Biomedical Engineering Reference
In-Depth Information
￿ Best estimate of the Hayflick limit is 44 to 50.
￿ About 1:5,000 progenitors do
apoptose.
￿ About 50 to 1,000 mature cells are made per progenitor (6 to 10 doublings).
￿ The entire differentiation pathway may be 17 to 20 doublings (soft fact).
￿ In vitro, about 10 to 30 million cells maximum can be made from a highly purified single
hematopoietic stem/preprogenitor cell.
Present order of magnitude calculations in constructing your decision. Also perform parameter
sensitivity analysis of each of the parameters that govern the hematopoietic process. How
important are the parameters that govern telomerase activity in determining the total number
of mature progeny produced over a person's lifetime?
2. At 1 pM concentration, how many molecules are found in a volume of liquid that is equal to
the volume of one cell (use a radius of 5 microns)?
3. Use the continuum approach (Eq. (4.3)) to show that in a steady state the number of cells
produced during a differentiation process that involves replication but no apoptosis (
not
a ¼
0) is
X out
X in ¼ e
m
=
d
and is thus primarily a function of the ratio
d
/
m
.
a ¼
0 is the completely undifferentiated state,
and
a ¼
1 is the completely differentiated state. What is X out if
m
and
d
are the same orders of
magnitude and if
is 10 times slower? Which scenario is a more reasonable possibility in a
physiological situation?
(Note that if the rates are comparable, only two mature progeny will be produced. On the other
hand, if the differentiation rate is 10 times slower than the replication rate (probably close to
many physiologic situations—i.e., 20 hr doubling time, and 200 hr
d
8 days differentiation
time), then about 1 million cells will be produced. Thus, the overall dynamic state tissue is
strongly dependent on the relative rates of the cellular fate processes.)
4. Kinetics of differentiation/continuous model.
i. Derive the first-order PDE that describes the population balance.
ii. Make time dimensionless relative to the rate of differentiation.
iii. Describe the two resulting dimensionless groups (call the dimensionless group for
apoptosis A, and the one for the cell cycle B).
iv. Solve the equation in steady state for A
¼
0.
v. Solve the equation(s) where A is nonzero for a portion of the differentiation process—that
is, between a 1 and a 2 .
vi. Solve the transient equation for A
¼
0.
5. Consider two cells on a flat surface. One cell secretes a chemokine to which the other
responds. Show that the steady-state concentration profile of chemokine emanating from
the first cell is
¼
R 2 F
C
ð
r
Þ¼
=
D
1
=
r
where R is the radius of the cell, F is the secretion rate (molecules/area time), and D is the
diffusion coefficient of the chemokine. The distance from the cell surface is r. Use the cell flux
equation to calculate the time it would take for the responding cell to migrate to the signaling
cell if there is no random motion (
m ¼
0) given the following values:
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