## SUMMARY

Although there is much data available on mammalian long-bone allometry, a theory explaining these data is still lacking. We show that bending and axial compression are the relevant loading modes and elucidate why the elastic similarity model failed to explain the experimental data. Our analysis provides scaling relations connecting bone diameter and length to the axial and transverse components of the force, in good agreement with experimental data. The model also accounts for other important features of long-bone allometry.

## Introduction

Mammalian long-bone allometry is commonly discussed in terms of the
allometric exponents *d* and *l* that relate bone diameter
*D* and length *L* to body mass *M via* the power laws:
(1)
where *M* varies over 6 orders of magnitude. McMahon's proposal, known
as the `elastic similarity model' (ESM), that Euler buckling is the constraint
determining the scaling of long-bone geometry, as well as other structural and
physiological variables (McMahon,
1973,
1975a), has set the direction
of much subsequent work. Although some experimental support was found in
ungulates and antelopes (McMahon,
1975b; Alexander,
1977), the predicted exponents are not in agreement with larger
data sets embracing a broader range of masses
(Alexander et al., 1979b;
Biewener, 1983a; Christiansen,
1999a,b;
Polk et al., 2000). While the
limitation of the elastic similarity model is well documented
(Alexander et al., 1979b;
Biewener, 1983a;
Economos, 1983;
Castiella and Casinos, 1990;
Christiansen,
1999a,b;
Currey, 2002), the physical
grounds for this remain unknown.

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*∝*D*^{2/3}, the deflection at the center δ
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
*L*^{1/2}, if δ/*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.

## Theory

In the project of a structure, engineers must know the physical properties
of the constituent materials and the forces which each part will endure. This
enables the calculation of the dimensions necessary to resist the applied
stresses. *In vivo* stresses in bone cannot exceed yield stresses,
since this leads to irreversible deformations. Indeed, several investigators
(Biewener, 1989,
1990,
1991;
Lanyon et al., 1979;
Biewener and Taylor, 1986)
have shown that maximum stresses *in vivo* maintain a safety factor to
yield of about three to four.

The compressive stress σ_{c} acting on a beam under pure
axial compression is:
(2)
where *A* is the cross-sectional area and *F*_{ax} is
the axial force. On the other hand, a transverse force *F*_{t}
produces a bending stress σ_{b} given by:
(3)
where *r* is the moment arm of the force, *y* is the distance
from the neutral plane of bending to the specified point and *I* is the
second moment of area. For a hollow cylinder of inner diameter
*d*_{inner}=*KD*, where 0<*K*<1,
*A*=(1–*K*^{2})π*D*^{2}/4 and
*I*=(1–*K*^{4})π*D*^{4}/64
(Gere and Timoshenko,
2000).

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)
(4)
where *E* is the elasticity modulus of the material. In Results and
Discussion, we perform some calculations in order to determine which of these
failure modes is relevant to long-bone allometry.

## The model

Our argument begins along the lines proposed by McMahon
(1973,
1975a), namely: (a) a long
bone can be described as a cylinder of length *L* and diameter
*D*; (b) long-bone allometry is determined by the elastic forces the
bone must bear; (c) mechanical properties such as elasticity modulus
(*E*) and tension- and compression-yield stresses (σ_{tens
yield} and σ_{comp yield}) are body-mass-independent.
Rather than focusing solely on elastic instability (buckling), our model is
based on the further hypotheses: (d) although the loading pattern of a long
bone is complex, there are only two modes relevant to mammalian long-bone
allometry: compression stress σ_{c} and bending stressσ
_{b}, caused, respectively, by an axial force
*F*_{ax} and a transverse force *F*_{t} (see
Fig. 1); (e) maximum tensile
and compressive stresses *in vivo*, which normally occur in bone's
midshaft during locomotion at top velocity, jumping, acceleration and other
strenuous activities, maintain a safety factor (*S*_{f}) to
yield stresses that are body-mass-independent. Euler buckling is avoided by
the same safety factor; (f) the ratio
*K*=*d*_{inner}/*D* is also
body-mass-independent.

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*∝*D*^{2} and
*I*∝*A*^{2}∝*D*^{4}. Indeed,
using the data of Selker and Carter
(1989), we find that
*A*∝*D*^{1.98} and
*I*∝*A*^{1.98}. In addition, Biewener
(1982) reports that
*I*∝*A*^{1.99}. (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.)

