# Digital Histology by Phase Imaging Specific Biomarkers for Human Tumoral Tissues Discrimination
## 2 Materials and Methods
### Phase Contrast Parameters  
The distribution of refractive index in a general turbulent medium requires a statistical description of the propagating beam. In this case, no temporal variation is expected, as we are dealing with turbid biological samples. Consequently, the analysis of phase contrast parameters must be based on the theory of random processes. The most relevant statistical characteristics are the first two moments of the refractive index spatial distribution: the mean value  
\[n_{0}\Big{(}\overrightarrow{r}\Big{)}=\Big{\langle}n\left(\overrightarrow{ r}\right)\Big{\rangle}_{M} \tag{5}\]  
and its covariance function  
\[B_{n}\left(\overrightarrow{r_{1}},\overrightarrow{r_{2}}\right)=\Big{\langle} \Big{[}n\left(\overrightarrow{r_{1}}\right)-n_{0}\Big{(}\overrightarrow{r_ {1}}\Big{)}\Big{]}\left[n\left(\overrightarrow{r_{2}}\right)-n_{0}\Big{(} \overrightarrow{r_{2}}\Big{)}\right]\Big{\rangle}_{M}. \tag{6}\]  
In a field propagating through a weakly scattering medium, the associated dispersion relation can be expressed as\[\langle k^{2}\rangle=n_{0}^{2}\beta_{0}^{2}\Bigg{(}1+\frac{\sigma_{n}^{2}}{n_{0}^{ 2}}\Bigg{)} \tag{7}\]  
where \(\langle k^{2}\rangle\) is the second-order moment of the wavevector, \(\langle k^{2}\rangle=\langle k_{x}^{2}\rangle+\langle k_{y}^{2}\rangle+ \langle k_{z}^{2}\rangle\), with angular brackets denoting ensemble averaging, \(n_{0}\) is the average refractive index, \(\beta_{0}\) is the wavenumber in vacuum, and \(\beta_{0}=\omega/c\), and \(\sigma_{n}^{2}\) is the spatial variance of the refractive index. The refractive index variance of a transparent sample can be calculated from the measured phase contrast image as [18]  
\[\sigma_{n}^{2}=\frac{1}{\beta_{0}^{2}}\langle|\
abla\phi|^{2}\rangle. \tag{8}\]  
Equation (8) shows that the refractive index variance can be extracted from the mean gradient intensity of the phase contrast images. The scattering coefficient (\(\mu_{s}\)) can be also estimated as a phase contrast parameter. It can be calculated using the scattering-phase theorem [17]  
\[\mu_{s}=\frac{\langle\Delta\varnothing^{2}(r)\rangle_{r}}{L} \tag{9}\]  
where \(\langle\Delta\varnothing^{2}(r)\rangle\) denotes the spatial variance of the phase and \(L\) is the thickness of the tissue sample. The anisotropy factor can be calculated, as biological tissues are intrinsically highly scattering media. The estimation employs the scattering-phase theorem and can be implemented by  
\[g=1-\frac{1}{2k_{0}^{2}}\frac{\langle|\
abla[\varnothing(r)]|^{2}\rangle_{r}}{ \langle\Delta\varnothing^{2}(r)\rangle_{r}^{2}} \tag{10}\]  
where \(k_{0}\) is the incident wave vector and \(\
abla[\varnothing(r)]\) is the phase gradient. Optical turbulence is usually explained by the presence of irregularities in the refractive index: the so-called \"turbulent eddies\" [16]. They are due to fluctuations in various physical properties, such as temperature, pressure, and inhomogeneous concentration of species. Different parts of the sample can be mixed by external or internal factors, such as cell growth or fluid transfer. This fact provokes energy transfer among eddies of different sizes. The largest possible size of an eddy is defined as the outer scale \(L_{0}\) of turbulence. Larger eddies break down into smaller ones until their size reaches the lower limit and energy dissipates. In order to study these characteristics in histological samples, the spectrum of index inhomogeneities is fitted to the equation [22]  
\[\Phi(\kappa)=\frac{4\pi\sigma_{n}^{2}L_{0}^{2}(m-1)}{\left(1+\kappa^{2}L_{0}^ {2}\right)^{m}} \tag{11}\]  
which is a Von Karman spectrum, with the exception of the exponent m, that can assume values other than \(4/3\). \(m\) is approximately equal to one-half of the measured slope in the range of power-law scaling. Its magnitude is related to the fractal dimension of a two-dimensional surface, \(d_{f}=4-m\). The spatial distribution of the main components of biological tissues has been shown to present self-similarity [23].