Ultrasound Structure Function for Quantitative Assessment of Steatotic Liver Disease Based on Total Severity Score

Abstract

Quantitative ultrasound (QUS) parameters offer effective tools to evaluate microstructural changes in liver tissue and noninvasively assess tissue health. We examined the relationship between steatotic liver disease severity, measured by the Total Severity Score (TSS), and the structure function (SF), a frequency-dependent ultrasound backscatter metric. TSS was calculated on a 0–14 scale from histological evaluation by a pathologist of fibrosis, inflammation, and steatosis in hematoxylin and eosin (H&E)-stained liver slides. A total of 274 regions of interest (ROIs) were examined from histopathology liver specimens acquired from 49 human subjects. SF was calculated for every ROI over a spatial frequency range of 3–80 MHz using histology-determined cell nuclei positions. Results revealed a strong relationship between SF and TSS across all analyzed ROIs, and distinct frequency-dependent patterns emerged across TSS groups. Statistical analysis showed significant differences between TSS 0 and TSS 1–3 for the frequency range of 4.5 to 9.0 MHz, and between TSS 1–3 and TSS 4–6 for the frequency ranges of 3.0 to 9.0 MHz and 32.5 to 80.0 MHz. TSS groups 4–6 and 7–14 did not differ significantly, indicating a possible plateau in ultrasound response at later disease stages. These findings help lay the foundation for establishing SF as a potential ultrasonic biomarker for identifying and distinguishing steatotic liver disease stages in vivo.

I. INTRODUCTION

Quantitative ultrasound (QUS) has shown potential to characterize tissues in various organs such as blood (Savery and Cloutier, 2005), breast (Argueta-Lozano, et al., 2023; Oelze and O’Brien, 2006), lung (Dashti et al., 2024), liver (Han et al., 2018a; Han et al., 2018b; Han et al., 2020a; Han et al., 2020b; Pirmoazen, et al., 2022), cervix (McFarlin, et al., 2023; McFarlin, et al., 2024), and lymph nodes (Mamou, et al., 2011). QUS techniques extract intrinsic features of the tissue by minimizing system- and operator-dependent effects from the radiofrequency (RF) ultrasound data (Han, et al., 2017; Yao et al., 1990). Among these, the backscatter coefficient (BSC) represents a fundamental parameter that contains rich information about tissue microstructure (Han, 2023; Wear, et al., 2022). Accurate modeling of the BSC could lead to further successful application of QUS techniques.

Within the context of discrete scatterer models, the BSC is determined by both the properties of individual scatterers modeled by the form factor (FF) (Insana, et al., 1990) and the spatial correlation of scatterers modeled by the structure function, SF (Savery and Cloutier, 2005; Twersky, 1987; Twersky, 1988; Fontaine, et al., 1999; Franceschini, et al., 2013; Franceschini and Guillermin, 2012; Franceschini, et al., 2014; Han and O’Brien, 2015; Han and O’Brien, 2016; Han, 2018; Tran, et al., 2024). FF describes how the shape and acoustic properties of individual scatterers contribute to the overall scattered signal. The SF describes how the interference among scatterers contributes to the overall scattered signal. SF-based ultrasound backscatter modeling has been evaluated across multiple pathological contexts beyond the liver, most notably in blood where red-cell aggregation alters spatial correlations (Savery and Cloutier, 2005), in tumor microstructure and cancerous lymph nodes (Oelze and O’Brien, 2006; Mamou, et al., 2011; Tran, et al., 2024) and directly from histology-derived nuclei maps (Han and O’Brien, 2016; Han, 2018). Collectively, these studies support SF as a biologically interpretable descriptor of tissue micro-architecture that complements FF analysis. However, to our knowledge, no prior work has explicitly tested whether histology-derived SF tracks a total hepatic disease severity metric like Total Severity Score (TSS). This study is the first to quantify and statistically validate a correlation between SF and TSS, demonstrating a link between frequency-dependent SF and clinically graded steatotic liver disease by a board-certified pathologist. Independently examining the contributions of the two components, FF and SF, may be helpful for elucidating the ultrasonic scattering mechanisms in tissue, which may provide theoretical foundations for the clinical application of QUS and guide development of more accurate and precise model-based QUS techniques for diagnosis.

