J. Anat. (2009) 214, pp362–373
doi: 10.1111/j.1469-7580.2008.01042.x
Cranial dimensions and forces of biting in the domestic
dog
Blackwell Publishing Ltd
Jennifer Lynn Ellis,1 Jeffrey Thomason,2 Ermias Kebreab,3 Kasim Zubair4 and James France1
1
Centre for Nutrition Modelling, Department of Animal and Poultry Science, University of Guelph, Guelph, ON, Canada
Department of Biomedical Sciences, University of Guelph, Guelph, ON, Canada
3
Department of Animal Science, University of Manitoba, Winnipeg, MB, Canada
4
Royal Canin Canada, Guelph, ON, Canada
2
Abstract
The purpose of this paper is to analyse the effects of cranial size and shape in domestic dogs (Canis familiaris) on
predicted forces of biting. In addition to continuous size-shape analysis, nine size-shape groups were developed
based on three skull shape categories and three skull size categories. Bite forces were predicted from measurements
made on dried skulls using two lever models of the skull, as well as simple models derived by regression analysis.
Observed bite force values were not available for the database used in this study, so only comparisons between
categories and models were undertaken. The effects of shape and size on scaled predicted bite forces were
evaluated. Results show that bite force increases as size increases, and this effect was highly significant (P < 0.0001).
The effect of skull shape on bite force was significant in medium and large dogs (P < 0.05). Significant differences
were not evident in small dogs. Size × shape interactions were also significant (P < 0.05). Bite force predictions by
the two lever models were relatively close to each other, whereas the regression models diverged slightly with
some negative numbers for very small dogs. The lever models may thus be more robust across a wider range of
skull size-shapes. Results obtained here would be useful to the pet food industry for food product development,
as well as to paleontologists interested in methods of estimating bite force from dry skulls.
Key words bite force; dogs; food mastication; modelling.
Introduction
The relationship between craniofacial morphology in
mammals and biting forces (BF) generated by the masticatory apparatus has two major foci of interest. The first is
the interaction between size-specific magnitudes of BF in
relation to trophic specialization among species within
higher mammalian taxa. Within Carnivora, for example,
the strength of the relationship between BF and diet outweighs phylogenetic constraints on craniofacial form
(Christiansen & Wroe, 2007). Divergent taxa of similar diet
showed convergent aspects of morphology and the capacity
to develop appropriate forces of biting. In comparing
crania of feline and saber-toothed cats, Christiansen (2008),
following Radinsky (1981), advocated that measurements
that are directly linked to biomechanical analyses be
included in phylogenetic analyses of mammals, particularly
when phylogenies prove difficult to resolve.
Correspondence
Jennifer Ellis, Centre for Nutrition Modelling, Department of Animal
and Poultry Science, University of Guelph, Guelph, ON, N1G 2W1,
Canada. T: + 1 519 8244120 ext.56683; F: + 1 519 836 9873;
E: jellis@uoguelph.ca
Accepted for publication 1 December 2008
The second major area of interest is the control of
craniofacial development by the genome and epigenetic
influences during ontogeny within species (Lieberman
et al. 2004; Herring et al. 2005). Within this category are
the broader biological questions of the interaction of
genetic and epigenetic factors, form-function interactions
in complex structures such as the skull, and medical issues
related to orthodontic problems in humans and their
treatment or prevention. A 6-month experiment in which
ferrets were fed either hard or water-softened pellets showed
differences in facial growth and cranial width between the
two groups (He & Kiliaridis, 2003). These authors observed
that in other species, similar dietary stimuli also resulted in
changes but not at the same locations in the skull, indicating
that cranial responses to loading are species-specific. In
humans, craniofacial height and masseter thickness are
among the variables to correlate positively (and presumably
causally) with force of biting (Charalampidou et al. 2008).
Differences in skull shape among canid species have
been associated with differences in jaw strength as a proxy
for forces of biting (Biknevicius & van Valkenburgh, 1996),
and the same has been observed anecdotally for breeds of
domestic dog (Case, 1999).
Domestic dogs (Canis familiaris) provide a unique model
for the study of the relationship between forces of biting
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
Food processing in dogs, J. L. Ellis et al. 363
and craniofacial form, in that breeding has produced a
diversity of forms within a single species (Miller et al. 1965;
Wayne, 1986). Genetic diversity is, therefore, constrained,
as is dietary variability to a large degree. As a result, the
interaction between forces of biting and cranial size and
shape may be explored, with the assumption that the influence
of other relevant factors will have a relatively small effect.
Such an exploration is the primary purpose of this study.
Central to any such study is a reliable evaluation of force
magnitudes. We have elsewhere (Ellis et al. 2008) calibrated
BF estimates generated from morphometric measurements
on dried skulls to forces recorded in vivo under anesthesia
for the same sample of 20 dogs of mixed breed and body
form. Three methods of BF calculation were used for the
ex vivo estimation, and compared. In the present work,
forces are estimated from dry skulls, so the effect of the
three estimation methods on the results is explored in
addition to the primary question of the effect of size and
shape. Forces of biting are estimated for two points on the
skull, approximating to the locations of a canine bite and
carnassial bite, which are characteristic of prey dispatch
and processing in carnivorous canids.
First we develop general relationships between forces
of biting and variables describing size and shape. Variables
include measurements of body weight (BW), skull length
(SL), basicranial length (bSL), skull width (SW) and indices
derived from them (e.g. ratios of facial length and skull
width to skull or basicranial length). The effect of sex on
these relationships is examined. Then we tease out the
interactions among size and shape by subdividing each
into three categories: size as small, medium and large, and
shape as brachy-, mesati-, and dolicho-cephalic (i.e. short,
medium, and long headed). The shape designations have
long been used in breed descriptions (Miller et al. 1965)
and in some morphometric analyses (Alpak et al. 2004);
they parallel similar terms used in the orthodontic literature (Pepicelli et al. 2005). We recognize that the boundaries defined between categories is arbitrary for both size
and shape, and that the results are likely to show some
effect of the location of the boundaries. The value of the
exercise, however, is in the expectation that interactions
do occur between size and shape in their effect on force
of biting, and that this method will demonstrate the
nature of the interactions. By defining the method for
locating the boundaries, it is possible for any other workers to repeat the analysis on a separate sample, and compare results.
