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acts as a limitation to the study of these models. Harrigan and Hamilton75 sought the origin of unstable
bone remodeling simulations mathematically, using strain-energy-based remodeling rules in an attempt
to assess whether the unstable behavior was due to the mathematical rules proposed to characterize the
process or to the numerical approximations used to exercise the mathematical predictions. The physio-
logic interpretation indicated that the instabilities that occur in some remodeling simulations are due,
at least in part, to the mathematical characterization of bone remodeling. In addition, the behavior of
the observed instabilities is not present in vivo. Consequently, the cause of this unstable behavior is most
likely not attributed to natural remodeling processes.
Carter et al.35 observed that when their simulation of femoral bone remodeling was allowed to progress
past the first few iterations, The method employed appeared to converge toward a condition in which
most elements will either be saturated & or be completely resorbed. Recently, Weinans et al.38,76 dem-
onstrated, confirmed by others,75,77 that previous bone remodeling implementations tend toward discon-
tinuous density patterns (Fig. 2.8). In the vicinity of the applied loads, elements predict alternating
patterns of high and low density, resembling the pattern of a checkerboard. Jacobs et al.,77 in an attempt
to eliminate the spurious near-field discontinuities, while maintaining anatomically correct far-field
discontinuities, implemented a node-based technique.
Fyhrie and Schaffler,78 in the same vein, sought to improve spatial stability via a revised phenomeno-
logical theory of bone remodeling. They cite that bone remodeling theories are often based on the
common assumption that the changes in bone structure in response to an error signal are adaptive, and
therefore bring about a reduction in error. They criticized that under these assumptions, the basic
formulation of the remodeling problem is to adapt the structure to make the error approach zero.
Consequently, this formulation will not converge to an optimal bone structure unless the error function
is specifically designed to do so. If, however, the optimality is defined as zero signal error at each point
in the bone, this formulation does result in an optimal solution. En route to the development of a new
remodeling theory, the following distinctions were made. The apparent density was identified as the
controlling variable, while the controlled variable was a function of the apparent strain, denoted M(E).
The controlling and controlled variables were defined as those which the bone cells can directly modify,
and those which measure the ability of bone to adapt to the current need, respectively. Although the
precise form of the function M(E) is not known presently, it is considered the homeostatic value of
apparent density attained by bone subjected to constant strain. The fact that the function is not necessarily
zero as the strain magnitude goes to zero accounts for the biological factors which prevent the total
disappearance of bone tissue. The fundamental character of the remodeling equation was exponential,
consistent with experimental observations of changes during disuse, after hip replacement surgery, and
during growth and aging. Fyhrie and Schaffler were able to demonstrate that the model is stable tempo-
rally, and more spatially stable than some models published previously.
Causal Mechanisms
The origin and function of adaptive remodeling have been debated extensively. The feedback mechanism
by which bone tissue senses the change in load environment and initiates the deposition or resorption
of bone is not understood.15 Undoubtedly, mechanical factors play an important role in remodeling;
© 2001 by CRC Press LLC
FIGURE 2.8 Predicted checkboard density distribution characteristic of the traditional element-based bone remod-
eling algorithms. Source: Jacobs, C.R. et al., J. Biomechanics, 28, 449, 1995. With permission.
inactivity results in widened Haversian canals and porotic bone, while stresses result in a more solid
compactum. Recent investigations have explored the biological response of bone to mechanical loading
at the cellular level, but the precise mechanosensory system that signals bone cells to deposit or resorb
tissue has not been identified.15
Numerous recorded observations suggest that bone cells in situ are capable of responding to mechanical
stimuli and do so in a predictable fashion (i.e., Wolff s law). Experimental limitations often hinder such
investigations at the cellular level. A major constraint of in vitro organ culture conditions is that the
cultured structures are complex and composed of heterogeneous cell populations.15 Although in vivo
loading conditions may be approximated, extracting satisfactory information from these models regard-
ing individual cell behaviors is laborious. Nonetheless, experimental procedures have implicated different
mechanisms for adaptive bone remodeling.
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