Scattering signatures of normal vs. abnormal livers with support vector machine classification

Abstract

Fifty years of research on the nature of backscatter from tissues has resulted in a number of promising diagnostic parameters. We recently introduced two analyses tied directly to the biophysics of ultrasound scattering; the H-scan based on a matched filter approach to distinguishing scattering transfer functions, and the Burr distribution for quantification of speckle patterns. Together, these analyses can produce at least five parameters that are directly linked to the mathematics of ultrasound in tissue. These have been measured in vivo in 35 rat livers under normal conditions and after exposure to compounds which induce inflammation, fibrosis, and steatosis in varying combinations. A classification technique, the support vector machine, is employed to determine clusters of the five parameters that are signatures of the different liver conditions. With the multiparametric measurement approach and determination of clusters, the different types of liver pathology can be discriminated with 94.6% accuracy.

INTRODUCTION

Assessment of the structural and functional state of the liver is a primary concern for diagnostic imaging (Taylor and Ros 1998; Ozturk et al. 2018). Ultrasound examination of the liver is an accessible and inexpensive tool across most of the world. There is a long and distinguished history of research in the ultrasound echoes returning from the liver, and their change with diseases, from early tissue characterization work (Chivers and Hill 1975; Gramiak et al. 1976; Bamber 1979; Lizzi et al. 1983; Campbell and Waag 1984; Insana et al. 1990; Zagzebski et al. 1993) to more recent investigations (Higuchi et al. 2014; Al-Kadi et al. 2016; Liao et al. 2016; Zhou et al. 2018; Lin et al. 2019; Tamura et al. 2020). At this point in time there are a variety of research techniques available and growing numbers of commercial scanners that offer new parameters related to echoes from tissue. Yet agreement on the precise properties of ultrasound from normal and diseased livers remains elusive. This situation is mirrored by uncertainty as to the most appropriate physical and mathematical models of scattering from the normal and diseased tissues.

However, there are numerous studies to characterize the diseases using ultrasound images or derived parameters (D’Souza et al. 2019) and, furthermore, these have been introduced into computer-aided diagnostic systems using machine learning. For example, the characterization of normal, cirrhotic, and hepatocellular carcinoma has been studied (Virmani et al. 2013a; Virmani et al. 2013b). The grade of liver fibrosis was determined by feature extraction and the support vector machine (SVM) (Yeh et al. 2003). Breast tumors were identified as benign or malignant using image processing including segmentation and feature extraction (Wu et al. 2012).

Almost all machine learning studies have used ultrasound images as the input, and used common image processing techniques including segmentation and feature extraction (Chang et al. 2010; Acharya et al. 2012). Some researchers employed parameters extracted from the images (such as contrast, signal-to-noise ratio, or standard deviation) (Sujana et al. 1996; Ogawa et al. 1998).

Although previous studies tried to include multiple features as the input for training, the majority are based on image processing metrics, which frequently are inter-related. In general, having more independent features results in more accurate classifications. In this study, we derive independent estimates of ultrasound first and second order statistics based on biophysical models.

Since pathology scores are based on fatty, cirrhotic and ballooning content of the liver, we first examine the theoretical models of scattering that are likely to be dominant in cases of normal, steatotic, fibrotic, and inflamed liver tissues. Secondly, the effect of these scattering models on the returning echoes are examined in terms of their first order statistics (the histogram of echo amplitudes) and second order statistics (the backscatter vs. frequency). Third, we examine liver echoes from rat livers using high frequency ultrasound scanner, with and without carbon tetrachloride (CCl₄) or concanavalin A (ConA) exposure which incites a hepatic response, including varying degrees of fat accumulation, fibrosis, and inflammation. Finally, a SVM was implemented to classify the clusters of pathological liver states in multiparametric space.

These results provide the beginning of a coherent framework for determining the signatures or clustering in multiparametric space, of the normal liver compared with steatotic, fibrotic, or inflammatory livers.

THEORY

Fractal branching theory for normal liver

The pioneering studies of ultrasound scattering from human and animal livers established a number of key results (Chivers and Hill 1975; Gramiak et al. 1976; Bamber 1979; Zagzebski et al. 1993). The frequency dependence of scattering was found to be a power law function of frequency, for example the average intensity rising as $f^{1.4}$ power (Campbell and Waag 1984). The speckle statistics of the returning echo amplitudes were found to be somewhat analogous to that of optical speckle (Burckhardt 1978). Most of the work on scattering theory postulated scattering from spheres or spherical correlation functions, mostly attributed to cell size and shapes (Lizzi et al. 1983; Insana et al. 1990). However, more recently we postulated that scattering from the normal liver is dominated by the weak acoustic impedance mismatch between the branching fractal structure of the fluid filled vascular bed, compared to the surrounding parenchyma comprised of mostly close-packed hepatocytes. In this theoretical framework the mathematics of speckle and scattering are not based on historical models of random points or spheres, but on the mathematics of scattering from cylinders and fractal branching structures (Parker 2019a; Parker 2019b; Parker et al. 2019) as illustrated in Figure 1. A key parameter in fractal analysis is the fractal dimension $D$ which in three-dimensional structures such as the liver vasculature is a measure of how the self-similar and multiscale elements progressively fill a three-dimensional volume. Measurements of fractal dimensions of vascularized tissues tend to estimate $D$ in the range of 2 – 2.5 (Carroll-Nellenback et al. 2020; Parker and Poul 2020b). A summary of the fractal structure’s key parameters are given in Table 1. Furthermore, the probability distribution function for speckle amplitudes from the fractal branching vasculature are dominated by a power law relationship related to $D$, specifically in the form of a classical Burr distribution (Parker and Poul 2020a).

Figure 1.

Schematic of dominant scattering structures from normal liver tissue in an abdominal ultrasound scan. Shown is a micro-CT contrast-enhanced 3D rendering of the vasculature within a mouse liver. In normal liver, the weak scattering from the fluid-filled vasculature is a major source of returning echoes. Mathematical models of speckle and scattering can account for the fractal nature of the vascular tree.

Table 1.

Interrelationship of fractal metrics in 3D.

