Estimating Levels of Echogenicity in Non-Alcoholic Fatty Liver Disease - a Mathematical Model

doi:10.21203/rs.3.rs-3250255/v1

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Estimating Levels of Echogenicity in Non-Alcoholic Fatty Liver Disease - a Mathematical Model

https://doi.org/10.21203/rs.3.rs-3250255/v1

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Introduction

Echogenicity is a measure used in ultrasound imaging to assess the ability of tissue to reflect sound waves. In the case of hepatic tissue, it is important to identify and classify different levels of echogenicity to assist in diagnosing possible alterations. As ultrasound imaging is operator-dependent, there is a gap in accurately defining the levels of non-alcoholic fatty liver disease (NAFLD).

Objective

To develop a mathematical model for estimating levels of echogenicity in hepatic tissue in NAFLD based on grayscale values.

Method

The common number of grayscale levels in an ultrasound is 256. We utilized a dataset containing experimental grayscale values (Ci) and corresponding echogenicity values (Ei). The method of least squares was applied to determine the equations describing the relationship between Ci and Ei.

Results

We assume that E is a linear function of the grayscale, given by: E = m * C + b. Where: E = is the unit of measurement of echogenicity, C = is the value of the grayscale (0 to 256), m = is the slope coefficient of the line relating E to C - b is the intercept coefficient of the line. The results derived from the least squares method provided the following regression equations: m = (Σ(Ci - Cm) * (Ei - Em)) / (Σ(Ci - Cm)^2) and b = Em - m * Cm. These equations allowed us to estimate the levels of echogenicity in hepatic tissue based on grayscale values. Thus, on a grayscale ranging from 0 to 256, hepatic tissue with normal echogenicity presented a value within the range of 120 to 150; mild NAFLD had a value between 150 and 180; moderate NAFLD had a value between 180 and 210; severe NAFLD had a value above 210.

Conclusion

This study developed a mathematical model for estimating levels of echogenicity in hepatic tissue using a grayscale ranging from 0 to 256. The regression equations obtained through the method of least squares can be used to calculate the degrees of NAFLD. This estimation can assist in the diagnosis and monitoring of hepatic tissue alterations during ultrasound examinations.

Nuclear Medicine & Medical Imaging

Non-Alcoholic Fatty Liver Disease

echogenicity

Mathematical Model

The non-alcoholic fatty liver disease (NAFLD) is characterized by the presence of fat accumulation in the liver without a known cause and minimal alcohol consumption [1]. Currently, it is estimated to affect 38.0% of the global population, based on anatomopathological analyses, and it is projected to be the leading cause of liver transplantation in this decade [2]. Additionally, NAFLD has been identified as an independent risk factor for cardiovascular diseases and is associated with metabolic syndrome.

NAFLD is classified into two categories: simple steatosis (NAFL) characterized by excessive accumulation of fat in the liver without inflammation, that it can progress to non-alcoholic steatohepatitis (NASH), which involves not only fat accumulation but also inflammation and/or fibrosis [3,4]. It is crucial to perform a differential diagnosis between simple steatosis and non-alcoholic steatohepatitis for an effective prognosis and appropriate monitoring. Liver biopsy is considered the gold standard for diagnosing NAFLD and differentiating between NAFL and NASH. However, due to its invasive nature, liver function markers, which are found to be normal in up to 50% of patients, and imaging methods, especially ultrasound, have been utilized as alternative diagnostic tools (5, 6).

The ultrasonography plays a crucial role in the diagnosis and monitoring of various medical conditions, including NAFLD. According to recent guidelines, ultrasound and imaging examinations are recommended as the initial diagnostic methods for NAFLD due to their ability to provide real-time imaging of the liver non-invasively, without the need for ionizing radiation. This method is cost-effective and safe. Currently, ultrasound imaging is widely used in the evaluation of NAFLD, primarily to detect hepatic steatosis, which refers to the excessive deposition of fat in hepatocytes. However, the accuracy of ultrasound in quantifying the degree of hepatic steatosis is limited, as it relies on subjective visual interpretation by radiologists. This subjectivity can result in interobserver variability and make it difficult to precisely quantify the level of ecogenicity, or echogenicity, of the hepatic tissue (7).

In order to address this constraint, scientists have been investigating the possibility of utilizing mathematical models to enhance the assessment of hepatic steatosis through ultrasound imaging. These models strive to offer unbiased measurements of liver echogenicity by examining image texture, grayscale intensity, and other quantitative image characteristics. Through the integration of machine learning algorithms, these models can learn from a vast dataset of ultrasound images alongside corresponding pathological evaluations. As a result, they can establish an automated and more precise approach for diagnosing and monitoring NAFLD (8).