Materials and methods
Database
Two independent dry skull collections were used to evaluate BF
in the modern dog. The canine skull collection at the Ontario
Veterinary College (University of Guelph, Guelph ON, Canada;
OVC) consisted of 29 specimens from a variety of breeds, sizes and
skull types, of unknown sex and age (though all but one skull had
complete cranial suture fusion). The second skull collection at the
Albert Heim Foundation (Natural History Museum of Berne,
Switzerland; SWISS) consisted of 98 samples from a variety of
breeds, sizes and skull types with information on sex and age. The
SWISS collection contained 46 known males and 49 known females,
seven known juveniles and 71 known adults, and was selected to
sample a range of body size and skull-shape types. The two sources
of skulls were pooled, and the database is summarized in Table 1.
Skull images
Skulls were digitally photographed from lateral, ventral, dorsocaudal
and dorsal views, and mandibles were photographed from lateral
and dorsal views. The lateral, dorsocaudal and ventral views are
illustrated in Fig. 1. Measurements on each skull were taken from
the scaled photographs using OPTIMAS (1999) software, and
measurements were combined into calculated variables that are
described in the following sections.
Bite force calculations
Forces of biting were estimated at two locations which combined
the characteristic bites used in killing and post-mortem processing
of prey with the ability to reliably reproduce measurements on all
skulls in the sample. The first point was immediately behind the
canine (contacting the first premolars, P1 and P1) and force estimates
for this location are termed canine bite forces (CBF). The second
location was at the junction of P4 and M1, on the maxilla and at the
junction of M1 and M2 on the mandible, giving molar bite forces
(MBF). These locations are the same as in Ellis et al. (2008). Bite
force was estimated from the scaled photographs of each skull
using two methods based on lever mechanics, and one derived
from regression modelling, as described in the next subsections.
All three modelling methods are based on the assumption of
maximal bilateral contraction of the jaw adducting musculature.
Usually the balancing side muscles are less active than those of the
biting side (Dessem, 1989). The necessity and limitations of the assumption are discussed in Ellis et al. (2008).
Lever models
Forces of biting at the canine and molar were estimated using two
models, which are based on the principle of lever mechanics. Both
involve making estimates of the force generated by the jaw
muscles (based on differing estimates of the effective area of
muscle cross section), and the leverage of that force about the
point of biting. lever model 1 is from Kiltie (1984) and lever model
2 from Thomason (1991):
lever model 1
CBF1 = (Lm × M + Lt × T)FPA/Oc
[1a]
MBF1 = (Lm × M + Lt × T)FPA/Om
[1b]
lever model 2
CBF2 = 2(MT × ML + TT × TL)FPA/Oc
[2a]
MBF2 = 2(MT × ML + TT × TL)FPA/Om
[2b]
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
364 Food processing in dogs, J. L. Ellis et al.
Table 1 Summary of dog database sorted by skull shape and size
Dog ID
Dog breed
Skull
Skull
shapea sizeb
Sexc Aged
Dog ID
Dog breed
Skull
shapea
Skull
sizeb
Sexc
Aged
13
22
89
95
19
25
12
33
71
72
73
74
10
18
85
1
63
115
113
47
91
127
42
53
97
98
21
93
68
30
48
5
4
3
31
80
17
120
46
90
43
44
126
50
94
56
109
54
62
107
108
7
77
39
40
76
75
96
125
105
110
Great dane
Irish wolfhound
Irish wolfhound
Mastiff
Newfoundland
St. Bernard
Boxer
Boxer
Boxer
Boxer
Boxer
Boxer
Bull mastiff
Chow chow
English bulldog
Boston terrier
Boston terrier
Boston terrier
Bulldog
Griffon bruxellois
Griffon bruxellois
Griffon bruxellois
King Charles spaniel
King Charles spaniel
King Charles spaniel
King Charles spaniel
Miniature poodle
Pekingese
Afghan hound
Collie
Saluki
Basset hound
Beagle
Bearded collie
Cocker spaniel
Cocker spaniel
Dachshund
Dachshund
Irish terrier
Irish terrier
Kerry blue terrier
Kerry blue terrier
Kerry blue terrier
Lhaso apso
Norfolk terrier
West highland white terrier
West highland white terrier
Whippet
Whippet
Whippet
Whippet
Chihuahua
Daschund
Dwarf dachshund
Dwarf dachshund
Dwarf dachshund
Dwarf Pomeranian
Maltese
Norfolk terrier
Pomeranian
West highland white terrier
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
−
−
M
M
−
−
−
F
F
M
M
M
−
−
F
−
M
F
F
M
F
F
F
F
M
M
−
M
M
F
F
−
−
−
F
M
−
M
F
M
F
F
M
M
F
M
F
F
F
M
F
−
M
F
F
M
M
F
M
F
F
35
67
34
66
2
41
79
78
83
84
36
81
82
14
92
45
52
20
28
102
26
27
6
116
11
8
32
69
16
15
38
86
119
87
88
104
58
59
103
57
101
100
99
55
106
118
117
64
65
114
70
121
122
123
124
37
51
49
29
60
61
Afghan hound
Afghan hound
Berner sennenhund
Berner sennenhund
Black lab retriever
Collie
Collie
Collie
German shepherd
German shepherd
Golden retriever
Golden retriever
Golden retriever
Great dane
Greyhound
Irish wolfhound
Lab retriever
Newfoundland
Rough collie
Siberian husky
St. Bernard
St. Bernard
American pointer
Berner sennenhund
Boxer
Cairn terrier
Cocker spaniel
Cocker spaniel
Dalmatian
English bull terrier
English bulldog
English bulldog
Greyhound
Kerry blue terrier
Kerry blue terrier
Pinscher
Shar-pei
Shar-pei
Shar-pei
Siberian husky
Siberian husky
West highland white terrier
West highland white terrier
Whippet
Whippet
Afghan hound
Akita inu
Border terrier
Border terrier
Boxer
Cairn terrier
Chihuahua
Chihuahua
Chihuahua
Chihuahua
English bulldog
Labrador retriever
Mastiff
Pomeranian
Pomeranian
Yorkshire terrier
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
F
M
F
F
−
M
M
−
M
M
F
M
M
−
M
F
F
−
−
M
−
−
−
F
−
−
M
F
−
−
F
F
F
M
M
M
F
F
M
F
M
M
M
F
M
F
M
F
F
M
M
F
F
F
F
F
F
M
−
M
M
A
A
A
A
−
A
A
−
A
A
A
A
A
−
A
A
A
−
−
A
−
−
−
−
−
−
A
A
−
−
A
A
A
A
−
A
A
A
−
J
A
A
A
A
−
−
J
A
A
−
A
A
A
−
A
A
−
−
−
A
A
L
L
L
L
L
L
M
M
M
M
M
M
M
M
M
S
S
S
S
S
S
S
S
S
S
S
S
S
L
L
L
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
S
S
S
S
S
S
S
S
S
S
−
−
A
−
−
−
−
−
J
A
A
A
−
−
A
−
A
A
−
A
A
A
A
−
A
A
−
−
J
A
A
−
−
−
A
A
−
A
J
−
A
A
A
A
A
A
A
A
A
−
A
−
−
A
A
A
A
A
J
J
A
a
Skull shape categorization determined by facial ratio, where B is brachycephalic, D is dolichocephalic and M is mesaticephalic skull shape.