Name	Symbol	Equation	Notes
fractal dimension	$D$	$N\left(l\right)~l^{-D}$	box counting with scale $l$; $D<3$
autocorrelation	$C\left(r\right)$	$C\left(r\right)~C_0/r^{\left(3-D\right)}$	$r$ is autocorrelation lag in spherical coordinates; $r>0$; $D<3$
3D spherical Fourier transform	${}^{3DS}\Im \left\{\right\}$	${}^{3DS}\Im \left\{1/r^{\left(3-D\right)}\right\}~1/q^D$	$q$ is spatial frequency; $1<D<3$
scattering differential cross section	${\sigma }_d\left(k\right)$	${\sigma }_d\left(k\right)~k^{\left(4-D\right)}$	$k$ is wavenumber, derived from Fourier transform of $C\left(r\right)$; $D<3$
speckle probability histogram	$p[A]$	$\frac{2A\left(b-1\right)}{{\lambda }^2{\left[{\left(\frac{A}{\lambda }\right)}^2+1\right]}^b}$	$p[A]$ is the probability of an echo of amplitude $A$, based on scattering from a fractal branching vasculature with power law parameter $b$ and scale factor $\lambda$, related to the classical Burr probability distribution

Rayleigh scattering for simple steatosis

The most elementary model for scattering from a liver with early steatosis is an additive model, where the base model for a normal liver still applies with the addition of the scattering from the accumulating fat. In early stages of fat accumulation in the liver, microvesicles and macrovesicles appear within hepatocytes. On pathology slides these can appear as small randomly positioned spheres (below 20 microns), and since they are comprised of triglycerides, these have a different speed of sound and density from the surrounding hepatocytes, and hence are a source of scattering. Classical models of random Rayleigh scatterers may apply with the long wavelength approximation of backscattered intensity increasing as $f^4$ power across conventional imaging frequencies. However, it may be too simple to consider only randomly positioned, single spherical scatterers as models of the inhomogeneous fat vesicles, particularly as the percentage of fat increases. As the steatosis progresses, the spatial distribution of fat can be heterogeneous, concentrating in periportal patterns (Schwen et al. 2016). Furthermore, the composition of fat in later stages can be shifted (Peng et al. 2015; Chiappini et al. 2017), so both the size distribution and the scattering strength may be a function of the stages of progression of simple steatosis. The clustering effect across a number of scales from the smallest microvesicles to the larger portal structures, may mimic a fractal clustering structure (Javanaud 1989; Shapiro 1992), which then comports with the general behaviors described in Table 1.

Thin septae for CC1₄ fibrosis

As shown in Figure 2, early fibrosis response following CCl₄ exposure is in the form of thin septae that extend around the portal triads within the liver. These can be under 20 microns in thickness and represent a sheet of increased density and compressibility compared to surrounding hepatocytes. Thus, these form a network of scattering sites which we model as additive to the baseline model for normal liver. In simplest theory, the one-dimensional convolution model for normal incidence on a thin sheet predicts a scattering transfer function proportional to $f^1$ (Macovski 1983; Parker 2016). This represents an upshift in the scattering amplitude transfer function as compared with baseline values of $f^{0.7}$ (or $f^{1.4}$ in intensity) by Campbell and Waag (1984).

Figure 2.

Histology images of liver sections stained with picrosirius red for fibrosis. Object scale and 10× object virtual magnification are shown. Left, middle, and right columns represent untreated control, vehicle control, and fibrosis induced by CCl₄ administration; fibrosis structures were stained picrosirius red and are shown in the right column images. Upper set of images denoted as “Day 29” were obtained 29 days after the start of dosing CCl₄ as a fibrosis inducer; similarly, lower set was obtained 56 days after the start of dosing. Control and vehicle groups appear to be unchanged, however the CCl₄ dosing group in the right column shows fibrosis growth from day 29 to 56.

Influence of inflammation

The effects of early inflammation include the presence of ballooning of cells, necrosis, and apoptosis (Lackner 2011), which could be modeled as a spherical impedance mismatch with respect to the surrounding hepatocytes, thereby serving as a source of Rayleigh scattering. This would contribute a scattering transfer function proportional to $f^4$ power in intensity ($f^2$ in amplitude) at long wavelengths. However, if the volume percent of swollen cells is low, and their Rayleigh scattering is weak, these additive contributions could be difficult to separate out from the stronger effects of the baseline scattering, plus steatosis, plus fibrosis if present.

Influence on H-scan, Burr parameters

The theoretical scattering models listed above are hypothesized to be additive to the baseline case of the normal liver. In our study the particular measurements employed are related to the H-scan analysis, a matched filter approach sensitive to scattering transfer functions. Also the analysis of speckle amplitude histograms are studied. The details of these analyses are given in the Methods section, and the analyses produce five estimated parameters. Our hypothesis, based on the scattering models, is that these metrics will be sensitive to changes in the hepatic scattering structures under conditions of fibrosis, steatosis, and inflammation. The trends are given in Table 2. Briefly, under the additive models most parameters are expected to increase with the addition of scattering structures. The exceptions are for ultrasound attenuation under fibrosis and inflammation, where the addition of low attenuating collagen and fluid, respectively, lowers the overall loss. However, we currently lack the accurate parameters (for example the size distribution and material properties of the fat vesicles) required for quantitative predictions. For that reason, the principal components analysis and clustering of classes using the SVM will be applied to the measured results. These are described in later sections.

Table 2.

Hypothesized changes in measured parameters based on additive models of scattering from fibrosis, fat, and inflammation. Arrow directions indicate increase or decrease.

PARAMETER \ GROUP	(H-scan) % blue: higher frequency scattering, structures	(H-scan) $\alpha$: attenuation	(H-scan) $I_{dB}$: brightness	(Burr) $b$: fractal branching number density	(Burr) $\lambda$: scale of echoes from fractal branching
Normal	50% by design	0.05 Np/cm-MHz	−15 dB measured	3 measured	350 measured
Fibrotic, low fat	↑	↓	↑	↑	↑
Fibrotic, high fat	dependent on distribution	↑	↑	↑↑	↑↑
Inflammation	weak ↑	weak ↓	weak ↑	weak ↑	weak ↑

