Stress analysis of a simplified compression plate fixation system for fractured bones

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STRESS ANALYSIS OF A SIMPLIFIED COMPRESSION PLATE FIXATION SYSTEM FOR FRACTURED BONES? EDWARD J. CHEAL,WILSONC. HAYES and AUGUSTUSA. WHITE, III Orthopaedic Biomechanics Laboratory, Department of O~hopaedic Surgery, Charles A. Dana Research

Instrtute, Beth Israel Hospital and Harvard Medical School, Boston, MA 02215, U.S A. and

STEPHANM.

PERREN

Laboratonum fur Experimentelle Chirurgie, Schweizerisches Forschungsinstitut

CH-7270 Davos, Schweiz

Abstract-A three-dimensional finite element model was generated of a plexiglass tube with an attached six-hole stainless steel compression plate to study the mecharncs of internal fixation of fractured long bones. To demonstrate the importance of the plate-bone interface, this interface was represented three different ways in the finite element model. A plated tube with a uniform transverse osteotomy gap was also examined to study the mechanics of plated fractured bones. To validate the model, the results for the intact plated tube were compared to composite beam theory and stram gauge data from an Instrumented physical model. Applications of the finite element model data included the prediction of screw failure modes, plate-induct osteopenia, and multi-axial strains in an interfragmentary region. The addition of sliding motion between the plate and tube resulted in a deviatton from composite beam theory and improved correspondence with strain gage data when compared to a model having the plate and tube securely bonded. Sliding motion resulted in a much smaller region of bone subjected to reduced axial stress levels, which may decrease the extent of plate-induced osteopenia. The complex nature of induced strains in an osteotomy gap was also demonstrated, along wtth the tendency for failure of the screws nearest the fracture site. I. INTRODUCTION One aim of fracture treatment is to re-establish the proper anatomical relationships between bone fragments and to support the injured region until bony union can occur[l]. In many instances, the fractured fragments cannot be adequately controlled in order to allow an early return to function, and surgical intervention with open reduction and internal fixation by a metal plate is necessary[2]. To achieve stable fixation of a fractured long bone, the fracture site is bridged with a rigid plate, fixed to the bone with four or more cortical bone screws. Stability may be improved by imposing static compression on the fracture site at the time of plate application[2], thus the term “compression plate fixation”. Internal fixation presents two different and seemingly contradictory requirements. In the early stages of fracture repair, the function of the internal fixation device is to rigidly immobilize the fracture fragments. This in turn allows bony union to proceed and encourages functional utilization of the injured extremity. In the later stages of fracture repair, however, a loss of bone mass may occur[3-51. Many researchers attribute this loss to decreased stresses in the bone tissue during functional loading of the plate-bone system. It has been demonstrated that the remodeling response of bone following the application of an internal fixation plate is at least in part a function of the plate mechanical characteristics. More rigid internal fixation, through increased plate elastic

tThts work was supported by NIH RCDA AM~749 and by a research grant from Synthes, Ltd.

modulus[&8], increased plate thickness and elastic modulus[9], or through the addition of an intracortical porous ingrowth layer [lo], generally results in greater bone loss. This loss of bone mass is termed *‘plate-induced osteopenia”, and appears morphologically as either an increase in cortical porosity, a decrease in cortical thickness, or both. The long term effect of rigid plate fixation is primarily cortical thinning, apparently as an adaptive response to restore physiologic stress levels. Gunst [ 1l] demonstrated, however, that plate application also interferes with the bone blood supply. Large ischemic areas were found directly beneath fixation plates one day following application to intact rabbit and sheep tibiae. A ten week period was required for the complete restoration of bone blood supply. Correspondingly, remodeling activity was first observed in the third week, and reached a peak at seven to eight weeks following plate application. At ten weeks, the metabolic activity, characterized as predominantly Haversian remodeling, continued at a reduced level, leaving a weakened and porous cortex. To explore the decreased stress hypothesis, investigators have used composite beam theory[l2, 131 and finite element models [ 14, 151to predict the cyclic bone stresses following plate fixation. The predicted stress resultants were then related to the observed remodeling response[l2, 161or a parametric analysis was performed to predict the remodeling response [ 151.An alternate approach was to measure directly the in viva locomotor strains and relate the strain reductions to the observed remodeling[l7]. There are limitations associated with the use of composite beam theory for predicting stress resultants in bone-plate systems. Composite beam the845

