On the scaling of mammalian long bones
Departamento de Física, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30123-970, Belo Horizonte MG, Brazil
* Author for correspondence (e-mail: jaff{at}fisica.ufmg.br)
Accepted 19 January 2004
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Summary |
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Key words: bone, allometry, mammals, stress, buckling, locomotion, muscle, force
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Introduction |
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![]() | (1) |
Mammals adopt several strategies to avoid the mechanical consequences of
large size. Biewener (1989,
1990
,
1991
) has shown that large
mammals keep bone stress constant through (i) a shift to a more upright
locomotor limb posture and (ii) an allometric increase in the moment arm of
antigravity muscles. Those artifices decrease joint moments relative to the
magnitude of ground forces, thus reducing mass-specific forces acting on
bones. It has also been realized that large mammals do not possess the same
locomotor agility of smaller ones, which is probably associated with reduced
bone loading and the maintenance of similar safety factors
(Biewener, 1991
; Christiansen,
1999a
,b
).
Nevertheless, buckling can suddenly occur even if stress levels are kept at
a safe margin. Euler buckling is an elastic instability that occurs when the
axial force acting in a rod overcomes a certain threshold. In this paper we
show that mammalian long bones are not slender enough to buckle, and that
long-bone allometry is governed by the need to resist bending and compressive
stresses. We propose a model, based on the requirement to maintain safety
factors to yield, which predicts scaling exponents in agreement with data and
elucidates various aspects of long-bone allometry, such as differential
allometry. Our work, in addition to papers by West et al.
(1997,
1999
), shows that allometric
laws in biology can be understood on the basis of the interplay between
geometric and physical constraints.
Note that, although the ESM was formalized in terms of end-loaded columns
that may fail in Euler buckling, McMahon
(1975a) derived the same
scaling relations for a beam subject to pure bending. He considered a rod
supported on its extremities and subject to bending by a force proportional to
its weight, and showed that if different-sized columns maintain
L
D2/3, the deflection at the center
divided by the length L is kept constant
(McMahon, 1975a
). In this
sense, the ESM holds that elastic deflections of long bones are self similar
across different sizes. This second derivation of the elastic similarity
scaling, however, is not consistent with the experimental observation that
maximum stresses in mammalian long bones are body-mass-independent (Biewener,
1989
,
1990
,
1991
), since the beam
described above will be submitted to stresses proportional to
L1/2, if
/L is kept constant. As Currey
(2002
) states, `McMahon's
basic idea was that organisms are designed so that the deflections they
undergo are what is controlled, not the stresses they bear'. Since this
derivation of the ESM is not in agreement with experiment, the hypothesis that
remains to be tested is the possibility of Euler buckling.
Currey (2002) investigated
this possibility. His analysis indicates that certain long bones are liable to
buckling if highly loaded in compression. However, Currey considered solely
axial compression, not taking into account that mammalian long bones are
subject to a high degree of bending
(Biewener, 1991
; Rubin and
Lanyon, 1982
,
1984
). As mentioned above, we
show that, under axial compression plus bending, mammalian long bones are not
slender enough to be vulnerable to Euler buckling.
It is important to observe that, besides Euler buckling, a cylindrical
beam, such as a long bone, may also fail due to local buckling. This is
characterized by deformation of a small part rather than the deformation of
the whole structure, which is what happens in Euler buckling. It occurs when
the walls are so thin relative to the diameter that the shape of the structure
does not support the wall sufficiently to prevent it from bending in an easy
direction (see Currey, 2002).
Currey and Alexander (1985
)
investigated the possibility that mammalian and avian long bones failed in
local buckling. They found that the ratio R/t of midradius
of the wall (R) to thickness (t) in mammalian long bones is
on average 2.0, which is far below the threshold (R/t=14)
above which long bones would be liable to local buckling.
The balance of this paper is organized as follows. In the next section we provide the mathematical expressions which will be used in our stress analysis. The hypotheses of our model are then presented (The model). In Results and Discussion, we explore the consequences of those hypotheses and compare our predictions with reported experimental values. Finally, we draw our conclusions in the last section.
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Theory |
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The compressive stress c acting on a beam under pure
axial compression is:
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![]() | (3) |
A different failure mechanism that must also be avoided is the elastic
instability known as Euler buckling. This occurs when the axial force applied
to a pillar overcomes a certain threshold. For a biarticulated beam, this
threshold is given by the Euler estimate
(Gere and Timoshenko, 2000)
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The model |
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Our hypotheses are all supported by experimental data. Assumption (c)
agrees with measurements suggesting that bone material properties are
size-independent (Biewener,
1982,
1991
). Hypothesis (d) is
supported by in vivo measurements, which show that, in most cases,
bending is the main loading mode of long bones and that the principal stresses
are almost parallel to the bone longitudinal axis
(Biewener, 1991
; Rubin and
Lanyon, 1982
,
1984
). Assumption (e) is
corroborated by the experimental observation that maximum tensile and
compressive stresses measured in vivo are approximately 1/3 the bone
tensile- and compressive-yield stresses and occur in the midshaft (Biewener,
1989
,
1990
,
1991
;
Lanyon et al., 1979
;
Biewener and Taylor, 1986
).
