Effects of Cyclic Loading and Binding to Extracellular Matrix on Macromolecular Transport in Soft Biological Tissue - Progress in Organic and Physical Chemistry: Structures and Mechanisms - page 87

(17), (20)) are transformed to dimensionless variables, as has been proposed by Mauck
et al. [2]:

u
h

x
h

N
c 0

u c F , B

c 0

D

x ′ =

,

′

,

h 2 t ,

,

′

(21)

= ′ N =

c F , B = ′

t =

After substitution of these variables, the governing system of equations reduces to:

u
∂ ′

u
∂ ′

⎧

∂ ′

2

′

⋅ ∂

− R g = 0

⎪
⎪ ⎪

t x 2

u
∂ ′

2

∂ ′ c F

∂ ′

+ ∂ ′

c B
∂ ′

= ∂

c F
∂ ′

′

+ ∂ ′

∂ ′ c F

∂ ′

⎨

(22)

x 2

t t t x

⎪
⎪
⎪

∂ ′ c B

∂ ′

′ ′ ′ ′

= k 1 ⋅ c F ( N − c B ) − k 2 ⋅ c B

t

⎩

c 0 h 2
D

k f ⋅

h 2
D

k r ⋅

H A ⋅ k

where R g , k 1 = , are three non-dimensional parameters.

k 2 =

=

D

Following Mauck et al. [2], it is useful to note, that R g represents the ratio of char-
acteristic velocity of fluid in the gel under load H A ⋅

k

to the characteristic diffusive

h

velocity of solute relative to the fluid D

h . k 1 is the ratio of the characteristic diffusion

1

time h 2

D to the characteristic binding time

on the edge of the gel. k 2 is the ratio

k f ⋅ c 0

of the characteristic diffusion time h 2

1
k r D to the characteristic unbinding time

, which

is independent of concentration.
The boundary and initial conditions respectively transform to:

u (0,

⎧ ′ ′

t ) = 0

⎪
⎪ ⎪

u (1,

′ ′ ′

R g ⋅ ′

t ) = ε 0 ⋅ sin(2 π ⋅ f ⋅ t )

⎨

c F (1, ′ t ) ′ = 1

(23)

⎪
⎪ ⎪

∂ ′ c F

∂ ′

0 =

0

x

⎩ x ′ =

and

⎧
⎨ ⎪
⎩ ⎪

c F ( ′ x ,0) ′ = 0

′ ′ (24)

c B ( x ,0) = 0

u (

′ x ,0) ′ = 0

where the non-dimensional frequency