The Transport of Insulin-Like Growth Factor through Cartilage - Porous Media: Applications in Biological Systems and Biotechnology - page 426

where the subscript i refers to each of the six IGFBPs, and the subscript j =1
refers to IGF-I and j = 2 to IGF-II.

11.4.2.1 Law of Mass Action with Competitive Binding

The chemical reactions between two growth factors (i.e., IGF-I and -II) and
two IGFBP functional groups can be described by

k −ji c ji −

k + ji c s BPi c j ,i =1 , 2

2

dc s BPi
dt

= (11.50a)

j =1

dc 1 i
dt

= k +1 i c f

1 c s BPi −

k − 1 i c 1 i ,

i =1 , 2 (11.50b)

dc 2 i
dt

= k +2 i c f

2 c s BPi −

k − 2 i c 2 i ,

i =1 , 2 (11.50c)

where c f
1

and c f

2 are the volume-based mobile (free) IGF-I and IGF-II concen-
tration, respectively; c 1 i and c 2 i are their immobile complexes attached to two
functional groups (i.e., c s BP 1 and c s BP 2 ); k + ji is the association rate constant
of IGF-I and -II with each two IGFBP functional group; and k −ji is the disso-
ciation rate constant of IGF-I and -II with each two IGFBP functional group.
Summing equations (11.50a-c) leads to

d ( c s BPi + c 1 i + c 2 i )
dt

=0 ,i =1 , 2 (11.51)

Thus, c s BPi ( t )+ c 1 i ( t )+ c 2 i ( t )= m i . Similar to equation (11.34), the inte-
gration constants m i can again be obtained from the initial condition, such
that,

c s BPi ( t )+ c 1 i ( t )+ c 2 i ( t )= c s BPi (0) + c 1 i (0) + c 2 i (0) ,i =1 , 2

where

c s BPi ( t =0)= c s BPi 0 , c 1 i ( t =0)= c 1 i 0 and c 2 i ( t =0)= c 2 i 0 (11.52)

Substituting equation (11.52) into (11.50a) leads to

dc 1 i
dt

1
k +1 i

= c f ( K D 1 i + c f

1 ( c s BPi 0 + c 1 i 0 + c 2 i 0 −

c 2 i )

1 ) c 1 i ,i =1 , 2 (11.53a)

−

dc 2 i
dt

1
k +2 i

= c f ( K D 2 i + c f

2 ( c s BPi 0 + c 1 i 0 + c 2 i 0 −

c 1 i )

2 ) c 2 i ,i =1 , 2 (11.53b)

−

where

K D 1 i = k − 1 i

k +1 i ,K D 2 i = k − 2 i k +2 i ,i =1 , 2

(11.54)