Tasuku KIMURA - JST

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Transcript of Tasuku KIMURA - JST

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人類誌, J. Anthrop. Soc. Nippon79(4):323-336 (1971)

Cross-Section of Human Lower Leg Bones

Viewed from Strength of Materials

Tasuku KIMURA

Department of Anthropology, Faculty of Science The University of Tokyo

Abstract The bones of the lower leg were examined from the viewpoint of strength of materials. The area, the moment of inertia of area and the polar

moment of inertia of area of the cross-section at the middle of the lower leg bones were calculated. The resistance of the bone against the normal force, against

the bending moment and against the torsion can be shown by these properties of the cross-section. The properties of the shape of the bones do not correlate with the age of the specimen. The sexual dimorphism is clear. The fibula is

very much weaker than the tibia. The index of cross-section has no direct cor- relation with the strength of bones nor with the curvature of tibia shaft.

INTRODUCTION

The mechanical strength of the long

bone has already been discussed from the

viewpoint of the strength of materials.

The long bone can be regarded as the

beam on which external forces are being

applied. The shape of the transverse cross-

section of the beam is related with the

strength of the beam. The forces acting

on the beam are mainly the normal force,

the bending moment and the torsion. The

area of the cross-section shows the resis-

tance against the normal force. The mo-

ment of inertia of the area shows the re-

sistance against the bending moment. The

polar moment of the inertia of the area

shows the resistance against the torsion.

In this study the cross-section of the

middle of the human lower leg bones was

examined. The area of the all kinds of

human long bones was studied by AOJI

et al. (1959). The report on the polar mo-

ment of inertia on the cross-section of the

bone has not been appeared except in the

paper by FRANKEL and BURSTEIN (1965).

The strongest working force on the long

bone is the bending moment as stated by

PAUWELS (1948), especially so on the tibia

(KIMURA, 1966). The moment of inertia

of area must be examined to know the

strength of the long bones. The reports

on the moment of inertia have been not

many. KNESE et al. (1954) reported on

many sections of all the long bones, but

their number of samples is small. JERN-

BERGER (1970) examined the minimum mo-

ment of inertia on many sections of five

tibiae. FRANKEL and BURSTEIN (1965)

calculated the moment of inertia on the

tibia at the mid-shaft and at the fractured

sites. I read the unpublished data on the

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324 T. KIMURA

long bones and compared them with mine

through the courtesy of Dr. Hiromi SUZU-

KI.

MATERIALS

Materials in this study consisted of

pairs of tibia and fibula of seventeen Jo-mon males, twenty-two recent Japanese

males and twenty recent Japanese females

(Table 1). All the bones, so far as is known, appeared to be normal. All recent

samples were left side. The right and left

bone were mixed in the Jomon samples.

The recent Japanese skeletons were ex-

amined through the courtesy of the De-

partment of Anatomy, Faculty of Medi-

cine, The University of Tokyo.

Table 1, Recent specimens.

The skeletons of the Jomon man (prehi-

storic food gatherer in Japan more than

two thousands years ago) were excavated

in the Honshu Island. All of them were

nearly in a perfect state of preservation.

They were stored in the Section of An-

thropology, University Museum, The Uni-

versity of Tokyo. The Jomon samples

were not taken at random. The flat tibiae,

which are rare in the recent ones, were

picked up purposely. The Jomon samples

were used as supplementary in this study.

The ages of the recent specimens were

from 40 to 82 years in the male and from

53 to 84 years in the female. It means

that these specimens were of relatively

old age.

METHODS

The bi-articular length in this study is

MARTIN'S condylo-talar length of the tibia.

This length can roughly be regarded as

the physiological length of the lower leg

and the tibia. The physiological axis pas-

ses through the centers of the upper and

lower articular surfaces of the tibia. The

frontal plane of the lower leg in this stu-

dy consists of the center of the upper

medial articular surface, the center of the

upper lateral articular surface and the

center of the lower articular surface.

The middle of the bi-articular length of

the lower leg is cut horizontally to show

the cross-section. A glass plate with a 1

mm mesh is put on this section and a

photograph is taken. The area and the

moment of inertia of the cross-section are

examined on the photograph by the nu-

merical method. Only the compact subs-

tance is regarded as the bone material in

this study. The spongy substance was

included in the marrow. The mechanical

strength of the spongy substance is much

weaker than that of the compact subs-

tance. The area of the spongy substance

is very small in the middle of the body.

