## 4 Results and Discussion  
Figure 1a shows cine-tagging images using a 16 mm sagittal grid in a 56-year-old male patient with fibrosis scores equal to F0. The first image (Frame 1) obtained at the maximum inspiratory phase shows the grid without distortion. The last image (Frame 9) obtained at the maximum expiratory phase shows the grid with evident distortion. The coordinate values of a given grid intersection on the first image (\(x,y\)) and those of the corresponding grid intersection on the last image (\(x_{i}\), \(y_{i}\)) were determined by a radiologist. The BE value calculated using these images was 2.52, and its corresponding DPS value was \(p_{1-9}(x,y)=42-10=32\), as shown in Figure 6a. Figure 1f shows a 72-year-old male patient with fibrosis scores equal to F4. The grid in the first image has no distortion, whereas the last image obtained at the maximum expiratory phase shows that the grid shifted slightly upward, but there is no distortion with the grid. The BE value calculated using these images was as low as 0.57, and its corresponding DPS value was \(p_{1-9}(x,y)=42-28=14\), as shown in Figure 6b. It is evident that the value of the non-rigid deformation is high if the value of BE/DPS is high. On the other hand, the deformation is low if the value of BE/DPS is low. The reason for setting an LM at the time of inhalation versus the time of expiration is that the difference in the strength of deformation between the normal liver and the abnormal liver conspicuously varies when the BE/DPS value is calculated when the volume of deformation of the liver is the largest. The shifts of both BE/DPS values are generally invariant to evaluate the degree of distortion or deformation of a non-rigid object. Rigid movement of the liver by breathing as an affine transform is ignored in the assessment, and only the non-rigid transformation caused by the liver itself is calculated. Although DPS is not rotationally invariant, the rotated element is very small in our datasets because the breathing of patients only makes the liver move up and down. Such properties would make it possible to use BE/DPS values to evaluate the quantitative stiffness of the liver. Figure 6: Difference in the power spectral (DPS) value within a \(5\times 5\) ROI between the first frame and the ninth frame of the MR tagging image in the 2D frequency domain is relatively large on the normal liver (**a**) and small on the abnormal, cirrhotic liver (**b**). We assessed that a 12 mm spacing was too small to clearly identify crossing points in the BE/DPS analysis and a 24 mm spacing was too large to have a sufficient number of coordinate samples, especially in small cirrhotic livers. Thus, we chose 16 and 20 mm stripe spacing and sagittal and coronal planes, resulting in four different types of cine-tagging images: 16 mm sagittal, 20 mm sagittal, 16 mm coronal, and 20 mm coronal grids (Figure 1a-d). Comparing the four different types of grids, our preliminary study concluded that the 16 mm sagittal grid with BE features showed the best performance in the diagnosis [19]. For the DPS feature, the performance on these four types of images is shown in Figure 7. The average DPS values in normal and abnormal livers listed in Table 1 indicate that the 16 mm sagittal DPS method also has the best performance, which is identical to the results obtained from the BE method. The performances based on the bending energy and power spectrum methods shown in Figure 8 are different. Neither of the methods are able to distinguish the abnormal liver from the normal liver. However, the distinction between the normal liver and the abnormal liver becomes much clearer by combining the two methods, as shown in Figure 9. There are many overlap cases in Figure 9a between automatically placing the LMs based on Hough transformation [26] and the automatic DPS value calculation from the original image. Figure 9b is the result of manually placing LMs as the gold standard versus DPS from the segmented liver image.