Liver fat quantification is a prominent recent success in the clinical application of QUS. Human studies have demonstrated strong correlations between the BSC and liver fat content (Lin, et al., 2015; Han et al., 2020b) as well as the repeatability and reproducibility of practical in vivo measurement of the BSC (Han et al., 2018a; Han, et al., 2018b; Han, et al., 2019); however, the mechanisms for the correlation between the BSC and fat content remain undefined. A clear understanding of the mechanisms by which the emergence and growth of liver fat droplets as well as other biological changes (e.g., inflammation and fibrosis) affect the BSC may not only guide the effective use of the QUS technique in determining the severity of fatty liver disease but may also lead to further refinement of the technology.

It has been hypothesized that two independent mechanisms contribute to the correlation between the BSC and liver fat fraction (Wu, et al., 2020): 1) fat droplets are acoustic scatterers in the liver and higher fat content causes an increase in scattering, and 2) hepatocyte nuclei are additional acoustic scatterers. Fat droplet deposition in hepatocytes displaces the nuclei within the hepatocytes, changing the spatial distribution of the nuclei, which alters the SF of the nuclei and hence changes the BSC. Other pathological changes, such as acute and chronic inflammation with infiltration by inflammatory cells into the liver, also affect the spatial distribution of hepatocyte nuclei (Zhou, et al., 2020; Sergi, 2024). They may also contribute additional changes in SF due to their unique physical characteristics and thereby produce additional backscatter effects.

While both mechanisms, FF and SF, may be important, this study focuses on investigating the second mechanism, examining how liver fat droplets and other histological features change the SF of hepatocyte nuclei. Digitized hematoxylin and eosin (H&E)-stained histopathological liver slides were used in this study. The images were systematically evaluated by a pathologist and were independently analyzed to obtain the SF of hepatocyte nuclei. The comparison between the histopathological score and the SF provides insight into the connection between histology and ultrasound.

II. MATERIALS AND METHODS

Important nomenclature note:

In June 2023, a multi-society group recommended nomenclature changes, suggesting “Steatotic Liver Disease” (SLD) as the overarching term for all forms of hepatic steatosis and “Metabolic dysfunction-Associated Steatotic Liver Disease” (MASLD) as the replacement term for Nonalcoholic Fatty Liver Disease (NAFLD) (Rinella, et al., 2023). Prior to June 2023, NAFLD and Nonalcoholic-related Steatohepatitis (NASH) were the recommended nomenclatures. Since June 2023, MASLD and Metabolic dysfunction-Associated Steatohepatitis (MASH) are the recommended nomenclatures. Also, NAS (NAFLD Activity Score) (Kleiner, et al., 2005) was assessed by the pathologists. Herein, we will adhere to the newly recommended MASLD and MASH terminology.

A. Histology Assessment and Grading

Routine hematoxylin and eosin (H&E)-stained slides (5-μm thick sections embedded in paraffin, digitized at 40X magnification) of needle core liver biopsies of the right lobe were collected from deidentified patients with MASLD/MASH (Table I). The Human Subjects Institutional Review Board (IRB) at the University of California, San Diego determined the research protocol did not meet the definition of human subjects research since the data were not traceable to the patients, thus did not require IRB review. Nonalcoholic Steatohepatitis Activity Scores (NAS) of varying severity, were assigned for each slide by qualified MD pathologists as part of the patients’ clinical care. The slides were separately evaluated by the research pathologist (MAW) and scored using four modified and expanded criteria from the accepted MASLD/MASH scoring system (Kleiner, et al., 2005; Onyekwere, et al., 2015). For each needle biopsy slide, 500 μm × 500 μm square Regions Of Interest (ROI) with no slide processing artifacts and completely occupied by hepatic tissue (i.e., no “empty space”), were selected for scoring by a member of the research team (WDO). The research pathologist who was not part of the original clinical pathologist group providing the NAS, evaluated a total of 274 ROIs obtained from 49 slides with NAS of 0 (72 ROIs), 1 (19 ROIs), 2 (52 ROIs), 3 (33 ROIs), 4 (48 ROIs), 5 (36 ROIs), and 6 (14 ROIs).

TABLE I.

Study Enrollment Information.

