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An approach to the mechanical properties of single osteonic lamellae
J Biomrchmrcs.
19-3. Vol. 6. pp. 223-135.
Pergmon
Press.
Printed inGreat
Bnmn
AN APPROACH TO THE MECHANICAL PROPERTIES OF SINGLE OSTEONIC LAMELLAE*t ANTONIO
ASCENZI, ERMANNO BONUCCI and ARIEL SIMKIN: Istituto di Anatomia Patologica, University of Rome, 00161 Rome, Italy
Abstract-A technique is presented for examining the role played by orientation of fiber bundles in the mechanical behavior of single osteonic lamellae. Cylindrical osteon samples are loaded perpendicular to their axis and, when required, the direction of loading can be changed continually. The main results are: (1) In osteon samples whose fiber bundles change direction in successive lamellae through an angle of about 90” (Type I), circular fractures appear in lamellae whose fiber bundles have a marked longitudinal spiral course, while lamellae whose fibers have an almost transversal spiral course are unaffected. (2) In osteon samples whose flber bundles have a marked longitudinal spiral course in successive lamellae (Type II), fractures spread radially from the central canal toward the periphery of the osteon, until all the lamellae are affected. (3) These findings are independent of the degree of calcification; they go to strengthen the view that the compactness of osteonic bone is strengthened by the presence of lamellae whose fiber bundles have an almost transversal spiral course. (4) The first fractures to appear are running between collagen fib&. indicating that the interfibrillar substance is considerably less resistant than the fibrils themselves. (5) In osteon samples of Type I. the fractures which appear in lamellae whose fibers have a marked longitudinal spiral course are circular, and this makes it possible to isolate lamellae whose fiber bundles have an almost _ transversal spiral course. ISTRODUCTIOS
on the micromechanical properties of bone have been carried out on osteons, i.e. structures of the second order in compact bone by Ascenzi and Bonucci (1964. 1965, 1967, 1968. 1971, 1972) and by Ascenzi. Bonucci and Checcucci (1966). As far as we are aware, TischendorfT( 195 1, 1952. 1954) is the only investigator who has tried to make a direct analysis of the mechanical behavior of bone structures of the third order. His specimens were very thin, elongated rectangular prisms, obtained from fresh tibia1 shafts and cut parallel to the long axis of the intact bone. The apparatus consisted of two supports for fixing specimens; a bending load was applied to each specimen INVESTIGATIONS
*Receired
I5 September
midway between the supports. By controlling the experiment microscopically TischendorfT was able to follow and measure deformations within lamellae step by step, and to measure the size of reversible deformations. The lack of any other research on structures of the third order has encouraged us to develop a technique for investigating the role played by the orientation of fiber bundles and crystallites in the mechanical behavior of single lamellae within isolated osteons. MATERIAL
AND METHOD
Theoretical premisses The distribution of stresses in a circular ring compressed by two opposite forces acting along a diameter is a very complicated
1972.
‘This work was supported by the grant No. 115.195 70.00907.04of the National Research Council of Italy. $Ariel Simkin. M.Sc.. Dept. of Orthopaedic Surgery. Hadassah University Hospital. Jerusalem. Israel. 7-J?
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A. ASCENZI, E. BONUCCI and A. SIMKIN
mechanical problem. Timoshenko (1922), analysing this problem as if it were a twodimensional one, obtained a solution for the case of a ring with an outer-inner dia. ratio of 1: 1. Frocht (1948) treated the problem from a photoelastic point of view and used as representative the photoelastic stress pattern induced in a bakelite ring by concentrated compressive diametral loads. As far as we are aware, the theory of stress distribution in a circular ring under diametral compression has only been developed for rings made of isotropic material. No theory has yet been advanced for a ring made of anisotropic material. The case of most immediate interest is that of a ring with a lamellar structure. In engineering terms this may be described as a composite laminate: it closely approximates to the structure of an osteon (see Ambartsumian, 1966; Stawsky and Hoff, 1969). From general considerations, it would seem reasonable to expect that in circular, cross-sectioned osteon samples, compressed by two opposite forces applied at the ends of a diameter, the distribution of stresses in the osteonic ring will be such that fractures will appear preferentially where bone texture is weakest; and further, that when the direction of diametral compression is changed continually by rotating the sample, a new stresspattern will appear, producing new fractures at other points of low resistance within the osteonic ring. By subjecting isolated osteons to such pressures, it was thought, pertinent data may be acquired about the pattern of stresses within osteons and about the mechanical properties of single osteonic lamellae. of osteon and selection Preparation samples. To test these theoretical premisses, cross-sections 30-40 p thick were prepared by grinding human femoral shafts. Every precaution was taken to avoid heating the material. Our samples were obtained from crosssectioned osteons, using a specially designed device which has been described in detail elsewhere (Ascenzi and Bonucci, 1968).
This consisted of a very thin. accurately sharpened needle eccentrically inserted on a dentist’s drill. When the drill turns, the tip of the needle describes a circle with a diameter of about 180-200 p, i.e. the average diameter of an osteon. When the rotating axis of the needle coincides with the axis of an osteon, and this osteon lies perpendicular to the surfaces of a bone section, the tip of the needle cuts out an almost complete osteon sample. The sample will then be cylindrically shaped and will have walls of uniform thickness. The osteons tested by us differed both in degree of calcification and in orientation of collagen bundles. The degree of calcification was determined microradiographically, our aim being to select fully calcified osteons or osteons at the initial stage of calcification. Among the various arrangements produced by differences in fiber bundle direction in successive lamellae, those characteristic of two types of osteon were chosen. In the first (called here Type I) the fibers in one lamella have a marked longitudinal spiral course, while in the next the fibers have an almost transversal spiral course so that the fibers in two successive lamellae make an angle of nearly 90”. Under the polarizing microscope osteons of this type reveal an alternation of dark and bright lamellae in cross-section. In the second type (called Type II) fibers have a marked longitudinal spiral course, with the pitch of the spiral changing so slightly that the angle of the fibers in one lamella is practically the same as that of the fibers in the next lamella. Under the polarizing microscope osteons of Type II appear uniformly dark in cross-section, although they are often bordered by a bright lamella both at the periphery and around the central canal. In every case the peripheral lamella was removed in cutting the osteon samples. Although every attempt was made to select osteon samples bounded by an intact peripheral lamella, this optimum was hardly ever attainable, because osteons only rarely have a
MECHANICAL
PROPERTIES
OF SINGLE
regular cylindrical shape, and lamellae do not always form a regular uninterrupted layer concentrically arranged around the central canal. The material used in the present investigation was obtained from femoral shafts of seven human subjects aged between 18 and 3 1. showing no apparent skeletal defects. Sixty osteon samples were tested in all. The temperature was held at about 20°C and the samples were kept wet by hydration with saline solution. Before and after testing, a series of photographs were taken of each osteon in ordinary and in polarizing light. Procedure for testitlg osteon snmples. The samples were subjected to direct compression perpendicular to their axis and, when required, the diametral points where pressure was applied were continually changed by rotation. This was done by using a glass slide on which an 18 x 18 mm coverslip was firmly fixed with Canada balsam (Fig. 1). One edge of the coverslip. which was 160 p thick, functioned as stopper during the pressure loading. Each cylindrical osteon sample was put on the slide with its curved surface touching the edge of the coverslip tangentially. The osteon samples were placed in position and turned by hand, and pressure was applied by pressing a very I
OSTEONIC
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small spatula against the side of the osteon opposite the coverslip. The whole process was observed under a light microscope. Special care was taken to avoid bending the samples. Electron microscopic excminntion qf the tested osteons. For thorough investigation of
the changes produced by compressive stress in osteon samples. the samples were fixed after loading in 4 per cent formalin, buffered to pH 7.2 according to Millonig (1962). dehydrated and embedded in .Araldite or glycol methacrylate. Specimens were sectioned with a Porter-Blum microtome fitted with glass or diamond knives. To avoid decalcification (Boothroyd, 196-l) ultrathin sections were in no case left in water for longer than 3 min. Sections 1/2-l p thick were examined under the light microscope either unstained or stained with Azure II-Methylene blue. Some osteon samples were examined after decalcification. This was carried out both on unembedded osteon samples, using 2 per cent formic acid and after embedding. by floating the ultrathin sections either on Z per cent phosphotungstic acid or on formic acid, the latter being followed by uranyl acetate and lead citrate staining. These procedures allowed us to study the ultrastructure of the osteonic organic matrix at sites of fracture.
