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In vivo surface strain and stereology of the frontal and maxillary bones of sheepImplications for the structural design of the mammalian skull.

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THE ANATOMICAL RECORD 264:325–338 (2001)
In Vivo Surface Strain and Stereology
of the Frontal and Maxillary Bones of
Sheep: Implications for the Structural
Design of the Mammalian Skull
JEFFREY J. THOMASON,1* LAWRENCE E. GROVUM,1
ARMAND G. DESWYSEN,2 AND WARREN W. BIGNELL1
1
Department of Biomedical Sciences, University of Guelph, Guelph, Ontario, Canada
2
Université Catholique Louvain, Faculty of Agricultural Science,
Louvain-la-Neuve, Belgium
ABSTRACT
Does the skull of the sheep behave as a tube or as a complex of independent bones linked
by sutures? Is the architecture within cranial bones optimized to local strain alignment? We
attempted to answer these questions for the sheep by recording from rosette strain gauges on
each frontal and maxillary bone and from single-axis gauges on each dentary of five sheep
while they fed on hay. Bone structure was assessed at each rosette gauge site by stereological
analysis of high-resolution radiographs. Structural and strain orientations were tested for
statistical agreement. Ranges of strain magnitudes were ⫾1200 ␮⑀ on the mandible, ⫾650 ␮⑀
on the frontals, and ⫾400 ␮⑀ on the maxillae. Each gauge site experienced one strain signal
when on the working (chewing) side and a different one when on the balancing (nonchewing)
side. The two signals differed in mode, magnitude, and orientation. For example, on the
working side, maxillary gauges were under mean compressive strains of –132 ␮⑀ (S.D., 73.3
␮⑀), oriented rostroventrally at 25°–70° to the long axis of the skull. On the balancing side,
the same gauges were under mean tensile strains of ⫹319 ␮⑀ (S.D., 193.9 ␮⑀), at greater than
65° to the cranial axis. Strain patterns on the frontals are consistent with torsion and bending
of the whole skull, indicating some degree of tube-like mechanical behavior. Frontal and
maxillary strains also showed a degree of individual loading, resulting from modulation of
strains across sutures and local effects of muscle activity. The sheep skull seems to behave as
a tube made of a complex of independent bones. Structural orientation was in statistically
significant agreement with the orientation of working-side compressive principal strain ⑀2,
even though principal tensile strains may be as large or larger. Cranial bone architecture in
sheep is not optimized to both strain signals it experiences. Anat Rec 264:325–338, 2001.
©
2001 Wiley-Liss, Inc.
Key words: sheep; skull; bone architecture; bone strain; stereology; optimization
The cranial skeleton in mammals has proved to be extremely plastic in its ability to adapt to the demands of the
numerous functions it performs. Cranial form has been
modified, for example, under the demands of dietary specialization (Smith and Savage, 1959), variation in brain
size (Radinsky, 1968), and rotation of the orbits for widefield or stereoscopic vision (Cartmill, 1970). This plasticity
suggests that cranial form within a species is optimized
for the parameters of each function that is relevant to the
species.
There has long been interest from zoologists and physical anthropologists in analyzing the natural design of the
skull in individual species and comparatively. Analyses
fall into two broad areas, which may be termed functional
design and structural design.
©
2001 WILEY-LISS, INC.
Most analyses of functional design have investigated
how the skull performs a given function. Many recent
Dr. Armand Deswysen passed away on September 18, 2001.
This paper is dedicated to his memory.
Grant sponsor: NSERC; Grant numbers: OGP0138214;
OGP0002377; Grant sponsor: NATO.
*Correspondence to: Dr. Jeff Thomason, Department of Biomedical Sciences, University of Guelph, 50 Stone Road East,
Guelph, Ontario, Canada N1G 2W1. Fax: (519) 767-1450.
E-mail: jthomaso@ovc.uoguelph.ca
Received 3 May 2000; Accepted 22 July 2001
326
THOMASON ET AL.
efforts derive from the seminal work of Smith and Savage
(1959), who compared the lever mechanics of the jaws of
generalized carnivores and herbivores. Other pertinent
issues are how well the skull performs a function and how
it has been adapted for the function. For example, Kiltie
(1982) considered the jaws and masticatory muscles of
several species of suid as lever systems, calculated the
biting forces produced by them, and related the resulting
forces to the hardness or consistency of the diet of each
species.
Structural design analyses focus on how the skull is
constructed to resist the forces acting on it (Thomason,
1991), how the sutures function to transmit force between
bones and absorb impact (Jaslow, 1990; Herring and Teng,
2000), and whether the internal architecture of individual
cranial bones is aligned with the predominant directions
of stress or strain (Buckland-Wright, 1978). Aspects of all
three issues are addressed in the present work, which is
an analysis of structural design in the cranium of the
sheep (Ovis aries).
The work has several aims, the first of which is to add to
the number of species for which in vivo cranial strain data
are available. Strains are recorded from rosette gauges on
the maxillae and frontal bones and from single-axis
gauges on the mandibles of adult sheep during unrestrained feeding. Previous workers have reported strains
from the cranial bones of miniature pigs (Teng and Herring, 1998; Herring and Teng, 2000), the mandibles of
opossums (Crompton and Hylander, 1989; Crompton,
1995), and facial bones and/or mandibles of the galago,
macaque, and owl monkey among the primates (Hylander
and Johnson, 1997; Hylander et al., 1998; Ravosa et al.,
2000b).
The second aim is to use the strain records from the
separate bones to address the question of whether the
mammalian skull behaves mechanically as a single unit or
as a complex of parts linked by sutures. Greaves (1985,
1995) has proposed that artiodactyl and canid skulls behave as short beams and that unilateral biting will load
the skull primarily in torsion. Under torsional loading,
stress isobars should form helices at 45° to the skull’s long
axis, and the postorbital bar should form a strut resisting
the compression isobars (Greaves, 1985). Interpreting the
skull as a solid beam has been challenged by Herring and
Teng (2000), who recorded from cranial bones and across
sutures in miniature pigs while feeding. Their results
suggest that local loading, from whichever of the masseter
and temporalis muscles were active, was a stronger determinant of bone and suture strain than global loading of
the skull via the teeth. They concluded that skulls with
patent sutures did not behave as solid beams but as complexes of independent parts.