Dog size categorization determined by skull length, where S is small, M is medium and L is large skull size.
c
Sex, where M = male, F = female.
d
Age, where A = adult, J = juvenile.
b
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
Food processing in dogs, J. L. Ellis et al. 365
Fig. 1 Measures utilized to calculate forces of biting (BF) using lever model 1 (Kiltie 1984), lever model 2 (Thomason 1991) and the regression models
of Ellis et al. (2008), as seen from a lateral view of the skull (a), lateral view of the mandible (b), dorsocaudal view of the skull (c) and ventral view of
the skull (d). * indicates the centroid of the relevant muscle, Lm is the length of the masseter origination scar on the zygomatic arch; Lt is the height of
the coronoid process above the jaw condyle; M is the area of a rectangle calculated as the product of the length and width of the masseter origination
scar on the zygomatic arch in ventral view, and T is the area of the temporalis origination scar calculated as the product of the length and height of
the temporalis fossa in lateral view; MT is the cross-sectional area of the masseter and medial pterygoid muscles in ventral view; ML is the lever arm
of the masseter and medial pterygoid combination about the jaw joint (measured from the midpoint of the jaw joint to the centroid of the combination,
parallel to the basicranial axis); TT is the cross-sectional area of the temporalis muscle in dorsocaudal view; SW is skull width; TL is the lever arm of the
temporalis about the jaw joint (measured from the centroid of the temporalis to the projection of the midpoint of the jaw joint onto the plane of part
1c); Oc and Om are the distances from the jaw joint to the canine and second molar, respectively.
where CBF1 and CBF2 are the calculated force of biting in Newtons
(N) at the canines predicted by lever model 1 and 2, as indicated
by the subscripts. MBF1 and MBF2 are the corresponding molar
forces; FPA is the force per unit area of muscle which was taken as
300 MPa after Weijs & Hillen (1985); and all model variables are
defined and illustrated in Fig. 1.
Initial results from the lever models were adjusted using values
recorded in vivo, during muscle stimulation of dogs under general
anesthesia, as described in Ellis et al. (2008; adjustment method
#1). This method consisted of plotting observed vs. predicted bite
force at the canine and molar, where the resultant regression
equation provided an ‘adjustment’ equation for the two lever
models, correcting bias and deviation of the regression slope from
unity of predicted vs. observed BF values:
All BF estimates presented in this paper for lever model 1 and 2
are adjusted values.
Adjustment for lever model 1
MBFR = −1892(± 331.2) + 15.15(± 6.677) × BW + 909.9(± 185.8)
× Lt + 0.7611(± 0.2439) × T
[5b]
Adj. CBF1 = 1.781 × CBF1 + 36.94
[3a]
Adj. MBF1 = 3.504 × MBF1 − 696.3
[3b]
Adjustment for lever model 2
Adj. CBF2 = 1.440 × CBF2 + 98.10
[4a]
Adj. MBF2 = 2.776 × MBF2 − 320.9
[4b]
Regression models
Ellis et al. (2008) used multivariate regression analysis to evaluate
the utility of a suite of cranial variables and BW in predicting
forces of biting, independent of any lever model. From a large
number of possible regression equations, one was selected for
each bite location (canine and molar) on the basis of minimum root
mean square prediction error (RMSPE) (Bibby & Toutenburg, 1977).
See Ellis et al. (2008) for a detailed description of this procedure:
CBFR = −555.5(± 238.1) + 88.45(± 18.75) × Oc
[5a]
Skull variables are illustrated and defined in Fig. 1. The regression
models of Ellis et al. (2008) were based on observed bite force
values of sedated dogs, where the masticatory muscles of the
head were maximally stimulated and bite forces recorded.
Relationships between skull measurements and observed bite
forces were recorded, and regression equations developed in
PROC REG of SAS (SAS, 2000).
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
366 Food processing in dogs, J. L. Ellis et al.
Measurement of skull size and shape
Statistical analysis
Three measures of size were considered: SL (cm), estimated BW
(kg), and maximal SW (cm) across the zygomatic arches. It was
recognized that all three include components of shape. Bodyweight
was not known for skulls in the collection, and estimates were taken
from book values based on breed (Pugnetti, 1980).
Four methods of evaluating skull shape were used. Firstly, the
Miller index (MI) was used to evaluate the width-to-length ratio
of each skull (Miller et al. 1965):
For evaluation of skull size and shape as continuous variables
across the skull database, regression equations were developed in
PROC REG in SAS (SAS, 2000) based on BW, SL, SW, and FR, MI and
MI′, for the three methods of estimating BF (Eqs 3 –5). Regression
equations were evaluated based on parameter significance and
residual variance (RV) values. Residual variance values have the
same unit as the variable being measured (BF, N), and therefore
give an indication of how well each equation developed fitted the
data. Residual variance is the variance left unexplained by the
model, and lower RV values indicated better model fit.