METHODS

Study design and animals

An in vivo study was designed as illustrated in Figure 3 to investigate normal and diseased liver in rats. All animals were maintained according to the NIH standards established in the “Guide for the Care and Use of Laboratory Animals.” The Pfizer Animal Care and Use Committee (IACUC) approved all experimental protocols. Rats were pair-housed, had free access to water and were fed a standard commercial laboratory certified rodent diet 5002 (PMI Feeds, Inc.). The testing facility maintained 12-hr light/dark cycle, with controlled temperature, humidity and air changes. A total of 35 male rats were analyzed for this study, 31 Sprague-Dawley (SD, Charles River Laboratories, Wilmington, MA, USA) and 4 TAC NIH: rnu (nude, Taconic Biosciences, Inc., Rensselaer, NY, USA). A CCl₄ (Sigma Aldrich, St. Louis, MO, USA) model was used to induce fibrosis with varying fat, and a ConA model was used to induce inflammation. Con A has been shown to induce acute hepatitis (Heymann et al. 2015), and CCl₄ induces fibrosis and varying degrees of steatosis as shown in Table 3. ConA was dosed IV once each week at dose of 20 ml /kg in sterile PBS, and CCl₄ was dosed PO three times per week (Monday, Wednesday, and Friday) at a dose of 1 ml/kg in 1:1 mixture with vehicle of olive oil. To monitor the livers over time, rats were ultrasound imaged at baseline, that is before dosing, and every two weeks after dosing began. A Vevo 2100 (VisualSonics, Toronto, Canada) was used to image the rats with a 21 MHz center frequency linear transducer (MS 250), and RF data of liver were collected. The liver echoes were used for signal processing to estimate tissue parameters, including H-scan classification, attenuation estimation, B-scan intensity, and ultrasound speckle statistics using Burr distribution.

Figure 3.

Study design and data acquisition of ultrasound. All 35 enrolled rats were scanned by ultrasound every two weeks. 26 rats were dosed to induce liver diseases. At the termination of the study, rats were euthanized, and then histology and assessments were performed.

Table 3.

Description of the four liver groups. This study was designed to use fibrosis inducers to cause liver disease. The total of 35 rats includes 31 SD and 4 nude rats, and all nude rats were confirmed as fibrotic with low fat. The four confirmed states by pathology are considered as desired classes in SVM classifier. Each rat has approximately 30 ultrasound scan frames, resulting in 998 frames for training set.

Dosing	Confirmed state	Oil Red O stain area [%, mean ± SD]	Number
controls or vehicle controls (olive oil PO)	Normal	1.2 $\pm$ 1.5	9
CCl4	Fibrosis with low fat*	2.9 $\pm$ 1.9	11
CCl4	Fibrosis with high fat*	17.3 $\pm$ 11.2	6
ConA	Inflammation**	0.7 $\pm$ 1.6	9

^*Fibrosis with low and high fat are classified by Oil Red O stain area compared to tissue stained area (%). Fibrosis with low and high fat have the ratio less than 6.5% and greater than 9%, respectively.

^**Inflammation group has no fibrosis and very low fat, which is comparable to normal group.

For scanning, rats were anesthetized with 1–3% isoflurane and euthanized by CO₂ inhalation, followed by necropsy and collection of liver tissues for histology analysis. The time points for euthanasia were 4, 6 and 8 weeks after dosing for 7, 12, and 16 rats, respectively. Blood serum panels were obtained at baseline and every 2 weeks for liver biomarker analysis (ALT, AST, GLDH, Albumin, Globulin, ALP, Glucose, Insulin, bilirubin, GGT). Fibrosis was assessed by Trichrome stain and Picrosirius Red. Oil Red O stain was used to detect lipid and the area was compared to the tissue stained area and provided a ratio (%) of Oil Red O stain area to tissue area as an indicator of fat content. With these measures, the liver states were categorized post-mortem by an expert pathologist as described in Table 3.

H-scan analysis

A summary of the H-scan method is given in this section, additional details can be found in Parker and Baek (2020). Fundamentally, the H-scan is a matched filter analysis that models the pulse-echo phenomenon as a power law transfer function in the frequency domaFin. In general, smaller structures have higher power law transfer functions and these are encoded as blue on the visual display of the H-scan output. However, frequency dependent attenuation effects can accumulate over depth and require compensation if an accurate analysis is required.

Attenuation estimation and correction within the H-scan analysis

Ultrasound imaging systems commonly employ a pulse with a round-trip impulse response that can be approximated by a bandpass Gaussian spectrum of $e^{-{\left(f-f_0\right)}^2/2{\sigma }^2}$ with a center or transmit frequency of $f_0$ and a bandwidth of $\sigma$. When considering the frequency and depth dependent attenuation of $e^{-\alpha fx}$, the frequency spectrum is described by:

$$S\left(f\right)=e^{-\frac{{\left(f-f_0\right)}^2}{2{\sigma }^2}}\cdot e^{-\alpha fx}$$ (1)

where $\alpha$ is attenuation coefficient in np/cm/MHz, f is frequency of ultrasound in MHz, and $x$ is depth in cm. The attenuation makes the peak frequency of the spectrum decrease, which can be estimated by taking the first partial derivative with respect $f$ and finding $0$ at peak frequency $f_p$ given by:

$${\frac{\partial S}{\partial f}|}_{f=f_p}=\left(\frac{f_p-f_0}{{\sigma }^2}+\alpha x\right)\cdot S\left(f_p\right)=0.$$ (2)

We obtain the attenuation coefficient in the form

$$\widehat{\alpha }\left(x\right)=-\frac{f_p\left(x\right)-f_0}{x\cdot {\sigma }^2},$$ (3)

where $f_0$ and $\sigma$ represent properties of the transducer related to transmit frequency and designed bandwidth, respectively. Therefore, by measuring peak frequency along with depth of $f_p\left(x\right)$, the attenuation coefficient can be estimated according to eqn (3). This approach assumes homogeneous (or stationary) distribution of scatterers within the ROI, but has the advantage of being independent of amplitude fluctuations related to system gain.

As an example of this approach, Figure 4(a) is a B-scan of a rat liver with region of interest (ROI) for H-scan processing, and Figure 4(b) is H-scan colormap. By averaging the color values over all scanlines within the ROI, representative H-scan color levels along with depth $x$ can be calculated as shown in Figures 4(c) and (d); the color levels can be converted into peak frequencies in Figure 4(f) by pseudocolor given in Figure 4(e). The color levels from 1 to 256 are mapped to frequencies ranging from 8.7 MHz to 20.3 MHz. The measured peak frequency in Figure 4(f) is used to estimate attenuation coefficient in eqn (3), and the obtained attenuation coefficients are averaged over depth. In summary, the attenuation coefficient can be obtained using H-scan results within assumptions of a Gaussian bandpass pulse and attenuation of the form $e^{-\alpha fx}$.

