Biomedical Engineering Reference
In-Depth Information
3.3. Parameter Estimates
All of the parameters in the model can be estimated from published data in
the literature. For example, salivary lobules are of the order of 100
m in diame-
ter, and branching occurs approximately every 8 hours (7). These provide esti-
mates of the length
L
and time
T
scales for nondimensionalization. Contractility
is readily estimated or measured in cell-populated collagen gels, and can give us
an estimate for the clefting force
f
. The quality of these estimates is variable,
since most applicable experiments have not been performed with this model in
mind. In one prominent counterexample, contractility of various dissociated
mesenchymal cells, including those of lung and submandibular gland, was
measured on artificial collagen gels (47), specifically to test the mesenchymal
theory of branching morphogenesis. One important factor in estimating the cleft-
ing force is that the younger the tissue, the stronger the contractility (52). Since
we are concerned in this chapter with embryonic tissues, we rely on the higher
published estimates of contractility.
The size of a branching rudiment and time scale of branching morphogene-
sis are well established, as is the viscosity of water (Table 1). We assume that if
the outer "fluid" is mesenchyme, then its viscosity is within an order of magni-
tude of the viscosity of the epithelium inside. If the outer fluid is a collagen gel
rather than a tissue, its viscosity may have a very wide range, depending on the
exact composition, including pH and most importantly water content. Unfortu-
nately, no one has reported direct measurements of the viscosity of the embed-
ding clots in the mechanically mesenchyme-free culture systems. In some cases,
even the composition of the embedding medium is not specified well enough for
us to estimate its viscosity. Hence depending on the particular mesenchyme-free
experiment we are modeling, the appropriate external viscosity, N
+
, could range
from that of water to that of mesenchyme (or even more viscous). However, in
general, we assume that the viscosity of the collagen gel in the mesenchyme-free
experiments is much smaller than the viscosity of mesenchyme.
Foty and colleagues (14) and Forgacs et al. (13) measured the viscosity and
surface tension of several types of embryonic tissues, and found each tissue to
be mechanically very similar, within an order of magnitude of the others. As far
as we know, there are no measurements reported of the tissue viscosity or sur-
face tension of the specific components of any embryonic branching tissue, but
we will assume that their values lie within the narrow range of the five tissue
types measured by Foty et al.
The basal lamina of a clefting epithelium affects the epithelio-mesenchymal
interactions (2) and more rapidly turns over on the lobules than in the clefts (6).
This implies that there may be a reduction in the surface tension of the epithe-
lium in the lobules, caused by faster dissipation of stresses in the lobular lamina
than in already-formed clefts. However the magnitude of any surface tension
from the basal lamina has not been measured, so we choose to ignore the basal
lamina mechanically.
