## 4 FF-OCT and Biomechanics
### Static Elastography Based on Digital Volume Correlation (DVC)  
The principle of the static approach is to register a volumetric image before and after mechanical solicitation of the sample [20]. From these two sets of images, we can estimate the 3D local displacement field, \(\vec{u}\), inside the sample using a finite element 3D digital image correlation algorithm developed by [PERSON] and his team [91; 92]. From the optically measured 3D displacement field, we can calculate the 3D strain tensor \(\overline{\overline{\overline{\overline{\epsilon}}}}\):  
\[\overline{\overline{\overline{\epsilon}}}=\frac{1}{2}(\vec{grad}(\vec{u})+ \prescript{t}{g\overline{\overline{\overline{\epsilon}}}}{d}(\vec{u})) \tag{7}\]  
However, biological samples can be very complex and mechanically heterogeneous. This complexity usually prevents direct access to the local stress tensor and so, in most cases, methods based on digital image correlation are limited to strain contrast and do not provide quantitative stiffness information. This is the main limitation of this type of approach but already the strain information recreates a sort of \"virtual local palpation\" that can highlight mechanical heterogeneity at the micrometer scale given by FF-OCT or OCT. From the local strain tensor, it is possible to extract different parameters that give access to mechanical information such as strain magnitude or strain anisotropy. * **Strain magnitude** In order to quantify the strain magnitude, we chose to use an equivalent Von Mises strain defined as: \[\forall\vec{x},\,\epsilon_{eq}(\vec{x})=\sqrt{\frac{2}{3}}tr(\overline{ \overline{\overline{\overline{\epsilon}}}}(\vec{x})^{2})\] (8) This parameter is independent of the base in which the strain tensor is expressed and allows quantification of the strain amplitude in the case of incompressible media such as biological samples. To illustrate this, we present strain amplitude maps calculated for a human breast tissue in Figure 3A,B, which highlight mechanical differences. * **Strain anisotropy** The full 3D strain tensor also allows calculation of eigenvalues and eigenvectors of the strain tensor that gives access to information on the mechanical anisotropy of the sample. One should keep in mind that, since the compression of the sample is generally performed along the optical axis, the local stress field is inherently anisotropic. In order to display and to quantify strain anisotropy, we mainly use two parameters. The first parameter is the principal strain direction. For each pixel, the eigenvector corresponding to the largest eigenvalue of the strain tensor is calculated. This vector gives the principal strain direction. For example, Figure 3C is a cross-sectional slice from the volumetric FF-OCT image of a rat heart. In the FF-OCT image, it is difficult to evaluate the fiber direction, and the two different fiber orientations are not visible. However, if the projection angle of the principal strain direction vector on the plane perpendicular to the compression (see Figure 3D) is plotted, it is clear that there are two different fiber orientations. Another parameter used to quantify the strain anisotropy is the fractional anisotropy _FA_. In a similar manner to classical diffusion tensor imaging (DTI) in MRI [93,94] or ultrasound [95], we define the fractional anisotropy as: \[FA=\sqrt{\frac{3}{2}}\frac{\sqrt{(\varepsilon_{1}-\overline{\varepsilon})^{2}+( \varepsilon_{2}-\overline{\varepsilon})^{2}+(\varepsilon_{3}-\overline{ \varepsilon})^{2}}}{\sqrt{\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+ \varepsilon_{3}^{2}}}\] (9) where \(\varepsilon_{1}\), \(\varepsilon_{2}\) and \(\varepsilon_{3}\) are the strain tensor eigenvalues and \(\overline{\varepsilon}\) is the mean of the absolute value of the strain tensor eigenvalues. Using this parameter, it is possible to differentiate isotropic from anisotropic strain.