Figure 4.

Attenuation estimation using H-scan.

Once the attenuation parameter is estimated for a given ROI in the liver, a depth-dependent inverse filter can be applied to correct for losses, at least to the limit of the noise floor (Parker and Baek 2020), and proceed with the matched filter analysis. The final outputs for each liver ROI are estimates of attenuation ($\alpha$ in dB/cm-MHz), echogenicity or brightness (dB), and percent of blue. The attenuation coefficient was estimated by averaging eqn (3) over depth within the ROI. Brightness was calculated from log-compressed data where 0 dB is set to the same brightness level for all scans. When calculating percentage of blue, color levels obtained from H-scan ranging from 1 to 256 as shown in the color bar in Figure 4(b) were used; the pixels with color levels of [1, 128] and [129, 256] are red $\left(i\in R\right)$ and blue pixels $\left(i\in B\right)$, respectively. Data normalization was performed by converting the color levels from 1 to 256 into the normalized color levels I from −1 to 1 in sequence, then percentage of blue is defined by:

$$\%\text{of}\text{blue}=\frac{\frac{1}{n_B}{\sum }_{i\in B}\left|I_i\right|}{\frac{1}{n_B}{\sum }_{i\in B}\left|I_i\right|+\frac{1}{n_R}{\sum }_{i\in R}\left|I_i\right|}\times 100\%,$$ (4)

where $i$ is the index of each pixel in B-scan, $I_i$ is normalized color level value for the pixel $i$, and $n_B$ and $n_R$ are the number of blue and red pixels, respectively.

First order statistics of speckle and the Burr distribution

Consistent with the framework shown in Figure 1 and Table 1, the normal liver’s speckle pattern results from the fractal self-similar network of fluid vessels (Parker 2019b; Parker et al. 2019; Parker and Poul 2020a). Our analysis assumes a broadband pulse interacts with fluid-filled, branching, self-similar set of long vessels in the tissue, whose number density as a function of radius $a$ is described by a power-law behavior as $N\left(a\right)=N_0/a^b$, with the key power law parameter, $b$ governing the branching behavior of the vasculature over a wide range of scales. By finding the dominant echoes from the 3D convolution model, the histogram of echo amplitude is derived and after normalization, can be expressed as a PDF:

$$N_n\left[A\right]=\frac{2A{\left(a_{\text{min}}\right)}^{b-1}\left(b-1\right)}{A_0^2{\left[{\left(\frac{A}{A_0}\right)}^2+a_{\text{min}}\right]}^b}.$$ (5)

Eqn (5) is a three-parameter PDF describing the distribution of the echo amplitude $A$ with $A_0$ and $a_{\text{min}}$ related to the system gain and minimum size of scattering vessels, respectively. This equation reduces to a two-parameter PDF by change of variables as $\lambda =A_0\sqrt{a_{\text{min}}}$:

$$N_n\left[A\right]=\frac{2A\left(b-1\right)}{{\lambda }^2{\left[{\left(\frac{A}{\lambda }\right)}^2+1\right]}^b}.$$ (6)

The two-parameter PDF of eqn (6) happens to be a Burr type XII distribution which was derived in the 1940s without any consideration of ultrasound (Burr 1942). The speckle distribution has also been shown to be in reasonable consistency with the parameters of Lomax distribution when fitted to the intensity of echoes and also with the logistic distribution parameters when fitted to the natural log of echo amplitudes, by employing the general transformation principle (Parker and Poul 2020a).

The two parameters of $b$ and $\lambda$ may be sensitive to the change in the scattering structures of soft tissues. The two parameters of Burr distribution are estimated using MATLAB (Mathworks, Inc., Natick, MA, USA) nonlinear least square minimization of errors when fitted to the normalized distribution of the speckle amplitude data from the liver ROIs. In order to place a reasonable bound on the parameters of $b$ and $\lambda$ in curve fitting, an additional step is made. We calculate the mode and median of the speckle histogram and compare these to the theoretical formulas for the Burr distribution:

$$\text{Mode}={\left(\frac{1}{2b-1}\right)}^{\frac{1}{2}}\cdot \left(\lambda \right)$$ (7)

$$\text{Median}={\left(2^{\left(\frac{1}{b-1}\right)}-1\right)}^{\frac{1}{2}}\cdot \left(\lambda \right)$$ (8)

Using eqn (7) and (8) for mode and median of the Burr distribution which both depend on $b$ and $\lambda$, a system of equations is solved for each image frame’s ROI to obtain frame-by-frame estimates for these two parameters. These data are used as bounds with ± 10% to ± 20% intervals for the histogram Burr fitting estimation of $b$ and $\lambda$. This additional step assures that the parameters lie in a trimmed, middle range. This process is done for all the frames for each of the selected 35 rat liver data and the results of the $b$ and $\lambda$ parameters presented as boxplots, separating the sensitivity of parameters in four groups of rat livers.

SVM classifier

The classification of liver pathology states was performed by the SVM, which results in decision planes for the classification in the five-dimensional parameter space: H-scan, attenuation, B-scan, and Burr $b$ and $\lambda$. The schematic of the proposed classifier is shown in Figure 5. For each of the 35 rats scanned, the liver ROIs were manually set to select relatively uniform liver appearance with an absence of artifacts such as shadowing or reverberations. Each case has approximately 30 frames, including the liver, therefore a total of 998 ultrasound images were enrolled as a training set; instead of using the images directly as the machine learning input, this study uses the five measurements within the ROIs as the input, which are the percentage of blue from H-scan, the attenuation coefficient $\alpha$ (dB/MHz/cm), the intensity $I_{\text{dB}}$ (dB) from the B-scan, and $\lambda$ and $b$ (dimensionless) from Burr histogram analysis. In other words, the five measurements define the five input features of the machine learning procedure, and the total number of features is 998:

$${\left\{\left(x^{\left(n\right)},y^{\left(n\right)}\right)\right\}}_{N=1}^{N=998},\text{where}{x}^{\left(n\right)}=\left(\text{\% of blue,}\alpha ,I_{\text{dB}},\lambda ,b\right)\in {ℝ}^5,$$ (9)

and where $y^{\left(n\right)}$ is a desired class confirmed post-mortem pathology, and there are four classes: normal, fibrosis with low fat, fibrosis with high fat, and inflammation. Ideally, the five metrics for each class form clusters in the five dimensional-space, whereby the goal of this machine learning study is constructing decision planes to distinguish the liver states based on the parameters. Further details of our implementation including the training protocol are found in Appendix A.

