The Value of Fractal Analysis in Ultrasound Imaging: Exploring Intricate Patterns

Abstract

Fractal analysis is a mathematical approach employed to study and describe complex patterns and structures across various disciplines, including mathematics, physics, computer science, biology and finance. Introduced by mathematician Benoit Mandelbrot in the 1970s, fractals are intricate, self-similar patterns that repeat at different scales, exhibiting consistent structures upon magnification or reduction. This analysis involves generating fractals through iterative processes or recursive equations, resulting in highly detailed and complex formations. Fractal analysis enhances medical images by removing noise while preserving details and improving diagnostic quality in magnetic resonance and computed tomography scans. However, there is a lack of comprehensive studies on its application in ultrasound imaging, prompting this narrative review to investigate its use and methodology in this context. Selected papers on the use of fractal analysis in ultrasound imaging were analyzed. Out of 186 records screened, 60 duplicates were removed and 28 were discarded. The text content of 98 potentially eligible papers was checked, with 65 not meeting inclusion criteria. Finally, 33 studies were included in the review. Fractal analysis enhances ultrasound imaging by providing detailed tissue texture characterization, aiding in the diagnosis of conditions like breast and lung cancer, osteoporosis and hypertensive disorders in pregnancy. It quantifies biological structure complexity and improves diagnostic accuracy and reliability. This technique supports clinicians in making informed decisions by offering critical insights into various medical conditions.

1. Introduction

Fractal analysis is a mathematical approach used to study and describe complex patterns and structures found in various fields, including mathematics, physics, computer science, biology and finance. Fractals are intricate, self-similar patterns that repeat at different scales, meaning that they exhibit similar structures when zoomed in or out. The concept of fractals was introduced by mathematician Benoit Mandelbrot in 1970 [1]. He coined the term “fractal” to describe geometric shapes or sets that have infinite complexity yet possess certain recurring patterns or motifs. Fractals can be generated by applying iterative processes or recursive equations, resulting in intricate and detailed structures [1].

Several methods are employed in fractal analysis to determine fractal dimensions and other properties. One common method is the box-counting method, where an image is covered with a grid of boxes of varying sizes and the number of boxes containing part of the fractal pattern is counted. The fractal dimension is then calculated based on the relationship between the size of the boxes and the number of boxes needed to cover the fractal [2]. Another method is the Hausdorff dimension, which is a more theoretical approach involving the concept of metric spaces that measures how a fractal scales differently than ordinary geometric shapes [3].

In medical imaging, fractal analysis has become a valuable tool for physicians. By analyzing the fractal properties of medical images, physicians can gain deeper insights into the texture, shape and organization of tissues and organs. This approach is particularly useful in distinguishing between normal and abnormal tissues and evaluating, for example, tumor growth patterns by segmenting images into meaningful regions and enhancing image quality. The practical applications of fractal analysis in medical imaging are vast and diverse, making it an invaluable asset in modern diagnostic radiology [4].

Fractal analysis facilitates the characterization of texture properties in medical images by quantifying the fractal dimension of textures. This provides insights into the complexity and heterogeneity of tissue, aiding in the differentiation of tissue types, identification of abnormal regions and diagnosis of disease. For instance, in mammography, fractal analysis has been used to evaluate breast tissue texture, potentially indicating the presence of breast cancer [4]. Moreover, fractal analysis proves instrumental in assessing tumors and their growth patterns. Tumors often exhibit irregular, complex shapes, and their fractal dimensions can reveal critical information regarding their aggressiveness or response to treatment [5]. By quantifying the fractal properties of tumor boundaries or vascular patterns, physicians can distinguish between benign and malignant tumors and monitor disease progression or regression [5].

Therefore, fractal-based algorithms enhance image segmentation, delineating structures or regions of interest with greater accuracy. By identifying self-similar and recurrent patterns in various tissue types, fractal analysis allows for more precise and automated segmentation, saving time and improving the efficiency and reproducibility of image analysis tasks [6]. Fractal-based compression techniques can be applied to medical images, significantly reducing storage requirements without compromising diagnostic information. Given that medical images often contain regions with similar patterns or textures, fractal encoding efficiently represents these areas, resulting in compact storage and faster transmission, beneficial for large-scale imaging databases and telemedicine [7].

Additionally, fractal analysis is also employed for image enhancement and denoising. By leveraging the self-similar properties of medical images, fractal-based algorithms effectively remove noise while preserving crucial details and structures. This enhances the clarity and diagnostic quality of medical images, such as magnetic resonance and computer tomography, assisting physicians, particularly radiologists, in making accurate interpretations and diagnoses [8].

Despite the clear advantages in the use of fractal analysis in medical imaging, there is a notable gap in the literature concerning a comprehensive overview of the methodology and use of fractal analysis in the ultrasound (US) imaging context or in association with US imaging. Therefore, this narrative review has the main purpose of investigating and understanding the use and methodology of fractal analysis in the US imaging field.

