For homogeneous, isotropic, elastic, and infinite-type media, the wave-based OCE method can be used to determine the relationship between Young's modulus
E and the shear wave velocity
cs based on the classic physical model:
\begin{eqnarray}{c_s} = \;\sqrt {\frac{E}{{2\rho \left( {1 + \nu } \right)}}} ,\end{eqnarray}
where ρ and ν are the density and Poisson's ratio, respectively, and ρ is 1062 kg/m
3 for the cornea. However, the layered thin-plate structure of the cornea, the different media on the anterior and posterior surfaces, and the viscoelastic properties all change the boundary conditions for shear wave propagation. Therefore, other mechanical wave models, such as Lamb waves, which depend on the sample structure, have been proposed to accurately characterize the corneal biomechanics. These models differ from the shear wave velocity model because the velocity varies depending on the frequency, that is, velocity dispersion. In the wave-based OCE model, the zero-order antisymmetric Lamb wave mode is consistent with the most common mode of vibration propagation in corneal tissue, and the numerical solution can be described as
44,45:
\begin{eqnarray}
&&4k_L^3{\beta _L}\cosh \left( {{k_L}h} \right)\sinh \left( {{\beta _L}h} \right) - {(k_s^2 - 2k_L^2)^2} \nonumber \\
&&\times {\rm{\;sinh}}\left( {{k_L}h} \right)\cosh \left( {{\beta _L}h} \right) \nonumber \\
&&- k_s^4\cosh \left( {{k_L}h} \right)\cosh \left( {{\beta _L}h} \right) = 0\end{eqnarray}
where
\({\beta _L} = \sqrt {k_L^2 - k_s^2} \),
\({k_L} = \frac{{2\pi f}}{{{c_L}}}\) is the wavenumber of the Lamb wave,
f is the frequency of the Lamb wave, and
cL is the phase velocity of the Lamb wave. In addition,
\({k_s} = 2\pi f\sqrt {\rho /U} \) is the wavenumber of the shear wave;
h is the half-thickness of the cornea; and
U is the viscoelasticity of the cornea.
cL can be calculated using the following equation:
\begin{eqnarray}{c_L} = \;\sqrt {\frac{{2\pi \times f \times 2h \times {c_s}}}{{\sqrt 6 }}} .\end{eqnarray}