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First published online September 19, 2008
Journal of Experimental Biology 211, 3085-3094 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.019042
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Frontal sinuses and head-butting in goats: a finite element analysis
Department of Anatomical Sciences, Stony Brook University, NY 11794-8081, USA
Present address: Raymond M. Alf Museum of Paleontology, Claremont, CA 91711-2199, USA (e-mail: afarke{at}webb.org)
Accepted 21 July 2008
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Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX 1References |
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Key words: Bovidae, Capra hircus, finite element analysis, goats, head-butting, paranasal sinus, pneumaticity
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INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX 1References |
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). In bighorn
sheep (Ovis canadensis) the impact force may exceed 3400 N, and the
sudden deceleration of impact occurs over less than 300 ms
(Kitchener, 1988). These
intense impacts so near critical cranial organs [associated with
coup/contrecoup injuries to the brain in humans, among other effects (e.g.
Gurdjian, 1975)] have invited
numerous lines of speculation on the structures that may be responsible for
`shock absorption' during combat (e.g.
Geist, 1966;
Jaslow and Biewener, 1995;
McDonald, 1981).
Shock absorbers dampen sudden accelerations by converting applied kinetic
energy into another form (e.g. heat), through deformation over a period of
time. These principles have been used to develop football helmets and other
protective devices, which protect the human skull and brain by attenuating
impact energy through deformation of the helmet in place of the head and
reducing the force directly transmitted into the skull itself and/or spreading
the force over a larger area (Levy et al.,
2004). Similarly, an idealized biological shock absorber in the
skull experiences elastic deformation (as opposed to brittle fracture or
plastic deformation) and concentrates this deformation away from delicate
cranial organs. A longer period of deformation, related to the elasticity of
the deforming object, reduces strains (and hence potential damage) on brain
tissue and blood vessels due to inertia. Several structures within a sheep or
goat skull could contribute to shock absorption. For example, the keratin
sheath of the horns is considerably more elastic than bone, allowing a
relatively greater amount of deformation
(Kitchener, 1987;
Kitchener, 1988), and the
position of the sheaths localizes this deformation away from the immediate
area of the brain and other cranial organs. Several cranial sutures that are
located near the horns (hence, near the area of impact), exhibit high strains
during impact loads (Jaslow and Biewener,
1995), and thus may also function as a sort of `crumple zone'
during impacts. Finally, the frontal sinus system is another candidate for
shock absorbers.
The frontal sinuses of bovids (Fig.
1C–E) are air-filled spaces that originate from the nasal
cavity, located wholly within the vaulted (expanded) frontal bone. The sinuses
are sandwiched between two layers of cortical bone: one at the outer table of
the skull (hereafter referred to as the `external cortex') and one forming
part of the surface of the endocranial cavity (`internal cortex') – and
may extend into the horncores. Bony struts (usually numbering between four and
six on each side in goats, with a typical thickness of 1 mm or less) may
divide the sinuses into a series of interconnected chambers. The idea of
frontal sinuses as `shock absorbers' is often repeated in the literature (e.g.
Geist, 1966;
Schaffer and Reed, 1972) and
has even been used, by analogy, to reveal the function of sinuses in extinct
dinosaurs such as Triceratops
(Molnar, 1977;
Forster, 1996). Despite this,
the idea of sinuses as protective structures remains completely untested.
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), the outer walls of the sinus or the struts within
the sinus could deform during impact, in place of deformation of the
endocranium. Thin walls of bone would be more deformable than solid bone. A
related possibility is that the walls direct the forces away from biologically
sensitive structures and towards sutures, where further deformation takes
place.
Any discussion of sinus function must also consider the broadly accepted
idea of the sinuses as a by-product of bony remodeling (e.g.
Weidenreich, 1941;
Edinger, 1950;
Witmer, 1997). This mechanism
follows models of bone adaptation discussed by Roux
(Roux, 1881) and others, in
which the unpneumatized skull contains areas of bone that are not necessary
for mechanical support. Osteoclasts associated with pneumatic diverticula from
the nasal cavity remove the structurally unnecessary bone, producing a sinus
within a more `optimized' skull (Witmer,
1997). Under this hypothesis, the frontal bone that contains the
sinus, and not the sinus itself, is the more important structure. The frontal
bone could have its current morphology for any number of reasons, such as
structural support of the horns, but the shape of the sinus itself reflects
only the loads placed upon the skull
(Preuschoft et al., 2002).