I
RESULTS
Fig. 1. Diagram in two projections illustrating the technique used in compressing single osteon samples perpendicularly to their a.xis. A, slide: B, coverslip: C, osteon: D, spatula.
When an osteon sample of Type I. i.e. an osteon with fiber bundles in one lamella making an angle of nearly 90” with the fiber bundles in the next, is pressure-loaded perpendicular to its axis- i.e. along one of its diameters-arc-shaped, concentrically distributed cracks appear. These are concentrated within the central portion of the anular sample and are found within all the four quadrants obtained constructing two orthogonal lines one of them representing the direction of loading. More exactly, each series of cracks is grouped round this line of loading and occupies a segment which subtends an
230
A. ASCENZI,
E. BONUCCI
angle of between 20 and 50” (Fig. 2). The polarizing microscope reveals that the cracks involve dark lamellae, i.e. lamellae whose fibers have a marked longitudinal spiral course, while bright lameliae, i.e. lamellae whose fibers have an almost transversal spiral course, are unaffected. When an osteon sample or Type I is pressure-loaded perpendicular to its axis, and the direction of loading is then changed continually by rotation, the cracks grow longer, join up and eventually become circular. Even after such rotation the cracks only alfect lamellae whose fibers have a marked longitudinal spiral course (black under the polarizing microscope), while lamellae whose fibers have an almost transversal spiral course (bright under the polarizing microscope) are unaffected (Figs. 3 and 13-16). The results reported above were obtained with low or fairly low loads. With higher loads lamellae whose fibers have almost transversal spiral course are severely damaged too, and radial and tangential cracks are seen in them. The worst damage and cracks are seen in the lamella which lies round the haversian canal. Damage falls off rapidly in the successive lamellae of the same type, so that the peripheral lamellae of this type are almost undamaged. When the compressive load applied to lamellae whose fibers have a marked longitudinal spiral course is increased, the circular cracks become more conspicuous, but no other effect is apparent. In osteon samples of Type I, the circular cracks produced in lamellae whose fibers have a marked longitudinal spiral course are actually deep fractures which extend to the full depth of each lamella. This has the advantage of enabling one to isolate individual lamellae whose fibers and crystallites have an almost transversal spiral course, using a steel needle for microscopic dissection. The first step in this process is shown in Fig. 4. Here an osteon sample of Type I is shown as seen, (a) under an ordinary microscope and (b) under a polarizing microscope. A lamella which
and A. SIMKIN
appears bright under the polarizing microscope, because its fiber bundles have an almost transversal spiral course, has been partly separated from the next lamella of the same type. Under the ordinary light microscope it is easy to see that pieces of the adjacent cracked lamella whose fibers have a marked longitudinal spiral course are still attached to it on both sides. In Figure 5a a bright lamella which has been completely isolated is seen under an ordinary microscope. It is worth mentioning that the thickness of the lamella specimen (5 /.L) is only l/6 of the height (30 p)-the same height as that of the osteon sample itself. Figure 5b shows another isolated lamella, this time under the polarizing microscope. It is not only possible to isolate individual bright lamellae; it is also possible to cut and open lamellae at any point on their circumference, as seen in Fig. 6. Figure 6a shows a small osteon sample consisting of seven lamellae; cracks are found in all the lamellae whose fiber bundles have a marked longitudinal spiral course. Figure 6b shows the same osteon under the polarizing microscope during separation of its bright lamellae. In this figure two findings merit attention. First, each bright lameila has attached to it, on at least one side, a thick layer, or all, of the adjacent dark lamella whose fibers and crystallites have a marked longitudinal spiral course. Second, the outermost lamella has straightened out, and only its thickest parts are now birefringent; the parts which now lie flat on the lateral surface no longer appear bright, because they are now insufficiently thick. Figure 7 shows one of these isolated, opened, flat-laying lamellae viewed under ordinary microscope. The electron microscope offers detailed information about how cracks are formed in lamellae whose fibers have a marked longitudinal spiral course. In osteons subjected to low-level, rotating compression, very small ultrastructural cracks appear in these lamellae. Some are irregularly shaped, and some form a
Fig. 9. Fully calcified osteon sa!!:le 30 of Type I pressure-loaded perpendicularly to its axis. The mows indicate the direction of loading. Arc-shaped cracks are visible. X 29%
Fig. 3. Fully calcified osteon sample 9 of Type 1 (a) before loading: Co) after pressure-loading, with continual change in direction of loading, circular cracks are visible: Cc) under the polarizing microscope the cracks do not appear to affect the bright lamellae. Y 220.
Fig. 4. (a) A partially removed lamella from the fully calcified osteon sample 9 of Type I. (b) Under the polarizing microscope the same lamella is bright. X 50.
?g. 5. (a) An isolated lamella. x 320. (b) An isolated bright lamella as seen under the polarizing microscope. x 500.
a _I.
Fig. 6. (a) The fully calcified osteon sample 7 of Type I after loading. lb) Two bright lametlae
of the same sample have been opened and partially isolated. Semipolarized light. x 300.
Fig. 7. An isolated and opened lamella is spread out.
x
450.
Fig. 8. Fully calcified osteon sample 18 of Type 11 after pressure-loading with continual change in direction of loading. x 230.
Fig. 9. .-I fully calcified osfeon of Type II as seen under polarizing microscope. X 230.
Fig. 10. (a) Fully calcified osteon sample 52 of Type II deprived of its internal bright iamella and pressureloaded perpendicular to its axis. The direction of compression is the same as in Fig. 2. (b) The same sample after pressure-loading with continual change in direction of loading. x 230.
Fig. 13. A section of the fully calcified osteon samr rie 50 ofType I. Before loading. X 1100.
‘ig. 11. After gf .^ _..:_
:-loading perpendicular _^^ :^ ,a:_^^I:-..
Fig. 15. After strong pressure-loading perpendicular . ” . to the axls and with contmual change m direction ot loadmg.
Fig. 16. The strongly compressed sample as seen under the polanzmg microscope.
Fig. I’. Electron micrographs of an ultrathin section from decalcified ostwn 41 of T:.pe I. (T) Lamella whose fibers have a transversal spiral course: CL) lamella u hose fibers have a longitudinal spiral course. The lamella whose fibers have a longitudinal spiral source lhow A frx:s:e (F) with many isolated fibrils. X 11.000. Fig.
18.Some
separated
fibriis surrounded
by fringes of tom cement sabstancs.
T 7h.W).
