Timothy David, Centre
for Bioengineering, University of Canterbury, New
Zealand The Challenge of Multiple Scales in the Biological
Sciences: Applications in Cerebro-vascular Perfusion
In line with architectural
advances in supercomputing science and engineering
have each been posing more and more complex problems
which are defined on complex geometric physical
spaces. These physical spaces are themselves defined
over vast ranges of scale lengths. In order to solve
problems whose scale lengths vary substantially
there are two possible solutions. Either discretise
down to the smallest scale with the possibility
of producing such large data sets and numbers of
equations that the memory requirements become too
large for the machine or divide the problem into
a subset of appropriate length scales and map these
discretised sub-domains onto appropriate machine
architectures. The definition of "appropriate"
here is determined on a case-by-case basis at present.
There are a significant number of problems that
exhibit a large range of physical scales but none
so prominent in the 21st Century as that exemplified
within the biological sciences. In the major arterial
networks the blood flow dynamic scales are of the
order of 1mm (cerebral vessels) up to 25mm (ascending
aorta). Downstream of any major vessel exists a
substantial network of arteries, arterioles and
capillaries whose characteristic length scales reach
the order of 10-20 microns. Within the walls of
these cylindrical vessels lie ion channels consisting
of proteins (100 nanometers and smaller) folded
in such a way as to allow only certain molecules
through the membrane. One can now of course ask
the question as to why all these scales should be
integrated into a single model.
To investigate the way in which the brain responds
to variations in pressure and yet maintains a virtually
constant supply of blood to the tissue numerical
models need to be able to have a representation
of not only the vascular tree but also a dynamic
model of how the small arteries constrict and dilate.
Simulating this phenomenon as a "lumped"
connection of arteries is insufficient since different
parts of the arterial tree respond differently.
Thus we have a range of scales from the major arteries
down to the arteriolar bed. The combination of a
3D model taken from MR data coupled with an autoregulation
model with a fully populated arterial tree able
to regulate dynamically remains a relatively unexplored
field. This particular talk will outline the reasons
for investigating multiple scales and their particular
constraints with special reference to the autoregulation
of blood in the cerebro-vasculature and outline
a possible solution.