This study focuses on analyzing tomographic images from the LiTS (Liver Tumor Segmentation Challenge) and CHAOS (Combined CT-MR Healthy Abdominal Organ Segmentation) datasets, comprising 40 patients, half with hepatic neoplasms(LiTS) and the rest healthy liver donors (CHAOS). The primary aim is to employ fractal dimension as a tool for characterizing hepatic organ morphology. The first step of analysis is a preprocessing 11,311 images, with a technique of pixel intensity weighting for effective liver segmentation. Binary classification categorized pixels, indirectly capturing organ contrast distribution. This preprocessing enriched subsequent fractal dimension analysis. Original images underwent pixel scanning and mean calculation for thresholding, this for transforming them into binary images. The calculation of the fractal dimension followed, the images with black pixels were filtered, resulting in 3,412 pathological images and 2,289 non-pathological images. Statistical analysis revealed a significant difference in fractal dimension between patient groups (p: 4.965e-39), indicating varying liver morphology in the groups. A box plot visually represented fractal dimension density, highlighting lower values for patients with hepatic pathologies (mean = 1.25) compared to those without (mean = 1.36). Additionally, a comprehensive database of images was compiled. This database includes the medical images with the segmented liver as well as their original versions. To facilitate future research and contribute to diagnosis and classification of hepatic diseases, this database will be made available online to the scientific community.
According to the World Health Organization (WHO), cancer stands as the second leading cause of
mortality worldwide. In 2015 alone, this devastating disease claimed the lives of 8.8 million
individuals, with liver cancer alone accounting for 788 thousand deaths. Many of these fatalities
can be attributed to the lack of timely diagnosis and treatment, resulting in the detection of
tumors at advanced stages. Consequently, there exists a pressing need to identify novel
techniques founded on biomarkers to aid in early detection of this condition. Such biomarkers
not only encompass clinical blood chemistry analysis but also encompass the incorporation of
liver lesion images that manifest during the initial stages of the disease [
In addition to the profound loss of human life, cancer also exacts a considerable toll on global
economies. Estimations reveal that the total cost associated with cancer in 2010 amounted to a
staggering $1.16 trillion. Disturbingly, as of 2017, merely 26% of low-income countries reported
the provision of pathology services in their public health systems, which serve the general
population. Comparatively, over 90% of high-income countries deliver cancer treatment services
to their patients, with the corresponding figure dwindling to below 30% for low-income countries
[
Given the dire implications of inadequate cancer care and management, researchers must
address the ongoing research problem of developing innovative techniques rooted in the
identification of biomarkers. It is imperative that these biomarkers enable prompt and accurate
diagnosis, facilitating effective treatment strategies. Moreover, these biomarkers ought to extend
beyond conventional clinical blood chemistry analysis, embracing the incorporation of liver
lesion images that may manifest during the primary stages of the disease [
The liver, a vital organ located in the right hypochondrium, stands out as the largest viscus
within the human body. Given its position in the abdominal cavity, the liver plays essential roles
in the regulation and maintenance of various metabolic and physiological processes. It is vital
because it not only plays a crucial role in digestion and nutrient storage but also actively
participates in detoxifying and filtering harmful substances. However, the significance of the liver
is also marked by its susceptibility to a wide range of pathologies. These conditions can range
from local disorders directly affecting the liver to systemic issues involving its function in
conjunction with other organs and systems of the body. The diversity of liver diseases
underscores the vulnerability and clinical importance of this organ in the context of human health
[
Within the spectrum of malignant lesions affecting the liver, Hepatocellular Carcinoma stands
out as an entity of considerable clinical significance. This form of hepatic neoplasia ranks sixth
globally in terms of frequency and represents the third leading cause of cancer-related mortality.
