Biomedical Engineering Reference
In-Depth Information
In the experimental component of the work, cells were placed at the bottom of a
laminar flow chamber where a fluid flow with a shear stress of 0.25 Pa (corresponding
to a flow rate of 500
l/min) was applied with a syringe pump [
62
]. Along the length
of the ciliium, it is safe to assume that the laminar flow velocity increases linearly
with distance from the bottom surface. The force is thus approximated as the 2D
planar drag force acting on a cylinder in low Reynolds number Stokes flow
µ
2
ˀ
h
˄
Q
(
h
)
=
)
,
(5.4)
ln
(
L
/
2
r
where
is the wall shear stress,
h
is the distance between the cilium and the wall,
r
is
the diameter, and
L
is the contour length of the primary cilium. The distance
h
˄
(
s
)
can
−
0
d
s
, where
H
is the height from the
tip of the cilium to the cell surface. Equation
5.4
is the drag force acting on a beam
that is slightly deformed. When the elastic beam is highly deformed, the slender-
body theory (see Sect.
5.3.2
) can better capture the drag force [
63
]. In addition to
the above closed-form approach they also conducted a computational study using
the commercial finite-element code COMSOL to simulate the interaction between
a viscous fluid and a cylinder (a model for a section of a cilium) in a low Reynolds
number planar flow [
62
]. The angle between the cylinder and the vertical was veried
from 0
ⓦ
to 80
ⓦ
and total drag was determined at each angle and fit with a polynomial
regression. Convergence was verified by creating a refined mesh of 85,000 elements
and observing less than a 15 % change in predicted drag.
The flexural rigidity
E
I of the axoneme is a parameter of this model. [
62
] developed
a numerical algorithm to find the
E
I that best fit experiment data. First, the nonstressed
configuration and
s
cos
s
))
be computed as
h
(
s
, ʸ)
=
H
(ʸ(
ʸ
base
were calculated from the experimental observation. They
then calculated the
E
I that would give the best-fit solution to the equilibrium profile
with the 2D analytic drag approximation, and the drag from the 3D finite element
simulation was used to refine the estimate for
E
I. The nonlocal differential Eq.
5.3
was solved numerically. In this model the basal body support is not included. Instead
the basal angle at equilibrium from the experiments is used ito find the best-fit
equilibrium cilium profile [
62
]. This assumption will be relaxed in the slender-body
model (Sect.
5.3.2
).
5.3.2 Slender-Body Theory
()
The ratio
of axonemal radius
r
to length
L
of primary cilia is often in the range
10
−
2
10
−
1
, consequently, the relevant physics underlying the primary cilium
dynamics under flow is the bending of a supported elastic slender filament under a
hydrodynamic load. The elastic slender-body formulation [
48
,
50
,
71
,
72
] is adopted
to model the bending of the ciliary axoneme under flow [
63
]. First, the force distri-
bution along the slender-body centerline is decomposed (as
F
≤
≤
F
t
)
t
F
n
=
(
s
+
(
s
)
dž
n
)
t
and the normal
in the tangential
n
directions with
s
dž
∈[
s
0
,
s
e
]
the arclength.
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