Examination of optimal moments as input parameters for evaluation of liver fibrosis based on multi-Rayleigh model

When many scattered points are distributed randomly and homogeneously, such as in normal liver tissue, the PDF of the echo amplitude can be approximated by a Rayleigh distribution. The Rayleigh distribution is given by

$$\begin{equation} p(x) = \frac{2x}{\sigma^{2}}\exp \left(-\frac{x^{2}}{\sigma^{2}}\right), \end{equation} \tag{ 1 }$$

where x and σ² are the echo amplitude and the variance of the echo amplitude, respectively.

On the other hand, in an inhomogeneous medium, such as a fibrotic liver, the PDF of the echo amplitude deviates from the Rayleigh distribution. It is considered that a fibrotic liver is composed of various types of tissue, such as normal and fibrotic tissues. We proposed a multi-Rayleigh distribution model that is a combination of Rayleigh distributions with different variances.

The multi-Rayleigh model with two components²⁴⁾ is given by

$$\begin{equation} p_{\text{mix2}}(x) = (1 - R_{\text{m}})p_{\text{low}}(x) + R_{\text{m}}p_{\text{high}}(x), \end{equation} \tag{ 2 }$$

where p_low(x) is the Rayleigh distribution with a low variance (normal tissue), $\sigma _{\text{low}}^{2}$, and p_high(x) is the Rayleigh distribution with a high variance (fibrotic tissue), $\sigma _{\text{high}}^{2}$. R_m is a mixture rate of Rayleigh distributions with high variances. By approximating the PDF of echo envelope using the multi-Rayleigh model, the fibrotic mixture rate R_m, which is an indicator of the amount of fibrotic tissue, and the fibrotic variance ratio $R_{\text{v}} = \sigma _{\text{high}}^{2}/\sigma _{\text{low}}^{2}$, which is an indicator of the fibrosis progressive ratio, can be estimated.²⁸⁾

Because the multi-Rayleigh model with two components has two degrees of freedom, model parameters can be estimated by using the combination of two moments. Moments are indicators of the shape of the PDF. The n-th moment around the average value, M_n, for normalized echo envelope is calculated as

$$\begin{equation} M_{n} = E\left[\frac{(x - \mu)}{\sigma^{n}}^{n}\right], \end{equation} \tag{ 3 }$$

where x is the echo amplitude. μ and σ are the average value and the standard variance of the echo amplitude, respectively.

We present the estimation procedure. First, the echo amplitude x is normalized such that the power of echo signals becomes 1. Then, two moments (M_n1 and M_n2) of normalized echo amplitudes are calculated, and the multi-Rayleigh model parameters $(R_{\text{m}},R_{\text{v}})$ are obtained by resolving reversely a look-up table of moments-model parameters. A look-up table contains the n-th order moments of the multi-Rayleigh model, p_mix2(x), with various model parameters, $(R_{\text{m}},R_{\text{v}})$. The n-th order moments are calculated numerically by changing R_m (from 0.05 to 0.7 by 0.001) and R_v (from 1.2 to 8 by 0.01). By resolving reversely this look-up table, the model parameters $(R_{\text{m}},R_{\text{v}})$ can be calculated stably at high speed. The order n was changed from 0.05 to 3 in steps of 0.05. When the moment is larger than order 3, even a slight deviation at high amplitude causes a large change in the moments.²⁸⁾ Thus, the upper limit of moment order was set to 3. The moment orders in which the relationship between the moment values and the model parameters does not satisfiy a one-to-one relation were exclued. When moment values (M_n1 and M_n2) and model parameters $(R_{\text{m}},R_{\text{v}})$ satisfy the one-to-one relation, a stable estimation of the inverse problem can be achieved.

