Development of a method for quantitative stress analysis in bones by three-dimensional photoelasticity.код для вставкиСкачать
THE ANATOMICAL RECORD 206:227-237 (1983) Development of a Method for Quantitative Stress Analysis in Bones by Three-Dimensional Photoelasticity ROSSELLA BIANCHI, PAOLO CLERICI, LAURA VIZZOTTO, A N D ALBERT0 MIAN! Istituto di Anatomia Umana Normale, Uniuersita di Milano (R.B., L. K, A.M.), and Dipartimento di Meccanica, Politecnico di Milano (f? C.), Italy ABSTRACT Comparative critical examination of methods suitable for studying stress in bones have shown that the three-dimensional photoelastic method is one of the most reliable. Described herein is the method for obtaining, by fusion, full-scale models in epoxy resin, that are exactly equivalent to external shape of the prototypes. This technique offers the advantages of being applicable without variation to any bone segment and of enabling a large number of additional resin castings to be made from the same mould. Hence it is possible to produce a very large number of copies of the same bone segment that will be suitable for comparative studies of different load situations. As a n example, quantitative data expressing both surface and internal tension trends in the proximal third of a normal human femur are given. It is well known that a bone exposed to mechanical stress becomes elastically deformed and develops corresponding internal tensions which, influencing its biodynamics, improve the bone structure, making it mechanically better suited for the stresses applied. The internal structure of bones thus is in direct relation to the mechanical stresses exerted on them. The correlation between mechanical requirements and bone structure, with particular reference to the arrangement and orientation of cancellous bone trabecules, has been known since early in the last century. Ward (1838) was the first to provide detailed description of cancellous bone in the femoral neck, by identifying the three main trabecular systems; furthermore, comparing the construction of femur epiphysis to that of a crane, he identified a zone subjected to compression along the medial cortical bone and a zone of tension along the lateral cortical bone. These results were later confirmed by Wyman (1857). Humphry (1858) observed that in the femoral neck the main trabecular systems cross each other at right angles, and that the trabeculae are arranged perpendicularly in relation t o the articular surface of the femur head. In 1867 von Meyer published, according to data of Culmann (1866), 0 1983 ALAN R. LISS, INC. his theory regarding cancellous bone architecture, with a definition of trabecular trajectories arranged along the principal stress lines. On the basis of Meyer’s and Culmann’s data, the close relationship between the function and architecture of cancellous bone was demonstrated by Wolff (1892), and developed and confirmed by Roux (1893) in his theory of functional adaptation. More recently the now undisputed link between shape, structure, and mechanical function of bone has been verified in a series of studies which have used more sophisticated and accurate methods of investigation. Among these, the most reliable are: finite elements (theoretical calculation method), electric strain gauges, brittle coatings, holographic interferometry, and photoelasticity methods (all experimental methods). The finite elements method applied to bones (Zinkiewicz, 1971; Scholten, 1976, on the human femur; Thresher and Saito, 1973; Farah et al., 1973; Lavernia et al., 1981, on the human tooth; Chand et al., 1976, on the human knee joint; Knoell 1977, on the human Received December 9,1982; accepted March 11, 1983. 228 R. BIANCHI, P. CLERICI, L. VIZZOTTO. AND A. MIANI mandible; Hayes et al., 1978, on the human tibia) makes it possible to determine the stresses produced by applied loads, but only after having determined the law that governs the relations between stresses and strains (the law of constitution) and geometrical shape of the bone. The simplest law of constitution assumes that stresses are proportional to strains, but in the case of bones, proportionality may be different among different sectors of the same bone, such as, for example, between cortical and cancellous bone. These factors are important as regards calculations complexity. The method also involves a description of the geometrical shape of the bone by means of a model formed by solid two- or threedimensional elements (representing the bone structure more accurately). Generally these are closer together in instance of more irregular surfaces. This is, in fact, the case for bones. With the finite elements method it is also possible to consider bone material anisotropy, but in this case calculation costs would be very high. For all these reasons the reliability of results may need experimental verification. On the other hand, electric strain gauge analysis determines directly the strains on the bone surface (Evans, 1953, on the canine tibia; Hirsch and Brodetti, 1956a,b; Indong and Harris, 1978; Jacob and Huggler, 1980, on the human femur; Fisher et al., 1976, on the swine skull; Wright