- Email: [email protected]
Mechanics of oblique fracture fixation using a finite-element model
.I. B~omechamc~. 1977, Vol. 10. pp. 141-148.
Pergamon Press
Pnnted m Great Bntam
MECHANICS OF OBLIQUE FRACTURE FIXATION USING A FINITE-ELEMENT MODEL* E. F. RYBICKIand F. A. SrMoNmt Applied Solid Mechanics Section. Battelle’s Columbus Laboratories, Columbus, OH 43210. U.S.A. Abstract-An investigation of the mechanics of a compression-plated oblique fracture using a finite-element stress analysis model is described. Three aspects of compression plating are considered: the effect of tension in the plate on stresses at the fracture site, the effects of placing bone screws across the fracture surface at different angles to the plane of the fracture, and the effect of a uniform end loading to represent weight bearing on the plated bone. In addition to stress distributions and deformations, the model predicts the contact area between the fractured surfaces. In one case, the predicted area shows good comparison with a range of values for a related case reported in the literature. Results of the model show that stress distributions at the fracture surfaces were not uniform and influenced by the tension in the plate, the bone screws, and the forces applied to the bone. The regions of contact between the fracture surfaces and the orientations of stress trajectories in terms of how these are influenced by plate tension, placement of bone screws, and applied loads are also
shown. Clinically related aspects of the results are discussed.
INTRODUCTION Bones are living weight-supporting structures. When a bone is fractured, it loses its weight-supporting capacity and there is a need to maintain the fragments in a position that will permit union. In clinical situations, various procedures are used to encourage an environment for rapid reliable union. During the past two decades, internal fixation by compression plating has received increased attention as a means of attaining reliable union of fractured bones. In addition to maintaining contact between the fractured surfaces, the compression can increase the structural stability of a plated bone. The popularity of this technique is largely due to the developments by the Swiss A-O (Arbeitsgemeinshaft ftir Osteosynthesefragen) as presented for example by Miiller et al. (1965). In compression fixation, a biocompatible plate (usually metal) is “stretched” across the fracture site, placing the plate in tension and the fractured bone in compression. Early mobilization is possible with this technique. The structural characteristics of the plate and its mode of attachment to the bone are critical elements in obtaining early mobilization because the forces acting on human bones can be large. During even the most basic movements such as standing on one leg or walking, the forces acting on the head of the human femur are estimated to be 2.54.5 times body weight as reported for example by Paul (1971) and Rybicki, Simonen and Weis (1972). These forces are not only large, but repetitive in nature. Thus, fatigue, as well as overstressing, is of concern for the plate and the method of securing the bone.
* Work supported by Battelle Institute. Grant No. B-1353-0102. t Structures and Mechanics Section, Battelle’s Pacific Northwest Laboratories. Richland. Washington. Ml
The function of internal fixation is then to provide an environment that promotes healing and remains stable in the presence of repetitive loadings due to mobilization. Thus, it becomes important to have an understanding of the force interactions between the bone and bone plate. In particular, a lack of understanding of the force interaction can result in implant failure due to high stresses or fatigue failure due to the repetitive loading. Another aspect of bone fixation related to stresses and the mechanics of fixation is that there appears to be a relationship between the amount of compressive stress at the healing site and the degree of healing. Qualitative studies of this relationship have been reported by Eggers, Shindler and Pomerat (1949). Friedenberg and French (1952) described a force range that produced the best healing in the ulnae of dogs. In a more quantitative study, Hassler et al. (1974) reported a range of compressive stress that produced optimal healing in rabbit calvaria. Thus, a better understanding of the stresses on bones during fixation is important in understanding bone healing during fixation. Another reason, and equally important motivation for understanding the mechanics of fixation devices, is that without such an understanding, it becomes difficult to introduce innovations into methods of fracture treatment. For all of these reasons, the efforts of several investigators have been directed toward obtaining a better understanding of the mechanics of compression-plated fixation. A review of early uses of bone plates is given by Miiller et al. (1965). Rybicki and coworkers (1974) describe some later studies and also present mathematical models and experimental studies on the mechanics of plated transverse fractures. In other studies, Askew et al. (1975) reported a mathematical and experimental study to predict the intraosseous stress distribution for compression-plated transverse
142
E. F.
RYBICKI and
fractures. In other related studies, Woo, Simon and Abeson (1975) have been examining the effects of rigidity of fixation plates on long bone remodeling and Turner, Mills, and Gabel (1975) have placed in uivo strain gages on plated intact equine long bones and reported strains during recovery and walking. Most studies to date have been concerned with a plated transverse fracture or plating on intact bone. While such studies provide a basic understanding of the mechanics of bone plate fixation, many fractures in clinical situations are more complex, often involving spiral fracture surfaces or comminutions. As a logical step from the transverse fracture studies toward the more realistic cases, the oblique fracture is considered here. In this paper, the mechanics of a compression plated oblique fracture is investigated. Using a finite element model, three aspects of compression plating are considered: the effect of tension in the plate on the stresses at the fracture site, the effects of placing bone screws across the fracture at different angles with respect to the fracture surfaces, and the effect of weight bearing on the plated bone. The following sections contain a description of the finite element model, numerical results of the study, and a discussion of the clinically related aspects of the results. FINITE
ELEMENT
REPRESENTATION
OF COMPRESSION
PLATING
In a previous study, Rybicki et al. (1974) found good agreement between strains predicted by a composite beam mathematical model for a bone and bone plate and laboratory measurements. It is important to describe the relationship between the composite beam mathematical model and the finite element model to place the finite element in proper perspective on the topic of comparisons with laboratory data. The composite beam model of the reference by Rybicki et al. (1974) consists of two beam structures: one for the bone plate and one for the bone. The beams have bending and stretching stiffnesses equal to those of the plate and bone. Two deformation modes are permitted in each beam: a uniform deformation to represent stretching and a linear variation of deformation through the depth of the beam to represent bending. The finite element model permits a more sophisticated representation of the deformation pattern by assembling many elements, each containing the basic beam deformation modes. However, the finite element representation can be equivalent to the beam model by reducing the number of finite elements through the depth to four, two for the bone plate and two for the bone. Thus, the important point is that similar good agreement between the composite beam model and laboratory measurements of the reference by Rybicki et al. (1974) could have been obtained by comparing a simple finite element representation (two elements for the bone plate and two for the bone) with laboratory measurements. Further-