2. Materials and Methods

This study is a narrative review of the evidence surrounding the use of fractal analysis in US imaging. We searched the existing literature, focusing on PubMed and Web of Science from inception to July 2024 due to the high availability of published data on the topic. Furthermore, the references of the included papers were thoroughly checked to find eligible publications. The MeSH keywords used were as follows: “Fractal Analysis”, “Fractal Dimension”, “lacunarity”, “Ultrasound Imaging”, “Ultrasonography”, “Ultrasound” and “Ultrasound examination”. The search strategy set for the topic was the following: (“Fractal Analysis”) OR (“Fractal Analysis” AND “Ultrasound Imaging”) OR (“Fractal Analysis” AND “Ultrasonography”) OR (“Fractal Analysis” AND “Ultrasound”) OR (“Fractal Analysis” AND “Ultrasound examination”) OR (“Fractal Dimension”) OR (“Fractal Dimension” AND “Ultrasound Imaging”) OR (“Fractal Dimension” AND “Ultrasonography”) OR (“Fractal Dimension” AND “Ultrasound”) OR (“Fractal Dimension” AND “Ultrasound examination”) OR (“Lacunarity”) OR (“Lacunarity” AND “Ultrasound Imaging”) OR (“Lacunarity” AND “Ultrasonography”) OR (“Lacunarity” AND “Ultrasound”) OR (“Lacunarity” AND “Ultrasound examination”). All relevant English-language publications were examined for potential inclusion. References were further screened by selecting the papers that we believed to provide a fair and accurate depiction of the use of fractal analysis for US imaging and that were not yet fully understood but not too speculative. The selection of studies was guided by predefined criteria for inclusion, ensuring relevance, methodological quality and consistency with the objective of our review. The inclusion criteria were as follows: (1) original studies involving the use of ultrasound imaging combined with fractal analysis; (2) studies with appropriate methodological rigor and transparent reporting of both ultrasound imaging protocols and fractal analysis techniques. Exclusion criteria were meticulously applied to filter out peripheral content, thereby concentrating on primary research efforts. The specific exclusion criteria were as follows: (1) papers that did not transparently report the use of fractal analysis combined with ultrasound imaging; (2) papers that did not use a quantitative or standardized approach for fractal analysis or that lacked sufficient methodological details; (3) papers that did not address the use of US imaging for diagnosis; and (4) papers that were not published in English. We conducted an initial screening of all studies by title and abstract, followed by a thorough review of the full texts of the eligible studies. Additionally, references within these studies were examined to identify any further relevant publications to be included. The literature search was conducted by one reviewer (N.P.) and subsequently verified by a senior researcher (C.P.). Any discrepancies were resolved through consensus among the authors (Figure 1).

3. Results

Selected papers on the use of fractal analysis in ultrasound imaging were analyzed. The agreement between the authors on the inclusion of articles was perfect (Cohen’s k = 0.92). Out of 186 records screened, 60 duplicates were removed and 28 were discarded. The text content of 98 potentially eligible papers was checked, with 65 not meeting the inclusion criteria. Finally, 33 studies were included in the review. Figure 1 depicts the study selection flow diagram.

3.1. Bone Texture Analysis

Some studies have utilized fractal analysis on radiographic images to evaluate trabecular bone pattern and density, which are crucial for assessing bone health and osteoporosis risk [9,10]. For instance, dental radiographs, including panoramic and cone-beam computed tomography (CBCT) images, have been extensively analyzed using fractal dimension to distinguish between healthy and osteoporotic patients. These analyses revealed that higher fractal dimension values are associated with denser and more complex trabecular patterns, typical of healthier bone structures. Systematic reviews and meta-analyses have consolidated data from various studies [10,11,12,13] to establish the reliability of fractal dimension in diagnosing osteoporosis. These studies often involved comparing fractal dimension values from radiographic images of healthy controls and osteoporotic patients. The findings consistently show significant differences in fractal dimension values, supporting the use of fractal analysis as a noninvasive diagnostic tool [10]. Combining fractal analysis with other diagnostic measures, such as ultrasound broadband attenuation T-scores, enhances the overall assessment of bone health. This combination allows for a more detailed evaluation of bone microarchitecture and structural integrity, providing a comprehensive diagnostic approach [9]. This integrated method not only improves the detection of osteoporotic changes but also helps monitor the progression of disease and evaluate the effectiveness of treatment interventions [9] (Figure 2).

