Various Methods for Measuring the Geometrical Properties of the Long

人類誌.J. Anthrop. Soc. Nippon 90 (1):1-16 (1982)

Various Methods for Measuring the Geometrical Properties

of the Long Bone Cross Section with Respect to Mechanics

Banri ENDO and Hideo TAKAHASHI Department of Anthropology Faculty of Science The University of Tokyo

Abstract Owing to the recent development of various kinds of equipment for the measurement of irregular figures, papers dealing with the long bone cross section related to mechanics are increasing in the field of anthropology. However, there have been few explanations of the actual methods, equipments and equations for measuring these properties. This paper explains briefly the geometrical properties of the cross section in relation to mechanics as well as their actual use in functional morphology related to anthropology and shows various methods and equations for the actual measurement of these geometrical properties. Some of the equations presented in this paper have never or rarely been published before.

for example, by KOCH (1917), PAUWELS INTRODUCTION (1950, 1954, 1968) and KIMURA (1971, 1974). Use has been made of circumference, Manual measurement, however, is very sagittal and transverse diameters, maxi- time-consuming work. Owing to the recent mum and minimum diameters etc. as the development of various kinds of equip- geometrical properties of the cross section ment for measuring irregularly shaped of long bones in historical morphology. planes, many researchers, such as MINNS These properties, however, are not availa- ct al (1976), LoVEJOY et al (1976), KIMURA ble for the analysis in biomechanics or (1980), MILLER and PURKEY (1980), BURR et functional morphology. The geometrical al (1981) and TAKAHASHI (in press), have properties of the cross section, such as begun to measure them. Nevertheless, few area, moment of inertia, position of cen- explanations of the methods and equations troid, principal axes, radius of gyration for measuring them with various equip- and section modulus are useful in asse- ments have been done, except those by ssing the complex relationship between PAUWELS (1968), by NAGURKA and HAYES structure and function of bone. (1980) and by KIMURA (1980). Moreover, The measurement of these geometrical they are rather specialized in using certain properties of the long bone cross section specific equipment. has sometimes been manually carried out, This report aims to explain briefly the Article No. 8141 geometrical properties related to mecha- 2 B. ENDO and H.TAKAHASHI

nics as well as their application to bone The closed plane has a centroid, which morphology and to show various methods is designated as C in Fig. 1. Let the Car- and equations for using different equip- tesian coordinates have their origin at the ments in order to obtain the geometrical centroid, as shown in Fig. 1. The moment properties of the long bone cross section of inertia about the x axis is defined as related to mechanics in a more generaliz- Ix = *y2dA, ed form. Some of the equations have the same moment about the y axis as never or only rarely been described in ly=*x2dA published papers and even in textbooks on strength of materials. The methods desc- and the product of inertia as ribed in this report are limited to those Ixy =*xydA.

based on the Cartesian coordinate system If the coordinates rotate, these moments which is easy to use with the various become larger or smaller. At a certain devices. angle of rotation the former two become GEOMETRICAL PROPERTIES OF simultaneously maximum and minimum THE CROSS SECTION RELATED respectively. Let them at this angle be TO MECHANICS designated as II and III, then The cross section of long bones, as shown in Fig. 1, is a kind of the closed plane. The area of the closed plane is defined as They are called principal moments of A = *dA. inertia. According to Fig. 1, they are also defined as