The third aim is to address the question of whether the
architecture of cranial bones in sheep is aligned with
predominant strain directions. Buckland-Wright (1978)
recorded strains on dried cat skulls during simulated biting. He then examined the architecture within the bones
using high-resolution radiographs taken of the sagittally
split skulls. Thickenings and channels within the bones
showed a strong tendency to align with surface strains.
Denser regions of bone were seen in adjacent bones on
either side of sutures. Buckland-Wright called these “continua” and suggested that stresses were preferentially
transmitted between and through bones along these
denser regions. Continua, therefore, would represent an
intermediate interpretation between the beam hypothesis
and that of the complex of parts. A difficulty with this
interpretation is that strains recorded in vivo from pigs
and primates vary widely in orientation direction during
the chewing cycle and depend on which side the bolus is
being chewed. The work has not been reexamined for the
frontal bones and maxillae on species other than cats. To
address this issue, radiographs are taken postmortem of
the bone underlying the rosette gauge sites. Predominant
orientations of channels and cortical thickenings within
the facial bones at each gauge site are assessed stereologically and compared with orientations of principal strains.
The final contribution of this work is in comparing the
structural design of the cranium and upper jaw with that
of the mandible, which is relatively slight. Upper and
lower jaws experience equal and opposite forces through
the teeth. Are the strains in each of comparable magnitude? An exact answer will not be provided because the
uniaxial gauges used on the mandible do not necessarily
provide maximal strain magnitudes. But the comparison
of strains between upper and lower jaw bones will provide
some insight as to the relative strength of each.
MATERIALS AND METHODS
Five adult sheep were used for this experiment (four
ewes, one wether; Arcott or Suffolk crossbreeds; weight
range, 55–71 kg). All had been maintained in a research
herd at the University of Guelph and fed daily on hay,
postweaning, until the experiment.
Strain Data Collection
Strain gauge implantation. A rosette strain gauge
(N32-FA-2-120-11, Showa Measuring Instruments Co.,
Tokyo, Japan), with 40-cm lead wires and connector already attached, was glued to the surface of each frontal
bone and each maxilla at the approximate sites shown in
Figure 1. Three single-element gauges (N11-FA-2-120-11,
Showa) were glued to the ventral border of each dentary
approximately equally spaced along the length of the premolar-molar grinding battery (Fig. 1). Locations of each
mandibular gauge were homologous among sheep, within
half a tooth width.
Strain gauges were implanted under general anesthesia
using sterile techniques by modifying for this species and
body region general procedures that have been previously
described (Biewener, 1992). All procedures used here were
preapproved by the University of Guelph’s Animal Care
Committee in accordance with Canadian federal regulations on animal use in research.
The external jugular vein was catheterized and anesthesia was induced with 25 mg of sodium pentobarbital/kg
of body weight injected into the vein. A cuffed endotracheal tube was inserted to maintain a patent airway and
prevent aspiration of saliva and rumen contents. Anesthesia was maintained by sodium pentobarbital introduced
through the catheter via peristaltic pump, at an average
dose rate of 16 mg/min. The dose rate was adjusted as
necessary, under continuous monitoring of the animal for
the duration of the surgery.
Each gauge site was exposed in turn by sharp and blunt
dissection. A region of periosteum large enough to accommodate the footprint of the gauge was scraped off the
bone, and this area was cleaned with acetone and alcohol
and allowed to dry. The gauge was attached with VetBond
cyanoacrylate adhesive (3M Animal Care Products, St.
Paul, MN). A sigmoid bend was put in the lead wires for
STRAIN AND STEREOLOGY OF SHEEP SKULL BONES
327
Fig. 1. Approximate locations of rosette
strain gauges on the left and right maxillae
(LMax and RMax, respectively) and frontal
bones (LFront and RFront) and of single-element gauges on the mandibles (only gauge
sites for the right mandible are shown: RMand).
The LMax gauge is depicted in this and subsequent figures as being visible through the
skull. Skull axis (SA) is positioned as described
in the text. Insets show the sign convention for
angle ␣ (between principal strain ⑀2 and SA)
used in Table 1.
strain relief and sutured to the subcutaneous fascia. All
lead wires were brought subcutaneously to the top of the
head between the horn buds and resutured at that point.
The lightweight connectors were tied loosely to the wool at
the nape.
Following surgery the animal was administered Banamine (2.2 mg/kg) for analgesia and was allowed to recover
in a small, soft-walled enclosure under continuous surveillance. Banamine (1.1 mg/kg) was given again that night
and on the next morning. A companion sheep was present
in an elevated cage and visible to the experimental animal
at all times.
Recording strain data. Strains were recorded the
day of surgery and on the day after. The animals were
monitored as they recovered from anesthesia until they
regained their feet, at which point they each began to look
for food and water. As soon as an animal accepted food by
hand, the strain gauges were connected to main leads
from the strain conditioning amplifiers (DiCaprio and
Thomason, 1989) and the circuits were brought close to
balancing. The amplifiers were calibrated using a shunt
resistor so that an output of 1 V represented 1000 ␮⑀
(microstrain) with a full-scale deflection of ⫾10 V
(⫾10,000 ␮⑀). Noise was minimal compared with signal
peaks, so filtering was limited to the low-pass value of
5000 Hz, which is standard for the amplifiers. Digital
filtering was applied later merely to improve the appearance of the traces.
A recording trial started when the animal was presented with a sample of food by hand, took a mouthful,
and began to chew. The recording equipment was triggered manually and collected strains associated with
chewing for 22.5 sec. Strains were recorded digitally at
66.7 Hz/channel— by an A-D converter (Metrabyte DASH16, Keithley Instruments Inc., Taunton, MA) mounted in
a generic PC—and were saved to disk.
The sheep were presented three food types in separate
trials: alfalfa, chopped hay, and concentrate in pellet form.
For the purposes of the present work, we considered only
the strains recorded when feeding on hay, because this
was the feed the animals had received for most of their
lives. We reasoned that any relationships observed between bone strain and structure would be most relevant
for this feed type. The reason for giving three food types
was to compare the strains and strain energy per chew in
reducing feeds of different consistency and nutrient and
roughage content, which will be presented elsewhere. All
details of the protocol pertinent to the present work are
included here.
Up to 24 trials were recorded for each animal on each of
the two recording days. This number was necessary because the three food types were presented separately and
because we could record only 15 of the 18 total electrical
channels at one time. Trials were repeated after swapping
in or out one rosette or three single-strain channels until
all gauges had been recorded at least twice for each food
328
THOMASON ET AL.
type. This swapping of gauges and the loss of some during
the experiment produced the variability in number (n) of
peaks presented in Table 1 of the Results.