Figure 5.

Overall block diagram for SVM classifier to train and predict liver states. Ultrasound scan of liver area where ROI is contoured. Five measurements are obtained by H-scan analysis, attenuation estimation, B-scan intensity estimation, and Burr histogram analysis with two parameters. Thus, each input image has five features that are assigned as the input of the SVM classifier. During training with a train set, SVM constructs decision planes for the four groups: normal, fibrosis with low fat, fibrosis with high fat, and inflammation. To visualize the training set and decided hyper planes in 3D plane, principal component analysis was performed to reduce number of features from five to three.

RESULTS

B-scan, H-scan, and attenuation parameters

The enrolled 35 rats were scanned every two weeks, and we attempted to contour the ROIs consistently over time by using vessels or skin layer as landmarks, whereby the ROIs for the same rat are located in the relatively same position near the biomarkers over time. The selected B-scan and H-scan results are in Figure 6; ROIs for the processing are indicated using the red boxes. The H-scan of low fatty fibrosis and inflammation classes appear more blue than those of normal livers, but high fatty fibrosis cases show more red compared to normal cases. Selected histology results are in Figure 2 and fibrosis structures were stained in red color. CCl₄ exposed rats show an increase in fibrosis from 29 days to 56 days after the start of dosing, while untreated and vehicle controls remain unchanged over time. The thin fibrotic septae seem to divide the sinusoids, driving the H-scan results for fibrosis cases with more blue colors, indicating the addition of relatively small scattering sites.

Figure 6.

B-scan and H-scan images of (a) normal, (b) fibrosis with low fat, (c) fibrosis with high fat, and (d) inflammation.

The investigation over time and the statistical plots are shown in Figure 7. H-scan results of low fatty fibrosis and inflammation in Figures 7(a) and (b) show an increase in percentage of blue over time, and low fatty fibrosis has more blue percentage than inflammation, which is also presented in Figure 6; low fatty fibrosis in Figure 6 (b) shows more increased blue than the case of inflammation in Figure 6 (d). As for the high fatty fibrosis group in Figure 7 (a), the effect of fibrosis growth is likely to mainly appear first until 4^th week as the increase in percentage of blue; but fat accumulation effect appears later in 6^th and 8^th week, showing the decrease in percentage of blue. According to statistics in Figure 7 (b), H-scan can distinguish the four groups from each other, although there are overlapped distributions between the groups. The following notations are used for the statistics: ns (no significance) p > 0.05; * p < 0.05; ** p < 0.01; *** p < 0.001; and **** p < 0.0001. Figures 7(c) and (d) represent the results of attenuation estimation. Attenuation for normal and inflammation groups remains unchanged over time. However, attenuation for fibrosis with low fat decreases over time. Attenuation for fibrosis with high fat tend to decrease until sixth week but increase later, suggesting that the dominant effects of fat appear later, which is consistent with the H-scan trend for fibrosis with high fat. According to statistics of attenuation results in Figure 7 (d), attenuation can separate the four groups except for one case of comparison: normal versus high fatty fibrosis. Figures 7(e) and (f) show the results of B-scan. Fibrosis shows the increase in brightness over time, while normal and inflammation remain unchanged. According to the brightness of B-scan in Figure 7 (f), B-scan have significant differences when comparing the four groups. Although normal and inflammation have overlap, B-scan can statistically separate the two groups; however, B-scan can show better separation between normal and fibrosis. According to the data distribution in half-violin plots in Figures 7 (b), (d), and (f), the three analyses play essential roles to separate some cases from others; but they work well for different separations. To be specific, B-scan shows better separation between normal and fibrosis, although it has more overlap between fibrosis with low and high fat; however, H-scan and attenuation can differentiate low and high fat. Furthermore, H-scan is the method that can show the best separation between normal and inflammation. Therefore, combining the results can provide the potential to discriminate each case from the others.

Figure 7.

H-scan, attenuation, and B-scan result plots. Left column: Investigation over time, including progression of diseases. Right column: Statistics at the final time points for 35 rats; rats were euthanized at different time points ranging from 4 to 8 weeks after the start of dosing.

Burr parameters

For each image frame from the 35 rat livers of this study, the analysis is performed on a well-defined ROI and the underlying Burr statistical properties of the liver speckle are derived from the envelope of the RF signal. The ROIs for the Burr study are located in the same region as the ones used in the H-scan and attenuation studies in this work to ensure that the analyses are applied consistently. Due to the sensitivity of the Burr parameters to the presence of large inhomogeneous regions, the ROIs were adjusted slightly if necessary to avoid any major vessel or area including large nodules or artifacts.

Figure 8 shows the B-scan with the selected ROI and also the histogram of normalized echo amplitude, backscattered from speckles, fitted to the Burr distribution for one sample frame of a rat liver representing each of four groups. The corresponding $b$ and $\lambda$ parameters of Burr distribution for each case are presented along with the results of fitting goodness as $R^2$ and RMSE in Table 4. The plots of histogram of amplitudes in all four groups show that the Burr model gives a close fit of the pathology of liver tissues in normal, fibrosis with low fat, fibrosis with high fat and also inflammation conditions. Also, the $b$ and $\lambda$ parameters are found to be sensitive to changes in pathological conditions of the liver. The clear trend is increasing in $b$ and $\lambda$ when rat liver condition shifts from normal to abnormal conditions due to fibrosis, fat increase (steatosis) or inflammation. To go into more detail, the $b$ and $\lambda$ increase with the presence of a high stage of fibrosis and also with increase of fat inclusion.

Figure 8.

Top: B-scan image, bottom: Burr fitting to the histogram of normalized echo amplitude for (a) normal rat liver, (b) rat liver with inflammation, (c) low-fat fibrotic rat liver and (d) high-fat fibrotic rat liver. The selected ROIs for Burr analysis are shown as dashed boxes.

Table 4.

Burr-fitting parameters of a rat liver sample from each of four groups along with the goodness of fit parameters.