In the Mexican context, a significant increase of 14% in mortality associated with this pathology
was observed during the period between 2000 and 2006 [
Human carcinogenesis, the process of cancer formation, is a complex phenomenon that
initiates with a single cell and is characterized by uncontrolled cellular growth. This intricate
process is a result of the interaction between external and internal factors, which provoke
irreversible changes in cellular function. External agents, such as environmental carcinogens and
radiation, intertwine with internal processes involving genetic and epigenetic alterations. This
combination of factors triggers biochemical and molecular events that drive the transformation
of normal cells into cancerous ones, promoting unrestricted cell growth, which is a defining
characteristic of malignant tumors. Understanding this molecular sequence is crucial for
effectively addressing the development and progression of cancer [
The concentrations of carcinogens or their metabolites assessed in tissues or bodily fluids play
a crucial role as indicators, known as cancer biomarkers. These biomarkers not only provide a
direct window into exposure to carcinogenic substances but also serve as revealing signals of
fundamental biological processes. Among these biological processes are xenobiotic metabolism,
which addresses the transformation of foreign compounds in the body; DNA repair, which
counteracts genetic damage; cell proliferation, which regulates cell growth; apoptosis, which
controls programmed cell death; and the immune response, which plays a vital role in defense
against abnormal cells. The comprehensive monitoring of these biomarkers provides a holistic
view of exposure to carcinogens and associated biological responses, thus contributing to a more
complete and accurate assessment of cancer risk [
The significance of these biomarkers lies in their ability to provide early prognosis regarding the likelihood of developing cancer. This prognosis becomes an invaluable tool that enables medical professionals to make informed decisions about the initiation of specific treatments or appropriate interventions. By obtaining early information through the assessment of biomarkers, the door is opened to more proactive and personalized medical strategies, thus optimizing the effectiveness of therapeutic measures. Ultimately, the early identification of these cancer biomarkers not only enhances treatment prospects but can also be instrumental in prevention or early detection, significantly elevating the quality of healthcare provided to patients.
In mathematics, the Hausdorff dimension is a measure of roughness first introduced in 1918
by the mathematician Félix Hausdorff. The Hausdorff dimension is an integer according to the
usual sense of dimension, also known as topological dimension. Nevertheless, formulas have been
developed to calculate the dimension of less simple objects, where, relying solely on their scale
and self-similarity properties, it is concluded that objects, including fractals, do not have integer
Hausdorff dimensions. Due to the significant technical advances made by Abram Samoilovitch
Besicovitch, allowing the calculation of dimensions for highly irregular or "rough" sets, this
dimension is also commonly known as the Hausdorff-Besicovitch dimension [
The fractal dimension, a measure characterizing the geometric and topological complexity of
fractal sets, is often defined using the Hausdorff-Besicovitch dimension. While there are various
ways to define the fractal dimension, one of the most common is precisely the
HausdorffBesicovitch dimension. The most popular method today for calculating the fractal dimension is
the box-counting method. This method involves counting the number of boxes N(ε) of size ε
needed to cover the entire object in the image. As the value of ε increases, the number of boxes
needed to cover the object decreases. The fractal dimension is obtained by calculating the slope
of the best-fit line from a graph of log(N(ε)) against log(1/ε) [
The Hausdorff–Besicovitch dimension (D) is defined for fractal sets F as follows Where: D: is the Fractal dimension N(ε): Is the minimum number of assemblies (ε-cover) required to completely cover fractal assembly F with elements of diameter not greater than ε. ε: This is the size of the element of the ε-cover.
D = log (1/ε) ((ε)) (1)
In simpler terms, the fractal dimension provides valuable information about the variation in
the number of smaller sets needed to cover a fractal set as the size of the covering elements is
adjusted. When applied to the analysis of medical images, this methodology has proven its utility
in various disciplines, from the detailed examination of coronary branching to the comprehensive
assessment of neurodegenerative diseases and dementia through ultrasound images. These
advances, guided by key dates and notable achievements, have transformed the understanding
and application of the fractal dimension in various scientific fields [
This study adopted an observational approach based on a series of cases, utilizing data from the
LiTS - Liver Tumor Segmentation Challenge (LiTS17) and the CHAOS - Combined (CT-MR)
Healthy Abdominal Organ Segmentation dataset. It included 130 computed tomography’s from
pathological patients, with their corresponding liver-segmenting masks, obtained from LiTS17,
and 40 computed tomography’s from healthy patients, with 20 volumes of masks from CHAOS.