Age (Mean ± SD)	Gender	Ethnicity
52.0 ± 14.4 years	Male: 17 Female: 22 Not Reported: 10	Asian: 5 White: 24 Black/African American: 1 Unknown/Not Reported: 19
Total	49	49

The histopathology scoring system developed for this evaluation was based on a refinement of four basic histologic features of MASLD/MASH from the Kleiner system (Kleiner, et al., 2005). The four criteria - (extent of steatosis, acute inflammation, chronic portal inflammation, and fibrosis) were hypothesized to have significant impacts on ultrasound SF in relation to the development of MASH. After assigning a severity score (for each of the four criteria, a Total Severity Score (TSS) was determined by adding the four criteria scores for the ROI (Table II). For example, a ROI with a steatosis score of 1 (<5% hepatocytes with lipid vacuoles), an acute inflammation score of 2 (neutrophils in 5–10%), a portal inflammation score of 2 (5–10%), and a fibrosis score of 1 (<5%) would have a TSS of 6 (1+2+2+1).

TABLE II.

Study scoring parameters for calculating Total Severity Score (TSS) within each Region Of Interest (ROI) to determine the impact of steatosis, inflammation and fibrosis on hepatic Structure Function (SF).

Total Severity Score (TSS) = steatosis + acute inflammation + chronic inflammation + fibrosis scores added to yield TSS
Score	Steatosis- Extent of steatosis within ROI
0	<5% hepatocytes with lipid vacuoles
1	5–33% hepatocytes with lipid vacuoles
2	33–67% hepatocytes with lipid vacuoles
3	>67% hepatocytes with lipid vacuoles
Score	Acute Inflammation- Extent of neutrophils infiltration within an ROI
0	No inflammation
1	<5% of the ROI involved
2	5–10% of the ROI involved
3	10–25% of the ROI involved
4	25–50% of the ROI involved
5	>50% of the ROI involved
Score	Chronic Portal Inflammation- Extent of infiltration by mixed inflammatory cells (lymphocytes. macrophages and neutrophils) infiltrating portal areas and extending into the hepatic lobule
0	No inflammation
1	<5% of the ROI involved
2	5–10% of the ROI involved
3	10–25% of the ROI involved
4	25–50% of the ROI involved
5	>50% of the ROI involved
Score	Fibrosis – Extent of the ROI deposition collagen with or without fibroblasts and inflammatory cells present
0	No fibrosis
1	<5% of the ROI involved
2	5–10% of the ROI involved
3	10–25% of the ROI involved
4	25–50% of the ROI involved

QuPath^® imaging (version 0.2.3; an open-source software for bioimage analysis; https://qupath.github.io/) was used to map nuclei coordinates within each ROI, from which SF was calculated. As shown in Fig. 1, for each 500 μm × 500 μm square ROI, QuPath^® yields the centroid coordinates of each detected nucleus encircled in red. For the 274 ROIs, the number of detected nuclei (N) ranged between 357 and 2107 for the 49 needle biopsy slides. The coordinates of the detected nuclei were used to calculate SF from the histology within the ROI.

FIG. 1.

Red outlines surrounding nuclei are QuPath determined. Lower left scale equals 10 μm.

B. SF vs Frequency Curve

A SF versus frequency curve was calculated for each ROI within the frequency range of 3–80 MHz. The SF of $\text{N}$ scatterers within an isotropic medium is defined as (Han, 2023)

$$\mathit{\text{SF}}\left(\mathbf{K}\right)=\frac{1}{N}E\left[{\left|{\sum }_{m=1}^N{e}^{-j\mathbf{K}\cdot {\mathbf{r}}_m}\right|}^2\right]$$ (1)

where $E\left[\right]$ represents ensemble averaging, ${\mathbf{r}}_m$ is the centroid position of the $m^{\mathit{\text{th}}}$ scatterer and $\mathbf{K}$ is backscattering vector whose magnitude is given by $\left|\mathbf{K}\right|=2k\text{sin}(\theta /2)$, where $k$ is the propagation constant ($2\pi f/c$ where $f$ is frequency and $c$ is the propagation speed), $\theta$ is the scattering angle ($\theta =\pi$ for backscattering). The 2-D form of Equation 1 is used to numerically calculate the SF for each ROI based on the nuclei centroid positions obtained using QuPath (Fig. 1). The SF describes the phase interference in acoustic scattering caused by the spatial distribution of discrete scatterers. A unity SF represents uncorrelated random scatterer positioning. SF values greater than unity represent constructive interferences, and the values less than unity represent destructive interferences due to the scatterer positioning (Han and O’Brien, 2016).

