Biomechanical consequences of scaling
Concord Field Station, Department of Organismic and Evolutionary Biology, Harvard University, Old Causeway Road, Bedford, MA 01730, USA
e-mail: biewener{at}fas.harvard.edu
Accepted 31 January 2005
Summary
To function over a lifetime of use, materials and structures must be
designed to have sufficient factors of safety to avoid failure. Vertebrates
are generally built from materials having similar properties. Safety factors
are most commonly calculated based on the ratio of a structure's failure
stress to its peak operating stress. However, yield stress is a more likely
limit, and work of fracture relative to energy absorption is likely the most
relevant measure of a structure's safety factor, particularly under impact
loading conditions characteristic of locomotion. Yet, it is also the most
difficult to obtain. For repeated loading, fatigue damage and eventual failure
may be critical to the design of biological structures and will result in
lower safety factors. Although area:volume scaling predicts that stresses will
increase with size, interspecific comparisons of mammals and birds show that
skeletal allometry is modest, with most groups scaling
(ld0.89) closer to geometric similarity
(isometry: l
d1.0) than to elastic
similarity (l
d0.67) or stress similarity
(l
d0.5). To maintain similar peak bone and
muscle stresses, terrestrial mammals change posture when running, with larger
mammals becoming more erect. More erect limbs increases their limb muscle
mechanical advantage (EMA) or ratio of ground impulse to muscle impulse
(r/R=
G/
Fm). The increase in limb EMA
with body weight (
W0.25) allows larger mammals to match
changes in bone and muscle area (
W0.72-0.80) to changes in
muscle force generating requirements (
W0.75), keeping bone
and muscle stresses fairly constant across a size range 0.04-300 kg. Above
this size, extremely large mammals exhibit more pronounced skeletal allometry
and reduced locomotor ability. Patterns of ontogenetic scaling during skeletal
growth need not follow broader interspecific scaling patterns. Instead,
negative allometric growth (becoming more slender) is often observed and may
relate to maturation of the skeleton's properties or the need for younger
animals to move at faster speeds compared with adults. In contrast to bone and
muscle stress patterns, selection for uniform safety factors in tendons does
not appear to occur. In addition to providing elastic energy savings, tendons
transmit force for control of motion of more distal limb segments. Their role
in elastic savings requires that some tendons operate at high stresses (and
strains), which compromises their safety factor. Other `low stress' tendons
have larger safety factors, indicating that their primary design is for
stiffness to reduce the amount of stretch that their muscles must overcome
when contracting to control movement.
Key words: bone, muscle, tendon stress, elastic savings, safety factor
Introduction
Scale effects on the biomechanical design of organisms depend strongly on
the properties of the biomaterials of which they are built, requirements for
avoiding failure over a lifetime of their use, and on physiological demands
for performance over a range of activities. For animals and plants constructed
of similar materials, geometrical effects on the scaling of biomechanical
function predominate. For structural support, design requirements depend
mainly on the operating stresses (; or strains,
) that structural
elements experience relative to their yield, or failure strength
(Wainwright et al., 1976
). The
ratio of failure stress to peak functional, or operating, stress is most
commonly used to define a structure's `safety factor'. However, it is likely
that lower safety factors based on yield stress more often apply to the
functional integrity of a structure. Repeated loading characteristic of
locomotion may also cause fatigue damage and eventual failure at stresses well
below those that cause failure in a single or few loading cycles
(Carter and Caler, 1983
;
Wang et al., 1995
). For this
reason, a structure's `loading history' is important, in terms of design for
an adequate safety factor, as well as for remodeling and repair
(Carter, 1987
). For the
dynamic loading conditions of locomotion, it is also likely that the amount of
strain energy that must be absorbed during a loading cycle relative to the
work of fracture of a material (e.g. bone or tendon) is the most critical
determinant of a structure's safety factor. Because the distribution and
amount of strain energy absorption in structural elements are difficult to
measure during locomotion, energy absorption is not often used to evaluate a
structure's safety factor relative to its mechanical use.