For categorical analysis, PROC MIXED in SAS (SAS, 2000) was used
to evaluate the effects of the fixed variables: size, skull shape,
size × shape and sex on bite force (estimated in three ways, Eqs 3–
5, at each of the canine and molar locations). PROC MIXED is a procedure that fits a variety of mixed linear models to data where the
data are permitted to exhibit correlation and non-constant variability. It is an analysis of variance with random and fixed factors
plus interactions. Size-shape BF means were statistically compared
to each other using a Tukey test within the LS means statement of
the PROC MIXED procedure. Significance was declared at P ≤ 0.05.
MI = SW × 100/SL
[6]
Miller et al. (1965) stated that the mean value for brachycephalic,
mesaticephalic and dolichocephalic skulls was 81%, 52% and
39%, respectively.
The second method was developed as a variation of the MI,
where SL was replaced with bSL. The resultant ratio (MI′) normalizes
skull width to a measure of cranial size accepted to be independent
of facial length (Jaslow, 1987). It was not possible to measure bSL
from many skull photographs because the suture defining its
rostral boundary was not always visible. So a repeatable measure of
facial length was devised, from the rostral-most point of the skull
to the caudal edge of the upper third molar. This latter point is
closely adjacent to the vomero-basisphenoid suture in the choana,
which is the anterior boundary of bSL, yet is caudal to the most
caudal position of the bite point (Christiansen & Adolfssen, 2005).
For our purposes, bSL was taken to be skull length minus facial
length. Facial length and SL are illustrated in Fig. 1.
The third index is the facial ratio (FR), which is the ratio of facial
length to SL. It is approximately the complement of the ratio of bSL
to SL, and therefore provides a means of identifying the degree of
facial elongation around the braincase. For skulls where bSL was
available, it was found that there was a highly significant correlation
between the bSL/SL ratio and FR (P < 0.0001), so the two ratios are
assumed to be equivalent, and FR is used here.
Placement of skulls into shape and size categories
For the categorical analysis, three categories of size were based on
SL – small, medium, and large – and three on shape – dolicho-,
mesati-, and brachy-cephalic. (SL was used as the primary measure
of size in this exercise.)
For SL analysis, the sample of dogs in the database (containing
skulls from Chihuahuas to Great danes) was assumed to be
representative of the population, and the difference between the
minimum SL and maximum SL, equally divided into three categories,
was used to determine the boundaries between small, medium
and large size dogs. The small-to-medium boundary was placed at
13 cm, and the medium-to-large boundary at 19.9 cm.
The methods outlined above used to quantify skull shape (FR,
MI and MI′) aim to objectively assign animals a numerical value
based on a measure of facial elongation relative to the brain case
or SW. For categorical analysis, FR was selected as the measure of
skull shape.
The mean FR for the skull database was 0.58 (± 0.0125 SD).
When FR was in the range 0.578–0.588 (i.e. within 3/8 of 1 SD of
the mean), a skull was designated as mesaticephalic. When FR fell
below the range, the skull was brachycephalic, and above it the
skull was dolichocephalic. This method allowed placement of all
skulls, including those that were not easy to place on simple
inspection or on breed definitions.
Results
Skull size and shape as continuous variables
Skull size and shape were examined as continuous variables
across all skulls using regression analysis. Regression
equation parameters from all models are presented in
Table 2, where predicted BF values from Eqs 3–5 were
regressed against the measurements SL, BW, and SW, and
indices FR, MI, and MI′ to develop these equations.
Table 2 shows that BF increases at both molar and
canine bite locations as measures of body size (BW, SL, SW)
increase, for all models. All slope parameter estimates
on these measurements are positive and significantly
different from 0 (P ≤ 0.05; Table 2, Fig. 2). Ranking the
skull size variables, the BF vs. SW regression resulted in
the lowest average RV value (353 N), followed in order
by the BF vs. SL regression (385 N) and the BF vs. BW
regression (394 N). Regressions of BF vs. SL are illustrated
in Fig. 2.
For the skull shape regressions (BF vs. FR, MI or MI′), all
parameter estimates, except for one within the BF vs. FR
regressions, were significant (P ≤ 0.05) (Table 2). All skull
shape regressions resulted in negative slope parameters,
indicating BF decreases as the face elongates relative to
the braincase. Ranking the skull shape variables, the BF vs.
MI′ regression resulted in the lowest average RV value (602
N), followed in order by the BF vs. MI regression (653 N)
and the BF vs. FR regression (676 N). Regressions of BF vs.
FR are illustrated in Fig. 2.
Average RV values for the shape regressions (range 602–
676) are higher than for the size regressions (range 353–
394), suggesting that size is more strongly related to BF
than size-specific indices of shape.