MECHANICAL
PROfERTIES
OF SINGLE
regular curve; all of them tend to take on a circular orientation. As compression is gradually increased, the cracks lengthen, until their ends join up, so resulting in a smaller number of longer cracks. Sometimes the first cracks to form occur in the central or paracentral zone of these lamellae. In this case an almost complete circular crack is produced, dividing the lamella into two unequal parts, each of them still in contact with the adjacent lamella, whose fibers have an almost transversal spiral course. In other cases the cracks spread along the boundaries between two adjacent lamellae, leaving fib& bordering a lamella with a longitudinal spiral course on a side and fibrils bordering a lamella with a transversal spiral course on the other. A much rarer pattern is that of a crack which takes on a circular orientation and then penetrates into a lamella whose fibers have an almost transversal spiral course. The crack then spreads through the outermost fibrils of this lamella, without, however, penetrating any deeper. The impression is thus given that the lamellae whose fibers have an almost transversal spiral course reject cracks, at most allowing them to develop at their extreme periphery, and only on very rare occasions. All these data help to explain the morphology of isolated lamellae whose fibers have an almost transversal spiral course, when seen under ordinary light microscopemost pertinently the fact that these lamellae have pieces or layers of at least one adjacent lamella with longitudinal fibers, or even one whole lamella of this type attached to them. So far we have discussed electron microscopic findings above a certain minimum frequency. Brief mention should also be made of an exceptional finding- that of fiber bundles which run like a bridge connecting two successive lamellae whose fibers have an almost transversal spiral course. When cracks start in these bundles they sometimes spread deeply into the lamellae on either side, producing radial fractures.
OSTEONIC
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23 1
The electron microscope also yields interesting data on the earliest changes in collagen fibrils where cracks start. As shown in Fig. 17, the crack in the central portion of the lamella whose fiber bundles have a marked longitudinal spiral course clearly involves the separation of fibrils, many of which are no longer attached to each other. Photographs of this phenomenon can be obtained at very high magnification, as in Fig. 18. There is evidence here that fibrils are not cut because around and, sometimes, between them, fringes of other material are seen - material probably consisting of tom cement substance. We will now consider changes produced in osteons of Type II i.e. osteons with fiber bundles which have a marked longitudinal spiral course. It may be worth recalling at this stage that in these osteons the innermost lamella usually consists of fibers and crystallites which have an almost transversal spiral course. When a low compressive load is applied to osteons of Type II perpendicular to their axis but with continual rotation, a circular crack is produced at the outer boundary of the innermost lamella. and this crack then spreads into the adjacent lamella whose fibers have a marked longitudinal spiral course. If the compressive load is raised, the innermost lamella becomes severely deformed and complete radial fractures appear in it. Some of these then branch outwards from the original circular crack, and spread toward the periphery. They occasionally reach the very edge of the osteon sample, but this is a rare occurrence, so that an unbroken layer is usually seen bordering the osteon. Figure 8 shows a loaded osteon sample of Type II as seen under an ordinary microscope. The lamella round the central canal has been fractured at four points, and irregular radial cracks are seen spreading from the original cracks toward the periphery. Such radial cracks often follow an irregular or zigzag course. This is because- as shown in Fig. 9-osteons of Type II sometimes con-
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tain small fiber bundles which have a transversal course and appear bright under the polarizing microscope. It would hardly be surprising if these fibers were responsible for deviating the radial course of such cracks; and this view is borne out by electron microscopic data. An attempt was made to investigate thoroughly the behavior of osteons of Type II, from which the innermost bright lamella has been removed. When a sample of this kind is pressure-loaded along one only of its diameters without rotation, radial cracks appear starting at the two opposite points of the internal border lying on the loaded diameter, where tension is greatest. Figure 10a shows an osteon sample so prepared and loaded along its vertical diameter. Two cracks have developed along this diameter, each starting from the internal border and lengthening radially. When the internal lamella is removed from osteons of Type II and a compressive load is then applied perpendicular to their axis. with continual rotation, the cracks increase in number (Fig. lob). The situation is very like that of intact osteon samples of Type II after the internal bright lamella has been fractured. It must therefore be concluded that the circular crack seen at the outer boundary of the innermost bright lamella is attribuable to the presence of that lamella. A comparison between the results for osteon samples of Types I and II suggests that the circular cracks or fractures produced in lamellae whose fibers have a longitudinal spiral course in osteons of Type I, are due to the presence of neighbouring lamellae whose fibers have an almost transversal spiral course. No lamellae of the latter type are present in osteons of Type II, except for the innermost and outermost lamellae and fragmentary lamellae consisting of rare, very thin fiber bundles with an almost transversal spiral course; this would explain why cracks in these osteons have a strong tendency to follow a radial course. The stage reached by calcification-initial
and A. SIMKIN
or far advanced-had no the results for osteons of Circular fractures develop ones in Type II, whatever fication.
significant effect on either Type I or II. in Type I and radial the degree of calci-
DISCUSSION
The main focus of interest provided by the results of the present investigation is the difference in behavior between osteon samples of Types I and II. When fairly low compressive loads are applied perpendicular to the axis of osteon samples of Type I, circular cracks appear, but only in lamellae whose fibers have a marked longitudinal spiral course, while lamellae whose fibers have an almost transversal spiral course are unaffected. This difference is particularly marked in lamellae which lie near the central canal. Cracks also appear in peripheral lamellae whose fibers have a marked longitudinal spiral course, but only when osteon samples are strongly loaded, and then some of the innermost lamellae whose fibers have an almost transversal spiral course may be radially fractured. While the compressive load is still low the electron microscope reveals ultrastructural cracks found exclusively in lamellae whose fibers have a marked longitudinal spiral course. Some of these cracks are irregular in shape, and other form a regular curve: all tend to take a circular orientation. As the load is increased, the cracks lenghten and join up. In some cases cracks develop in the central or near-central zone of lamellae, as well as near the boundary with neighbouring lamellae whose fibers have a transversal spiral course. In all cases the cracks run between. not across, collagen fibrils, separating them and indicating that the interfibrillar substance is considerably less resistant than the fibrils themselves. Unlike osteon samples of Type I, samples of Type II show radial cracks running from the central canal toward the periphery after pressure loading perpendicular to their axis.
MECHANICAL
PROPERTIES
OF SINGLE
To find an explanation for these differences between the two types of osteon sample. the stress distribution pattern in a ring of isotropic, elastic material was analysed using the solutions of Timoshenko (1922). The problem is treated as a two-dimensional one. with polar coordinates, p, 0 (Fig. 11). For each point there are three stress componentsnormal radial S,.,, normal circumferential SHH and circumferential shearing stress T,.+ The stresses depend on the ratio between hole radius and ring radius. r/R and the formulae involved are of formidable difficulty so that a computer had to be used to calculate them for r/R = 0.5. The highest value of all the stresses is obtained for SHHat p/R = 0.j and 0 = 0” or 180”. that is. at the circumference of the hole and on the diameter of the applied load. If a ring of isotropic material is so compressed, it does in fact fracture at these points. These fractures occur in osteon sample of Type II after removal of the innermost lamella consisting of fiber bundles which have a transversal spiral course. Osteon samples of this kind. as seen in Fig. 10a can, in fact, be considered isotropic, if the problem is considered in two dimensions only. The normal radial and shear stresses vanish at the outer and inner surfaces and rise inside the ring, with the highest values for both S,., and T,.~ for H between 20 and 25” and
OSTEONIC
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LAMELLAE
p/R between O-6 and 0.75 (Fig. 12). These values are about 24 and 5 per cent of the highest Seti for rrH and S,, respectively. A ring of an elastic isotropic material which has. however. a low tensile strength in a radial direction and/or a low shear strength in the circumferential direction, will fracture first in this region. along circumferential lines. The fracture lines seen in the compressed osteon samples of Type I are circumferential too, though they initially occupy larger angular ranges than those calculated, as seen in Fig. 2. This may be due to a different stress distribution pattern brought about by the presence of lamellae, but it is very probable that these fractures start to form at sites of high circumferential and radial tensile stress. Lamehae whose fiber bundles have a longitudinal spiral course can be expected to have low resistance to such stresses, because the cracks can find many weak interfaces between fib&, as the electron microscope clearly shows. This last explanation is indirectly supported by Dempster and Liddicoat’s investigation ( 1952) on the effect of direction of loading on fractures produced by compressive strength. They used cubes of com-
\ \ 5
%---r I
I -
5
’
1I
-----
‘ai j/ ---. _---
x // \\ /
-__
_
/I
1\\ 1 I \
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Fig. I I. Definition of stresses in polar coordinates for a ring under diametral compression.
