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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Life Science, Biophysics".

Deadline for manuscript submissions: closed (30 January 2026) | Viewed by 10254

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Dr. Ivan Zaletel

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Institute of Histology and Embryology “Aleksandar Đ. Kostić”, Faculty of Medicine, University of Belgrade, 11000 Belgrade, Serbia
Interests: neuroscience; image analysis; morphometry; fractal theory; medical education

Special Issue Information

Dear Colleagues,

Ever since Benoit Mandelbrot coined the term fractal and paved the way for the development of fractal geometry and fractal analysis, researchers have found a way to utilize this non-linear analysis method in diverse areas of natural sciences. During the previous two decades, there has been a tremendous increase in the number of published papers that relied on the use of fractal analysis to measure the complexity of a certain geometrical figure. The complexity is expressed as a fractal dimension, which depicts the space-filling properties of different irregularly shaped objects that can be found in various living and non-living objects. In the vast field of biomedical sciences these objects usually represent various forms of cells, their ultrastructural parts, such as the nucleus, chromatin, and organelles, or specific cell protrusions like dendrites, but also tissue elements or anatomical units of organs. Traditional quantitative, semiquantitative, and qualitative techniques of analysis of these structures provide results of limited value since these conventional methods are not able to detect subtle differences in histological images. However, fractal analysis has now proven to be a useful tool for the detection of those changes in biological objects obtained from both physiological and pathological conditions. In addition to objects obtained from histological images, fractal analysis is increasingly used in the analysis of various images obtained using different imaging methods and biological signals, such as the complexity-based analysis of brain and muscle electrical activity. Although fractal analysis has found its applicability in various fields of biomedical sciences, neuroscience research remains a major field of its utilization, due to its ability to quantify complex and branching structures, such as dendritic tree arborizations. Its simplicity, sensitivity, and cost-effectiveness allow researchers numerous possibilities to implement this non-linear method of analysis in their investigative work.

This Research Topic will welcome diverse types of articles including original research articles, review articles, technical note articles, and perspective articles related to the application of fractal analysis in different biomedical fields.

Topics of interest include diverse biomedical research areas as follows:

Fractals and fractal theory in biomedical sciences;
Fractal analysis of cell and tissue histological images;
Application of fractal analysis methods in neuroscience research;
Fractal dimension as a complexity discrimination method between physiological and pathological conditions;
Fractal analysis of various images and signals in medical diagnostics;
Multi-fractal analysis of biomedical images;
Utilization of related non-linear analysis methods in biomedical research.

Dr. Ivan Zaletel
Guest Editor

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Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

fractal analysis
fractal dimension
complexity
self-similarity
neuroscience
morphometry
cell and tissue analysis

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Published Papers (6 papers)

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Research

14 pages, 1004 KB

Open AccessArticle

Optimization of Region-of-Interest Configuration for Fractal Analysis of Peri-Implant Bone on Panoramic Radiographs

by Devrim Deniz Üner, Bozan Serhat İzol, Remzi Boynukara and Nezif Çelik

Fractal Fract. 2026, 10(4), 215; https://doi.org/10.3390/fractalfract10040215 - 26 Mar 2026

Viewed by 406

Abstract

Objective: The aim of this study was to determine the optimal region-of-interest (ROI) pixel size for fractal dimension analysis on panoramic radiographs that best reflects implant stability assessed by resonance frequency analysis (ISQ) and to investigate whether implant stability can be directly estimated from radiographic images. Materials and Methods: This retrospective cross-sectional study included 65 patients for whom panoramic radiographs and resonance frequency analysis measurements were available. All panoramic images were converted to TIFF format and standardized to a resolution of 2627 × 1646 pixels. All radiographic images were obtained using the same panoramic imaging device and standardized acquisition protocol. Exposure parameters were adjusted within the manufacturer’s recommended range to ensure optimal image quality while maintaining methodological consistency across patients. During ROI selection, care was taken to avoid cortical bone margins, overlapping anatomical structures, and radiographic artifacts in order to ensure that the analyzed regions represented trabecular bone adjacent to the implant surface. Fractal dimension analysis was performed in the cervical peri-implant bone region, starting from the first bone–implant contact and extending apically, using three different ROI configurations. The ROI size was defined as 30 pixels apically and 10 pixels horizontally for FMD1, 30 × 20 pixels for FMD2, and 30 × 30 pixels for FMD3. Implant stability was assessed using ISQ values. Data distribution was evaluated using the Shapiro–Wilk test. Associations between ISQ and fractal dimension measurements were analyzed using Pearson and Spearman correlation analyses. Multiple linear regression models adjusted for age and sex were constructed to assess independent associations. Results: The mean age of the participants was 50.0 ± 9.9 years, and the mean ISQ value was 78.6 ± 5.9. The mean fractal dimension values were 1.466 ± 0.055 for FMD1, 1.595 ± 0.031 for FMD2, and 1.655 ± 0.046 for FMD3. No significant association was found between ISQ and FMD1 or FMD3. A weak positive correlation was observed between ISQ and FMD2; however, this association did not remain statistically significant after correction for multiple comparisons. In multiple linear regression analysis, ISQ was identified as an independent predictor of FMD2, but not of FMD1 or FMD3. Age and sex had no significant effect on fractal dimension measurements. Conclusions: Fractal dimension measurements derived from panoramic radiographs showed a weak association with implant stability that was dependent on the selected ROI pixel size. Among the evaluated configurations, the 30 × 20-pixel ROI at the cervical peri-implant region demonstrated the strongest association with ISQ values, suggesting that this ROI configuration showed the most consistent association with ISQ values among the tested ROI sizes. Full article

