The Transport of Insulin-Like Growth Factor through Cartilage - Porous Media: Applications in Biological Systems and Biotechnology - page 427

11.4.2.2 Steady-State Growth Factor Uptake

At steady state, equations (11.53a-b) reduce to

K D 1 i + c 1 c 1 i =0 ,i =1 , 2

c 1 ( c s BPi 0 + c 1 i 0 + c 2 i 0 −

c 2 i )

−

(11.55a)

K D 2 i + c f

2 c 2 i =0 ,i =1 , 2

c f

2 ( c s BPi 0 + c 1 i 0 + c 2 i 0 −

c 1 i )

− (11.55b)

Solving equations (11.55a) and (11.55b) for c 1 i and c 2 i we obtain

c s BPi 0 K D 2 i c f

c 1 i =

K D 11 K D 21 + c f

2 + K D 21 c f

1

,i =1 , 2 (11.56a)

1

c s BPi 0 K D 1 i c f

c 2 i =

K D 11 K D 21 + c 2 + K D 21 c 1

2

,i =1 , 2 (11.56b)

The results are a pair of competitive Langmuir sorption isotherms [77].
The total growth factor uptake ratio ( R ui ) (analogous to definitions in
equation (11.47)) can be expressed as

⎛ ⎞

2

1
c i 0

⎝ c i +

c ij

⎠ ,i =1 , 2

R ui = (11.57)

j =1

where c i 0 are IGF-I and -II concentration in synovial fluid at the outer surface
of cartilage, respectively.
At steady state, the IGF concentration within cartilage is equal to that in
synovial fluid (i.e., c i = c i 0 ). Further, substituting equations (11.56a,b) into
(11.57) leads to

2

c b BPi 0 K D 2 i

K D 11 K D 21 + c f

2 + K D 21 c f

R u 1 =1+ (11.58a)

i =1

1

2

c b BPi 0 K D 1 i

R u 2 =1+

K D 11 K D 21 + c f

2 + K D 21 c f

(11.58b)

i =1

1

11.4.2.3 Model Calibration

The six unknown parameters in Equations (11.58a,b), namely, K D 11 , K D 12 ,
K D 21 , K D 22 , c b BP 10 , and c b BP 20 , can be determined by fitting the experimental
data of Bhakta et al. (2000) (shown in Figure 11.13) using a nonlinear least