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E. J. CHEALet al.

ory cannot account for different plate-bone interface characteristics. The assumption of plane sections remaining plane is valid only if the plate and bone are bonded together securely. Therefore, composite beam theory may only be accurate for plates having a porous surface which allows the ingrowth of bone. Static stress fields due to a preload in the plate and screws must be determined using more complex models. Static stress fields are also difficult to monitor over long periods of time using in oivo strain gage techniques. In addition, to properly relate the complete resultant stress fields to the observed remodeling responses. both circumferential and longitudinal variations in stress and remodeling must be established. Thus, only a three-dimensional analysis can completely relate altered stress resultants to the remodeling response. The histological pattern of fracture healing with compression plate fixation has been studied by many workers[ 188221. With relative immobilization of the fracture fragments provided by the plate, there is direct cortical reconstruction of the fracture site and no periosteal callus is observed. This pattern is referred to as primary bone healing, in contrast to secondary bone healing, a process characterized by the production of external callus. Perren [23] recently suggested that the controlling factor for the histological pattern of fracture healing may be the local strains in the healing tissues. The hypothesis is that healing tissues can exist only under conditions where the local strains do not exceed a limiting strain level. This limit may be either the elongation at rupture or the elongation at yield. These limiting strains can be expected to vary over wide ranges depending on the tissues involved. Limiting strains of about lOOSi may be possible with granulation tissue whereas values of less than 2q; may be allowed for fully mineralized cortical bone. Thus, the concept of interfragmentary strain would indicate that interfragmentary motion must be limited so that strain values of less than 2”,; occur if cortical bone is to form. Several mechanisms are available for this reduction in interfragmentary strain to occur. The fracture fragments can be immobilized, either by the presence of a compression plate or by the production of a mineralized external callus. In addition. interfragmentary strain levels can be reduced by the resorption of the fragment ends. Both mechanisms result m a reduction in the strain levels m the mterfragmentary region. Perren’s hypothesis, if proven valid, may greatly increase our understanding of the relationship between the mechanical environment and the process of tissue differentiation during fracture healing. The objective of this investigation is to develop an accurate, three-dimensional finite element model of a geometrically simplified compression plate fixation system as a precursor to more complex models of plated fractured bones. The model is used to study the mechanics of plate-induced osteopenia and the strain” as a control concept of “interfragmentay variable for the histological pattern of bone healing. Because of the complexities of musculoskeletal structures. it is convenient to separate material and geometric complexities and to analyze first a simpler, representative system consisting of a six-hole selfcompressing stainless steel plate applied to an intact

plexiglass tube. Model development, strain gage validation, and a comparison with composite beam theory are presented here. The results for a range of applied loads, using several different conditions of plate-tube contact, are then examined as specifically related to plate-induced osteopenia. A fimte element model of a plated tube with a uniform osteotomy gap is also developed and used to explore the strains in the interfragmentary region. 2. ANALYTICAL AND EXPERIMENTAL METHODS