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
Food processing in dogs, J. L. Ellis et al. 367
Table 2 Regression equation parameters for predicted bite force (BF) vs. (a) measurements of size: skull length (SL), bodyweight (BW) and skull width
(SW); and (b) indices of facial shape: facial ratio (FR), Miller index (MI) or normalized Miller index (MI′). For each of the regressions no observed BF values
were available. Values of BF were separately predicted by the regression model of Ellis et al. (2008), lever model 1 or lever model 2 at the molar (M)
or canine (C) for all skull samples, and then these predicted BF values were regressed against SL, VW, SW, FR, MI and MI′
Regression
(a)
BF vs. SL
BF vs. BW
BF vs. SW
(b)
BF vs. FR
BF vs. MI
BF vs. MI’
Equationa
Intercept
SEM
P value
Slope
SEM
P value
n
RVb
regression model (M)
regression model (C)
lever model 2 (M)
lever model 2 (C)
lever model 1 (M)
lever model 1 (C)
Average RV
regression model (M)
regression model (C)
lever model 2 (M)
lever model 2 (C)
lever model 1 (M)
lever model 1 (C)
Average RV
regression model (M)
regression model (C)
lever model 2 (M)
lever model 2 (C)
lever model 1 (M)
lever model 1 (C)
Average RV
−2544
−517
−958
−90.7
−1030
−99.1
−
−146
132
867
257
743
210
−
−3896
−724
−2669
−512
−2255
−410
−
205
22.3
233
55.9
194.5
59.2
−
102
28.7
98.3
19.8
86.8
20.1
−
314
68.9
204
54
228
74.7
−
< 0.0001
< 0.0001
< 0.0001
0.107
< 0.0001
0.0907
−
0.1529
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
−
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
−
243.6
57.7
165
37
163
36
−
69.7
13
43.4
11.7
43.3
12.7
−
571
123
471
110
418
96.5
−
12.1
1.32
13.2
3.23
11.2
3.48
−
3.61
1.02
3.39
0.687
3.04
0.712
−
33.1
7.26
20.9
5.59
23.6
7.86
−
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
−
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
−
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
−
118
117
103
111
104
115
−
118
117
103
111
104
115
−
118
117
103
104
111
115
−
679
73
634
166
563
192
385
701
198
626
129
571
137
394
762
165
412
490
116
175
353
regression model (M)
regression model (C)
lever model 2 (M)
lever model 2 (C)
lever model 1 (M)
lever model 1 (C)
Average RV
regression model (M)
regression model (C)
lever model 2 (M)
lever model 2 (C)
lever model 1 (M)
lever model 1 (C)
Average RV
regression model (M)
regression model (C)
lever model 2 (M)
lever model 2 (C)
lever model 1 (M)
lever model 1 (C)
Average RV
6575
1336
10179
2251
6637
1439
−
4070
1125
2833
744
3187
814
−
5413
1450
3876
971
3968
997
−
2245
434
1647
387
1729
425
−
485
95.4
417
93.4
389
97.1
−
507
93.3
447
104
413
108
−
0.0041
0.0067
< 0.0001
< 0.0001
0.0002
0.001
−
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
−
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
−
−8931
−1583
−14287
−2973
−8462
−1637
−
−44.3
−11.7
−16.7
−3.68
−25.3
−5.42
−
−2779
−713
−1434
−313
−1599
−353
−
3846
829
2819
664
2952
728
−
7.73
1.52
6.88
1.5
6.38
1.55
−
341
62.8
309
70.7
284
72.6
−
0.022
0.0586
< 0.0001
< 0.0001
0.005
0.0265
−
< 0.0001
< 0.0001
0.0167
0.0161
0.0001
0.0007
−
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
−
118
117
103
111
104
115
−
118
117
103
111
104
115
−
118
117
103
111
104
115
−
1407
303
904
227
950
262
676
1270
250
984
240
919
254
653
1147
211
920
227
862
243
602
a
Results from the lever models have been adjusted as per the methodology of Ellis et al. (2008).
RV = residual variance, a measure of overall fit of the model to the data.
b
Effect of skull size on bite force
Because SL may be a more objective estimator of size
than BW and is an observed variable in the current
database, it was used for categorical analysis of size effects
on BF. Bite force averages for size categories, using each
of the equations, are presented in Table 3. Regardless of
model used, predicted BF of large dogs was greater than
that of medium size dogs, which was greater than that
of small size dogs at both the canine and molar teeth
(Table 3). Accordingly, PROC MIXED analysis of the data
showed size to be highly significantly (P < 0.0001)
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
368 Food processing in dogs, J. L. Ellis et al.
Fig. 2 Predicted bite force (BF) at the molar (right) or canine (left), for the regression model (blue closed diamonds), lever model 2 (pink open squares)
and lever model 1 (green closed triangles) vs. skull length (SL, cm) (top) or facial ratio (FR) (bottom).
Table 3 Summary of bite force (N) predictions where skull type is placed by facial ratio and size by skull length
Size
Small
Medium
Large
Skull Type
Location
Equationa
Mean
SEM
n
Mean
SEM
n
Mean
SEM
n
Brachycephalic
Molar
Regression model
lever model 2
lever model 1
Regression model
lever model 2
lever model 1
Regression model
lever model 2
lever model 1
Regression model
lever model 2
lever model 1
Regression model
lever model 2
lever model 1
Regression model
lever model 2
lever model 1
−392
834
352
25.1
292
216
294
876
782
89.8
380
374
−184
512
715
40.9
230
262
88.4
80.6
56.6
22.5
18.8
14.8
361
150
125
56.7
85.5
105
151
88.0
157
33.7
14.1
20.7
13
8
8
13
11
13
16
8
11
15
11
15
10
8
8
9
8
8
2087
2713
2228
527
708
600
1408
1914
1667
454
533
466
1006
1345
1395
377
396
400
238
238
195
28.4
63.5
52.7
118
146
121
23.8
31.7
30.7
86.9
78.4
68.3
26.7
16.3
12.5
9
8
7
9
9
8
23
22
21
23
23
23
20
20
20
20
20
20
4468
3833
3909
946
1042
1063
2749
2576
2450
755
693
670
1837
1579
1638
661
472
486
144
165
167
34.6
45.6
49.3
147
144
151
19.8
32.6
33.8
149
36.6
182
15.2
19.6
37.1
6
6
6
6
6
6
22
22
22
22
22
22
3
3
3
3
3
3
Canine
Mesaticephalic
Molar
Canine
Dolichocephalic
Molar
Canine
a
Results from the lever models have been adjusted as per the methodology of Ellis et al. (2008).
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
Food processing in dogs, J. L. Ellis et al. 369
Table 4 Summary of PROC MIXED analysis of skull type, size and sex effects on bite force predictions
Location
Equationa
Skull type
Skull size
Skull type × size
Sexb
Molar
Regression model
Lever model 2
Lever model 1
Regression model
Lever model 2
Lever model 1
< 0.0001*
< 0.0001*
< 0.0001*
0.0015*
< 0.0001*
0.0002*
< 0.0001*
< 0.0001*
< 0.0001*
< 0.0001*
< 0.0001*
< 0.0001*
< 0.0001*
0.0006*
< 0.0001*
0.0053*
< 0.0001*
< 0.0001*
0.0595
0.0835
0.0451*
0.0488*
0.0279*
0.0165*
Canine
a
Results from the lever models have been adjusted as per the methodology of Ellis et al. (2008).
