Biomedical Engineering Reference
In-Depth Information
Figure 6
. Magnification of a deepening cleft over time. Surface tension C = 0.01, nondimen-
sional clefting force G = 1.25. Same initial conditions, same epithelial viscosity, same clefting
force, same final depth of cleft. (
a
) Relatively firm mesenchyme. Viscosity ratio B = 100,
nondimensional time
t/T
= 0 to 200. (
b
) Relatively soft mesenchyme or ECM. Viscosity ratio B
= 1, nondimensional time
t/T
= 0 to 15. The epithelium embedded in a lower-viscosity material
has taken less time to form its cleft, and the cleft is narrower than the epithelium embedded in a
more viscous material. Reprinted with permission from Lubkin and Li (2002) (37).
ferent rates, and with slightly different morphology. No one has done that ex-
periment.
Our simple model is able to make many of these predictions, and they are
qualitatively robust. However, because the model is so simple, quantitative pre-
dictions are probably at best approximations. For instance, we assumed that each
tissue was viscous, not viscoelastic, and had a spatially and temporally uniform
viscosity, contrary to what is already known. In particular, we know that preex-
isting clefts have abundant collagen in them naturally making the clefts more
resistant to deformation, and mesenchyme condenses around the branches,
which should also make the lobe region stiffer and/or more viscous than the sur-
rounding mesenchyme. We ignored the mechanical role of the basal lamina, for
want of adequate information about its thickness, mechanics, and spatial and
temporal features. The basal lamina is likely to be most important mechanically
as a barrier to expansion (growth), rather than as a regulator of clefting. Since
we focused in this study on clefting only, separate from growth, we expect that
the omission of a term for the basal lamina is reasonable.
The most serious limitation of our model is restricting ourselves to two di-
mensions due to computational constraints. A two-dimensional model requires
us to artificially model the clefting force as a point force, and it probably affects
the quantitative observations by at least a factor of two. Other recent mechanical
models of morphogenesis have either been confined to two dimensions (9,58)
or, if in three dimensions, have been on computational domains much smaller
than ours (8), or axisymmetric (25,36), or using a mechanical model that was
easier to solve accurately (10). Methods to accurately solve multi-fluid flow
problems in 3D are still being developed.