Importantly, this concept is not mutually exclusive of other functions for the
sinuses or the vaulted frontal bone, such as shock absorption.
The idea of sinuses as somehow producing `optimal' structures (greatest strength with least materials) remains extraordinarily difficult to test with conventional experimental methods. A similar problem plagues the shock absorption hypothesis. Thus, finite element modeling (FEM) was used in this study to test the effects of cranial sinuses in the domesticated goat Capra hircus.
Hypotheses
Finite element models of variable morphology – including skulls with
strutted sinuses, unstrutted sinuses, sinuses filled with trabecular or
cortical bone, and skulls completely lacking sinuses and a vaulted frontal
bone (Fig. 1A–E) –
were constructed in order to test three (not mutually exclusive)
hypotheses:
).
Strain energy, used here as a proxy for shock absorption in static analysis
(see Materials and Methods), should be elevated in the frontal bones for
skulls with sinuses relative to skulls without, reflecting the fact that this
energy then would not be transferred into the endocranial cavity. A second
aspect of shock absorption could be effected by sinuses directing the strain
that results from forces placed on the horns away from the endocranial cavity
(Schaffer and Reed, 1972). A
skull with sinuses that is otherwise identical in shape to a skull that lacks
sinuses (compare Fig. 1E with
Fig. 1B) should experience
higher strains within the endocranial cavity, simply because the removal of
bone reduces the overall stiffness of the frontal bones and results in greater
overall deformation. But if the removal of bone to form a sinus results in a
structure that directs the force of impact away from the endocranial cavity
(or towards different regions), the skull with sinuses still should experience
lower strains within the bone surrounding the endocranial cavity, as well as
exhibit a different spatial pattern of strain distribution. The struts within
the sinus may also play a role in this regard, with similar predictions. |
MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX 1References |
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) (see Appendix 1). Model geometry was based on
CT scans of the skull of a young male goat, with an in-plane pixel resolution
of 0.49 by 0.49 mm and interslice spacing of 2.5 mm. Visual inspection of
renderings of the data indicated that this slice spacing adequately portrayed
the cranial structures of interest in this study. The CT slices were imported
into the modeling package SolidWorks 2003 (SolidWorks Corp., Concord, MA,
USA), in which bone and keratinous horn sheath were traced manually on each
slice, and the tracings were lofted into a three-dimensional solid model
(Fig. 2A). In order to simplify
the geometry of the model, the foramina of the basicranium (except for the
foramen magnum) and the paroccipital and styloid processes were not
incorporated into the model. Because the primary area of interest in the model
was the cranial roof and not the cranial base, these simplifications were
considered appropriate.
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;
Herring and Teng, 2000;
Jaslow, 1990). Jaslow and
Biewener (Jaslow and Biewener,
1995) even suggested that the sutures play a major role in shock
absorption in goats. Thus, the models were constructed with lines of compliant
elements for the interfrontal and fronto-parietal sutures. These sutures are
closest to the sinuses (the area of interest) and the horncores (the area of
primary loading) and thus are most likely to affect the results. Importantly,
the sutures themselves were not modeled here, but rather a sutural zone that
included both the suture and surrounding bone. For this reason, it was not
appropriate to give the suture zones material properties for sutures that are
reported in the literature (e.g.
Radhakrishnan and Mao, 2004).
Instead, the suture zones were given properties intermediate between cortical
bone and sutural material [elastic modulus (E)=0.400 GPa, Poisson's
ratio (
)=0.28, density (
)=1.13 g cm–3], reflecting
the composite nature of the suture zone. These properties were determined by
iteration of a test model, until it displayed strains across the sutures
similar to those reported by Jaslow and Biewener
(Jaslow and Biewener, 1995)
for their in vitro experiments on goat heads.
Four model geometries were constructed in order to test the functional effects of different morphologies of the frontal (Figs 1 and 2). The first model (`unvaulted frontal') lacked sinuses, and the contour of the frontal was smoothed to a thickness approximately equal to that of the combined inner and outer tables of bone in the unmodified skull (Fig. 1A). The resulting geometry was reminiscent of some bovids that lack enlarged frontal sinuses, such as members of the genus Gazella. The second model consisted of a skull completely lacking the frontal sinuses; the space within the vaulted frontal normally occupied by the sinuses was filled with bone (Fig. 1B,C). Two variants of this model were considered in the analysis, by setting the material properties of the bone occupying the location of the sinus to that of cortical bone (`vaulted cortical bone-filled frontal') or trabecular bone (`vaulted trabecular bone-filled frontal'; see below). The third model (`unstrutted sinus') comprised a skull with a simple unstrutted sinus cavity (Fig. 1D). Here, the boundaries of the frontal sinuses were simplified to eliminate all struts within the sinus, except for the midline strut at the interfrontal suture. The final model incorporated a complex frontal sinus, with all struts included (Fig. 1E). This model (`strutted sinus') represented the original goat skull. All models were otherwise identical.