Figure 1 shows an example of parameter estimation. Using echo image simulation with the setting model parameters $(R_{\text{m}},R_{\text{v}})$, PDF of echo data was obtained, as shown in Fig. 1(a). From the echo data, arbitrary order moments of data can be calculated. If the selected moments are 0.15th and 1.5th moments, M_0.15 and M_1.5 are calculated to be 0.860 and 0.477, respectively, as shown in Fig. 1(a). Figure 1(b) shows the relationship between (0.15th, 1.5th) moments pair and model parameters $(R_{\text{m}},R_{\text{v}})$. This relationship can be obtained numerically by using Eqs. (2) and (3). In this figure, the red line and blue dotted line correspond to the fibrotic mixture rate and variance ratio contours, respectively. The crossing point at M_0.15 = 0.860 and M_1.5 = 0.477 is plotted in Fig. 1(b). Then, we can estimate the parameters by resolving reversely the look-up table. In the look-up table, we can find the setting value whose distance between the theoretical moments and the moments of echo data is minimum and that of the setting value is estimated as the model parameters. In this case, the estimated model parameters $(R_{\text{m}},R_{\text{v}})$ are $(0.415,3.0)$. The two components in the estimated multi-Rayleigh model are shown in Fig. 1(c).

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Figure 2 shows the relationship between the (0.25th, 2.75th) moments pair and model parameters $(R_{\text{m}},R_{\text{v}})$. The red line and blue dotted line correspond to the fibrotic mixture rate and variance ratio contours, respectively. In this figure, the model parameters' coordinate is folded over and overlaps in the moments' plane. Using this moments pair, it is impossible to uniquely determine the model parameters. This moments pair was excluded from the look-up table to solve the inverse problem. Similar moment pairs that do not satisfy a one-to-one relation with model parameters were also excluded from the look-up table in advance. Using this look-up table, a stable estimation of the inverse problem can be achieved.

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Moments of echo envelope are used to estimate the multi-Rayleigh model parameters, and clinical data moments have statistical variation. The statistical variation causes the variation of estimation results.²⁹⁾ The estimation variation depends on the combination of input moments; therefore, the combination of moments used in the estimation algorithm affects the estimation accuracy of multi-Rayleigh model parameters. Thus, the selection of the optimal combination of moments is important to estimate the multi-Rayleigh model parameters stably.

From the scatterer distribution model, an ultrasound image was calculated using Field II, which is a tool for ultrasonic simulation.^30,31) The scattered points are distributed randomly and the scattered density is set to be 11 points per point spread function (PSF). The scatterer distribution model was located 10 to 40 mm from the transducer. The center frequency was 7.5 MHz. The focal position was moved dynamically from 5 to 80 mm at 1 mm intervals to realize a uniform ultrasonic beam width in an image; therefore, the effect of beam width on the quantitative estimation of liver fibrosis can be ignored.³²⁾ The lateral size of the scatterer distribution model was 15 mm. The B-mode images of normal and fibrotic tissues shown in Figs. 3(a) and 3(b), respectively, can be made by changing the reflection strength of the scatterer. The B-mode image of the tissue model which has a multi-Rayleigh distribution with two components, can be made by combining the normal and fibrotic tissues marked with a red frame. The result is shown in Fig. 3(c). The size of the red frame is determined by the fibrotic mixture rate. The mixture rate and variance ratio in Fig. 3(c) are 0.3 and 3, respectively.

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To improve the estimation accuracy of the multi-Rayleigh model, we try to determine the input moments by focusing on the distribution in the multi-Rayleigh model parameters' space.