Dosing	$b$	$\lambda$	R 2	RMSE
Normal	3.06	253	0.997	0.051
Liver with inflammation	3.40	358	0.997	0.043
Low-fat fibrotic	4.60	804	0.995	0.004
High-fat fibrotic	5.41	1013	0.997	0.003

To summarize the results of all the analyses for the 35 rat livers considering the parameter estimates for all the frames for each rat liver, the results are shown as two separate half-violin plots in Figures 9(a) and 9(b) for $b$ and $\lambda$, respectively. The median of the results in each group is marked as a horizontal line in the box and the variation outside the quartiles is shown as dashed whiskers. The significance of the difference between each pair was analyzed by the p-value from the one-way analysis of variance (ANOVA) and the multiple comparison test. The p-value is indicated by asterisk as in Figure 7. Comparing different groups on the boxplots in Figure 9, we observe that the power law parameter $b$ distinguishes between the four groups of rat livers, with the high-fat fibrotic group having the highest values of $b$ than the low-fat fibrotic group and both groups show significant increases in $b$ in comparison to the normal and inflammation groups of rat liver. On the other hand, when looking at the $\lambda$ results, the same trend is observed and the $\lambda$ parameter can discriminate five pairs out of six pairs, however the difference is less obvious due to presence of a few outliers in the low-fat and high-fat fibrotic group. One of the rats in the low-fat fibrotic group shows unusually high values of $\lambda$ in comparison to the other rats in this group. This specific case produced most of the elevated outliers in Figure 9(b). From the histology of this case a hemorrhage in its lung was noted and this might be an indication of additional complications.

Figure 9.

The summary of the Burr parameter estimation results for 35 different rat livers. (a) the boxplot of the power law parameter $b$ and (b) the boxplot of $\lambda$ for four groups of livers. **** indicates statistically there is significant difference, and ns indicates that the difference is not statistically significant.

SVM-based classifier

The SVM-based liver state classifier was implemented, as described in more detail in Appendix B. To build the classifier, the two parameters of box constraint (C) and $\sigma$ were decided according to accuracy and shape of hyperplanes; we found that optimal values of C and $\sigma$ are approximately 50 and 0.7, respectively. The final results with classification accuracy of 94.6% is shown in Figure 10. The details of the SVM optimization procedure are further examined in Appendix B.

Figure 10.

View of clusters and SVM classification. (a), (b), and (c) show groups in 2D parameter space: (a) results of Burr analysis; (b) H-scan and attenuation measures. (c) and (d) show the first two and three principal components (PC) derived from the five parameter analysis: Burr $\lambda$, $b$, H-scan, attenuation, and intensity. (e) and (f) show hyperplanes to separate the liver states in 3D principal component space defined by the SVM-based liver state classifier that were optimized and implemented in this work. (g) represents the misclassified cases. Classification accuracy is 94.6% for the implemented SVM-based liver classifier of this work. All fibrosis and fatty cases were correctly classified. 2 cases of inflammation and 5 of normal cases have misclassified frames; each liver scan has around 30 frames.

To visually examine the hyperplane shapes or clusters of data set, reduced dimensions were considered since the employed features have five dimensions that cannot be visualized in 3D space. The five dimensions were reduced into two or three dimensions using principal component analysis (PCA), as depicted in Figures 10(c) and (d). In order to derive the principal component analysis, a uniform scaling was performed by modifying min-max normalization features (Han et al. 2011). Further details of this normalization and analysis are provided in Appendix C.

DISCUSSION

Our SVM with the optimized parameters classified 998 image frames with the extracted five scaled features, whereby the decision planes that define liver states were produced with 94.6% classification accuracy. Figure 10(g) shows the misclassified cases among 998 data; there are a total of 35 rats, and each rat has around 30 frames. Although two and five rats in inflammation and normal group have misclassified frames, some frames were misclassified, but the others were correctly classified; for example, the rat #1 in normal group has 4 misclassified frames, but 26 frames are correctly classified. In accordance with SVM theory, our SVM classifier allows a small percentage of misclassified training data, resulting in smoother decision boundaries while avoiding overfitting. As shown in Figures 11(c) and (d), we can design a SVM classifier with 100% accuracy, but that would work only for the input of this study; whereas, the purpose of this work is to propose a liver state classifier for any liver, hence we have optimized the classifier until 94.6% of accuracy without overfitting before reaching 100 % of accuracy.

To visualize the clusters of the input data and classification with decision boundaries, PCA reduced the measured five features into two or three parameters since five dimensions cannot be displayed. Therefore, Figure 10 shows view of clusters with hyperplanes, which have loss of information compared to five-dimensional analysis that have been mentioned in previous sections. When using only the two and three components in Figure 10(c) and (d), classification accuracies of SVM are 90.5% and 91.9%, respectively; these are lower than the 94.6% accuracy of five-dimensional analysis due to information loss by PCA. The reduced parameters help us to visualize the real five-dimensional results, but are not sufficient by themselves for the highest accuracy.

Figures 10(a) and (b) show clusters derived by scattering models related to the histogram analysis of echo amplitudes and frequency analysis, respectively: (a) Burr parameters; (b) H-scan and attenuation. Figure 10(a) shows that we can visually separate this space into three regions of normal/inflammation, low-fat fibrotic, and high-fat fibrotic regions with a distinctive area. It is noted that the inflammation and the normal groups seem to have considerable overlap regions when applying only the two-parameter Burr analysis. Although these two are not well-separable visually in Figure 10(a), frequency-dependent studies of H-scan and attenuation estimation in Figure 10(b) can provide better separation between normal and inflammation data. Moreover, Figure 10(b) tends to show separable four clusters with mild overlaps. Since Figure 10(a) tends to more clearly distinguish fibrosis cases from normal than frequency analysis and Figure 10(b) is more likely to discriminate each case from the other groups, combining the results can take advantages of the different methods and compensate for their drawbacks; furthermore, the conventional B-scan intensity is also added as a feature. By including all five measurements, Figure 10(c) shows clusters in reduced 2D space generated by PCA. As expected, based on Figures 10(a), (b), and (c) shows a better separation between liver state groups compared to Figure 10(a) or (b) alone. Furthermore, when considering the first three principal components in Figure 10(d), each cluster is better distinguished from the other clusters compared to 2D space; since three first principal components have 97.7% information from the raw data, but two first principal components have 93.5%. Therefore, it is expected that using 100% information on the raw data can have better discrimination than the view of clusters that is visualized in Figure 10. The decision boundaries in Figures 10(e) and (f) were defined by information from the first three principal components, with 91.9% of classification accuracy. However, the hyperplanes to classify the liver states in 5D space have 94.6% accuracy, whereby these provide better separation between liver states than 3D space; among 998 frames investigated, the founded frames in the bar graph in Figure 10(g) are the only misclassified frames.