2.1 Physical model We analyzed a six-hole, 316L stamless steel plate distributed by SYNTHES. Ltd. The plate was 10.3 cm long, 1.2 cm wide, and 0.38 cm thick, fixed by six standard stainless steel cortical screws. The screws passed through the plate and both sides of a plexiglass cylinder. The compression plate was applied to the center of a 29cm long plexiglass tube with an inside diameter of 1.90cm and an outside diameter of 2.54 cm. The plate used in this analysis is a Dynamic Compression Plate (DCP) due to the selfcompressing features[2]. The screw holes are sloped. such that as each screw is inserted, the plate undergoes a longitudinal displacement relative to the underlying bone. The fracture site is thus reduced and compressed upon application of the plate. 2.2 Finite element model The ADINA finite element modeling package (ADINA Engineering Inc.) implemented on a VAX I l/780 was used to perform the finite element analysis. One quarter of the symmetric model was analyzed with the mesh shown in Fig. 1. Symmetry was maintained by displacement constraints on the nodes in the planes of symmetry. The tube and plate were represented by 138 twenty-node isoparametric solid elements[24], and the screws by 48 two-node beam elements, for a total of 963 nodal points. Only linear elastic isotropic elements were used. An elastic modulus of 3.1 GPa and Poisson’s ratio of 0.2 were used for the plexiglass tube, which represents properties for low to moderate load under static conditions For the stainless steel compression plate and fixation screws an elastic modulus of 196 GPa and a Poisson’s ratio of 0.3 were used. The compression plate and plexiglass tube interact indirectly through the screws and directly through

Fig. I. Three-dimenwonal finite clement mesh wrth hiddenlme and perspectrve opttons.

Stress analysis of a simplified compression plate fixation system

plate-tube contact along the outer edges of the plate. In the first model, to be referred to as the “directcontact” model, nodai point constraint equations were used to directly connect the plate surface to the tube surface. This eliminated the possibility of platetube separation and allowed the transfer of normal and shear stresses. In the second model, to be referred to as the “linear-truss” model, stiff linear trusses were used along the line of contact between plate and tube. The trusses represented sliding contact, such that only axial forces (normal to the tube surface) were transmitted. This assumes that the shear forces transmitted through plate-tube friction are negligible. In the third model, to be referred to as the “bi-linear truss” model, bi-linear trusses, with a low elastic modulus in tension, were used, making “stress-free’ separation possible. The addition of the nonlinearity requires an iterative solution technique, greatly increasing the solution complexity. For this reason, the mesh density was decreased by approximately three-quarters (Fig. 2). From the use of test examples, it was found that with the Broyden-Fletcher-Goldfar~Shanna (BFGS) method of equilibrium iterations[2.5] it was possible to use trusses with a tensile elastic modulus a factor of 100 less than the compressive modulus. To examine the resultant strains in the interfragmentary region, the linear-truss finite element model was further modified, to create an “osteotomygap” model. The mesh was altered such that a 1 mm thick layer of elements, representing a 2mm thick osteotomy gap, was beneath the center of the plate. The material properties were changed to correspond approximately to granulation tissue or cartilage. The elastic modulus of this material was decreased by a factor of 1000, from the elastic modulus of plexiglass, while the Poisson’s ratio was increased to 0.45. Linear isotropy was still assumed. This assumption is discussed in a later section. In the clinical application of compression plates, tensile stresses are generated in both plate and screws in order to create the desired compressive stresses at the fracture site. To simulate plate pretension in the models having truss-interface elements, applied longitudinal displacements were used at the center of the

plate. It was not possible to realistically simulate plate pretension using applied displacements with the direct-contact model. The nodai point constraints between the plate and the tube result in excessive

Rg. 2. Coarse

mesh

used with elements.

