## 4 FF-OCT and Biomechanics
### Description of the Different Mechanical Parameters and Approximations  
Before describing how it is possible to optically measure tissue biomechanics, we briefly summarize here the numerous different mechanical parameters that can be found in the extensive literature on tissue biomechanics. We will emphasize the relationship between these parameters, and their relevance for diagnosis.  
Most studies in MRI, as well as in ultrasound or optics, focus on the elastic properties of tissues, meaning the reversible linear deformation \(\overline{\varepsilon}\) caused by a small mechanical stress \(\overline{\overline{\sigma}}\). In the elastic regime, and neglecting viscosity, the generalized [PERSON]'s law gives:  
\[\overline{\overline{\sigma}}=\overline{\overline{\overline{\overline{C}}}}. \overline{\varepsilon} \tag{1}\]  
where \(\overline{\overline{\overline{C}}}\) is a fourth order tensor, named stiffness tensor, that depends on the mechanical parameters of the sample. All the other parameters that we will describe are elements of this fourth order tensor, and are usually obtained when the dimensions of the generalized problem are reduced. For example, Young's modulus \(\mathrm{E}\) corresponds to the linear coefficient that relates a purely axial stress, and its associated deformation along the stress axis. In this case, the problem is reduced to a single dimension:  
\[\mathrm{E}=\frac{\varepsilon}{\sigma} \tag{2}\]  
We can add that Young's modulus is qualitatively the main measurement that is performed with manual palpation. Basically, the first information that one can get with manual palpation is how the tissue resists to an axial deformation or stress, which is given by the Young's modulus.  
From Equation (1), it is easy to realize that a given sample is not only deformed in the direction of the stress, but can also be deformed in the transverse direction, with a preferential axis for anisotropic material. We will describe in Section 4.2 an example metric to quantify the local anisotropy of a sample.  
In addition to these parameters, many mechanical measurements rely on shear wave speed measurement that we will describe more carefully in Section 4.3. The mechanical parameters on which the propagation of mechanical waves depend are simply defined as a result of the wave equation in a medium of a given stiffness. Inside biological media, two waves can propagate [89]: A longitudinal wave that corresponds to compression and traction of the medium with a mechanical displacement parallel to the direction of propagation. Additionally, shear waves can propagate, which correspond to perpendicular mechanical displacements with respect to the direction of propagation. The two waves propagate at speeds \(\mathrm{c}_{L}\) and \(\mathrm{c}_{shear}\) respectively that are controlled by the mechanical parameters of the medium. We can therefore define the bulk modulus \(\mathrm{K}\), and the shear modulus \(\mathrm{\mu}\) with:  
\[\mathrm{c}_{L}=\sqrt{\frac{\mathrm{K}}{\rho}} \tag{3}\]  
\[\mathrm{c}_{shear}=\sqrt{\frac{\mathrm{\mu}}{\rho}} \tag{4}\]  
where \(\rho\) is the medium local density. These parameters are related to [PERSON]'s modulus [89]:  
\[\mathrm{E}=\mathrm{\mu}\frac{3K}{K+\frac{1}{3}.\mathrm{\mu}} \tag{5}\]  
Finally, for soft tissues, which represent most biological tissues, except bones, the bulk modulus is much larger than the shear modulus. Bulk moduli of biological samples range from 1 GPa to 10 GPa, while shear moduli range from 0.1 kPa in the brain to 0.1 GPa in epidermis and cartilage, and up to 10 GPa in bones [90]. When considering Equation (5) for soft tissues, the [PERSON]'s modulus becomes dependent on the shear modulus only:  
\[\mathrm{E}\simeq 3.\mathrm{\mu}\simeq 3\rho\mathrm{c}_{shear}^{2} \tag{6}\]  
With such approximations, we have shown that it was possible to locally measure the [PERSON]'s modulus by measuring the propagation speed of shear waves.