To address the diversity of medical formats, the versatile capabilities of Python were employed
[
The datasets were provided in .nii and DICOM formats, and to ensure equality in the number of patients in both populations, 20 volumes were selected from the first dataset. Subsequently, slices were extracted from each volume and exported to JPG image files (Figure 1). This approach allowed for the standardization of data, facilitating consistent comparison and analysis across both patient groups.
This approach of integrating the mask and the original image serves as a robust method to precisely highlight and delineate hepatic regions (Figure 2, section d). The procedure is based on the binary nature of a mask, where pixels are classified into two distinct categories: 0, representing the background or areas to be omitted, and 1, indicating the foreground or areas of interest. By assigning these values, we precisely establish which regions of the original image should be preserved (in the case of the mask with a value of 1) and which should be suppressed or excluded (when the mask has a value of 0). This approach proves to be an efficient method to accurately highlight and isolate anatomical structures of interest.
With the set of JPG images from each group, we performed the mathematical analysis of the fractal dimension. In this process, the first step involves scanning the pixels and calculating the mean to use as a threshold in the transformation from original to binary images. The binarization threshold is crucial because pixels are classified into two categories: those belonging to the object of interest and those that do not (Table 1, column 2). As a second step, the fractal dimension is calculated on that binarized image, and each dimension is stored in a list for each group of images.
The algorithm we used to compute the fractal dimension is the following. 8. Result Recording: • Record the number of boxes needed to cover the fractal set for each box size (Table 1). 9. Logarithmic Analysis: • To analyze the relationship between box size and the number of boxes needed, take the logarithm of both sides. This is done to visualize the relationship on a logarithmic scale (Table 1). 10. Result Visualization: • Visualize the results graphically, often using a log-log plot where the x-axis represents the logarithm of the box size, and the y-axis represents the logarithm of the number of boxes needed (Figure 3). 11. Fractal Dimension Estimation: • The slope of the line in the log-log plot provides an estimate of the fractal dimension of the set. FD 1.562 1.560 1.547 1.456
In Table 1, we can see the binarization of images from some patients, both pathological and
non-pathological. The Box-Counting Chart shows the log-log plot of box counting. Each point on
the graph represents the box size (on the x-axis) and the total number of boxes needed to cover
the image (on the y-axis). In this way, we can observe a more homogeneous structure in terms of
binarization in non-pathological patients. The above observation validates that the lower
complexity of the fractal dimension in images of pathological patients (Table 1) may be related to
metabolic alterations that would be indirectly observed using the weighted pixel intensity
threshold. This could provide a pathway for in-depth analysis in future projects. A healthy liver
tends to show a uniform and efficient distribution of contrasts, while in a liver affected by cancer,
changes in blood flow could result in noticeable alterations in the uptake and dispersion of
contrast, influencing the interpretation of medical images, such as tomography (Figures 3 and 4)
[
With the stored fractal dimensions, we filter out any slice that represents an image with completely black pixels, i.e., slices from the tomography where the liver is not present, mainly the first and last slices of the study. It is this reduced filtered list to which the distribution study and the non-parametric test of mean difference (Mann-Whitney U) were applied.
