## SUMMARY

We studied how the ratio (*K*) of the internal:external diameter of
human femora follows the biomechanical optima derived earlier by other
researchers for marrow-filled tubular bones with circular cross section and
minimum mass designed to withstand yield and fatigue, or stiffness, or bending
fracture, or impact strengths. With evaluation of radiographs of 107 femora
from 57 human mummies the values of *K* were measured. We found that
*K*_{posterior}=0.498±0.085 for the posterior
radiographic view, and *K*_{medial}=0.589±0.070 for the
medial view with *K*_{min}=0.345 and
*K*_{max}=0.783. The theoretical optima for *K* depend
on the ratio (*Q*) of the marrow:bone density. Accepting the assumption
of earlier authors that *Q*=0.50, our data show that human femora are
optimised to withstand bending fracture, or yield and fatigue strengths. There
were no sex-, age- and length-specific differences in *K*, and the
means of *K* of the right and left femora of individuals were
statistically not significantly different. The biomechanical optimization for
*K* of human femora is not finely tuned. Compared with fox femora,
*K* of human femora follows the biomechanical optimum to a much lesser
extent. Although the relative wall thickness *W*=1–*K* of
human femora are optimised, the very low relative mass increment due to
deviation of *K* from the optimum and the considerable intraspecific
variance of *K* make it probable that an accurate optimization of the
relative wall thickness is irrelevant in humans.

## Introduction

The diaphysis of human femora is a hollow tube with a nearly circular cross
section filled with marrow. One of its characteristic variables is the ratio
(*K*) of the internal:external diameter. Pauwels
(1980), Currey
(1982), Alexander
(1968,
1982,
1983,
1996) and Currey and Alexander
(1985) derived different
biomechanical optima for *K* of tubular bones with a circular cross
section. The optimum value of *K*, which allows the minimization of the
mass of a marrow-filled bone, depends on whether the bone is selected
principally for yield and fatigue strength, for ultimate strength, for impact
strength, or for stiffness. Currey and Alexander
(1985) showed that the change
in mass as a function of relative wall thickness *W*=1–*K*
was quite gradual, suggesting that natural selection would not act strongly
against relative wall thicknesses that were not very close to the optimum.

Legs have to be accelerated and decelerated in every step. Optimal
*K*-values for leg bones will allow bones to have sufficiently thick
walls to maintain mechanical integrity, while remaining sufficiently thin so
as to moderate the energetic costs of limb acceleration
(Pauwels, 1980;
Currey, 1982; Alexander,
1968,
1982,
1983,
1996;
Currey and Alexander, 1985;
Lieberman et al., 2003).
According to the biomechanical optimization theory of Alexander
(1982), Currey
(1982), Currey and Alexander
(1985), the optima for
*K* depend on the ratio *Q* of the marrow to bone density (see
Equations 1,
2,
3,
4 in the Materials and methods of
the present work). Unfortunately, the exact values of *Q* are unknown.
The density of human cortical bone ranges from 1700 to 2100 kg
m^{–3}, the density of yellow (fatty) marrow is about 930 kg
m^{–3} (Ashman,
1989; Currey,
2002), suggesting that *Q* ranges between 0.44–0.55.
Alexander (1982,
1996) assumed
*Q*=0.50.

To test their biomechanical optimization theory, Alexander
(1982), Currey
(1982) and Currey and
Alexander (1985) surveyed the
*K*-values of 240 long bones from single individuals of 70 species.
They found that the interspecific variation of *K* was high, most
*K*-values ranged from 0.4 to 0.8, and there was a general
correspondence between theoretical predictions and real life. In general, they
examined only one or two bones from any species, and therefore had no estimate
of within-species variation. To say something about the force of selection, it
was necessary to determine the mean (*K*_{mean}) and standard
deviation (σ* _{K}*) of

*K*of leg bones within a species. The first species in which

*K*

_{mean}andσ

*of a given bone type was measured is the red fox,*

_{K}*Vulpes vulpes*. With evaluation of radiographs of 62 femora of adult foxes, Bernáth et al. (2004) found that in fox femora

*K*=0.68±0.036 with

*K*

_{min}=0.59 and

*K*

_{max}=0.74. Accepting the assumption of earlier authors that

*Q*=0.50, Bernáth et al. (2004) found that the fox femora are optimised for stiffness. The mass increment, μ, relative to the minimum mass of fox femora was smaller than 5% under all four mentioned mechanical conditions for

*Q*=0.50. Currey (2002) has argued that such small differences are selectively important.

