Biomedical Engineering Reference
In-Depth Information
where
m
is the mass of the inner wall,
V
is the velocity of the inner wall on
y
direction,
F
flow
is
the flow of the aqueous humor applied to the wall, and
F
spring
is the spring force applied on the
wall through the trabecular mesh.
The velocity of the wall can be determined by the displacement of the wall as
y
2
y
0
Δt
v5
and:
dv
dt
The new displacement of the inner wall determines the spring force in the form of
v
2
v
0
Δt
5
F
spring
5 k
spring
ðy2y
0
Þ
Therefore, the discrete form of the equation for the inner wall motion can be rearranged, and we
can get the expression for the inner wall displacement as
m
Δt
ðv
0
Þ 1
m
Δt
2
F
flow
1
ðy
0
Þ
y5
m
Δt
2
1 k
spring
F
flow
5pðxÞL
IOP
ðtÞ 2pðxÞ
E
y
2
y
0
y
0
52
where
p
(
x
) is the local pressure in Schlemm's canal and
E
is the spring stiffness of the canal.
For each time step (
Δt
), the location of the inner wall (mesh changes) can be determined
based on the equation derived above, which depends on the IOP(
t
), the local pressure
p
(
x
), and
the size of the time step.
Furthermore, the flow rate of aqueous humor can be determined by
p
(
x
):
dp
dx
5
12
μQðxÞ
ωhðxÞ
3
where
is the aqueous humor viscosity,
w
is the depth of the canal, and
h
(
x
) is the local height
of the canal, which depends on y and initial conditions.
μ
13.3 BUCKINGHAM PI THEOREM AND DYNAMIC SIMILARITY
Many real flows cannot be solved by analytic methods alone, but even for those analytic
models, the use of full-sized prototypes/simulation geometries may not be practical. In
order to relate the actual flow conditions with the simulation, they must be linked by
some scaling factors. When the simulations and the real conditions match, they are said to
be similar based on dimensional scaling. The Buckingham Pi Theorem is a mathematical
approach that allows the formation of a relationship between a quantity of interest
between the model and the real scenario. As an example, let us consider the drag force on
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