3.2. Breast Cancer Detection

The integration of fractal analysis into computer-aided diagnosis (CAD) systems has significantly enhanced the detection and classification of breast lesions in ultrasound images [14,15,16,17]. CAD systems employ techniques such as fractal Brownian motion and k-means clustering, which, when combined with preprocessing methods like histogram equalization, improve the accuracy of distinguishing between benign and malignant tumors. The use of these techniques allows for detailed texture analysis, thus increasing the precision of cancer detection. Studies have demonstrated that CAD systems incorporating fractal analysis achieve high accuracy rates, validating the efficacy of these methods in breast cancer diagnostics. Specifically, deep learning approaches, such as transferable texture convolutional neural networks (TTCNNs), have been shown to outperform traditional methods, achieving accuracies as high as 97.49% in classifying mammograms [18]. Recent advancements in high-definition microvasculature imaging (HDMI) have introduced novel quantitative biomarkers that significantly enhance breast cancer detection. These biomarkers include microvessel fractal dimension (mvFD), Murray’s deviation, bifurcation angle and spatial vascularity pattern. The detailed analyses of tumor microvessel structures provided by these biomarkers have increased the sensitivity and specificity of diagnostic models [14,15,16,17]. Studies [14,15,16,17] have indicated that the inclusion of mvFD and other biomarkers in diagnostic frameworks improves the accuracy of differentiation between benign and malignant breast lesions. This precise characterization of tumor vasculature is crucial for early detection and effective treatment planning [14,15,16,17]. mvFD is particularly useful in identifying the structural complexity of tumor vasculature. Malignant tumors often exhibit higher mvFD due to the irregular and chaotic nature of cancerous blood vessel growth compared to the more orderly vessel structures found in benign tumors. This differentiation is vital for accurate diagnosis and treatment planning. The higher mvFD in malignant tumors reflects their aggressive nature and helps predict tumor behavior and potential metastasis. By incorporating mvFD into imaging analyses, clinicians can achieve more accurate and detailed assessments of tumor pathology, thus enhancing the overall effectiveness of breast cancer management [14,15,16,17].

The integration of fractal analysis and advanced imaging techniques such as HDMI in CAD systems has revolutionized breast cancer diagnostics. These methods enhance the accuracy of detecting and classifying breast lesions and provide detailed insights into tumor structure and behavior. The high precision and reliability of these techniques underscore their importance in clinical practice, improving early detection, treatment planning and ultimately patient outcomes (Figure 2).

3.3. Lung Cancer Detection

Endobronchial ultrasound (EBUS) is a minimally invasive procedure used primarily for the staging and diagnosis of lung cancer. EBUS combines traditional bronchoscopy with US technology to provide detailed images of the lungs and surrounding lymph nodes. This method allows for real-time needle aspiration of lymph nodes and masses, aiding in the accurate staging of lung cancer and other thoracic diseases [19]. Fractal dimension analysis has been applied to EBUS images to differentiate between malignant and benign mediastinal nodes. This technique measures the complexity of structures within the US images, with malignant nodes typically exhibiting lower fractal dimension than benign nodes [20]. This distinction is due to the irregular and chaotic nature of cancerous tissue growth, which is less complex geometrically than benign tissues [20]. Studies have shown that fractal dimension analysis significantly improves the ability to differentiate between malignant and benign nodes. In one study, the fractal dimension was found to be significantly lower in malignant nodes, with an area under the ROC curve of 0.76, indicating good diagnostic performance [19,20]. EBUS–TBNA (transbronchial needle aspiration) is less invasive than traditional surgical methods, reducing patient risk and recovery time. It provides a high diagnostic yield with minimal complications, making it a preferred method for lung cancer staging [21]. The combination of EBUS with advanced image processing techniques, including fractal dimension analysis, enhances the visual and quantitative assessment of lymph nodes. This allows for better targeted biopsies and more accurate diagnoses [19,20,21]. EBUS is particularly useful in staging non-small cell lung cancer (NSCLC) by assessing the involvement of mediastinal and hilar lymph nodes. Accurate staging is crucial for determining the appropriate treatment plan and prognosis [20]. The ability to distinguish between malignant and benign nodes helps to identify metastatic disease, which is critical for comprehensive cancer management [20]. The implementation of fractal dimension analysis in routine EBUS procedures could standardize and improve diagnostic accuracy, offering a reliable tool for thoracic oncologists [19,20,21] (Figure 2).

3.4. Salivary Glands

Elastography is a noninvasive imaging technique that measures tissue stiffness and provides valuable insights into various pathological conditions. It is particularly useful for assessing the submandibular glands, where changes in tissue elasticity can indicate the presence of diseases, such as inflammation, sialadenitis and tumors [22]. Fractal analysis applied to elastographic images helps to identify pathological changes in the submandibular gland by quantifying the complexity of tissue structures. Higher fractal dimension values in elastographic images are indicative of greater structural complexity due to inflammation or other pathological processes. This advanced analysis enhances the diagnostic accuracy of elastography, making it a powerful tool for noninvasive diagnosis of salivary gland disease [22]. Elastographic studies have shown that higher fractal dimension values correspond to areas of increased stiffness and complexity in inflamed submandibular glands. This correlation helps diagnose conditions like acute or chronic sialadenitis by distinguishing between healthy and diseased tissues [23]. Fractal analysis can differentiate between benign and malignant tumors in the submandibular gland. Malignant tumors generally exhibit more chaotic and complex structures, reflected in higher fractal dimension values compared to the more regular structure of benign tumors. This distinction aids in accurate diagnosis and appropriate treatment planning [23]. Moreover, elastography combined with fractal analysis is effective in assessing the involvement of salivary glands in Sjogren’s syndrome [23,24]. Chikui et al. [23] reported that patients with Sjogren’s syndrome exhibited higher fractal values in their salivary glands, indicating significant structural changes due to the autoimmune process.