II=*2dA

III= *2dA

The angle thus rotated is

The rotated axes (I and II in Fig. 1) are called principal axes. The polar moment of inertia is defined as Ip =*r2dA. The r of the polar coordinate system can Fig.1. Cross section of a femoral shaft with be obtained from the Cartesian coordinate its centroid (C), coordinates through cent- system as x2*y2. Therefore, roid (x and y) and principal axes (I and II). Hatched area is integrated. Measurement of geometrical properties of bone 3 and also is fairly similar in shape. =II+IIIIp . When the external forces act on a beam, The radii of gyration about the x and reactions to cope with them occur in it. y axes are If this beam is virtually cut normally to its longitudinal axis to make a couple of The principal radii of gyration are cross sections facing one another, forces appear on each of the cross sections, also Let the maximum distance from the coupling with one another. These forces margin of the closed plane normal to the may be called the internal forces or the axis through the centroid be e. This exists cross-sectional forces. The internal forces on both sides of the axis. They are usual- are usually classified into three kinds: ly designated as e1 and e2 (see Fig. 3). The axial force (normal force), shearing force section modulus (also called the modulus and bending moment. The internal forces of resistance) is defined on each side of are the resultant of the forces continuous- the axis as ly distributed over the cross section. The force per unit area of the distributed force is called stress. The stress is either the It should be noted that the moment of normal stress or the shearing stress. inertia of the cross section may also be When a beam is acted on by the exter- defined even about any arbitrary axis which does not pass through the centroid. However, the moment of inertia takes a certain value which is proper to the cross section, when the axis passes through the centroid. Therefore, the term "moment of inertia" is generally used with respect to the axis through the centroid.

MECHANICAL SIGNIFICANCE OF THE GEOMETRICAL PROPERTIES OF THE CROSS SECTION

The relationship between geometrical

properties of the cross section and their Fig. 2. A beam acted on by three external mechanical significance is based on the forces (P1, P2. and P3) in equilibrium as a beam theory. The beam means a slender simplified model of long bone acted on by but not too long bar, the cross section of joint forces and muscular force (top) and intensity diagrams of internal forces dist ri- which is always the same or gradually buted in the beam (bottom). The equilibrium changes along its longitudinal axis. equations are P1 cos *-P2 cos *- P3cos *=0, Therefore, the beam theory is approxi- P1 sin *-P2 sin *+P3 sin *=0 and l1P2 sin * mately applicable to the long bone which P3sin *=0. l 4 B. ENDO and H. TAKAHASHI nal forces, the internal forces are distributed over the whole length of the beam in a certain manner which is related to the condition of the external forces. Fig. 2 shows an example of the internal force distribution.. As concerned with a cross section at an arbitrary position on the beam, the axial force produces the normal stresses uni- formly distributed over the cross section. Designating the axial force as P and the normal stress as *, the intensity of the normal stresses is Fig.3. Cross section of a tibia for explanation of normal stress at dA in bending about the The bending moment also produces normal principal axis (I). stresses, but their intensity is not uniform. When the axis of the bending moment (M) and the principal axis of the cross section are the same, the stress at dA is

in the case of Fig . 3. Intensity of stress depends on the distance between the axis and the dA. The maximum intensity of stress appears on each extremity of the axis as

If the axis of the bending moment is inclined counterclockwise from the principal axis at an angle of * as shown in

Fig. 4, the intensity of the stress at the dA Fig.4. Cross section of a tibia for explanation is of normal stress at dA in bending about an axis inclined from the principal axis. I and II, principal axes; *, inclination angle of the and the inclination of the neutral axis of axis of bending moment; *,inclination angle bending is of the neutral axis of bending; * and *,dis- tances of dA from the I and II axes respectively. Measurement of geometrical properties of bone 5

to the torsional force which also produces the shearing stresses, but this relation is limited to the approximately circular cross section (solid or hollow).