Fig. 11.Regions of high tensile radial and circumferential shear stresses in a ring under diametral compression.
234
A. ASCENZI.
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pact bone and tested them along their long, radial and tangential axes. They were able to show that the oblique rolling, sliding type of failure in cubes of bone under lateral compression suggests a special weakness in shear parallel to fibers. Similar results are given by compression tests on hollow cylinders of bone loaded longitudinally by Carothers, Smith and Calabrisi (1949), who reported longitudinal splitting and shear. In the same way Rauber’s tests (1876) clearly show that the shearing strength of compact and spongy bone is considerably greater when loaded perpendicular to the direction of fibers than when loaded parallel with fibers. The results obtained during the present investigation for the mechanical properties of single osteonic lamellae are supported by those of our previous investigations on the tensile, compressive and shearing properties of single osteons along their axis (Ascenzi and Bonucci, 1964, 1965, 1967, 1968, 1971, 1972; Ascenzi, Bonucci and Checcucci, 1966). The ultimate compressive strength and the modulus of elasticity to compression are greatest for osteons having transversally oriented fiber bundles in successive lamellae, while the ultimate tensile strength and the modulus of elasticity to tension are greatest for osteons having a marked longitudinal arrangement of fiber bundles in successive lamellae. When osteons are loaded by shearing along their axis, those having a marked longitudinal spiral course of the fiber bundles in successive lamellae are least able to support stress. All these data and in particular those mentioned in the last paragraph corroborate the view that in osteons the compactness of bone is strengthened by the presence of lamellae whose fiber bundles have a marked transversal spiral course. On the other hand the fiber bundles which have an almost transversal spiral course form a system of rings which prevent bending. These twin concepts may help to explain the function of various types of bone in mechanical terms.
and A. SIMKIN
For instance, the main function of the human femoral dyaphysis is to support compressive loads, and among the osteons found in this type of bone those of Type I predominate. The results obtained here apply both to fully calcified osteons and osteons at the initial stage of calcification. The last topic to be discussed is the possibility of applying the techniques used here to future quantitative investigations on the tensile properties of isolated lamellar samples. It appears self-evident that it will be necessary to use osteons of Type I i.e. osteons with fiber bundles in one lamella making an angle of nearly 90” with the fiber bundles in the next, because only these are suitable for the isolation of single lamellae. Furthermore only lamellae whose fiber bundles have a marked transversal spiral course can be isolated because those whose fiber bundles have a marked longitudinal spiral course become severely damaged and fractured when subjected to compressive stress perpendicular to the osteon axis. The samples to be used will have a circular, viz anular, or rectangular shape. The last type will be obtained by opening the anular samples and applying a radial section to them. At the time of writing no final decision has been taken as to the method to be followed in carrying out this research program, but we hope to achieve results in the future. AcXno~rledgement-The authors are deeply grateful to A. Benvenuti for excellent technical assistance during the course of the present investigation. REFEREXCES Ambartsumian, S. A. (1966) Some current aspects of the theory of anisotropic layered shells. In Applied Mechanics Surceys. (Edited by H. N. Abramson, H. Liebowitz, J. M. Crowley and S. Juhasz) pp. 3013 14. Spartan Books, Washington. Ascenzi,. A. and E. Bonucci (7964) The ultimate tensile strength of single osteons. rlcrn annr. 58. 160-l 83. Ascenzi, A. and E. Bonucci (1965) The measurements of the tensile strength of isolated osteons as an approach to the problem of intimate bone texture. In Calcified Tissues. Proc. 2nd European Symposium on Calcified Tissues. (Edited by L. J. Richelle and M. J. Dallemagne) pp. 325-335. University of Liege.
MECHANICAL
PROPERTIES
OF SINGLE
Ascenzi. A. and E. Bonucci (1967) The tensile properties of single osteons. ,-lnnr. Rec. U&375-386. Ascenzi. A. and Bonucci. E. ( 1968) The compressive propenies of singleosteons. rlnur. Rec. 161. 377-392. Ascenzi. A. and Bonucci. E. (1971) A micromechanic investigation on single osteons using a shearing strength test. lsrnel J. ‘bled. Sci. 7.47 l-472. Ascenzi. A. and Bonucci. E. (1972) The shearing properties of single osteons. Anat. Rec. 172.499-5 10. Ascenzi. A., Bonucci E. and Checcucci A. (1966) The tensile properties of single osteons studied using a microwave extensimeter. In Studies on the Anatom,v curd F!rnction of Bone und Joints. (Edited by F. G. Evans) pp. I2 l-i4 I. Springer, Heidelberg. Boothroyd. B. (1961) The problem of demineralisation in thin sections of fullv calcified bone. J. Cell Biol. 20. 165-173. Carothers. C. 0.. Smith. F. C. qnd Calabtisi. P. (1949) The elasticity and strength of some long bones of the human body. Naval Medical Research Institute. ProjesrNXlOOl 056.02.13. Dempstcr. W. T. and Liddicoat, R. T. (1952) Compact bone as a non-isotropic material. Am. J. Ancx 91. 33 l-36’.
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Frocht, M. M. (1948) Photoelasricity. Wiley. New York. Millonig. G. (1962) Further observattons on a phosphate buffer for osmium solutions in fixation. In Electron &licroscop.v.
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of the jrh
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(Edited by S. S. Breese) Vol. 2. p. 8. Academic Press, New York. Rauber. A. A. (1876) Ektsricit‘it and Festiukeir der Knochen. Engelmann, Leipzig. Stawsky. Y. and Hoff, N. J. (1969) Mechanics of composite structures. In Composite Engineering Laminates. (Edited by A. G. H. Dietz) pp. 5-59. The MIT Press, Cambridge Mass. Timoshenko, S. (1922) On the distribution of stresses in a circular ring compressed by two forces acting along a diameter. Phil. .Clap. 44. 1014-1019. Tischendorff, F. ( 195 I) Das Verhalten der haverschen Systeme bei Belastung. Rorc.r’.4rch. 115.3 18-331. Tischendorlf, F. (1952) Quantitative Beobachtungen ueber das Verhalten der haverschen Lamellen bei Belastung. Ro~r.r’.-Irch. 146, l-20. TischendorK F. (1954) Die mechanische Reaktion der haverschen Systeme und ihrer Lamellen auf experimentelle Belastung. Rwr Arch. 116.66 I-704. tron
.Clicroscup~
19-3. Vol. 6. pp. 223-135.