Hypothesis (f) is confirmed in various experimental reports. Currey and
Alexander (1985
) have made a
large compilation of values of K for mammals. Analysing these data,
we find that K does not correlate with body mass and that its average
value is 0.57±0.08. Moreover, if K is a constant, we expect to
find A
D2 and
I
A2
D4. Indeed,
using the data of Selker and Carter
(1989
), we find that
A
D1.98 and
I
A1.98. In addition, Biewener
(1982
) reports that
I
A1.99. (These scaling relations for
A and I were calculated by least squares regression. If
reduced major axis (rma) analysis were used, no significant differences would
have arisen since the correlation coefficients were always above 0.98.)
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Results and Discussion |
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The maximum axial force
is found substituting
in Equation 7, which provides
.
Since
A=
comp yield/Sf,
the maximum axial force acting in bone is given by:
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We define the dimensionless parameter f as:
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We will now determine if mammalian long bones are in the region
f<1, where yield stress is the primary concern, or in the interval
f>1, for which buckling is the real threat. Substituting Equations
8 and 9 in 10, we find:
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We have seen in the Theory section that, for a hollow cylinder of inner
diameter dinner=KD, we have
A=(1K2)D2/4 and
I=(1K4)
D4/64.
Substituting these values in Equation 11, we find that the `critical'
L/D ratio is:
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![]() | (13) |
We can now understand why the elastic similarity model fails to explain the
experimental data. We have analyzed a large amount of data available in the
literature (Alexander et al.,
1979b; Biewener,
1983a
; Bertram and Biewener,
1992
; Christiansen,
1999b
) and found that long bones seldom have
L/D>26. Femura, humerii and tibiae are never more slender
than L/D=26. Only two of a total of 117 radii are more
slender than (L/D)cr. On the other hand, ulnae and fibulae
are found to exceed this limit often (27 in a group of 68 ulnae examined, and
35 fibulae in a total of 47 exceeded L/D=26). This, however,
does not necessarily imply that Euler buckling determines the allometry of
those bones; it probably simply reflects their non-load-bearing condition in
some animals (Christiansen,
1999a
,b
).
It is important to note that the uncertainty in the value of
(L/D)cr is quite large as a consequence of the variation
observed experimentally in the physical (E, tens
yield and
comp yield) and geometrical (K)
properties of bone. Nevertheless, the discussion above is still correct even
if we choose the smallest estimate for (L/D)cr, namely,
(L/D)cr=18.
Determining the scaling exponents d and l
Let us derive the allometric exponents d and l defined in
Equation 1. As shown in Fig. 1,
we describe the resultant force acting on half-bone by two components, namely
an axial component Fax and a transverse component
Ft. There is no a priori reason to assume that,
at maximum loading, the components
and
are proportional to each
other. Therefore, we consider that each component scales with its own
allometric exponent, i.e.
and
.
Below, we show that generally
ax
at. The exponents
ax and at will be deduced from
experimental data on the scaling of muscle force, ground reaction force and
direct measurements of the forces acting on a long bone.
We now show how the scale-invariance of bone mechanical properties, safety
factor and ratio K lead to the power-law dependence of bone
dimensions on body mass (Equation 1). For f<1, the bone fails when
the maximum stresses reach the yield limit. Since yield stresses and safety
factors are body-mass-independent [assumptions (c) and (e)], equation 8
implies that .
Substituting
A
D2
M2d, we find
the scaling relation 2d=ax [here we have used
assumption (f)].
Equation 3 implies that the maximum transverse force acting on a bone is
.
From Equation 7 we have that the maximum bending stress
in bone is
, which is
body-mass-independent. Consequently
and, since
,
we have our second scaling relation, which is
3dl=at. Therefore the scaling
exponents for non-gracile bones are:
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In order to estimate d and l, we use the experimental
values of ax and at. Although McMahon
assumed that
Fbuckling=FaxM
(ax=1) (McMahon,
1973
,
1975a
), the loading situation
of a long bone is not so simple. The usual procedure
(Alexander, 1974
;
Alexander et al., 1979a
;
Biewener, 1983b
) to evaluate
the forces acting on bones using force platform recordings is to equate the
moments exerted by muscle force (Fmuscle) and ground
reaction force (Fground):
Fmuscler=FgroundR,
where r and R are the moment arms defined in
Fig. 2. The forces exerted in a
bone can then be written as
Fax=Fmusclecos(
m)+Fgroundcos(
g)
and
Ft=Fmusclesin(
m)+
Fgroundsin(
g), where
m
and
g are measured with respect to the bone longitudinal
axis. Since muscle forces are almost parallel to the bone axis
(
m
10°) and
Fmuscle>>Fgroundcos(
g),
because cos(
g)<1 and, in general,
r<R, we assume
Fax
Fmuscle and
Ft
Fgroundsin(
g).
|
The scaling of Fax is determined in three different
ways. First, it appears that maximum muscle stress is approximately
independent of body mass (Schmidt-Nielsen,
1990), which implies that muscle force is proportional to muscle
area, so that
Fmuscle
Amuscle
Ma.