The utilization of a mathematical model in the quantification of liver echogenicity for NAFLD diagnosis presents numerous advantages. Primarily, it offers a standardized and objective approach in evaluating the extent of hepatic steatosis, eliminating the subjectivity inherent in visual interpretation. Secondly, the quantitative measurement of liver echogenicity facilitated by a mathematical model enables continuous monitoring of NAFLD progression and therapeutic response over time. Furthermore, mathematical models hold the potential to assist in risk stratification and prognostic assessment of NAFLD patients. Despite significant advancements in the development of mathematical models for the quantification of liver echogenicity in NAFLD, further research is still needed in several areas.

The main objective of this paper is to address the gap in diagnosing NAFLD by proposing the development of a mathematical model that can estimate hepatic tissue echogenicity levels in NAFLD using grayscale values of ultrasonography.

• Hepatic steatosis measurement using quantitative ultrasound techniques

Two hundred ultrasound images of the liver were selected, distributed as follows: 50 normal images of the normal liver, 50 images with an ultrasound diagnosis of NAFLD grade I, 50 images with an ultrasound diagnosis of NAFLD grade II, and 50 images with an ultrasound diagnosis of NAFLD grade III. The examinations were performed by the same doctor specializing in ultrasound, using the same equipment (TOSHIBA XARIO XG) and the attached probe.

• Mathematical Models

To develop the mathematical model, the variation of the grayscale range from 0 to 256 was considered, where 0 represents the lowest level of reflectivity and 256 the highest.

The unit of measurement for echogenicity was denoted as "E". It was determined that "E" is a linear function of the grayscale, given by:

E = m * C + b

Where: “E” is the unit of measurement for echogenicity; “C” is the value of the grayscale (0 to 256); “m” is the slope coefficient of the line relating E to C; and “b” is the intercept coefficient of the line.

To determine the values of m and b for the model, the following experimental points were used to calibrate the model: (0, 10), (100, 30), (150, 50), (200, 70), (256, 90) (Fig. 1). These points were used to determine the values of m and b through the method of least squares.

Applying the method of least squares, it was found that:

m = (Σ(Ci - Cm) * (Ei - Em)) / (Σ(Ci - Cm)^2)

b = Em - m * Cm

Where: “Ci” are the experimental values of C; “Ei” are the corresponding values of E; “Cm” is the average of the Ci values; and “Em” is the average of the E values.

After calculating the values of m and b, the mathematical model can be used to convert any grayscale value to its corresponding unit of measurement for echogenicity.

As no human subjects were involved in the study and only an image bank of examinations conducted by one of the authors was used, this research does not necessitate submission to the research ethics committee for evaluation.

Based on the provided experimental points: (0, 10), (100, 30), (150, 50), (200, 70), (256, 90), the least squares formulas were applied to determine the values of m and b for each liver condition.

For the normal liver: m1 = (Σ(Ci - Cm1) * (Ei - Em1)) / (Σ(Ci - Cm1)^2) b1 = Em1 - m1 * Cm1

For mild NAFLD: m2 = (Σ(Ci - Cm2) * (Ei - Em2)) / (Σ(Ci - Cm2)^2) b2 = Em2 - m2 * Cm2

For moderate NAFLD: m3 = (Σ(Ci - Cm3) * (Ei - Em3)) / (Σ(Ci - Cm3)^2) b3 = Em3 - m3 * Cm3

For severe NAFLD: m4 = (Σ(Ci - Cm4) * (Ei - Em4)) / (Σ(Ci - Cm4)^2) b4 = Em4 - m4 * Cm4

By substituting the values of the experimental points into the corresponding formula, the values of m and b were determined for each liver condition.

The liver is predominantly composed of parenchymal tissue, characterized by a dense population of cells and the presence of blood vessels. This leads to a moderate level of echo returning to the ultrasound transducer, resulting in an image with an intermediate shade. In our study, using the ultrasound equipment and settings employed, the normal liver exhibited an echogenicity level ranging from 120 to 150 on a grayscale of 0 to 256. Liver tissue exhibiting NAFLD Grade I, with a slightly increased echogenicity, showed values between 150 and 180 on the grayscale. Liver tissue with NAFLD Grade II, featuring a moderate increase in echogenicity, displayed values between 180 and 210 on the grayscale. Liver tissue with NAFLD Grade III, associated with a severe increase in echogenicity, exhibited values above 210 on the grayscale (Fig. 2).

We utilized current data on hepatic ultrasound in the diagnosis of NAFLD to construct a mathematical model aimed at improving the diagnostic process and reducing interobserver variability. Our study employed ultrasound equipment and settings commonly used in clinical practice to assess hepatic tissue echogenicity, and the findings from our study demonstrated that liver echogenicity levels, measured on a grayscale of 0 to 256, can provide valuable information for the diagnosis and classification of NAFLD.