It may be possible to exclude this subs-

tance in this part of the bone. The area,

A, is the area of the compact substance

in this study. The total area, Atot, means

the sum of A and the area of the marrow,

Am.

The axis parallel to the maximum an-

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Cross-Section of Human Lower Leg Bones 325

Fig. 1, Cross-section of the lower leg bones showing the axes. Sag : Sagittal axis of the lower leg. Front : Frontal axis. X : X-axis of the tibia. Y : Y-axis. Max : Principal axis of the fibula in the direction of the maximum mo-ment of inertia. Min : Principal axis in the direction of the minimum moment of inertia.

O: Centroid. N: Physiological axis of the low-er leg. a: anterior side, p: posterior side. m: medial side. 1: lateral side.

tero-posterior diameter (Y-axis) is used

as the principal axis of the tibia for the

simplification of integration (Fig. 1). The

transverse axis, X-axis, forms a right an-

gle with the Y-axis. The nearly maximum moment of inertia with respect to the X-

axis, Ix, is in the direction of Y-axis. The

nearly minimum moment of inertia with

respect to the Y-axis, Iy, is in the direction

of X-axis. This simplification can be al-

lowed as shown by the results in this

study. The maximum and the minimum

moment of inertia of the fibula, Imax and

Imin with respect to the principal axis of

the cross-section are determined by the

numerical integration. The polar moment

of inertia, Ip, is the sum of Ix and Iy in the

tibia or of Imax and Imin in the fibula.

When the normal force (P) works on

the beam, the stress (*) will be

*=P/A A : area of the cross-section

The area shows the resistance against the

normal force.

The moment of inertia of an area from

the neutral axis z (Iz) is given by

Iz =* Ay2dA y : distance from z

when the pure bending is applied on a

beam, the maximum stress (*x)max on a

cross-section of the beam in the axial (x)

direction appears on the most outside part

of the cross-section from z ; that is,

(*x)max=Mh/I2=M/ Z

M : bending moment

h : maximum height on the cross-section

from z

Z : section modulus for z

The larger the moment of inertia or the section modulus is, the greater is the re-

sistance of the beam against the bending

moment.

The maximum shearing stress (*max) of

the circular shaft which is produced by the torsion is

Mt : torsional moment

*max=Mtd/2Ip d: diameter Ip : polar moment of inertia

The polar moment of inertia shows the

resistance of the circular shaft against the

torsion. The long bone can be considered

as the circular hollow shaft. The lower

leg bones, however, are not exactly the

circular shaft. The problem of the torsion

of the non-circular shaft is complicated,

due to the warping of the cross-section.

The polar moment of inertia of the lower

leg bones will show a rough standard of

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the strength against the torsion. Further

details of the strength of the beam can be

found in the textbook dealing with the

strength of materials.

RESULTS

The personal records of the specimen

and the data on the tibia and fibula at

the middle of the bi-articular length of

the lower leg are shown in Tables 1 to 3.

The correlation coefficients between two

properties of them are shown in Table 4

and Fig. 2. The data on each the recent

specimen are shown in Appendices 1 to 3.

At first, the influence of the sampling

must be commented on. There is no sig-

nificant correlation between the index of

the cross-section and the area of the tibia.

Table 2. Tibia.

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Cross-Section of Human Lower Leg Bones 327

Table 3, Fibula.

The moment of inertia and the polar mo-

ment of inertia have almost no correlation

with the index (Fig. 2-1). For these a-

bove reasons, discussions will be made also

on the Jomon specimen. The recent speci-

mens in this study were of old age. The

age of the specimen, however, is correlated

significantly neither with the area, with

the moment of inertia (Fig. 2-2) nor with

the polar moment of inertia.

The bones of the female have a small

area, moment of inertia and the polar mo-

ment of inertia compared with the bones

of the male (Fig. 2). The difference of the

mean area between the male and the

female is greater than the standard devi-

ation of each in case of both the tibia and

the fibula. The differences of the moment

of inertia and of the polar moment of in-

ertia between the male and the female are

also larger than the standard deviations

of each.

The fibula has a very small area, mo-

ment of inertia and polar moment of in-

ertia compared with the tibia. The mean

of the area of the fibula is less than thirty

percent of that of the tibia. The mean

moment of inertia and the mean polar

moment of inertia are less than ten per-

cent. The standard deviations of the mo-

ment of inertia and polar moment of in-

ertia of the fibula are relatively larger

than those of the tibia.