For each region of interest, we assume isotropy in the lateral plane and estimate a one-dimensional, angle-averaged SF that depends only on the wavenumber magnitude. We first recenter the region at its centroid and apply a circular window whose radius is 250 $\mu m$, retaining only the points inside this disk to reduce directional edge effects. We then select a grid of spatial frequencies between the defined minimum (3 MHz) and maximum (80 MHz) and convert each frequency to the corresponding two-way (round-trip) acoustic wavenumber using the assumed speed of sound (1540 m/s). For each wavenumber, we uniformly sample a set of angles (720 angles uniformly sampling $\left[\mathrm{0,2}\pi \right))$ to define plane-wave directions around the full circle. Along every combination of wavenumber and angle, we compute the complex spatial Fourier sum of the point set in that direction and form its power spectrum by taking the squared magnitude normalized by the number of retained points. Because the biopsy section represents a random orientation of the underlying 3D tissue relative to the ultrasound beam, any given direction in the 2D histological plane may correspond to a backscattering configuration in vivo. Therefore, rather than selecting a single direction, we compute SF over a dense set of angular directions (720 angles) and average the results to obtain an orientation-independent and more robust estimate of the spatial correlation properties. The use of multiple angles is intended to account for the unknown orientation of tissue microstructure relative to ultrasound insonification and to improve the statistical stability of the SF estimates. Under the isotropy assumption, the SF at a given wavenumber is obtained by averaging these power spectrum values over all angles.

C. Statistics

Statistical analyses were performed on SF versus frequency data 1) to test whether groups of SF versus frequency curves were statistically significantly different from each other, and 2) find the frequency range within which various groups of SF curves are distinguishable from each other. The data being analyzed, the SF versus frequency, represents a unique type of data due to its functional nature (SF is a function of spatial frequency). Therefore, the analyses of this type of data required specialized statistical tools that properly consider the functional relationship. A method that is capable of handling functional data (Park and Han, 2026) was used to perform these statistical analyses. Specifically, the statistical method considered not only the magnitude of the SF versus frequency curves but also their first- and second-order derivatives to capture the trajectories of the SF curves. This inference method, based on SF trajectories, provides p-value functions, and we apply a significance level of 0.05 to draw an inferential conclusion. This approach is preferred over statistical tests based on only the magnitude of the SF. Further details about this method are described in Park and Han (Park and Han, 2026).

III. RESULTS

Fig. 2 provides a generalized understanding of the relationship between SF and TSS from the 274 ROIs, for which the SF versus frequency curves shown for 10 TSS curves, 9 of them representing individual TSS values between 0 and 8 and the tenth representing the average TSS 9–14. An important observation is the relatively strong SF relationship in the ~20–60 MHz frequency range, whereby the SF values tend to increase as the TSS values decrease, suggesting a relationship between SF and TSS.

Fig 2:

Average SF of all the patients in each of the TSS levels against frequency. TSS levels are: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 1 group: 9–14.

To further illustrate this SF-TSS relationship, the SF versus frequency curves have been grouped into four TSS ranges: TSS=0 (TSS 0), TSS between 1 and 3 (TSS 1–3), TSS between 4 and 6 (TSS 4–6), and TSS between 7 and 14 (TSS 7–14) (Fig. 3). Visual inspection suggests that the four TSS groups yielded different SF curves, and the SF appeared to decrease as TSS increased within the approximate frequency range of 20 to 60 MHz. Formal statistical testing showed a statistically significant difference between TSS 0 and TSS 1–3 for the frequency range of 4.5 to 9.0 MHz, and between TSS 1–3 and TSS 4–6 for the frequency ranges of 3.0 to 9.0 MHz and 32.5 to 80.0 MHz (Table III). No statistical difference was found between TSS 4–6 and TSS 7–14, possibly due to the relatively smaller sample size of the TSS 7–14 group.

Fig. 3.

Average SF versus frequency for four TSS groups: <0>, <1–3>, <4–6>, and <7–14>.

TABLE III.

Results of statistical testing for pairs of TSS groups: The left column shows the pairs tested, and the right column shows the frequency ranges within which a statistically significant difference was found.