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Press.
Printed inGreat
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AN APPROACH TO THE MECHANICAL PROPERTIES OF SINGLE OSTEONIC LAMELLAE*t ANTONIO
ASCENZI, ERMANNO BONUCCI and ARIEL SIMKIN: Istituto di Anatomia Patologica, University of Rome, 00161 Rome, Italy
Abstract-A technique is presented for examining the role played by orientation of fiber bundles in the mechanical behavior of single osteonic lamellae. Cylindrical osteon samples are loaded perpendicular to their axis and, when required, the direction of loading can be changed continually. The main results are: (1) In osteon samples whose fiber bundles change direction in successive lamellae through an angle of about 90” (Type I), circular fractures appear in lamellae whose fiber bundles have a marked longitudinal spiral course, while lamellae whose fibers have an almost transversal spiral course are unaffected. (2) In osteon samples whose flber bundles have a marked longitudinal spiral course in successive lamellae (Type II), fractures spread radially from the central canal toward the periphery of the osteon, until all the lamellae are affected. (3) These findings are independent of the degree of calcification; they go to strengthen the view that the compactness of osteonic bone is strengthened by the presence of lamellae whose fiber bundles have an almost transversal spiral course. (4) The first fractures to appear are running between collagen fib&. indicating that the interfibrillar substance is considerably less resistant than the fibrils themselves. (5) In osteon samples of Type I. the fractures which appear in lamellae whose fibers have a marked longitudinal spiral course are circular, and this makes it possible to isolate lamellae whose fiber bundles have an almost _ transversal spiral course. ISTRODUCTIOS
on the micromechanical properties of bone have been carried out on osteons, i.e. structures of the second order in compact bone by Ascenzi and Bonucci (1964. 1965, 1967, 1968. 1971, 1972) and by Ascenzi. Bonucci and Checcucci (1966). As far as we are aware, TischendorfT( 195 1, 1952. 1954) is the only investigator who has tried to make a direct analysis of the mechanical behavior of bone structures of the third order. His specimens were very thin, elongated rectangular prisms, obtained from fresh tibia1 shafts and cut parallel to the long axis of the intact bone. The apparatus consisted of two supports for fixing specimens; a bending load was applied to each specimen INVESTIGATIONS
*Receired
I5 September
midway between the supports. By controlling the experiment microscopically TischendorfT was able to follow and measure deformations within lamellae step by step, and to measure the size of reversible deformations. The lack of any other research on structures of the third order has encouraged us to develop a technique for investigating the role played by the orientation of fiber bundles and crystallites in the mechanical behavior of single lamellae within isolated osteons. MATERIAL
AND METHOD
Theoretical premisses The distribution of stresses in a circular ring compressed by two opposite forces acting along a diameter is a very complicated
1972.
‘This work was supported by the grant No. 115.195 70.00907.04of the National Research Council of Italy. $Ariel Simkin. M.Sc.. Dept. of Orthopaedic Surgery. Hadassah University Hospital. Jerusalem. Israel. 7-J?
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A. ASCENZI, E. BONUCCI and A. SIMKIN
mechanical problem. Timoshenko (1922), analysing this problem as if it were a twodimensional one, obtained a solution for the case of a ring with an outer-inner dia. ratio of 1: 1. Frocht (1948) treated the problem from a photoelastic point of view and used as representative the photoelastic stress pattern induced in a bakelite ring by concentrated compressive diametral loads. As far as we are aware, the theory of stress distribution in a circular ring under diametral compression has only been developed for rings made of isotropic material. No theory has yet been advanced for a ring made of anisotropic material. The case of most immediate interest is that of a ring with a lamellar structure. In engineering terms this may be described as a composite laminate: it closely approximates to the structure of an osteon (see Ambartsumian, 1966; Stawsky and Hoff, 1969). From general considerations, it would seem reasonable to expect that in circular, cross-sectioned osteon samples, compressed by two opposite forces applied at the ends of a diameter, the distribution of stresses in the osteonic ring will be such that fractures will appear preferentially where bone texture is weakest; and further, that when the direction of diametral compression is changed continually by rotating the sample, a new stresspattern will appear, producing new fractures at other points of low resistance within the osteonic ring. By subjecting isolated osteons to such pressures, it was thought, pertinent data may be acquired about the pattern of stresses within osteons and about the mechanical properties of single osteonic lamellae. of osteon and selection Preparation samples. To test these theoretical premisses, cross-sections 30-40 p thick were prepared by grinding human femoral shafts. Every precaution was taken to avoid heating the material. Our samples were obtained from crosssectioned osteons, using a specially designed device which has been described in detail elsewhere (Ascenzi and Bonucci, 1968).
This consisted of a very thin. accurately sharpened needle eccentrically inserted on a dentist’s drill. When the drill turns, the tip of the needle describes a circle with a diameter of about 180-200 p, i.e. the average diameter of an osteon. When the rotating axis of the needle coincides with the axis of an osteon, and this osteon lies perpendicular to the surfaces of a bone section, the tip of the needle cuts out an almost complete osteon sample. The sample will then be cylindrically shaped and will have walls of uniform thickness. The osteons tested by us differed both in degree of calcification and in orientation of collagen bundles. The degree of calcification was determined microradiographically, our aim being to select fully calcified osteons or osteons at the initial stage of calcification. Among the various arrangements produced by differences in fiber bundle direction in successive lamellae, those characteristic of two types of osteon were chosen. In the first (called here Type I) the fibers in one lamella have a marked longitudinal spiral course, while in the next the fibers have an almost transversal spiral course so that the fibers in two successive lamellae make an angle of nearly 90”. Under the polarizing microscope osteons of this type reveal an alternation of dark and bright lamellae in cross-section. In the second type (called Type II) fibers have a marked longitudinal spiral course, with the pitch of the spiral changing so slightly that the angle of the fibers in one lamella is practically the same as that of the fibers in the next lamella. Under the polarizing microscope osteons of Type II appear uniformly dark in cross-section, although they are often bordered by a bright lamella both at the periphery and around the central canal. In every case the peripheral lamella was removed in cutting the osteon samples. Although every attempt was made to select osteon samples bounded by an intact peripheral lamella, this optimum was hardly ever attainable, because osteons only rarely have a
MECHANICAL
PROPERTIES
OF SINGLE
regular cylindrical shape, and lamellae do not always form a regular uninterrupted layer concentrically arranged around the central canal. The material used in the present investigation was obtained from femoral shafts of seven human subjects aged between 18 and 3 1. showing no apparent skeletal defects. Sixty osteon samples were tested in all. The temperature was held at about 20°C and the samples were kept wet by hydration with saline solution. Before and after testing, a series of photographs were taken of each osteon in ordinary and in polarizing light. Procedure for testitlg osteon snmples. The samples were subjected to direct compression perpendicular to their axis and, when required, the diametral points where pressure was applied were continually changed by rotation. This was done by using a glass slide on which an 18 x 18 mm coverslip was firmly fixed with Canada balsam (Fig. 1). One edge of the coverslip. which was 160 p thick, functioned as stopper during the pressure loading. Each cylindrical osteon sample was put on the slide with its curved surface touching the edge of the coverslip tangentially. The osteon samples were placed in position and turned by hand, and pressure was applied by pressing a very I
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small spatula against the side of the osteon opposite the coverslip. The whole process was observed under a light microscope. Special care was taken to avoid bending the samples. Electron microscopic excminntion qf the tested osteons. For thorough investigation of
the changes produced by compressive stress in osteon samples. the samples were fixed after loading in 4 per cent formalin, buffered to pH 7.2 according to Millonig (1962). dehydrated and embedded in .Araldite or glycol methacrylate. Specimens were sectioned with a Porter-Blum microtome fitted with glass or diamond knives. To avoid decalcification (Boothroyd, 196-l) ultrathin sections were in no case left in water for longer than 3 min. Sections 1/2-l p thick were examined under the light microscope either unstained or stained with Azure II-Methylene blue. Some osteon samples were examined after decalcification. This was carried out both on unembedded osteon samples, using 2 per cent formic acid and after embedding. by floating the ultrathin sections either on Z per cent phosphotungstic acid or on formic acid, the latter being followed by uranyl acetate and lead citrate staining. These procedures allowed us to study the ultrastructure of the osteonic organic matrix at sites of fracture.