For skeletal support elements, such as animal limb bones, design to
maintain an adequate safety factor and avoid failure over a lifetime of use is
most critical. However, for elastic elements, such as tendons and ligaments,
the capacity for strain energy savings to reduce muscle work and lower the
energy cost of locomotion also has selective value. Consequently, a trade-off
exits in the design of tendons for adequate strength versus achieving
high-energy savings, which depends on the tendons experiencing high strains
(strain energy per unit volume of material is 2 or,
equivalently,
2;
Alexander, 1988
;
Biewener, 2003
). Morphological
evidence suggests that most tendons are designed to have high safety factors,
in excess of what is required for effective strain energy savings and
necessary strength (Ker et al.,
1988
). This indicates that maintaining sufficient stiffness
(force/displacement) for control of length and thus, position and movement of
limb segments, may be an overriding design requirement for many tendons. Such
tradeoffs in functional design therefore can result in quite distinct
operating stresses and safety factors.
In this article, I focus on the biomechanical consequences of scaling in the limbs of terrestrial vertebrates, principally birds and mammals, comparing design requirements for three locomotor elements: bones, muscles and tendons. While other structural elements (e.g. cartilage) are clearly of key importance to the biomechanical function of these animals, less is known about the scaling implications for the design of cartilage in the joints of small versus large animals. Although my focus is on vertebrate animals, general principles concerning design tradeoffs for differing functions likely apply broadly to other organisms. Scaling comparisons across vertebrate taxa also reflect my conviction that size effects predominate over historical evolutionary constraints with respect to their influence on skeletal design and interpretations that may be drawn from scaling patterns. Although historical constraints clearly underlie the biomaterials of which diverse groups of organisms are built, over this broad diversity body size varies enormously and, as a result, plays a dominant role.
Scale effects on the vertebrate skeleton
Because stress (and strain, which is proportional to stress based on a
material's elastic modulus; (Biewener,
2003; Wainwright et al.,
1976
) depends on the ratio of force applied and transmitted over a
certain cross-sectional area of a material (
=F/A),
peak functional stresses are predicted to increase with size. By geometric
similarity (or isometry), in which all lengths (l) and diameters
(d) of similar structural elements scale to the animal's weight,
l
d
W1/3, stress is predicted
to scale
W1/3, given that force scales
W1 and A
W2/3. This
has profound implications for how vertebrate animals are built and function,
and in the terrestrial environment counter gravitational and inertial forces.
Because of the strong increase in stress with size predicted by geometric
similarity, early work dating from Galileo
(Galilei, 1638
) predicted that
skeletal elements in different sized animals would exhibit strong positive
allometry (i.e. becoming increasingly robust with size). McMahon
(1973
,
1975b
) published two
influential papers arguing that animals scale with elastic similarity
(l
d2/3
W1/4) to
maintain similar elastic deformations under equivalent loading conditions.
Although elastic similarity reduces the scaling effect of size on stress and
strain, it predicts an increase:
W1/4 for
axial stress and
W1/8 for stresses due to
bending and torsion. Only under conditions of static stress similarity
(McMahon, 1975b
) can stress
remain constant across size based on geometry alone.
Although ungulates, and bovids in particular
(McMahon, 1975a), have limb
bones that scale close to elastic similarity, over a much broader size range
mammalian (Fig. 1) and avian
taxa scale closer to geometric similarity
(Alexander, 1979
;
Biewener, 1982
;
Maloiy et al., 1979
;
Selker and Carter, 1989
), with
the prediction that peak bone stresses scale
W0.28.
