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f
f
g
f
′
=
(25)
k
h
2
-the characteristic frequency of the gel (or “gel diffusion”
frequency [3]), that represents the frequency of loading at which the matrix of the gel
is deformed throughout the full thickness
h
.
H
A
⋅
is normalized by
f
g
=
7.3.7 Numerical Solution
The matrix laboratory (MATLAB) software is employed to solve numerically the gov-
erning set of partial differential equations (PDE). The ordinary differential equations
(ODEs) resulting from descritization in space are integrated to obtain approximate
solutions at times specified. The spacial domain
0
1
is descritized into 1,000
equally sized increments. The time steps during PDE solving are chosen by solver,
and for output the time span is defined by the user so that there are no less than thirty
time increments per loading cycle. The displacement
′
≤
x
′
≤
u
(
t
)
is calculated separately
by solving the first PDE in the system (22), which is independent of other equations.
Then the set of two PDEs for solute concentration is solved with implementation of
the available
′
x
,
′
′
u
(
t
)
function.
To get the solution for unknowns
′
′
′
x
,
u
(
x
,
′
t
) ,
′
t
)
,
′
′
t
)
, it is required to
′
c
F
(
′
x
,
′
′
c
B
(
x
,
specify seven non-dimensional parameters:
R
g
,
k
1
,
k
2
,
f
. To de-
¿ ne the appropriate values for these parameters the ranges of values of
H
A
,
k
,
k
f
,
k
r
,
D
, and N for different types of tissues, biological gels and solutes found in the litera-
ture are summarized (see Table.1).
From these data it may be estimated that the value of
R
g
lies in the range 10-1,000,
and will be varied during our analysis by the order of 10. It is assumed that porosity
φ =
N
,
,
ε
0
, and
φ
′
′
1
. Although for cartilage this approximation is not reasonable, for other biological
gels its difference from unity can be neglected.
ε
0
range from 0.05 to 0.005. The fre-
quency considered is between 10 Hz and 0.01 Hz as this range is physiological: blood
pulse frequency is 1 Hz the mean frequency of muscles contraction is about 0.01-10
Hz depending on type of action (walking, standing up, chewing, etc.).
7.4 DISCUSSION
Using the present model, the solute transport enhancement by cyclic deformation of
the tissue has been investigated. It is useful to compare our predictions for two cases
of solute transport: without binding to the matrix and transport influenced by solute
binding.
As stated in results, for the case with no binding the gain does not depend on
frequency in the sense that the same gain is achieved at different times for different
frequencies. The gain is roughly proportional to the deformation amplitude. Binding
of the solute to the matrix can increase the gain (relative to the gain for non binging
solute) by few orders of magnitude, but this effect depends on parameters of reaction,
frequency, and their ratio. The model predicts that there are optimal reaction constants,
ratio of the binding sites concentration to the bath concentration and frequency, for
which maximum gain in solute uptake can be achieved.
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