I
RESULTS
Fig. 1. Diagram in two projections illustrating the technique used in compressing single osteon samples perpendicularly to their a.xis. A, slide: B, coverslip: C, osteon: D, spatula.
When an osteon sample of Type I. i.e. an osteon with fiber bundles in one lamella making an angle of nearly 90” with the fiber bundles in the next, is pressure-loaded perpendicular to its axis- i.e. along one of its diameters-arc-shaped, concentrically distributed cracks appear. These are concentrated within the central portion of the anular sample and are found within all the four quadrants obtained constructing two orthogonal lines one of them representing the direction of loading. More exactly, each series of cracks is grouped round this line of loading and occupies a segment which subtends an
230
A. ASCENZI,
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angle of between 20 and 50” (Fig. 2). The polarizing microscope reveals that the cracks involve dark lamellae, i.e. lamellae whose fibers have a marked longitudinal spiral course, while bright lameliae, i.e. lamellae whose fibers have an almost transversal spiral course, are unaffected. When an osteon sample or Type I is pressure-loaded perpendicular to its axis, and the direction of loading is then changed continually by rotation, the cracks grow longer, join up and eventually become circular. Even after such rotation the cracks only alfect lamellae whose fibers have a marked longitudinal spiral course (black under the polarizing microscope), while lamellae whose fibers have an almost transversal spiral course (bright under the polarizing microscope) are unaffected (Figs. 3 and 13-16). The results reported above were obtained with low or fairly low loads. With higher loads lamellae whose fibers have almost transversal spiral course are severely damaged too, and radial and tangential cracks are seen in them. The worst damage and cracks are seen in the lamella which lies round the haversian canal. Damage falls off rapidly in the successive lamellae of the same type, so that the peripheral lamellae of this type are almost undamaged. When the compressive load applied to lamellae whose fibers have a marked longitudinal spiral course is increased, the circular cracks become more conspicuous, but no other effect is apparent. In osteon samples of Type I, the circular cracks produced in lamellae whose fibers have a marked longitudinal spiral course are actually deep fractures which extend to the full depth of each lamella. This has the advantage of enabling one to isolate individual lamellae whose fibers and crystallites have an almost transversal spiral course, using a steel needle for microscopic dissection. The first step in this process is shown in Fig. 4. Here an osteon sample of Type I is shown as seen, (a) under an ordinary microscope and (b) under a polarizing microscope. A lamella which
and A. SIMKIN
appears bright under the polarizing microscope, because its fiber bundles have an almost transversal spiral course, has been partly separated from the next lamella of the same type. Under the ordinary light microscope it is easy to see that pieces of the adjacent cracked lamella whose fibers have a marked longitudinal spiral course are still attached to it on both sides. In Figure 5a a bright lamella which has been completely isolated is seen under an ordinary microscope. It is worth mentioning that the thickness of the lamella specimen (5 /.L) is only l/6 of the height (30 p)-the same height as that of the osteon sample itself. Figure 5b shows another isolated lamella, this time under the polarizing microscope. It is not only possible to isolate individual bright lamellae; it is also possible to cut and open lamellae at any point on their circumference, as seen in Fig. 6. Figure 6a shows a small osteon sample consisting of seven lamellae; cracks are found in all the lamellae whose fiber bundles have a marked longitudinal spiral course. Figure 6b shows the same osteon under the polarizing microscope during separation of its bright lamellae. In this figure two findings merit attention. First, each bright lameila has attached to it, on at least one side, a thick layer, or all, of the adjacent dark lamella whose fibers and crystallites have a marked longitudinal spiral course. Second, the outermost lamella has straightened out, and only its thickest parts are now birefringent; the parts which now lie flat on the lateral surface no longer appear bright, because they are now insufficiently thick. Figure 7 shows one of these isolated, opened, flat-laying lamellae viewed under ordinary microscope. The electron microscope offers detailed information about how cracks are formed in lamellae whose fibers have a marked longitudinal spiral course. In osteons subjected to low-level, rotating compression, very small ultrastructural cracks appear in these lamellae. Some are irregularly shaped, and some form a
Fig. 9. Fully calcified osteon sa!!:le 30 of Type I pressure-loaded perpendicularly to its axis. The mows indicate the direction of loading. Arc-shaped cracks are visible. X 29%
Fig. 3. Fully calcified osteon sample 9 of Type 1 (a) before loading: Co) after pressure-loading, with continual change in direction of loading, circular cracks are visible: Cc) under the polarizing microscope the cracks do not appear to affect the bright lamellae. Y 220.
Fig. 4. (a) A partially removed lamella from the fully calcified osteon sample 9 of Type I. (b) Under the polarizing microscope the same lamella is bright. X 50.
?g. 5. (a) An isolated lamella. x 320. (b) An isolated bright lamella as seen under the polarizing microscope. x 500.
a _I.
Fig. 6. (a) The fully calcified osteon sample 7 of Type I after loading. lb) Two bright lametlae
of the same sample have been opened and partially isolated. Semipolarized light. x 300.
Fig. 7. An isolated and opened lamella is spread out.
x
450.
Fig. 8. Fully calcified osteon sample 18 of Type 11 after pressure-loading with continual change in direction of loading. x 230.
Fig. 9. .-I fully calcified osfeon of Type II as seen under polarizing microscope. X 230.
Fig. 10. (a) Fully calcified osteon sample 52 of Type II deprived of its internal bright iamella and pressureloaded perpendicular to its axis. The direction of compression is the same as in Fig. 2. (b) The same sample after pressure-loading with continual change in direction of loading. x 230.
Fig. 13. A section of the fully calcified osteon samr rie 50 ofType I. Before loading. X 1100.
‘ig. 11. After gf .^ _..:_
:-loading perpendicular _^^ :^ ,a:_^^I:-..
Fig. 15. After strong pressure-loading perpendicular . ” . to the axls and with contmual change m direction ot loadmg.
Fig. 16. The strongly compressed sample as seen under the polanzmg microscope.
Fig. I’. Electron micrographs of an ultrathin section from decalcified ostwn 41 of T:.pe I. (T) Lamella whose fibers have a transversal spiral course: CL) lamella u hose fibers have a longitudinal spiral course. The lamella whose fibers have a longitudinal spiral source lhow A frx:s:e (F) with many isolated fibrils. X 11.000. Fig.
18.Some
separated
fibriis surrounded
by fringes of tom cement sabstancs.
T 7h.W).