F. A. SIMONEN more, it is emphasized that the finite element models of the previous study and those described here represent more sophisticated deformation patterns than the beam model can and, thus, provide an improved representation for the mechanics of the bone and bone plate. The beam model was chosen for the previous study because it is a well-known and basic model for stress analysis. Also, for the purpose of demonstration, certain numerical results can be conveniently and efficiently obtained from the beam model. The finite element representation for the bone plate with an oblique fracture is an extension of the finite element model of the transverse fracture used in the study by Rybicki et al. (1974). The purpose of the present study is to examine the influence of several factors, characteristic of the compression-plating procedure, on the stress distribution at the fracture site. In particular, the effects of tension in the plate, bone screws, and an axial force on the bone are considered via the results of the finite element model. Since the study is based on the results of a mathematical model, it is important to describe the model and its inherent assumptions in order to place the results of the study in proper perspective. Figure 1 shows the finite element grid selected to represent a 5 in. length of bone containing a 45degree oblique fracture at midlength. The width of the bone is 1.09 in. The bone plate is 2.5 in. long and is connected to the bone at four locations to simulate screw attachments. There is a span of 1.25 in. between the two center screws. The cross sectional geometry and modulus of the bone were obtained from the reference by Koch (1917). The properties of the bone plate were selected to represent a commercially available plate. The bone and plate were assumed to be isotropic. The model is two dimensional and permits deformations only in the plane of Fig. 1. Although the model is two dimensional, the noncircular cross section of the bone is included by varying the effective stiffness of the elements representing the bone. That is, the elements on any line transverse to the axis of the bone in Fig. 1 were assigned different stiffness properties to represent the variation in the width of bone in the direction perpendicular to the plane of the page. For example, the actual bone at one location has a dimension going into the plane of the page equal to TI and the dimension at another location is TZ. The stiffness referred to is the product of the modulus and the thickness. The stiffnesses for these two locations are assigned as follows. At location 1, the stiffness of the bone in the finite element model is assigned a value equal to EBoNET1,and at location 2, it is assigned the value of EBONET2.This procedure was followed for all elements in a line transverse to the axial direction. One cross sectional shape was used for all sections along the axis of the bone. The bone plate is straight as are the top and bottom surfaces of the bone. Perfect alignment of the bone segments is assumed. Compressive stresses can be trans-
143
Mechanics of oblique fracture fixation BonePIOIB
bne
s,n?= k”X %’ Modulus=26 Poisson’s Length
Area= i IO’pei
Ratlo=
3
= 2 5”
775”
Moments
of lnert~a=0.067
Modulus:
2.5 x IO’pri
,n4
Langttl=5”
Fig. 1. Finite element model of fractured bone with compression plate.
mitted across the fracture site, but, of course. tension cannot. The fractured surfaces are assumed to be sufficiently rough so that relative sliding is prevented. A sequence of loading conditions for the model was selected to represent the attachment of the bone plate to the bone. Subsequently, end forces were applied to represent weight bearing conditions. The following sections describe the results for a sequence of three loading conditions. First, tension is introduced into the plate. In this step, no screws are placed across the fracture, and it is assumed that the plate is attached to the bone at four locations. Next. forces to represent two bone screws are placed on the model. One set of forces was selected to model a bone screw that crosses the fracture line. The final loading is a uniform axial compressive stress applied to represent the bone and plate in a weight-supporting condition. The results of the finite element stress analysis for these three loadings are contained in the following three sections.
MODEZING THE EFFECT OF TENSION IN THE PLATE
In modeling this condition, it is assumed that the plate has been stretched and secured to the bone at four locations. The tensile force in the plate was 75 lb. Figure 2 shows the obtained stress distribution on the fracture site and the area of contact for the 75 lb tensile force in the plate. The value of 75 lb is cu. 201b below the force in the plate as measured by Rybicki et al. (1974) and below the range of forces reported by Perren et al. (1969). However, the loading simulates the type of forces imposed on the fractured bone during fixation by compression plating. In terms of the mathematical analysis, this is a nonlinear problem because the area of contact is an unknown in the solution. An iterative procedure requiring only three consecutive solutions was used to determine the contact area. To start the iterative pro-
cedure, the element nodes along the fracture surfaces are connected. As stated earlier, the fracture surfaces cannot support tension. Thus, the stresses along the fracture surface were monitored to determine if any tension stresses occur. If tension occurs in an element, the nodes are disconnected and a subsequent stress analysis is carried out while again monitoring the stresses for tensile values. This procedure is continued until no tensile stresses occur across the part of the fracture that is in contact. Displacements of the nodes on the fracture surface were also monitored to determine if unconnected nodes “overlap” and thus should be connected. Figure 2 shows a small contact area and an opening of the surfaces over most of the fracture site. The length of contact along the fracture surface is 0.135 in. The projection of this length along the direction perpendicular to the long axis of the bone is 0.095 in. This value compares very well with the range 0.072-O. 11 in. measured and obtained by Askew et al. (1975) from laboratory experiments similar to but not exactly like this model. This comparison is discussed in the Summary and Discussion Section and provides
Fig. 2. Contact stress distribution for fractured bone with an initial force of 75 lb in the compression plate.
144
E. F.
A
RYBICKI and
F. A. SIMONEN
.\\ 408
PII
/
Fig. 3. Contact stress distribution for fractured bone with applied forces by bone screws and initial force of 751b in the compression plate. 661b force in bone screw. a check on the model to predict the results of laboratory tests. EFFECT OF BONE SCREWS Next, the placement of bone screws was modeled. In surgical fixation, proper placement of the screws attaching the plate to the bone can assist in closing. the fracture site. Figures 3-5 show the effect of superimposing a range of attachment screw forces to the 75 lb tensile preload from the plate. It was assumed that the screws passed through both cortices normal to the axis of the bone. Thread engagement was taken to occur only in the cortex on the side of the bone opposite from the plate. The screw forces were represented by pairs of equal and opposite point loadings shown in the Figs. For each pair of forces, one force was applied to the plate just above the bone as indicated by the arrows. The other force of each pair was applied to the plate along the same line of action as shown in the Figs. The intent here was not to model the screws precisely, but to impose representative forces and examine the effects on the stress distribution at the fracture site. A maximum value of 583 lb for the screw load was based on the pull-out strength of screws used for surgical fixation as reported by Ray et al. (1971). Several
Fig. 4. Contact stress distribution for fractured bone with applied forces by bone screws and initial force of 751b in the compression plate. 3OOIb force in bone screw.
Fig. 5. Contact stress distribution for fractured bone with applied forces by bone screws and initial force of 751b in the compression plate. 583 lb force in bone screw.
load values between zero and the maximum value were considered. At each load, the iterative procedure was required to determine the area of contact between the bone surfaces at the fracture site. Figure 3 shows that a force on the screw of 661b closed the fracture along 0.61 in. of this model. The length of the 45” fracture is 1.54in. Increasing the force to 3001b produced a closure over 1.25 in. of the fracture surface as shown in Fig.4. Figure 5 shows that a further increase in screw force, up to 583 lb, caused the fracture site to reopen again slightly to l.lOin. The model shows that the maximum amount of closure is obtained for a force in the screw less than the pull-out values reported by Ray et al. (1971). Forces between this value and the pull-out value appear to offer no apparent advantage with respect to closing the fracture area and may reduce residual strength of the bone and plate assemblage because the stress interaction between the screw and the bone is nearer the pull-out value. Figure 6 shows the contact area and stress distribution with the screw placed at a 67.5 degree angle to
Fig. 6. Contact stress distribution for fractured bone with applied forces by bone screws and initial force of 751b in the compression plate. Screw loading across fracture = 583 lb at a 67.5”, angle with respect to the long axis.