We have collected and calculated averages of muscle-area allometric exponents
from several sources. The results are as follows: a=0.77 for
antelopes (Alexander, 1977
),
a=0.83 for insectivores and rodents
(Castiella and Casinos, 1990
),
a=0.78 for rodents (Druzinsky,
1993
) and a=0.80 and 0.81 for mammals as a whole
(Alexander et al., 1981
;
Pollock and Shadwick, 1994
).
We note that mammals of very different body masses, such as rodents and
antelopes, exhibit similar behavior, with muscle area scaling on average as
M0.80 (individual muscle exponents range from 0.65 to
0.92). Second, measuring the effective mechanical advantage
(EMA=r/R
M0.26) and using his
previous result that Fground
M in small
mammals, Biewener (1989
)
reported that Fmuscle
M0.74 and
predicted maximum muscle stress to scale as M0.06,
a prediction that has yet to be confirmed. (Note that this result is
consistent with the scaling of muscle force and area mentioned above). Third,
the only direct estimate of ax that we are aware of was
given by Rubin and Lanyon
(1984
) and, although based in
a small sample (5 species), provides a value (ax=0.69)
consistent with the scaling of muscle force. These results allow us to
predict:
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|
|
The experimental exponents presented in
Table 1 were taken or
calculated from Christiansen
(1999b). We chose these data
for two reasons: (i) they represent the most extensive sample, and (ii)
animals with similar locomotor modes are included. (Note that Christiansen's
data has an inconvenience, namely, animals of mass <1 kg are not included.)
The agreement of the predicted value of d with the experimental
exponents reinforces that long-bone allometry is governed by the need to
resist compressive and bending stresses. Notice that the correlation
coefficients are much higher when we consider only non-gracile ulnae and
fibulae.
In contrast to the assumption of McMahon
(1973,
1975a
) that bone mass is
proportional to body mass (D2L
M),
Christiansen (2002
) has
recently shown that bone mass scales with slight positive allometry (on
average, bone mass scales as M1.06 using the rma method).
Indeed, the assumption D2L
M
together with our result d
0.37 leads to a poor prediction of the
bone length exponent (l
0.26) in comparison to the experimental
value (l
0.30) (Table
1). Therefore the positive scaling of long-bone mass, although
weak, cannot be ignored. This point has already been noted by Hokkanen
(1986
).
Here we make a digression regarding the pioneering work of Prange and
collaborators (1979) on the
scaling of mammalian skeletal mass (Mskeletal). Since
their work was published, it has been widely cited as evidence that mammalian
skeletal mass scales with positive allometry (for instance, see
Schmidt-Nielsen, 1984
). Their
data, however, is not entirely conclusive. Among the 49 mammals used in the
study, only the elephant has a body mass above 70 kg. Moreover, it seems that
man and dog have skeletal masses above the values expected for their body
masses. Fitting their data for the 44 mammals with masses less than 12 kg
using the least-square regression method, we find
Mskeletal=0.061M1.02,
r=0.993. (Note that rma analysis would not change this result
appreciably due to the high correlation coefficient.) In agreement with this
result, Bou and Casinos (1985
)
found that Mskeletal=0.04225M1.0143,
r=0.993, in insectivores and rodents. Therefore, experimental data
indicates that skeletal mass is proportional to body mass for mammals smaller
than 12 kg. It is necessary to collect more data in the gap between 67 kg
(man) and 6600 kg (elephant) in order to obtain a more reliable equation for
the whole group of mammals. Finally, we note that different bones scale with
different allometric exponents. While long-bone masses scale with significant
positive allometry (Bou and Casinos,
1985
; Christiansen,
2002
), the masses of other bones, such as the skull, scale with
significant negative allometry (Bou and
Casinos, 1985
).
It was relatively easy to estimate ax. By contrast, the
exponent at is more difficult to evaluate because it
depends on the scaling of ground reaction force (Fground)
and experimental reports for this are scarce. As stated above (The model),
maximum stresses may occur in different activities, such as galloping at top
speed, jumping and acceleration. Here we evaluate the exponent
at only during top speed locomotion, since we did not find
any experimental data for the scaling of Fground in
jumping nor in accelerating. Nevertheless, this does not seem to be a
shortcoming, since maximum tensile stresses during top speed locomotion
maintain the same safety factor to yield that are kept by compressive stresses
(Biewener, 1989,
1990
,
1991
;
Lanyon et al., 1979
;
Biewener and Taylor, 1986
).
