Question 13.1: Figure 13.9 demonstrates the drainage route of aqueous human...
Figure 13.9 demonstrates the drainage route of aqueous human based on John and Kamm’s model. Trabecular mesh was modeled as a porous elastic body, shown in the figure as springs. Schlemm’s canal is modeled in two dimensions, with the outer wall (top) and the inner wall (bottom) being parallel to each other. The inner wall is permeable (so aqueous humor leaks through), and it deforms as the trabecular mesh work stretches. The local height of the canal is a function of position x and intraocular pressure (IOP). Under certain disease conditions (such as glaucoma), the size of Schlemm’s canal may decrease due to the increase in IOP; therefore, aqueous humor drainage resistance increases. We can use FSI to model the interface of the moving inner wall and aqueous humor.
First of all, we need to develop the mesh motion for our model. According to Newton’s second law, the acceleration of the inner wall is proportional to the net force acting on it. In differential form, the equation to describe the motion of the inner wall is
m\frac{d \upsilon}{dt}= F_{flow} – F_{spring}
where m is the mass of the inner wall, V is the velocity of the inner wall on y direction, Fflow is the flow of the aqueous humor applied to the wall, and Fspring 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
\upsilon= \frac{y – y_{0}}{\Delta t}
and:
\frac{\upsilon – \upsilon_{0}}{\Delta t}= \frac{d\upsilon}{dt}
The new displacement of the inner wall determines the spring force in the form of
F_{spring}= k_{spring}(y – y_{0})
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
y= \frac{F_{flow} + \frac{m}{\Delta t}(\upsilon_{0}) + \frac{m}{\Delta t^{2}}(y_{0})}{\frac{m}{\Delta t^{2}} + k_{spring}}
Fflow = p(x)L
\frac{IOP(t) – p(x)}{E}= – \frac{y – y_{0}}{y_{0}}
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):
\frac{dp}{dx}= 12\frac{\mu Q(x)}{\omega h(x)^{3}}
where μ 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.