Consistent with their geometrically similar form, mammals and birds of
different size move in dynamically similar fashion
(Alexander and Jayes, 1983
; and
see Alexander p. 1645, this issue) at equivalent points in gait. Nevertheless,
because the fracture strengths of mammalian and avian bones are generally
uniform (Biewener, 1982
;
Currey, 2002
), measurements of
compressive stresses operating in the limb bones of chipmunks and ground
squirrels during galloping in the range of -31 to -86 MPa (mean for both
species: -57 MPa; Biewener,
1983
) would predict (using
W0.28)
that no terrestrial mammals or birds should exist with similar ability to run
and gallop at a size much greater than 20 kg in body mass, without exceeding a
compressive fracture strength of -240 MPa
(Currey, 2002
)!
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By adopting more erect postures when they run, large mammals are able to
reduce the moment arm (R) of the ground reaction force (G)
relative to the muscle's moment arm (r,
Fig. 2). This decreases the
magnitude of joint moments produced by the ground reaction force (=R
xG), which are the principal moments that the muscles and bones
of the limb must withstand during ground contact (moments due to segment
inertia and segment weight are much smaller and most important during limb
swing; Biewener, 2003). The
`external' moments produced by the ground force are counteracted by muscle
moments, so that at each joint Fm
xr=G xR (ignoring segment inertia and
weight). By measuring changes in G and R at different
instances in time during ground contact, and assuming a fixed agonist moment
arm (r), weighted by the muscle agonists' fiber-areas, changes in
muscle force required to support ground reaction moments can be determined.
These are averaged over the period of limb support by determining the ratio of
ground impulse (
G) to muscle impulse (
Fm),
to calculate the agonist muscles' effective mechanical advantage (EMA) at a
particular joint. By running with more upright postures, larger terrestrial
mammals increase their effective limb's effective mechanical advantage
(EMA=r/R=
G/
Fm), reducing
the magnitude of muscle force transmitted to bone elements and tendons. Over a
size range from mice to horses, when averaged for the major fore and hind limb
joints, EMA scales
W0.25
(Fig. 3A,B). Given that peak
ground forces generally match changes in body weight (G
W1), this implies that weight-specific muscle forces
(Fm/W) are expected to vary
W-0.25. It also predicts that terrestrial mammals of
different size produce the same force per unit gram (volume) of muscle
recruited to support their body weight when running
(Biewener, 1989
). This follows
from the fact that mammalian muscle fiber lengths generally scale
W0.23 (Alexander et
al., 1979
), so that force per active muscle volume (=
weight-specific force x fiber length) generally scales
W-0.25 xW0.23, or close to
W0. However, it is worth noting that muscle fiber-length and
fiber-area scaling exponents can vary substantially between different muscle
groups (Pollock and Shadwick,
1994
), so that the scaling of muscle stress and force per unit
volume within different muscles may also be expected to vary. The functional
implications of such variation would be interesting to explore.
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The scaling of mammalian limb EMA also matches the general scaling of force
production observed by Marden and Allen
(2002) (and see also Marden p.
1653, this issue) for both animate (muscle, myosin, dyenin, kinesin) and
inanimate (winches and linear actuators) translational motors, as applied to
dragonfly flight muscle (Schilder and
Marden, 2004
). Marden and Allen
(2002
) found that force output
of animate and inanimate translational motors scales
W0.67, but when the leverage (moment arm, r)
of dragonfly flight muscles is measured, overall `leveraged force output' of
the dragonfly flight muscles to move their wings scales
W1.0. Although this scaling pattern obeys geometric
similarity (i.e. force
W0.67 and r
W0.33), it parallels closely the pattern of muscle
force output and limb mechanical advantage observed for terrestrial mammals,
which follow 3/4 and 1/4 power scaling.