(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)

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9 pages, 680 KB

Open AccessArticle

Limitations of Panoramic Radiograph-Based Fractal Dimension Analysis in Detecting Mandibular Trabecular Changes in Type 2 Diabetes Mellitus

by Oya Törün, Nihat Laçin and Fatih Cabbar

Fractal Fract. 2026, 10(3), 144; https://doi.org/10.3390/fractalfract10030144 - 26 Feb 2026

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Abstract

Background: Fractal dimension (FD) analysis has been widely applied to medical and dental images to quantify trabecular bone complexity. Panoramic radiographs are routinely used in dental practice; however, the capability of two-dimensional panoramic imaging combined with FD analysis to detect subtle mandibular trabecular alterations associated with systemic diseases such as type 2 diabetes mellitus (T2DM) remains uncertain. Methodology: This retrospective cross-sectional study included 106 individuals, comprising 53 patients diagnosed with T2DM and 53 non-diabetic controls. Fractal dimension values were calculated using a standardized box-counting algorithm from four anatomically defined mandibular regions of interest (anterior, premolar, molar, and condylar) on panoramic radiographs. Intergroup comparisons of FD values were performed to evaluate the sensitivity of panoramic radiograph-based FD analysis in detecting diabetes-related trabecular differences. Results: No statistically significant differences in fractal dimension values were observed between the T2DM and control groups across all evaluated mandibular regions (p > 0.05). These findings highlight methodological limitations related to image dimensionality, projection geometry, and regional trabecular heterogeneity. Conclusions: Fractal analysis remains a valuable quantitative tool; however, its application to panoramic radiographs should be interpreted cautiously when used to assess systemic bone alterations. Full article

(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)

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16 pages, 1508 KB

Open AccessArticle

Fractal and Multifractal Analysis as Methods of Quantifying Dendritic Complexity Changes in the Traumatic Brain Injury Model

by Rada Jeremić, Nemanja Rajković, Sanja Peković, Sanja Dacić, Irena Lavrnja, Ivana Bjelobaba, Marija Jeremić, Vladimir Baščarević, Predrag Brkić, Nebojša T. Milošević and Ivan Zaletel

Fractal Fract. 2025, 9(9), 590; https://doi.org/10.3390/fractalfract9090590 - 9 Sep 2025

Viewed by 1603

Abstract

Background: Traumatic brain injury (TBI) disrupts hippocampal neurogenesis and dendritic structure. Objective: The objective was to assess whether fractal and multifractal analyses can sensitively quantify dendritic complexity changes in newly formed dentate gyrus neurons following TBI and hyperbaric oxygen therapy (HBO). Methods: Adult rats underwent sham surgery with HBO (SHBO), lesion-induced TBI (L), or lesion-induced TBI with HBO (LHBO). Dendritic morphology was evaluated using Euclidean, monofractal, and multifractal metrics. Results: Lesioned animals exhibited marked reductions in dendritic complexity across multiple metrics compared to both HBO-treated groups. HBO treatment partially restored complexity to near-sham levels, with multifractal spectra revealing subtle structural differences between SHBO and LHBO. Conclusions: Fractal and multifractal analyses provide sensitive tools for detecting TBI-induced morphological changes and therapeutic effects. Our findings support HBO as a potential neuroprotective intervention and demonstrate the utility of mathematical modeling in evaluating therapeutic efficacy in neurotrauma. Full article

