Biomedical Engineering Reference
In-Depth Information
The surface remodeling rate coefficients and the reference values of strain
are phenomenological coefficients of the bone surface and can only be deter-
mined experimentally. To simplify the analysis, Cowin [11] assumed that the
surface remodeling rate coefficients
C
ij
are not site dependent; that is, they
are independent of the position of the surface point
Q.
He also postulated
that
C
ij
are independent of strain. Equation (2.1) gives the normal velocity of
the surface at
Q
as a function of the existing strain state at the point
Q.
If the
right-hand side of Equation (2.1) is positive, the surface will be growing by
the deposition of material. If the right-hand side of Equation (2.1) is negative,
the surface will be resorbing.
It should be mentioned that Equation (2.1) by itself does not constitute
the complete theory. The theory is completed by assuming that bone is a
linearly elastic material. Thus, the complete theory is a modification of
linear elasticity in which the external surfaces of the bone move according
to the rule prescribed by Equation (2.1). A boundary value problem can be
formulated in the same manner as a boundary value problem in linear elas-
tostatics, but it is necessary to specify the boundary conditions for a specific
time period. As the bone evolves to a new shape, the stress and strain states
will be varying quasistatically. At any instant, the bone will behave exactly
as an elastic body, but moving boundaries will cause local stress and strain
to redistribute themselves slowly with time.
2.2.3 Internal Bone Remodeling
As described in Cowin [11], the small theory of internal bone remodeling
is an adaptation of the theory of equilibrium of elastic bodies. The theory
models the bone matrix as a chemically reacting porous elastic solid. The
bulk density ρ of the porous solid is written as the product of γ and ν:
ρ = γ ν
(2.2)
where γ is the density of the material that composes the matrix structure, and
ν is the volume fraction of that material.
Both γ and ν can be considered field variables, by the same arguments as
employed by Goodman and Cowin [22]. Let ξ denote the value of the volume
fraction ν of the matrix material in an unstrained reference state. The density
γ of the material composing the matrix is assumed to be a constant; hence, the
conservation of mass will give the equation governing ξ. It is also assumed
that there exists a unique zero-strain reference state for all values of ξ. Thus,
ξ may change without changing the reference state for strain. The change in
the volume fraction from a reference volume fraction ξ
0
is denoted by
e
:
e
= ξ − ξ
0
(2.3)
The basic kinematic variables and also the independent variables for the
theory of internal bone remodeling are the six components of the strain