Based on 95 skulls for which sex was known.
*Implies significance (P ≤ 0.05).
b
related to predicted BF regardless of the prediction
equation (Table 4).
Bite force at the molar location was consistently larger
than at the canine when predicted by the lever models,
which is not surprising given the longer out-lever arm
from the jaw joint to the canine compared to the molar for
these equations (Table 3). The regression model produced
some non-sensible negative numbers in small dogs at the
molar location (Table 3). Other than these two instances
where bite force at the molar was lower than at the canine
(because it was negative), the same pattern of increasing
BF with increasing size was evident for the regression
model. Unlike the lever models, where the equations differ primarily in the out-lever arm length between canine
and molar locations, the regression models for canine and
molar are completely unrelated to each other (Eq. 5a and
5b) and so some inconsistencies may occur.
Effect of skull shape on bite force
Table 3 shows that BF increases as skull shape moves from
dolichocephalic to mesaticephalic to brachycephalic, and
PROC MIXED analysis showed that skull type has a significant effect on shape-size category means (Table 4). Table 5
examines whether skull shape category means from
Table 3 are significantly different from each other within
a size category. Results indicate that most of the shape
means within medium and large dog categories are significantly different from each other, whereas all but one are
nonsignificantly different in small dogs. This suggests there
may be a size × shape interaction for BF, and that shape
may not be a significant factor in determining BF in small
dogs. The PROC MIXED analysis agrees with this observation,
and shows that size × shape interaction is significant (Table 4).
Figure 3 illustrates this observation, as it shows a non-linear
relationship between skull shape means in small dogs.
Effect of sex on bite force
PROC MIXED analysis in SAS (SAS, 2000) was also used to
test the effect of sex on BF (Table 4). At the canine tooth,
BF was significantly different between the sexes (P ≤ 0.05)
Fig. 3 Bite force at the molar (MBF) (top) and canine (CBF) (bottom) (N)
predicted by lever model 1 (Kiltie 1984) (
), lever model 2
(Thomason 1991) (
), or the regression model of Ellis et al. (2008)
( ) vs. skull type, where B is brachycephalic, M is mesaticephalic, and
D is dolichocephalic skull type; for small (S), medium (M) and large (L)
dogs.
regardless of BF prediction equation. At the molar, sex
effects were only significant for one equation (Table 4). In
all cases, P values were not exceptionally low. Numerically,
BF for females was always lower than for males [BF (N) at
the canine = 486 ± 42 and 375 ± 23 for males and females,
respectively; and BF (N) at the molar = 1606 ± 170 and
1217 ± 95 for males and females, respectively, averaged
across BF prediction equation]. The number of known
males and females in the database was roughly equal, and
in forming the database, efforts were made to distribute
males and females evenly across skull shape and size. It is
thus not suspected that the above results had any major
influence on the other results reported.
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
370 Food processing in dogs, J. L. Ellis et al.
Table 5 Summary of least square mean statistic results for skull shape
category means of different size and bite location
Size
Location
Equationa
Contrastb
S
M
L
Molar
Regression model
B vs. M
M vs. D
B vs. D
B vs. M
M vs. D
B vs. D
B vs. M
M vs. D
B vs. D
B vs. M
M vs. D
B vs. D
B vs. M
M vs. D
B vs. D
B vs. M
M vs. D
B vs. D
NS
NS
NS
NS
NS
NS
*
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
*
*
*
*
*
*
*
NS
*
NS
NS
*
*
*
*
*
NS
*
*
NS
*
*
*
*
*
NS
*
*
NS
*
*
*
*
*
NS
*
Lever model 2
Lever model 1
Canine
Ellis et al. (2008) Eq. 19
Thomason (1991)
Kiltie (1984)
a
Lever models are ADJUSTED as per Ellis et al. (2008).
Using the LS MEANS procedure of SAS PROC MIXED to perform
a Tukey test to determine whether the two category means are
significantly different from each other, where B is brachycephalic,
M is mesaticephalic and D is dolichocephalic skull shape.
*Implies significance (P ≤ 0.05), NS implies nonsignificance and n/a
indicates that there were not enough data points in the group to
perform the analysis.
b
mean BF values were obtained from the regression model
for the molar BF (Table 3). This suggests that the current
database challenged this equation outside of the range of
values on which it was developed and it should be used
with caution when being applied to the extremes of facial
shapes. The regression model (Ellis et al. 2008) may be
more viable for dogs closer to the mean, or when the
complex measurements required for the lever models
cannot be taken.
For the results presented in Table 2, within the BF vs. SL
regression, the Ellis et al. (2008) Eq. 19 also resulted in a
substantially lower RV value compared with the other
equations used in this regression. This is not surprising as
this Ellis et al. (2008) equation is based on Oc, the out-lever
from jaw joint to the canine tooth, and is thus strongly
related to SL. Thus the lower RV value for this equation
must not be interpreted as predicting BF substantially
better, it is more that the BF estimate and SL measure are
confounded and are not independent of each other. Using
the BF vs. SL regression equation from this equation is not
recommended. Similarly, the Ellis et al. (2008) Eq. 15
includes BW as a predictor, and thus should not be used in
combination with the general BF vs. BW regression in
Table 2.
From Table 2, in general, for predicted BF at the canine
and molar teeth locations, lever model 1 resulted in the
lowest RV values on average (across regression) [RV = 597,
497 and 437 for the Ellis et al. (2008) regression model,
lever model 2 and lever model 1, respectively]. These
results do not imply that lever model 1 is a better predictor
of BF, but rather that the BF predictions using lever model
1 coincide with the regressed variables in Table 2 better, as
no observed BF values are available.
