How does the relative wall thickness of human femora follow the biomechanical optima? An experimental study on mummies
1 Department of Anthropology, Hungarian Natural History Museum, H-1083
Budapest, Ludovika tér 2, Hungary,
2 Department of Biological Physics, Eötvös University, H-1117
Budapest, Pázmány Péter sétány 1,
Hungary
3 Department of Anatomy and Histology, Faculty of Veterinary Science, Szent
István University, H-1078 Budapest, István u. 2,
Hungary
* Author for correspondence (e-mail: gh{at}arago.elte.hu)
Accepted 22 December 2004
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Summary |
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Key words: marrow-filled tubular bones, optimum bone-wall thickness, human femora, mummies, bone mechanics
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Introduction |
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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
m3, the density of yellow (fatty) marrow is about 930 kg
m3 (Ashman,
1989
; Currey,
2002
), suggesting that Q ranges between 0.440.55.
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 (Kmean) and standard
deviation (
K) of K of leg bones within a
species. The first species in which Kmean and
K of a given bone type was measured is the red fox,
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 Kmin=0.59 and
Kmax=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.
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Materials and methods |
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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 07 years; `infans II' 814 years; `juvenis' 1517 years; `adultus' 1839 years; `maturus' 4059 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 020 years (20 individuals; number of femora: Nfemale=18, Nmale=15); (2) adults, age 2150 years (14 individuals; number of femora: Nfemale=18, Nmale=10); (3) old people, age above 50 years (23 individuals; number of femora: Nfemale=16, Nmale=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 marrow to
bone density
bone by
Q=
marrow/
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:
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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) |
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Results |
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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=(Kleft+Kright)/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).
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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 (Kfemale=0.560) and male (Kmale=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
Kmean of subadult femora should differ significantly from
that of adult and old-people femora, but this was not observed.
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Discussion |
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(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 Kopt, so selection will not be acting strongly on the value of Kopt.
(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
Kmean±
K=0.68±0.036,
Kmin=0.59 and Kmax=0.74
(Bernáth et al., 2004
).
Using the same method as Bernáth et al.
(2004
), in this work we
established that
,
,
Kmin=0.379 and Kmax=0.783 of adult
human femora (Fig. 3,
Table 2). The lack of sex-,
age- and length-specific as well as rightleft 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=1K. 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 (07 years), three to the infans II age-group (814 years), and two 15 year old females to the juvenis age-group (1517 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
1.441.94-times higher than that of adult fox femora
(
K=0.036). The maximal difference in K of adult
human femora is
K=KmaxKmin=0.404,
which is 2.7-times as high as
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 (KY, KS,
KF, KI;
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
Kopt may be best described as a range
±
K of values around Kopt. However,
at present there is no reliable estimation of the range
±
K encompassed by the flat portions of each
m(K) curve.
The biological relevance of optimization of the relative wall thickness W=1K 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.
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Acknowledgments |
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References |
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