Pairs	Frequencies showing a significant difference
TSS 0 vs. TSS 1–3	4.5 – 9.0 MHz
TSS 1–3 vs. TSS 4–6	3.0 – 9.0 MHz and 32.5 – 80.0 MHz
TSS 4–6 vs. TSS 7–14	None

Inflammation causes substantial alterations in normal hepatic morphology that may alter SF. In acute inflammation, neutrophils accumulate in small to large aggregates among injured and dead hepatocytes. Neutrophils are substantially smaller than hepatocytes, with indistinct cell borders and small, dense convoluted nuclei. In chronic inflammation, lymphocytes are often abundant and have small, spherical, highly condensed nuclei and little cytoplasm, while macrophages have somewhat larger, ovoid nuclei with dispersed chromatin and sparse cytoplasm. Fibroblasts, collagen-producing cells that accompany chronic inflammation, have elongated nuclei and sparse cytoplasm. Unlike acute inflammation, in which neutrophils predominate and in mild cases infiltrate rather than distort the lobule, chronic inflammation usually radiates outward from portal triads and progressively impinges on the lobule from the periphery, often accompanied by the deposition of collagen (fibrosis). These three processes may impact SF in different ways, due to their differing cellular and stromal composition. Fig. 4e illustrates the disruptive impact portal inflammation has on morphology while Fig. 4f illustrates the disruption of normal morphology due to both severe portal inflammation and fibrosis.

Fig. 4.

a) TSS 0 and b) TSS 0 for both steatosis and inflammation: Both a) and b) are normal livers composed almost entirely of relatively uniformly spaced hepatocytes of uniform size containing abundant cytoplasm. The hepatocytes contain circular, basophilic nuclei of uniform morphology (black arrows). Stroma is sparse and not generally discernible. Sinusoids surrounding plates of hepatocytes are represented by clear spaces (yellow ellipses). Hepatocyte nuclei (black arrows) are identified in all six panels c) TSS 2: with a mixture of large (red stars) and small (red arrows) vacuoles (clear spaces) only affecting some hepatocytes, a small degree of nuclear displacement and minimal hepatocellular distortion. d) TSS 3: A moderately steatotic liver (with numerous large vacuoles) and mild acute inflammation in the TSS 0–3 group for inflammation (black circles). Small clusters of acute inflammatory cells with small irregular nuclei and sparse indistinct cytoplasm are scattered among the hepatocytes. Despite the steatosis and scattered clusters of inflammatory cells that add to the total number of nuclei, overall hepatic structure is relatively intact. e) TSS 6: Moderately inflamed liver with minimal steatosis but substantial alteration in regular hepatic structure, illustrating chronic portal inflammation (green polygon) and characterized by numerous chronic inflammatory cells running diagonally within the central portion of the ROI. These cells greatly expand the normally small and indistinct portal regions. The numerous densely packed inflammatory cells have sparse cytoplasm, indistinct cell borders and small dark condensed round or oval nuclei typical of chronic inflammatory cells. Bile ducts are trapped within these areas (red ellipses). f) TSS 14: Severe chronic inflammation (blue circle) and severe fibrosis (yellow polygon) with entrapped dilated bile ducts (red ellipses) are noted in this ROI. Normal hepatic arrangement is severely disrupted and there is loss of hepatocytes and their nuclei, replaced by more numerous inflammatory, biliary and fibroblastic cells.

Typical morphologic alteration for ROIs in the TSS 1–3 groups are shown in Figs. 4(c)–4(d). In ROIs with TSS 1–3, there generally was variable steatosis. Acute inflammation was present in all ROIs and in the vast majority was minimal to mild (TSS 1). No chronic inflammation or fibrosis was observed. Steatosis was variable and generally mild. Fig. 4c (TSS 2) illustrates a subtly inflamed, mildly steatotic liver whereas Fig. 4d (TSS 3) shows a more substantial steatosis and mild acute inflammation.

In TSS 4–6 ROIs, as typified by Fig. 4e, acute and chronic inflammation scores increased as TSS increased, with higher severity scores (TSS 5 & 6). Portal inflammation was absent from most ROIs in the TSS 4 groups, but increasingly present in the TSS 5 groups and frequent and extensive in the TSS 6 groups. Fibrosis scores showed an almost identical pattern of increasing severity among TSS groups as chronic portal inflammation. Steatosis was variable but relatively high (Steatosis scores 2–3) in slightly over half the ROIs from the TSS 4 & 5 groups, but the reverse was true for the ROIs with TSS scores of 6, where slightly more than half the ROIs had steatosis scores of 0–1. The morphological changes in this TSS grouping may have a significant effect on SF.