The model geometries were imported into the finite element modeling package ALGOR FEMPro (v.20.0; Algor, Pittsburgh, PA, USA). Within FEMPro, the model was meshed using mixed brick elements. Mesh density was set to allow for multiple element thickness across the regions of experimental interest, such as the cranial vault, to achieve a more realistic solution. Owing to the size and complexity of the models, it was not feasible to conduct convergence tests with varying mesh densities. The model with an unvaulted frontal had 152,750 nodes and 382,972 elements; the models with vaulted solid frontals had 134,709 nodes and 363,407 elements; the model with unstrutted sinuses had 119,748 nodes and 309,790 elements; and the model with strutted sinuses had 119,436 nodes and 311,503 elements.
All models used identical material properties. Because properties for goat
cranial bone have not been published, it was assumed that values based on
averages of measurements from primate cranial vaults were appropriate
substitutes (E=14.550 GPa, =0.28, =1.725 g
cm–3) (Wang et al.,
2006). Properties of trabecular bone within the frontals (for
models with vaulted, trabecular bone-filled frontals) were based on published
measurements from human tibiae (E=0.637 GPa, =0.28, =0.2634
g cm–3) (Ashman et al.,
1989). With the exception of Poisson's ratio, to which an
arbitrary value of 0.30 was applied, properties for the horn sheath were
adapted from those published for other bovids (E=3.900 GPa,
=0.30, =1.3 g cm–3)
(Kitchener, 1991). Linear
isotropic material properties were assumed, but cranial bone may have a high
degree of anisotropy, potentially affecting analysis results
(Strait et al., 2005).
However, the present study is largely focused on patterns, not on precise
values. Several studies have found that isotropic material properties, if
appropriately selected, may be adequate for broadly characterizing most
aspects of strain distribution within a skull
(Metzger et al., 2005;
Strait et al., 2005). Thus,
anisotropic properties were not considered a major concern. For the same
reasons, and because appropriate data are not available for goats, uniform
material properties were used across the skull (except for the areas of
trabecular or cortical bone within the frontal for certain models, keratin and
sutures).
Static versus dynamic models
Shock absorption implies a dynamic component – deformation of a
structure over time. Dynamic finite element analyses incorporating skull
deceleration during impact are technically feasible, but require a set of data
(duration of impact, motion of skull during impact, etc.) not currently
available, as well as massive computing power for the resolution of models
considered here. Strain energy, which is the potential energy stored within a
deformed material (Hibbeler,
1997), presents a relevant proxy for examining shock absorption in
static models. Briefly, an object under load deforms and converts the kinetic
energy of impact into strain energy. This conversion reduces the accelerations
acting on the skull. Strain energy stored in one region of a structure (the
walls of the frontal bone and frontal sinuses) is not transferred elsewhere
(e.g. as vibrations in the walls of the endocranial cavity).
Loading conditions
The skulls were modeled at the moment of peak force during impact, as
estimated from values provided in previous studies
(Jaslow and Biewener, 1995).
Five loading cases of equal total magnitude (1088 N) were tested here, in
order to examine the effects of different locations and directions of loading
upon the skull (Fig. 1F,G). In
the first loading case (double-horn loading), equal loads of 544 N were
applied to the anterior surfaces of both horn sheaths at their proximal ends,
for a total force of 1088 N. The forces were angled so that their combined
vector passed through the center of the foramen magnum. Biologically, these
loading conditions were considered to be a realistic representation of
head-butting behavior in the goat (Jaslow
and Biewener, 1995). Additionally, this loading condition was
comparable with that used for in vitro experiments on goat crania by
Jaslow and Biewener, allowing for comparison of the model with their data
(Appendix 1). In the second loading case (single horn front loading), a load
of 1088 N was applied to the anterior surface of the base of the right horn
sheath only (at the same angle as in the double-horn loading condition). In
the third loading case (single horn lateral loading), a medially directed load
of 1088 N was applied to the lateral surface of the base of the right horn
sheath only. In the fourth loading case (single horn tip loading), a load of
1088 N was applied to the rostral surfaces of the right horn sheath
approximately two-thirds of the way distally along its length. In the fifth
and final loading case (frontal loading), a load of 1088 N was applied to the
dorsal surface of the middle of the frontal bone, rostral to the base of the
horncores. The angle relative to the base of the skull was identical to that
used in double-horn loading; the only difference was in the location of the
load relative to the base of the horn.