non-linear

interface

847

shear stresses in the region of the applied displacements. As an alternative, linear isotropic thermoelastic elements were used. These elements were utilized in the plate to apply a uniform, negative, prestrain. For all models the resultant longitudinal tensile load in the plate was approx. 490 N. The beam elements representing the screws were defined using the corresponding nodal points from the twenty-node soIid elements. Two additional nodes, yielding one additional internal element, were used adjacent to the plate-tube interface for each screw. Tensile stresses in the screws were generated bq compressing this element. Resulting tensile axial screw forces of approx. 200 N at the plate-tube interace were used. Thus, the beam elements were equivalent to surgically applied fixation screws threaded through both cortices. The applied loads were chosen to provide a general set of representative loading conditions. Three external load cases were examined, including uniform compressive axial loading on the tube (load case A), four-point bending such that the plate is on the tensile aspect of the tube (load case B), and compressive axial loading on the tube highly skewed away from the side of plate application (load case C). Both distributed axial loads were scaled to a resultant force of 342N, and the bending load was scaled to a resultant moment of 9.34N-m. The three external load cases (A through C) represent possible modes of applied cyclic load, whereas the internal load case (D) represents a static load resulting from a preload in the plate and screws. The linear superposition of load cases A through C with load case D was also possible, with the exception of the bi-linear truss model. For this model, the combined load cases were also solved explicitly. Pre- and post-processing were accomplished using FEMGEN and FEMVIEW (JAR Associates, Inc.), on the same VAX 1l/780 computer. The output from ADINA was converted to FEMVIEW compatible format using a software interface IFACE. FEMVIEW was then used to generate graphic plots of nodal displacements. stress contours, and principal stress vectors.

2.3 Strain gage analysis Strain gage experiments were designed to test the validity of the finite element model, using a plexiglass/compression plate system instrumented with multiple strain gages. The biaxial strain gages mounted on the side and bottom surfaces of the plexiglass tube were BLH type FAET-06C-3% 513EL. The uniaxial strain gage mounted on the top surface of the compression plate was a MicroMeasurements type EA-06125AD-120. Signal conditioning and amplification were provided by an ENDEVCO 2100 IO-channel signal conditioning system. The instrumented plate and tube were then tested in an INSTRON 1331 materials testing system in both axial compression and four-point bending (Fig. 3). In the test specimen, the screws were applied in the neutral position so that screw tightening resulted in minimal pre-tensioning of the plate, The experimental results were then compared to the finite element

load cases A and B.

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Fig. 3. Instrumented plexiglass tube and fixation plate in four-point bending load apparatus.

3. RESULTS

3.1 Model mlidution The finite element model results were compared to composite beam theory[M] for prediction of the axia’l, normal stress distributions on the tube crosssection beneath the center of the plate for four-point bending (Fig. 4). Composite beam theory (Fig. 4a) predicts a neutral axis in the plate, with the entire tube cross-section in compression. The direct-contact model (Fig. 4b) agrees very well with composite beam theory, with only a slight deviation from a linear stress distribution in the region directly beneath the plate. Good correspondence is expected between these models since both assume the plate and tube are securely bonded. The truss-interface model (Fig. 4c), however, predicts a smaller shift in the neutral axis, resulting in a “neutral zone” in the region beneath the plate. The stress gradient, in the linear portion, is greater for the truss-interface model than for composite beam theory or direct-contact models, indicating that in the truss-interface model, the tube carries a greater fraction of the bending load. Strain gage data and finite element results for load cases A and B are compared in Fig. 5. For both load cases, the maximum strain is axial and compressive and occurs on the surface of the tube opposite the plate (location B). The top of the plate (location A) exhibits tensile strain in the axial direction for both load cases. For axial compression (Fig. 5a), there is little variation in the results from the finite element models. The direct-contact and linear-truss models have a linear strain distribution, in contrast to the strain gage results. The bi-linear truss model has a less linear strain distribution, but not in a way which is approaching the strain gage results. For four-point bending (Fig. 5b), however, the addition of relative motion between the plate and the tube, through the use of truss interface elements, significantly improves the correspondence between the finite element and strain gage results. For both modes of external