Interestingly, in contrast to quadrupedal mammals that have been studied,
which show little consistent change in limb EMA across speed and gait change
(Biewener, 1989), human limb
EMA significantly decreases from walking to running
(Fig. 3B;
Biewener et al., 2004
). This
results primarily from the knee being extended during walking versus
flexed during running, consistent with the shift in limb mechanical function
from pendular exchange of center of mass potential and kinetic energy during
walking to spring-like energy savings during running
(Cavagna et al., 1977
). The
lower limb EMA during running results in a fivefold increase in force by the
knee extensors, compared with a twofold increase at the hip and no change at
the ankle. It also correlates with the higher energy cost of human running
compared with walking (Margaria,
1976
) and compared with quadrupeds of similar size
(Taylor et al., 1970
).
When rodents are compared (as a group) with mammals more generally there is no significant difference in EMA scaling (Fig. 3C,D). However, a limitation of this comparison is that most of the species (9 of 14) included in the `mammal' line are rodents. When goats (Capra hircus) are compared with capybaras (H. hydrochaeris), and adjusted for the same size, goats have a significantly greater forelimb EMA (P<0.01) and greater overall (fore and hind) limb EMA (P<0.05) than capybaras. Whether artiodactyls, and ungulates as a group, have generally more erect postures than rodents remains to be determined. A broader range of ungulates, particularly at smaller size, needs to be studied. A potential difficulty with this comparison is that capybaras are semiaquatic animals, which might be expected to affect their limb mechanics. As the largest rodent species, their values may disproportionately influence the regression line. However, the capybara values fall close to the line for rodent EMA scaling. When compared with values predicted based on least-squares regression of the other rodent species alone (capybara data excluded), the capybara hind limb EMA is 78% greater than predicted and the fore EMA is 5% lower than predicted.
Consequently, while general scaling effects of more erect posture on limb
EMA to reduce bone and muscle stresses appear to apply within rodents and
across mammals more generally, data for a broader diversity of land animals is
needed to improve our understanding of how evolutionary ancestry may influence
biomechanical constraints of scaling relative to other functional determinants
of limb and skeletal form. For example, in a comparison of different sized
macropoidids, Bennett and Taylor
(1995) found that ankle EMA
does not change with size (averaging 0.24), in contrast with the eutherian
mammals discussed above (Fig.
3D). This appears to reflect selection favoring increased elastic
energy savings (discussed below) in larger macropodids, at the risk of
increased operating stresses in their hind limb tendons, bones and muscles. In
addition, no data yet exist to compare avian bipeds across their size range in
terms of limb EMA and whether a scaling pattern similar to terrestrial
eutherian mammals emerges. In a study comparing skeletal allometry in
different carnivoran families with that observed in bovids and certaomorphs,
Bertram and Biewener (1990
)
found evidence for differential allometry across taxonomic groups at different
sizes (Fig. 1). In contrast to
carnivorans and other small to moderate-sized mammals
(Bou et al., 1987
;
Christiansen, 1999
), which
scale generally close to geometric similarity, bovids display elastically
similar scaling (McMahon,
1975b
). Whereas, at extremely large size ceratomorphs scale with
increasing skeletal robustness (Prothero
and Sereno, 1982
) close to static stress similarity
(Fig. 1). These patterns of
differential skeletal allometry within terrestrial mammals suggest different
biomechanical and physiological constraints may apply between taxonomic groups
that span different size scales. It is also the case that no study has
examined whether ontogenetic changes in limb EMA occur within a species.
Although size change during growth is often much less than across a diversity
of related species, postural changes in limb loading could well play a role in
terms of growth-related changes in skeletal form and function.