According to Alexander
(1982,
1983,
1996), the long bones of
mammals are optimum structures. Until now this hypothesis have been thoroughly
tested only in the case of fox femora
(Bernáth et al., 2004).
The aim of this work is to understand whether the relative wall thickness of
femora in humans (which may be subject to natural selection to a smaller
extent than wild animals) corresponds to a biomechanical optimum. In spite of
the intense study of human bones and bone mechanics (e.g.
Ruff and Hayes, 1983; Cowin,
1989,
2001;
Runestad et al., 1993;
Ohman, 1993;
Stock and Pfeiffer, 2001;
Currey, 2002), this problem
has not yet been investigated. In this work we present an experimental study
on *K* of femora of human mummies. With evaluation of the radiographs
of 107 human femora we measured the mean and standard deviation of *K*.
The measured *K*-values were compared with the four theoretical optima
for *K* derived by Currey and Alexander
(1985).

We chose mummy femora because they were easily available in large numbers
from the Anthropology Department of the Hungarian Natural History Museum. We
studied femora because, in humans, the femora have the most circular mid-shaft
cross section (Cubo and Casinos,
1998). Since the theoretical optima for *K* were derived by
Currey and Alexander (1985)
for circular cross sections of marrow-filled tubular bones, the femur is the
most appropriate bone to test the optimality of *K* in human long
bones. Since both sex and age of the investigated mummified persons were
known, we could investigate the possible dependence of *K* of human
femora on sex and age. For reason of duty towards the dead, femora of recent
dead persons could not be investigated. Conversely, the radiographs of femora
of living persons available from hospitals were not of appropriate quality for
our evaluation. Furthermore, the radiographs obtained from hospitals showed
anatomical changes (e.g. fractures, cracks, fissures, or pathological
alterations) and partly that is why they were inappropriate for our
biomechanical analysis.

Finally, we would like to emphasize that our major aim was only to test experimentally whether human femoral wall thickness matches one (or several) theoretical optima. Any speculation about bone adaptation governed by the loading conditions in human femora is beyond the scope of this work.

## Materials and methods

### Mummy femora and their evaluation

The investigated human femora originated from the mummy collection of the Department of Anthropology of the Hungarian Natural History Museum. The individuals came from the Dominican Church, Vác, Hungary, and were buried during 1731–1838. Although most of them were naturally mummified, approximately 30% of them were skeletonised. Contemporary written records of the parish register of the church are available for many individuals and include date of death, age, sex and name (Pap et al., 1997).

We examined 57 specimens (28 females, 29 males). Contemporary archives
enabled us to determine the age at death in the case of 48 individuals. In
anthropology, the standard way of classifying ages is the so-called Martin
method: `infans I' age-group 0–7 years; `infans II' 8–14 years;
`juvenis' 15–17 years; `adultus' 18–39 years; `maturus'
40–59 years; `senium' above 59 years. Fifteen (7 females, 8 males) of
the investigated mummies belonged to the infans I age-group, three (2 females,
1 male) to the infans II, and two 15-year old females to the juvenis. The
distribution of grown-ups was: three females belonging to the adultus
age-group, 14 specimens (8 females, 6 males) to the maturus, and 11 (4
females, 7 males) to the senium. For nine individuals (2 females, 7 males) age
records were not available. Their age at the time of death was estimated using
standard anthropological methods (1 adultus female, 5 adultus males, 1 senium
female, 2 senium males). For statistical analyses the original Martin
age-groups were drawn into the following three age-groups: (1) subadults, age
0–20 years (20 individuals; number of femora:
*N*_{female}=18, *N*_{male}=15); (2) adults, age
21–50 years (14 individuals; number of femora:
*N*_{female}=18, *N*_{male}=10); (3) old people,
age above 50 years (23 individuals; number of femora:
*N*_{female}=16, *N*_{male}=30).