Effect of equation used to calculate bite force
The equations used to estimate BF also influence the
results obtained. This study calculated BF using the two
lever models, which were previously adjusted to be on
scale with observed BF values (Ellis et al. 2008), and regression
models, which were developed from observed BF data
(Ellis et al. 2008). Means from each model are presented in
Table 3. These results show that while the two lever
models predict BF closely to each other, the regression model
diverges. Selection of the most accurate equation for
predicting BF is challenging due to lack of observed BF
values in this study. However, the regression models used
here were the best of eight models for canine biting,
and of seven for molar biting, based on RMSPE analysis
(Ellis et al. 2008). This evaluation of models was performed
on a much smaller database lacking the full range of
skull shapes and sizes present in the current database.
An important point to consider when selecting an
equation, other than the availability of input variables,
is the presence of negative non-sensible BF values. Particularly for the small brachycephalic skull group, negative
Discussion
The objectives of this study were (1) to examine differences
in predicted BF relative to skull size and shape considered
on a continuum, as well as for nine categories of shape
and size combined, and (2) to compare predictions of
the three bite force prediction models (Kiltie, 1984;
Thomason, 1991; Ellis et al. 2008) used to estimate BF.
Results from this study suggest that BF is strongly correlated with body size as measured by BW, SL and SW,
regardless of the effects of cranial shape. The categorical
analysis performed demonstrates interactions between
size and shape, whereby higher forces are predicted
for small mesaticephalic skulls than small brachy- or dolichocephalic skulls. This contrasts with the inverse relationship
seen in the medium and large size categories between
facial shortening and BF, as might have been expected.
The model used to predict BF is also a significant factor
in the results obtained, and all of the results are accompanied by the caveat that they are derived from ex vivo
models.
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
Food processing in dogs, J. L. Ellis et al. 371
Skull size and shape: continuous variables
Skull size and shape: categorical analysis
Regardless of model used, BF of domestic dogs increased
with increasing size. However, whereas SL and SW were
measured directly and easily off of the skull images, BW
was required to be estimated based on book breed
averages. It therefore likely contained more variation and
error. An obese dog might be expected to have a stronger
BF than a non-obese dog of the same breed and stature.
However, the extra adipose tissue on the obese dog
conveys no additional force of biting advantage above its
leaner counterpart, and BF would be over-predicted for
this animal. While either method could be used to scale
size, it appears that using SL or SW may be the more
reliable and objective methodology. As well, SL and SW
are easily measured from dry skull images, and this
measurement may also be obtained easily from live
animals.
Caution is needed when interpreting the regression
of BF on the indices FR, MI and MI′, partly because
ratios tend to obscure aspects of size-shape relationships
by oversimplification, and because BF was calculated
from measurements that are not in all cases independent
of those used in the ratios. In general the average RV
values were higher for the skull shape indices than the
size indices. Aside from this, these regressions have
implications that warrant further study. Bite force is
inversely related to all three ratios, at both canine and
molar locations. While this relationship was expected
for FR at the canine – a long face increases the out-lever
of the bite – it was not for the molar (Greaves, 1983, 1988;
Christiansen and Adolfsen, 2005). Greaves (1983, 1988)
used geometric models to show that the location of the
carnassial ought to be constrained to be in the same relative position in carnivoran jaws of different sizes and
shapes, and therefore to exert similar bite forces with
respect to body size. If this finding is not an artifact of
the modelling process, it might indicate that the
breeding process had circumvented the physical principles involved in biting.
It is also not immediately intuitive for BF to show inverse
relationships with MI or MI′, the ratios of SW to SL and bSL,
respectively. Even though the absolute force of biting
increases with skull width, it decreases relative to skull
length. The probable reason is that SW is negatively isometric with respect to both SL and bSL, i.e. skull width at
the zygomas does not generally increase at the same rate
as either measure or skull length. Least-squares regression
gave slopes of 0.188 for SW on SL (r 2 = 0.6346) and 0.569
for SW on bSL (r 2 = 0.6366), which are well below the
isometric slope of 1. (We did not calculate scaling
regressions, using logarithmic variables, because the size
range did not warrant doing so, based on a couple of test
examples which showed lower value of r 2 than for nontransformed data.)
Interactions exist between skull size and shape (Table 4),
such that differences in BF caused by skull shape may be
more evident in large skulls than in small skulls (Table 5;
Fig. 3). It is apparent from Fig. 3 that divergence between
the shape categories occurs as size increases. The lack of
significant differences in BF between the shape groups in
small dogs could be caused by several factors. In small
brachycephalic skulls, such as that of the Chihuahua, the
calvaria is disproportionately large in relation to facial
structures. A larger brain case impinges on the space
available for the masseter muscle to occupy, and thus the
size of the masseter may be limited in these dogs and result
in lower BF values than expected. This disproportionate
brain case size is not as evident in larger brachycephalic skulls,
and Wayne (1986) suggested it was a result of paedomorphosis in the smallest dogs. At the other extreme, although
brain case would not be expected to impinge on masseter
muscle size in an elongated dolichocephalic skull, the
longer out-lever arm for this animal decreases the overall
BF. The larger brain case, which may cause predicted bite
forces to be lower than expected in small brachycephalic
dogs, may be countered by the long out-lever arm in the
small dolichocephalic skull, making the two groups nonsignificantly different from each other in terms of BF. As
brain case does not seem to impinge on the space for the
masseter muscle in larger brachycephalic dogs, the bite
forces of these dogs are higher and more significant differences between shape categories are evident. Although
our database does not have enough small animal samples
of all three skull shapes to test this hypothesis accurately,
it could be easily examined by measuring masseter muscle
size and brain case volume in a variety of dogs and seeing
how these proportions of the two values vary with increasing
size and changing shape.
While it was attempted to compile a database of skulls
that would span all size-shape categories, given the objective methodology used to place skulls in categories, some
categories ended up with low sample numbers (Table 3).
Averages obtained from low sample numbers must be
approached with caution, as the overall variability within
some groups was high (Table 3). It is likely that variability
is high because the models cannot capture all aspects of
skull morphology important to bite force.