and Hayes, 1979, on the bovine tibia and metatarsals). These measurements have also been carried out “in vivo” on various animals (Lanyon, 1972, on the sheep vertebrae; Lanyon, 1973, on the sheep calcaneus; Baggott and Lanyon, 1977, on the sheep and goat radius; Carter et al., 1980, on the canine radius and ulna; Caler et al., 1981, on the canine radius and femur). This method is based on variation in the electric signal of transducers (electric strain gauges) secured to the bones, this variation being proportional to the strains which the transducers undergo when the bone is loaded. This technique gives very accurate data on the state of stress but is limited by the fact that the strain gauge, being of rather small dimensions, reveals only what happens on a relatively small surface, and thus provides only local information. The brittle coatings are lacquers which are sprayed or spread onto the bone surface (Gurdjian and Lissner, 1945, 1946, 1947, on the human skull; Evans and Lissner, 1948; Evans et al., 1953; Pedersen et al., 1949; Kalen, 1961, on the human femur) and break orthogonally in the direction of maximum strain when the local deformation limit-value is exceeded. This method gives a total view only of the surface stress state and does not specify by how much the deformation threshold value is exceeded. Holography is a method that is not as well documented in the literature on bone biomechanics (Fuchs and Schott, 1973, on the human skull; Hewitt, 1977, on the primate skull; Kragt, 1979, on the human skull). It is based on interferometry and optical diffraction. This technique makes it possible to register clearly only the displacements normal to the surface, while the total displacement, and thus the stresses, can be obtained only by very difficult experimental processes. Photoelasticity is based on the properties of certain plastic materials which are transparent to light and become birefringent when loaded and observed in a polarized light. It is possible to see (with a polariscope) lines (or fringes) of differing optical intensity which express the trajectories and the entity of the stress in the element. With this method, photoelastic coatings are applied to the bone (in which case the results are essentially quantitative, particularly in the most irregular areas) or models (two- or three-dimensional) are made from special plastic resin. We think that of all these experimental methods, photoelasticity applied to models is overall the most satisfactory. In fact, this method provides a quite accurate overall view of the stresses, both on the surface and inside the bone, while other methods either yield only local results (electric strain gauges), even if they are very accurate, or overall results (photoelastic coatings, brittle coatings), which are only superficial and of low accuracy, or overall results of high accuracy (holography) but that are restricted to the external surface. One of the first applications of the photoelasticity technique to bones was that of Milch (19401, who utilised it to study bone shape on two-dimensional models. Photoelastic studies were later carried out on biological models, both with two-dimensional models (Pauwels, 1951, 1955, 1965, 1973; Kummer, 1956; Fessler, 1957, on the human femur; Maquet et al., 1966; Maquet and Pelzer, 1977, on the human knee) and with three-dimensional models (Knief, 1967, on the human EXPERIMENTAL STRESS ANALYSIS IN BONES femur; Johnson et al., 1968; Farah et al., 1973, on the human tooth; Yoshizawa, 1969, on the human spine; Chand et al., 1976, on the human knee). The most common photoelastic method reported in the literature employs two-dimensional models obtained directly from a plate of plastic resin. This method is, therefore, simple, rapid, and inexpensive. Fundamental to this field are the studies carried out by Pauwels that have extended and elaborated what was already known about bone biomechanics. However, the two-dimensional photoelastic method has a significant limitation, for real structures (such as bones) are usually three-dimensional and are loaded spatially. Quantitative studies of stresses on two-dimensional models may produce results not fully applicable to real biological situations. Thus it appears that photoelastic analysis with three-dimensional models is the most reliable method, because it provides a generally accurate picture of stress distribution a t both the external surface and inside the bone. It is, of course, necessary that the real conditions of the actual bone be simulated on the model, and that the different characteristics of resin and bone tissue be considered. Three methods of analysis by three-dimensional photoelasticity are available: the sandwich method, the diffused-light method, and the stress-freezing method. In the sandwich method, a transparent sheet of photoelastically active resin is inserted in a transparent three-dimensional body of very low photoelastic activity. Thus the study is of a two-dimensional type as only the inserted sheet is photoelastically active. It is possible to apply successive and different load systems to the “sandwich,” which is useful, but