The longitudinal modulus
M′ of a material can be expressed in terms of its standard Young's modulus
E′ via the Poisson's ratio σ, that is,
M′ =
E′(1 −
σ)/(1 +
σ)(1 − 2
σ). In Brillouin measurements, the modulus
M′ is probed in a hypersonic frequency range of 5 to 10 GHz. It has been shown empirically that the Brillouin-measured longitudinal modulus
M′ is related to the conventional Young's (or shear) moduli
E′ through a log–log linear relationship: log(
M′) =
a log(
E′) +
b, where
a and
b are material-dependent coefficients.
32 From the log-log relationship, the change of elastic modulus induced by a CXL protocol can be written as:
ΔM′/
M′ =
aΔE′/
E′. From this relationship, it follows that the relative change of Brillouin modulus of two different procedures,
ΔM2/
ΔM1, is equal to the relative change of Young's modulus,
ΔE2/
ΔE1: that is,
The CSI is defined as a quantitative measure of the mechanical outcome of a specific CXL procedure (denoted by a subscript X) in direct comparison to the traditional Dresden protocol:
The CSI, therefore, compares the increase in corneal stiffness with respect to the standard CXL method. By definition, the CSI of the Dresden protocol is 100. Any procedures that resulted in a smaller modulus change would have CSI less than 100. For example, CSI is 50 for a procedure that increases elastic modulus half as much as the Dresden method. Interestingly, CSI allows universal comparison between mechanical tests that use very different methods to assess mechanical properties, for example Brillouin microscopy, stress-strain tests, and shear rheometry.