Constraints
The models were constrained at the occipital condyles from translation and
rotation in all planes. This followed an assumption of several previous
studies, that the occipital region is held relatively steady by the vertebrae
and cervical musculature during impact
(Schaffer and Reed, 1972). All
models had identical constraints.
Comparison of results
Results were extracted for two areas: (1) the bony surface of the
endocranial cavity; and (2) the frontal bone, excluding the surface of the
endocranial cavity (Fig. 3). In
order to evaluate hypotheses related to the role of the sinuses in protecting
the brain and associated structures, values for principal strains were
extracted for nodes of elements lining a region of the bony surface of the
endocranial cavity. These nodes were located in the dorsal half of the cavity,
exclusive of the region of the cribriform plate, rostral to the frontoparietal
suture, and excluding values from the interfrontal suture
(Fig. 3B). This region was
selected because it was closest to the frontal sinuses (hence most likely to
be affected by any changes in their morphology) and furthest from the the
occipital condyles (a region likely to have elevated values of stress and
strain due to the constraints there). Graphical plots of strain were also
generated, in order to visualize its distribution of across the surface of the
endocranial cavity.
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Interpretation of results for the frontals was complicated. For example, it
was expected a priori that average stress magnitudes should be
greater throughout the model with sinuses than in the model with a solid
vaulted frontal, because there was less bone to dissipate forces. The relative
distribution of the stresses was considered to be more informative. Thus,
histograms were used to visualize the distribution of nodal magnitudes for von
Mises stress within the frontal, as has been commonly done elsewhere in the
modeling literature (e.g. Van Rietbergen
et al., 2003). Because the models differed in the numbers of
elements in the frontal (reflecting whether the frontal was filled with bone
or struts, or unvaulted), the nodal results were downsampled to 100,000 nodal
values each (by random sampling without replacement) in order to create
comparable histograms.
Strain energy was a bulk measure of total energy within the entire frontal bone, so no further corrections were necessary. The value was calculated as the sum of the strain energies of all of the elements of the region of the frontal bone outlined in Fig. 2 and Fig. 3A (the same region used for the analysis of von Mises stress). Models were loaded identically (the same force vector for each loading case, regardless of model morphology), so the value allowed a direct evaluation of which models stored the most energy from the applied load within the frontal.
The 50th (median) and 95th percentile values were calculated for each of the above samples (except for strain energy); mean and maximum values were not determined directly, because of the occurrence of rare extreme outliers. These extreme values occurred at sharp steps in the model geometry (an occasional byproduct of the process of generating the geometry from CT data in SolidWorks) or due to sharply shaped elements generated by the automated meshing routine. For example, for the model with strutted sinuses under the double-horn loading condition, the 95th percentile value for maximum principal strain within the body of the frontal bone was 858 N m–2, whereas the maximum value was 24,454 N m–2. Thus, the latter value is a modeling artifact that could exert extreme leverage on the mean. Instead, interquantile means (referred to hereafter as `means', for simplicity) were calculated for all values between the 5th and 95th percentiles. All statistical calculations and histograms construction were completed in R (v.2.6, R Foundation for Statistical Computing, Vienna, Austria).
The models were validated by comparison to results from previously
published in vitro experiments
(Jaslow and Biewener, 1995).
These comparisons are presented in Appendix 1.
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RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX 1References |
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Principal strains on the endocranial surface
For models with vaulted frontals under double-horn loading, the greatest
magnitudes of principal strains (mean, median and 95th percentile values) were
seen in the models with sinuses, followed by the models with trabecular
bone-filled frontals and then cortical bone-filled frontals (Tables
2 and
3). The overall range of values
across models typically spanned between 75 and 200 µ
. The model with
an unvaulted frontal varied in rank relative to the other models, but
typically had magnitudes less than or approximately equal to the models with
sinuses.