the study of internal stresses is limited to the region where the material is photoelastically active. Problems of adhesion among the parts and the duration of usefulness sometimes make this method rather difficult. Furthermore, the model as a whole requires final processing after the glueing of the parts and so it is practically impossible to reproduce very irregularly shaped elements with a high degree of accuracy. The diffused-light method is based on the principle that, in a transparent field (specimen) crossed by a beam of light, a light is emitted, plane polarized, perpendicular to the incident beam. Thus, the specimen acts as analyser and polarizer. The optical fringes that appear (oriented perpendicular to the 229 incident beam) are due to the sum of the photoelastic effects along the path of the same incident beam; to obtain the single local stress values, it is necessary to perform some analytical and trigonometrical operations, which sometimes makes the method difficult and inaccurate. The stress-freezing method, described below, makes it possible to visualize the state of stress of the model point by point, both on the external surface and inside the bone, even taking into account that it is practically impossible to faithfully reproduce the organization of cancellous bone. The last technique (stress-freezing) has been chosen, for our study, because in our judgment it is the best of the experimental methods. It also has certain advantages over the finite elements method, which, it will be recalled, also takes three-dimensionality into account. In fact, as already pointed out, the finite elements method does make it possible to provide a good representation of a structure which is as spatial and irregular in shape as a bone, especially if the elements are three-dimensional and locally very dense. However, it presents disadvantages of preparation time, and construction costs are considerable, and the division into elements must be begun again with each shift in bone segments studied. Once techniques have been perfected for the preparation of three-dimensional photoelastic models to be frozen, the stress-freezing method presents no such difficulties when one moves from one bone to another, and models can have the same external shape a s the actual bone segments. It has one disadvantage: Each model allows the study of only one loading situation, while the finite elements method permits a variety of loads to be applied. Thus both methods, in our view, are of comparable value; the choice of the experimental method was prompted by the desire to have a n accurate method that, once refined, might become almost routine for subsequent comparative studies and any kind of bone. In this work, therefore, our aim has been 1)to perfect a technique for the preparation of three-dimensional models of any bone, not by manual shaping from blocks of resin (which may require considerable geometrical approximations) but with a technique which reproduces the original external shape faithfully, in full scale; and 2) to carry out a n accurate three-dimensional quantitative 230 R. BIANCHI, P. CLERICI, L. VIZZOTTO, AND A. MIANI analysis of stress distribution in the proximal third of the normal human femur. The subject was chosen because this particular bone segment has been thoroughly discussed in the literature and so our data can be compared with those of earlier workers. We see the technique as suitable for further biomechanical investigations on other bones. MATERIALS AND METHODS The full-size model of a normal human femur was obtained by placing the bone in a suitable aluminum container into which was poured a n elastomer formed from two liquid components (RTV-M 533 with T 35 Wacker hardener) which, when mixed in appropriate proportions, solidify. This produced a negative surface shape or mould of the bone, having a very high level of precision. Later, a thermosetting epoxy resin cast (two components: CIBA Araldit B 46 + HT 903) was made in the same mould. The femur model so obtained may be seen in Figure 1. It should be noticed that the elastomer must have a thermal expansion coefficient very close to that of epoxy resin, so as to avoid separation phenomena between the two materials during cooling. The method followed for the study of the stresses in the loaded model is known as the “stress freezing process” (Avril, 1974). The model was heated in a oven at a temperature of 140°C; a suitable load was applied; then it was slowly cooled, still under load, to room temperature. In such a way, the mechanical stresses typical of the loaded condition were locked (“frozen”) in the model, to obtain a permanent effect of optical birefrangence (fringes) proportional to such mechanical stresses. As the elastic and optical characteristics of epoxy resin vary with temperature, the same thermal cycle was applied to a calibration disc so as to,obtain the two fundamental parameters for the material of the model: Young’s elastic modulus (E) and the fringe value (F). A very significant static condition was experimentally reconstructed-i.e., that corresponding to erect position with the body weight on one limb