GEOMETRICAL PROPERTIES OF BONE CROSS SECTION IN RELATION TO FUNCTIONAL MORPHOLOGY

Bone is a part of the locomotor system. Therefore, a bone is acted on externally by joint forces and muscular forces during body movement or body support. These forces produce internal forces as already mentioned. Although the axial force is strong in the bone, the bending moment produces more intensive stresses which tend to concentrate to certain localities of bone. Thus the bending moment has usu- Fig. 5. Cross section of a tibia for explana- ally far larger influence on the strength tion of shearing stress at dA. I and II, of bone. Consequently, the moment of principal axes; b,breadth; *,height; bi and breadth and height at the level of dA; *,*i, inertia of the bone cross section plays a distance between dA and II axis; *, angle very important role in terms of the resis- between the II axis and the line from dA tance to bending. to the point at which the tangent of the KOCH (1917) was the first to calculate margin at the level of *i intersects the II axis. the stress distribution in the femur, by measuring the geometrical properties of When the axial force and the bending the cross sections from top to bottom. The moment simultaneously act, results of his calculations are shown in Fig. 6. Apparently the bending moment plays the most important role, according to his results. However, since the condition The equation of the shearing force and of the external forces seemed considerably shearing stress is more complicated, but in different from the actual condition, this case the shearing force (S) direction is kind of analysis has been continued by orthogonal to the principal axis, the she- many researchers and is still being carried aring stress * at the dA in Fig. 5 is out now. HIRSCH (1895) considered that the sagit- taly long shaped cross section of the tibia The polar moment of inertia is related is due to the strong bending moment 6 B,ENDO and H.TAKAHASHI

Fig. 7. Cross section of a tibia at the mid- point with a radial diagram of magnitude of section modulus (top) and a simplified demonstration of the same cross section showing distantly concentrated hone mass for resisting to the bending moment (bottom). S,centroid; H1 and H2,principal ales; FE, strongest direction against the bending moment; NF, neutral axis of bending through the centroid clue to the above bending moment. (From PAUWELS, 1950)

this strong bending moment and that this fact is related to the reason for the sagit- tally long cross section of the tibia, as

Fig. 6. Stress distribution (magnitude and seen in Fig.7. Along this line of investi- trajectories) in a femur loaded on the head gation, KIMURA (1974) studied extensively calculated from the geometrical properties the relationship between the geometrical of a successive series of the cross section. properties of the lower leg bones and the Numerals are in Ib/sq. in.. +,compressive; strains produced in them during walking. - , tensile. (From KOCH,1917) He indicated the adaptation of the tibial produced mainly by the soleus muscle in geometry to locomotion. He made an locomotion. PAUWELS (1950) showed that analysis of geometrical properties as the tibia had a larger moment of inertia well as an experimental stress analysis about the transverse axis to cope with and compared both the results. LOVEJOY Measurement of geometrical properties of bone 7 et al (1976) studied the biomechanical platycnemia in Paleolithic Sapiens. Ana- significance of the platycnemia and con- lyses of these problems seem to have sidered that the strengthening of the platy- already started in part. cnemia is specialized in the resistance Functional interpretations of the various to the sagittal deformation, i. e., the bend- characteristics of limb bones of the recent ing moment about the transverse axis, people having different life activities may which is assumingly produced by specific also be of interest, because they may offer patterns of locomotion which are common reliable basis of the diversity of bone in Paleolithic people. form. As for the general significance of the PRINCIPLE OF MEASUREMENT OF geometrical properties of the long bone THE GEOMETRICAL PROPERTIES cross section, PAUWELS (1948) has already stated that the bending moment has a In the case of the irregularly shaped pla- causal relationship to the bone form and ne, such as the cross section of long bones, especially to the form of the bone cross the position of the centroid is not easily section, i. e., its geometrical properties detected, unlike beam and column of the related to mechanics. He (1968) showed artificial structure. Therefore, the meas- as a proof the fact that ricket femur has urement must always start from an arbi- different geometrical properties of the trarily selected coordinate system, as cross section to cope with different bending moment due to bone deformation. According to his paper (1954), the Linea aspera of the femur is a structure also to cope with the bending moment by enlarg- ing the moment of inertia of the cross section. Based on the results and interpretations as mentioned above, many problems of the limb bones concerned with the human evolution may be solved in future by analysing the measured geometrical properties related to mechanics and by using other biomechanical analyses. For exam- Fig. 8. Cross section of a femur with arbi- ple, the peculiar features of the femur of trary coordinates and additional axis used at Australopithecine, the rather flat femur the start of uniaxial measurement to deter- with the Crista medial is and the Crista mine the cent roid, coordinates through the centroid and principal axes. X and Y, arbi- ateralis in Sinanthropus, the round l cross trary coordinates; W, additional axis; C, section of the Neanderthal limb bones, the centroid; x and y, coordinates through the formation of platolenia, platymeria and centroid; I and II,principal axes. 8 B.ENDO and H.TAKAHASHI shown in Fig. 8 and 9. where A', M' and I' are all-included mo- To begin with, there are uniaxial measu- ments, A", M" and I" are moments of rement and biaxial measurement. In the closed space, i is X, Y or W and j is the former case, the measurement is made number of closed space. about a single axis, whereas in the latter The position of the centroid (x, y) is case it is made simultaneously about two obtained as axes. In the uniaxial measurement, another axis is needed besides the coordinates. In Thus the axes of moment of inertia paral- the case of Fig. 8, the third axis W passes lel to the arbitrarily chosen axes can be through the origin at an angle of */4 to the drawn as the lines passing through the other axes (this axis may be most useful centroid as x and y in Fig. 8. for the actual measurement). Then three The moments of inertia about these sets of 0th, 1st and 2nd moments must be axes are measured, i. e., about each of the axes, as Ix=Ix-y2A (9) y=Iy-x2A (10) I Iw=Iw- w2A, (11) where w = Mw/A. The product of inertia for calculating the principal moment of inertia cannot be obtained in the uniaxial measurement, but the moment of inertia about the third axis (Iw) is available instead of it. In the case of Fig. 8 the W axis has an angle of /4 to the other axes. Therefore, *