3.5. Pancreatic Cancer Delineation

Enhanced endoscopic ultrasound (EUS) imaging is a vital tool in the diagnosis and characterization of pancreatic lesions. This advanced imaging technique combines traditional EUS with fractal analysis and other advanced methods to improve lesion delineation, aiding in more accurate diagnosis and effective clinical decision making [25]. Fractal analysis has been integrated into EUS to quantify the complexity of pancreatic lesion structures. Malignant lesions tend to exhibit higher fractal dimensions due to their irregular and chaotic growth patterns compared to the more orderly structures of benign lesions. Spadaccini et al. [26] showed that fractal geometry can effectively quantify surface roughness, which helps distinguish different types of pancreatic tissue [26,27]. The integration of fractal analysis, contrast enhancement and elastography into endoscopic ultrasound imaging significantly enhances the diagnostic capabilities for pancreatic lesions.

3.6. Prostate Cancer Detection

In the domain of prostate cancer detection, an advanced analytical approach has been developed that combines fractal analysis with discrete Fourier analysis of ultrasound radiofrequency (RF) [28]. This innovative technique utilizes a set of features derived from the frequency spectrum of RF signals, significantly improving the accuracy of prostate cancer diagnostics. By leveraging the discrete Fourier transform (DFT) of ultrasound RF time series, six features representing the frequency spectrum were extracted and used in conjunction with neural network classifiers to detect cancerous regions as small as 0.03 cm² [28]. Previous studies have established that the fractal dimension of these RF signals correlates strongly with tissue microstructure. Building on this, the new approach incorporates additional spectral features to enhance diagnostic performance. In practical applications, this method has achieved remarkable results, demonstrating a mean sensitivity of 92% and a specificity of 90% in detecting prostate cancer, as validated by detailed pathology results from human prostate specimens [29]. This technique’s efficacy is underscored by its ability to identify malignancies in very small tissue regions, which is a critical factor in early and accurate cancer detection. The robustness of this method offers promising advancements in the field of medical imaging and prostate cancer diagnostics, providing a noninvasive, reliable and highly sensitive tool for identifying cancerous tissues [29]. The combination of discrete Fourier and fractal analyses of ultrasound RF time series represents a substantial advancement in prostate cancer diagnostics. Its high accuracy and noninvasive nature make it a valuable tool for clinicians, offering the potential to improve early detection and treatment outcomes for prostate cancer patients [30].

3.7. Skin and Wound Healing

High frequency ultrasound (HFU), combined with fractal analysis, offers a sophisticated approach to evaluating the healing process of pressure sores. By extracting parameters that relate to the echographic structure and attenuation properties of tissue, researchers have developed a quantifiable healing function that facilitates monitoring the progression of wound generation and healing [31]. This quantitative method provides a powerful tool for the early detection and continuous monitoring of pressure sores, which is particularly advantageous for patients with impaired mobility. The integration of HFU and fractal analysis allows for a detailed assessment of skin and tissue changes at the microscopic level. Fractal analysis, a method used to evaluate complex geometric patterns, can detect subtle variations in tissue structure that traditional imaging techniques might miss. This capability is crucial in tracking the intricate process of wound healing, which involves multiple phases, including inflammation, tissue formation and remodeling [31]. Mirpuri et al. [31] reported that HFU was used to analyze skin changes in pregnant women, revealing significant variations in skin thickness and structure between hypertensive and nonhypertensive pregnancies based on skin structure changes detected by HFU, thus offering a predictive value for potential complications [31]. The combination of HFU and fractal analysis represents a promising advancement in the field of wound care. This method not only enhances the early detection and monitoring of pressure sores, but also offers a detailed understanding of the healing process, ultimately improving patient outcomes and quality of life [31].

3.8. Pregnancy

The integration of advanced fractal analysis and Doppler ultrasound for evaluating umbilical artery blood flow has significantly enhanced the accuracy of fetal health monitoring [32]. Doppler ultrasound remains a noninvasive, reliable method for assessing vascular perfusion within the umbilical artery, which is critical for fetal well-being. By emitting ultrasound waves that reflect off moving red blood cells, the Doppler system measures shifts in frequency proportional to blood flow velocity, offering detailed insights into fetal circulation. Recent advancements in spectral analysis methods, including the use of the Hurst exponent for the calculation of fractal dimensions, have further refined the assessment of umbilical artery Doppler signals [32,33]. Latifoğlu et al. [33] explored the efficacy of these fractal dimension curves in comparison to traditional Doppler indices, demonstrating the superior sensitivity of the PSDHURST index in detecting blood flow changes during pregnancy. Doppler signals were collected from 20 pregnant women with normal pregnancies between 18–24 weeks and 32–40 weeks of gestation. Data acquisition was conducted using a Doppler ultrasound unit with a consistent insonation angle and preset ultrasound settings to ensure data integrity. Signal processing involved denoising using wavelet soft thresholding, spectral analysis through AR methods and extraction of maximum frequency envelopes. The traditional Doppler indices-resistance index (RI), pulsatility index (PI) and systolic-to-diastolic ratio (S) were calculated from these envelopes. Additionally, fractal dimension curves were derived using the Hurst exponent, providing a novel metric for evaluating umbilical artery blood flow [33]. Fractal analysis was performed to quantify the Doppler blood flow signal’s complexity. The Hurst exponent was used to estimate fractal dimensions together with the variance fractal dimension (VFD). Power spectral density (PSD) graphics were generated using autoregressive methods and the Hurst exponent values from PSD curves (PSDHURST) were evaluated to monitor changes in blood flow velocity with gestational age. However, the PSDHURST index demonstrated higher sensitivity in detecting blood flow changes with ROC curve analysis, showing an area under the curve (AUC) of 0.97 for the PSDHURST index, compared to 0.931, 0.959 and 0.938 for RI, PI and S. The findings suggest that fractal dimension analysis and PSDHURST indices provide more robust metrics for evaluating fetal and placental circulation [33]. These methods offer a significant advancement over traditional Doppler indices, facilitating the earlier detection of potential complications in fetal development.