Reduction of strain data. A program custom written in GAUSS (Aptech Systems Inc., Maple Valley, WA)
calculated magnitudes of maximum and minimum principal strains (⑀1 and ⑀2) and the orientation (␾) of ⑀2 with
respect to the rosette’s axis. It extracted peak values of ⑀2
for each chewing cycle and values of ⑀1 and ␾ at peak ⑀2.
(As preliminary steps, the program applied a low-pass
filter to all channels of raw data at 10 times the primary
frequency, i.e., at approximately 25 Hz, and corrected for
drift of the baselines from zero.)
The strain output of the mandibular gauges was found
to indicate unequivocally the side of chewing: strain was
tensile (⑀1 ⬎ ⑀2) on the working side and compressive
(⑀2 ⬎ ⑀1) on the balancing side (Fig. 2), as has been
previously noted for opossums and primates (Crompton,
1995; Hylander et al., 1998). The program was, therefore,
modified to segregate strains from working- and balancing-side chewing cycles.
Orientation of strains to skull axis. The sheep
were euthanatized (with 340 mg of intravenous sodium
pentobarbital/kg) following the second recording day, and
the gauge sites were exposed. The central axis of each
rosette gauge was marked by drilling 0.5-mm-diameter
holes where the axis met the edges of the gauge. The
skulls were cleaned in a dermestid colony, degreased,
lightly bleached, and dried. The axis of each skull (Fig. 1,
SA) was defined as a midsagittal line parallel to the ventral borders of the external auditory meatus and the tip of
the premaxilla. Bone was removed from the ventral aspect
of the brain case and secondary palate to allow a thin
metal rod to be physically placed along this line. The axis
of each rosette was determined with respect to reference
line SA, from digital images taken perpendicular to the
plane of the gauge with a video camera (CCD-72, DageMTI, Inc., Michigan City, IN). Image-analysis software
(Optimas, Bioscan Inc., Edmunds, WA) was used to determine the gauge angle with respect to the projection of
reference line SA onto the plane of the gauge. Angles of
orientation of the peak principal strains (␾), which were
initially calculated with respect to the central axis of the
rosette, were now expressed with respect to the skull’s
axis as angle ␣. Strain peaks from each chew (with associated orientations for the rosette data) were individually
assigned codes for sheep, food type, gauge site, and side of
chewing and were saved for statistical analysis.
Structural Analysis
Radiography. The skulls were split midsagittally with
a band saw, and the maxillary and ethmoid conchae were
removed. Each skull half was radiographed twice in a cabinet radiography system (Faxitron 8050, Hewlett-Packard,
Field Services, McMinnville, OR), with the frontal and maxillary gauge sites in turn pressed against high-resolution
film (Industrex AA 400, Kodak, Rochester, NY). The degree
of exposure was controlled automatically by a feedback loop
in the machine, while the rate was set at low by the operator
to enhance resolution.
Stereological principles. The stereological procedure and analysis that followed was based on one developed over a number of years for use in metallurgy and the
study of bone structure (the development is reviewed by
Turner (1992)) and is now in use by other workers (e.g.,
Teng and Herring, 1995). The principle is that if a grid of
parallel lines is rotated from 0° to 180° with respect to an
axis on the structure (in this case the skull axis), the
average length of portions (intercepts) of the grid lines
overlying trabecular bone varies with the angle (Fig. 3a
and b). If the average length, termed the mean intercept
length (MIL), is squared and inverted and is plotted
against angle on polar coordinates, the theoretical result
is an ellipse (Harrigan and Mann, 1984). The orientation
of the ellipse’s major axis gives the predominant orientation of trabeculae with respect to the external reference
line, and the ratio of the lengths of the two axes of the
ellipse gives a measure of the degree to which trabeculae
are oriented in the predominant direction (Fig. 3c). The
grid also has points in a square pattern, so the number of
points overlying bone rather than space can be counted.
This number is used in the calculation of MIL and of
trabecular spacing and number.
Stereological procedure. A region of each radiograph was enlarged and positioned on the monitor of the
video digitizing system so that a circular overlain grid had
a diameter equivalent to 1 cm of bone, centered on the
middle of a selected gauge site. The grid had lines at a
scaled spacing (d) of 0.78 mm and a total number of test
points (P) of 135 on a square pattern of the same spacing.
Our application of stereology differs from previous work
in using radiographs rather than thin slices of trabecular
bone. The rationale for using radiographs is that the
gauges were positioned on bone less than 2 mm thick and,
therefore, that the orientation of strains would vary little
through the thickness of the bone. Features visible on the
radiographs were thickenings in the cortical bone, channels within it, and the struts bracing the two tables of the
diploë. Buckland-Wright’s (1978) work on cats demonstrated the correspondence of strain orientation to structure visible in high-resolution radiographs and was a basis for this rationale. The main caveat about the approach
is that the resolution of boundaries between bone tissue
and space is definitely worse than in thin sections, even
though the spaces that are marrow filled in life are air
filled and radiolucent in our postmortem specimens. Identifying boundaries consistently on the radiographs required some care on the part of the operator, and all
stereological data were collected by one person (W.W.B.).
The procedure followed the clear description in Teng
and Herring (1995). The grid was overlain on the image,
and the observer counted the number of times (ix␪) the
grid lines crossed the boundaries between radiodense bone
and more radiolucent channels (Fig. 3a and b). The number of hits of the grid points on bone (hx␪) were also
counted. The angle ␪ between the grid lines and the skull
axis was varied from 0° to 170° in 10° increments, and
both counts were made at each position. The MIL was
calculated from an equation in Turner (1992) at each
value of ␪ as follows:
MIL␪ ⫽ 2.d.hx␪ /ix␪
(1)
where d is the spacing between grid lines and points.
Squared values of MIL␪ were inverted and plotted against
␪ on polar coordinates (Fig. 3c). The plots approximated
ellipses described by:
共MIL␪2 兲⫺1 ⫽ M11 .cos2 ␪ ⫹ M22 .sin2␪ ⫹ 2.M12 .cos␪.sin␪
(2)
329
STRAIN AND STEREOLOGY OF SHEEP SKULL BONES
TABLE 1. The mean (with standard deviation) and absolute maximum (円Max円) strains recorded at each gauge
site for each sheep while masticating hay
ε1
Side
Sheep
Frontal, working
Right
2
3
4
5
Left
1
2
3
4
5
Pooled
Frontal, balancing
Left
2
3
4
5
Right
2
3
4
5
Pooled
Maxilla, working
Left
1
2
3
4
5
Right
2
3
4
5
Pooled
Maxilla, balancing
Right
2
3
4
5
Left
2
3
4
5
Pooled
ε2
Mean
S.D.