The bi-articular length of the tibia cor-

relates with the maximum moment of in-

ertia and the polar moment of inertia.

The length correlates with the area of the

tibia when the sexes are not considered.

The stature and the body weight correlate

with the area, the moment of inertia and

the polar moment of inertia of the tibia

and the fibula in case of not considering

the sexes. When considering by sexes,

they do not correlate well especially in

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Table 4. Correlation coefficients.

case of the female.

The moment of inertia of area correlates

with the diameters in both the tibia and

fibula. The index of the cross-section at

the middle of the tibia correlates with Iy/

Ix. The index of the cross-section of the

fibula correlates with Imin/Imax.

The tibia with a wide area has also a

large moment of inertia or polar moment

of inertia (Fig. 2-3). The ratio of the ef-

fective area, A divided by Atot, becomes

large when the area become wide (Fig. 2-

4). The area of the marrow shows almost

no correlation with the total area.

The area of the tibia significantly cor-

relates with that of the fibula. But the

moment of inertia of the tibia and that of

the fibula do not show a good correlation.

The polar moment of inertia of the tibia

correlates with that of the fibula.

The index of the curvature of the tibia

using the centroid, the index being the

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Cross-Section of Human Lower Leg Bones 329

Fig. 2. Scatter-diagramm showing the correlations.

(2-1) Between Ix and the index at mid-shaft of the tibia.

(2-2) Between the age of specimens and A of the tibia.

(2-3) Between Ix and A of the tibia.

(2-4) Between A and the ratio of the effective area of the tibia.

distance from the centroid of the cross-

section to the physiological axis divided

by the bi-articular length, shows no sig-

nificant correlation with the index of the

cross-section. The centroid of the tibia

in this section is placed forward of the

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330 T. KIMURA

physiological axis and about 1mm back- ward of the mid-point of the maximum

antero-posterior diameter.

The principal axis on the cross-section

of the tibia diverges from the Y-axis only

from eight degrees laterally to ten degree

medially at the maximum. The difference

between Ix and Imax of the tibia is less

than two percent and between Iy and Imin

is less than six percent in this study.

The principal axis diverges from the sa-

gittal axis from 18 to 42 degrees laterally.

DISCUSSION

The area of the tibia and the fibula re-

ported by AOJI et al. (1959) is the average

of the right and left bones. The area of

the tibia reported by them is slightly

smaller than that of the present study.

The moment of inertia of the European

bones (KNESE et al., 1954; JERNBERGER,

1970) is rather large than that of the Ja-

panese in this study.

AOJI et al. (1959) believed that the

sectional area diminishes after sixty years

in the female and eighty years in the male.

The age of the specimens does not corre-

late with the area, with the moment of

inertia nor with the polar moment of in-

ertia in the present study, though the

specimens are rather old ones. It is im-

possible to find the correlation between the area of the bones and the age in the

data by Aoji et al. which include the

specimens of the Japanese in the twenties

and thirties. YAMADA (1970, p. 20, 255)

reported that the mechanical properties of

bones decrease in the old age group. The

present study is concerned only with the

mass and shape of the bone and the me-

chanical properties are not being con-

sidered.

Compared with the data reported by the

Nutrition Section, Ministry of Health and

Welfare in 1967, the mean stature of the

present recent samples is slightly lower

within the same range of age, and the

body weight is much lower. Since none

of the recent specimens died accidentally,

they may have suffered from the disease

for some period of time. This may be one

of the reasons why the body size does not

show a good correlation with the area nor

with the moment of inertia. On the other

hand, the length of the tibia is not affec-

ted by the disease and may show to some

extent the body mass of the specimen.

The long bone shows a large resistance

especially against the bending. The bone

which is strong against the normal force

is also strong against bending and torsion.

The area of the compact bone becomes

very wide when the total area becomes

wide. The bone with a wide area has the

thick compact substance. In other words,

the shape of the cross-section is not simi-

lar in the bones with a wide and a small

area. The marrow of the bone with a

wide total area is not necessarily large.

The sexual dimorphism in the cross-

section of the long bones is very clear.

The female has very weak bones compared

with the male, though considering her

small body size. The male bone is not

only large in size externally, but also has

a thick compact substance.

