Liver cancer is one of the leading causes of cancer-related deaths globally, with incidence and mortality rates steadily increasing (1, 2). Hepatocellular carcinoma (HCC) is the most common form of primary liver cancer, accounting for 90% of cases, while intrahepatic cholangiocarcinoma (ICC) makes up 10-15% (1). Compared to HCC, ICC is more aggressive and has a higher potential for metastasis, resulting in differences in treatment and prognosis (3, 4). On dynamic contrast-enhanced computed tomography (CT) or magnetic resonance imaging (MRI), ICC can present imaging features similar to those typical of HCC, making differentiation complex and time-consuming, even for experienced specialists (5, 6). Additionally, the use of contrast agents is costly and may be contraindicated in certain patients (7). A biopsy is also a common method for distinguishing pathological subtypes of liver cancer. However, this approach is invasive and has several drawbacks, including low patient compliance and a high rate of complication. Thus, the development of a non-invasive technique to accurately differentiate pathological subtypes of liver cancer remains a significant challenge.

DWI, the pioneering diffusion imaging technique employed in clinical practice, quantifies the extent of restricted diffusion motion of water molecules through the quantitative parameter known as the apparent diffusion coefficient (ADC) (8). Study has demonstrated that ADC can serve as a potential surrogate imaging biomarker for distinguishing HCC, ICC, and metastatic cancer (9). However, DWI assumes a Gaussian distribution of the diffusion motion of water molecules in biological tissues, which makes it unable to fully capture the non-Gaussian characteristics of water diffusion in complex subcellular microstructures, which in turn leads to limitations in the accuracy of ADC value assessment (10). To obtain more accurate information on water diffusion and to map tissue microstructure, researchers have developed non-Gaussian mathematical models based on high b-value DWI, such as the continuous-time random walk (CTRW) (11, 12). The CTRW provides three parameters: diffusion coefficient (D), temporal diffusion heterogeneity (α), and spatial diffusion heterogeneity (β). The D describes the non-Gaussian diffusion behavior in biological tissues, while α and β are related to temporal and spatial diffusion heterogeneity, respectively. Both parameters may reflect different aspects of tissue structural heterogeneity within the voxel (13). Currently, CTRW has demonstrated significant potential in distinguishing benign from malignant breast lesions, identifying pathological subtypes, and evaluating prognosis (14–16). However, reports on the use of CTRW to assess the severity of liver fibrosis and liver cancer metastasis are limited (12, 17). Various DWI techniques encounter the challenge of significant overlap between diffusion signals due to the mixed signals from intracellular and extracellular water molecule diffusion (18). A novel diffusion-weighted magnetic resonance imaging technique, known as restriction spectrum imaging (RSI), addresses this issue (19). In the three-compartment model, each compartment represents a different water molecule diffusion, f₁ indicates the signal fraction of restricted diffusion, reflecting the tumor’s cellular composition; f₂ represents the signal fraction of hindered diffusion, indicating delayed water molecule passage around cellular obstacles; and f₃ represents the signal fraction of free water diffusion, reflecting microcirculation perfusion (20). RSI can isolate areas of truly restricted diffusion by separating and removing hindered diffusion signals, offering a more direct measurement of tumor cells compared to other diffusion-weighted methods (19). RSI has achieved significant breakthroughs in distinguishing benign from malignant tumors and has demonstrated potential in differentiating cancerous from non-cancerous tissues in studies involving lungs, prostate, and breast (18, 20, 21). However, the relative value of CTRW and RSI in distinguishing pathological subtypes of liver cancer has not been compared. Histogram is a classical analysis method based on image voxel values, which not only has high reproducibility and consistency, but also can provide additional quantitative indicators (22–24). Therefore, if histogram analysis is applied to RSI and CTRW, it is expected to mine richer image information and thus provide more evidence for ICC and HCC identification.

The objective of this study was to compare the value of RSI, CTRW and DWI and their associated histograms in differentiating HCC from ICC, with the aim of finding an accurate, non-invasive imaging marker to guide clinical decision making.

This study represents a significant and pioneering effort in evaluating the diagnostic capabilities of two advanced, non-Gaussian diffusion-weighted models, CTRW and RSI, in distinguishing HCC from ICC. The results demonstrated that both the CTRW and RSI models effectively distinguish between HCC and ICC. Notably, the RSI model exhibited superior diagnostic performance compared to the CTRW, with the signal fraction of restricted diffusion (f₁) from RSI showing greater potential for differentiating HCC from ICC. Moreover, combining histogram parameters or integrating multiple diffusion models significantly enhanced diagnostic accuracy. The combined histogram parameter model demonstrated the highest diagnostic efficacy among all approaches tested. This study establishes the potential of these models to improve diagnostic accuracy and provides a foundation for future research into their broader clinical application, particularly in the context of liver cancer subtyping.

A substantial body of research suggests that the ADC values derived from the mono-exponential model are based on the assumption that water molecule diffusion follows a Gaussian distribution (28). However, the actual microenvironment within biological tissues, especially in heterogeneous tumors, is far more complex. In such environments, the movement of water molecules is restricted by cellular structures and membranes, resulting in non-Gaussian diffusion behavior (11). Consequently, ADC values fail to account for the non-Gaussian characteristics of water diffusion within these intricate subcellular microstructures, limiting their diagnostic efficacy (12). This aligns with our study’s findings, which indicate that ADC values are suboptimal for distinguishing between pathological subtypes of liver cancer.