兩Max兩
n
Mean
S.D.
兩Max兩
n
Mean
S.D.
⌬1
⌬2
359
2
17
5
58
43
248
82
128
271
155
412
198
515
259
351
367
227
306
37.7
15.6
107.9
49.8
61.5
37.7
44.3
41.7
58.7
107.9
356
166
504
251
635
308
492
427
345
635
359
2
17
5
58
43
248
82
128
⫺115
⫺92
⫺325
⫺220
⫺469
⫺419
⫺235
⫺407
⫺210
⫺277
115.9
9.9
93.7
55.5
56.2
61.6
26.3
46.6
51.8
127.5
⫺358
⫺99
⫺412
⫺276
⫺575
⫺494
⫺339
⫺472
⫺312
575
359
2
17
5
58
43
248
82
128
⫺8
⫺2
⫺8
⫺9
24
23
9
29
9
5.7
1.4
0.7
0
0.8
15
1
5
1.2
⫺62.9
87.8
80.7
80.3
⫺61.2
⫺72.9
⫺83.3
⫺73.1
⫺86
27.1
⫺2.2
⫺9.3
⫺9.7
28.8
17.1
6.7
16.9
4
43
145
172
58
359
93
17
29
40
13
12
46
242
46
203
13
77
17.2
6.9
6.6
12.1
29.5
10.6
49.6
4
84.4
143
62
33
67
296
74
247
21
296
43
145
172
58
359
93
17
29
⫺214
⫺145
⫺88
⫺17
⫺229
⫺129
⫺146
⫺50
⫺127
38.4
17.1
12.6
4.5
34.9
22.4
30.2
9.5
76.5
⫺263
⫺202
⫺111
⫺26
⫺309
⫺175
⫺169
⫺61
309
43
145
172
58
359
93
17
29
82
⫺13
33
25
⫺52
13
⫺53
⫺38
24.7
2.6
80.9
6.9
2.9
5.3
3.6
12.1
⫺13.9
74.7
⫺69.1
⫺70
73.1
⫺77.2
35.7
51.3
76.1
⫺15.3
20.9
20
⫺16.9
12.8
⫺54.3
⫺38.7
58
43
248
172
128
231
93
17
29
49
163
93
95
106
76
57
37
61
82
9.1
42
17.8
18.4
21.6
12.7
11.7
6.6
14.8
36.0
73
305
145
143
145
106
86
50
91
305
58
43
248
172
128
231
93
17
29
⫺63
⫺318
⫺77
⫺108
⫺179
⫺96
⫺95
⫺108
⫺140
⫺132
22
51.7
14.9
23.7
51.2
19.2
11.8
34.3
45.2
73.4
⫺121
⫺379
⫺126
⫺169
⫺278
⫺166
⫺118
⫺145
⫺207
379
58
43
248
172
128
231
93
17
29
⫺26
50
58
43
71
⫺62
⫺23
⫺59
⫺57
12.7
5.9
9
10.3
3.9
21.7
22.8
11.5
5.5
⫺0.8
79.8
⫺87.4
74.3
⫺69.1
56.9
⫺81.9
⫺77.6
⫺85.7
⫺90.8
⫺10.2
2.6
⫺15.7
20.9
⫺33.1
8.1
12.4
4.3
2
248
172
128
359
93
17
29
104
204
479
280
711
105
417
248
319
11.3
21.2
65.4
65.5
92.3
17.3
114.7
49.1
208.4
112
257
593
424
955
150
532
316
955
2
248
172
128
359
93
17
29
⫺140
⫺170
⫺32
⫺96
⫺29
⫺98
⫺15
⫺52
⫺79
4.9
25.2
7.2
23.6
15.5
18.5
4.9
16.7
55.5
⫺143
⫺379
⫺49
⫺143
⫺147
⫺161
⫺21
⫺84
379
2
248
172
128
359
93
17
29
⫺5
29
⫺2
26
18
⫺25
⫺1
⫺21
26.9
1.1
1.4
2.7
3
3.9
4.5
4
⫺66.1
⫺29.9
⫺20.6
⫺2.7
47.8
9.6
30.3
18.9
23.9
60.1
69.4
87.3
⫺42.2
⫺80.4
⫺59.7
⫺71.1
Rostral
Side
Sheep
Mandible, working
Right
2
3
4
5
Left
2
3
4
5
Pooled
Mandible, balancing
Left
1
2
3
4
5
Right
2
3
5
Pooled
␣
n
n
Mean
Middle
S.D.
兩Max兩
n
Caudal
Mean
S.D.
兩Max兩
n
Mean
S.D.
676
607
854
464
331
630
182
507
854
128
91
17
29
43
248
90
70
357
172
895
379
232
⫺10
500
259
348.0
70.4
109.8
210.3
97.4
120.8
30.1
82.9
62.6
260.67
529
333
1110
511
384
⫺90
618
367
1110
58
41
103
172
128
359
93
24
⫺570
⫺668
⫺700
⫺907
⫺400
⫺859
⫺15
⫺421
⫺567.5
65.4
102
64.9
119.8
110.7
138.5
31.5
114.2
310.72
⫺699
⫺863
⫺853
⫺1059
⫺637
⫺1205
60
⫺572
⫺1205
219
91
17
29
43
248
90
70
782
91
434
54
724
759
312
⫺187
371
173.9
13.3
92.3
13.2
98.5
91.5
68.5
91.6
343.9
1209
112
581
69
362
962
423
⫺307
1209
128
91
17
29
43
248
90
70
534
472
668
346
640
458
640
325
510
63.3
79.2
148.5
99.1
111.1
60.9
104.7
83.3
198.9
0
41
103
172
128
359
93
24
..
⫺748
⫺155
⫺668
⫺65
⫺674
⫺619
⫺123
⫺436
..
118.8
18.5
90.2
18.4
121.4
118.8
30.8
307.6
0
⫺951
⫺193
⫺797
⫺104
⫺984
⫺839
⫺182
984
..
41
103
172
128
268
93
24
..
⫺485
⫺609
⫺764
⫺541
⫺780
⫺480
⫺309
⫺567
..