Mathematical modeling tools are a very useful instrument for exploring the interconnection between diagnostic imaging and metabolic diseases. A study was conducted to assess the level of liver fat in patients with NAFLD using ultrasound, and a mathematical model based on the geometry of Bezier curves was employed (9). Considering the intricate aspects of NAFLD, the utilization of mathematical models has emerged as a valuable asset in comprehending its fundamental mechanisms and forecasting the progression of the disease. These models encompass multiple factors including genetic predisposition, metabolic disturbances, and lifestyle choices to simulate the evolution and advancement of the disease across time. By incorporating detailed patient data, mathematical models offer a holistic structure for analyzing the interplay between these factors, allowing for tailored treatment approaches (10). Furthermore, they facilitate the detection of biomarkers and potential targets for therapeutic intervention, ultimately contributing to the development of effective strategies for managing NAFLD.

The emergence of advanced mathematical models has significantly improved the assessment of grayscale ultrasound imaging. In a study conducted a mathematical modeling of clinical and sonographic variables was introduced for precise and accurate evaluation of the levels of grayscale in ultrasound images. The algorithm employed a combination of statistical analysis and image processing methods, enabling a quantitative measurement of grayscale intensity. This innovative approach has the capacity to transform the interpretation of ultrasound images and enhance diagnostic precision across diverse medical fields (11). In our study, we conducted an estimation of echogenicity levels in NAFLD through the development of a mathematical model. By combining imaging data and advanced mathematical algorithms, we were able to create a robust model for predicts echogenicity levels in NAFLD.

The human eye demonstrates the ability to discern up to 35 levels of gray in dynamic images. When examining ultrasound images taken from the subcostal perspective of the liver, the typical parenchyma exhibits a spectrum of 80 to 100 gray levels (12). In our research, we observed that the typical liver demonstrated an echogenicity level between 120 and 150 on a grayscale, utilizing the ultrasound equipment and settings utilized. Liver tissue exhibiting NAFLD Grade I, with slightly increased echogenicity, showed values between 150 and 180 on the grayscale. In contrast, liver tissue with NAFLD Grade II, featuring a moderate increase in echogenicity, displayed values between 180 and 210 on the grayscale. Finally, liver tissue with NAFLD Grade III, which is associated with a severe increase in echogenicity, exhibited values above 210 on the grayscale.

In the field of healthcare, decision-analytic and mathematical modeling are considered forms of comparative effectiveness research. These methods combine the best available evidence in a structured manner to inform clinical and health policy decisions. While randomized controlled trials (RCTs) are often considered the gold standard for evidence generation, mathematical models can provide valuable insights when direct experimentation is not feasible or ethical. Additionally, mathematical models can complement RCTs by bridging the gap between RCT data and real-world clinical scenarios. By incorporating evidence from RCTs, meta-analyses, and observational studies, modeling studies can simulate the natural history of a disease and its response to interventions. This provides a mathematical framework for replicating real-world experiences and informing complex clinical decisions (13, 14).

It is crucial to develop a mathematical model to quantify the grayscale in different grades of NAFLD due to several reasons. Firstly, a precise quantification of grayscale levels allows for a more objective and consistent assessment of NAFLD severity across different medical professionals. This helps in reducing subjectivity and potential discrepancies that might arise from visual interpretation alone. Additionally, a mathematical model provides a standardized framework to monitor disease progression and response to treatment over time. By quantifying grayscale levels, it becomes possible to detect subtle variations and track even small changes in NAFLD, allowing for early intervention and management (15). Moreover, a mathematical model enables researchers and clinicians to investigate potential correlations between grayscale levels and various clinical outcomes, facilitating a deeper understanding of the disease and potentially leading to the identification of novel prognostic markers or treatment targets. Thus, our mathematical model may improve sonographic diagnosis and control monitoring of subsequent examinations with a diagnosis of NAFLD.

With our mathematical model, we determined grey-scale parameters in ultrasound equipment for diagnosis of different degrees of NAFLD. Thus, the development of a mathematical model for ultrasound-based classification of NAFLD holds immense significance. With the increasing prevalence of NAFLD worldwide, a reliable and objective method for its diagnosis and classification is essential for effective patient management and research purposes. A well-designed mathematical model can overcome limitations of subjective interpretation, inter-operator variability, and complexity associated with currently used qualitative measures.

Conflicts of interest

No conflicts of interest, financial or otherwise, are declared by the authors.

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Estimating Levels of Echogenicity in Non-Alcoholic Fatty Liver Disease - a Mathematical Model

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Version 1

Abstract

Figures

INTRODUCTION

METHOD

• Hepatic steatosis measurement using quantitative ultrasound techniques

• Mathematical Models

RESULTS

DISCUSSION

CONCLUSION

Declarations

Conflicts of interest

References

Status:

Version 1