Bone Tissue - Meshless Methods in Biomechanics - page 238

of the bone apparent density, as well as the microstructural designs characterizing
both anisotropy and bone surface area density, consistent with the real bone
apparent density distribution.

6.3.5 Proposed Adaptation of Carter's Model

In the present topic it is considered the remodelling algorithm presented by
Belinha and co-workers [ 27 , 28 ], which assumes that the mechanical stimulus,
suitably described by stress and/or strain measures, acts as the principal driving force
in the bone tissue remodelling process. The local density and material orientation is
dependent on the stress/strain field caused by the mechanical load. The presented
remodelling algorithm is an adaptation of Carter's model for meshless methods.

6.3.5.1 Model Description

In this subsection a bone remodelling nonlinear equation is introduced and com-
bined with the meshless method procedure. The expression can be presented as a
differential equation, in which a temporal-spatial based functional, q app ð x ; t Þ ,is
minimized with respect to time,

oq app ð x ; t Þ
ot

Dq app ð x ; t Þ
Dt

¼ ð q model
app

Þ t j ð q model Þ t j þ 1 ¼ 0 ð 6 : 15 Þ

app

d þ 1

Being q app ð x ; t Þ : defined for the temporal one-dimension and the

spatial d-dimensions. The analysed problem must be discretized in space and time.
It is assumed that the d-dimensional spatial domain is discretized in N nodes:
X ¼ x 1 ; x 2 ; ... ; x N

R 7! R

ffi 2 X,

f g 2 X, leading to Q interest points: Q ¼ x 1 ; x 2 ; ... ; x Q

d . The temporal domain is discretized in iterative fictitious time steps

being x i 2 R

t j 2 . The medium apparent density for the complete model domain is

defined by ð q model

R , with j 2 N

app Þ t j at a fictitious time t j . Within the same iterative step, the
medium apparent density of the model, q model
app

, can be determined with,

¼ Q 1 X

Q

q model
app

ð q app Þ i ð 6 : 16 Þ

i¼1

being and ð q app Þ I the infinitesimal apparent density on interest point x I defined by
q I ¼ g ð r I Þ . The functional g ð r I Þ :

3

R 7! R is defined by,

g ð r I Þ ¼max ðf r 1 ð q I Þ; r 1 ð q I Þ; r 1 ð q I ÞgÞ ð 6 : 17 Þ

1 2 3