This means that, although the magnitude of ground reaction forces may be
larger during acceleration or jumping in comparison to top speed locomotion,
the allometric exponent at is probably the same for these
three vigorous activities.
Large ground reaction forces occur during top speed locomotion and are
known to scale as M/ß, where ß is the duty factor (fraction
of the stride during which a foot touches the ground). Alexander et al.
(1977) reported that in
ungulates, ß
M0.11
(rß=0.79) for the fore feet and
ß
M0.14
(rß=0.78) for the hind feet. When analyzing
allometric data, the least-square regression (lsr) method is not expected to
be the most appropriate, since it assumes that error is present only in the
dependent variable. Reduced major axis (rma) analysis is to be preferred
because it takes the uncertainties of both variables into account
(Christiansen,
1999a
,b
;
Sokal, 1981). Reanalysing Alexander's data using rma, we obtain the exponents
0.14 and 0.18, for fore and hind feet, respectively. Since there
is no apparent posture change in large mammals (Biewener,
1989
,
1990
), we assume that the
angle
g is constant; then at
0.84 for
these animals. This estimate implies that
l=3dat
0.27 in large mammals,
in reasonable agreement with the experimental data (see
Table 2). On the other hand,
small mammals change posture from a crouched to a more upright position
(Biewener, 1989
,
1990
), and consequently the
angle
g diminishes with increasing body mass
(
g
M0.07 in small mammals at
the trotgallop transition speed;
Biewener, 1983a
). As already
mentioned, Biewener reported that
Fground
M1.0 in this group at top
galloping speed. Thus, considering that
g scales at top
velocity in the same manner as at the trotgallop transition speed, one
predicts that at
0.93 in small mammals. This result,
however, does not agree with the experimental data. The calculation of
at for small mammals needs further study, as discussed
below.
Selker and Carter (1989)
found that at=3dl
0.80 using
their data for bone dimensions of artiodactyla, and Biewener's of mammals.
Knowing that muscle force scales approximately as M0.80,
they concluded that the transverse component of force is proportional to
muscle force (Ft
Fmuscle).
However, this conclusion is in contrast with the widely accepted analysis
(Alexander, 1974
;
Alexander et al., 1979a
;
Biewener, 1983b
) of the
loading situation in legs, which led us to the conclusion that the transverse
component of force (Ft) is proportional to ground reaction
force, not muscle force. Moreover, if we accept
Ft
Fmuscle, we would conclude
that ax=at and so
l=d
0.37. Although this is a reasonable result for small
mammals (see Table 2), large
mammals do not follow this relation. In order to solve this puzzle, more data
are needed on the scaling of maximum muscle stress, bone mechanical properties
and duty factor to confirm if they are mass-independent or if they exhibit a
small, but relevant, variation with size. It is also very important to measure
at experimentally, as Rubin and Lanyon
(1984
) did for
ax, and also to improve the measurement of
ax, presently based on strain data for only five species
(see discussion above). We recognize that those experiments are difficult,
because rosette strain gauges can only be used to record strains from bones of
a certain size very small bones cannot be studied in this way.
Nevertheless, the arguments presented here show that, in order to completely
describe long-bone allometry, one needs to determine which are the forces
applied on bone and their scaling with body mass.
Finally, the model presented here accounts for two further important
aspects of bone allometry not explained by McMahon's elastic similarity
(McMahon, 1973,
1975a
). First, it has been
realized that long-bone allometry exhibits different scaling regimes for small
and large mammals (Table 2) and
that this should be related to a posture change found mainly in small mammals
and to the reduced locomotor performance of large mammals
(Economos, 1983
; Biewener,
1989
,
1990
;
Bertram and Biewener, 1990
;
Christiansen,
1999a
,b
).
Our model confirms this distinction between regimes by coupling the allometric
exponents with ground reaction forces, and angles of force to bone, both of
which are body-mass dependent. (Note that this coupling makes it possible to
study the forces involved in the locomotion of extinct species, such as
dinosaurs, using bone-allometry data.) Second, Christiansen reported that
large mammals develop progressively shorter limb bones as a means of
reducing bending stress, rather than proportionally thicker bones
(Christiansen, 1999b
). This
fact is a direct consequence of our analysis. We have shown that
Fax
Fmuscle and that
muscleforce allometry does not distinguish small and large mammals.
Thus Equation 14 implies that d must have similar values for all
mammals and, therefore, differential scaling can only appear in differences of
l.
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Conclusions |
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Acknowledgments |
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