The centroid of the cross-section at the

mid-shaft of the tibia is situated forward

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Cross-Section of Human Lower Leg Bones 331

of the physiological axis and backward of

the mid-point of the maximum antero-

posterior diameter. When the compressive

normal force is applied to the physio-

logical axis, the load on the cross-section

at the mid-shaft becomes eccentric. The

entire body of the tibia can be seen as a

curved column as discussed by KIMURA

(1966). If the posterior part of the bone stretches out and forms a buttress, the

curvature of the shaft can become small.

The index of the cross-section, however,

has no correlation with the curvature.

The flatness of the tibia does not increase

the strength against the normal force.

The significance of the tibia shaft has

been discussed by many investigators. As

shown by the correlation coefficient in

Table 4 and in Fig. 2-1, the flat tibia is not

absolutely strong. The index of the cross-

section of the tibia shows the ratio of the

strength in the antero-posterior direction

to that in the transverse direction. The

index of the cross-section of the fibula also

shows the ratio of the strength between

the principal axes. The bending force

acting on the tibia is mainly in the antero-

posterior direction because of the move-

ment of the muscles and articulation. The

strength in the antero-posterior direction

will be more important than that in the

transverse direction. On this point, the

flatness would be regarded as a suitable

form of the tibia.

The strength of the tibia correlates with

that of the fibula. The strength of fibula

is not related closely with the body size.

The variation of the properties of the fi-

bula is relatively large as seen by the

standard deviation. One of the reasons for

this variation will be that the fibula does

not bear a large portion of the strength

of the lower leg. The area and the mo-

ment of inertia of the fibula are much

smaller than those of the tibia. The tibio-

fibular connections are not rigid ones.

The forces acting on the lower leg may be

sustained mainly by the tibia. It is not

possible to consider the size and shape of

the cross-section of the fibula only from

the mechanical viewpoint of the fibula

alone.

The strength against the pure bending

is shown by the section modulus. The

maximum section modulus of the mid-

shaft of the tibia appears at the posterior

side which is in the direction of the

maximum moment of inertia. It is because

the anterior side is pointed and has a large

height from the neutral axis compared

with the posterior side. The minimum

section modulus is at the lateral side

where it is high because of the existance

of the crista interossea. Usually the lar-

ger the moment of inertia is, the greater

is the section modulus on both sides of

the axis. For simplification, in this study

the moment of inertia is used to show the

resistance against the bending.

The bone is heterogeneous and aniso-

tropic. An experimental study is necce-

sary to clarify the mechanical properties

of the bone. The strength of the bone,

however, could be shown to a certain deg-

ree by the shape of the cross-section in

this study.

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332 T. KIMURA

SUMMARY

The area, the moment of inertia of area

and the polar moment of inertia of area

of the cross-section at the middle of the

human lower leg bones were examined to

learn about the strength of the bone from

the viewpoint of the strength of materials.

The specimens were obtained from twenty-

two recent Japanese males, twenty recent

Japanese females and seventeen Jomon ma-les.

The results are shown in Tables 1 to 4

and Fig. 2 and summarized as follows :

1) The age of the specimens does not cor-

relate with the area, with the moment of

inertia nor with the polar moment of in-

ertia ; 2) The bones of the female have

distinctly a small area, moment of inertia

and polar moment of inertia compared

with the bones of the male; 3) There will

be a tendency that a large body mass is

associated with a large strength of the

bone ; 4) The shape of the cross-section

with a wide area is not similar to that

with a small area ; 5) The strength of the

fibula is much smaller than that of the

tibia and the deviation of the properties

of the fibula is greater than that of the

tibia ; 6) The index of the cross-section

at the middle has no direct correlation

with the strength of the bones nor with

the curvature of the tibia, and the index

of the tibia shows a ratio of strength be-

tween the antero-posterior and transverse

directions ; and 7) The principal axis at

the mid-shaft of the tibia is parallel to

the maximum antero-posterior diameter.

ACKNOWLEDGMENT

This study is indebted to Professor Ta-

dahiro OOE, Associate Professor Ichiyoh

ASAMI and Dr. Toshiro KAMIYA of the

Department of Anatomy, Faculty of Me-

dicine, the University of Tokyo for allow-

ing and assisting me to examine the recent

specimens. Thanks are also expressed to

Mr. Hisao BABA and Mr. Yasuo FUKUSHIMA

of the Department of Anthropology, Facul-

ty of Science, the University of Tokyo for

their assistance of this study.