To avoid the difficult transport of whole mummies and to minimize their damage, we tended to select skeletonised bodies, from which the femora could be separated. Taking radiographs from such detached femora was much easier. Both left and right femora of individuals were examined, if it was possible.

Detachable femora were individually packed and transported to the Department and Clinic of Surgery and Ophthalmology of the Faculty of Veterinary Science of the Szent István University in Budapest, where lateromedial and anteroposterior radiographs were taken from every femur using EUREKA Diamond 150 (CEA OGA, green sensitive). After chemical development, the radiographs were digitized using an AGFA Arcus 1200 scanner with a resolution of 400 dpi. The evaluation of the radiographs for the majority of the investigated human femora was as described in detail by Horváth (2001) and Bernáth et al. (2004). Our method is partly similar to the evaluation procedure of computer tomographs used by Spoor et al. (1993) to determine the thickness of human enamel and cortical bone. Biplanar radiographs are commonly used to obtain dimensions of limb bones: Ruff and Hayes (1983), Runestad et al. (1993), Ohman (1993) and Stock and Pfeiffer (2001), for example, have developed and applied such a technique.

After the evaluation we obtained the ratio *K* of the internal to
external diameter of the bone at the selected mid-section for both the
lateromedial and anteroposterior radiographic views. The reliability of this
method was tested by comparison of computationally obtained *K*-values
with data measured directly by a caliper on bone cross sections. Our method
based on the evaluation of radiographs of tubular bones can measure the
*K*-value with an accuracy of ±1%
(Bernáth et al.,
2004).

Because some mummies had residual marrow or exhibited porous bone
structure, the automatic evaluation of some bones was impossible. In these
cases the following modification of the evaluation was necessary. The selected
rectangular area on each radiograph (see the areas demarcated by white line in
Fig. 1) was divided into five
small rectangular horizontal zones. In each zone, lines were fitted to the
inner and outer bone walls visually and manually. The computer program
determined the distance between the appropriate lines in each row of the zone
and calculated the *K*-value for the zone. The final *K*-value
was calculated as the arithmetical mean of the *K*-values of the five
zones. This method was compared with the automatic procedure on bones suitable
for both kinds of evaluation. The differences were very small, and not biased
in a particular direction. We could not evaluate the medial radiograph of a
few subadult and old-people femora, because these radiographs were so
contrast-poor (usually due to osteoporosis) that the bone walls could not be
recognized computationally or visually.

To examine the differences between the measured *K*-values and the
four theoretical optima for *K*, a two-tailed single *t*-test
was used. The possible correlation between *K* and the bone length
*L* was tested by calculating Pearson correlation coefficients for the
*L*- and *K*-values obtained for the anteroposterior and
lateromedial views. The difference between the mean *K*-values obtained
for the anteroposterior and lateromedial views of the femora was confirmed
using associated two-tailed paired *t*-test. Since the theoretical
optima for *K* were derived by Currey and Alexander
(1985) for circular cross
sections of marrow-filled tubular bones, further statistical analysis was
performed using the average of the *K*-values obtained for the
anteroposterior and lateromedial views of the femora. A few incomplete bones
with missing epiphyses were excluded from these tests. The difference between
the *K*-values of the left and right femora of individuals was examined
using two-tailed paired *t*-test. The possible differences between the
*K*-values of femora of women and men were tested using a two-tailed
unassociated *t*-test. To avoid pseudoreplication, only single femora
of individuals were involved in the statistics. The possible differences
between the *K*-values measured in the three age groups were tested
using one-way ANOVA. Statistical tests were performed with the statistical
software StatSoft STATISTICA 6.1.