If the closed plane has closed spaces in it, such as the medullary cavity and vessel holes of the bone cross section, and In the case of the biaxial measurement if these spaces are included in the process as shown in Fig. 9, measurements are done of measurement, these moments must be about arbitrary coordinates as separately measured and subtracted from all-included moments, i. e., Measurement of geometrical properties of bone 9

VARIOUS METHODS FOR ACTUAL MEASUREMENT

For the purpose of obtaining the cross section, the long bone is cut normally to its longitudinal axis. The cross section is copied on a sheet of paper or film either by taking a photograph or by stamping. It is also useful to take a thin slice from the bone by using a diamond cutter and to put it on a sheet of photographic paper Fig. 9. Cross section of a femur with arbi- or film for making a direct copy. Then trary coordinates used at the start of biaxial measurement to determine the centroid, arbitrary coordinates are drawn on the coordinates through the centroid and copied sheet. principal axes. X and Y, arbitrary coordi- In both the cases of uniaxial and biaxial nates; C, centroid; x and y, coordinates measurements, various methods, apparatu- through the centroid ;I and II, principal axes. ses and equations are available. One can choose any of these measurement methods according to the apparatus available in one's laboratory. (A) to (D) in Fig. 10 are methods of uniaxial measurement and (E) to (I) are methods of biaxial measurement. If the correction calculation for the closed In order to obtain principal axes and spaces is necessary, the equations (4), (5) moments using the uniaxial measurement, (6) must be used. The position of the the measurement must be repeated three centroid is also times about three axes as X, Y and W in x=MY/A (15) Fig.8. y=Mx/A. (16) Devices for the data input are roughly The moments of inertia and the product divided into four kinds, as seen in Fig, 11. of inertia about the axes through the cent When the brightness and position sensor roid and parallel to the arbitrarily chosen is used, the cross section must be distin- axes are guished from its surrounding and inner Ix=Ix-y2A (17) spaces, i, e., a binarized signal using the Iy=IY- x2A (18) brightness sensor is needed. In the case Ixy=IxY -xyA. (19) of the position sensor, tracing of the con- For obtaining the principal moments and tour of the cross section with a sensor axes the equations of (1), (2) and (3) are head (or pointer) with an automatic signal available. input at a constant interval is desirable, but the device only with manual input 10 B, ENDO and H, TAKAHASHI