REFERENCES

AOJI, O., MOTOJIMA, T, and BANDO, T., 1959:

On the effective sectional areas and maximum compressive loads of diaphysis of human long

bones. J. Kyoto Pref. Med. Univ., 65: 979-983

(Japanese with English summary). FRANKEL, V. H, and BURSTEIN, A.H., 1965;

Load capacity of tubular bone. Biomechanics

and Related Bio-Engineering Topics (Ed. R.

M. KEN E DI), pp. 381-396. Oxford.

JERNBERGER, A., 1970: Mesurement of stability of tibial fractures. Acta Orthop. Scand., Sup-

pl. No.135. KIMURA, T., 1966: An experimental study of

the form of the human tibia from the bio-

mechanical point of view. J. Anthrop. Soc.

Nippon., 74: 119-227. KNESE. K-H., HAHNE, 0. H, and BIERMANN, H.,

1954: Festigkeitsuntersuchungen an mensch-

lichen Extremitatenknochen. Gegenbauers

Morph., 96: 141-209.

厚生省公衆衛 生局栄養課(編),1969:国 民栄養の現

状,昭 和42年 度 国民栄養調査成績.東 京.

(Nutrition Section, Public Health Bureau, Ministry of Health and Welfare, The report

of the national nutrition survey in Japan,

1967. Tokyo. In Japanese)

PAUWELS, F., 1948: Die Bedeutung der Baup-

rinzipien des Stutz- and Bewegungsapparates

fur die Beanspruchung der Rohrenknochen.

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Cross-Section of Human Lower Leg Bones 333

Z. Anat. Entwickl. Gesch., 114: 129-166.

*, 1950: Die Bedutung der Muskelkrafte

fur die .Regelung der Beanspruchung des

Rohrenknochens wahrend der Bewegung der

Glieder. Z. Anat. Entwickl. Gesch., 115: 327-

351.

YAMADA, H., 1970: Strength of biological ma-

terials (Ed. F. G. EVANS). Baltimore.

(Received June 3, 1971)

材 料 力 学 的 に 見 た ヒ トの 下 腿 骨 横 断 面

木 村 賛

東京大学理学部人類学教室

長骨の強 さは断面形 において材料力学的にお しはか ることができる.長 骨 に加 わる外力は主 として軸力,曲

げモー メ ン ト,〓 りの三つで ある.断 面の面積の大 きさは軸力 に対 す る抵抗 の大 きさを示す.断 面二次 モーメ

ントは曲げに対す る抵抗 の大 きさを ほぼ示 している.断 面二次極モーメ ン トは丸軸 の〓 りに対す る抵抗 の大 き

さを示す.

本研究では現代 日本人男性22側,女 性20側,縄 文時代人 男性17側 の下腿骨中央 断面において面積,断 面二次

モー メン ト,断 面二次極モー メン トの三種の数値が調べ られた.骨 の持つ機械的性質の違いについて は考慮せ

ず,緻 密質部分の形 状によってわか る強さのみが論 じられている.骨 の力学的性質の解明には実験 的研 究が不

可 欠で ある.

断面 の形状に関 して下記の結果が得 られ た.1)資 料の年令 は下腿骨の断面 積,断 面 二次モーメ ン ト,断 面

二次極 モー メン トの大 きさのいずれ とも相関がない.2)女 性骨 は男性骨 と比較 し三種 の 数値がすべて非 常に

小 さい. 3)体 の大きさ と骨 の丈夫 さとには関係があ るよ うで ある.4)断 面積 の大 きさが異な る骨 の断面 の

形状 は相似形でない.大 きな骨の緻密質は厚 くなる.5)腓 骨 は脛骨 と比べて三種の数値が著 しく小 さ く,そ

のば らつきの程度が比較 的大であ る.6)断 面係数 は骨 の強 さと直接の相関はない.脛骨 の弯 曲と も相関がない.

断面係 数は前後方向及び横方 向の 断面二次モーメ ン トの比 と相関 している.7)脛 骨中央断面 の 主軸は最大前

後径 と平行で ある.

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334 T. KIMURA

Appendix 1. Recent Japanese specimens.

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Cross-Section of Human Lower Leg Bones 335

Appendix 2. Tibia of recent Japanese.

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336 T. KIMURA

Appendix 3, Fibula of recent Japanese.