In recent studies of the ontogenetic growth and skeletal biomechanics of
goats (Main and Biewener,
2004a) and emu (Dromaius novaehollandiae;
Main and Biewener, 2004b
),
locomotor bone strains were found to increase in the goat radius and in the
emu tibiotarsus (Fig. 4A,B) and
femur (not shown) with growth and age, in contrast to the generally uniform
peak strain patterns found in the chick tibiotarsus during growth
(Biewener et al., 1986
). This
increase appears to result from isometric scaling of peak ground reaction
forces (G) at different ages during growth relative to negatively
allometric scaling of bone area and second moment of area in the goat radius
and isometric scaling of bone geometry in the emu tibiotarsus
(Fig. 4C,D) and femur (not
shown). The finding of a relative thinning of the radius in goats and to a
certain degree in the emu tibiotarsus and femur is surprising, but may be
explained by the need of smaller young animals to run at relatively faster
speeds to keep up with larger adults (Main
and Biewener, 2004a
). In an earlier study, Carrier
(1983
) found that jackrabbits
(Lepus californicus) also exhibit negative ontogenetic growth
allometry in the tibia and femur, which may offset the reduced mineralization
(and stiffness) of their bone at an early age and small size. As no difference
in mineralization has been found in goats and emu from 2 weeks of age to adult
(Main and Biewener, 2004b
), it
would be interesting to know whether jackrabbit bone strains are maintained
uniformly during growth as they are in the chicken tibiotarsus
(Biewener et al., 1986
). These
results show how other factors (such as an ecological pressure for group
movement to reduce predation risk in younger animals) may influence the
biomechanics and scaling of ontogenetic growth in ways that may differ
considerably from those found more broadly across species of different size.
Additional studies of musculoskeletal growth trajectories in relation to
locomotor performance are needed for a greater understanding of how such
factors may interact with size to influence biomechanical design strategies
during ontogenetic growth.
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Scale effects on muscle function
The scaling of limb mechanical advantage in terrestrial eutherian mammals
suggests that peak stresses developed by limb muscles at equivalent levels of
physical activity are generally similar. This follows from the scaling of
muscle force based on limb EMA W0.75
(Biewener, 1989
), which fairly
closely matches the scaling of muscle cross-sectional area
W0.80, based on the results of Alexander et al.
(1979
). To date no study has
formally tested the prediction that peak muscle stresses are size-invariant,
but reported values of muscle stresses developed in various terrestrial
species during moderate to strenuous activity to date generally support this
conclusion (Table 1). Although
peak muscle stresses vary fourfold, there is no significant correlation with
the ratio of muscle area to tendon area:
Am/At;
Fig. 5), or with body size.
Much of the variation in peak muscle stresses reflects differences in
measurement technique and differing performance of the animals studied (e.g.
kangaroo rat hopping versus jumping,
Table 1). In one study
(Perry et al., 1988
) that
compared muscle stresses acting in the ankle extensors of quadrupedal rats
(Rattus norvegicus) versus bipedal kangaroo rats
(Dipodomys spectabilis) moving at their preferred speed and gait (rat
gallops at 1.48 m s-1; kangaroo rat hops at 1.46 m s-1),
similar peak muscle stresses (70 and 69 kPa) were found.
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Whereas in vivo methods provide direct measurements of
muscle-tendon stress, they are limited to muscles having sufficiently long
tendons that enable direct measurements. By contrast, indirect methods based
on inverse dynamics (linked-segment joint moment analysis that accounts for
external, inertial and gravitational forces acting at a joint; e.g.
Winter, 1990) and free-body
analysis, rely on assumptions about muscle force distribution among muscle
agonists. For more proximal joints (such as the hip and knee), such approaches
also cannot resolve agonist muscle forces due to force transmission by
two-joint muscles, resulting in an indeterminant condition. Nevertheless, the
highly conservative nature of muscle contractile proteins and their
organization in vertebrate skeletal muscle fibers (other than for differing
distributions of muscle fiber type and associated mitochondrial and capillary
densities for energy supply among different species that show important
allometric patterns (Hoppeler et al.,
1981a
,b
;
Mathieu et al., 1981
;
Weibel, 1985
), indicates that
maximal stresses within the skeletal muscles of different sized animals are
likely fairly constant.
However, if two species have differing capacities for increasing their
locomotor performance beyond rest, peak muscle stresses at preferred levels of
activity might also differ. In comparison to metabolic scope (see Bishop p.