loading the measured plate surface strain is less than that predicted if a linear strain gradient were assumed across the entire cross-section. This confirms that in the physical model relative motion does occur between the plate and the tube. 3.2 Resultant screw ,fbrces For the beam elements used to represent each screw, the finite element analysis yields the axial force, the transverse shear force (in the z-direction), and the bending moment about the r-axis. The other shear and bending forces are identically zero due to symmetry conditions. Tables 1-3 show the internal force resultants (for the outer, middle, and inner screws) for the bending-closed external load, load case B. Data are taken from the plate-tube interface, since this is, in general, the location of maximum load on the screws. The data is contrasted for all four of the finite eiement models to further demonstrate the importance of the plate-bone interface characteristics and the effects of the introduction of a fracture gap. For the intact models (direct-contact, linear-truss, and bi-linear truss), the outermost screw is most highly loaded. The direct-contact model demonstrates an approximately seven-fold increase in transverse shear force and a ten-fold increase in bending moment for the outermost screw in comparison to the middle screw. The linear-truss model demonstrates three-fold and two- to four-fold increases in transverse shear and bending, respectively. The addition of sliding motion between the plate and tube results in greatly increased resultant screw forces. The axial, shear, and bending forces are all increased about one order-of-magnitude for the linear truss model. Lowering the tensile elastic modulus of the trusses further increases the resultant screw forces. The screw forces for the bi-linear truss model are approx. loo/;, greater than the screw forces in the linear-truss model. Surprisingly, the screw axial forces are decreased in magnitude for the four-point

849

Stress analysis of a simplified compression plate fixation system

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Fig. 5(a-b). A comparison of axial strains for three locations at the center of the plate and tube as measured on the physical mode1 and as predicted by the finite element models. The location 12.7 mm corresponds to the side of the tube and 29.2mm corresponds to the top surface of the plate.

Fig. 4.(a-c). Axial normal stress distribution for load case B as predicted by composite beam theory (top) and the direct-contact (middle) and linear-truss (bottom) finite element models.

bending loads. The uniform and skewed axial loads result in modest increases in screw axial forces. The addition of an osteotomy gap most notably affects the loads on the innermost screw, the screw closest to the fracture site. This is especially true for the transverse shear and bending forces. The total resultant screw forces on the innermost screw are fiveto ten-times greater for the osteotomy gap model than for the linear-truss model.

Table 1. Screw axial force resultants (Newtons)

Direct-Contact Line~TIUaS Bi-Linear Truss Oeteotcry-Gap

-4.2

-14.7

3.8

-12.0

-18.4

17.4

-8.6 -50.2

-18.5 10.7

-5.6 34.0

Pour-Point Berdlng: I4= 9.34 N-m I = 1lm.r sorew; U c middle sorw;

0 = outer *orew.

E. J. CHEALet al.

850

Table 2. Screw shear force resultants (Newtons) Finite

Element Hodel

I

Direct-Contact Linear-Truss B1-Linear

Truss

Osteotcmy-Gap

Four-Point I = inner

Table

screu;

no_

-0.4

-1.0

-13.9

-25.4

-86.0

-283.2

-27.2

-98.4

-2% .4

133.0

-42.7

-256.5

Bending:

H = 9.34 N-m

H = middle screw;

3. Screw bending

moment

0 = outer

resultants

SIX-BY.

(Newton-

meters x IO-‘) Finite

Element Model

I

Direct-Contact Linear-Truss El-Linear

Truss

Osteotcmy-Gap

screv;

0 -60.2

35.4

108.8

255.9

41 .a

134.1

299.7

0.4

224.1

-455.9

Four-Point I E inner

H -6.5

2.2

Bending:

H E 9.34 N-m

I4 = middle screv;

0 = outer

screw.