Under a load applied to the frontal, a similar ranking of magnitudes of principal strains occurred for models with vaulted frontals (Tables 2 and 3). Again, the model with an unvaulted frontal fell within the range exhibited for models with vaulted frontals (and had a smaller magnitude than the models with sinuses, except at the 95th percentile).
No major differences were found between models in the distribution patterns of strain across the bone lining the endocranial cavity (Fig. 4) under the double-horn loading condition. Sutural zones were under relatively high strain for both double-horn and frontal loading conditions. Under the frontal loading condition, the model with an unvaulted frontal was unique among the model geometries in the distribution of strain across the surface of the endocranial cavity (Fig. 4F,P). At the rostral end of the endocranial cavity, there was a zone of high principal strain magnitudes (Fig. 4F,P) (this was primarily a difference in spatial distribution rather than overall magnitude; Tables 2 and 3). This contrasted with the regions of lower strain immediately caudal to that zone. A very different pattern occurred in all four of the models with vaulted frontals, in which maximum principal strains were more evenly distributed across the surface of the endocranial cavity (Fig. 4G–J) and where greatest magnitudes of minimum principal strain were confined to the lateral edges of the region of interest (Fig. 4Q–T). The model with a cortical bone-filled frontal had consistently low magnitudes of strain across the endocranial region of interest for minimum principal strain, with a slight elevation of maximum principal strain at the caudal end (Fig. 4G,Q).
Von Mises stress in the frontal bone
Under the double-horn loading condition
(Fig. 5A–E), the model
with unvaulted frontals and the model with cortical bone-filled,vaulted
frontals showed a greater concentration of low-magnitude values than seen in
the models with sinuses (Fig.
5A,B,D,E). A peak of low-stress elements in the model with
trabecular bone corresponded to elements with material properties of
trabecular bone (Fig. 5C).
Under the frontal loading condition, the models with solid vaulted frontals
and unvaulted frontals (Fig.
5F–H) all had prominent peaks of low stress elements in the
frontal loading condition. Stress values were more evenly distributed in all
models with sinuses (Fig.
5I,J).
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Deformation
Overall deformation of the models was relatively consistent under loads
applied to the horns. The bulk of the deformation occurred caudal to the base
of the horncore, especially at the frontoparietal suture, regardless of
frontal morphology (Movies 1 and 2 in the supplementary material). Most bony
deformation occurred in the bone immediately rostral and immediately caudal to
the suture, caused by the cranial vault warping outwards. Under the frontal
loading condition, the models with vaulted frontals experienced little
deformation of the endocranial cavity. Instead, the external cortex of the
frontal exhibited the bulk of the deformation as the cortex was pushed inwards
(Movie 3 in the supplementary material). Deformation in the model with the
unvaulted frontal occurred as the bone at the front of the endocranial cavity
was pushed inwards (Movie 4 in the supplementary material).
Results from other loading conditions
Results for additional loading conditions are summarized here. Ranks of
strain energy magnitude for loads applied to a single horn (regardless of
direction) broadly followed those seen in the model under the double-horn
loading condition (see Table S1 in the supplementary material). Models with
sinuses always had greatest magnitude, and the model with a vaulted cortical
bone-filled frontal always had the least magnitude. The model with vaulted
frontals had between 46 and 92% of the magnitude of strain energy seen in the
model with vaulted strutted sinuses. For principal strain magnitudes in the
surface of the endocranial cavity or within the external frontal cortex, the
rank order of maximum, minimum or median values differed slightly in some
cases, but there were few exceptions to the general patterns described above
(Tables S2 and S3 in the supplementary material). Generally, magnitudes of
principal strain for the model with unvaulted frontals fell within the range
seen for or occasionally slightly higher than (by no more than 120 µ,
but typically much less) seen in models with vaulted frontals.
Contrasting with the double-horn loading condition, patterns of principal
strain were not symmetrical across the endocranial cavity when the load was
applied only to a single horn (see Fig. S1 in the supplementary material).
Strain magnitudes dropped greatly across the interfrontal suture in this case
[consistent with the results of Jaslow and Biewener
(Jaslow and Biewener, 1995)].
Patterns of distribution were quite similar across model geometries within
each loading condition.
Just as in the frontal loading condition (and the double-horn loading condition, to a lesser extent), histograms of von Mises stress for the models with an unvaulted frontal and vaulted solid frontals showed prominent peaks of elements under low stress (Fig. S2 in the supplementary material).