1695, this issue), factorial scopes for muscle force and power output (for
increased speed or acceleration) have not been examined across species.
Species with a greater ratio of top speed to preferred speed might be expected
to operate with lower stresses at their preferred speed of locomotor movement.
Nevertheless, changes in limb mechanical advantage and relative limb length
might offset such speed-related influences on peak muscle stress. In the case
of large macropodids (kangaroos and wallabies), selection for effective tendon
elastic energy savings favors higher operating stresses in the animal's
tendons (Alexander, 1988;
Bennett and Taylor, 1995
).
Stresses in the muscles of tammar wallabies determined from tendon
force-buckle recordings (Biewener and
Baudinette, 1995
) are also quite high (227 to 262 kPa at 6 m
s-1) compared with peak muscle stresses found in other species,
such as goats, domestic cats, guinea fowl and kangaroo rats, based on similar
force-buckle recordings (Table
1). Kangaroo rat stresses are low during steady speed hopping but
this allows them to increase muscle and tendon stresses dramatically when
jumping. Muscle stresses in dogs and rock wallabies are similarly high during
jumping, being in the range of peak isometric stress for skeletal muscle.
Muscle stresses calculated to act in horses during steady speed trotting and
galloping also reach high levels, which likely reflects their role in favoring
tendon energy savings. It would be interesting to know how these stress levels
compare with those developed when horses jump.
Before discussing biomechanical constraints on tendon design it is worth
noting the important influence of size on the rate of muscle contraction and
speed of limb movement. The rate of muscle shortening scales inversely with
size (W-0.18; Rome
et al., 1990
) in a manner that is generally inversely related to
limb length. This corresponds to the fact that larger animals move their limb
more slowly than small animals to move at a given speed. As a result, when
compared at equivalent speeds (trot-gallop transition speed, preferred
trotting and galloping speeds) Heglund and Taylor
(1988
) found that stride
frequency scales
W-0.15, close to that predicted by
the scaling of maximum shortening velocity. The higher stride frequencies,
shorter ground contact times and faster muscle contraction rates of smaller
animals results in the use of more metabolic energy to generate a given amount
of force to support their body weight
(Kram and Taylor, 1990
), which
helps to explain their greater cost of transport compared with larger animals
(Taylor et al., 1982
;
Taylor et al., 1970
). As noted
before, because of the scaling of limb EMA
(Fig. 3;
Biewener, 1989
) quadrupeds of
different size generally recruit the same volume of muscle per unit body
weight, so changes in force do not account for the scaling of metabolic rate
with speed and size.
Scale effects on tendon design and function
A structure's safety factor and its probability of failure depend not only
on the average magnitude of loading to which it is subjected, but also on how
variable such loads are over a period of use
(Alexander, 1981). For a given
median operating stress, a structure's risk of failure increases with more
variable loading. As a result, structures subjected to more variable loads of
a given median magnitude may be expected to have greater safety factors than
structures subjected to more predictable loads. Whereas bones may experience a
broad range of loads resulting from shifting limb orientation and muscle
contractile patterns, the loads that tendons bear are limited by the strength
of their muscles. Tendons simply needn't bear forces greater than the maximum
their muscles can produce. This would suggest that tendons might operate with
particularly low safety factors.
However, in a study of muscle-tendon architecture in different mammalian
species, Ker et al. (1988)
found that most mammalian tendons appear to be built with safety factors that
likely exceed 8 or more, based on an estimated peak isometric force that their
muscles can transmit (= 0.3 MPa isometric stress x the muscle's fiber
area, Am). Only highly specialized muscle-tendon systems,
such as those found in the digital flexors of horses and other ungulates, and
in kangaroos and wallabies, have such slender tendons that they would be
expected to operate with safety factors as low as two to three. Ker et al.
