Biomedical Engineering Reference
In-Depth Information
gle capillary can supply with oxygen and remove waste products from. The
number of service units scales ~ M 3/4 (2,3), which implies that the volume of a
service unit scales as ~ M 1/4 . If we assume that a service unit is spherical, its ra-
dius ( R ) will scale ~ M 1/12 .
Now to connect the WBE model with immunology, we assume that the ser-
vice volumes for the blood circulation are also the service volumes for immune
surveillance, that is, the capillaries allow lymphocytes to exit the circulatory
system and explore regions of tissue for foreign molecules and cells, collectively
called antigens. This implies that if each clone of B cells or T cells contains at
least ~ M 3/4 cells it can be represented in each of the ~ M 3/4 service volumes.
One of the essential ideas in the WBE model is that the capillary that sup-
plies a service volume is universal in its properties, such as its diameter. This
implies that the amount of blood delivered to the service unit, per unit of time, is
independent of M . From the point of view of immune surveillance, most anti-
gens will enter the service unit in the blood. In other words, the number of anti-
gens that enter into a service unit per unit of time is independent of M .
2.2. The Time to Find an Antigen in a Service Unit
Consider a single antigen and one specific lymphocyte, of some clone that
is specific for the antigen, both located in the same service unit. This lympho-
cyte will crawl within the service unit in a more or less random fashion, search-
ing for antigen. How long does it take until it makes first contact with the
antigen?
If one describes the random walk of the lymphocyte as spatial diffusion
with a diffusion coefficient D , then one can show that T , the average time until
first contact between the lymphocyte and antigen, is given by (4)
3
R
t
=
A ,
[2]
3
D
where A is the sum of the radii of the lymphocyte and the antigen, and R is the
radius of the service unit.
If the diffusion coefficient D is independent of M , then as R ~ M 1/12 , T ~ M 1/4 .
Thus, if there were only one lymphocyte per clone present in the service unit,
the antigen could go undetected by that lymphocyte for a period of time (~ M 1/4 )
that increases with animal size. Since the search time for each clone should scale
in the same manner, this result applies to all clones. In order to keep the time
until detection a fixed value (smaller than the time during which the antigen
could proliferate significantly) the organism has to put ~ M 1/4 copies of the lym-
phocyte into this service unit; this would reduce the mean time until first detec-
tion by a factor of ~ M 1/4 to a value independent of M . We conclude that if D is
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