75.6
57.4
101.4
147.7
114.8
87.6
81
272.3
..
⫺623
⫺746
⫺906
⫺863
⫺1088
⫺625
⫺417
1088
兩Max兩
For maxillary and frontal rosette gauges, values of the two principal strains ε1 and ε2 are presented with the angle of
orientation ␣ of ε2 to the central axis of the skull. For mandibular gauges, strain values are presented for the rostral, middle
and caudal gauges on each mandible. All strains are in microstrain (␮ε) and angles are in degrees in a range of ⫾90°. See
Figure 1 for the sign convention for angles from each rosette.
Missing data are indicated by .. .
330
THOMASON ET AL.
Predominant orientation ␤ ⫽ arctan共H2/H1兲
Degree of anisotropy ⫽ H2/H1 .
(4)
Comparison of Strains With Structure
Quantitative comparison. The differences (⌬1 and
⌬2) between structural angle (␤) and the mean angles of
orientation of principal strains ⑀1 and ⑀2 were calculated
for each gauge site for each sheep, considering working
and balancing sides separately. Angles ⌬1 and ⌬2 express
how closely the strains and intraosseous structures are
aligned.
Statistical analysis. Our first null hypothesis was
that the predominant orientation of principal strains at
the site of each rosette gauge would not equate to the
predominant structural orientation at that site, i.e., that
⌬1 and ⌬2 would not equal zero. The test performed was
for agreement between angles ␣ (orientation of ⑀2) and ␤
(structural alignment). For agreement, a regression of (␣ –
␤) on (␣ ⫹ ␤) produces a nonsignificant slope and intercept
(P ⬎ 0.05; Bland and Altman, 1986). Agreement was also
tested between the orientations of ⑀1 (␣ ⫹ 90) and ␤.
Regressions were performed using Procedure REG in SAS.
The second null hypothesis was that the degree of structural anisotropy (H2/H1) would not be related to principal
strain magnitudes ⑀1 and ⑀2. We tested for significance of
correlations (at P ⬍ 0.05) using Procedure CORR in SAS.
Fig. 2. A model of mandibular bending (after Crompton (1995)). The
working-side dentary is under 4-point bending from the muscular force
M, the reaction force from the bolus B, force S transmitted from the
balancing side at the symphysis, and a reaction force J at the jaw joint.
The net bend on the body of the dentary under the tooth row is upwards,
which puts the ventral border into tension. On the working side, there is
a 3-point bend between the reaction force at the jaw joint J, the muscular
force M, and the reaction at the symphysis –S (equal and opposite to S).
The net downward bend on the dentary puts the ventral border into
compression.
where M11, M22, and M12 are coefficients that were derived by nonlinear regression using the known values of
MIL␪ and ␪. Procedure NLIN in SAS (SAS Institute, Cary,
NC) calculated the three coefficients for bone at each
gauge site. The coefficients were expressed as a tensor M
(in which M21 ⫽ M12):
M⫽
冏 MM
11
21
M12
M22
冏
(3)
and a fabric tensor H was calculated as M–1/2, as defined
by Cowin (1985). The eigenvalues H1 and H2 of H were
calculated, where H2 is the eigenvalue of the major axis of
the ellipse (Fig. 3c). All matrix computations were done by
custom-written programs in GAUSS.
Stereological properties of bone at gauge sites.
The direction of predominant orientation (␤) of bony structures in the radiograph of each gauge site and the degree
of anisotropy were calculated as:
RESULTS
Strain Magnitudes and Directions
A typical trace of one trial is in Figure 4, which clearly
shows the change from left- to right-side chewing.
All strains recorded fell in the range –1200 ␮⑀ compressive strain to ⫹1200 ␮⑀ tensile strain, with the extremes
both coming from the mandible (Table 1). Mandibular
strains may not represent peak values, because the singleelement gauges may not have been aligned with the largest principal strain. Similar patterns of strain were seen
at each gauge among the five sheep, with working-side
gauges being clearly distinguishable from balancing-side
gauges (Table 1, Figs. 4 and 5).
Mandible. When right or left mandibular gauges were
on the working side, they registered tensile strain (positive values) and compressive strain (negative) while on
the balancing side (Table 1, Fig. 4). The only exceptions
were the caudal gauge of sheep 3 and the rostral gauge of
sheep 5, both on the left dentary, which showed compression during left-side chewing. Mean working-side strains,
pooled for the mandibles of all five sheep, were 371, 510,
and 348 ␮⑀, from rostral to caudal. Comparable values on
the balancing side were – 436, –567, and –567 ␮⑀. There
was a tendency for gradients of strain from rostral to
caudal, but in opposite directions in different sheep. For
example, when the right mandible was on the balancing
side, sheep 2 showed strains increasing rostrally (– 674,
–780, and – 859 ␮⑀), while values decreased rostrally in
sheep 3 (– 619, – 480, and –15 ␮⑀).
Maxilla. Working-side gauges on the maxilla registered principal compressive strains ⑀2 as large as –379 ␮⑀
(Table 1). Individual means ranged from – 63 to –318 ␮⑀,
and the pooled mean was –131 ␮⑀ (S.D., 73.3 ␮⑀). Corresponding tensile values of ⑀1 were always smaller, reach-
STRAIN AND STEREOLOGY OF SHEEP SKULL BONES
331
Fig. 3. Stereological analysis of bone architecture. a, b: A grid of points and lines is superimposed over a bone
sample at various angles ␪. The number of points overlying bone (circled) are counted as hx␪. MILs are the heavy
lines. The number of ends of these lines is counted as ix␪. c: Values of (MIL␪2)–1 are calculated from hx␪ and ix␪,
and an ellipse is fitted to these data. H1 and H2 are eigenvalues of the axes of the fitted ellipse.
ing a maximum of 300 ␮⑀, with individual means from
37–163 ␮⑀, and a pooled mean of 82 ␮⑀ (S.D., 36.0␮⑀).
Principal strains were oriented such that ⑀2 was inclined
down toward the nose at an angle ␣, usually between 25°
and 70° to the cranial axis (Fig. 5).
On the balancing side, tension predominated in the
maxilla, usually making ⑀1 absolutely larger than ⑀2. The
maximal value for ⑀1 of 955 ␮⑀ was on the right maxilla of
sheep 2. Individual means for ⑀1 were in the range of
100 –700 ␮⑀, with a pooled mean of 319 ␮⑀ (S.D., 193.9 ␮⑀).