(1988
) interpreted the greater
than expected thickness of most mammalian tendons as reflecting a need to
maintain adequate stiffness, rather than sufficient strength. The high safety
factors of most tendons suggest that minimizing in-series compliance (stretch)
of the tendon relative to the operating length range of the muscle's fibers
may be of overriding importance. Reducing tendon compliance likely favors
increased control of position and length change of the muscle-tendon unit as a
whole. Therefore, control of limb segment movement, rather than strength and
weight savings for an adequate safety factor, appears to be the critical
design of these tendons.
By contrast, the `high stress' tendons found in the distal limbs of
ungulates, canids, macropodids, and other terrestrial animals that make
effective use of elastic energy savings, must operate with low safety factors.
Indeed, the operating stresses in the tendons of wallabies and kangaroos
during steady, preferred speed hopping (range: 32-84 MPa,
Table 1;
Alexander and Vernon, 1975;
Bennett, 1987
;
Biewener and Baudinette, 1995
;
Ker et al., 1986
) are
generally high. This prevents these animals from achieving the factorial
increase in tendon stress that smaller animals, such as heteromyid kangaroo
rats, benefit from when jumping or accelerating to escape predation
(Biewener and Blickhan, 1988
).
Rapid acceleration simply is not feasible or safe for animals that have
evolved slender tendons for achieving effective elastic energy storage and
return at preferred speeds of movement. The strong positive allometry of
energy savings in macropodids
W1.73
(Bennett and Taylor, 1995
),
which results (in part) from a size-invariant low ankle EMA, is evidence of
this.
These results make clear that the design of tendons in terms of failure
strength and loading predictability for ensuring a minimum, but adequate,
safety factor is the exception rather than the rule. Instead, most tendons
appear to have higher than expected safety factors, reflecting their need to
be stiff enough for muscular control of position and movement. Other tendons
specialized for elastic savings operate at much lower safety factors, but this
likely limits the factorial increase in locomotor performance that these
animals can achieve. As a result, peak tendon stresses vary much more
(>20-fold) than stresses developed in muscles
(Table 1) and, in contrast to
peak muscle stresses, they exhibit a significant change with differences in
muscle-tendon architecture (Fig.
5). Studies of the scaling of the Achilles tendon in mammals
(Peterson et al., 1984;
Pollock and Shadwick, 1994
)
indicate that Achilles tendon energy storage is also scale-dependent
(
W1.28) in quadrupedal mammals, but not as strong as
in macropodids (Bennett and Taylor,
1995
).
Fatigue damage in tendons and bones
Skeletal support tissues are subjected to repeated loading as a consequence
of locomotor activity. Because of this, they are liable to fatigue failure.
Repeated cyclic loading can lead to fatigue damage accumulation, with the
result that a structure, or the material of which it is built, fails at a
stress that may be much lower than its failure stress for a single or very few
cycles of loading (Ashby, 1999;
Currey, 2002
). Among other
factors, the rate of damage accumulation depends on the number and frequency
of loading cycles, as well as their stress magnitude and amplitude (and
whether stresses reverse from compression to tension). Unless fatigue damage
is repaired, cyclic loading will therefore reduce a structure's safety factor
and eventually lead to its fatigue failure. However, because remodeling and
repair are common features of many biological tissues, it has been generally
believed that, except for extreme loading circumstances (such as those that
new army recruits or elite athletes experience), failure resulting from
fatigue damage may not be a significant feature of the mechanical design of
most organisms.
Carter and colleagues (Caler and
Carter, 1989; Carter and
Caler, 1983
; Carter and Hayes,
1976
) carried out early work on the fatigue properties of
mammalian compact bone and found that compact bone from various sources,
tested under different loading conditions, generally fails from creep at
faster rates when loaded at higher stresses, and when subjected to cyclic
loading fails at >106 loading cycles for stress amplitudes in
the range of 50-100 MPa. (Currey,
2002
) discusses these data and speculates that a migrating animal
might sustain this number of loading cycles over a period of month, but at
much lower stress levels than those that operate during fast locomotion. Given
that damage accumulation is likely to be repaired, it would appear that under
only the most strenuous physical conditions is fatigue failure a problem for
maintaining the structural integrity of a bone. It remains unclear if wild
populations of animals undergoing natural activity patterns incur skeletal
fatigue fractures.
