## 3 Results
### Proof of Concept Application of the Framework for the Quantification of Cardiac Fibrosis  
The capability of the analysis framework to detect and distinguish different fibrosis patterns was showcased on a dataset of human cardiac tissue samples. The mean value and the standard deviation of the mean for the collagen-detection thresholds, manually set on a subgroup of cardiac sample images, were \(t_{col}\) = 0.152 = \(\pm\) 0.004. The standard deviation of the thresholds was 0.017, indicating a low dispersion of the selected threshold values and a general coherence of choice. The threshold \(t_{col}\) was used for automatic processing of all the images in the dataset. Over the dataset, the section optical thickness varied within the range 0.13-0.46. The validity of the approximation \(\sin^{2}\frac{\delta}{2}\ll 1\), used in Equation (14) for correcting against thickness variations, was checked for the threshold \(t_{col}\) = 0.152. For the unnormalized birefringence images in Equation (13), the threshold is given by \(t_{col}\left(\mu_{(r)}L^{i}\right)^{2}\), which yielded a maximum value of 0.032 \(\ll\) 1 for the largest optical thickness of 0.46. The amplitude of the unnormalized signal was in general inferior to 0.1. Figure 6 shows representative examples of the method application in two patients with different arrhythmic profiles. In the exemplary SR patient (left panels), the red stain in the bright-field image occupied limited areas (Figure 6a). This suggested the presence of a small amount of fibrosis, which was confirmed by the low intensity of the processed birefringence signal (Figure 6c). The binary classification performed by the algorithm on the birefringence signal yielded a value of 3.4% (Figure 6e). In the exemplary AF patient (right panels), both the red stain and the processed birefringence signal were much more intense and occupied a much larger area (Figure 6b,d), resulting in an estimated collagen fraction of 18.9% (Figure 6f). The overall results of the application of the analysis framework to the cardiac sample database are shown in Figure 7. The method detected significant differences between the amount and heterogeneity of mid-wall fibrosis in patients characterized by different arrhythmic profiles. SR patients displayed low (fibrosis fraction of 7.8 \(\pm\) 3.8 %) and almost stable (fibrosis heterogeneity of 2.1 [1;9; 2.5] %) fibrotic content in the mid-wall. Conversely, patients with AF displayed a significantly higher amount of fibrosis (14.5 \(\pm\) 6.0 %, \(p\) < 0.05)  
Figure 5: **Correction for sample thickness variability. ** The robustness of the algorithm against sample thickness variability is examined comparing a slice of average optical thickness = 0.36 (left column) with a nearby slice of average optical thickness = 0.21 (right column): **(a,b)** The sample transmitances \(T_{i}^{s}\) evidences the different tissue thickness between the two samples; **(c,d)** Original dark-field images \(A_{c}^{i}\) (windowed 0–32,768); **(e,f)** RGB birefringence signal images \(S_{thr}^{i}\) (windowed 0–0.5); **(g,h)** Greyscale collagen signal images \(S_{col}^{i}\) (windowed 0–0.5). The collagen mask for a threshold \(t_{col}\) = 0.152 is contoured in yellow. The estimated collagen fractions were \(c^{i}\) = 8.1% and 11.1%, respectively. and higher variability among slices (3.6 [2.4; 3.7] %, \(p<0.05\)), which may be indicative of a structural remodeling of the atrial tissue. The results of the sensitivity analysis performed on the effects of threshold variations on fibrosis amount results are shown in Figure 8. As it can be appreciated, the progressive increase of the threshold from 0.12 to 0.18 made collagen detection more and more restrictive and a progressively smaller collagen amount was detected. However, the decrease was similar and consistent in the two patient groups, so that the relative difference in collagen content was maintained and remained statistically significant (\(p<0.05\)) for all the considered thresholds.