MECHANICAL
PROfERTIES
OF SINGLE
regular curve; all of them tend to take on a circular orientation. As compression is gradually increased, the cracks lengthen, until their ends join up, so resulting in a smaller number of longer cracks. Sometimes the first cracks to form occur in the central or paracentral zone of these lamellae. In this case an almost complete circular crack is produced, dividing the lamella into two unequal parts, each of them still in contact with the adjacent lamella, whose fibers have an almost transversal spiral course. In other cases the cracks spread along the boundaries between two adjacent lamellae, leaving fib& bordering a lamella with a longitudinal spiral course on a side and fibrils bordering a lamella with a transversal spiral course on the other. A much rarer pattern is that of a crack which takes on a circular orientation and then penetrates into a lamella whose fibers have an almost transversal spiral course. The crack then spreads through the outermost fibrils of this lamella, without, however, penetrating any deeper. The impression is thus given that the lamellae whose fibers have an almost transversal spiral course reject cracks, at most allowing them to develop at their extreme periphery, and only on very rare occasions. All these data help to explain the morphology of isolated lamellae whose fibers have an almost transversal spiral course, when seen under ordinary light microscopemost pertinently the fact that these lamellae have pieces or layers of at least one adjacent lamella with longitudinal fibers, or even one whole lamella of this type attached to them. So far we have discussed electron microscopic findings above a certain minimum frequency. Brief mention should also be made of an exceptional finding- that of fiber bundles which run like a bridge connecting two successive lamellae whose fibers have an almost transversal spiral course. When cracks start in these bundles they sometimes spread deeply into the lamellae on either side, producing radial fractures.
OSTEONIC
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23 1
The electron microscope also yields interesting data on the earliest changes in collagen fibrils where cracks start. As shown in Fig. 17, the crack in the central portion of the lamella whose fiber bundles have a marked longitudinal spiral course clearly involves the separation of fibrils, many of which are no longer attached to each other. Photographs of this phenomenon can be obtained at very high magnification, as in Fig. 18. There is evidence here that fibrils are not cut because around and, sometimes, between them, fringes of other material are seen - material probably consisting of tom cement substance. We will now consider changes produced in osteons of Type II i.e. osteons with fiber bundles which have a marked longitudinal spiral course. It may be worth recalling at this stage that in these osteons the innermost lamella usually consists of fibers and crystallites which have an almost transversal spiral course. When a low compressive load is applied to osteons of Type II perpendicular to their axis but with continual rotation, a circular crack is produced at the outer boundary of the innermost lamella. and this crack then spreads into the adjacent lamella whose fibers have a marked longitudinal spiral course. If the compressive load is raised, the innermost lamella becomes severely deformed and complete radial fractures appear in it. Some of these then branch outwards from the original circular crack, and spread toward the periphery. They occasionally reach the very edge of the osteon sample, but this is a rare occurrence, so that an unbroken layer is usually seen bordering the osteon. Figure 8 shows a loaded osteon sample of Type II as seen under an ordinary microscope. The lamella round the central canal has been fractured at four points, and irregular radial cracks are seen spreading from the original cracks toward the periphery. Such radial cracks often follow an irregular or zigzag course. This is because- as shown in Fig. 9-osteons of Type II sometimes con-
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tain small fiber bundles which have a transversal course and appear bright under the polarizing microscope. It would hardly be surprising if these fibers were responsible for deviating the radial course of such cracks; and this view is borne out by electron microscopic data. An attempt was made to investigate thoroughly the behavior of osteons of Type II, from which the innermost bright lamella has been removed. When a sample of this kind is pressure-loaded along one only of its diameters without rotation, radial cracks appear starting at the two opposite points of the internal border lying on the loaded diameter, where tension is greatest. Figure 10a shows an osteon sample so prepared and loaded along its vertical diameter. Two cracks have developed along this diameter, each starting from the internal border and lengthening radially. When the internal lamella is removed from osteons of Type II and a compressive load is then applied perpendicular to their axis. with continual rotation, the cracks increase in number (Fig. lob). The situation is very like that of intact osteon samples of Type II after the internal bright lamella has been fractured. It must therefore be concluded that the circular crack seen at the outer boundary of the innermost bright lamella is attribuable to the presence of that lamella. A comparison between the results for osteon samples of Types I and II suggests that the circular cracks or fractures produced in lamellae whose fibers have a longitudinal spiral course in osteons of Type I, are due to the presence of neighbouring lamellae whose fibers have an almost transversal spiral course. No lamellae of the latter type are present in osteons of Type II, except for the innermost and outermost lamellae and fragmentary lamellae consisting of rare, very thin fiber bundles with an almost transversal spiral course; this would explain why cracks in these osteons have a strong tendency to follow a radial course. The stage reached by calcification-initial
and A. SIMKIN
or far advanced-had no the results for osteons of Circular fractures develop ones in Type II, whatever fication.
significant effect on either Type I or II. in Type I and radial the degree of calci-
DISCUSSION
The main focus of interest provided by the results of the present investigation is the difference in behavior between osteon samples of Types I and II. When fairly low compressive loads are applied perpendicular to the axis of osteon samples of Type I, circular cracks appear, but only in lamellae whose fibers have a marked longitudinal spiral course, while lamellae whose fibers have an almost transversal spiral course are unaffected. This difference is particularly marked in lamellae which lie near the central canal. Cracks also appear in peripheral lamellae whose fibers have a marked longitudinal spiral course, but only when osteon samples are strongly loaded, and then some of the innermost lamellae whose fibers have an almost transversal spiral course may be radially fractured. While the compressive load is still low the electron microscope reveals ultrastructural cracks found exclusively in lamellae whose fibers have a marked longitudinal spiral course. Some of these cracks are irregular in shape, and other form a regular curve: all tend to take a circular orientation. As the load is increased, the cracks lenghten and join up. In some cases cracks develop in the central or near-central zone of lamellae, as well as near the boundary with neighbouring lamellae whose fibers have a transversal spiral course. In all cases the cracks run between. not across, collagen fibrils, separating them and indicating that the interfibrillar substance is considerably less resistant than the fibrils themselves. Unlike osteon samples of Type I, samples of Type II show radial cracks running from the central canal toward the periphery after pressure loading perpendicular to their axis.
MECHANICAL
PROPERTIES
OF SINGLE
To find an explanation for these differences between the two types of osteon sample. the stress distribution pattern in a ring of isotropic, elastic material was analysed using the solutions of Timoshenko (1922). The problem is treated as a two-dimensional one. with polar coordinates, p, 0 (Fig. 11). For each point there are three stress componentsnormal radial S,.,, normal circumferential SHH and circumferential shearing stress T,.+ The stresses depend on the ratio between hole radius and ring radius. r/R and the formulae involved are of formidable difficulty so that a computer had to be used to calculate them for r/R = 0.5. The highest value of all the stresses is obtained for SHHat p/R = 0.j and 0 = 0” or 180”. that is. at the circumference of the hole and on the diameter of the applied load. If a ring of isotropic material is so compressed, it does in fact fracture at these points. These fractures occur in osteon sample of Type II after removal of the innermost lamella consisting of fiber bundles which have a transversal spiral course. Osteon samples of this kind. as seen in Fig. 10a can, in fact, be considered isotropic, if the problem is considered in two dimensions only. The normal radial and shear stresses vanish at the outer and inner surfaces and rise inside the ring, with the highest values for both S,., and T,.~ for H between 20 and 25” and
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p/R between O-6 and 0.75 (Fig. 12). These values are about 24 and 5 per cent of the highest Seti for rrH and S,, respectively. A ring of an elastic isotropic material which has. however. a low tensile strength in a radial direction and/or a low shear strength in the circumferential direction, will fracture first in this region. along circumferential lines. The fracture lines seen in the compressed osteon samples of Type I are circumferential too, though they initially occupy larger angular ranges than those calculated, as seen in Fig. 2. This may be due to a different stress distribution pattern brought about by the presence of lamellae, but it is very probable that these fractures start to form at sites of high circumferential and radial tensile stress. Lamehae whose fiber bundles have a longitudinal spiral course can be expected to have low resistance to such stresses, because the cracks can find many weak interfaces between fib&, as the electron microscope clearly shows. This last explanation is indirectly supported by Dempster and Liddicoat’s investigation ( 1952) on the effect of direction of loading on fractures produced by compressive strength. They used cubes of com-
\ \ 5
%---r I
I -
5
’
1I
-----
‘ai j/ ---. _---
x // \\ /
-__
_
/I
1\\ 1 I \
/k/ / ;
Fig. I I. Definition of stresses in polar coordinates for a ring under diametral compression.