In recent work Ker and colleagues (Ker
et al., 2000; Wang and Ker,
1995
; Wang et al.,
1995
) have examined the fatigue rupture of different types of
tendons, based on cyclic loading and creep rupture under in vitro
conditions (Fig. 6). They found
that tendons fail after a surprisingly few loading cycles (40,000 or less) at
test stresses (40 MPa) well below the failure stress of tendon (
100-120
MPa for a single loading cycle). When subjected to a constant stress that was
50% or less of their failure stress, the tendons also ruptured over fairly
short periods of time. When `low stress' tendons (e.g. tail and digital
extensor tendons) of a wallaby were compared with `high stress' tendons (e.g.
plantaris and digital flexor tendons), Ker et al. found that the `low stress'
tendons failed over much shorter times (as low as 1 h or less) at a given test
stress (50 MPa) than the `high stress' tendons (up to 22 h;
Fig. 6B). However, when each
tendon was subjected to its `stress-in-life' (calculated based on the peak
isometric force that the tendon's muscle would transmit to the tendon = 0.3
MPa x Am/At; see
Table 1), Ker et al. found that
the different tendons all failed at generally the same time:
5.5 h under
a constant creep load (Fig.
6C). Despite the apparently high safety factors of most tendons
(Ker et al., 1986
), the
surprisingly short fatigue life of the different tendons suggests that the
tendons, like bones, must undergo ongoing repair of damage that accumulates as
a result of cyclic loading during locomotor activity. Otherwise, fatigue
failure may result. It seems probable that under in vivo conditions,
with the tendon cells alive and the local matrix biochemistry intact, damage
would not accumulate at a rate as rapid as the in vitro tests
suggest. Nevertheless, these dramatic results indicate that fatigue damage
likely affects the safety factors of tendons and bones, and needs to be
considered in relation to skeletal structures are loaded over longer periods
of use.
|
Given the occurrence of fatigue damage, it seems likely that scaling
effects on longevity that influence lifetime loading histories may affect the
relative importance of design for fatigue failure in different sized animals.
Many factors influence a species' lifespan, but in general larger animals live
longer than small animals, with lifespan generally scaling
W1/4 (Calder,
1984
; and see Speakman p. 1717, this issue). This suggests that
fatigue damage accumulation may be increasingly prevalent in larger,
longer-lived animals. Small animals, such as rodents, with less determinant
growth may also benefit from continued growth and modeling of their skeletal
tissues, with the consequence that any damage resulting from repeated loading
might be removed as an ongoing feature of growth. Because of this, larger
animals may operate with smaller safety factors than small animals when
fatigue damage over lifetime use is considered. However, the increased
longevity of larger animals might be offset by the fact that they take fewer
strides over a lifetime of activity (reflecting their longer limbs and lower
stride frequency). Similar to the general scale invariancy of the number of
lifetime heart beats and breaths, animals of different size may also, on
average, take the same number of total steps in a lifetime, offsetting the
possible size effect on fatigue failure.
One way to test this might be to compare the relative fracture incidence in
natural populations of small versus large animals, with the
prediction that smaller animals would suffer a lower incidence of limb bone
fractures. Unfortunately, other than for certain primates, including humans,
horses, and a few other vertebrate species, such data are difficult to obtain
(Alexander, 1981), and other
factors that affect safety factors (e.g. loading variability) are also are
likely to complicate these comparisons and the interpretations that may be
drawn from them.
Acknowledgments
Much of the author's work reflects the efforts and abilities of his students and colleagues, which has been supported by grants from the NIH (AR-047679) and NSF (IBN-9306793).
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