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DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX 1References |
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If sinuses (or a vaulted frontal) have a major role in storing energy, models with these structures should show elevated strain energy in the frontal. Strain energy was always greatest for models with sinuses, appearing to support their suggested role in shock absorption. But a vaulted frontal alone did not always confer an advantage – under double-horn loading in particular, models with solid vaulted frontals stored a much lower proportion of the strain energy within the frontal than did the model with unvaulted frontals. Thus, a sinus in combination with a vaulted frontal, rather than a vaulted frontal alone, was needed to maximize the shock absorption potential of the frontal bone.
Additionally, sinuses did not reduce strains on the surface of the endocranial cavity during most loads applied to the horns relative to a model with unvaulted frontals. In nearly all cases of horn loading, the model with frontals filled with cortical bone had the lowest overall magnitudes of strains (as measured by the median values), and the model with strutted sinuses (similar in morphology to real goat skulls) had the highest magnitudes (Tables 2 and 3; Tables S2 and S3 in the supplementary material). Most importantly, the overall patterns of strain distribution across the bone lining the surface of the endocranial cavity were quite similar across most models (Fig. 4; Fig. S1 in the supplementary material) (exceptions detailed below).
The model with trabecular bone-filled frontals generally behaved quite similarly to the models with sinuses (either strutted or unstrutted), when comparing the distribution of strains across the endocranial cavity. This was expected, in light of the fact that the elastic modulus for trabecular bone is much lower than that for cortical bone.
The frontal loading case was the only situation in which sinuses or a
vaulted frontal seemed to have a major beneficial effect on strains on the
surface of the endocranial cavity or strain energy within the frontal bone,
over the model with unvaulted frontals. Models with sinuses or a vaulted
frontal distributed the load more evenly across the endocranial bone for loads
to the frontal. The advantages of this are clear: by distributing deformation
across a wide area, the risk of damage to dural sinuses or meningeal arteries
(relatively delicate structures carrying blood to and from the brain) is
minimized. Strains were highly concentrated at the rostral end of the
endocranial cavity (beneath the point of load application) in the model with
unvaulted frontals, and strain energy magnitudes were also much lower than for
models with vaulted frontals (except for the model with a vaulted, cortical
bone-filled frontal). Yet, frontal loading does not appear to be a
particularly realistic loading condition for goats. Extensive behavioral
observations on wild ibex (Capra pyrenaica, congeneric with the
Capra hircus modeled here and also sharing its general frontal
morphology) do not report any instances of blows applied to the frontals
(Alvarez, 1990). Thus, the
biological relevance of this loading condition is questionable for goats
(although certainly important for animals with extensive frontal sinuses that
do butt frontals, such as Bison).
In summary, sinuses (or more specifically, the bone surrounding the space of the sinuses) appeared at first glance to have shock-absorbing potential under most loading conditions. This may be due to the fact that the thin walls of the sinus deform slightly during impact, creating a sort of `crumple zone', and morphologies without this thin wall (or a thin wall of cortical bone against a relatively deformable layer of trabecular bone) do not allow such deformation. Yet, magnitudes of strain energy within the model with unvaulted frontals often approached magnitudes seen in models with sinuses, and patterns of principal strains also conflict with the idea of sinuses as shock absorbers.
The extra bone associated with sinuses or vaulted frontals seemingly would
offer significant benefits for strain reduction. Why, then, do these
structures fail to offer much greater shock absorption than seen in unvaulted
frontals or a greater strain reduction in the walls of the endocranial cavity,
especially when compared to dramatically different skull morphologies with
unvaulted frontals? The most likely explanation is that the vaulted frontal
and its sinuses are poorly placed to offer much protection from the typical
dorsoventral and rostrocaudal loads applied to the horns in life. Thus, the
loads are directed away from the thickest part of the frontal, and directly
into the thinner part of the braincase, where most of the deformation occurs.
A biomechanically `better' design (at least in light of desired endocranial
strains and deformation) would place the horns atop or in front of the vaulted
frontal, instead of at the rear of this structure. Instead, the caudal
placement of the horns in goats and many other bovids, along with
reorientation of the basicranium and other structures, may serve to reduce
torque about the foramen magnum during head-butting
(Schaffer and Reed, 1972;
Jaslow, 1987). The morphology
of the goat skull is clearly a trade off between multiple structural
considerations.