Mechanics of oblique fracture fixation the bone axis. The angle of 67.5 degrees is half way between a line transverse to the bone and a line perpendicular to the fracture surface. This gave a greater component of force directed across the fracture site than the case shown in Fig. 5. As a result, Fig. 6 shows an increase in the peak contact stress. Figure 5 shows contact over 1.08 in. of the fracture length while placing the screw at 67.5 degrees in Fig. 6 decreased the area slightly to 1.04 in. One factor contributing to the decreased contact length is that the screw in Fig. 6 crosses the fracture nearer to the plate than in the case shown in Fig. 5. MODEL
FOR WEIGHT-BEARING
LOADS
Next, an axial force to indicate the effects of weight bearing was considered. Figure 7 shows the effect of an axial compressive load superimposed on the compression plated bone containing four screws. The magnitude and direction of the forces acting on the screws is also shown in Fig. 7. An applied force of 400 pounds was selected to represent a 2001b person in a one-legged stance. This force acting on the 0.775 in? area of the bone gave a stress of 516 psi. As expected, the effect of the axial load was to increase both the contact stresses and the area of contact at the fracture site. In Fig. 7, the bone surfaces are in contact for 1.39 in. of the 1.54in. fracture length. It is of interest to examine the stress trajectories for some of the loadings considered. The results shown thus far have concentrated on the stress distribution along the fracture surfaces. The stress trajectories give a picture of the overall stress pattern and
145
show the near and far-field effects. Stress trajectories indicating directions of principal stresses (in this case compressive stress) are shown for three cases in Figs. 8-10. Figure 8 shows the stress trajectories for the case where there is a 75 lb tensile force in the plate and no screws across the fracture. The contact stress distribution for this case is shown in Fig. 3. Figure 8 shows two “bulbs” of stress corresponding to the two center locations where the plate was attached to the bone. In the model, most of the stretching of the plate takes place bekween the center two screws. The “bulbs” indicate the regions in the bone influenced by the load transfer from the plate to the bone. While the region of high stress, shown where the lines are closer together, is localized, the area affected by the screws extends beyond the length of the plate. Figure 9 shows the stress trajectories for the case of two 583 lb screw loadings that is also shown in Fig. 5. In contrast to Fig. 8, Fig. 9 shows trajectories that cross the fracture line at nearly 90 degrees. This is due to the compressive stresses imposed by the screw. The region influenced by the screws can be seen as the family of lines between the arrows. The trajectories just under the center of the plate represent a combination of the effects of the tension in the plate and closing of the fracture. The trajectories that are close together, particularly across the fracture, show an area of high stress. Again, it is seen that the region of the model that is affected by the applied forces is localized to a length of bone that is close to. but greater than, the length of the plate. Figure 10 shows the stress trajectories resulting from a compressive end loading, forces to represent the two screws and tension in the plate. The end loading extends the tra-
516 PSI
Fig. 7. Contact stress distribution for fractured bone with uniform end compression. Applied forces by bone screws and initial force of 75 lb in the compression plate.
E. F. RYBICKIand F. A. SIMONEN
Fig. 8. Stress trajectories for fractured long bone with compression plate. Force in plate = 75 lb. No screw loading.
,583
lb
I583 tb
,583
lb
I 583
ID
Fig. 9. Stress trajectories for fractured long bone with compression plate. Initial force in plate = 75 lb. Screw loading across fracture = 583 lb perpendicular to long axis. Axial load = 400 lb.
503 lb
583
583lb
t
lb
583 lb
Fig. 10. Stress trajectories for fractured long bone with compression plate. Initial force plate = 75 lb. Screw loading across fracture = 583 lb at a 67.5” angle with respect to the long axis.
jectories to the entire section of bone in the model. The nearly straight and equally spaced trajectories at the ends of the bone show that the stress pattern is very close to the applied stress. However, the deformation pattern does reflect the additional stiffnesses contributed by the plate in that a larger deformation was found on the side opposite the plate than on the side under the plate. SUMMARY AND DISCUSSIONS
Internal fixation by compression plating is one of the clinical techniques for promoting the union of fractured bones. In the literature, attention has been given to studies directed toward understanding the mechanics of plated transverse fractures. In clinical
situations, oblique fractures, spiral fractures, and comminuted fractures are more common than transverse fractures. As a logical step toward examining the mechanics aspects of more complex fractures, a 45 degree oblique fracture was considered here. A finite element representation for a bone containing a through 45 degree fracture and secured by compression plating was the basis for this study. While the representation was two dimensional, the noncircular cross-sectional shape of the bone was included in the model. Two checks for the model were described to provide a degree of confidence in its ability to represent the mechanics of fracture fixation by compression plating. First, in a previous study, Rybicki et al. (1974) found good agreement between values for strains, forces, and bending moments of a plated in-
Mechanics of oblique fracture fixation tact equine metacarpal obtained from laboratory measurements and values predicted from a simple beam-bending model. While a finite element model of this situation was not used, the results did show that a mathematical model could be useful for representing some of the mechanics aspects of the plated intact bone. Furthermore, the finite element representation provides more sophisticated representation than the beam-bending model can. Thus, similar good correlation between the predicted and measured values would be obtained if the finite element model had been used. As a second check, a more direct comparison between the results of the model and laboratory data was made. This comparison is concerned with the length of the fracture closed by attaching a plate with a tensile force of 75 lb to the bone. The finite element representation shown in Fig. 2 predicted a closure of 0.136 in. along the 45” fracture surface. This means the end of the closure was 0.096 in. directly below the plate. As a reasonable approximation, one might expect this value of 0.096in. to occur for a transverse fracture. Askew et al. (1975) reported a range 0.072-O. 11 in. for experiments representing a transverse fracture with 1201b tension in the plate and that the amount of closure changed very little over large range of forces. This latter result was also experienced with the model presented here. While differences may exist between the finite element model presented here and the experiments reported by Askew et al. (1975), the agreement of the length of 0.096 in. predicted by the model and the range 0.072-0.11 in. reported by Askew et al. (1975) does provide a degree of confidence that the model can predict a closure of the same order of magnitude of that which was measured. In terms of the percent area of the fracture surface that is in contact, Askew er al. (1975) report g-14”/, for their study and the finite element model presented here predicts ca. 9%. Thus, this finite element model displays an ability to predict the amount of contact area which is an important quantity to check the model with. Several aspects of the results obtained here can be discussed in terms of clinically related aspects of compression plating. These include the effects of the bone screws, overstressing, distraction of the fracture surfaces, the amount of stress protection afforded to the bone by the plate, and a comparison of the mechanics of plated transverse and oblique fractures. First, the tightness of the screws is considered. Miiller et al. (1970) recommends that screws should not be tightened ‘to the very end. This is supported by the contention presented here that tightening the screws may reduce the overall strength of the assemblage and also by the results of the model indicating the contact area reached a maximum for a given screw tightness. While the model predicted that further tightening would decrease the contact area by a small amount, a more extensive study is needed to make a quantitative evaluation of the clinical significance of this.