Among the TSS 7–14 ROIs (see Fig. 4f), acute inflammation scores ranged from 0–5, but only a few had scores of 0. The ROIs herein also had high chronic portal inflammation. Fibrosis scores followed chronic portal inflammation scores with regards to TSS distribution. For steatosis, ~60% of the scores were 0–1 and the rest 2–3. The severe alterations in morphology in this grouping could have a substantial impact on SF.

IV. DISCUSSION

SF is affected primarily by spatial distribution of scatterers, and the FF is primarily determined by nuclei size, morphology and composition. The underlying structural basis for this is not entirely clear but may be related to the general biochemical/biophysical composition of the nuclei, as manifested by cellular characteristics on H&E-stained slides. Nuclei are filled with highly negatively charged, long-chain macromolecules, mainly DNA and mRNA. Furthermore, the DNA that is not being actively transcribed (i.e., unwound), is tightly condensed and compact, and visible histologically as intensely basophilic (dark blue) chromatin. The nucleolus, where DNA is transcribed, is packed with mRNA, and is also negatively charged. This negatively charged environment is highly attractive to hematoxylin, which is positively charged (Kiernan, 2018). The densely packed nature of large nuclear macromolecules is presumed to be sufficient to produce significant ultrasound backscatter and contribute to a characteristic SF curve. The cytosol, on the other hand, as well as organelles such as mitochondria, secretory granules, microfilaments, microtubules, cilia and other structural elements, are primarily proteinaceous, and overall tend to be positively charged. Hence, the negatively charged histologic dye, eosin, readily binds to them (Chan, 2014). Unlike DNA, these proteins are usually much less densely packed and more dispersed, and exist suspended within an aqueous milieu, the cytosol, presumably contributing less to backscatter and minimal impact on SF.

While the above discussion highlights the role of nuclei as a dominant and quantifiable component in this modeling framework, it is important to note that ultrasound backscatter in liver tissue is influenced by multiple structural features. Pathological processes such as steatosis, inflammation, and fibrosis alter hepatocyte size, intracellular composition, spatial organization, and extracellular matrix structure, all of which can affect acoustic scattering. In this study, nuclei are used as a practical and quantifiable estimation for scatterer spatial distribution derived from histological images. Therefore, the resulting structure function (SF) should be interpreted as reflecting the combined effects of these microstructural alterations, rather than being attributed solely to nuclear properties.

Structure Function (SF) and the Total Severity Score (TSS) show a statistical correlation in this study. The SF represents underlying alterations in tissue microstructure that impact ultrasonic backscatter, while the TSS score records progressive tissue degeneration. Our findings support the idea that SF is sensitive to the cellular and architectural abnormalities that accompany hepatic disease by showing that higher TSS groups consistently exhibit lower SF values, especially in the mid to high frequency ranges (30–80 MHz). Specific changes in tissue organization, such as inflammatory cell infiltration and collagen deposition, are visible in the histological pictures and TSS-based classifications. Higher SF values are found in tissues with lower TSS (such as TSS 0), where the cellular architecture is mostly unaltered. On the other hand, the buildup of fat, inflammatory/immune cells, and fibrotic material likely lead to increased ultrasound backscattering and attenuation but decreased SF values. Consequently, results imply that SF may be a valuable ultrasound biomarker for assessing liver pathology and that it provides information about the density and distribution of scatterers like nuclei and vacuoles. For inflammation/fibrosis, the accumulation of inflammatory cells theoretically should have a significant impact on QUS SF scores. Inflammatory cells – neutrophils in acute inflammation and lymphocytes, macrophages and fibroblasts in chronic inflammation, have different nuclear characteristics and far less cytoplasm than hepatocytes. Neutrophils have small, very irregular very condensed nuclei and very indistinct cytoplasm; lymphocytes have small spherical and highly condensed nuclei with little cytoplasm; macrophages have larger (but still relatively small) ovoid nuclei with less condensed chromatin and sparse cytoplasm with indistinct borders; fibroblasts have elongated nuclei and sparse cytoplasm. Inflammatory cells also tend to cluster in variably sized aggregates; in the case of acute inflammation, generally within the hepatic lobule, while in chronic inflammation, in portal triad areas (but they may extend between lobules). Mild acute inflammation appears to have minimal impact on SF, most likely because it occupies a small portion of the ROI, the aggregates are small and unless extensive, do not distort overall lobular spatial arrangements. Chronic inflammation, on the other hand generally involves more cells in larger aggregates that often impinge on surrounding lobules, impacting SF curves. Furthermore, chronic inflammation by its nature generally involves deposition of collagen (fibrosis), especially with extensive and prolonged inflammation. Collagen deposition has an effect on SF as well, replacing hepatocytes in regular lobular array with more condensed collagenous stroma; in chronic cases, even inflammatory cells are excluded, and the affected tissue becomes relatively acellular. The progressive chronic inflammation/fibrosis effect has a substantial impact on SF in the higher TSS scoring groups. Steatosis has a significant although less clear-cut effect on SF. Its impact appears to be most substantial in moderately affected ROIs, that is, ROIs with some degree of moderate inflammation (i.e., the TSS groups 4–6 severity cluster.

