The Transport of Insulin-Like Growth Factor through Cartilage - Porous Media: Applications in Biological Systems and Biotechnology - page 417

reaction is reversible), we can express the interaction as.

IGF-I + IGFBP k +1

k − 1

IGF/IGFBP (11.33)

where k +1 is the rate constant for the association reaction (binding) and k − 1
is the rate constant for the dissociation reaction. Ignoring the possibility of
diffusion and advection of fixed IGFBPs and complexes, the law of mass action
[68] can be used to quantitatively describe the IGF-I and IGFBP interaction
and leads to the following set of differential equations.

dc I
dt

= k +1 c I c b BP − k − 1 c I (11.34a)

dc b BP
dt

k +1 c I c b BP + k − 1 c I = − (11.34b)

Summing equations (11.34a) and (11.43b) it is easily confirmed that when
there is no net production/degradation of binding proteins

dc b BP
dt

+ dc I
dt

= 0 (11.35)

Thus, c b BP ( t )+ c I ( t )= m . The integration constant, m , can be obtained
using the initial condition, and leads to

c b BP ( t )+ c I ( t )= c b BP 0 + c I 0 , where c b BP ( t =0)= c b BP 0 , c I ( t =0)= c I 0

(11.36)

Finally, equation (11.36) is substituted into (11.34a) to obtain

= k +1 c b BP 0 + c I 0 − c I c I −

dc I
dt

k − 1 c I (11.37)

Equation (11.37) provides a relationship linking c I and c I .

11.3.2.2 Model of Solute Transport and Binding in

a Deformable Cartilage

Now that IGF-I is able to bind to IGFBPs on the solid phase, the solute
conservation law derived previously needs to be modified and an additional
expression needs to be introduced for solute conservation on the solid phase.
Specifically the conservation of mass of solute in the free and bound state can
be described by

∂ φ f c I
∂t

c I + φ f v f c I = −s −φ f D

+ ∇• ∇ (11.38)

∂ 1 φ f c I

∂t

−∇• 1 φ f v s c I = s

−

− (11.39)