## Results and Discussion

### Euler buckling vs. yield stresses: which is the failure mechanism of mammalian long bones?

We consider a cylindrical beam loaded as in
Fig. 1. The beam can fail in
two different ways: it will be permanently deformed as soon as yield stresses
are reached, and, if the beam is gracile enough, it will buckle before the
yield limit. The critical ratio (*L*/*D*)_{cr}
separating these failure regimes can be estimated as follows. Adopting the
convention that tensile stresses are positive and compressive stresses
negative, the total stress at point A (Fig.
1) is given byσ
_{A}=–σ_{c}–σ_{b},
where σ_{c} is given in Equation 2 andσ
_{b}∝*F*_{t}*LD/I* as given in
Equation 3. Similarly, the total stress at point B isσ
_{B}=–σ_{c}+σ_{b}. We
consider here only the maximum values of the stresses developed in long bones,
so that our assumption (e) implies that, at maximum loading,σ
_{A}=σ_{comp yield}/*S*_{f} andσ
_{B}=σ_{tens yield}/*S*_{f}.
Defining *c*=σ_{tens yield}/σ_{comp yield},
it follows (note that σ_{comp yield} and *c* are negative
with these definitions) that:
(5)
(6)
and
(7)
Using the experimental value *c*=(128±11
MPa)/(–180±13 MPa)=–0.71±0.11
(Currey, 2002), this result
implies that the bending component (σ_{b}) accounts for
approximately (1–*c*)/2=86±6% of the maximum compressive
stress (σ_{A}) on the bone. To show this, note that|σ
_{b}/σ_{A}|=(1–*c*)/2=0.855±0.055,
while|σ
_{c}/σ_{A}|=(1+*c*)/2=
0.145±0.055. This prediction is in excellent agreement with rosette
strain gauge data for tibia, which show that σ_{b} represents
84.4% of the total stress in dogs and 83.5% in horses during locomotion
(Rubin and Lanyon, 1982), and
with the values for buffalo (81%) and elephant (89%) obtained through analyses
of films of galloping animals (Alexander et
al., 1979a).

The maximum axial force
is found substituting
in Equation 7, which provides
.
Since σ_{A}=σ_{comp yield}/*S*_{f},
the maximum axial force acting in bone is given by:
(8)
In hypothesis (e), we assume that damage due to buckling is prevented by the
same safety factor. Thus, the maximum axial force acceptable is:
(9)

We define the dimensionless parameter *f* as:
(10)
i.e. the ratio of the axial component of force when yield stress are reached
to the axial force that causes buckling. For *f<*1, we have
,
so that yield stresses are reached before buckling occurs and bone fails due
to undesirable permanent deformations. On the other hand, if *f*>1,
bone buckles before the yield limit. Therefore *f=*1 is the boundary
that separates these two failure regimes.

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:
(11)
The experimental values of the above parameters are *E*=22±5
GPa, σ_{comp yield}=–180±13 MPa,
*c=*–0.71±0.11
(Currey, 2002). Thus we find
that Euler buckling is avoided provided that
*I*/*AL*^{2}>1.2×10^{–4}.
Unfortunately experimental reports are usually limited to bone length
*L* and diameter *D*, and seldom provide cross-sectional area
*A* and second moment of area *I*. Exceptionally, Selker and
Carter (1989) list *A,
I* and *L* for 40 long bones of 12 species of artiodactyls. In
their data, there is no bone in the buckling regime, and the minimum value of
*I*/*AL*^{2} is 3.3×10^{–4}, which
is 2.75 times larger than the boundary value. It is worth noting that Biewener
(1982) also reports direct
measurements of *A* and *I*. Nevertheless, since *L* is
not given in this study, Equation 11 could not be used to determine if those
bones are liable to Euler buckling.

We have seen in the Theory section that, for a hollow cylinder of inner
diameter *d*_{inner}=*KD*, we have
*A*=(1–*K*^{2})π*D*^{2}/4 and
*I*=(1–*K*^{4})π*D*^{4}/64.
Substituting these values in Equation 11, we find that the `critical'
*L*/*D* ratio is:
(12)
which corresponds to *f=*1. Substituting the experimental result
*K*=0.57±0.08 and the mechanical properties of bone related in
the previous paragraph, we find that
(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*, σ_{tens
yield} and σ_{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 *F*_{ax} and a transverse component
*F*_{t}. 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
*a*_{x}≠*a*_{t}. The exponents
*a*_{x} and *a*_{t} 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*∝*D*^{2}∝*M*^{2d}, we find
the scaling relation 2*d*=*a*_{x} [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
3*d*–*l*=*a*_{t}. Therefore the scaling
exponents for non-gracile bones are:
(14)

In order to estimate *d* and *l*, we use the experimental
values of *a*_{x} and *a*_{t}. Although McMahon
assumed that
*F*_{buckling}=*F*_{ax}∝*M*
(*a*_{x}=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 (*F*_{muscle}) and ground
reaction force (*F*_{ground}):
*F*_{muscle}*r*=*F*_{ground}*R*,
where *r* and *R* are the moment arms defined in
Fig. 2. The forces exerted in a
bone can then be written as
*F*_{ax}=*F*_{muscle}cos(α_{m})+*F*_{ground}cos(α_{g})
and
*F*_{t}=–*F*_{muscle}sin(α_{m})+
*F*_{ground}sin(α_{g}), where α_{m}
and α_{g} are measured with respect to the bone longitudinal
axis. Since muscle forces are almost parallel to the bone axis
(α_{m}≤10°) and
*F*_{muscle}≫*F*_{ground}cos(α_{g}),
because cos(α_{g})<1 and, in general,
*r*<*R*, we assume
*F*_{ax}∝*F*_{muscle} and
*F*_{t}∝*F*_{ground}sin(α_{g}).

