Biomedical Engineering Reference
In-Depth Information
Stokes fluids
. There is substantial precedent for considering the morphogenetic
behavior of embryonic tissues in terms of a fluid ((38,49,50) deal with a small
subset of these, while (12,15,25) consider single cells behaving as fluids). There
is excellent experimental calibration of the fluid model of embryonic mechanics
from Malcolm Steinberg and his colleagues (13,14).
Our desire is to keep all aspects of the model "as simple as possible, but not
moreso" (A. Einstein). For example, the epithelium may or may not initially be
hollow. In the salivary gland, a lumen is created as the epithelium matures, but
is not present when the branches are created. For simplicity, we ignore lumens.
We model a branching rudiment as a uniform epithelium inside either a uniform
mesenchyme or a uniform acellular fluid (of the consistency of water or a colla-
gen gel). It has been shown that growth-suppressed salivary gland rudiments can
still branch once, though they do not grow enough to generate subsequent
branches (42). Again, in the interest of simplicity, we focus on the single step of
the creation of one cleft in a branching rudiment, so we choose to model the
tissue as not growing.
Finally, also in the interest of simplicity, we do not try to explicitly model
the force that causes the deformation. Instead, we simply apply a localized sur-
face force at several points on the epithelial surface, and/or modify the local
surface tension. Since our goal is just to understand the relationship between
force and deformation in branching morphogenesis, this artificial force will
serve our purposes fine.
The geometry of our model is simple. We have an epithelium-shaped region
of fluid surrounded by a second fluid representing either mesenchyme or culture
medium. The Stokes equations apply in each fluid:
2
p
=
N
u
,
[1]
¸
u0
,
=
[2]
where
p
is the pressure,
u
is the velocity vector, and N is the viscosity, which we
shall assume to be constant within each fluid.
On the interface between the fluids, there is a surface tension H, which can
in principle vary in space, and which will be the only agent driving shape
changes in this model. We can write the boundary condition between the fluids
in terms of the jumps [
"
] of two quantities across the interface:
[
p
2
N
(
¸ ¸
un n
)
]
=
HL
,
[3]
s
H
[(
N
¸ ¸
un
)]
U
=
s
,
[4]
U