3.3 Plate-induced osteopenia To examine the question of bone loss associated

with reduced bone stress levels, the axial normal stress distributions are shown for the cross-section beneath the center of the plate for load case C (Fig. 6). Included are the predicted stress distributions for the unplated tube and for the plated tube as predicted by the direct-contact finite element model. The most obvious effect of plate application is an upward shift of the neutral axis toward the plate-tube interface. This effectively reduces the magnitude of the longitudinal stresses, especially in the region directly beneath the plate. This “stress protection”, measured as a percentage of the stress level generated by the same load case applied to the unplated tube, represents an 91% decrease for this load case. Note that for this location the results for the linear-truss model (not shown) are not markedly different from those of the direct-contact model. To further examine the question of stress protection, the axial, normal stress contours at the inner, middle, and outer screw locations are presented for the skewed axial load (load case C) as predicted by the direct-contact model (Fig. 7). Figure 6(a) represents the reference condition for the stress in the unplated tube. Near the outer end of the plate, the tube carries a greater proportion of the total load, as evidenced by the downward shift of the location of the neutral stress state. This results in tensile stresses at the outer screw location reaching a peak of approx. 7Of, of the corresponding tensile stress on the unplated tube. Contour plots for the same stress component as predicted by the linear-truss finite element model, are shown in Fig. 8. At the inner screw location, there is still a SO”, reduction in stress level compared to the unplated tube. At the middle screw location. how-

Fig. 6(a, b). Axial normal stress distribution for load case C on an unplated tube (top) and plated tube as predicted by the direct-contact finite element model (bottom).

ever, there is a sharp horizontal gradient reaching a local maximum which exceeds that predicted for the unplated tube. At the outermost screw, there is a three-fold increase in axial stress, indicating a significant stress concentration. This is consistant with the high level of loading in the outermost screw for the linear-truss model. To further examine the transfer of load from the tube to the plate, vector plots of the principal stresses in the plexiglass tube were generated. Figure 9 corresponds to the maximum principal stresses (Pl) for the skewed axial load (load case C) for the region around the outermost screw and the end of the plate. The results are contrasted to those for the directcontact model, the linear-truss model, and the bilinear truss model. Fig. 9(a), from the direct-contact model, emphasizes the rapid transfer of load which occurs from the tube to the plate in this region. High magnitude (5.5-6.5 MPa) tensile stresses occur directly beneath the outer end of the plate, oriented primarily in an axial direction. with a small vertical component. Figure 9(b), from the linear-truss model, displays high magnitude axial tensile stresses at the screw locations, with the magnitude increasing from the innermost to the outermost screws. At the outermost contact point between plate and tube, there is a high magnitude (7.0 MPa) tensile stress directed approx. 30” off vertical, indicating a tendency for

Stress

bin=

-4

A=-4 F=-15

053 IG=-I

Mm = -5 355 A=-5 IS=-4 F=O

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analysis of a srmplified compression plate fixation system

Mox z-0 0308’ I~=-05

1

jC=-3

ID=-2

Wm =-4 4=-4 -: (

282 18=-3 IG=2

IC=-2

ID=-1

Max=245 IE-0

1

Max = I 23C IEz_1

1G.I

Fig. 7(ac). Axial normal stress drstribution for load case C at the inner (top), middle (center), and outer (bottom) screw locations as predicted by the direct-contact model.

Fig. S(a-c). Similar to Fig. 7, as predicted by the lineartruss model.

from the tube surface. The bi-linear truss model, as shown in Fig. 9(c), also displays high magnitude tensile stresses at the screw locations. At the end of the plate, however, the vertical stress component resulting from a tendency for plate separation is greatly reduced. Directly beneath the outer end of the plate, there is a 2.5 MPa tensile stress directed approx. 60” off vertical. From Fig. 9(b), the surface boundary conditions are violated for a small region for load case C, for the linear-truss finite element model. The resultant stresses are not oriented tangential to the external surface for several nodes on the upper surface of the tube located one element length beyond the end of the

plate. Beneath the plate, higher magnitude tensile stresses occur which also are not oriented tangential to the surface. However, these stress components are accounted for by the linear truss elements which simulate the sliding contact between the plate and the tube. The finite element method used in this analysis is displacement based and the element interpolation functions need only satisfy geometric boundary conditions. The geometric boundary conditions are specified at the boundaries of the structure, in this case the two planes-of-symmetry, and the interelement continuity conditions are maintained. A numerical consequence is the occurance of stress re-

plate separation

E. J. CHEAL ef al.

852

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190 IB=-~ IG*l 5

Max=2 13; jE=-I 5 1

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\\\ I Ftg. lO(a. b). Axial normal stress distributions for load case D as predicted by the direct-contact (top) and linear-truss (bottom) models

