Page 409 - Advances in Biomechanics and Tissue Regeneration

21.2 BONE REMODELING MODEL 407

8 9
1 ð < ε xx x I k =
ðÞ
ðÞ σ yy x I k
U x I k σ xx x I k ðÞ σ xy x I k ε yy x I k dΩ I (21.2)
2
ðÞ
ðÞ
ðÞ ¼
Ω I : ;
xy x I k
γ ðÞ
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 2 2
1
ðÞ σ 2 x I k
ðÞ σ 1 x I k
σ x I k σ 1 x I k ðÞ þ σ 2 x I k ðÞ þ σ 3 x I k ðÞ (21.3)
ðÞ σ 3 x I 2
2
ðÞ ¼
In Eq. (21.3), σ i (x I ) k is the principal stress of the integration point, x I , of load case k, being i={1,2,3}. Lastly, at each time
instant, j, the variable fields obtained for each load case, k, are weighted using the following expression:
j
j
j
j

X l m k u , ε , σ , U , σ j
u , ε , σ , U , σ (21.4)
j j j j j
k¼1 X l
¼
m s
s¼1
being l the number of load cases and s the corresponding number of load cycles.
21.2.3 Remodeling Points
A particular feature of this model is the fact that only a portion of integration points is selected to suffer bone remo-
deling. Therefore, with an optimization algorithm, only the integration points with SED values are belonging to the
following chosen interval:
M x I 2 U min , U min þα 4U _½ŠU max β 4U, U max Š, 8M x I 2 ℝ (21.5)
ðÞ
ðÞ ½
in which U min =min(U), U max =max(U), and 4U=U max U min . This approach then considers that only integration
points that are under extreme levels of mechanical stimulation will be actively remodeled. The ratio of integration
points included in the low stimulus group is given by α, while β defines the ratio of integration points composing
the high stimulus group. The low stimulus group is named the “resorption group” since the bone apparent density
of the points in that group will potentially decrease. In turn the high stimulus group is denominated “formation
group” since its points will potentially increase their apparent density.
21.2.4 Phenomenological Law
In this step, each of the selected remodeling points will update their bone apparent density using the phenomeno-
logical law proposed by Belinha et al. [7]. Belinha’s law is a mathematical proposal capable to describe the experimen-
tal results obtained by Lotz et al. [15] and Zioupos et al. [16]. Correlating bone apparent density, ρ, with the ultimate
c
c
compression stress in the axial, σ axial , and transverse, σ trans , directions, a new value of ρ is determined by back substi-
tution in Eqs. (21.6), (21.7).
σ c axial ¼ X 3 j¼0 a j :ρ j (21.6)
σ c X 3 b j :ρ j (21.7)
trans ¼ j¼0
A new ρ field leads to an update of the mechanical properties of the chosen remodeling points. Thus, using again the
anisotropic Belinha’s law, the elasticity modulus in the axial, E axial , and transverse, E trans , directions is determined with
the following expressions:
8
X 3 j 3
> c j :ρ if ρ 1:3g=cm
j¼0
<
E axial (21.8)
3
X
d j :ρ if ρ > 1:3g=cm
> j 3
:
j¼0
X 3 j
e j :ρ (21.9)
j¼0
E trans ¼
c
c
3
In Belinha’s law the apparent density is expressed in g/cm , while E axial , E trans , σ axial, and σ trans are expressed in MPa.
The values of the coefficients in Eqs. (21.6)–(21.9) are presented in Table 21.1.
The final step is to determine the mean apparent density of the computational model, ρ med , as described in the fol-
lowing expression:
II. MECHANOBIOLOGY AND TISSUE REGENERATION