Biomedical Engineering Reference
In-Depth Information
[mMol]
glucose
t
=
12 [h]
2.5
2
1.5
1
construct
porous filter
(a)
(b)
Figure 14.1
(a) Schematic view of a bioreactor system designed for growing articular cartilage tissue (b) result
of a numerical calculation of the concentration of glucose in the tissue engineered construct. In the
analysis it is assumed that the glucose concentration in the medium surrounding the construct is
constant (essential boundary condition) and that glucose is consumed by the cells in the construct
(sink term). Because of symmetry, only the right half of the construct is modelled. Adapted
from [
17
].
In the human body diffusion is very important, but also in in vitro experimen-
tal set-ups in the laboratory, for example in tissue engineering applications (see
Fig.
14.1
).
Example 14.2
Another completely different process which is also relevant in biomechanics is
steady state one-dimensional heat conduction with a source term:
d
dx
dT
dx
λ
+
f
=
0,
where
λ
is the Fourier coefficient of heat conduction and
f
a heat source term.
Further,
T
(
x
) represents the temperature. The boundary conditions might be for-
mulated as: prescribed temperature at
u
and prescribed heat flux at
p
. It should
be noticed that the mathematical structure of this heat conduction problem is fully
equivalent to the structure of the diffusion problem.
Example 14.3
A third example of a completely different diffusion type problem is the uniax-
ial tension or compression of a bar as introduced in Chapter
6
, governed by the
equation:
EA
du
dx
d
dx
+
f
=
0,
where
E
is the Young's modulus,
A
the cross section of the bar and
f
a dis-
tributed force per unit length. The unknown
u
(
x
) is the displacement field of the
bar (also see Eq. (
6.20
)). Boundary conditions may be imposed as a prescribed
displacement on
u
and a prescribed force at
p
.