Fig. 9(a-c). Principal stress PI for load case C beneath the outer end of the plate as predicted by the direct-contact (top), linear-truss (middle), and bi-linear truss (bottom) models. which may not satisfy the boundary conditions, especially when extrapolated to the nodal points immediately adjacent to externally applied loads. In this instance, rather than an external load, the stress resultant from the truss interface element at the end of the plate is responsible. It should be noted that the occurance of this anomaly does not influence the stress distributions in other areas, such as in the tube beneath the center of the pfate. s&ants

3.4 Plate and screw pretension The distribution of axial normal stresses resulting from static pretension in the plate and screws (load case D) is shown in Fig. IO. The results from the direct-contact model (Fig. loaf, using thermo-elastic

brick elements in the plate to produce a prestrain, are generally similar to the results from the linear-truss model (Fig. lob), using applied displacements on the plate. Both produce high magnitude compressive stresses directly beneath the plate and tensile stresses opposite the plate. The bending component confirms the tendency for fracture site distraction for a straight plate applied to a straight bone. To contrast the two techniques for generating a preload in the plate, contour plots of the von Mises stress for the same cross-section are shown in Fig. 11. The direct-contact model exhibits peak effective stress directly beneath the outer edge of the plate, whereas the peak occurs beneath the center of the plate for the linear truss model. For the direct-contact model this is due to the isotropic nature of the applied prestrain. The applied displacements method is preferable since it avoids the more complex thermo-elastic elements and applies a non-isotropic prestrain to the plate. AS previously mentioned, however, this method is not applicable to the direct-contact model. 3.5 Interfragmentary strains To examine the strains which occur in the gap region of an osteotomized bone, principal strain data was generated for the osteotomy-gap modei. Figure 12 shows the three principal strain components for the skewed compressive axial load. The strains in the

Stress analysis of a simplified compression plate fixation system

0 18584 lE=2 IG=?

p=4 11=9

853

Max=10 3E IE=5 IJ=IO

Fig. Il(a, b). Similar to Fig. 10 for von Mises stress.

gap region are very large in comparison to the strains in the rest of the mesh since these elements are highly compliant. The largest principal strains (in magnitude) are compressive, directed axially, and of highest magnitude at the bottom of the tube, as expected. This is the strain component which can be predicted with a one-dimensional model of the system. The maximum tensile principal strains, shown in Fig. 12(b), range in magnitude from 60 to 90% of the maximum compressive component. These are generally oriented in a radial direction. The primary gradient is in the vertical direction, with the highest magnitude occuring at the bottom, opposite the plate. In addition there is a local minimum along the central circumference, with the strain magnitude increasing in both directions to the inner and outer surfaces. The remaining principal strain component (Fig. 12c), are compressive and oriented circumferentially. The magnitude of these strains are about one-third the magnitude of the other components, and display a radial gradient, with the local maximum occuring at the inner surface and the local minimum occuring at the outer surface. 4. DISCUSSION

A three-dimensional finite element model of an idealized system was used to study the mechanics of compression plate fixation. The results for different plate-tube interface characteristics were compared to

Fig. 12(a-c). Principal strains P3 (top), Pl (middle), and P2 (bottom) for load case C as predicted by the osteotomygap model.

composite beam theory and to strain gage measurements. The finite element models were used to predict the resultant screw forces, to study the mechanics of plate-induced osteopenia, and to predict the resultant strains in an interfragmentary gap region. The finite element model suggest features about internal force resultants in the screws which may be of clinical importance in the compression plating of fractures. If a fracture gap is present, the loads on the innermost screws are greatly increased. After the fracture site regains stiffness, the outermost screws carry the most load. The innermost screws therefore are more susceptable to static or quasi-static failure, which can occur during the early stages of weight