The role of struts
In goats, the struts within the sinuses did not appear to have a major role
in absorbing shocks or distributing loads applied to the frontals. In
comparing the model with strutted sinuses to the model with unstrutted
sinuses, very few differences were evident. The histogram distributions were
virtually identical in most cases (Fig.
5; Fig. S2 in the supplementary material). Similarly, there were
no appreciable or consistent differences between the two models when
considering patterns of endocranial strains
(Fig. 4; Fig. S1 in the
supplementary material), or when considering strain energy within the frontal
itself (Table 1; Table S1 in
the supplementary material).
Do the struts have any function, then? One possibility is that the struts
are just a by-product of sinus formation – bone that was `accidentally'
left behind by osteoclasts. The midline strut separating the right and left
frontal sinus (a structure that was included in all models) may be retained
due to interactions between sutures (the interfrontal suture, in this case)
and pneumatic epithelia (Farke,
2007). Another possibility is that the struts improve the overall
strength of the frontal. However, given the extremely thin nature of some of
these struts in Capra, this hypothesis appears doubtful.
Additionally, many other bovids (e.g. Alcelaphus and
Damaliscus) have sinuses with very few struts
(Farke, 2007), with apparently
little consequence. By contrast, bighorn sheep (Ovis canadensis) and
Cape buffalo (Syncerus caffer) are notable for extensive strutting
within their frontal sinuses [hundreds of struts, far beyond that seen in
Capra (Schaffer and Reed,
1972); A.A.F., unpublished]. These species, and many of their
close relatives, engage in extremely vigorous head-butting. It is quite
possible that the numerous struts of such taxa do play a role in structural
support and shock absorption, whereas struts are less important in goats.
Comparative quantitative analyses of sinus morphologies are necessary in order
to determine whether there is a correlation between the number of struts and
behavior. Further finite element modeling may also prove useful.
Sinuses and `structural efficiency'
Areas of bone that experience only low magnitude stresses are not used in
load transmission or shock absorption, so such regions could be thought of as
unnecessary for the structural support of the skull. Thus, an efficient
structure eliminates such elements. The histograms of von Mises stress
produced for these models were consistent with this hypothesis
(Fig. 5; Fig. S2 in the
supplementary material). This was well-illustrated by models under the
double-horn loading condition (Fig.
5B–E). The model with cortical bone-filled frontals had a
relatively larger peak of low-magnitude stresses (and fewer high magnitude
stresses) when compared with the models with sinuses or the model with
trabecular bone-filled frontals. Replacing the cortical bone in the center of
the frontal with trabecular bone or a sinus eliminated this peak of elements
under low stress.
Why a vaulted frontal?
Although `structural efficiency' and gains of shock absorption potential
may explain the presence of a frontal sinus, it does little to explain why the
frontal should be vaulted in the first place (when compared with other bovids,
such as gazelles, that lack frontal vaulting). Certainly, in terms of strain
energy, the vaulting does offer an advantage in shock absorption in some
cases. Further study is needed to determine if this advantage is biologically
meaningful or relevant over evolutionary time scales. A second possibility is
that the vaulted frontal is needed as a base of support for the large
horncores. The large base of support would increase the second moment of area,
and hence strengthen the horns against applied loads. This is supported in
part by the high proportion of frontal bone elements under low stress
(Fig. 5A,F; Fig. S2A,F,K in the
supplementary material) in skulls with unvaulted frontals. This indicates that
a small proportion of elements is bearing a significant proportion of the load
(especially considering that there is less bone volume when compared with any
of the other skull morphologies). Alternatively, the enlarged frontals may
increase the apparent `size' of the skull (useful for visual display). A whole
host of possibilities exist, many of which are untestable using the current
techniques.
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CONCLUSIONS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX 1References |
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;
McHenry et al., 2006;
Rayfield, 2005) have compared
skulls of related animals with varying morphologies. Although a comparative
taxonomic approach may also offer useful information (and be desirable for
answering some questions), a clear interpretation of results may be muddied by
disparate morphologies unrelated to the investigation at hand. Thus, digital
manipulation of morphology is an often underutilized strength of finite
element modeling for vertebrate skulls (e.g.
