| |
The goal of this
study was to prove that the load carrying capacity of a bone with an
osteolytic defect is determined by the weakest cross-section in the bone
and that the determination of the load carrying capacity using composite
beam theory to calculate the axial, bending and torsional rigidity at
each transaxial cross-section integrates the site, size and location of
the defect and material properties of the host bone. We accomplished
this aim by conducting an ex-vivo experiment on adult cadaver femurs and
a case-control in-vivo study on children with benign osteolytic
tumors of the appendicular skeleton. For the ex-vivo experiment,
simulated lytic defects of varying size were created at either the
antero-medial or postero-lateral aspect of the femur at the
intertrochanteric level. The fracture force for an applied load
configuration simulating single legged stance was measured
experimentally and compared to the predicted failure load using a
simplified curved beam, plane strain model of the femur that assumed
failure to occur at the weakest cross-section through the bone
determined using CT-based structural analysis. For the in-vivo
study, we studied a unique patient population of children under clinical
evaluation for benign osteolytic tumors of the appendicular skeleton
without associated systemic abnormalities and derived a fracture risk
probability model based on the reduction in bending and torsional
rigidity induced by the tumor relative to the homologous region of the
contralateral normal limb. We also studied ex-vivo the variation
in bilateral symmetry for the geometric, densitometric and structural
properties of the adult femur to establish the lower limit for
significant differences in these properties when using the contralateral
limb as an intra-subject control. The high accuracy of the ex-vivo
experiment and the clinically derived fracture risk probability
model obviated the need to perform a parametric finite element analysis
that we had originally proposed as part of our contingency plan.
 Results
Measurement of the out of plane forces and in plane moments validated
our representation
of the proximal femur as a plane stress, curved beam model with a
torsional spring at the knee to resist varus/valgus moments when loaded
in single legged stance. The out-of-plane force, Fx, was less
than 5% of the in-plane forces and the in-plane moments (Fx < 2.4
±
2.27), My and Mz, were less than 5% of the out of
plane moment, Mx (My and Mz < 4.0
±
3.7% of Mx).
The defects created were
equivalent and spanned a similar range of reduced structural properties
for both anteromedial (AM) and posterolateral (PL) locations: maximal %
reduction at defect = 28±7%
for AM vs 36±11%
for PL (t=-1.32, p=0.22). There were no significant differences between
femurs with AM and PL defects with respect to the age, weight, height,
and body mass index of the donor. The measured failure load was
unaffected by defect location (AM: 7.3±1.3
kN vs PL: 6.9±2.0
kN; t=0.77, p=0.32) however the predicted failure loads tended to be
smaller for AM defects (assuming either tensile or compressive failure),
but did not quite reach statistical significance because of the
relatively small number of specimens in each group (predicted tensile
fracture - AM: 3.7±1.4
kN vs PL: 5.6±1.4
kN; t=-2.19, p = 0.06, predicted compressive fracture - AM: 6.2±1.7
kN vs PL: 9.1±2.7
kN; t=-2.05, p = 0.07). This is not unexpected, since defects of
equivalent size would (theoretically) be expected to have more a more
profound reduction in the load carrying capacity of the femur when
located anteromedially at the calcar femoralis. The location of the
minimum predicted fracture force calculated using the compressive
failure strain consistently predicted the actual location of fracture.
Tension consistently under-predicted the fracture load; compression
tended to overestimate the fracture load, but was not significantly
different from the actual measured failure load (Figure C.1.4.2). The
average of the calculated fracture loads using tensile and compressive
failure strains was not different from the measured fracture load
(predicted =6.17 ±
1.82 kN vs. measured = 7.14 ±
1.61kN; t=1.34, p=0.20) and came closest to predicting the measured
fracture force (absolute error = 2.1±1.2
kN).
Discussion
This study demonstrates that a simplified plane stress curved beam model
of the proximal femur can be used to predict the fracture load at the
proximal femur. The cross-sectional structural properties incorporate
the effect of the size and location of the defect and quality of the
host bone. Fracture occurred at the level of the minimum predicted
failure load calculated using a compressive failure strain of 1%. The
fracture force we measured exceeds the elastic yield load and includes
post-yield plastic deformation prior to catastrophic failure. Composite
beam theory was developed to predict the elastic deformation of
axisymmetric beams. Since bone fails at a constant strain independent of
density, we have demonstrated that composite beam theory can be extended
to predict the yield load (the force at which the material deformation
departs from linear elastic behavior to non-linear plastic deformation)
for bones with lytic defects. The femur is a materially heterogeneous,
irregularly shaped, non-axisymmetric bone that is subjected to both
tensile and compressive strains when loaded in single legged stance. We
have demonstrated for the simplified load case of single legged stance,
that the location of the minimum predicted yield force calculated using
a plane stress, curved beam, composite material model of the femur and a
compressive failure strain of 1% correctly identified the site of
fracture. The best estimate of the fracture load at that location was
given by the average of the calculated yield forces for the model using
a tensile failure strain of 0.8% and compressive failure strain of 1%.
|