The smaller corresponding values of ⑀2 had a maximum of
–379 ␮⑀, individual means in the range of –15 to –170 ␮⑀,
and a pooled mean of –79 ␮⑀ (S.D., 52.7 ␮⑀). The large
tensile strains were inclined steeply with respect to the
cranial axis SA, all falling within 30° to either side of a
perpendicular to SA (Fig. 5).
Frontal bone. On the working side, frontal gauges
showed strain maxima for ⑀1 and ⑀2 of comparable magnitude: 635 and –575 ␮⑀, respectively (in sheep 1). Individual means ranged from 155 to 515 ␮⑀ for ⑀1, and –92 to
– 469 ␮⑀ for ⑀2, and the pooled means were 306 ␮⑀ (S.D.,
107.9) for ⑀1, and –277 (S.D., 127.5) for ⑀2 (Table 1).
Compressive principal strains ⑀2 were inclined toward the
working tooth row (Fig. 5) and were within 2° and 29° of
the cranial axis, SA.
On the balancing side, frontal strain peaks were lower:
296 and –309 ␮⑀ for ⑀1 and ⑀2, respectively. Individual
mean tensile strains ⑀1 ranged from 12–242 ␮⑀, with a
pooled mean of 79 ␮⑀ (S.D., 85.7 ␮⑀). Individual means of
compression strain ⑀2 varied from –17 to –229 ␮⑀, with a
pooled mean of –127 ␮⑀ (S.D., 69.1 ␮⑀). Compressive
strains ⑀2 were mostly inclined toward the active dentition (Fig. 5), as on the working side, but were at more
variable angles: within 13° and 82° of the cranial axis.
Bone Structure
Bone structure visible on the radiographs showed
strong individual variability (Fig. 6). The structural results are summarized in Table 2 for individual sheep and
in Figure 7 for all of the animals pooled. In all cases the
bone under the gauge sites was anisotropic: the measure
of anisotropy (H2/H1) varied from 1.32– 6.85, with a trend
toward higher values for the frontal bone sites than for the
maxillae (this trend was not tested statistically). The predominant direction of structural orientation for the maxillae was inclined ventrally and rostrally (Fig. 7, upper
panel), at angles ␣ varying among sheep from 29°–70° to
the cranial axis SA. In the frontal bones the predominant
orientation was close to the structural axis (Fig. 7, lower
panel), with all but one value of ␣ being within 12° of SA.
Comparison of Strains With Structure
Angles of strain and structural orientation.
When the mean principal strains were superimposed over
the predominant structural orientations at each gauge
site for each sheep (Figs. 8 and 9), the maxillary gauges
showed a clear trend for the compressive strain ⑀2 on the
working side to be quite closely aligned with the structural
orientation (Fig. 8). Angle ⌬2 (between ⑀2 and the structural orientation) was within 33° for both maxillae when
on the working side (Table 1), except for the left maxillary
gauge in sheep 1, where tensile strain ⑀1 was closely
aligned with the bone structure. Ignoring this outlier, the
mean angle ⌬2 for the working-side maxillae of all sheep
pooled was 13.4° (S.D., 9.81°). For the working-side frontal
bones (which had no outliers), the mean ⌬2 was similar:
13.3° (S.D., 9.36°), with a maximum of 29° (Table 1, Fig. 9).
For the balancing-side maxilla, ⑀1 was more closely
aligned with the bony structure than ⑀2, but the alignment was weaker than on the working side (Fig. 8). Angle
⌬1 had a mean of 28° (S.D., 20.7°) and a maximum of – 66°.
332
THOMASON ET AL.
Fig. 4. Typical strain trace, from one trial of
sheep 5 masticating hay. Maxillary and frontal
bone records are principal strains; mandibular
records are the uniaxial strain.
For the balancing-side frontal bone, ⑀2 was closer than ⑀1
to the structural orientation, but again the alignment was
weak (Fig. 9). Angle ⌬2 had a mean of 39° (S.D., 20.7°) and
a maximum of 66°.
The statistical tests of agreement indicated that compressive strain ⑀2 on the working side was significantly
aligned with the predominant orientation of bone architecture (P ⬎ 0.05; Table 3). For agreement, this test requires P ⬎ 0.05 for slope and intercept of the regression.
Orientations of balancing-side ⑀2 and all orientations of ⑀1
were not in significant agreement with structural alignment (P ⬍ 0.05 for slope or intercept or both).
When correlating the magnitudes of ⑀1 and ⑀2 with
trabecular anisotropy (H2/H1), only one correlation was
significant (P ⬍ 0.05) when sorting by gauge and side of
chew: ⑀1 with H2/H1 for gauge left maxillary on the balancing side.
DISCUSSION
All gauge sites (rosette or single axis) show two strain
signals, depending on the side of chewing (Figs. 4 and 5).
For the mandible, peak strains tend to be of higher absolute value on the balancing side, possibly indicating
greater force generation by the balancing-side muscula-
ture. This is not necessarily true, because we cannot be
sure that the single-element gauges recorded peak principal strains. The leverage of the muscles is different on
balancing and working sides (Fig. 2), which means that
the same force might produce different strains on each
half of the mandible. It is certain, however, that significant balancing-side muscular activity occurs during the
power stroke in sheep, and this is in accord with work on
the few other species that have been studied.
Two patterns emerge from the combined data of this
and previous studies. First is that the distribution of
strains in cranial bones is dependent on the relative degree of recruitment of balancing- and working-side muscle
components. Macaques and owl monkeys appear to have
high balancing-side recruitment, as seen in sheep, while
galagos have less (Hylander et al., 1998). The difference
among the primates has been suggested to correlate with
the strengthening or fusion of the mandibular symphysis in anthropoids (Hylander et al., 1998). This is not the
case in sheep in which the symphysis is mobile. Perhaps
the necessity for maintaining occlusion of shear surfaces in sheep by rotating the mandible during the
power stroke takes precedence over bracing the symphysis against force transmission from the balancing
STRAIN AND STEREOLOGY OF SHEEP SKULL BONES
333
Fig. 5. Peak principal strains ⑀1 and ⑀2 at the four rosette sites and single-axis strains for the middle right
mandibular site, pooled and averaged for all chews and for all sheep while feeding on hay. Length of each
bar represents strain magnitude. Gauge sites abbreviated as in Figure 1. Shaded ellipse represents the bolus
on the working side. B, balancing side; W, working side.
side, as has been suggested for goats (Lieberman and
Crompton, 2000).
