Appendix A
When a hollow organ is distended, its thin mucosal layer has a maximum circumference. The cross sectional area of the lumen
is also maximal. When the organ contracts, the area of the lumen decreases to zero. The circumference, however, cannot
approach zero as the elasticity of the mucosa is not infinite. We can get a numerical handle on these circumstances by
noting that the area of the mucosal layer remains constant despite distention or collapse of the lumen.
If one assumes that as much mucosa is squeezed into a given cross section as is squeezed out, then the cross sectional area of the mucosal layer
will be the same before and after contraction of the gut. Before contraction (Figure 1) the mucosal area is the difference between the areas of
the inner and outer
circles:

After contraction (Figure 2) the mucosal area is . . . 
As the mucosal area is unchanged by contraction, these expressions
can be set equal to each other and solved for r_{2} resulting in
the expression . . . 


The circumference of the free margin of the mucosa after contraction,
C_{2}, is a fraction of its precontraction length. The decimal
fraction C_{2}/C_{1} is equal to E, the elasticity of the
mucosal tissue. It is the elasticity which determines how much the mucosa
can shrink. The fully contracted circumference, in leiu of shrinking to
zero, must form folds. 
From Figure 3, the mucosal length taken up by one fold is twice the distance
from the center of the organ to the depth of the interfold valley or . . .
An expression for N_{f}, the expected number of postcontraction
folds, can now be written. 


The dimensions are all postcontraction, however, from the above, we
know that C_{2} = C_{1}E and , from Figure 1, C_{1}
= 2(r_{1}t_{1}). Moreover, r_{2} was calculated
in equation 3. We can also calculate that t_{2} C_{2}
= t_{1}C_{1} or

Performing the indicatedsubstitutions yields a formula for fold numbers
in terms of the inital values of thickness, organ radius and mucosal elasticity.
The subscripts, therefore, can be dropped giving . . .


Last Updated July 31 2007 by David PJ Stiennon