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bearing, whereas the outermost screws are more susceptable to fatigue failure, which can occur if the plate is left on for a long period of time. The model also demonstrates the importance of the plate-bone interface characteristics. The interface characteristics are especially important for external bending loads, as demonstrated by the comparison with strain gage data. Under four-point bending, the models with truss interface elements demonstrated significantly improved correspondence with the strain gage data, whereas under axial compression the interface characteristics made little difference. The relatively poor correspondence for all finite element models with the strain gage data for axial loading may be due to a bending component in the load applied to the physical model. Pure axial loading is difficult to achieve and may be a poor choice for model validation. The plate-bone interface has significant clinical importance. The transfer of load from the tube to the plate, for external loads, occurs very rapidly when the plate is securely bonded to the tube surface. as in the direct-contact model. This results in high magnitude stresses occuring in the tube at the end of the plate. With sliding contact, however, the load is transferred primarily through the screws. The transfer of load is more gradual and. as a result, the region of the tube in which the stress levels are reduced is smaller. If plate induced-osteopenia is related to a reduction in cyclic axial stresses, the extent of bone which is affected will be highly dependent on the plate-bone interface. This is supported by the results of Pilliar et al. [lo], wherein metal plates with a porous ingrowth surface resulted in significantly greater bone resorption than similar plates having a smooth surface. In this investigation, the addition of the bi-linear interface characteristics was not highly productive. The minor improvements over the linear-truss model do not justify the decrease in element mesh density necessitated by the required equilibrium iterations. In future models of actual plate-bone systems, lacking the symmetry conditions. nonlinear interface elements will not be used. The simulation of a static preload demonstrated the high magnitude axial compressive stresses which may occur directly beneath the plate. Models of plate fixation used for the study of plate-induced osteopenia must consider these static stresses. The question remains as to the relative importance of the applied cyclic loads from daily activities and the static stresses resulting from plate pretension in the long term remodeling response to internal fixation. The model of the plated osteomized tube demonstrates the multi-axial nature of the strains which may occur in the interfragmentary region. In a plated fractured bone, it can be expected that interfragmentary relative motion is extremely complex. In addition, as healing progresses, the material properties of the tissue in this region can be expected to change dramatically. The present model demonstrates that, even in the simplified situation of a uniform osteotomy gap of constant width, high magnitude radial and circumferential strains may be expected, in addition to axial strains. Further limitations of the present model of the interfragmentary region should be pointed out. The level of apphed loading produced peak strains of 30%

or even higher. These occur only in a small region of the interfragmentary gap, for this arbitrary level of load. Clearly here the assumption of linearity is invalid. More refined element types should be employed, such as elastic-plastic or strain-hardening, to represent the gap region. Also. the mesh should be altered such that the element mesh density is much higher in the gap region, and lower m the rest of the model. 5. CONCLUSION The objective of this investigation was to develop a three-dimensional finite element model of a simplified fracture fixation system. A six-hole stainless steel compression plate applied to an intact plexiglass tube was represented using three different plate-tube interface characteristics. The mechanics of a plated fractured bone were examined using a similar mode1 with a uniform transverse osteotomy gap. The following clinically relevant conclusions may be drawn for this study: (I) The screws nearest the fracture site are most susceptable to failure during the early stages of healing. The outermost screws are most susceptable to failure if the plate is not removed for a long period of time. (2) The presence of sliding motion between the plate and bone greatly increases the resultant screw loads, decreases the extent of bone subjected to reduced axial stress levels, and decreases the stress on the bone at the outer ends of the plate. (3) A static tensile preload in the plate results in high magnitude compressive stresses in the adjacent bone, which should be considered in models of plate-induced osteopenia.

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