An elastic analysis to determine fracture gap motions occurring in the osteotomized and plated canine femur was performed using the finite element program NASTRAN. The femur was idealized as a hollow right cylinder, and transverse anisotropy was assumed for the elastic properties of the bone. A 3-D 360 degree model consisting of 224 isoparametric quadrilateral hexahedral and 11 beam elements was created. A range of plate stiffnesses was tested by varying the modulus of elasticity of the plate from 207 GPa to 1 GPA. Moments were applied in the plane of the plate, about the axis of the plate, and in the plane of the screws. Results showed that, for plates of typical geometry and elastic modulus under 10 GPa, the contribution to fracture gap motion occurring due to deformation in the bone was negligible compared to that contribution from deformation in the plate

Cooke, F. W.

Vannah, W. M.

English

NASA Technical Reports Server

AN ELASTIC ANALYSIS OF A PLATED BONE TO 
DETERMINE FRACTURE GAP MOTION 
Francis W. Cooke and William M. Vannah 
Bioengineering Department, Clemson University 
SUMMARY 
An elastic analysis to determine fracture gap motions occuring 
in the osteotomized and plated canine femur was performed using 
the finite element program NASTRAN. The femur was idealized as a 
hollow right cylinder, and transverse anisotropy was assumed for 
the elastic properties of the bone. A 3-D 360 degree model 
consisting of 224 isoparametric qaadrilateral h~xahedral and 11 
beam elements was created. A range of plate stiffnesses was tested 
by varying the modzlus of elasticity of the plate from 207 GPa to 1 
GPA. Moments were applied in the plane of the plate, about the 
axis of the plate, and in the plane of the screws. 
Results showed that, for plates of typical geometry and 
elastic modulus under 10 GPa, the contribution to fracture gap 
motion occuring due to deformation in the bone was negligible 
compared to that contribution from deformation in the plate. 
Fracture gap motion for bone-plate systems using plates in this 
modulus range could therefore be estimated using beam theory. 
Deformations occr.rring in the bone were relatively unaffected by 
plate stiffness. The rel-ationship of gap motion to plate stiffness 
was non-linear above 10 GPa due to thy decreasing contribution of 
the plate's deformation. 
INTRODUCTION 
The use of a compression plato as a method of treating severe 
bone fractures is a currently accepted practice in orthopedics. In 
this design (Figure l), the plate is applied in such a manner that 
it locks in a small amouat of press-fit between the fragments; the 
applied force and close fit of the fragments induces the bone to 
grow faster. However, the demand for this close, solid fit means 
the compression plate must be stiff - so stiff, in fact, that the 
plate continues to carry much of the load on the bone even after 
the fracture has healed. Most bones neei strains imposed on them to 
maintain their size and strength; in the absence of these strains, 
they atrophy in a manner similar to unused muscles. It has been 
shown experimentally (8,9,10,11) that the application of a stiff 
bone plate to an intact bone will generally letd to a long-term 
https://ntrs.nasa.gov/search.jsp?R=19850017571 2020-03-22T19:17:10+00:00Z
loss in the volume of bone under the center of the plate. The 
resulting decreased wall thickness under the plate midspan makes 
the bone prone to refracture after plate removal. 
To solve the problem of stress shielding, a number of 
researchers have experimented with an increasingly more flexible 
series of plates. The m o s ~  obvious lower bound to plate flexibility 
occurs when the plate permits so much motion of the fragments that 
it is no longer possible for the fracture gap to be bridged. As the 
plate stiffness approaches the non-union point, the tire required 
for the fracture to heal increases to infinity. In mcch of this 
research, the osteotomized canine femur has been used as a model. 
Selected results of these tests, in terms of elapsed time before 
union of the fracture versus plate stiffness, are shown in Table I. 
We were interested in the possibility of a resorbabie plate 
but, since the resorbable material we were working with was fairly 
flexible, we were unsure whether we could make a plate sciff enough 
to be above the non-union bound. While estimates of the value of 
this bound could be made from the data of other researchers, we 
wished to define it as accurately as possible before making oar own 
plates. Before starting a series of implantation studies in dogs to 
define this bound, we performed a mechanics analysis to determine 
the characteristics of the bone-plate system, This analysis is the 
subject of this report. 
METHODS 
It was assused that motion at the fracture gap was the 
governing variable determining whether or not a fixed fracture 
would go to union. To determine the total deflection at the gap, it 
was necessary to determine how much of the total deflection was due 
to plate deformation and how much was due to bone deformation. 
While it was easy to estimate the plate bending using beam theory, 
defining the bone deformation required a complex mechanics analysis 
easiest done, and perhaps only possible, with the finite element 
method. It was expected that the application of loads from the 
relatively narrow plate to the bone might cause the bone to deflect 
significantly in the area of the plate. This was especially true 
because odr plate design created a line contact betwezn the plate 
and bone. Since this effect could not be simulated with a two 
dimensional (2D) ~odel, it was decided to create a three dimen- 
si~nal ( 3 0 )  model with the bone composed of solid quadrilateral 
elements, and the plate and screws of one dimensional (ID) beam 
elements. 
The dog femur was modelled as a hollow right cylinder. The 