The scaling of *F*_{ax} 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
*F*_{muscle}∝*A*_{muscle}∝*M*^{a}.
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
*M*^{0.80} (individual muscle exponents range from 0.65 to
0.92). Second, measuring the effective mechanical advantage
(EMA=*r*/*R*∝*M*^{0.26}) and using his
previous result that *F*_{ground}∝*M* in small
mammals, Biewener (1989)
reported that *F*_{muscle}∝*M*^{0.74} and
predicted maximum muscle stress to scale as *M*^{–0.06},
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 *a*_{x} that we are aware of was
given by Rubin and Lanyon
(1984) and, although based in
a small sample (5 species), provides a value (*a*_{x}=0.69)
consistent with the scaling of muscle force. These results allow us to
predict:
(15)
since *a*_{x}≈0.74. This is in good agreement with
experimental values, as shown in Tables
1 and
2. Notice that even if we
choose the highest (*a*_{x}≈0.80) or the lowest
(*a*_{x}≈0.69) estimates for the allometric exponent of
axial force, the predicted value for *d*, namely
0.69≤2*d*≤0.80, is still in the experimental range.

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 (*D*^{2}*L*∝*M*),
Christiansen (2002) has
recently shown that bone mass scales with slight positive allometry (on
average, bone mass scales as *M*^{1.06} using the rma method).
Indeed, the assumption *D*^{2}*L*∝*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 (*M*_{skeletal}). 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
*M*_{skeletal}=0.061*M*^{1.02},
*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 *M*_{skeletal}=0.04225*M*^{1.0143},
*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 *a*_{x}. By contrast, the
exponent *a*_{t} is more difficult to evaluate because it
depends on the scaling of ground reaction force (*F*_{ground})
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
*a*_{t} only during top speed locomotion, since we did not find
any experimental data for the scaling of *F*_{ground} 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 *a*_{t} 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*/β, where β is the duty factor (fraction
of the stride during which a foot touches the ground). Alexander et al.
(1977) reported that in
ungulates, β∝*M*^{–0.11}
(*r*β=0.79) for the fore feet andβ∝
*M*^{–0.14}
(*r*_{β}*=*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 α_{g} is constant; then *a*_{t}≈0.84 for
these animals. This estimate implies that
*l*=3*d*–*a*_{t}≈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 α_{g} diminishes with increasing body mass
(α_{g}∝*M*^{–0.07} in small mammals at
the trot–gallop transition speed;
Biewener, 1983a). As already
mentioned, Biewener reported that
*F*_{ground}∝*M*^{1.0} in this group at top
galloping speed. Thus, considering that α_{g} scales at top
velocity in the same manner as at the trot–gallop transition speed, one
predicts that *a*_{t}≈0.93 in small mammals. This result,
however, does not agree with the experimental data. The calculation of
*a*_{t} for small mammals needs further study, as discussed
below.

Selker and Carter (1989)
found that *a*_{t}=3*d*–*l*≈0.80 using
their data for bone dimensions of artiodactyla, and Biewener's of mammals.
Knowing that muscle force scales approximately as *M*^{0.80},
they concluded that the transverse component of force is proportional to
muscle force (*F*_{t}∝*F*_{muscle}).
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 (*F*_{t}) is proportional to ground reaction
force, not muscle force. Moreover, if we accept
*F*_{t}∝*F*_{muscle}, we would conclude
that *a*_{x}=*a*_{t} 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
*a*_{t} experimentally, as Rubin and Lanyon
(1984) did for
*a*_{x}, and also to improve the measurement of
*a*_{x}, 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
*F*_{ax}∝*F*_{muscle} and that
muscle–force 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*.

## Conclusions

In summary, we propose a model that predicts scaling exponents in agreement with experiment, and that also accounts for the other important features of mammalian long-bone allometry. Those results have not been explained by any previous model. In particular, we elucidate why McMahon's elastic similarity model is not obeyed, a long-standing puzzle in this field. Our model sets the direction for the description of avian and reptile long-bone allometry and provides a means to study the problem of terrestrial locomotion of extinct and extant species.

## ACKNOWLEDGEMENTS

We thank Professor R. McNeill Alexander for generously providing his data on long-bone allometry, Professor Andrew A. Biewener for an extensive and fruitful discussion of our work and Ronald Dickman for a careful reading of the manuscript. Funds for this work came from the Brazilian agencies CNPq (Conselho Nacional de Desenvolvimento Científico e tecnológico) and Fapemig (Fundacao de Amparo à Pesquisa do Estado de Minas Gerais).

- © The Company of Biologists Limited 2004