Dear All,

Following Marco Viceconti's contribution to the discussion on "Topic
1: Boundary conditions in bone stress analysis", we would also like to
share our views with the list members.

In general, in formulating a stress analysis problem, it is necessary
to specify the surface values of the three components, either of the
stress vector, the displacement vector or a mixture of both, at each
point on the surface of the analysed structure.

For a body of a given shape and mechanical properties, different
boundary conditions lead to different distributions of internal stress
and strain.

In the most simple example, under a certain load configuration the
distributions of bending moment, stress and strain throughout a beam
depend critically on the end supports - whether free, simply supported
or built-in. Furthermore, the distributions of bending moment, stress
and strain depend critically on the way loads are applied: concentrated
or distributed. The same is true of parts of the musclulo-skeletal
sytem, such as the human femur.

For a body as complex as the femur, with difficult geometry and
mechanical properties, simplification is desirable in order to reduce
the computational problem and to ease interpretation of results. This
is particularly true when iterative solutions are being sought, as in a
bone remodelling study, when the stress analysis has to be repeated
many times. However, as computers get bigger and faster, it is possible
to be more ambitious. More attention needs to be paid to boundary

Simplification has to be justified either by showing that results from
the simplified model accord well with experiment (i.e. calculated
strain values should be similar to those measured in vivo, calculated
force values should be similar to those measured with instrumented
implants in vivo, relative displacement of selected points should
accord well with RSA measurements). Comparison of the FE model with an
experiment which embodies the same simplifications - either in its
loading or in its displacement boundary conditions - can hardly justify
the simplifications. With this type of analysis one can validate the FE
mesh and show that it is accurate enough and can be used to model
another problem with sufficiently small error. Alternatively, a
simplified model can be justified by showing that it yields results not
significantly different from a more complicated model.

When analysing the femur, it seems clear now that adequate modelling of
the applied muscle forces is necessary. The recent work by our group
(Polgar et al., to be published) shows that stress/strain distributions
in the vicinity of muscle attachment areas differ significantly when
the effects of concentrated loads are compared with the effects of
distributed loads. This finding might sound obvious, but in previously
published studies of the human femur this issue has hardly ever been
addressed and the effect of muscle actions were accounted for as a
concentrated force applied in one node representing the muscle
attachment area centre. When muscles with large attachment areas (large
area could be defined as, at least in one dimension, greater than the
cortical thickness of the bone) are included in the model, St Venant's
Principle should be applied with caution.

There are further questions which arise when modelling the intact femur
and loads are specified at the hip - ie. what are the appropriate
boundary conditions at the knee?
Bending moments at the knee are transmitted by a combination of tension
forces in the muscles and ligaments and compression forces between the
articular surfaces.
Since six displacement constraints should be specified, these could be:
- simulating the constraints applied by the tibial plateaus and
the patella: zero normal displacements at the two most distal points on
the condyles and at a point on the trochlea;
- zero displacement at their femoral attachment points along the
lines of three of the four main ligaments (ACL, MCL and PCL maybe).
Even this is a simplification since it ignores the movements associated
with tissue deformation. It is possible that treating the distal femur
as encastre would be appropriate for some purposes but this would have
to be demonstrated by comparative calculations.

It may be that an appropriate experiment designed to fatigue test the
cement mantle around a hip stem could comprise a proximal femur fixed
just below mid-shaft with a hip stem loaded slightly eccentrically and
no muscle simulation or considering the abductors only, but it would be
necessary to show (similarly to Duda et al., Internal forces and
moments in the intact femur during walking. J Biomechanics 30, 1997,
933-941) that the associated bending moment distribution along the
shaft agreed reasonably well with that produced by the physiological
load case. A recent study addressing this issue has been carried out
by Stolk et al. (Stolk et al., Hip-joint and the abductor-muscle forces
adequately represent in vivo loading of a cemented total
reconstruction. J Biomechanics 34, 2001, 917-926).

On the other hand, what might be an appropriate simplification of
muscle loading for one type of study (for instance, modelling cemented
total hip replacements) could be totally inappropriate for an other
(bone remodelling simulations of the intact femur, stresses on the
bone-cement interface in case of long stems, etc). In the latter
case, considering physiological loading (including all muscles,
distributing forces over their muscle attachment areas) might be
necessary in order to obtain realistic stress or strain results.

All the above, of course, is in agreement with the opinion of Ton van
der Bogert and others who said that there is an appropriate model
complexity for each question...

Best regards,

Krisztina Polgar
Richie Gill
John O'Connor
Nuffield Department of Orthopaedic Surgery/OOEC
University of Oxford
Nuffield Orthopaedic Centre, Oxford OX3 7LD, UK

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