*Optima for* K *of marrow-filled tubular bones with
given* Q

Let us designate the ratio of the marrow density ρ_{marrow} to
bone density ρ_{bone} by
*Q*=ρ_{marrow}/ρ_{bone}. If the cross section
of the diaphysis remains approximately circular when a marrow-filled tubular
bone is bent, the biomechanical optima for the ratio *K* of the
internal to external diameter of the diaphysis under different mechanical
strengths are the following (Currey and
Alexander, 1985; Bernáth
et al., 2004).

**Stiffness.** The optimum value for stiffness is:
1

**Yield and fatigue.** The optimum value for *K* for a bone of
minimum mass for yield strength and fatigue strength is
2

**Impact.** The optimum for impact loading is:
3

**Bending fracture.** If the bone is strong enough not to fracture,
under the greatest bending moments likely to act on it, the optimum
*K*-value is:
4
These expressions were derived from the equations described by Currey and
Alexander (1985). If the
*K*-value of a marrow-filled tubular bone is equal to one of these four
optima, the total mass of bone and marrow is minimal under the above-mentioned
mechanical conditions. *Q*=0.50 was assumed by Alexander
(1982,
1996).
Table 1 contains the optimum
*K*-values calculated from Equations
1,
2,
3,
4 for *Q*=0.50.

## Results

Fig. 2 shows the frequency
of the *K*-values of all investigated human femora from posterior and
medial views. Fig. 3 represents
the frequency of the *K*-values of the femora of adults for both views.
Table 2 contains the means
*K*_{mean}, standard deviations σ_{K}, minima
*K*_{min} and maxima *K*_{max} of the femora of
subadults, adults and old people.
of (all
investigated) human femora for posterior view is significantly smaller than
for medial
view (paired *t*-test: *t*=–8.93; d.f.=35;
*P<*0.001). The difference between
and
is smallest for
subadult femora (Δ*K*_{mean}=0.036) and greatest for
adult femora (Δ*K*_{mean}=0.127). The standard deviationσ
_{K} of *K* is 0.085 and 0.070 for posterior and medial
views, respectively. Depending on the age and radiographic view,σ
_{K} ranges between 0.052 and 0.093, and *K* changes
between the extrema *K*_{min}=0.345 and
*K*_{max}=0.783. Due to this variation of *K*, within
the sample of human femora, there were several individuals that exhibited
*K*-values that were identical to each of the different theoretical
optima *K*_{Y}, *K*_{S}, *K*_{F},
*K*_{I} given in Table
1.
is nearest
to *K*_{F}=0.50 (optimum for bending fracture load), while
is nearest to
*K*_{Y}=0.63 (optimum for yield and fatigue strength).
does not
differ significantly only from *K*_{F}=0.50 (single sample
*t*-test: *t*=–1.11263, d.f.=47, *P=*0.272).
is nearest to
*K*_{Y}=0.63, but it differs significantly from
*K*_{Y} (single sample *t*-test:
*t*=–3.40844, d.f.=36, *P=*0.00162).
differs from
*K*_{F}, *K*_{S}, *K*_{I} with
significance levels lower by several orders of magnitude. Hence, the
investigated human femora seem to be optimised either for bending fracture
load or yield and fatigue strengths.

To reveal a possible difference in *K* between the left and right
femora, we selected those mummies, in which both the left and the right femora
could be investigated. Table 3
contains the mean, standard deviation, minimum and maximum of *K* of
these femur pairs, for which the average
*K*=(*K*_{left}+*K*_{right})/2 of the
*K*-value of the left and right femur was calculated. Using paired
*t*-test, we found that the means of *K* of the right and left
femora of individuals were not significantly different (paired
*t*-test: *t*=0.961; d.f.=35; *P=*0.343).