147
Another question of interest is where to put the screws. In this study, it was shown that screws placed across the fracture surface will reduce distraction of the surfaces and close the gap created by the tension force in the bone plate. This is seen by comparing Figs. 2 and 4. Of course, the question of how much of a gap is clinically acceptable cannot be investigated with a mathematical model alone. Miiller et al. (1970) report that for a plated transverse fracture. healing occurred on the cortex side away from the plate in the presence of distraction of the bone surfaces, but the amount of distraction was not reported. In addition to distraction of the fracture surfaces, the question of overstressing a local region of the fracture site is one that may be of concern if the stress remains too high for a period of time. Based on the findings of Hassler et al. (1974), there is a level of stress that retards the healing. It is emphasized that this stress level must be maintained for a prolonged period of time. In the case of a metal plate, the stiffness is very high and, thus, it may be difficult to maintain a constant force in the plate over a long period of time as was shown by the experiments of Miller er al. (1970). The point to be made here is that while there is a need to maintain the fragments in proximity and to place a stress level on the surfaces that is helpful to healing, laboratory studies coupled with an understanding ofstresseson the fracture surfaces can help to define proper stress levels. The finite element model described here appears to be a good candidate for obtaining this understanding of the stresses on the fracture surfaces. Concerning alleviating the distraction of the fracture surfaces due to tension in the plate, one concept for doing this is prebending the plate so that it will tend to close the distraction when stretched and secured to the bone. Askew et al. (1975) reported that attempts to do this have not been successful for the transverse fracture. It appears that to date. quantitative techniques for doing this have not been developed. Models such as the finite element representation described here may contribute to designing tests for establishing a better understanding of the benefits and practicality of prebending a bone plate in terms of the mechanics and stability of such a system. The amount of stress protection afforded to the bone by the bone plate is of interest from a bone remodeling point of view. Stress protection occurs because part of the force is transmitted by the plate and, thus, a localized portion of the bone supports less stress than it would if the plate were absent and the bone intact. Miiller et al. (1970) have reported porotic bone adjacent to the bone plate where the bone is protected from stress by the plate. The finite element model offers a means to obtain a better understanding of the stress protection problem. A comparison of the mechanics of the plated transverse fracture (Rybicki et al., 1974) and the oblique fracture considered here reveals, as one would expect,
148
E.
F. RYBICKI
and F. A.
different stress distributions on the fracture surfaces for those two cases. The reaSons for this are twofold. First, the likelihood of placing bone screws across the fracture line is greater for the oblique fracture. Thus, the localized effect of the screws is present in the oblique case. Secondly, the response to an axial end loading is to cause a sliding deformation along the fracture surfaces for the oblique fracture. This does not happen in the transverse fracture. Resisting this sliding motion and the fact that the fracture plane is oblique to an axial loading causes additional stress distributions such as shear stress for the oblique case. Finally, it is important to emphkize that the results obtained here are based on the finite element model presented and not intended to be generalized for all cases of fracture fixation. The stress distribution along the fracture site is a result of the loading conditions, the geometry of the bone and plate, and the mechanical Dronerties. However. the finite element method _ _ of stress analysis provides a way of including these quantities in the model and appears to offer a means for obtaining a better understanding of the stress interactions due to fracture fixation by compression plating. Acknowledgements-The authors wish to acknowledge C. R. Hassler of Battelle’s Columbus Laboratories and E. B. Weis, Jr., of Ohio State’s Department of Orthopaedic Surgery for their helpful discussions.
REFERENCES
Askew, M. J., Mow, V. C., Wirth, C. R. and Campbell, C. I. (1975) Analysis of the intraosseous stress field due to compression plating. J. Biomechanics 4, 203-212.
SIMONEN
Eggers, G. W. N., Shindler, T. 0. and Pomerat, C. M. (1949) The influence of the contact-compression factor on osteogenesis in surgical fractures. .I. Bone Jnt Surg. 31A, 693. Friedenberg, Z. B. and French, G. (1952) The effects of known compression forces on fracture healing. Surg. Gynec. Obstet. 94, 74.
Hassler, C. R., Rybicki, E. F., Simonen. F. A. and Weis, E. B. (1974) Measurements of healing at an osteotomy in a rabbit calvarium: The influence of applied compressive stress on collagen synthesis and calcification. J. Biomechanics 7. 545-550.
Koch, J. C. (1917) The laws of bone architecture, Am. J. Anal. 21, 177. Miiller, M. E.. AllgGwer, M. and Willenegger, H. (1965) Internal Fixation of Fracture. Springer, Berlin. Miiller, M. E., AllgSwer, M. and Willenegger. H. (1970) Manual of Internal Fixdon. Springer, Berlin. Paul, J. P. (1971) Load actions on the human femur during walking and some stress resultants. Exp. Mech. 121-125 Rav. D. R.. Ledbetter. W. B.. Bvnum. D. and Bovd. C. i: (1971)‘A parametric anaiy& of done fixation-plates on fractured equine third metacarpal. J. Biomechanics 4, 163-174. Rybicki, E. F., Simonen, F. A., Mills, E. J., Hassler, C. R., Stoles, P., Milne, D. and Weis, E. B. (1974) Mathematical and experimental studies on the mechanics of plated transverse fractures. .I. Biomechanics 7, 377-387. Rybicki, E. F., Simonen, F. A. and Weis, E. B. (1972) On the mathematical analysis of stress in the human femur. J. Biomechanics 5, 20%215. T umer. A. S., Mills, E. J. and Gabel. A. A. (1975) In oiuo measurements of bone strain in the horse. Am. J. Vet. Res. 1573-l 579. Woo, S. L. Y., Simon, B. R. and Abesen, W. H. (1975) An interdisciplinary approach to evaluate the rigidity of internal fixation plate on large bone remodeling. 28th Ann. Conf. on Eng. in Mech. Biol., Sept. 2&24 in New Orleans, LA, Paper Cl. 10. Zienkiewicz, 0. C. and Cheung, Y. K. (1967) The Finite Element Method in Strucrural and Continuum Mechanics. McGraw-Hill, NY.