Figure 4(i) shows the distribution of two different moments of echo envelopes, the PDF of which can be approximated using the multi-Rayleigh model with the setting model parameters (R_m = 0.3, R_v = 3). In this figure, the red line and blue dotted line correspond to the fibrotic mixture rate and variance ratio contours, respectively. Each red dot in the figure is plotted using one B-mode image. The trial of B-mode simulation was iterated 1000 times. The 1000 B-mode images following the multi-Rayleigh model with two components were made, and 1000 points are plotted in Fig. 4(i) using the calculated moments of echo PDF as a scatter diagram. The boundary of the area, which statistically contains 95% of the data, is shown in black. The black boundary is converted into the multi-Rayleigh model parameters' space as shown in Fig. 4(ii). The distribution area of the multi-Rayleigh model parameters' space is different with different combinations of moments. When the distribution area shown in Fig. 4(ii) becomes smaller, the estimation accuracy of the multi-Rayleigh model parameters becomes higher; therefore, we should choose the combination whose distribution area of the multi-Rayleigh model parameters' space is small. To determine the optimal input parameters that have high estimation accuracy, the deviation of mixture rate and variance ratio with different combinations of two moments $(m_{1},m_{2})$, $U(R_{\text{m}},R_{\text{v}},m_{1},m_{2})$, are calculated. $U(R_{\text{m}},R_{\text{v}},m_{1},m_{2})$ is given as

$$\begin{align} U(R_{\text{m}},R_{\text{v}},m_{1},m_{2}) &= \frac{R_{\text{m}} - R_{\text{m,min}}(m_{1},m_{2})}{R_{\text{m,max}}(m_{1},m_{2}) - R_{\text{m,min}}(m_{1},m_{2})} \\ &\quad + \frac{R_{\text{v}} - R_{\text{v,min}}(m_{1},m_{2})}{R_{\text{v,max}}(m_{1},m_{2}) - R_{\text{v,min}}(m_{1},m_{2})}, \end{align} \tag{ 4 }$$

where R_m and R_v are those of setting values, respectively. R_m,max and R_m,min are the maximum and minimum values of estimated mixture rate on the projected contour in Fig. 4(ii), respectively. R_v,max and R_v,min are the maximum and minimum values of estimated variance ratio, respectively. A small U value means a high estimation accuracy.

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To determine the optimal combination of moments, $U(R_{\text{m}},R_{\text{v}},m_{1},m_{2})$, for different combinations of two moments, m₁ and m₂, the orders of which were set from 0.05 to 3 at 0.05 intervals, were calculated. As mentioned in Sect. 2, the upper limit of moment order for stable estimation was determined to be 3 using preliminary results. The results are shown in Fig. 5 when R_m = 0.3, R_v = 3. The minimum and maximum values of U in Fig. 5 are marked with a circle and a cross, respectively. The combinations of moments that cannot determine the multi-Rayleigh model parameters uniquely are excluded. The excluded area is shown with a white color in Fig. 5. We can see that the estimation accuracy is higher when $U(R_{\text{m}},R_{\text{v}},m_{1},m_{2})$ is smaller; therefore, the combination moments m₁ and m₂ with high estimation accuracy can be determined by the smallest $U(R_{\text{m}},R_{\text{v}},m_{1},m_{2})$. Results in Fig. 5 show that the multi-Rayleigh model parameters can be estimated with the highest estimation accuracy by using the 0.15th and 1.65th moments (open circle). By using the 0.95th and 3rd moments (cross mark), the multi-Rayleigh model parameters can be estimated with the lowest estimation accuracy. Figures 4(ii)-(a) and 4(ii)-(b) are the figures for the highest and lowest cases, respectively.

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In previous studies, we used the 1st and 3rd moments around zero for the estimation of multi-Rayleigh model parameters. Figure 6 shows the comparison between the highest (a), the lowest (b), and the previous (c) estimation cases. Figure 6(a)(iii) shows the histogram of fibrotic parameters estimated using the 0.15th and 1.65th moments. To compare with the results shown in Figs. 6(b) and 6(c), the histogram of fibrotic parameters estimated using the 0.95th and 3rd moments and that using the 1st and 3rd moments around zero, which have worse concentration, are shown in Figs. 6(b) and 6(c), respectively. From these results, we can see that the multi-Rayleigh model parameters can be estimated with the highest estimation accuracy by using the 0.15th and 1.65th moments when R_m = 0.3, R_v = 3.

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