The statistical analysis verifies that variations in SF between TSS groups are noteworthy in both the high-frequency (32.5–80 MHz) and low-frequency (3–9 MHz) regions. Pairwise comparisons between the early and intermediate TSS groups (TSS 0 vs. TSS 1–3, and TSS 1–3 vs. TSS 4–6), particularly at the previously described frequency bands, demonstrated statistically significant differences in SF, as indicated in Table III. This implies that SF can distinguish between different stages of disease progression in addition to being sensitive to the presence of pathology.

Furthermore, information about the underlying tissue characteristics can be observed from the shape of the SF vs. frequency curve itself. A nonlinear, U-shaped response was displayed by all TSS groups, with the SF decreasing to a minimum at 20–30 MHz and then increasing at higher frequencies. The magnitude of separation in the 3–9 MHz frequency range is smaller compared to higher frequencies, which may limit visual discrimination in Fig. 2 and lead to the impression of reduced sensitivity. This behavior is consistent with the expected frequency dependence of scattering, where lower frequencies are less sensitive to fine-scale spatial organization due to the larger acoustic wavelength. In contrast, higher frequencies with much shorter wavelength can probe cellular-scale features more effectively, resulting in stronger differentiation between TSS groups.

Within this framework, SF provides a frequency-dependent tissue health indicator that quantitatively reflects cumulative morphological changes within the liver. SF can effectively distinguish between tissues that are normal, slightly altered, and seriously diseased. The potential to identify, noninvasively, these changes with ultrasound has significant ramifications for the evaluation of liver disease. For example, SF analysis may facilitate the early identification of inflammation or steatosis prior to the histological prominence of fibrosis, allowing for earlier treatment approaches and possibly allow monitoring of treatment response. The results suggest that SF is a promising quantitative marker for histopathological severity. Its relationship with TSS is both statistically significant and physiologically interpretable, providing a foundation for future applications in noninvasive liver disease assessment.

V. CONCLUSION

Our results demonstrate that SF can effectively encapsulate microstructural changes in liver tissue like steatosis, inflammation and fibrosis by showing a strong correlation between SF and TSS. Our findings highlight the potential of SF as a noninvasive tool for the diagnosis of steatotic liver disease.

APPENDIX

Functional data analysis:

To compare SF variations across four TSS groups and identify frequencies showing a significant difference among them, we treat individual SF as a functional observation and apply the domain selection method within the functional Analysis of Variance (fANOVA) framework (Zhang and Liang, 2014). Using the general notation, let $X_{i1}\left(t\right),X_{i2}\left(t\right),\dots ,X_{i{n}_i}\left(t\right),i=1,2,\dots ,k,$ denote k groups of functional trajectories collected over a given finite interval. We assume these trajectories are i.i.d. samples from $\mathit{\text{SP}}\left({\mu }_i,\gamma \right),$ a stochastic process with a mean function ${\mu }_i\left(t\right)$ and covariate function $\gamma \left(s,t\right)$. In functional data, discretized observations are regarded as realizations of smooth functions, thereby preserving smoothness and inherent continuum properties. This ensures that inference primarily reflects the underlying functional behavior without being misled by noise or discretized sampling. Under this setting, the equality of the k group location parameters is tested through fANOVA, where the null hypothesis is written as $H_0:{\mu }_1\left(t\right)={\mu }_2\left(t\right)=\cdots ={\mu }_k\left(t\right)$.