Pergamon Press
Pnnted m Great Bntam
MECHANICS OF OBLIQUE FRACTURE FIXATION USING A FINITE-ELEMENT MODEL* E. F. RYBICKIand F. A. SrMoNmt Applied Solid Mechanics Section. Battelle’s Columbus Laboratories, Columbus, OH 43210. U.S.A. Abstract-An investigation of the mechanics of a compression-plated oblique fracture using a finite-element stress analysis model is described. Three aspects of compression plating are considered: the effect of tension in the plate on stresses at the fracture site, the effects of placing bone screws across the fracture surface at different angles to the plane of the fracture, and the effect of a uniform end loading to represent weight bearing on the plated bone. In addition to stress distributions and deformations, the model predicts the contact area between the fractured surfaces. In one case, the predicted area shows good comparison with a range of values for a related case reported in the literature. Results of the model show that stress distributions at the fracture surfaces were not uniform and influenced by the tension in the plate, the bone screws, and the forces applied to the bone. The regions of contact between the fracture surfaces and the orientations of stress trajectories in terms of how these are influenced by plate tension, placement of bone screws, and applied loads are also
shown. Clinically related aspects of the results are discussed.
INTRODUCTION Bones are living weight-supporting structures. When a bone is fractured, it loses its weight-supporting capacity and there is a need to maintain the fragments in a position that will permit union. In clinical situations, various procedures are used to encourage an environment for rapid reliable union. During the past two decades, internal fixation by compression plating has received increased attention as a means of attaining reliable union of fractured bones. In addition to maintaining contact between the fractured surfaces, the compression can increase the structural stability of a plated bone. The popularity of this technique is largely due to the developments by the Swiss A-O (Arbeitsgemeinshaft ftir Osteosynthesefragen) as presented for example by Miiller et al. (1965). In compression fixation, a biocompatible plate (usually metal) is “stretched” across the fracture site, placing the plate in tension and the fractured bone in compression. Early mobilization is possible with this technique. The structural characteristics of the plate and its mode of attachment to the bone are critical elements in obtaining early mobilization because the forces acting on human bones can be large. During even the most basic movements such as standing on one leg or walking, the forces acting on the head of the human femur are estimated to be 2.54.5 times body weight as reported for example by Paul (1971) and Rybicki, Simonen and Weis (1972). These forces are not only large, but repetitive in nature. Thus, fatigue, as well as overstressing, is of concern for the plate and the method of securing the bone.
* Work supported by Battelle Institute. Grant No. B-1353-0102. t Structures and Mechanics Section, Battelle’s Pacific Northwest Laboratories. Richland. Washington. Ml
The function of internal fixation is then to provide an environment that promotes healing and remains stable in the presence of repetitive loadings due to mobilization. Thus, it becomes important to have an understanding of the force interactions between the bone and bone plate. In particular, a lack of understanding of the force interaction can result in implant failure due to high stresses or fatigue failure due to the repetitive loading. Another aspect of bone fixation related to stresses and the mechanics of fixation is that there appears to be a relationship between the amount of compressive stress at the healing site and the degree of healing. Qualitative studies of this relationship have been reported by Eggers, Shindler and Pomerat (1949). Friedenberg and French (1952) described a force range that produced the best healing in the ulnae of dogs. In a more quantitative study, Hassler et al. (1974) reported a range of compressive stress that produced optimal healing in rabbit calvaria. Thus, a better understanding of the stresses on bones during fixation is important in understanding bone healing during fixation. Another reason, and equally important motivation for understanding the mechanics of fixation devices, is that without such an understanding, it becomes difficult to introduce innovations into methods of fracture treatment. For all of these reasons, the efforts of several investigators have been directed toward obtaining a better understanding of the mechanics of compression-plated fixation. A review of early uses of bone plates is given by Miiller et al. (1965). Rybicki and coworkers (1974) describe some later studies and also present mathematical models and experimental studies on the mechanics of plated transverse fractures. In other studies, Askew et al. (1975) reported a mathematical and experimental study to predict the intraosseous stress distribution for compression-plated transverse
142
E. F.
RYBICKI and
fractures. In other related studies, Woo, Simon and Abeson (1975) have been examining the effects of rigidity of fixation plates on long bone remodeling and Turner, Mills, and Gabel (1975) have placed in uivo strain gages on plated intact equine long bones and reported strains during recovery and walking. Most studies to date have been concerned with a plated transverse fracture or plating on intact bone. While such studies provide a basic understanding of the mechanics of bone plate fixation, many fractures in clinical situations are more complex, often involving spiral fracture surfaces or comminutions. As a logical step from the transverse fracture studies toward the more realistic cases, the oblique fracture is considered here. In this paper, the mechanics of a compression plated oblique fracture is investigated. Using a finite element model, three aspects of compression plating are considered: the effect of tension in the plate on the stresses at the fracture site, the effects of placing bone screws across the fracture at different angles with respect to the fracture surfaces, and the effect of weight bearing on the plated bone. The following sections contain a description of the finite element model, numerical results of the study, and a discussion of the clinically related aspects of the results. FINITE
ELEMENT
REPRESENTATION
OF COMPRESSION
PLATING
In a previous study, Rybicki et al. (1974) found good agreement between strains predicted by a composite beam mathematical model for a bone and bone plate and laboratory measurements. It is important to describe the relationship between the composite beam mathematical model and the finite element model to place the finite element in proper perspective on the topic of comparisons with laboratory data. The composite beam model of the reference by Rybicki et al. (1974) consists of two beam structures: one for the bone plate and one for the bone. The beams have bending and stretching stiffnesses equal to those of the plate and bone. Two deformation modes are permitted in each beam: a uniform deformation to represent stretching and a linear variation of deformation through the depth of the beam to represent bending. The finite element model permits a more sophisticated representation of the deformation pattern by assembling many elements, each containing the basic beam deformation modes. However, the finite element representation can be equivalent to the beam model by reducing the number of finite elements through the depth to four, two for the bone plate and two for the bone. Thus, the important point is that similar good agreement between the composite beam model and laboratory measurements of the reference by Rybicki et al. (1974) could have been obtained by comparing a simple finite element representation (two elements for the bone plate and two for the bone) with laboratory measurements. Further-