Moreover, Miron et al. [34] highlighted the critical role of fractal analysis in the detection of microcalcifications in the placenta during pregnancy [34]. Fractal dimension was used as a key feature in the analysis due to its ability to characterize the complexity and self-similarity of structures within the image. In the context of this study, fractal dimension is computed to capture the fine structural details of the microcalcifications, which are essential for accurate detection. The fractal dimension is particularly useful in edge enhancement, which is a crucial preprocessing step in this methodology. By applying the Sobel and Laplacian of Gaussian (LoG) filters, the edges of the gray level image were detected and the fractal dimension was computed after these filters were applied. The box-counting method was used to calculate the fractal dimension of the 2D US images. This method provides fractal dimension values in the range of 1 to 2, indicating the roughness and detail of the image [34]. In this study, a high-pass filtering method was implemented, followed by contrast enhancement to ensure that the small details, such as microcalcifications, were more visible. The fractal dimension, when combined with other textural features, contributed to the overall improvement in classification accuracy. The proposed method, which included fractal analysis, showed superior performance in detecting microcalcifications compared to raw image analysis. The inclusion of fractal dimension in the feature set enhanced the ability to detect and classify microcalcifications more accurately, helping to distinguish between healthy and calcified regions by capturing the intrinsic texture and edge details that are otherwise challenging to detect with conventional methods [34] (Figure 2).

3.9. Heart and Blood Vessels

Mahon et al. [35] investigated the fractal correlation properties of R–R interval dynamics in asymptomatic relatives of patients with familial dilated cardiomyopathy (DCM). The authors utilized the short-term scaling component from detrended fluctuation analysis to measure heart rate variability (HRV) as a prognostic tool. The study compared 22 asymptomatic relatives with left ventricular enlargement (LVE), 24 DCM patients and 14 controls. The results showed that, while traditional HRV indices, such as the standard deviation of NN intervals (SDNN), were reduced only in DCM patients, the short-term scaling component index was significantly lower in both DCM patients and LVE relatives compared to controls. This suggests that fractal measures may be more sensitive in detecting early autonomic dysfunction in individuals at risk of DCM progression [35].

Looking at blood vessels, Moroni et al. [36] showed a new fractal analysis-based technique to quantitatively assess the irregularity of atherosclerotic plaque borders. This method is aimed at improving the prediction of plaque instability and correlating it with cardiovascular risk factors. The study involved 42 asymptomatic subjects with carotid stenosis. Carotid ultrasound evaluations were performed and blood samples were analyzed for lipid profiles, including HDL and triglycerides. Fractal dimension was used as a measure of the complexity and irregularity of the plaque surface and was computed using the box-counting via FraLac plugin. The fractal dimension was calculated for the main plaques (mFD) and globally across all plaques (gFD) for each patient [36]. The mean mFD was 1.136 ± 0.039 and the mean gFD was 1.145 ± 0.039. There was a significant inverse correlation between mFD and plasma HDL levels (r = −0.367, p = 0.02) and a positive correlation between mFD and the triglycerides-to-HDL ratio (r = 0.480, p = 0.002). Fractal dimension showed good reproducibility with high inter- and intra-observer agreement [36]. The study suggested fractal analysis as a feasible and reproducible method for quantifying plaque irregularity, which correlates with lipid profiles, indicating a potential marker for plaque instability [36]. Fractal analysis offers a quantitative and reproducible approach to evaluating plaque irregularity, which correlates with lipid profiles. This technique could aid in the identification of patients at higher risk of cardiovascular events due to plaque instability.

In the realm of cardiovascular medicine, the accurate diagnosis of coronary artery stenosis is pivotal for the effective management of patients with ischemic heart disease. Yong et al. [37] explored the diagnostic performance of intravascular ultrasound-based fractional flow reverse (IVUS–FFR) analysis utilizing a generative adversarial network (GAN) and bifurcation fractal law. The study was conducted with a retrospective design, including data from 87 patients with 108 vessels. The participants underwent both IVUS and invasive FFR measurements. The IVUS–FFR analysis was performed by analysts blinded to the invasive FFR values. The methodology involved segmentation of IVUS images using GAN (SegAN), followed by an IVUS–FFR calculation considering side branch flow through bifurcation fractal law [37]. The IVUS–FFR calculation, conducted in several steps, involved extracting the vessel’s centerline and cross-sectional area, splitting the centerline into segments, detecting stenosis and solving the Navier–Stokes equations to determine pressure distribution [37]. The bifurcation fractal law, specifically the HK model, was used to account for side branch blood flow, enhancing the accuracy of the single-tube coronary artery model. The study highlighted the effectiveness of IVUS–FFR analysis using SegAN for IVUS image segmentation and the bifurcation fractal law for side branch blood flow consideration. This method not only aligns well with invasive FFR but also reduces computation time, making it a viable alternative to traditional methods [37]. The HK model outperformed the Finet and Murray models in terms of diagnostic accuracy, making it the preferred choice for IVUS–FFR calculations. The IVUS–FFR analysis, based on a generative adversarial network and bifurcation fractal law, demonstrates excellent diagnostic performance and efficiency in assessing myocardial ischemia. This approach offered a significant advancement in the noninvasive diagnosis of coronary artery disease, providing accurate and timely results that can guide clinical decision making [37] (Figure 2).