The second pattern to emerge from comparative strain
data is that muscle action has strong local influence on
bone strain magnitudes and orientations. In miniature
pigs, combined electromyographic and strain studies have
shown that the force of the power stroke is primarily from
the ipsilateral masseter and contralateral temporalis
(Herring and Teng, 2000). This puts a torque onto the
cranium that strains the frontal and parietal bones in the
opposite direction than would be expected from the torque
owing to unilateral biting. The high tensile strain on the
balancing-side maxilla of sheep is also an example of local
muscle influence. Strains in the zygomatic bones of miniature pigs resulting from masseter contraction are independent of the chewing side (Herring et al., 1996). In this
case the local muscle activity masks the general asymmetry of loading of the whole skull.
Is the Skull Mechanically a Solid Tube or a
Complex of Independent Parts?
The present results indicate that the sheep skull (cranium and face) show elements of solid and composite
behaviour. Figure 10 shows the general pattern of forces
that may be inferred to cause the strains in Figure 5; no
attempt has been made to quantify these forces. On the
working side, the action of the temporalis and masseter
muscles produce the distributed force vectors T and M,
respectively (Fig. 10a), with M also representing activity
in the medial pterygoid muscle. The arrows illustrate the
approximate directions of vector components of the force
from different regions of each muscle and, even more
approximately, their relative magnitude. The vectors are
not linked with specific anatomical parts of each muscle.
These muscular forces pull the mandible toward the cranium, exerting force J at the temporomandibular junction
and force B at the point of biting (B will be distributed
along the grinding battery). The magnitude of J is likely to
be reduced by forces from the balancing side, while B is
likely to be similarly increased (Crompton and Hylander,
1989). Force B tends to rock the maxilla on the basicranium, rotating it upwards and backwards toward the frontal bones.
Strain ⑀2 on the working-side maxilla appears to be
largely the result of biting force B, as might be expected.
There is possibly some influence of masseteric force M adding to tensile strain ⑀1, because ⑀1 is often larger than would
be expected from Poisson’s ratio. (A unidirectional compres-
334
THOMASON ET AL.
Fig. 6. Radiographs of architecture in the maxillae of all five sheep. Upper panel shows whole facial
region of sheep 1 for reference. Area sampled was a 1-cm-diameter circle surrounding the two drilled holes
that identified the orientation of maxillary rosette. Middle row shows maxillary sample area for sheep 1, 2, and
3, lower row shows sheep 4 and 5. Scale: centers of drilled holes are 0.5 ⫾ 0.05 cm apart.
STRAIN AND STEREOLOGY OF SHEEP SKULL BONES
sive force will induce compressive strain in the same direction, and a perpendicular tensile strain because the material
tries to retain its volume. The magnitude of the tensile strain
TABLE 2. Stereological values for the bone at each
gauge site for which strain records were available
Gauge site
Right frontal
Left frontal
Right maxilla
Left maxilla
Sheep
H1
H2
H2/H1
Angle ␤
2
3
4
5
1
2
3
4
5
2
3
4
5
1
2
3
4
5
0.41
0.52
0.46
0.32
0.35
0.48
0.52
0.66
0.61
0.74
0.40
0.75
0.36
0.38
0.54
0.45
0.75
0.23
2.78
1.00
1.10
1.75
2.28
2.53
1.30
1.34
1.87
0.97
0.93
1.20
1.71
0.66
1.04
1.22
1.56
0.58
6.85
1.94
2.39
5.40
6.52
5.30
2.53
2.03
3.06
1.32
2.31
1.60
4.70
1.75
1.92
2.72
2.08
2.49
⫺35.1
0.2
1.3
0.7
⫺4.8
5.9
2.3
12.1
5.0
⫺28.9
⫺31.1
⫺71.4
⫺61.3
64.8
60.2
55.4
58.7
50.1
H1 and H2 are the eigenvalues of an ellipse describing the
predominant orientation of architectural features. Their ratio is
a measure of the degree of orientation. Angle ␤ is between the
direction of predominant orientation and the cranial axis SA.
Fig. 7. Mean orientation of the bone architecture (given by angle ␤) underlying each
gauge site for all of the sheep. Line length is
arbitrary. Gauge abbreviations as in Figure 1.
335
is less than the compressive strain parallel to the force, the
ratio of the two strains being termed Poisson’s ratio. It has a
value of 0.2– 0.4 for bone (Currey, 1984).
The working-side frontal bone appears to be distorted by
a compressive force at an acute angle to the skull’s axis,
with ⑀1 and ⑀2 being close in magnitude. These strains
indicate the effects of combined bending and torsion. Force
B pushes the maxilla up and back into the frontal bone,
effectively bending the skull. The force also twists the
skull, because it is asymmetrically placed, tending to drive
the maxilla across the dorsal midline.
On the balancing side, force B is absent, and force J may
be increased by the action of the jaw adducting muscles on
this side (Fig. 10b). On the maxilla, the large, tensile
strains appear (Fig. 5) to result from the tension exerted
by the active masseter on that side, pulling the bone
ventrally. In several cases the perpendicular compressive
strains are less than expected from Poisson’s ratio, suggesting that a second, separate tension of unknown cause
is also present.
Strains on the balancing-side frontal bone are similarly
oriented to those on the working side, but are of lower
magnitude (Fig. 5). They are in agreement with the suggestion made above that force B both twists and bends the
skull. They also give some indication of the modulating
effect of cranial sutures on the transmission of forces
between skull bones. If the skull were constructed of homogenous material as a single structural unit, the strains
on the two frontal bones would be expected to be almost
336
THOMASON ET AL.
Fig. 9. Trabecular alignment compared with that of the principal
strains for the left and right frontal (LFront and RFront, respectively)
gauges when on the working or balancing side for all five sheep.
Fig. 8. Trabecular alignment compared with that of the principal
strains for the left and right maxillary (LMax and RMax, respectively)
gauges when on the working or balancing side for all five sheep. Figure
is essentially an overlay of maxillary strains as in Figure 5, on architectural orientation as in Figure 7, but for all sheep not just means.
exactly the same. The reduction in strain magnitudes, of
30%– 60% from biting to balancing side, shows the effects
of the midline and nasomaxillary sutures in redistributing
and absorbing the force of biting.
The discussion above leads to the conclusion that the
sheep cranium does show solid behavior to a greater extent than does that of the miniature pig, in which muscle
activity dominates local strains (Herring and Teng, 2000).