Fig. 11.Regions of high tensile radial and circumferential shear stresses in a ring under diametral compression.
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A. ASCENZI.
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pact bone and tested them along their long, radial and tangential axes. They were able to show that the oblique rolling, sliding type of failure in cubes of bone under lateral compression suggests a special weakness in shear parallel to fibers. Similar results are given by compression tests on hollow cylinders of bone loaded longitudinally by Carothers, Smith and Calabrisi (1949), who reported longitudinal splitting and shear. In the same way Rauber’s tests (1876) clearly show that the shearing strength of compact and spongy bone is considerably greater when loaded perpendicular to the direction of fibers than when loaded parallel with fibers. The results obtained during the present investigation for the mechanical properties of single osteonic lamellae are supported by those of our previous investigations on the tensile, compressive and shearing properties of single osteons along their axis (Ascenzi and Bonucci, 1964, 1965, 1967, 1968, 1971, 1972; Ascenzi, Bonucci and Checcucci, 1966). The ultimate compressive strength and the modulus of elasticity to compression are greatest for osteons having transversally oriented fiber bundles in successive lamellae, while the ultimate tensile strength and the modulus of elasticity to tension are greatest for osteons having a marked longitudinal arrangement of fiber bundles in successive lamellae. When osteons are loaded by shearing along their axis, those having a marked longitudinal spiral course of the fiber bundles in successive lamellae are least able to support stress. All these data and in particular those mentioned in the last paragraph corroborate the view that in osteons the compactness of bone is strengthened by the presence of lamellae whose fiber bundles have a marked transversal spiral course. On the other hand the fiber bundles which have an almost transversal spiral course form a system of rings which prevent bending. These twin concepts may help to explain the function of various types of bone in mechanical terms.
and A. SIMKIN
For instance, the main function of the human femoral dyaphysis is to support compressive loads, and among the osteons found in this type of bone those of Type I predominate. The results obtained here apply both to fully calcified osteons and osteons at the initial stage of calcification. The last topic to be discussed is the possibility of applying the techniques used here to future quantitative investigations on the tensile properties of isolated lamellar samples. It appears self-evident that it will be necessary to use osteons of Type I i.e. osteons with fiber bundles in one lamella making an angle of nearly 90” with the fiber bundles in the next, because only these are suitable for the isolation of single lamellae. Furthermore only lamellae whose fiber bundles have a marked transversal spiral course can be isolated because those whose fiber bundles have a marked longitudinal spiral course become severely damaged and fractured when subjected to compressive stress perpendicular to the osteon axis. The samples to be used will have a circular, viz anular, or rectangular shape. The last type will be obtained by opening the anular samples and applying a radial section to them. At the time of writing no final decision has been taken as to the method to be followed in carrying out this research program, but we hope to achieve results in the future. AcXno~rledgement-The authors are deeply grateful to A. Benvenuti for excellent technical assistance during the course of the present investigation. REFEREXCES Ambartsumian, S. A. (1966) Some current aspects of the theory of anisotropic layered shells. In Applied Mechanics Surceys. (Edited by H. N. Abramson, H. Liebowitz, J. M. Crowley and S. Juhasz) pp. 3013 14. Spartan Books, Washington. Ascenzi,. A. and E. Bonucci (7964) The ultimate tensile strength of single osteons. rlcrn annr. 58. 160-l 83. Ascenzi, A. and E. Bonucci (1965) The measurements of the tensile strength of isolated osteons as an approach to the problem of intimate bone texture. In Calcified Tissues. Proc. 2nd European Symposium on Calcified Tissues. (Edited by L. J. Richelle and M. J. Dallemagne) pp. 325-335. University of Liege.
MECHANICAL
PROPERTIES
OF SINGLE
Ascenzi. A. and E. Bonucci (1967) The tensile properties of single osteons. ,-lnnr. Rec. U&375-386. Ascenzi. A. and Bonucci. E. ( 1968) The compressive propenies of singleosteons. rlnur. Rec. 161. 377-392. Ascenzi. A. and Bonucci. E. (1971) A micromechanic investigation on single osteons using a shearing strength test. lsrnel J. ‘bled. Sci. 7.47 l-472. Ascenzi. A. and Bonucci. E. (1972) The shearing properties of single osteons. Anat. Rec. 172.499-5 10. Ascenzi. A., Bonucci E. and Checcucci A. (1966) The tensile properties of single osteons studied using a microwave extensimeter. In Studies on the Anatom,v curd F!rnction of Bone und Joints. (Edited by F. G. Evans) pp. I2 l-i4 I. Springer, Heidelberg. Boothroyd. B. (1961) The problem of demineralisation in thin sections of fullv calcified bone. J. Cell Biol. 20. 165-173. Carothers. C. 0.. Smith. F. C. qnd Calabtisi. P. (1949) The elasticity and strength of some long bones of the human body. Naval Medical Research Institute. ProjesrNXlOOl 056.02.13. Dempstcr. W. T. and Liddicoat, R. T. (1952) Compact bone as a non-isotropic material. Am. J. Ancx 91. 33 l-36’.
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Frocht, M. M. (1948) Photoelasricity. Wiley. New York. Millonig. G. (1962) Further observattons on a phosphate buffer for osmium solutions in fixation. In Electron &licroscop.v.
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of the jrh
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(Edited by S. S. Breese) Vol. 2. p. 8. Academic Press, New York. Rauber. A. A. (1876) Ektsricit‘it and Festiukeir der Knochen. Engelmann, Leipzig. Stawsky. Y. and Hoff, N. J. (1969) Mechanics of composite structures. In Composite Engineering Laminates. (Edited by A. G. H. Dietz) pp. 5-59. The MIT Press, Cambridge Mass. Timoshenko, S. (1922) On the distribution of stresses in a circular ring compressed by two forces acting along a diameter. Phil. .Clap. 44. 1014-1019. Tischendorff, F. ( 195 I) Das Verhalten der haverschen Systeme bei Belastung. Rorc.r’.4rch. 115.3 18-331. Tischendorlf, F. (1952) Quantitative Beobachtungen ueber das Verhalten der haverschen Lamellen bei Belastung. Ro~r.r’.-Irch. 146, l-20. TischendorK F. (1954) Die mechanische Reaktion der haverschen Systeme und ihrer Lamellen auf experimentelle Belastung. Rwr Arch. 116.66 I-704. tron
.Clicroscup~