In summary, each measurement has its distinct role in distinguishing specific liver states. SVM plays a prominent role in effectively integrating the different five analyses into a combined classification. Consequently, SVM demonstrates that there exist the boundaries that can separate the liver states with 94.6% accuracy, meaning that each group has distinct quantitative characteristics based on the five measurements of this work.

CONCLUSION

In this study of an animal liver model in normal and abnormal states, we employed two relatively new analyses, the H-scan and the Burr distribution approaches. These produced five output parameters linked to ultrasound propagation and scattering models from physics. The parameters were sensitive to changes in liver structures, and formed clusters in five dimensional spaces that enabled a robust classification of individual livers into the diagnostic categories of: normal, fibrosis (low fat), fibrosis (high fat), and inflammation. A support vector machine classification approach was capable of discriminating between groups with a 94.6% accuracy. We believe that this supports the general argument that matching measures to biophysical models of tissues provides the strongest ability to discriminate and classify pathological conditions.

APPENDIX A

The SVM (Cortes and Vapnik 1995; Vapnik 1999; Bishop 2006) is one of the most widely used machine learning methods called a maximum margin classifier, which is used for this study due to several important properties. To maximize the margins between the classes, the SVM considers only subsets of data points located near the class boundaries, which are called as support vectors. It could be an advantage when there are not enough training data set, since the SVM does not care how many data points far from the class boundaries. Another promising property of the SVM is that it is a convex optimization problem, meaning that the solutions can never stuck in a local minimum; in contrast other methods including neural networks are not guaranteed to reach the global maximum or minimum. Lastly, the SVM can be a practical method by allowing some misclassified data for training, and then it minimizes the penalty of the data by introducing slack variables.

When we first consider the classification problem of two classes denoted as $y$ of $\left\{-1,+1\right\},$ let the training set is ${\left\{\left(x^{\left(n\right)},y^{\left(n\right)}\right)\right\}}_{n=1}^N$, where $x^{\left(n\right)}\in {ℝ}^D$ and $N$ is the number of training data. The hyperplane and margin can be described by

$$y^{\left(n\right)}\left(\overrightarrow{w}\cdot {\overrightarrow{x}}^{\left(n\right)}+b\right)-1\ge 0,$$ (A.1)

resulting in the margin size of $2/\Vert \overrightarrow{w}\Vert$. Maximizing the margin can be equivalent to minimizing ${\left(\Vert \overrightarrow{w}\Vert \right)}^2/2$, which is a convex optimization problem with the constraint of eqn (A.1). Now, we consider the misclassified cases in the training set, whereby eqn (A.1) is replaced with

$$y^{\left(n\right)}\left(\overrightarrow{w}\cdot {\overrightarrow{x}}^{\left(n\right)}+b\right)\ge 1-{\xi }_n,$$ (A.2)

where ${\xi }_n$ is called as the slack variables or the penalty. The eqn (A.2) and the condition of ${\xi }_n\ge 0$ become the constraints of the following new convex optimization problem, therefore it is to minimize

$$C\sum _{n=1}^N{\xi }_n+\frac{1}{2}{\Vert \overrightarrow{w}\Vert }^2,$$ (A.3)

where the box constant of $C>0$. The corresponding dual Lagrangian form is obtained by:

$$\underset{a}{\max }{\sum }_n{a}_n-\frac{1}{2}{\sum }_n{\sum }_m{a}_n{a}_m{y}_n{y}_m{\overrightarrow{x}}^{\left(n\right)}\cdot {\overrightarrow{x}}^{\left(m\right)},$$ (A.4)

under a constraint of $0\le a_n\le C$ for Lagrangian multipliers of $a_n$. In general, the set of $a_n$ is obtained from eqn (A.4) by setting the partial derivatives of Lagrangian with respect to $w$ and $b$ equal to zero to maximize eqn (A.4). Then, the decision function for unknown vector $\overrightarrow{u}$ can be described as:

$$\text{sign}\left({\sum }_n{a}_n{y}_n{\overrightarrow{x}}^{\left(n\right)}\cdot \overrightarrow{u}+b\right),$$ (A.5)

where $a_n$ is nonzero for support vectors, meaning that only support vectors contribute to determine decision planes. To consider linearly non-separable data set, kernel function $K\left({\overrightarrow{x}}^{\left(n\right)},\overrightarrow{u}\right)$ was introduced, and then eqn (A.5) can be rewritten as

$$\text{sign}\left({\sum }_n{a}_n{y}_{n}K\left({\overrightarrow{x}}^{\left(n\right)},\overrightarrow{u}\right)+b\right).$$ (A.6)

In this study, the multiclass SVM classifier was implemented by using two class SVM in MATLAB; “one-versus-the-rest approach” was used. In other words, four separate SVMs were constructed by setting one of the classes as a $\left\{+1\right\}$ class and the remaining three classes as a $\left\{-1\right\}$ class, and then training them. Feature scaling was performed for the different scaled inputs. Gaussian kernel was used for linearly non separable data set. Optimization of box constraint and kernel scale were performed.

To evaluate the performance of SVM, classification accuracy was defined as follows:

$$\text{Classification accuracy}=\frac{\text{Correctly classified \#}}{\text{Total \# of data}}\times 100\%,$$ (A.7)

where “Total # of data” is the total number of training data used for SVM with corresponding desired output, that is pathologically confirmed states of liver, and “Correctly classified #” is the count for the cases, of which results from SVM are the same with the pathological result.

APPENDIX B

A procedure including four steps to optimize the parameters is described in Table B.1.

The first step investigates the overall shape of the hyperplane by varying the two parameters in 2D reduced component space, which is shown in Figure B.1(a) and (b). As we can see, the cases on the right side tend to have higher accuracy than the left, but it goes to overfitting; otherwise, the hyperplanes in left side tend to represent under-fitting with low accuracy. By investigating the hyperplane shape dependency on parameters, a prospective optimal parameter set is selected, which is 50 and 0.7 for C and $\sigma$ in Figure B.1. The step 2 considers the classification accuracy using the five features that is the input of SVM in this study; since the five features cannot be visualized in 3D space, step 1 uses the reduced two dimensions.