To test a possible difference in *K* of femora of female and male
persons only the left or the right femur of a given individual was used. The
means of *K* of female (*K*_{female}=0.560) and male
(*K*_{male}=0.536) femora were not significantly different
(*t*-test for independent samples: *t*=1.053, d.f.=34,
*P=*0.299). Hence, we could not establish a sex-specific difference in
*K*. Similarly, the means of *K* of subadult, adult and
old-people femora were not significantly different [one-way ANOVA:
*SS*=0.0112, *MS*=0.0056, *F*(2, 33)=1.525,
*P=*0.233]. Note the higher *K*-value in the posterior view of
the subadult femora (0.549 for separate femora, and 0.541 for femur pairs)
compared with that of the adult (0.462, 0.463) and old-people (0.485, 0.483)
femora (Tables 2 and
3). In our opinion, this
statistically non-significant difference between the subadult femora and the
older ones is functionally not significant.

To test whether *K* is influenced by the bone length *L*, we
investigated the correlation between them. We obtained that neither
, nor
depends on
*L* (Pearson correlation between
and *L*:
*N*=33, *r*=0.019, *P=*0.917; while between
and *L*:
*N*=36, *r*=–0.18, *P=*0.28). This was expected,
because there were no age-specific differences in *K*. Among the
investigated bones only the subadult femora differed significantly in length.
If *K* were influenced by *L*, the mean
*K*_{mean} of subadult femora should differ significantly from
that of adult and old-people femora, but this was not observed.

## Discussion

According to Currey and Alexander
(1985), long bones can be
solid, or very-thin walled (having values of *K* from 1 to, say, 0.1).
However, except for those having special life styles (living in water, or
having no marrow) they do not span that range, but their median is about 0.63.
The reasons for this are the following.

(1) If the central bone cavity contains marrow, there will be an optimum
value for *K* that produces a bone of minimum mass. The precise value
of the optimum depends on what mechanical situation (or combination of them)
for which the bone is optimised. The optimal value of *K* for stiffness
is larger than that for bending strength, for instance.

(2) The curves of mass *m*(*K*) as a function of *K*
are rather flat near the optimum *K*_{opt}, so selection will
not be acting strongly on the value of *K*_{opt}.

(3) The examination of actual values of *K* for land mammals and
flightless birds shows them to be roughly where one would expect them to be,
with perhaps a bias towards strength rather than stiffness. Flying bird's
bones, if anything, seem to be appropriate for stiffness rather than strength.
The values of *K* for pterosaurs, marrowless bones of birds, and
water-living vertebrates, deviate in the expected directions.

(4) This suggests that the hollowness of bones is to produce values of minimum mass for the bones.

Since the incidence of osteoporosis and osteoarthritis becomes greater and greater in human populations, bone wall thickness and bone density have become important subjects of quantitative investigations. These studies are focused on medical aims rather than on evolutionary relationships. As far as we know, human bones were not involved in interspecific comparative studies on the biomechanical optimality of the relative wall thickness of tubular bones.

The *K*-value of the femora in terrestrial mammals and flightless
birds ranges from 0.26 (*Melursus ursinus*) to 0.73 (*Sorex araneus,
Pedetes capensis, Litocranius walleri, Struthio camelus*) with a median of
about 0.63 (Currey and Alexander,
1985). The mean, standard deviation, minimum and maximum of
*K* of adult fox (*Vulpes vulpes*) femora are
*K*_{mean}±σ_{K}=0.68±0.036,
*K*_{min}=0.59 and *K*_{max}=0.74
(Bernáth et al., 2004).
Using the same method as Bernáth et al.
(2004), in this work we
established that
,
,
*K*_{min}=0.379 and *K*_{max}=0.783 of adult
human femora (Fig. 3,
Table 2). The lack of sex-,
age- and length-specific as well as right–left differences in *K*
of human femora demonstrates well how robust and general are the biomechanical
design and the structure of marrow-filled tubular bones in humans.