3.10. Thyroid

Acharya et al. [38] showed a computer-aided diagnostic (CAD) system for the classification of thyroid nodules into benign and malignant categories using US images. Their system leveraged a combination of texture features, including fractal dimension, local binary pattern, Fourier spectrum descriptor and laws texture energy, to quantify local changes in texture characteristics. These features were then used to train various classifiers, such as a support vector machine (SVM), decision tree (DT), Sugeno fuzzy, Gaussian mixture model (GMM), K-nearest neighbor (KNN), radial basis probabilistic neural network (RBPNN) and naïve Bayes classifier (NBC). The system achieved 100% accuracy for high-resolution ultrasound (HRUS) images and 98.1% accuracy for contrast-enhancement ultrasound (CEUS) images [38]. A novel integrated index, the thyroid malignancy index (TMI), was proposed to help clinicians make more objective distinctions between benign and malignant nodules. The main innovation was the use of histogram analysis and segmentation-based fractal texture analysis (SFTA), which are not dependent on the orientation of the ultrasound probe. The system was tested on 40 thyroid nodules (20 malignant and 20 benign) and used features such as histogram parameters, fractal dimension and mean brightness value in different grayscale bands. The study achieved an overall accuracy of 92.42% using random forests and 94.64% using a support vector machine (SVM) with leave-one-out cross-validation, highlighting the limitations of fine needle aspiration (FNA) biopsy and suggesting that the CAD system could provide a valuable second opinion and thus reduce unnecessary surgeries [39]. Raghavendra et al. [40] developed a novel CAD system combining spatial gray-level dependence features (SGLDF) and fractal textures to analyze the intrinsic structure of thyroid lesions. The features were then subjected to marginal Fisher analysis (MFA) for dimensional reduction and ranking before classification [40]. The system achieved an average accuracy of 97.52%, a sensitivity of 90.32% and a specificity of 98.57% using the support vector machine (SVM) classifier. The study developed the thyroid clinical risk index (TCRI), a single numerical index, to differentiate between benign and malignant lesions effectively. The fusion of texture features with MFA significantly enhanced the classification performance, providing a robust tool for clinical use [40] (Figure 2).

3.11. Muscle

Moradi et al. [41] explored the application of fractal dimension analysis to high-frequency ultrasound RF time series for tissue characterization [41]. The key premise is that variations in RF echo intensity over time, recorded from a fixed tissue location, correlate with the tissue’s microstructure. This method leverages the fractal dimension as a feature to classify tissues based on their microstructural differences. RF signals were continuously recorded from fixed tissue samples using high-frequency ultrasound probes. The fractal dimension of these time series was calculated and used to classify different tissue types. The study used tissues such as bovine liver, pig liver, bovine muscle and chicken breast, capturing their unique microstructural features. The fractal dimension values from the RF time series effectively distinguished between tissues with different microstructures, achieving classification accuracies as high as 98% for segments as small as 20 µm. Statistical analyses confirmed significant differences in fractal dimension values between different tissue types, suggesting that RF time series capture microstructure-related information [41]. This approach showed promise for noninvasive tissue characterization, potentially aiding in the diagnosis of pathology conditions like cancer, where microstructural changes are pronounced [41]. Therefore, high-frequency ultrasound combined with fractal analysis could also be a diagnostic tool for distinguishing various tissue types in muscle tissue based on their microstructural properties.

Additionally, Miron Mombiela et al. [42] investigated the diagnostic performance of muscle echo intensity (EI) and fractal dimension derived from US images to identify frailty in older adults. Frailty, characterized by decreased muscle mass and strength, poses diagnostic challenges, which this study aimed to address using noninvasive US techniques. In this retrospective analysis of US scans, Miron Mombiela et al. [42] evaluated participants classified into nonfrail and frail groups based on Fried’s criteria. Echo intensity was measured from the rectus femoris muscle and the fractal dimension was calculated using D box-counting techniques. A positive correlation was found between muscle EI and fractal dimension (r = 0.38) and both parameters displayed distinct patterns when comparing nonfrail to frail individuals. In addition, EI showed significant diagnostic accuracy (AUC = 0.69) for categorizing frailty, with high intrarater and interrater reliability. While fractal dimension correlated with EI, it did not independently improve diagnostic accuracy over EI alone [42]. EI proved to be a useful marker for distinguishing nonfrail from frail individuals, while fractal dimension provided additional insights into the muscle architecture changes associated with frailty. Further validation and normative data are required for clinical application [42] (Figure 3).