F. A. SIMONEN more, it is emphasized that the finite element models of the previous study and those described here represent more sophisticated deformation patterns than the beam model can and, thus, provide an improved representation for the mechanics of the bone and bone plate. The beam model was chosen for the previous study because it is a well-known and basic model for stress analysis. Also, for the purpose of demonstration, certain numerical results can be conveniently and efficiently obtained from the beam model. The finite element representation for the bone plate with an oblique fracture is an extension of the finite element model of the transverse fracture used in the study by Rybicki et al. (1974). The purpose of the present study is to examine the influence of several factors, characteristic of the compression-plating procedure, on the stress distribution at the fracture site. In particular, the effects of tension in the plate, bone screws, and an axial force on the bone are considered via the results of the finite element model. Since the study is based on the results of a mathematical model, it is important to describe the model and its inherent assumptions in order to place the results of the study in proper perspective. Figure 1 shows the finite element grid selected to represent a 5 in. length of bone containing a 45degree oblique fracture at midlength. The width of the bone is 1.09 in. The bone plate is 2.5 in. long and is connected to the bone at four locations to simulate screw attachments. There is a span of 1.25 in. between the two center screws. The cross sectional geometry and modulus of the bone were obtained from the reference by Koch (1917). The properties of the bone plate were selected to represent a commercially available plate. The bone and plate were assumed to be isotropic. The model is two dimensional and permits deformations only in the plane of Fig. 1. Although the model is two dimensional, the noncircular cross section of the bone is included by varying the effective stiffness of the elements representing the bone. That is, the elements on any line transverse to the axis of the bone in Fig. 1 were assigned different stiffness properties to represent the variation in the width of bone in the direction perpendicular to the plane of the page. For example, the actual bone at one location has a dimension going into the plane of the page equal to TI and the dimension at another location is TZ. The stiffness referred to is the product of the modulus and the thickness. The stiffnesses for these two locations are assigned as follows. At location 1, the stiffness of the bone in the finite element model is assigned a value equal to EBoNET1,and at location 2, it is assigned the value of EBONET2.This procedure was followed for all elements in a line transverse to the axial direction. One cross sectional shape was used for all sections along the axis of the bone. The bone plate is straight as are the top and bottom surfaces of the bone. Perfect alignment of the bone segments is assumed. Compressive stresses can be trans-
143
Mechanics of oblique fracture fixation BonePIOIB
bne
s,n?= k”X %’ Modulus=26 Poisson’s Length
Area= i IO’pei
Ratlo=
3
= 2 5”
775”
Moments
of lnert~a=0.067
Modulus:
2.5 x IO’pri
,n4
Langttl=5”
Fig. 1. Finite element model of fractured bone with compression plate.
mitted across the fracture site, but, of course. tension cannot. The fractured surfaces are assumed to be sufficiently rough so that relative sliding is prevented. A sequence of loading conditions for the model was selected to represent the attachment of the bone plate to the bone. Subsequently, end forces were applied to represent weight bearing conditions. The following sections describe the results for a sequence of three loading conditions. First, tension is introduced into the plate. In this step, no screws are placed across the fracture, and it is assumed that the plate is attached to the bone at four locations. Next. forces to represent two bone screws are placed on the model. One set of forces was selected to model a bone screw that crosses the fracture line. The final loading is a uniform axial compressive stress applied to represent the bone and plate in a weight-supporting condition. The results of the finite element stress analysis for these three loadings are contained in the following three sections.
MODEZING THE EFFECT OF TENSION IN THE PLATE
In modeling this condition, it is assumed that the plate has been stretched and secured to the bone at four locations. The tensile force in the plate was 75 lb. Figure 2 shows the obtained stress distribution on the fracture site and the area of contact for the 75 lb tensile force in the plate. The value of 75 lb is cu. 201b below the force in the plate as measured by Rybicki et al. (1974) and below the range of forces reported by Perren et al. (1969). However, the loading simulates the type of forces imposed on the fractured bone during fixation by compression plating. In terms of the mathematical analysis, this is a nonlinear problem because the area of contact is an unknown in the solution. An iterative procedure requiring only three consecutive solutions was used to determine the contact area. To start the iterative pro-
cedure, the element nodes along the fracture surfaces are connected. As stated earlier, the fracture surfaces cannot support tension. Thus, the stresses along the fracture surface were monitored to determine if any tension stresses occur. If tension occurs in an element, the nodes are disconnected and a subsequent stress analysis is carried out while again monitoring the stresses for tensile values. This procedure is continued until no tensile stresses occur across the part of the fracture that is in contact. Displacements of the nodes on the fracture surface were also monitored to determine if unconnected nodes “overlap” and thus should be connected. Figure 2 shows a small contact area and an opening of the surfaces over most of the fracture site. The length of contact along the fracture surface is 0.135 in. The projection of this length along the direction perpendicular to the long axis of the bone is 0.095 in. This value compares very well with the range 0.072-O. 11 in. measured and obtained by Askew et al. (1975) from laboratory experiments similar to but not exactly like this model. This comparison is discussed in the Summary and Discussion Section and provides
Fig. 2. Contact stress distribution for fractured bone with an initial force of 75 lb in the compression plate.
144
E. F.
A
RYBICKI and
F. A. SIMONEN
.\\ 408
PII
/
Fig. 3. Contact stress distribution for fractured bone with applied forces by bone screws and initial force of 751b in the compression plate. 661b force in bone screw. a check on the model to predict the results of laboratory tests. EFFECT OF BONE SCREWS Next, the placement of bone screws was modeled. In surgical fixation, proper placement of the screws attaching the plate to the bone can assist in closing. the fracture site. Figures 3-5 show the effect of superimposing a range of attachment screw forces to the 75 lb tensile preload from the plate. It was assumed that the screws passed through both cortices normal to the axis of the bone. Thread engagement was taken to occur only in the cortex on the side of the bone opposite from the plate. The screw forces were represented by pairs of equal and opposite point loadings shown in the Figs. For each pair of forces, one force was applied to the plate just above the bone as indicated by the arrows. The other force of each pair was applied to the plate along the same line of action as shown in the Figs. The intent here was not to model the screws precisely, but to impose representative forces and examine the effects on the stress distribution at the fracture site. A maximum value of 583 lb for the screw load was based on the pull-out strength of screws used for surgical fixation as reported by Ray et al. (1971). Several
Fig. 4. Contact stress distribution for fractured bone with applied forces by bone screws and initial force of 751b in the compression plate. 3OOIb force in bone screw.
Fig. 5. Contact stress distribution for fractured bone with applied forces by bone screws and initial force of 751b in the compression plate. 583 lb force in bone screw.
load values between zero and the maximum value were considered. At each load, the iterative procedure was required to determine the area of contact between the bone surfaces at the fracture site. Figure 3 shows that a force on the screw of 661b closed the fracture along 0.61 in. of this model. The length of the 45” fracture is 1.54in. Increasing the force to 3001b produced a closure over 1.25 in. of the fracture surface as shown in Fig.4. Figure 5 shows that a further increase in screw force, up to 583 lb, caused the fracture site to reopen again slightly to l.lOin. The model shows that the maximum amount of closure is obtained for a force in the screw less than the pull-out values reported by Ray et al. (1971). Forces between this value and the pull-out value appear to offer no apparent advantage with respect to closing the fracture area and may reduce residual strength of the bone and plate assemblage because the stress interaction between the screw and the bone is nearer the pull-out value. Figure 6 shows the contact area and stress distribution with the screw placed at a 67.5 degree angle to
Fig. 6. Contact stress distribution for fractured bone with applied forces by bone screws and initial force of 751b in the compression plate. Screw loading across fracture = 583 lb at a 67.5”, angle with respect to the long axis.