only. The situation is the most stressful, for in these conditions a force F higher than the weight P of the body acts on the femur head (Fig. 2). This is true because the action line of the weight P has a lever arm relative to the rotation center of the ilium: the moment due to the weight P is u balanced by the action T of pelvis-trochanter abductor muscles. Figure 3 shows the model of femur with the load applied and the tie rods schematizing the abductor muscles. The choice of the load applied is dependent on the mechanical theory of the models (Wilbur and Norris, 1950); in fact, as the bone and the epoxy resin have different elastic moduli, the loads applied are different, if we wish to have the same deformation on each. The following condition must be respected: Fig. - 1. a) Normal human femur. b) Femur model in epoxy resin. 231 EXPERIMENTAL STRESS ANALYSIS IN BONES ’/ I I I T- P h k Fig. 2. Schematization of the load acting on the femur model where: P, = load applied to the model, P, = load applied to the prototype (bone), Em = Young’s modulus of the model, E, = Young’s modulus of the prototype, L, = linear dimension of the model, L, = linear dimension of the prototype. The stresses in the bone and in the model are in this ratio: where: Of course, in our case, the ratio LmL, is equal to 1because, as previously stated, the model is a full-size model of the prototype. From the calibration of the disc, the elastic modulus of the resin is equal to: Em = 19 N/ mm2, and as the bone elastic modulus corresponds to (10 + 20) lo3 N/mm2, in order to obtain the same deformations it is necessary to apply to the model a weight P approxiof the corremately equal to (1 + 2 ) sponding real body weight. The optical response of the loaded model is proportional to the mechanical stress according to the relationship: n = -u . s F a, = mechanical stress in the model, a, = mechanical stress in the prototype. in which n is the number of fringes and s 232 R. BIANCHI, P. CLERICI, L. VIZZOTTO, AND A. MIANI Fig. 3. Model of the bone under load and Calibration disc placed in the oven. represents the model thickness. It can be seen that n also depends on the fringe value F of the material, so that with the same ( T . s the number of fringes is inversely proportional to the value F. The calibration of the resin used gave a value F = 0.275 N/mm fr; with this value of F, the application of a weight P corresponding to the average body weight of a man would have produced too low an optical response, so we preferred to use a slightly higher load. Thus, we applied a load of 2.45 N (0.250 kg). Figure 4 shows the proximal third of the frozen model, inserted in the polariscope, in which appear the fringes caused by the stresses. Such fringes represent the average of the local stresses, which usually vary along the thickness of the model crossed by the single optical beam; so it is necessary to obtain rather thin “slices” (sections) along which the stresses may be held sufficiently constant. It is important to mention that such cuts, if carefully made with a machine tool, will not introduce optical disturbance signals. In this way the slices will maintain, completely unchanged inside them, the mechanical stresses caused by the load applied on the three-dimensional model. The analysis was carried out on two different frontal sections, one median and the other one displaced with respect to the median by 10 mm. The choice of frontal sections was made on the basis of the literature indicating them to be the most highly stressed. The more constant the state of stress (in both value and direction) along the thickness, the more accurate and reliable are the data obtained from the fringe readings, and thus the tension values. This is usually all the more so in thinner sections. Unfortunately, however, as we have seen before, the optical response (the number of fringes) is higher at the same stress in thicker sections. Thus there are two contrasting requirements. Therefore we considered the following as the best way to operate: obtaining, a t the beginning, slices of 8.2-mm thickness, noting the data, and then decreasing the thickness to 5.8 mm, noting further data, and finally decreasing it again to a thickness of only 3.6 EXPERIMENTAL STRESS ANALYSIS IN BONES 233 mm. The differences in data collected between the first slice and the subsequent ones express the stress values in the removed tract. The method allows us to obtain two kinds of information. If the slices are observed in circular polarized light, the observation of fringes gives the point-by-point difference between the maximum and minimum stress which acts on the plane of the slice. On the borders, moreover, one of the two stresses is usually known, after determining the position of the loads applied; thus it is possible to obtain the value of the single stress: (u = n F -1 S from the fringe order without any difficulty. If the slices are observed in polarized plane light, it is possible to see not only the preceding fringes but also some lines (isoclinics) which give the direction point of such maximum and minimum stresses in the plane. RESULTS AND CONCLUSIONS The