The RSI separates the diffusion of water molecules in tissues into restricted diffusion, hindered diffusion, and free water diffusion, and its parameters f₁, f₂, and f₃ represent the proportions in which the above three compartments are located, typically summing to 1. In this study, the f_{1 median}, f_{1 20th}, f_{1 40th}, and f_{1 60th} values were significantly higher in HCC patients compared to ICC patients, which may be related to the fact that HCC usually have higher cell densities and more tightly packed cells (29). Also in this study, we found that none of the differences in f₂ and f₃ between HCC and ICC were statistically significant. This is similar to the study by Xiong et al. (30), and we hypothesize that this may be related to the fact that the fitting of f₂ and f₃ values is susceptible to the number and size of b-values.

The CTRW model has three quantitative parameters, namely, D, α, and β. The rate parameter D is mainly used to reflect the speed of the diffusion of water molecules. The present work found that the D values (including D_mean, D_median, D_40th, D_60th, and D_80th) were significantly lower in HCC patients. One possible explanation is that HCC is characterized by a higher cell density and more complex microstructure, which restricts the free movement of water molecules. In contrast, the central region of ICC consists of loose fibrous tissue, with tumor cells predominantly located at the periphery, often arranged into adenoidal patterns. This structural arrangement in ICC facilitates the diffusion of water molecules (31). The α and β describe the potential of water molecules to be retained or released during diffusion. There were no significant differences in both α and β between HCC and ICC in this study, which is not consistent with previous study (16). We consider that this may be related to the small sample size in this study and the different fitted b values used between studies.

This investigation also explore the correlations between mean histogram-derived parameters from these advanced models. The Spearman correlation analysis revealed a strong negative correlation between D_mean and f_{1 mean}, a strong positive correlation between D_mean and ADC _mean, and a moderate negative correlation between f_{1 mean} and ADC _mean, indicating their association with tissue cellular structures (32). These findings support the consistency of these parameters in assessing tumor tissue characteristics. Theoretically, smaller α and β values suggest a more heterogeneous spatial environment (33), while f₁, f₂, and f₃ correspond to intracellular restricted water, extracellular restricted water, and freely diffusing water, respectively (20). Further analysis demonstrated a moderate negative correlation between α _mean and f_{1 mean}, as well as a moderate positive correlation between β_mean and f_{2 mean}. This suggests that as cellular density increases, water molecules encounter more obstacles or irregularities during their movement between cells, leading to a greater proportion of water molecules being confined to intracellular spaces. Consequently, this results in an increase in the fraction of restricted diffusion (f₁) and a decrease in the fraction of extracellular restricted water (f₂). The observed correlations between diffusion parameters not only support the theoretical associations between specific parameters in these different diffusion models, but also have the potential to provide markers for tumor characterization assessment, such as monitoring cell morphology and density changes during treatment by f1-f2-ADC alterations. However, there are few relevant studies yet, and further large sample size experimental observation is still needed in the future.

In this study, the results indicate that both CTRW and RSI outperform traditional DWI-ADC in differentiating between HCC and ICC, with RSI exhibiting the highest discriminatory power. This superiority is likely due to the non-Gaussian distribution characteristics of CTRW and RSI, which, unlike the Gaussian assumptions of the ADC model, accurately reflect the complex diffusion behavior of water molecules and the intricate microstructure of tumor tissues (13). In addition, the high diagnostic performance of RSI may be attributed to the ability to isolate genuinely restricted diffusion areas, thereby minimizing interference from extracellular diffusion signals (34). This allows RSI to more directly measure tumor cell density and tissue structure, making the difference in high cell density regions between HCC and ICC more pronounced. Although CTRW and RSI represent significant advancements in DW-MRI, no single imaging technique currently fulfills all diagnostic requirements for tumors. Our study demonstrates that both the combined models of CTRW and RSI, as well as the integrated models of histogram parameters showing statistically significant differences between the pathological subtypes of liver cancer, provide superior diagnostic performance compared to individual models or histogram parameters used in isolation. This enhanced performance is attributed to the comprehensive integration of multidimensional information, including cellular metabolism and water molecule diffusion, which provides valuable complementary insights. Therefore, where possible, adopting multimodal imaging approaches for lesion assessment is likely to yield the greatest diagnostic benefits. By leveraging the strengths of various imaging techniques, clinicians can obtain a more holistic understanding of tumor characteristics, ultimately leading to more accurate diagnoses and better patient management.

Firstly, it was a single-center and relatively small sample size study Secondly, no test-retest procedure was implemented for diffusion MRI within the same participants. The Quantitative Imaging Biomarkers Alliance advocates for such retest procedures to assess the repeatability and reproducibility of quantitative MRI techniques. Thirdly, the scans in this study were performed a single imaging vendor or scanner. These limitations suggest that while our findings are promising, caution should be exercised in generalizing the results until further research can provide more robust validation. In the future, we will include more patients and conduct multicenter studies at different institutions, and will further optimize the parameters, port the relevant scanning protocols to different devices, and conduct external validation of the relevant parameter measurements with a view to improving the fitness and reducing the bias.

In summary, both the CTRW and RSI models, along with their derived histogram parameters, demonstrated the ability to differentiate between pathological subtypes of liver cancer, whether utilized individually or in combination. Furthermore, whole-lesion histogram parameters provide richer statistical information compared to mean values, offering a quantitative approach to analyzing subtle changes in tumor voxels. This advancement has the potential to find an accurate, noninvasive imaging marker for the differentiation of HCC and ICC, which could help reduce unnecessary biopsies, especially in resource-limited settings, and in turn guide clinical decision-making.