(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)

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13 pages, 1582 KB

Open AccessArticle

Assessment of Mandibular Trabecular Bone Structure in Hypodivergent Growth Patterns Using Fractal Analysis

by Gizem Boztaş Demir, Rüveyda Doğrugören, Kübra Gülnur Topsakal, Gökhan Serhat Duran and Serkan Görgülü

Fractal Fract. 2025, 9(8), 517; https://doi.org/10.3390/fractalfract9080517 - 8 Aug 2025

Cited by 1 | Viewed by 2152

Abstract

The objective of this study is to evaluate the trabecular structure in hypodivergent individuals using fractal analysis, with a particular focus on specific mandibular regions. This study aims to assess the impact of hypodivergent growth patterns on bone microarchitecture. This research involved a methodological approach using panoramic radiographs to assess trabecular structure at specific regions of the mandible using fractal analyses. The dimensions of the fractals were calculated with the use of the box-counting technique by the software Image J (v1.53c; Bethesda, MD, USA, National Institutes of Health), while the statistical evaluations were carried out with the Jamovi Software (The Jamovi Project, version 2.3.21.0). The study found significant differences in fractal dimension values between hypodivergent individuals and the control group, particularly in the condyle and angulus regions, indicating a less complex trabecular structure in hypodivergent individuals. This study concludes that individuals with a hypodivergent growth pattern exhibit alterations in trabecular bone structure within the mandibular condyle and angulus regions, characterized by reduced complexity. These findings suggest that increased occlusal forces and mechanical stress associated with this growth pattern may contribute to changes in trabecular architecture. Understanding these variations is essential for orthodontic and maxillofacial diagnosis, treatment planning, and biomechanical considerations, particularly in cases requiring vertical dimension management or anchorage control. Full article

(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)

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14 pages, 2847 KB

Open AccessArticle

Linear and Non-Linear Methods to Discriminate Cortical Parcels Based on Neurodynamics: Insights from sEEG Recordings

by Karolina Armonaite, Livio Conti, Luigi Laura, Michele Primavera and Franca Tecchio

Fractal Fract. 2025, 9(5), 278; https://doi.org/10.3390/fractalfract9050278 - 25 Apr 2025

Cited by 1 | Viewed by 1079

Abstract

Understanding human cortical neurodynamics is increasingly important, as highlighted by the European Innovation Council, which prioritises tools for measuring and stimulating brain activity. Unravelling how cytoarchitecture, morphology, and connectivity shape neurodynamics is essential for developing technologies that target specific brain regions. Given the dynamic and non-stationary nature of neural interactions, there is an urgent need for non-linear signal analysis methods, in addition to the linear ones, to track local neurodynamics and differentiate cortical parcels. Here, we explore linear and non-linear methods using data from a public stereotactic intracranial EEG (sEEG) dataset, focusing on the superior temporal gyrus (STG), postcentral gyrus (postCG), and precentral gyrus (preCG) in 55 subjects during resting-state wakefulness. For this study, we used a linear Power Spectral Density (PSD) estimate and three non-linear measures: the Higuchi fractal dimension (HFD), a one-dimensional convolutional neural network (1D-CNN), and a one-shot learning model. The PSD was able to distinguish the regions in α, β, and γ frequency bands. The HFD showed a tendency of a higher value in the preCG than in the postCG, and both were higher in the STG. The 1D-CNN showed promise in identifying cortical parcels, with an 85% accuracy for the training set, although performance in the test phase indicates that further refinement is needed to integrate dynamic neural electrical activity patterns into neural networks for suitable classification. Full article

(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)

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16 pages, 6041 KB

Open AccessArticle

Relating the Morphology of Bipolar Neurons to Fractal Dimension

by Bret Brouse, Jr., Conor Rowland and Richard P. Taylor

Fractal Fract. 2025, 9(1), 9; https://doi.org/10.3390/fractalfract9010009 - 28 Dec 2024

Cited by 1 | Viewed by 2807

Abstract

By analyzing reconstructed three-dimensional images of retinal bipolar neurons, we show that their dendritic arbors weave through space in a manner that generates fractal-like behavior quantified by an ‘effective’ fractal dimension. Examining this fractal weave along with traditional morphological parameters reveals a dependence of arbor fractal dimension on the summation of the lengths of the arbor’s dendrites. We discuss the implications of this behavior for healthy neurons and also for the morphological deterioration of unhealthy neurons in response to diseases. Full article

(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)

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