By varying parameters of SVM, parameter versus accuracy plots are shown in Figure B.1(c), (d), and (e); the selected parameter set is also shown in the plots using red circles. When C and $\sigma$ have large and small values, respectively, it goes to overfitting. Thus, this step checks the red circle representing the prospective optimal parameter set and whether it is located before the saturation in varying C and before unusual rapid increase in accuracy in varying $\sigma$. Step 3 randomly generates train and test set by dividing the 998 sets into two groups; 70% and 30% of the data are set to train and test set. After training SVM using the train set, classification accuracies were calculated for train and test set. After training SVM with the train set, classification accuracies for train and test set were calculated, which was repeated 15 times. The accuracies of train set (94.48 % $\pm$ SD 0.82) and test set (93.86 % $\pm$ SD 1.21) are not significantly different (p = 0.1111), meaning that the selected parameters are reasonable without overfitting; if the accuracy of the test set is much lower than the train set, the training is over-fit to the used train set. As the last step, the shape of the hyperplane generated by the optimal parameters was visualized as shown in Figure 10(e) and (f) to check whether there are overfitted boundaries. These four steps were repeated until the hyperplanes in step 4 are reasonable and accuracies in step 3 are comparable. With the repeated procedure, this study selected the optimal parameter set for the SVM-based liver classifier.

Figure B.1.

SVM parameter (C and $\sigma$) optimization. (a) and (b) illustrate the overall shape of hyperplanes by varying parameters. This parameter selection in 2D space is the step 1 in Table 5. Cases in left and right side tends to be under- and over-fitting, respectively. C = 50 and $\sigma$ = 0.7 can be an optimal parameter set according to the shapes of boundaries. Reduced two dimensional components were obtained by principal component analysis. (c), (d), and (e) show SVM parameters versus classification accuracy plots according to step 2 in Table 5. (c) box constraint versus accuracy, (d) $\sigma$ in kernel function versus accuracy, (e) the two parameters versus accuracy. Classification accuracy was calculated over varying the parameters. Red circles are the selected optimal parameter set for SVM-based liver classifier: C = 50 and $\sigma$ = 0.7. Since the larger C and smaller $\sigma$ represent overfitting, the optimal parameters are located in the position before the accuracy saturation in large C and rapid increase over small $\sigma$.

Table B.1.

Optimal parameter selection for SVM. Box constraint (C) and $\sigma$ of Gaussian kernel is optimized in this procedure, which can result in proper shape of decision planes and high enough classification accuracy without under- or over-fitting (PC = principal components from principal component analysis).

procedure Optimal parameter selection for SVM

repeat

Step 1: (under 2 reduced PC)

- Investigate overall shape of hyperplanes by varying parameters

- Select an optimal set (C, $\sigma$)

Step 2: (under raw 5 features)

- Check the classification accuracy

Step 3: (under raw 5 features)

- for trial = 1 … N do

Randomly divide train (70%) and test set (30%)

Run SVM training

Calculate accuracies for train and test set

- Compare the two accuracies of train and test set

Step 4: (under 3 reduced PC)

- Check shapes of hyperplanes

until Hyperplanes in Step 4 are reasonable AND

Accuracies in Step 3 are comparable

APPENDIX C

The min-max normalization is defined as follows:

$$x_{\text{scale}}=\frac{x-\text{average}\left(x_{\text{NOR}}\right)}{\frac{1.0}{0.9}\cdot \text{max}\left[\left|\text{max}\left(x\right)-\text{average}\left(x_{\text{NOR}}\right)\right|,\left|\text{min}\left(x\right)-\text{average}\left(x_{\text{NOR}}\right)\right|\right]},$$ (C.1)

where $x$ is measured features and $\text{average}\left(x_{\text{NOR}}\right)$ is average of normal cases. The average of normal case in $x_{\text{scale}}$ is set to zero to see how different the abnormal cases are compared to normal cases. Scale is 1.0/0.9 times of our data set to generate decision planes that can cover the cases when the measurements are greater or less than our 998 data set. For instance, in Figure 5, this work determined the decision boundaries based on the data set, and then diagnosing or predicting can work although new measurements are 1.0/0.9 times greater or less. These scaled features from the five measurements become the components of vector X where $\text{X}={\left({\text{x}}_{\text{H-scan}}{\text{,x}}_{\text{Attenuation}}{\text{,x}}_{\text{B-scan}}{\text{,x}}_{\text{Burr}\left(\lambda \right)},{\text{x}}_{\text{Burr}\left(b\right)}\right)}^T$, which are assigned as an input of SVM and PCA. Mathematically, a vector of principal components ${\text{X}}_{\text{pc}}\text{=}{\left({\text{pc}}_{\text{1}}{\text{,pc}}_{\text{2}}{\text{,pc}}_{\text{3}}{\text{,pc}}_{\text{4}}{\text{,pc}}_{\text{5}}\right)}^T$ can be finally calculated by

$${\text{X}}_{\text{pc}}\text{=WX,}$$ (C.2)

where W is a transformation matrix of PCA. In this study, W is 5 × 5 matrix whose components are the weights of the original components when calculating the new principal components. Figure C.1(b) represents X_pc as the data after PCA, of which components have 73.7%, 19.8%, 4.2%, 1.3%, and 1% of retained variance; in other words, the first two principal components have 93.5% information of the input data. Figure C.1(c) shows W with % of contribution for original five measurements. The number written on the bar graph is the components of W. Since the sign represents the relative direction from the normal data in one dimensional space, the absolute values can show the contribution for the principal components. Thus, the percentage of absolute values of each component in W can show the contribution. The five measurements have similar contribution for the first principal component, which is approximately 20%. It is reasonable, since each measurement provides different contributions when distinguishing the four liver states; for instance, when separating normal and fibrosis cases, Burr $\lambda$ is the best; when separating normal and inflammation, H-scan and attenuation work; when separating low and high fat cases, H-scan, attenuation, and Burr b provide important information; when separating inflammation and low fatty fibrosis, B-scan and Burr $\lambda$ are critical, etc.

Figure C.1.

PCA (a) Measured features, of which components are obtained by H-scan, attenuation estimation, B-scan, and Burr histogram analysis. (b) Input features for SVM, which are calculated by PCA. The first principal component has 73.7 % of variance retained. (c) Contributions of the five raw measured data in (a) to generate PCA features in (b).