Torque on the whole skull appears to be present in galagos, but low strain values are associated with it in the
orbital region, which suggests that it is only one component of the mechanical behavior of the skull (Ravosa et al.,
2000a). We reiterate Herring and Teng’s caveat that considering the skull solely as a beam does not sufficiently
describe its mechanical behavior.
This brings us to the function of the postorbital bar in
sheep, which Greaves (1985) suggested is to brace the
compressive forces associated with torsion from unilateral
biting. Cartmill (1970) proposed that the presence of a
postorbital bar in many mammals was associated with the
reorientation of the orbits for stereoscopic vision. These
and other hypotheses concerning the postorbital bar have
been quite widely debated in the anthropological literature (Noble et al., 2000; Ravosa et al., 2000b). The present
results showed that frontal bone strain was not aligned
along a 45° helix as proposed by Greaves (1985). Indeed
there was a significant component of tensile strain aimed
toward the postorbital bar in most sheep (Fig. 5). This
indicates it may not act as a strut resisting compressive
strains due to torsion of the whole skull. Its function is
more likely to be as a local brace for the orbit, in which
case its stiffness may be more important than its strength,
as has been suggested for the primate postorbital bar
(Ravosa et al., 2000b).
Stereology of Cranial Bones in the Sheep and
the Question of Continua
The major finding from the combined strain and stereological analysis was that the predominant orientation of
the bone structure was in statistically significant agreement (P ⬎ 0.05) with the working-side compressive principal strain ⑀2. This was despite the fact that the tensile
principal strain ⑀1 was larger in magnitude on the maxilla
and of equivalent magnitude on the frontal bones.
337
STRAIN AND STEREOLOGY OF SHEEP SKULL BONES
TABLE 3. Results of the statistical tests of agreement between principal strain and
structural orientations
Maxilla
ε1
ε2
Slope
Intercept
Slope
Intercept
Frontal
Working
Balancing
Working
Balancing
0.865
0.002
0.865
0.392
0.014
0.001
0.014
0.745
0.74
0.004
0.74
0.159
0.003
0.407
0.003
0.62
The tests give values of the probability that the slopes and intercepts of regressions of (␣ ⫺ ␤) on
(␣ ⫹ ␤) are significant (P ⬍ 0.05). Only for the working side ε2 are both values greater than 0.05.
Buckland-Wright (1978) may not reflect the trajectories
followed by the larger strains in adjacent bone. It also
suggests a direction for further study on the mechanism
whereby the bone seems to selectively respond to one
mode of loading rather than the other.
Comparing Cranial and Mandibular Strains and
Construction in Sheep
Fig. 10. Reconstruction of the probable forces in the masticatory
muscles (M and T) and on the cranium. B, biting force; J, jaw joint
reaction forces; M, force components in masseter muscle; T, components in temporalis muscle.
Facing two strain signals is clearly a challenge for the
adaptive remodeling processes of bone, which attempt to
optimize bone architecture to best resist local loading
(Huiskes et al., 2000). In the sheep facial bones, the processes do not appear to attempt an optimization to both
strain signals, but to the compressive one selectively.
Even though the architecture does align well with the
compressive strain, there is no real correlation of the
degree of orientation (H2/H1) with strain magnitudes.
This finding suggests that the continua identified by
The upper part of the skull is a robust hollow tube in
sheep, made of many bones, while the mandible is a pair of
beams in a V shape with a mobile symphysis. They experience equal and opposite resultant forces and moments
through the teeth and jaw muscles. How the skull resists
those forces is mechanically quite different than how the
mandible does because of the differences in construction.
Both structures are of essentially the same material (allowing for variation in properties of bone of varying density and porosity). On this basis we might expect that each
would be built to achieve equal safety factors in the material, i.e., the same ratio of peak strain magnitudes with
respect to the strain at failure of the material (Biewener,
1993). The strain records indicate that this may not be the
case: strains are usually higher on the mandible by at
least 50% and up to 100% (Table 1). If the single-axis
gauges on the mandible did not record peak principal
strains, this difference would be exacerbated. Certainly,
the strains on all bones studied in the present work are
sufficiently low that none are in great danger from failing
during normal use. But the mandible seems to have a
lower safety factor than do the two cranial bones studied.
This implies that the skull is overbuilt, a suggestion that
has been previously raised for the macaques and galagos
(Hylander and Johnson, 1997; Ravosa et al., 2000b).
Among the possible reasons is that the skull has a number
of other functions as well as that of providing a lever
system for mastication (Smith, 1993; Thomason, 1991).
Enclosing the brain and nasal cavity gives the skull its
generally tubular shape and dictates the volume that has
to be enclosed within the tube. The tubular form certainly
confers strength and stiffness on the whole structure, even
if there is some mechanical independence of the bones
comprising it. The orbital regions provide support and
visual direction for the eyes, and the important property of
the bones surrounding the orbit may be their rigidity
rather than their strength (Ravosa et al., 2000a and b).
The sum total of all the extra functional demands on the
skull is probably at the root of the differences in mechanical behavior between it and the mandible. Certainly they
are the main impediment to fully understanding the structural design of the skull. Comparative and in vivo studies
have enabled rapid progress in analysis of cranial functional and structural design in the past few years. The
338
THOMASON ET AL.
complexity of the problem ensures that it will be several
more before a reasonably complete picture is available.
CONCLUSIONS
The main findings from this study are as follows:
1. Unilateral mastication results in markedly different
strain signals on working- and balancing-side cranial
bones of sheep, as has been previously shown for primates and for some bones in the miniature pig (but not
those in the zygomatic arches).
2. The mechanical behavior of the ovine skull shows aspects of behaving as a solid tube and of a complex of
independent bones.
3. The alignment of architectural features within the
frontal and maxillary bones of sheep statistically
agrees with the orientation of principal compressive
strains ⑀2 on the working side, despite the fact that
balancing-side principal tensile strains ⑀1 were of equal
or larger magnitude.
4. The ovine mandible appears to experience larger
strains than do the facial bones, indicating that the
skull is overbuilt in comparison to the mandible.
ACKNOWLEDGMENTS
We thank Dr. S. Young and Dr. M. Shoukri for advice on
the computer programming and statistical procedures, respectively. We thank the three reviewers for their constructive comments, which have improved the paper.
Grants from NSERC (OGP0138214 and OGP0002377)
were given to J.J.T. and L.E.G., respectively. A visiting
professorship grant from NATO was given to A.G.D.
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