4. Discussion

This narrative review is the first comprehensive attempt to investigate and understand the use and methodology of fractal analysis in the field of US imaging. Effective preprocessing is a cornerstone for the success of fractal analysis in medical imaging, particularly in ultrasound imaging. Preprocessing techniques, such as noise filtering, morphological operations and histogram equalization, are pivotal in ensuring that ultrasound mages exhibit consistent gray levels and enhanced contrast [43]. These preprocessing steps are crucial for accurate fractal dimension estimation, which is vital for reliable diagnostic outcomes. Various studies [14,15,16,17] have demonstrated that preprocessing methods, such as histogram equalization, significantly improve the accuracy of lesion classification in breast ultrasound images. This improvement in image quality influences the precision of medical diagnoses, facilitating better differentiation between normal and pathological tissues [44].

In addition to basic preprocessing methods, more advanced techniques, such as wavelet transforms and anisotropic diffusion filtering, have been developed to further enhance image quality. Wavelet transforms are useful for the multiresolution analysis of images, making it easier to identify patterns across different scales, while anisotropic diffusion filtering helps preserve edge information while reducing noise, which is especially beneficial in US imaging [44]. The refined preprocessing approaches are essential for the computation of fractal dimensions, which involve sophisticated methodologies like the box-counting method, the structure function method and multifractal detrended fluctuation analysis. These methods examine the spatial and structural relationships within the image, providing insights into the complexity and roughness of biological tissues [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. Advances in algorithms have further improved the precision of fractal analysis, making it a robust tool in medical diagnostics [44]. For instance, Higuchi’s algorithm is used to calculate the fractal dimension of radiofrequency (RF) echo time series, showing high accuracy in distinguishing tissue types based on their microstructural properties [44].

Advanced preprocessing methods not only improve the visual quality of US images but also play a critical role in enhancing the qualitative accuracy of subsequent analyses. For example, wavelet-based denoising techniques allow for the separation of important image features from background noise, which is crucial for maintaining the integrity of tissue boundaries and other diagnostically relevant structures. Furthermore, the use of adaptive filtering techniques helps tailor the preprocessing steps to the specific characteristics of different types of US images, thereby optimizing the analysis for diverse clinical scenarios. Anisotropic diffusion filtering, in particular, is effective in maintaining edge sharpness while suppressing unwanted speckle noise, which is a common challenge in US imaging. This balance between noise reduction and edge preservation is essential for improving the reliability of fractal dimension calculations, as it ensures that the fractal analysis is based on accurate and representative imaging data. The computation of fractal dimensions involves several advanced algorithms that allow for a detailed analysis of tissue structures. The box-counting method, for instance, is widely used due to its simplicity and effectiveness in estimating fractal dimensions by analyzing the spatial distribution of pixel intensities. The structure function method and multifractal detrended fluctuation analysis provide deeper insights into the heterogeneous nature of biological tissues by capturing variations in texture and complexity at multiple scales. These methods are valuable in differentiating between healthy and pathological tissues, as they can reveal subtle changes in tissue architecture that may not be apparent through conventional imaging techniques. Recent developments in computational power and algorithmic efficiency have further enabled real-time fractal analysis, making it feasible to integrate these sophisticated methods into routine clinical practice.

Moreover, the use of multifractal analysis enables a more detailed characterization of tissue heterogeneity. Techniques such as calculating the multifractal spectrum provide a comprehensive understanding of the spatial distribution of tissue properties, which is crucial for identifying pathological changes at an early stage [4,43,44]. The continuous development of preprocessing techniques and computational methods ensures that fractal analysis remains at the forefront of noninvasive US imaging technologies. By integrating fractal analysis into routine diagnostic workflows, healthcare professionals are empowered to make more informed decisions, ultimately leading to improved patient outcomes [43].

