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 84

where v C F is the free solute velocity relative to the fixed frame of reference and q is
the solute mass sink due to binding to the matrix.
The solute velocity relative to the À uid is described by Fick's law of diffusion.
Making use of Equation (6), the mass À ux of the solute relative to the solid phase is
given by:

D ∂

c F
∂

v
φ

c F ⋅ ( v C F − v s ) = c F ⋅ ( v C F − v x f ) + c F ⋅ ( v x f − v s ) =− − c F ⋅

(10)

x

The continuity equation for a bound solute reads:

∂φ ⋅ c B ∂

∂

( ) =− q

( v C B v s )

+ φ ⋅ c B ⋅ −

(11)

∂ t x

where v C B is the bound solute velocity relative to the fixed frame of reference, the sink
term q equals the sink term of free solute with the opposite sign, because binding

and unbinding are the only mechanisms of solute exchange between the two phases.
By neglecting the diffusion of a bound solute in a solid phase the mass flux term in
Equation (11) can be omitted.
Porosity of the gel generally depends on strain [10]:

−

φ + ε

.

φ =

0

1 + ε

From this expression follows that for φ 0 ≈ 1 and small deformations

0.05 , the de-

≤

with respect to x and t are practically zeros.
Combining the Equation (7, 9-11) and assuming that porosity

φ

rivatives of

φ 0 ≈ 1, yields:

2 c F
∂

∂ c F

∂

+ ∂

c B
∂

D ∂

∂
∂

(12)

( )

= + v ⋅ c F

t t x 2 x

Equation (12) is the governing equation for the free solute transport. To relate un-
knowns c F and c B to each other, the equation for a binding reaction must be pro-
posed.

7.3.3 Bimolecular Reaction
It is assumed that the free solute (F) can bind to the binding site on matrix (N). And the
bounded form of the solute (B) can dissociate from the matrix:

k f
↔

F + N B (13)

k r

where k f is the rate constant for the forward reaction of association and k r is the rate
constant for the reverse reaction. This reaction could be described in terms of bimolec-
ular reaction [12, 19]: the rate of increase of bound solute concentration is proportional
to the free solute and free binding sites concentrations, and the complex dissociates