wter radius of 9.5 mm and the inner radius of 6.9 mm were 
deterniined from inspecti~n of availhble femurs and the published 
data of other investigators, and are intended to represent dogs in 
the 18 to 27 kg ranga. The length of bone modelled was chosen to be 
thc length covered by the plate plus one bone diameter more past 
the end of the plate to allow point loads appl~ed at the end to 
spread out. 
The geometry of the plate (Pigcre 2 )  had been prsviously fixed 
as a preliminary to the manufacture of prototype plates. It was 
intended that factors such as orientation of the composite layers 
and percentages of resin would be varied to produce the desired 
stiffneses. This effect was simulated in the model by simply 
varying the modulus of elasticity of the beam elements. The high 
stiffness plate used the elastic modulus of 316L stainless steel. 
The geometry and size of our plate was apptoximately what would be 
chosen to fix a bone this size in clinical practice. The results 
for the high stiffness plate could therefore be considered to 
represent a typical case where stress shielding might be a problem. 
The modulus used for the medium stiffness plate was chosen to 
produce a stiffness slightly above the estiaated non-union bound. 
The modulus used for the low stiffness plate produced a stiffness 
below the non-union bound. 
Values for the elastic modulus of canine bone were available 
in the literature, but were taken under conditions inappropriate 
for the loadings anticipated in this study. The material properties 
used were those deternined for human cortical bone by Reilly and 
Burstein ( 7 ) :  these properties assume transverse anisotropy. The 
bone was idealized as being composed of purely cortical bone, no 
attempt was made to model cancellous bone. 
With these dimensions settled, a model was then created using 
the NASTRAN finite element program. The input deck to NASTRAN was 
generated using the PATRAN graphics pre- and post-processor. The 
model was made to be roughly the maximum size that would allow 
reasonable executicn times on the DEC VAX-11/780 computer. The 
model consisted of 224 solid isoparametric quadratic CIHEX2 
elements and 11 CBAR beam elements, and executed in eight hours cpu 
time. b smaller, faster executing 180 deqrze model was also 
constructed for use in those cases where it was possible to assume 
symmetry about the plane of the screws(Pigure 3). It was not 
pcssible to run a Kore finely divided model to test for conver- 
gence. The results snown in this report will be checked against 2 
similar physical model in an experimental program currently 
underway. Additlonally, a modcl si~ilar to the 180 degree model 
with nearly identical mesh but isotropic material properties was 
tested against a strain-gauged test model by Cheal et ai. ( 5 )  and 
shown to be accurate. 
In order to most efficiently utilize the available computer 
time, only those cases which would produce the most significant 
deflections were run. Axial compression was not applied since it 
was felt that, because loads were applied to the axis of the piate, 
this loa? case would result in on1.y minor deflection in the plate 
and a saall shear deflection in the bone. A force camins a moment 
in the plane of the screws, directed so as to open the osteotomy, 
was applied on the axis of the plate at the far end of the bone. 
This is referred to here as the bending open or BO load case. A 
moment was applied about the axis of the plate; this axial torque 
is referred to as the TA load case. A force causing a moment in the 
plane nornal to the axis of the screws was applied on the axis of 
the plate at the far end of the bone. This moment, which produced a 
bending against the plate's stiffest direction, is referred to here 
as the BA load case. 
In the final =load casem, an axial torque was again applied, 
and also a configuration change was made. The plate-bone contact 
was changed  fro^ a line contact at the axis of the plate to a two 
line contact. These lines were at the axis of the plate and the 
outer edge of the plate which would be driven into contact with the 
bone by an axial torque. This load/configuration case, which 
modelled a mwide-uheelbasem plate, is referred to as the UW load 
case. 
RESULTS 
The fracture gap motions resulting from the various load cases 
are presented in Table 11. The deformed shapes are presented in 
Figures 4a-d. 
DISCUSSION 
The bighest deflections were produced by the 00 and TA load 
cases. The BA load case produced deflections which were alnost 
negligibly small in comparison, 
For the BO load case, the deflection took place primarily in 
the plate. This was especially true for the lower stiffness plates. 
With the high stiffness plate, significant deflection (accounting 
for 34% of the mo~ion at the gap) occurred in the bone. This was 
not a whole-scale bending of the bone, but iiistead appeared to oe a 
local deformation under the plate, This was especially visible as a 
raised dimple under the screw closest to the fracture. A shallow 
depression under the fa: end of the plste vas also visible. This 
would seem to indicate that the area under the screws (where load 
transfer OCCUKS~ was highly stressed, consistent with the prolif- 
eration of bone seen in this area on X-rays. A possible hypothesis 
is that what really affects fracture gap motion is not che bone's 
moment of inertia as a cylinder, but the stiffness of the wall 
under the plate. However, this is not a critical point since, for 
the low stiffness plates, the motion at the gap was almost entirely 
a f'lnction of the plate stiffness. 
Deformation under axial torsion (TA) also took place primarily 
in two modes. The first node, responsible for the great majority of 
the motion at the gap, was twist of the plate in the span between 
the f~acture gap and the closest screw. The second mode was twist 