Fractal analysis has shown significant potential in various clinical fields, including bone texture analysis, breast cancer detection, lung cancer detection, salivary gland disease assessment, pancreatic lesion delineation, prostate cancer detection, skin and wound healing, pregnancy monitoring, cardiovascular assessments and thyroid nodule classification. For instance, in bone texture analysis, fractal analysis has been used on a radiographic image to evaluate trabecular bone patterns and assess osteoporosis risk [9,10]. Significant differences in fractal dimension values between healthy and osteoporotic patients suggest that this technique can be a reliable noninvasive diagnostic tool. Furthermore, integrating fractal analysis with other measures, such as ultrasound broadband attenuation T-scores, could enhance the assessment of bone health and provide a more comprehensive diagnostic approach. This integration is crucial in monitoring the progression of osteoporosis and in evaluating treatment effectiveness [9]. In breast cancer detection, integrating fractal analysis into computer-aided diagnosis (CAD) systems has enhanced the detection and classification of breast lesions in US images. The combination of fractal analysis with advanced techniques, such as Brownian motion, k-means clustering and deep learning models, has led to high accuracy in distinguishing between benign and malignant tumors [14,15,16,17,18]. The inclusion of novel biomarkers, such as microvessel fractal dimension (mvFD), has improved sensitivity and specificity in breast cancer diagnostics. The higher mvFD observed in malignant tumors reflects their aggressive nature, which is crucial for early detection and effective treatment planning. Similarly, in lung cancer detection, the application of fractal analysis to endobronchial ultrasound (EBUS) images has shown promise in differentiating between malignant and benign mediastinal nodes. The lower fractal dimension observed in malignant nodes compared to benign nodes indicates its potential utility in staging lung cancer and guiding targeted biopsies [19,20,21]. The integration of fractal dimension analysis in EBUS procedures could standardize and improve diagnostic accuracy, offering a reliable tool for thoracic oncologists. Fractal analysis has also demonstrated its value in assessing salivary gland diseases through elastography. By quantifying tissue stiffness and complexity, fractal dimension analysis enhances the diagnostic accuracy of elastography images, helping to distinguish between healthy and diseased tissues, such as in cases of sialadenitis or tumors [22,23,24]. This capability is further extended to autoimmune diseases like Sjogren’s syndrome, where fractal values indicate significant structural changes in the salivary glands [23]. In pancreatic cancer delineation, enhanced endoscopic ultrasound (EUS) imaging combined with fractal analysis has proven effective in quantifying the complexity of pancreatic lesions. Malignant lesions tend to exhibit higher fractal dimensions due to their irregular and chaotic growth patterns when compared to benign lesions [25,26,27]. This integration of fractal analysis with contrast enhancement and elastography in EUS imaging significantly enhances the diagnostic capabilities for pancreatic lesions. In prostate cancer detection, an advanced analytical approach that combines fractal analysis with discrete Fourier analysis of ultrasound radiofrequency (RF) has demonstrated remarkable accuracy. By leveraging features from the RF frequency spectrum, this method has achieved high sensitivity and specificity, providing a reliable noninvasive diagnostic tool [28,29,30]. In skin and wound healing, high-frequency ultrasound (HFU), combined with fractal analysis, offers a sophisticated approach to evaluating the healing process of pressure sores. Fractal analysis can detect subtle variations in tissue structure, providing crucial insights into wound healing phases, which is important for monitoring and improving patient care [31]. Fractal analysis has also shown utility in pregnancy monitoring. By integrating advanced fractal analysis with Doppler ultrasound, clinicians can achieve more precise assessments of fetal health, including monitoring umbilical artery blood flow and detecting structural changes in the placenta [32,33,34]. This approach enhances the accuracy of fetal health assessments, offering early detection of potential complications. In cardiovascular assessments, fractal analysis has been employed to evaluate the complexity of atherosclerotic plaques and heart rate variability, providing insights into plaque instability and autonomic dysfunction, respectively [35,36,37]. These applications underscore the potential of fractal analysis to improve risk assessment and early intervention in cardiovascular diseases. Finally, in thyroid nodule classification, computer-aided diagnostic (CAD) systems that incorporate fractal analysis have shown high accuracy in differentiating between benign and malignant thyroid nodules. The integration of fractal dimension with other texture features enhances diagnostic performance, offering a valuable tool for clinicians [38,39,40].

Future research and clinical trials will play a key role in validating and expanding the applications of fractal analysis in ultrasound imaging, solidifying its position as a vital tool in modern diagnostics. The exploration of novel biomarkers and advancements in high-definition ultrasound imaging techniques. such as dynamic ultrasound assessment and contrast-enhanced ultrasound, highlights the ongoing innovations and potential of fractal analysis in enhancing diagnostic capabilities. Additionally, integrating artificial intelligence and machine learning with fractal analysis holds significant promise for improving diagnostic accuracy efficiency. These emerging technologies can automate complex analytical processes, thus making advanced diagnostic tools more accessible and further elevating the role of fractal analysis in medical imaging. Proposed future application directions should also be aligned with specific clinical and analytical needs. For example, the integration of artificial intelligence and machine learning with fractal analysis could address the need for the automated identification of fractal features that correlate with specific disease markers, thereby streamlining the diagnostic process by US imaging. Additionally, the development of predictive models based on fractal characteristics and patient data could assist in early disease detection and personalized treatment planning. These directions underscore the importance of tailoring future advancements to address real-world clinical challenges, ultimately ensuring that the integration of fractal analysis continues to meet evolving diagnostic needs. Each of these applications demonstrates the versatility and potential of fractal analysis as a powerful tool across US imaging in diverse medical fields. By aligning these advancements with specific clinical needs, fractal analysis can be more effectively integrated into standard diagnostic workflows, ultimately leading to better patient care and outcomes.

5. Conclusions

Fractal analysis significantly enhances the diagnostic capabilities of ultrasound imaging by providing a detailed texture characterization of tissues. Its applications span various fields, including breast cancer detection, lung cancer diagnosis, pancreatic cancer detection, osteoporosis assessment, hypertensive disorders in pregnancy and wound healing. By quantifying the complexity of biological structures, fractal analysis aids in distinguishing between normal and pathological conditions, improving the accuracy and reliability of diagnoses. This technique is widely used in the ultrasound field due to its ability to provide critical insights into various conditions, thereby supporting clinicians in making informed decisions.