ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

where n p is the number of proliferating chondrocytes per unit time (cells/day) and
h max is the maximum height, in μm, of the fully mature hypertrophic chondrocytes.
d) Based on the mathematical model proposed by Garzón-Alvarado et al [12] the
growth rate can be written as:
where
proliferation
d and
proliferat ion hypertroph y
& , (4)
ε = d + d
hipertrophy
d are the strain rate tensors due to proliferation and
hypertrophy, respectively. The growth in the proliferative zone is due to cellular
mitosis, therefore it is given by
⎛ ⎞
proliferation
d ⎜
dn
= n ⎟n
⊗n
⎜ p ,
⎟
(5)
⎝ lp
⎠
where d n is the width of the chondrocyte in the preferential direction of growth n,
and l P is the width of the proliferative zone. Furthermore, the growth in the
hypertrophic zone is due to the elongation experienced by the cells during their
differentiation from the proliferative state (ovoid) to the hypertrophic state (quasispherical
shape) [12]:
⎛
⎞
hypertrpohy
d ⎜
2 ⎛ R ⎞
i(
t)
− s
= ⎟
⎜ ∑ ⎜
⎟
⎟
n ⊗n
, (6)
⎝ lh
i∈C ⎝ Δt
H i ⎠⎠
where Ri (t)
is the instantaneous radius of the i-th chondrocyte in the growth
direction once the cell begins to differentiate, s is the minor radius of the
proliferative chondrocyte ( d n = 2s
), l h is the width of the hypertrophic zone and
Δ ti
is the hypertrophy elapsed time. This time is limited by the maximum
chondrocyte maturation time t E [12].
e) According to Stokes et al [7] the final size of the hypertrophic chondrocytes is
mechanically modulated, so we assume that the instantaneous radius is a function of
the stress modulation over the preferential direction of growth Δ σ n :
where
n
R
( Δσ
) − s
( 1+
αΔσ
)
− s
Ri ( t)
= s + max
tE
n Δti
= s +
n
tE
f
max Δti
Δ σ = σ −σ
is the difference between the current stress over the preferred
n
f
n
f f
direction of growth σ n and the stress under physiological conditions σ n , Rmax is
the radius of the sphere representing the hypertrophic chondrocyte achieved under
physiological loading conditions ( Rmax hmax
2
f
= ) and α is the change in the
maximum size of chondrocyte per unit stress. α is calculated using experimental
data [6]
R
(7)