of the bone cylinder itself. This was less visible on the plots 
with flexible plates because the twist of the plate was greatly 
increased, making the bone twist appear relatively small (the 
deflections for plots were scaled by the computer for readibility). 
The torsional twist of the bone appears to be an action of the 
whole bone, rather than a local action as was the case under the BO 
load. 
The BO and TA load cases produced roughly the same amount of 
deflection at the fracture gap for equivalent applied moments. 
Bending in the plane of the plate took place mainly as a 
bending of the plate. The beam element bent against their stiffest 
direction. The fracture gap aotion produced was more than an order 
of magnitude less than the deflections produced by either the BO or 
TA load cases, on an equivalent applied moment basis. 
Changing the plats bone contact to the wide wheelbase config- 
station and applying an axial torsional load produced a noticeable 
flattenin? of the bone cylinder under the outer plate edge. The 
bone appeared to be more distorted in this case than in any other, 
yet fracture gap motion was only decreased by one-third. This leads 
to the conclusion that the wide wheelbase plate design may not be 
necessary if the screw/plate fit can be relied upon to provide 
rotational stiffness. 
The fracture gap motions predicted are quite high compared to 
what might be expected to be the maximum movement allowable for 
healing. Carter et al. ( I ) ,  implanted strain gauges on the femur of 
a 35 kg dog and studied the strains txder normal gait. Prom these 
strains, they calculated the loads at the instant of maximum net 
loading of the mid-femoral diaphysis as a 250 N axial compression, 
a 2 5 N-m bending moment directed to produce tension anteriorly, a 
1.4 N-m bending m o ~ e n t  producing tension laterally, and a torsional 
moment of 0.56 N-n where the vector points proximally. These loads 
produce hypothetical fracture gap motions oE 52 to 0.33 mm dae to 
bending moments, and 9.0 to 0.44 mm due to torsional moments, 
depen6ing on the stiffness of the plate used. With the medium 
stiffness plate, the predicted fracture gap deflections are 5.0 and 
1.1 ma, for bending and torsional loads respectively. A possible 
hypothesis is that these magnitudes are realistic, but thac the 
deflections in bending stay low because the plate is positioned so 
that it is usually in tension (i.e., bending to open the osteotomy 
doesn't occur), and khat the torsional displacements do occur and 
are the relevant concern in the design cf fixation plates. In cases 
of fractures, the interdigitation of the ragged fracture edges 
could be expected to lend support and there may be some stiffening 
due to the surrounding tissues; neither of which was considered in 
this model. The main conclusion from this result though is that 
unusual care probably would have to be taken of a fracture 
stabilized with a flexible plate. 
The total deflection of the system was a product of the 
deflections of the plate, the sctews in the bone, and the bone 
itself. In this analysis, only the plate stiffness was varied, and 
therefore the deflection in the bone and screws was relatively 
constant (altnough affected somewhat by the bracing action of the 
plate). The two ends of the deflection-stiffness curve, that is the 
respnse at very high and very low plate stiffnesses, could then be 
estimated. In the case of very flexible plates, the plate would 
deform so much that the defornation of the bone and screws would be 
negligble by comparison. In this case, the deflection would vary 
with the inverse of the stiffness and have a constant slope which 
could be calculated by beam theory. In the case of an infinitly 
stiff plate, the deflections present would all occur in the bone 
and screws. Therefore, in this case the deflection would be 
constant; that is, it would not vary (or would vary only a negli- 
gible amount) as the beam stiffness was varied. Using this 
reasoning to predict the shape of the tail-s, the general shape of 
the deflection curve can be predicted (Figure 5 ) .  
The calculated results show that the deflections varied in 
nearly exact proportion with the plate flexibility across the range 
of the two lower stiffness plates tested. Even as the plate 
stiffness greatly increased to the stiffness of a typical solid 
stainless steel plate, the bending in the bone contributed only a 
small portion of the fracture gap motion. This implies that, for 
flexible plates, the fracture gap motion is a function solely of 
the plate stiffness and can be directly calculated from beam 
theory. 
TABLE I. Materials evaluated using the canine femur model 
Investigator Material EI (Nm2) Results (weeks) 
Bradley (I) SS 16.80 
GEC 5.48 
GPC 1.45 
Brown (2) PA 0.49 
PBT 0.41 
PBT 0.23 
PBT 0.49 
Brown (3) PA 0.265 
PBT 0.221 
PP 0.239 
UHMPPE 0.161 
Zenker (11) GFMM 3.01 U 
NASTRAN model high 6.02 
med i urn 0.29 
1 ow 0.029 
Legend : 
GFMM - graphite fiber reinforced methyl methacrylate 
S S - stainless steel 
GEC - glass-epoxy resin composite 
GPC - graphite reinforced polysulfone 
PA - polyacetal 
PBT - polybutyleneteraphtalate 
PP - polypropylene 
UHMWPE - ultrahigh molecular weight polyetylene 
U - proceeded to union, no exact time given 
TABLE 11. Calculaked fracture gap motions 
................................................................... 
Load case :.late stiffness Slope (l/Nm) Fracture gap (mm/Nm) 
................................................................... 
high 
med i um 
low 
high 
medium 
low 
WW high 0.00440 0.167 
BO - bending to open the osteotomy 
TA - axial torsion 
WW - axial torsion, wide wheelbase plate 
BA - bending in the plane of the plate 
Figure 1. Derivation of Itlode1 
3.65 dl . .  
A l l  dimensions i n  m i l l l r t e r s .  
Figure 3. 180 tkqree rrPdel 
Figure 2. P l a t e  geawtry 

