Methods

1
Methods The expanded Hookes law in the Voigt
matrix notation s c e is (1) where s is a 6
x 1 matrix of stresses, e a 6 x 1 matrix of
strains, and c a 6 x 6 symmetric matrix of
elastic constants. The elements of c are related
to the components of the rank 4 elasticity tensor
in 3 spatial dimensions. The elements of s are
components of the stress tensor. The normal
strain components of e are elements of the strain
tensor the shear strain components of e are
twice the tensor shear strain components. The
relations between components of c and the
engineering constants (Youngs and shear moduli,
and Poissons ratios) depend on material symmetry
and are tabulated in many texts. Orthotropy is a
symmetry commonly ascribed to bone, for which 9
independent constants exist. Bone is frequently
assumed to possess even greater symmetry, e.g.,
transverse isotropy with 5 or isotropy with 2
independent constants. The expanded Hookes law
in the Kelvin notation
is (2) where the matrix c contains the
components of a rank 2 elasticity tensor in 6
dimensions. We wish to interpolate at an
intermediate point between known elastic
constants matrices c(1) and c(2). The
intermediate point is located a distance d from
point 1, defined so that the distance from point
1 to point 2 is 1 and 0 d 1. Simply
thinking, the relative distance d between the
first and intermediate points can be thought of
as a spatial (geometric) distance, but we propose
to define d in terms of radiographic density. In
addition, component-wise interpolation of the
elastic constants does not work, as shown
subsequently, so we propose an eigensystem-based
interpolation method instead, and one within
which we will use an interpolation distance d
based on the function that provides the best
correlation between combinations of the elastic
constants and radiodensity. The
eigensystem-based method borrows substantially
from a key work (Cowin Yang, J Elast 1997) on
averaging multiple measurements of elasticity
tensors. The matrices c(1) and c(2) are
determined from Eqn. 2, and their eigenvalues and
eigenvectors (3) respectively, (k 1-6) are
determined by standard methods of linear algebra.
Nominal weighted averages of the eigenvectors
are (4) Interpolated eigenvectors are
determined by minimizing the sum of the square of
the distances between them and the eigenvectors
at each point (5) where denotes the tensor
inner product. A set of interpolated eigenvalues
referred to the interpolated eigenvectors
is (6) The interpolated rank 2 elasticity
tensor in Kelvin notation is then (7) Finally,
an interpolated cINT matrix in the Voigt notation
can be determined from Eqns. 1 and 2, and the
interpolated engineering constants from tabulated
relations.