To further detect frequencies that differ significantly among groups, interval-wise testing (Pini and Vantini, 2017) is adopted within the fANOVA framework. The key idea is to obtain the p-value function p(t) from interval-wise fANOVA, where frequencies with $p\left(t\right)<\alpha =0.05$ are identified as regions of significant difference. In this detection procedure, both the original trajectories and their first derivatives are analyzed simultaneously to capture distinct group differences that may arise in either feature.

Interpreting variability and reliability:

In statistics, a reliable estimate typically refers to a consistent estimate that converges to the true parameter as the sample size increases, accompanied by a low standard error. And by the Central Limit Theorem, the pointwise sample is already known to be a consistent estimator. To the best of our knowledge, there is no objective criterion for determining the minimum sample size in estimation problems. However, in inferential contexts, the minimum sample size is often understood as the number of observations required to achieve a desirable level of statistical power. While power analysis for ANOVA in multivariate data settings is relatively well-established, analogous studies for functional data, particularly under the domain selection framework, have not yet been developed. Nevertheless, group-wise average functions, along with their 95% confidence intervals, provide valuable insight into how different sample sizes may influence inferential conclusions. We note that domains with nonoverlapping confidence intervals roughly indicate frequencies of significant distinction. In our data, mean functions of TSS4–6 and TSS7–14 are quite close, and their confidence intervals largely overlap. We conjecture that even with the narrower confidence interval (resulting from the increased sample size of TSS7–14), the two groups would not be clearly separated. Accordingly, the current sample size and resulting inferential conclusion appear reasonable when considering both the observed group differences and the associated confidence intervals.

Connecting SF to BSC Data:

A scattering model is used to link the measured backscatter coefficient (BSC) to the structure function (SF) through an effective scatterer description. The formulation includes extracting quantitative parameters from BSC using a fluid-filled sphere model-based form factor and effective acoustic concentration (EAC) (Han, 2023). The SF is calculated using processed patient Backscatter Coefficient (BSC), Effective Acoustic Concentration (EAC) and Effective Scatterer Diameter (ESD). The following scattering model is used to derive the SF values from BSC data.

$$\mathit{\text{BSC}}=\mathit{\text{Rayleigh}}\mathit{\text{Scattering}}*\mathit{\text{Form}}\mathit{\text{Factor}}*\mathit{\text{SF}}$$ (A1)

$$(1/9)*{(2*\pi *f/c)}^4*{(ESD/2)}^6*EAC$$ (A2)

$$\mathit{\text{Form}}\mathit{\text{Factor}}={\left[\frac{3}{2\mathit{\text{ka}}}{j}_1\left(2\mathit{\text{ka}}\right)\right]}^2$$ (A3)

$$a=\frac{\mathit{\text{ESD}}}{2}$$ (A4)

Ultrasound Data Acquisition:

The data used to generate Fig. 5 and Fig. 6 were acquired separately and is part of a separate study (Han et al., 2020b). All participants underwent US liver examinations by using a US system (Siemens S3000; Siemens Healthineers) with a 4C1 (1–4 MHz nominal) transducer and/or 6C1HD transducer (1–6 MHz nominal) by one or two of six registered diagnostic medical sonographers. To eliminate potentially confounding physiologic effects on quantitative US data, participants were asked to fast for 4 hours prior to the US examinations. Each participant underwent at least one but up to four same-day examinations. During each examination, a sonographer made at least 10 data acquisitions in the same location in the right liver lobe by using a lateral intercostal approach. Participants were positioned in the dorsal decubitus position with the right arm at maximum abduction. Before the first data acquisition, system settings were adjusted for each participant to optimize right hepatic lobe visualization and to identify a region of the parenchyma without major vasculatures. Settings remained constant for the subsequent acquisitions in the examination. Each acquisition consisted of a single operator button press that recorded a B-mode image and the underlying radiofrequency data. Acquisitions were repeated during separate shallow expiration breath-holds separated by about 15 seconds. After completion of the liver acquisitions, a calibrated reference phantom (CIRS, Norfolk, Va) with known AC and BSC was imaged by using the same method without changing the settings. The reference phantom technique was used to compute the BSC (Yao et al., 1990).

Fig. 5.

Processed BSC (1/cm-sr) data from the liver data of 36 patients.

Fig. 6.

Calculated SF data using processed liver data of 36 patients.

DATA AVAILABILITY

The data that supports the findings of this study are available from the corresponding author upon reasonable request.