Fig.10. Various methods for measuring the geometrical properties of the bone cross section related to mechanics. Measurement of geometrical properties of bone 11

Fig. 11. Various devices for measuring and various apparatuses for signal transfer and for computation as well as their various combination. switching by placing the sensor head dis- put are available. If the number is around continuously along the margin is also a hundred, even manual input to a micro- available. computer or a small programmable calc- Usually these devices put out a large ulator is possible. When the integrator or number of numerical data (words). When a measure and graph paper is used, a small the number is more than a thousand, an on- programmable calculator is sufficient. line minicomputer or an on-line microcom- The relationship between the methods puter with a large capacity is necessary. shown in Fig. 10 and the apparatuses for When it can be reduced to the hundreds, data input, transfer and calculation shown an ordinary microcomputer is sufficient. in Fig. 11 with the suitable equations are Both the on-line input and the off-line in- as follows: 12 B.ENDO and H.TAKAHASHI

Case (A) interval may usually be 1mm or more. An integrator or a position sensor having The equations are a position signal output at a small and constant interval of *x connected with a minicomputer or a large capacity microcomputer is available. The contour of the cross section is traced by the pointer of Case (D) the device. The equations for the inte- Only a sheet of graph paper, a measure grator are and a small programmable calculator are required for this method. The cross section is parallelly divided and simplified to a series of trapezia and triangles. The breadth (bi) and the height (yi) are measured and put into the calculator. The The equations for the position sensor are equations are

In this case dx or *x becomes positive or negative according to the sense of movement of the pointer. Case (B) If the division is rough, these values be- The brightness and position sensor is use- come more or less inaccurate. ful. One can use this method without a Case (E) large capacity computer. The breadth (bi) This method requires a position sensor is successively measured by the device with automatic position signal output at at *y intervals. *y must be small, i. e., 1 small and constant intervals of *x and mm at most. The equations are y as well as an on-line minicomputer * or an on-line large capacity microcomputer. The contour of the cross section is traced. x and *y become positive or negative* Case (C) according to the sense of movement of the A position sensor without the function to pointer. In Fig. 10 (E), leftward *x and put out automatic serial signals or manual upward *y are negative. The equations measurement using a sheet of graph paper for this method are is sufficient. A large interval between yi and yi+1 can be used when the contours of both sides are nearly parallel. The Measurement of geometrical properties of bone 13

Case (H) A position sensor without the automatic position signal output at a constant interval and a microcomputer are available, as in case (C). This method can be applied Case (F) even with a measure and a sheet of graph An automatic scanning type brightness paper. The interval between yi and yi+1 and position sensor is necessary with an need not be constant. The equations are on-line minicomputer. Points in the cross section and those in the space must be distinguished by binarization. The equations are

Case (I) Case (G) A position sensor with automatic position A position sensor with an automatic posi- signal output at an interval of constant tion signal output at a small interval of displacement or constant time and an on- y, as well as an on-line minicomputer *or line microcomputer are available. The microcomputer are necessary. The equa- contour of the cross section is traced in a tions are single sense (clockwise in the case of Fig. 10). The equations are