The major reasons for the statistically significant difference between
and
are that: (1)
the human femur is not exactly symmetrically circular; and (2) its wall
thickness is not exactly uniform. Since circular cross section and uniform
wall thickness are the prerequisites of the biomechanical optimization theory
of Currey and Alexander
(1985), the asymmetry of the
cross section of the human femur makes it difficult to test the predictions of
the theory for the optima of *K*. Until a more sophisticated theory is
developed, it is only possible to analyse human femora. However, our
conclusions remain valid in spite of the fact that the optima to which the
human *K*-values are compared are based on the assumption of circular
cross sections. Note that in comparison to other human long bones, the human
femora possess the most circular mid-shaft cross section
(Cubo and Casinos, 1998). More
detailed explanation and functional interpretation of our findings that
is significantly
smaller than could
be the task of future research.

The human femur has considerably smaller *K* than the fox femur.
Note that smaller *K* means greater relative wall thickness
*W*=1–*K*. According to Currey and Alexander
(1985), interspecific variance
of *K* can be high either because the different ways of life may demand
optimization for different mechanical loads and/or because of the biological
irrelevance of optimization of the relative wall thickness due to the too tiny
relative mass increments.

In our subadult group, 15 of the investigated mummies belonged to the infans I age-group (0–7 years), three to the infans II age-group (8–14 years), and two 15 year old females to the juvenis age-group (15–17 years). Thus, all subadult femora originated from subjects aged below 15 years, and the majority of the bones was not older than 7 years. Hence, these subadult bones were far from the borders of skeletal infancy and near-maturity, where considerable changes take place.

The standard deviation of *K* of adult human femora
(,
) is
1.44–1.94-times higher than that of adult fox femora
(σ_{K}=0.036). The maximal difference in *K* of adult
human femora isΔ
*K*=*K*_{max}–*K*_{min}=0.404,
which is 2.7-times as high as Δ*K*=0.15 of adult fox femora. This
relatively high variance in *K* in human femora explains why we could
find several human femora that had similar *K*-values to each of the
theoretical optima (*K*_{Y}, *K*_{S},
*K*_{F}, *K*_{I};
Table 1). With the assumption
of Alexander (1982,
1996) that *Q*=0.50,
from our data (
and ) we conclude
that the adult human femora are optimised to withstand bending fracture load,
or yield and fatigue strengths. By comparison, fox femora are optimised for
stiffness (Bernáth et al.,
2004).

Note that considerable deviations of *K* from the optimum value
result in only small mass increments
(Bernáth et al., 2004),
which could explain the relatively high variation of *K* in human
femora (Fig. 1, Tables
2,
3). Currey and Alexander
(1985) noted that the values
around the minima do not result in large changes in the bone mass
*m*(*K*), suggesting that each effective optimum value
*K*_{opt} may be best described as a range±Δ
*K* of values around *K*_{opt}. However,
at present there is no reliable estimation of the range±Δ
*K* encompassed by the flat portions of each
*m*(*K*) curve.

The biological relevance of optimization of the relative wall thickness
*W*=1–*K* of the diaphysis in tubular bones in a given
species should be reflected by low intraspecific variance of *K*. Since
the standard deviation of *K* in human femora is 1.44-1.94-times higher
than in fox femora, we conclude that the biomechanical optimization of
*K* in human femora is not finely tuned. Compared with fox femora,
*K* of human femora follows the biomechanical optimum to a lesser
extent. Although the relative wall thickness of the diaphysis in human femora
is optimised to withstand bending fracture load, or yield and fatigue
strengths, the very low relative mass increments due to deviation of
*K* from the optima and the relatively high intraspecific variance of
*K* make it probable that an accurate optimization of the relative wall
thickness is irrelevant in humans.

## ACKNOWLEDGEMENTS

This work was supported by the grant OTKA T-034982 from the Hungarian Science Foundation. The three-year István Széchenyi research fellowship from the Hungarian Ministry of Education to Gábor Horváth is acknowledged. We are grateful to Mária Kampó for taking and developing the radiographs. Many thanks are due to two anonymous Referees for their constructive comments.

- © The Company of Biologists Limited 2005