Mechanics of oblique fracture fixation the bone axis. The angle of 67.5 degrees is half way between a line transverse to the bone and a line perpendicular to the fracture surface. This gave a greater component of force directed across the fracture site than the case shown in Fig. 5. As a result, Fig. 6 shows an increase in the peak contact stress. Figure 5 shows contact over 1.08 in. of the fracture length while placing the screw at 67.5 degrees in Fig. 6 decreased the area slightly to 1.04 in. One factor contributing to the decreased contact length is that the screw in Fig. 6 crosses the fracture nearer to the plate than in the case shown in Fig. 5. MODEL
FOR WEIGHT-BEARING
LOADS
Next, an axial force to indicate the effects of weight bearing was considered. Figure 7 shows the effect of an axial compressive load superimposed on the compression plated bone containing four screws. The magnitude and direction of the forces acting on the screws is also shown in Fig. 7. An applied force of 400 pounds was selected to represent a 2001b person in a one-legged stance. This force acting on the 0.775 in? area of the bone gave a stress of 516 psi. As expected, the effect of the axial load was to increase both the contact stresses and the area of contact at the fracture site. In Fig. 7, the bone surfaces are in contact for 1.39 in. of the 1.54in. fracture length. It is of interest to examine the stress trajectories for some of the loadings considered. The results shown thus far have concentrated on the stress distribution along the fracture surfaces. The stress trajectories give a picture of the overall stress pattern and
145
show the near and far-field effects. Stress trajectories indicating directions of principal stresses (in this case compressive stress) are shown for three cases in Figs. 8-10. Figure 8 shows the stress trajectories for the case where there is a 75 lb tensile force in the plate and no screws across the fracture. The contact stress distribution for this case is shown in Fig. 3. Figure 8 shows two “bulbs” of stress corresponding to the two center locations where the plate was attached to the bone. In the model, most of the stretching of the plate takes place bekween the center two screws. The “bulbs” indicate the regions in the bone influenced by the load transfer from the plate to the bone. While the region of high stress, shown where the lines are closer together, is localized, the area affected by the screws extends beyond the length of the plate. Figure 9 shows the stress trajectories for the case of two 583 lb screw loadings that is also shown in Fig. 5. In contrast to Fig. 8, Fig. 9 shows trajectories that cross the fracture line at nearly 90 degrees. This is due to the compressive stresses imposed by the screw. The region influenced by the screws can be seen as the family of lines between the arrows. The trajectories just under the center of the plate represent a combination of the effects of the tension in the plate and closing of the fracture. The trajectories that are close together, particularly across the fracture, show an area of high stress. Again, it is seen that the region of the model that is affected by the applied forces is localized to a length of bone that is close to. but greater than, the length of the plate. Figure 10 shows the stress trajectories resulting from a compressive end loading, forces to represent the two screws and tension in the plate. The end loading extends the tra-
516 PSI
Fig. 7. Contact stress distribution for fractured bone with uniform end compression. Applied forces by bone screws and initial force of 75 lb in the compression plate.
E. F. RYBICKIand F. A. SIMONEN
Fig. 8. Stress trajectories for fractured long bone with compression plate. Force in plate = 75 lb. No screw loading.
,583
lb
I583 tb
,583
lb
I 583
ID
Fig. 9. Stress trajectories for fractured long bone with compression plate. Initial force in plate = 75 lb. Screw loading across fracture = 583 lb perpendicular to long axis. Axial load = 400 lb.
503 lb
583
583lb
t
lb
583 lb
Fig. 10. Stress trajectories for fractured long bone with compression plate. Initial force plate = 75 lb. Screw loading across fracture = 583 lb at a 67.5” angle with respect to the long axis.
jectories to the entire section of bone in the model. The nearly straight and equally spaced trajectories at the ends of the bone show that the stress pattern is very close to the applied stress. However, the deformation pattern does reflect the additional stiffnesses contributed by the plate in that a larger deformation was found on the side opposite the plate than on the side under the plate. SUMMARY AND DISCUSSIONS
Internal fixation by compression plating is one of the clinical techniques for promoting the union of fractured bones. In the literature, attention has been given to studies directed toward understanding the mechanics of plated transverse fractures. In clinical
situations, oblique fractures, spiral fractures, and comminuted fractures are more common than transverse fractures. As a logical step toward examining the mechanics aspects of more complex fractures, a 45 degree oblique fracture was considered here. A finite element representation for a bone containing a through 45 degree fracture and secured by compression plating was the basis for this study. While the representation was two dimensional, the noncircular cross-sectional shape of the bone was included in the model. Two checks for the model were described to provide a degree of confidence in its ability to represent the mechanics of fracture fixation by compression plating. First, in a previous study, Rybicki et al. (1974) found good agreement between values for strains, forces, and bending moments of a plated in-
Mechanics of oblique fracture fixation tact equine metacarpal obtained from laboratory measurements and values predicted from a simple beam-bending model. While a finite element model of this situation was not used, the results did show that a mathematical model could be useful for representing some of the mechanics aspects of the plated intact bone. Furthermore, the finite element representation provides more sophisticated representation than the beam-bending model can. Thus, similar good correlation between the predicted and measured values would be obtained if the finite element model had been used. As a second check, a more direct comparison between the results of the model and laboratory data was made. This comparison is concerned with the length of the fracture closed by attaching a plate with a tensile force of 75 lb to the bone. The finite element representation shown in Fig. 2 predicted a closure of 0.136 in. along the 45” fracture surface. This means the end of the closure was 0.096 in. directly below the plate. As a reasonable approximation, one might expect this value of 0.096in. to occur for a transverse fracture. Askew et al. (1975) reported a range 0.072-O. 11 in. for experiments representing a transverse fracture with 1201b tension in the plate and that the amount of closure changed very little over large range of forces. This latter result was also experienced with the model presented here. While differences may exist between the finite element model presented here and the experiments reported by Askew et al. (1975), the agreement of the length of 0.096 in. predicted by the model and the range 0.072-0.11 in. reported by Askew et al. (1975) does provide a degree of confidence that the model can predict a closure of the same order of magnitude of that which was measured. In terms of the percent area of the fracture surface that is in contact, Askew er al. (1975) report g-14”/, for their study and the finite element model presented here predicts ca. 9%. Thus, this finite element model displays an ability to predict the amount of contact area which is an important quantity to check the model with. Several aspects of the results obtained here can be discussed in terms of clinically related aspects of compression plating. These include the effects of the bone screws, overstressing, distraction of the fracture surfaces, the amount of stress protection afforded to the bone by the plate, and a comparison of the mechanics of plated transverse and oblique fractures. First, the tightness of the screws is considered. Miiller et al. (1970) recommends that screws should not be tightened ‘to the very end. This is supported by the contention presented here that tightening the screws may reduce the overall strength of the assemblage and also by the results of the model indicating the contact area reached a maximum for a given screw tightness. While the model predicted that further tightening would decrease the contact area by a small amount, a more extensive study is needed to make a quantitative evaluation of the clinical significance of this.