Pla te  F l ex ib i l i t y  
Figure 5. Predicted fracture gap motion 
REFERENCES 
1. Bradley, G. W., McKenna, G. B., Dunn, H. K., ~aniels, A. U. and 
Statton, W. 0. "Effects of Flexural Rigidity of Plates on Bone 
Healing." J. Bone and Joint Surgery, 61A:866-872. 1979. 
2. Brown, S. A., Merritt, K. and Mayor, M. B. "~nternal Fixation 
with Metal and Thermoplastic Plates." In: Current Concepts of 
Internal ~ixation of Fractures Uhthoff, H. K., ed., Berlin: 
Springer Verlag 1980. pg. 334-341. 
3. Brown, S. A .  and Vandergrift, J. "~ealing of   em oral 
Osteotomies with Plastic Plate Fixation." Biomat., Med. D~v., Art. 
Org., 9:27-35. 1981. 
4. Carter, D. R., Vasu, R., Spengler, D. M. and Dueland, R. T. 
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1980. 
5. Cheal, E. J., Hayes, W. C. and White A. A. "Stress Analysis of 
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7. Reilly, D. T. and Burstein, A. H. "The Elastic and Ultimate 
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1975. 
8. Tonino, A.  J., Davidson, C. L., Klopper, P. J. and Linclau, L. 
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9. Uhthoff, H. K. and Dubuc, F. L. "Bone Structure Changes in the 
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An Elastic Analysis of a Plated Bone to Determine Fracture Gap Motion

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