Perimeters
Counting techniques can be used to determine the length of perimeters
(and hence perimeter/area and surface/volume ratios if the three
dimensional structure is known). Some restrictions do apply as will be
evident after considering the following example, but the method is simple
and valuable.
There are a number of measures of the circle we can make using the grid.
-
We can count the number of points (P)
in the circle to determine the area.
If the width of the grid squares is w,
then the area will be given by:
A = Pw2
-
We can count the number of points across the centre, i.e.
the number across the diameter (nD).
The diameter will be given by:
D = nDw
and of course the length of the perimeter (circumference, C)
is then predictable using
C =
D
-
We can count the number of intersections between the perimeter
and the grid lines, i.e. the points shown in the next picture.
This number can be denoted ni.
Note that the number of points of intersection between the circle and the
grid is predictable. Consider the vertical lines. D/w lines will
cross the circle, each creating two intersections, so there will be 2D/w
intersections with the vertical lines, and the same for the horizontal lines,
or 4D/w intersections in all. i.e.
ni = 4D/w
or
D = niw/4
But as noted above, the circumference is also related to diameter, so
D = C/
so we can eliminate diameter and show that
C = niw/4
This conclusion is quite important, for the expression says we can determine
the perimeter length merely by counting the intersections.
Furthermore, the expression is independent of diameter, so the total perimeter
of a number of circular profiles can be determined by counting the
intersections they all make with the grid.
Perhaps even more importantly, the perimeter of a profile made up of a
number of circles can also be estimated in this way, i.e. a profile like:
which is made up of three circles. Any profile which is isotropic,
that is has no preferred orientation to its perimeter, can be measured
in this way.
Perimeter/area ratios
Since we now have counting methods to measure some forms of perimeters,
and their areas, we can equally calculate perimeter/area ratios.
These may enable estimation of surface to volume rations, but only
if we know the three-dimensional form of the structure our measured perimeters
represent.
Surface/volume ratios of cylindrical structures
If our circular profiles are perpendicular sections through cylinders,
then the perimeter/area ratio gives the surface/volume ratio of the
cylinder. The extension of the argument above to isotropic profiles other
than circles will also apply. This is a useful approach for elongated
cells like nerve and muscle, or intracellular tubules or the like.
The equation to give the surface/volume ratio (Sv) is
Sv = ni/4Pw
where P is the number of
points counted
within the profile.
©D.F. Davey,
Department of Physiology,
University of Sydney
Last updated 10 April 2002