In physiological conditions, we treat the tissue as rigid, and we compute a spatial pressure distribution
p0(
r,
z). We define a pressure averaged over the thickness of the retina as
\begin{equation}
\overline{p}_0(r)=\frac{1}{h_0}\int _0^{h_0} p_0(r,z) d z.
\end{equation}
We denote with
p(
r,
z) the value of the pressure in the presence of EME and with
\(\overline{p}(r)\) the corresponding depth-averaged value, defined as
\begin{equation}
\overline{p}(r)=\frac{1}{h(r)}\int _0^{h(r)} p(r,z) d z.
\end{equation}
We assume that the retinal thickness changes in response to the excess pressure
\(\overline{p}(r)-\overline{p}_0(r)\). In the numerical simulations, we use the following nonlinear law:
\begin{equation}
\overline{p}(r)-\overline{p}_0=\frac{2\kappa (r)\epsilon _{lim}}{\pi } \tan \left( \frac{\pi \epsilon }{2\epsilon _{lim}}\right),
\end{equation}
where ϵ = (
h −
h0)/
h0 is tissue strain, and ϵ
lim is the maximum admissible tissue strain, which we take equal to 1 since values above this (
h >
h0) are rarely observed clinically.
27 The parameter κ(
r) is the linear stiffness of the tissue (measured in Pa), which is taken to be proportional to the spatial density of MCs. We thus write
\begin{eqnarray}
\kappa (r) = n(r) \cdot \kappa _c
\end{eqnarray}
where
n represents the varying density of MCs along the
r-direction within the domain (measured in 1/m
2), while κ
c denotes the stiffness of an individual cell.
We model MCs as cylinders, and we assume that when a single cell is acted on by a force of magnitude
F in the longitudinal direction, it deforms according to
F = κ
c(
l −
l0)/
l0, where
l is the current length of the cell and
l0 is the reference one. The stiffness of a single cell is defined as
\begin{eqnarray}
\kappa _c =AE
\end{eqnarray}
where
E represents the Young’s modulus of the cell and
A is the cross-sectional area of a single cell. The values of
n,
A, and
E adopted are reported in the
Table, with the corresponding references. With these numbers, assuming the cell density of the parafoveal region, we obtain the value of κ = 132 Pa. This is of the same order of magnitude as Young’s modulus of the retina as measured by Chen et al.,
42 who found the values of 0.3 and 0.2 kPa for vertical and horizontal retinal strips, respectively. The value of ϵ
lim is chosen equal to 1 since values above this (
h > 2
h0) are rare, where
E represents the Young’s modulus of the cell and
A is the cross-sectional area of a single cell. The value of ϵ
lim is chosen equal to 1 since values above this (
h > 2
h0) are rarely observed clinically.
27