In the following table (Table 2), we can see the summarized analysis of the images, including the selection of slices that underwent fractal dimension calculation. A total of 20 tomography volumes were processed for each patient group, ensuring a representative sample from both sets. The segmentation of the tomography slices resulted in a significant number, initially extracting 11,437 for pathological patients and 2,874 for non-pathological ones. This was derived from the size of the slices used as configuration in the tomographic equipment. For the research, those slices without the presence of the hepatic organ were eliminated, meaning the masks that showed a completely black image, leaving us with a final number of images of 3,412 for pathological patients and 2,289 for non-pathological ones. Analyzing this group of images, we obtained a mean DF of 1.36 in "Non-pathological Patients," while in "Pathological Patients," it was slightly lower, with a mean of 1.25. In the following table (Table 2), we can see the summarized analysis of the images, including the selection of slices that underwent fractal dimension calculation.
without organ to be segmented 585
organ cuts
Now, these differences in populations alone are not sufficient for the classification of
pathological and non-pathological patients, as seen in Figure 6 where there is a significant
intersection between results for both cases (pathological and healthy). However, fractal
dimension can be integrated with other clinical features and biomarkers to improve the
robustness of classification models. This could result in more comprehensive and accurate
systems to distinguish between patients with pathological and non-pathological livers, as texture
is the most appropriate descriptor for mass detection in images [
In the previous graph (Figure5), the Y-axis shows the occurrence density of fractal dimension values, indicating the concentration of values in both study groups and providing a general idea of how they are distributed in amplitude, a part that shows differences in the studied groups.
In the follow Figure (Figure 6), we can observe a relative distance between the means and
medians of the studied data, leading us to theorize that by using fractal dimension as a variable
for classification models, a quantitative evaluation of tomographic images could be carried out.
This would allow for a more precise characterization of the complexity and irregularity of hepatic
structures, providing valuable information for diagnosis and treatment. By quantifying fractal
complexity over time in images and even at the cellular level, doctors could gain insights into the
evolution of pathology and adjust treatment strategies [
The above graph (Figure 6) provides a clear insight into the behavior of fractal dimensions in the analyzed patients. The X-axis contains the studied groups, while the Y-axis represents the values of the fractal dimension, here is where we can observe the variability of fractal dimensions among the groups and how they behave along this axis. The interquartile range (IQR) can be observed in the height of the boxes, and, in turn, the whiskers extending from the boxes upward and downward represent the dispersion of data outside the interquartile range, highlighting outliers, indicated as points outside the whiskers. Meanwhile, horizontal lines inside the boxes represent the medians presented in Table 2, which are slightly distant from each other. Additionally, a new feature is the horizontal lines outside the boxes; these dashed lines represent the mean (Table 2) of the data in each group, allowing for a comparison of the mean and median in each dataset.
Mann-Whitney U Test U-Test Statistic: 4701167.0 p-value: 4.965716675822758e-39
With this result, we conclude that there is a significant difference between the populations, suggesting a contribution of this information derived from the fractal analysis of images in the future generation of useful models in the classification of hepatic neoplasms.
This study presents relevant evidence concerning the description of liver texture and characteristics, which carries significant implications for the diagnosis and classification of liver diseases, particularly in the context of cancer. Through the application of an innovative approach utilizing pixel intensity weighting in medical images, precise segmentation of hepatic regions has been achieved. Empirical evidence has shown that this methodology is highly useful, as it has successfully revealed significant differences in fractal dimension between patients with and without liver pathologies. Considering the complex nature of hepatocarcinogenesis, the observed variation in pixel segmentation among these groups suggests the possibility that liver lesions may exhibit diverse blood flow patterns, thus influencing the absorption and distribution of contrast agents in imaging studies. Pathological patients demonstrate visually smoothed regions in their images, indicating that a less complex pixel-level structure is being analyzed, as illustrated in Table 1.
The findings of this study provide an informative basis for future research endeavors, such as collaboration with specialists to improve segmentation techniques and to undertake more diverse computational analyses. in addition to advancing clinical understanding, the image dataset generated from this study, which includes the segmented organ with its corresponding original image extracted from the tomographic volume, holds tremendous potential in serving as a foundation for further investigations. This dataset can facilitate more advanced analyses that take into consideration fractal features, weighted thresholds, and clinical data, potentially leading to enhanced models for the classification of liver pathologies.
Organization",» 2022, https://www.who.int/es/health