Biomedical Engineering Reference
In-Depth Information
20 cm
2
/sec-M
w
(C)
¼
10
6
cm
2
/sec
D
¼
R
m
Production rate
¼
5
m
5,000 molecules/cell/sec
Hint: Show that the constitutive equation for J reduces to v
¼
¼ w
dC/dr, where v is the velocity
of the cell.
6.
The flux (F) of a molecule present at concentration C through a circular hole of diameter d on a
surface that is adjacent to a fluid that it is diffusing in is given by
F
¼
4DdC
The total that can be transferred is the per pore capacity times the number of pores formed.
Calculate the flux allowed though each pore if the diffusion coefficient is 10
6
cm
2
/sec, the
concentration is 1 mM, and the pore diameter is 1.5 nm. Discuss your results, and try to
estimate how many pores are needed to reach meaningful cell-to-cell communications. With
the per-pore flux just estimated, derive the time constant for transfer of a metabolite from a
particular cell to a neighboring cell. Assume that these are two epithelial cells whose geometry
can be approximated as a box and that the two adjacent boxes are connected with transfer
occurring through
pores.
7.
If the cellularity in cartilage is about 1 million cells per cc, estimate the average distance
between the cells. Discuss the characteristics of this microenvironment.
8.
Use a one-dimensional analysis of the diffusion of oxygen into a layer of adherent cells to show
that the maximum oxygen delivery per unit area (N
ox
max
) in Example Problem 6.8 is given by
N
max
ox
n
DC
=
R
where C
*
is the saturation concentration of oxygen and R is the thickness of the liquid layer.
9.
Consider a neuron growth cone that is being influenced by a chemoattractant produced by a
target cell. The geometry of the model system is shown in the attached figure. The target cell
secretes a chemoattractant at a rate, P
r
, which diffuses into a three-dimensional volume, with a
diffusivity D. The governing equation for mass transport for a spherical source is
¼
@
C
@
t
¼
D
1
r
2
@
@
C
@
r
2
r
@
r
a.
What are the boundary conditions for the system?
b.
Considering a steady state, derive the concentration profile as a function of
It has been
found that the growth cone senses a target cell when the concentration difference across
the growth cone is higher than 2 percent. It is believed that growth cones develop filopodia
that extend radially out of the cone as a means to enhance their chemosensing ability. Let
r.
b
1
be the angle that a filopodia makes with the center radius line, R, and
b
2
the angle made by
a filopodia extending in the diametrically opposite direction. Filopodia can extend radially
from the growth cone surface, except where the cone connects to the axon as defined by
a
.
c.
What are the appropriate limits of
b
1
and
b
2
?
d.
What are
r
1
and
r
2
as a function of
b
1
?
Continued
