝐼−𝑋 = 𝜃𝐼 − 𝑋
The orientation of the bite force to the occlusal plane (θBF) is calculated:
tan 𝜃𝐵𝐹 =
4
𝐵𝐹𝑉
𝐵𝐹𝐻
So, the proportion of bite force directed along the long axis of the lower incisor (IBF) is:
𝐼𝐵𝐹 = 𝐵𝐹 × cos(𝜃𝐵𝐹 − 𝜃𝐼−𝑋 )
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:
(𝑎(𝑣 2 + 𝑤 2 ) − 𝑢(𝑏𝑣 + 𝑐𝑤 − 𝑢𝑥 − 𝑣𝑦 − 𝑤𝑧))(1 − cos 𝜃) + 𝑥 cos 𝜃 + (−𝑐𝑣 + 𝑏𝑤 − 𝑤𝑦 + 𝑣𝑧) sin 𝜃 ,
(𝑏(𝑢2 + 𝑤 2 ) − 𝑣(𝑎𝑢 + 𝑐𝑤 − 𝑢𝑥 − 𝑣𝑦 − 𝑤𝑧))(1 − cos 𝜃) + 𝑦 cos 𝜃 + (𝑐𝑢 − 𝑎𝑤 + 𝑤𝑥 − 𝑢𝑧) sin 𝜃 ,
( (𝑐(𝑢2 + 𝑣 2 ) − 𝑤(𝑎𝑢 + 𝑏𝑣 − 𝑢𝑥 − 𝑣𝑦 − 𝑤𝑧))(1 − cos 𝜃) + 𝑧 cos 𝜃 + (−𝑏𝑢 + 𝑎𝑣 − 𝑣𝑥 + 𝑢𝑦) sin 𝜃 )
References
Druzinsky RE (2010) Functional anatomy of incisal biting in Aplodontia rufa 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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