Biomedical Engineering Reference
In-Depth Information
Table 1
. Expected Ranges of Parameter Values
Parameter Symbol Units Range Reference(s)
Time scale
T
s
10
4
-10
5
Bernfield et al. 1984 (7)
Rudiment size
L
m
10
-4
Bernfield et al. 1984 (7)
Epithelial viscosity
N
-
poise
10
4
-10
6
Foty et al. 1994 (14),
Forgacs et al. 1998 (13),
Phillips et al. 1977, 1978 (49,50)
Mesenchymal
N
+
poise
10
4
-10
6
Forgacs et al. 1998 (13),
viscosity
Phillips et al. 1977, 1978 (49,50)
Viscosity of em-
N
+
poise
10
0
-10
6
Nogawa & Takahashi 1991 (48),
bedding gel
Barocas et al. 1995 (4)
Viscosity of water
N
+
poise
10
-3
—
Epithelial surface
H
0
N/m
10
-3
-10
-2
Foty et al. 1994 (14),
tension
Forgacs et al. 1998 (13)
Clefting force
f
N
10
-7
-10
-6
Rappaport 1977 (51),
Kolodney & Wysolmerski 1992 (28)
The clefting force is difficult to estimate. We can take estimates for single
cells in very different contexts (e.g., Rappaport's (51) cleaving oocytes or fibro-
blasts on collagen gel (28), and estimate that an epithelial region about to cleft
could generate 10-100 times the force of a single oocyte or fibroblast. This
gives an estimate for
f
of 10
-7
to 10
-5
N.
Hence the possible range of B is from 10
-9
to 10
1
, of C from 10
-1
to 10
3
, and
of G from 10
-1
to 10
2
.
3.4. Methods
The fluid equations were solved by a finite-difference method (35) specifi-
cally designed to simulate two fluids of different viscosities separated by an
interface. Because our major interest is in examining the effects of the viscosity
of the outside fluid (mesenchyme or growth medium), the equations were solved
in two space dimensions.
We performed the following numerical experiments:
1. To test the effect of the viscosity ratio B on the time course of deforma-
tions, we deformed identical 3-lobed "rudiments" at the centers of the far ends
of their lobes, with the same normal force, until each lobe was nearly cleft in
two. One rudiment was embedded in a material (gel or mesenchyme) of viscos-