Rayfield et al., 2007;
Strait et al., 2007). The present study finds mixed support for the frontal sinuses of the goat in protecting the skull against blows. Future work might investigate the shock-absorptive role of the keratinous sheaths of the horns; studies on attenuation of impact vibrations in horse hooves (e.g. Willeman et al., 1999) certainly warrant parallel investigations in goats. If the bone of the horncores is significantly less stiff than the bone of the cranial vault, the horncores themselves could play an important role in energy absorption during impact. Measurement of material properties within the goat skull would be important in this regard. Although vaulted frontals with sinuses offer shock absorbing potential under certain loading conditions, skulls with unvaulted frontals often show high shock absorption potential, too. Furthermore, frontal sinuses or even just a vaulted frontal bone do little to change the patterns of bone strain across the endocranium for most loading conditions. All of these supposed `protective' structures are poorly placed for protecting the skull from most loads. Struts within the sinuses also seem to have little effect, at least for goats. Results are consistent, however, with the idea of sinuses removing `unnecessary' bone from the frontal.
Several questions remain. If vaulted frontals and frontal sinuses are so poorly placed to deal with blows to the horns, why do so many head-butting taxa have such morphologies? Are sinuses more important in taxa (such as bison) that butt frontals directly? Are these morphologies indeed correlated with head butting? What other factors lead to enlargement of the frontal bone? The static models considered here present only one proxy for investigating the effects of head-butting on the brain. How is the brain tissue itself affected during the dynamic motions of head-butting? Further experiments, modeling and comparative anatomy may answer such questions.
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APPENDIX 1 |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX 1References |
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). Their study placed strain gauges on bone at
several points across goat heads under in vitro impact loads. Strain
gauge locations from the experimental study that were comparable to the FE
models included the anterior surface of the base of the horncore, posterior
surface of the base of the horncore, and the external aspects of frontal and
parietal bones [fig. 1 of Jaslow and Biewener
(Jaslow and Biewener, 1995)].
Due to inevitable differences in skull morphology and precise strain gauge
placement as compared to the experimental results, strain gauge locations had
to be approximated on the FE model. At each site of comparison, a set of nine
nodes in a roughly rectangular pattern (three nodes in a triangular pattern
for the posterior horncore, because of the limited exposed area) were selected
for further analysis. Multiple nodes were sampled in order to mitigate the
potential effects of any poor-quality elements.
Strain gauges only measure strain in two dimensions at most, so the
three-dimensional strain vectors provided by FE models were not directly
comparable. In order to measure the principal strains in the plane of the
surface of the skull, the coordinate system was redefined separately for each
area of the skull to coincide with the nodes sampled at each validation point.
Then, the normal and shear strain tensors in the plane of the sampling region
were extracted, and the magnitude and direction of planar maximum and minimum
principal strains were calculated using standard engineering equations
(Hibbeler, 1997).
Jaslow and Biewener (Jaslow and
Biewener, 1995) published mean values and s.d. for each strain
gauge site; modeled strains and strain orientations were considered realistic
if they fell within two standard deviations of the experimental values. Ratios
for maximum and minimum principal strain were also compared, as another
measure of model performance. Results were validated in the model with
strutted sinuses (reflecting real-life skull geometry) for the condition of
double-horn loading.
Validation results
All principal strain magnitudes fell within two s.d. of experimentally
determined values, and most fell within one standard deviation (see
Table A1). This suggests that
the finite element models produced a realistic picture of strain magnitudes
relative to in vitro loading conditions, at least at the external
surfaces of the frontal, parietal and horncores. Only two out of five sampled
locations (on the left frontal and right parietal) produced angles of maximum
principal strain within one s.d. of experimental values (and the rest were
greater than two s.d. in difference). On a positive note, even the
`noncomparable' angles were within at least 26 deg. or less of the average of
the experimental values. Ratios of maximum principal strain to minimum
principal strain (reflecting the proportion of tensile strain to compressive
strain) were quite similar across most locations. This indicated another broad
level of similarity in the behavior of the computer model to in vitro
experiments (regardless of differences in magnitude).
|
Differences between the experimental results and the finite element model may be due to subtle differences in shape or loading conditions between the model and experimental skulls. The material properties selected for the finite element model certainly affect results; strain magnitudes in the model were consistently lower than in vitro, suggesting that a less stiff elastic modulus would close the gap of strain magnitudes from two s.d. down to much less than one s.d. Application of anisotropic material properties could also bring the angles of principal strain closer to experimental observations.
LIST OF ABBREVIATIONS
1
2
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Acknowledgments |
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Footnotes |
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References |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX 1References |
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