The Transport of Insulin-Like Growth Factor through Cartilage - Porous Media: Applications in Biological Systems and Biotechnology - page 418

TABLE 11.2
IGF-I, IGFBP Association/Dissociation Rates Parameters Used
in the Model

Parameter Value References

10 5 M − 1 sec − 1 Association rate constant ( k +1 ) 3 . 67 × [69]

0.001 sec − 1

Dissociation rate constant ( k − 1 ) [69]

where s is the solute sink due to the binding of free solute to binding proteins
attached to the solid phase.
The addition of equations (11.38) and (11.39) leads to

∂ φ f c I
∂t

+ ∂ 1

φ f c I
∂t

c I + φ f v f c I

−

φ f D + ∇• − ∇

−∇• 1 φ f v s c I = 0 − (11.40)

Using equation (11.10) and performing some algebraic manipulations [33],
we obtain the general transport governing equation for the solute in the
deformable cartilage with consideration of binding of solute to the solid matrix.

φ f ∂c I
∂t

+ 1 φ f ∂c I

2 c I + φ f v s

−∇ φ f D •∇

φ f D I ∇ c I −

∂t −

− κ ∇ p

+ 1 φ f v s •∇

c I = 0 − (11.41)

With consideration of above assumptions, the governing equations (11.12)
and (11.41) can be rewritten in radial coordinates as

κ r ∂ 2 p

= 0

v r
r

+ ∂v r
∂r

+ ∂ε z

∂r 2 + 1

∂p
∂r ∂t −

(11.42)

r

∂ 2 c I
∂r 2

φ f ∂c I
∂t

∂c I
∂r

+ 1 φ f ∂c I

+ 1
r

φ f D I −

∂t −

∂c I
∂r

+ φ f v r − + 1 φ f v r ∂c I

∂r

κ r ∂p
∂r

− = 0 (11.43)

In the radial IGF-I transport equation (11.43), the first term represents
the change of concentration of IGF-I with respect to time; the second term
represents the IGF-I transported by diffusion, and the third and fourth terms
describe the contribution of mechanical loading and advection in the deform-
ing porous media.