147
Another question of interest is where to put the screws. In this study, it was shown that screws placed across the fracture surface will reduce distraction of the surfaces and close the gap created by the tension force in the bone plate. This is seen by comparing Figs. 2 and 4. Of course, the question of how much of a gap is clinically acceptable cannot be investigated with a mathematical model alone. Miiller et al. (1970) report that for a plated transverse fracture. healing occurred on the cortex side away from the plate in the presence of distraction of the bone surfaces, but the amount of distraction was not reported. In addition to distraction of the fracture surfaces, the question of overstressing a local region of the fracture site is one that may be of concern if the stress remains too high for a period of time. Based on the findings of Hassler et al. (1974), there is a level of stress that retards the healing. It is emphasized that this stress level must be maintained for a prolonged period of time. In the case of a metal plate, the stiffness is very high and, thus, it may be difficult to maintain a constant force in the plate over a long period of time as was shown by the experiments of Miller er al. (1970). The point to be made here is that while there is a need to maintain the fragments in proximity and to place a stress level on the surfaces that is helpful to healing, laboratory studies coupled with an understanding ofstresseson the fracture surfaces can help to define proper stress levels. The finite element model described here appears to be a good candidate for obtaining this understanding of the stresses on the fracture surfaces. Concerning alleviating the distraction of the fracture surfaces due to tension in the plate, one concept for doing this is prebending the plate so that it will tend to close the distraction when stretched and secured to the bone. Askew et al. (1975) reported that attempts to do this have not been successful for the transverse fracture. It appears that to date. quantitative techniques for doing this have not been developed. Models such as the finite element representation described here may contribute to designing tests for establishing a better understanding of the benefits and practicality of prebending a bone plate in terms of the mechanics and stability of such a system. The amount of stress protection afforded to the bone by the bone plate is of interest from a bone remodeling point of view. Stress protection occurs because part of the force is transmitted by the plate and, thus, a localized portion of the bone supports less stress than it would if the plate were absent and the bone intact. Miiller et al. (1970) have reported porotic bone adjacent to the bone plate where the bone is protected from stress by the plate. The finite element model offers a means to obtain a better understanding of the stress protection problem. A comparison of the mechanics of the plated transverse fracture (Rybicki et al., 1974) and the oblique fracture considered here reveals, as one would expect,
148
E.
F. RYBICKI
and F. A.
different stress distributions on the fracture surfaces for those two cases. The reaSons for this are twofold. First, the likelihood of placing bone screws across the fracture line is greater for the oblique fracture. Thus, the localized effect of the screws is present in the oblique case. Secondly, the response to an axial end loading is to cause a sliding deformation along the fracture surfaces for the oblique fracture. This does not happen in the transverse fracture. Resisting this sliding motion and the fact that the fracture plane is oblique to an axial loading causes additional stress distributions such as shear stress for the oblique case. Finally, it is important to emphkize that the results obtained here are based on the finite element model presented and not intended to be generalized for all cases of fracture fixation. The stress distribution along the fracture site is a result of the loading conditions, the geometry of the bone and plate, and the mechanical Dronerties. However. the finite element method _ _ of stress analysis provides a way of including these quantities in the model and appears to offer a means for obtaining a better understanding of the stress interactions due to fracture fixation by compression plating. Acknowledgements-The authors wish to acknowledge C. R. Hassler of Battelle’s Columbus Laboratories and E. B. Weis, Jr., of Ohio State’s Department of Orthopaedic Surgery for their helpful discussions.
REFERENCES
Askew, M. J., Mow, V. C., Wirth, C. R. and Campbell, C. I. (1975) Analysis of the intraosseous stress field due to compression plating. J. Biomechanics 4, 203-212.
SIMONEN
Eggers, G. W. N., Shindler, T. 0. and Pomerat, C. M. (1949) The influence of the contact-compression factor on osteogenesis in surgical fractures. .I. Bone Jnt Surg. 31A, 693. Friedenberg, Z. B. and French, G. (1952) The effects of known compression forces on fracture healing. Surg. Gynec. Obstet. 94, 74.
Hassler, C. R., Rybicki, E. F., Simonen. F. A. and Weis, E. B. (1974) Measurements of healing at an osteotomy in a rabbit calvarium: The influence of applied compressive stress on collagen synthesis and calcification. J. Biomechanics 7. 545-550.
Koch, J. C. (1917) The laws of bone architecture, Am. J. Anal. 21, 177. Miiller, M. E.. AllgGwer, M. and Willenegger, H. (1965) Internal Fixation of Fracture. Springer, Berlin. Miiller, M. E., AllgSwer, M. and Willenegger. H. (1970) Manual of Internal Fixdon. Springer, Berlin. Paul, J. P. (1971) Load actions on the human femur during walking and some stress resultants. Exp. Mech. 121-125 Rav. D. R.. Ledbetter. W. B.. Bvnum. D. and Bovd. C. i: (1971)‘A parametric anaiy& of done fixation-plates on fractured equine third metacarpal. J. Biomechanics 4, 163-174. Rybicki, E. F., Simonen, F. A., Mills, E. J., Hassler, C. R., Stoles, P., Milne, D. and Weis, E. B. (1974) Mathematical and experimental studies on the mechanics of plated transverse fractures. .I. Biomechanics 7, 377-387. Rybicki, E. F., Simonen, F. A. and Weis, E. B. (1972) On the mathematical analysis of stress in the human femur. J. Biomechanics 5, 20%215. T umer. A. S., Mills, E. J. and Gabel. A. A. (1975) In oiuo measurements of bone strain in the horse. Am. J. Vet. Res. 1573-l 579. Woo, S. L. Y., Simon, B. R. and Abesen, W. H. (1975) An interdisciplinary approach to evaluate the rigidity of internal fixation plate on large bone remodeling. 28th Ann. Conf. on Eng. in Mech. Biol., Sept. 2&24 in New Orleans, LA, Paper Cl. 10. Zienkiewicz, 0. C. and Cheung, Y. K. (1967) The Finite Element Method in Strucrural and Continuum Mechanics. McGraw-Hill, NY.