In the uniaxial measurement further 14 B.ENDO and H.TAKAHASHI calculation is continued with the equa- 506. tions from (4) to (6) if necessary, then MILLER, G. and W. PURKEY, Jr.,1980: The geo- metric properties of paired human tibiae. with the equations from (7) to (14). In J. Biomechanics, 13: 1-8. the biaxial measurement it is continued MINNS, R., G. BREMBLE and J. CAMBELL, 1975: also with the equations from (4) to (6) The geometrical properties of the human if necessary, then with the equations from tibia. J. Biomechanics, 8 : 253-255. (15) to (19) and from (1) to (3). NAGURKA, M. and W. HAYES, 1980: An in- teractive graphic package for calculating cross-sectional properties of complex shape. REFERENCES J. Biomechanics, 13: 59-64. BURR, D., G. PIOTROWSKI and G. MILLER, 1981: PAUWELS, F., 1948: Die Bedeutung der Bauprin- Structural strength of the macaque femur. zipien des Stutz- und Bewegungsapparates Am. J. Phys. Anthrop., 54: 305-319. fur die Beanspruchung der Rohrenknochen. HiRSCH, H.,1895: Die mechanische Bedeutung Z. Anat. Entwickl. Gesch., 114: 129-166. der Schienbeinform. Springer, Berlin . ,1950: Die Bedeutung der Muskel- KIMURA, T., 1971: Cross-section of human krafte fur die Regelung der Beanspruchung lower leg bones viewed from strength of des Rohrenknochen wahrend der Bewegung materials. J. Anthrop. Soc. Nippon, 79: 323- der Glieder. Z. Anat. Entwickl. Gesch., 115: 336. 327-351. 1974: Mechanical characteristics of 1954: Die statische Bedeutung der human lower leg bones. J. Fac. Sci. Univ. Linea aspera. Z. Anat. Entwickl. Gesch., 117: Tokyo, Sect V,4: 319-393. 497-503. _,1980: Influence of the gravity force ,1968: Beitrag zur funktionellen An- on the growth. Biomechanism 5 : 48-55 , Univ. passung der Corticalis der Rohrenknochen. Tokyo Press (in Japanese). Untersuchung an drei rachitisch deformier- KOCH, J.,1917: The law of bone architecture . ten Femora. Z. Anat. Entwickl. Gesch., 127: Am. J. Anat., 21: 171-298. 121-137. LOVEJOY, C., A. BURSTEIN and K. HEIPLE,1976 : TAKAHASHI, H., (in press) : Geometrical pro- The biomechanical analysis of bone perties of the femoral shaft. Biomechanism strength: A method and its application to 6, Univ. Tokyo Press (in Japanese) . platycnemia. Am. J. Phys. Anthrop., 44: 489- (Received August 28, 1981) Measurement of geometrical properties of bone 15

長骨骨体断面形態の力学的諸特性値測定のためのさまざまな方法

遠藤萬里・高橋秀雄東京大学理学部人類学教教室

長骨を力学的に解析する場合,その断面形態の力学的諸特性値を知ることが重要である。しかし,それらを手で測定することは容易でない。近年,計測機器や計算機器の発達にともない,この測定作業が容易になってきた。それとともに,長骨骨体断面の力学的諸特性値を論ずる論文が急速に増加し,人類学領域でも多く見られるようになった。しかし,その測定法を具体的に論述してある報告は非常に少ない。したがって,あらたに測定しようとする者にとっては不便な状況にある。本報告は長骨骨体断面形態の力学的諸特性値とその機態形態学的意義について解説し,それを測定計算するさまざまな方法とそれに要する諸機器について具体的に述べたものである。長骨骨体の断面のような不規則図形の測定は工学や物理学ではあまり必要がないので,本報告に類するものがない。したがって,本報告で述べている方法や計算式の一部は今までどの分野でも公表されていないものである。断面形態の諸特性値とは図心点位置,面積,断面2次モーメント,主軸位置,主断面2次モーメント,極モーメント,回転半径,断面係数などである。しかし,面積,図心点,断面2次モーメントが得られれば ,他の特性値は導出できる。上記の特性値を得るには,任意の直交座標をつくり,その軸まわりの0次,1次,2次のモーメントを測定演算する。この場合単軸まわりの測定法と直交2軸まわりの測定法がある。単軸まわりの測定から主軸や主断面2次モーメントを求めるには,特別な演算式を使う。任意の軸まわりのモーメントを求める場合,測定法はさまざまである。本報告では測定法,測定器,演算式の各種の使いやすい組合せを考えた。