Animal breeding
We introduced the possibility above that breeding might
have circumvented the physical principles involved in
biting, and it is fair to expect that this artificial circumstance
has not followed the path that natural selection would
have proscribed. The rate of morphological change by
selective breeding is impressive: a sample of skulls from
adult St. Bernard dogs spanning only 120 years showed
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
372 Food processing in dogs, J. L. Ellis et al.
increases in width and ventral rotation of the facial region
relative to the basicranium (Drake & Klingenberg, 2008).
Both changes relate to characteristics deemed desirable
in the breed. The causal relationship between the
magnitudes of bite forces needed to dispatch prey of
varying size has presumably been over-ridden by the
desires of breeders for conformational and functional
attributes shown by the breeds. Work by others has shown
that the rapid morphological changes in breeds of domestic
dogs have only been possible because of genetic mechanisms that appear to be for the specific purpose of allowing such changes while maintaining a version of Cuvier’s
pre-Darwinian ‘law of correlation of parts’. Alpak et al.
(2004) demonstrated a series of correlations within different breeds among measurements of cranial and
postcranial bones, thus providing circumstantial evidence
for a genetic linkage between the masticatory and locomotory regions of the body. (Unfortunately they did not
correlate the cranial measurements with each other, so a
direct comparison with the present results cannot be
made). Chase et al. (2002), in an elegant and sophisticated
procedure using one canine breed, used the eigenvectors
from a principal components analysis of cranial and
postcranial measurements as phenotypic characters, and
correlated them with the presence of specific alleles at
nine gene loci known to specify skeletal form. The results
strongly hint at a genetic basis for the correlations found
by Alpak et al. (2004). Confirmation would require further
studies of other breeds. Still to be added to that picture is
the effect of epigenetic influences, such as the feedback
between masticatory force and craniofacial growth
(Herring et al. 2005). We have demonstrated in this paper
some aspects of the end result of these processes.
Effect of model choice on absolute absolute estimates
of bite force
Many previous authors interested in broader biological
questions involving BF, such as those related to trophic
specialization (Christiansen & Wroe, 2007), have wisely
used relative rather than absolute values of BF. Our objective in this and a previous paper (Ellis et al. 2008) was to
attempt to provide methods for accurate estimation of
absolute BF. The models used here contain and contribute
variation to the overall results. However, they represent
useful tools to estimate BF when live samples are not
available, as is often the case. It is possible that the variables
the equations use to predict BF do not capture all aspects
of skull morphology important in determining BF, and this
may contribute to high variation in predicted BF within a
size-shape category.
Although the adjusted lever models both closely predict
BF, the regression model results diverge slightly and
produced a few non-sensible negative averages at small
sizes (Table 3; Figs 2 and 3). The evaluation of Ellis et al.
(2008) showed the regression models to perform best on
the database of in vivo bite forces. This database, however,
did not completely span the size and shape extremes
present in the current database. For example, the range of
SL values from the Ellis et al. (2008) live dog database
spanned 12.8–23.9 cm, whereas the current database
spanned 6.23–26.7 cm. In terms of FR, Ellis et al. (2008) had
a range of 0.582–0.673, whereas the current database had
a range of 0.464–0.610, covering more extreme brachycephalic skull shapes at the low end. The regression model
would have to be adjusted based on a larger database
spanning a wider variety of dog skulls to accurately
estimate BF of these more extreme dogs. These regression
models would likely be satisfactory for average dogs,
and may have input variables easier to obtain than for
the lever models. Given that the inputs required for the
lever models are available, these models seem to give
more reliable and sensible results than the regression
model.
Although it is difficult to confirm results without observed
values against which to compare predicted values, several
recommendations can be extracted. Firstly, whereas any
size-shape classification method could be used, the SL × FR
classification method was used here for categorical
analysis because the variables are objective and easy to
measure.
This paper also presents three estimates of BF for each
size-shape classification, based on the three prediction
models. One model may be selected over the other two based
on the ease of collecting the input variables required;
however, due to the appearance of negative BF values
with the regression model for extreme animals, the two
lever models may be more desirable. The size-shape
averages can be extracted from Table 4 for these equations,
the equations themselves can be used to estimate BF with
the adjustment of Ellis et al. (2008), or the new regression
equations presented in Table 2 can be used.
When attempting to apply the category averages
presented here, one must keep in mind that the prediction
equations were calibrated using observed values on
maximally stimulated sedated dogs (Ellis et al. 2008).
Thus, the values presented may not represent average BF
values, but rather maximum values. The relationship
between average and maximum BF is unknown to the
authors.
Overall, results indicate that BF increases as size
increases, and differences due to skull shape may be less
apparent in small skulls. In medium and large skulls, it
appears that the skull of the brachycephalic dog conveys a
greater BF advantage, resulting in larger bite force values
for these dogs. Results also suggest that while the regression
models of Ellis et al. (2008) may be easier to use, the lever
models by Kiltie (1984) and Thomason (1991) produce less
non-sensible numbers and may be more sensitive to differences in BF due to skull shape.
© 2009 The Authors
Journal compilation © 2009 Anatomical Society of Great Britain and Ireland
Food processing in dogs, J. L. Ellis et al. 373
Conclusions
Force of biting in domestic canids is strongly related to
size, as quantified by measures of BW, SL and SW. The
effects of cranial shape interact with those of size, particularly in small dogs, in which brachycephalic breeds
appear to have lower bite forces relative to short-faced
dogs of larger size. We present three choices of model for
estimating absolute forces of biting from dried skulls, and
demonstrate the variability among values obtained from
each model on a sample of dried skulls encompassing much
of the size range and several breeds of domestic dogs.
Acknowledgements
The authors wish to thank Dr. Jim Atkinson, Warren Bignell, Dr.
Marc Nussbaumer and Elisabeth Schaeublin for their efforts and
contributions to this project. Also, thank you to the Albert Heim
Foundation at the Natural History Museum of Berne, Switzerland,
for granting us access to their dry skull collection. Funding was
provided by MARS Canada Inc. and in part through the Canada
Research Chairs Program. The constructive comments of two
reviewers stimulated improvements in the manuscript and are
appreciated.
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© 2009 The Authors
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