Bone Tissue - Meshless Methods in Biomechanics - page 235

Harrigan and Hamilton [ 40 - 42 ]. Based in the idea that the remodelling process is
not just influenced by the momentary strain state, Bucháˇek [ 43 ] developed an
extended version of the adaptive elasticity model of Cowin where bone density and
material orientation are dependent on the whole bone strain history. It was
assumed that the difference between the actual and optimal density and the vari-
ation between the actual and optimal material orientation, with both depending on
the strain state, serve mutually as the adaptive stimuli and have a time-fading
effect on the remodelling process. In this manner, at a given time t any past strain
event will have some effect on the remodelling process, however because it is
considered an exponential time-fading function the bone reactions are dominated
mainly by the recent strain state.

6.3.3 Carter's Model

A mathematical formulation for the functional adaptation of trabecular bone based
on a self-optimization concept was proposed by Carter and co-workers [ 44 - 46 ]. In
this model, as in Pauwels's model, it was also assumed, within the bone tissue, that
a mechanical stimulus S n must be present in order to maintain a quasi-stationary
state in the bone remodelling process. Carter suggested that this stimulus should be
constant for the entire bone and proportional to the effective stress.

S n / X

l

m i r i

ð 6 : 12 Þ

i¼1

In this formulation distinct load cases can be considered, i ¼ 1 ; ... ; l, and each one
is weighted by the corresponding number of load cycles, m i . The influence of the
magnitude of the correspondent stress state is considered by the exponent j. The
effective stress r i ð r i ; q app Þ depends on the local stress state r i of load case i and
on the local apparent density q app . Regardless the considered bone material law or
the biological basis of bone remodelling algorithm, with this model it is assumed
that the functional adaptation provides the bone with the ability to maximize its
structural integrity with the least amount of bone mass. Which is the same as
assuming that the induced stress from bone remodelling acts as an optimization
tool, minimizing an objective function [ 47 ]. Instead of using the local effective
stress, it is possible to use the strain energy density (SED) approach. The use of the
strain energy is connected with the idea that the bone tissue is attempting to
maximize its stiffness while the use of a failure stress criterion is related with the
material strength optimization. The SED approaches lead to a correlation between
the apparent density q app and the local strain energy, U.