Biomedical Engineering Reference
In-Depth Information
p
=
c
du
dx
=
1at
x
=
1,
0. Compute the FEM solution (array
sol
)and
compare the nodal values with the exact solution.
(b) Use the array
pos
to extract the nodal solutions within the second
element from the array
sol
.
(c) Determine the solution for the third node using the array
dest
.
14.8 One of the major problems in the tissue engineering of cartilage is to make
thick constructs. Nutrients coming from the surrounding medium have to
reach the cells in the middle of the construct by diffusion, but cells close
to the edge of the construct may consume so much nutrients that there is
nothing left for cells in the middle. Consider the experimental set-up as
given in the figure below, representing a schematic drawing of a bioreactor
to culture articular cartilage.
The construct is fixed between two highly permeable membranes allowing
free contact with the culture medium. The thickness
t
is a trade-off between
the diffusion coefficient
c
, the consumption rate of the cells
f
(both cannot
be influenced) and the amount of the molecules that can be supplied via the
medium, which is usually bound to a maximum.
The diffusion problem can be considered as a one-dimensional problem.
The current analysis is meant to determine the glucose concentration
u
(
x
), which is an essential nutrient for the cells. The following proper-
ties are given:
c
=9.2
with
c
=
1 and
f
=
10
−
7
[Mol hour
−
1
cm
−
3
]. Low glucose medium is used, concentration = 5
10
−
6
[cm
2
s
−
1
] and
f
×
=
56
×
10
−
3
×
[Mol
litre
−
1
].
(a) Assuming the consumption rate
f
is constant and the medium is
refreshed continuously, a stable glucose concentration as a func-
tion of the location in the construct will be reached after a while.
Give the differential equation and boundary conditions describing this
process.
(b) Give the analytical solution of this problem by integrating the
differential equation.
(c) Adjust
demo_fem1d
to solve the problem with the finite element
method. Solve the problem for
t
=
1, 2, 3, 4 and 5 [mm].
(d) At which thickness do the cells appear to die in the middle of the
construct?
14.9 To determine the material properties of a skeletal muscle a uniaxial tensile
test is performed as shown in the figure. The muscle is clamped on one side
and a force
F
=
10 [N] is applied on the other side. The muscle has a total