results obtained from the fringe readings provide a variety of information: the Fig. 5. Isochromatic fringes in the median frontal section of the femur model. ~ i 4, ~Interference , fringes in the three-dimensional model. toto,, femur isochromatic fringes (Fig. 5) on the borders of the sections show that negative stresses (compressions), in agreement with the qualitative data in the literature, are in modulus greater than the positive ones (tensions) in all the observed sections. This is valid not only in median sections but also in the posterior ones. Figure 6 shows the stress values both on the borders and inside the median sections; as can be seen, their numerical values in the corresponding geometrical points, according to the different thicknesses of the median section, are not constant even if the qualitative trends are still of the same kind. In particular, the maximum values, in the median slice, increase when moving from the greatest thickness to the intermediate and then to the minimum one. This means that the state of stress is different in the subsequent points examined at the same level. The maximum variations are of about 15%. In the median slice it is possible to see that stresses tend toward zero in the trochanter zone except for the muscle insertion points. 1,7.10-‘ +12,4.10-’ +10,1.10-2 +6,7.10-2 t3.4.10-2 t + 3.7 ‘ 10-2 +2,1 .10-2 + 5,3.10-2 b Fig. 6. Stresses on the borders and inside the median frontal section of the femur model for decreasing thicknesses (a-c). a C EXPERIMENTAL STRESS ANALYSIS IN BONES t 1,5.16' - 3.8. lo-2 +38.1Ci2 +78.10' + d,4.16' -6.9.16' - 1,6 * lo-' t13,O.16* 235 flow in the central slice, surveyed by 10" in 10" per increasing angle clockwise. The critical analysis of the results obtained with three-dimensional photoelastic method shows that the values of stresses obtained vary in relation to slice thickness; this indicates that the state of stress is variable along the thickness, which implies that schematization with a two-dimensional model is not completely reliable, if not only qualitative but also quantitative results are needed. The isoclinic fringes show that stress trajectories in the three-dimensional model coincide with the orientation of trabeculae, as is shown by the extensive data in the literature on this subject. Similar and equally valid results can be obtained also by applying the previously mentioned finite elements method, as underlined earlier, which requires a calculation schematization necessarily different for each bone. Even though the technique for the preparation of the three-dimensional model 3,6mm Fig. 7. Stresses on the borders and inside a frontal section displaced by 10 mm with respect to the median one. where the stresses, purely at a local level, reach values corresponding to about half the maximum values found in the diaphysis tensile area. It is also interesting to note that, in the loading situation studied, the neutral axis is closer to the tensile area than to the one compressed in the diaphysis, while going up the femur head it continually moves toward the compressed side, as is shown in Figures 6 and 7. The ratio between the surveyed maximum compression, in modulus, and maximum tension is about 1.3 for the slices of maximum thickness, which then decreases toward 1.1 as the slice thickness decreases. The second kind of information which was obtained from the fringe reading indicated the direction of maximum and minimum stresses (principal stresses) which operate in the plane of the sections studied; Figure 8 shows a n example of a 20" isoclinic of the median slice of the 8.2-mm-thick model. On the other hand, Figure 9 shows the isoclinics Fig. 8. 20" Isoclinic fringes in the frontal median section of the femur model. 236 R. BIANCHI, P. CLERICI, L. VIZZOTTO. AND A. MIANI Fig. 9. Complete course of isoclinics in the frontal median section of the femur model. requires high technical competence, it makes possible the obtaining of models identical in external shape to the prototypes and, for this reason, we consider it the most valid at this stage of our knowledge of materials. Among the main advantages that this experimental methodology offers, one should mention the possibility of its application to practically any bone segment without the need for any methodological variation, and the fact that the elastomer from which the mould is made can withstand many further resin casts which harden both when cold and when hot, so that it is possible to reproduce a very large number of copies of the same bone segment which can be used for comparative studies in different loading situations. LITERATURE CITED Avril, J. (1974) Encyclopedie Vishay d’analyse des contraintes. Vishay-Micromesures, Pans, pp. 129-152. Baggott, D.G., and L.E. Lanyon (1977) An independent “post-mortem” calibration of electrical resistance strain gauges bonded to bone surfaces “in vivo”. J. Biomech., 103315-622. Caler, W.E., D.R. Carter, and W.H. Harris (1981) Techniques for implementing an “in vivo” bone strain gage system. J. Biomech., 14t503-507. Carter, D.R., D.J. Smith, D.M. Splenger, C.H. 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