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Article

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

by
Rada Jeremić
1,†,
Nemanja Rajković
2,†,
Sanja Peković
3,
Sanja Dacić
4,
Irena Lavrnja
3,
Ivana Bjelobaba
3,
Marija Jeremić
5,
Vladimir Baščarević
6,
Predrag Brkić
1,
Nebojša T. Milošević
2 and
Ivan Zaletel
7,*
1
Institute of Medical Physiology “Richard Burian”, Faculty of Medicine, University of Belgrade, 11000 Belgrade, Serbia
2
Department of Biophysics, Faculty of Medicine, University of Belgrade, 11000 Belgrade, Serbia
3
Department of Neurobiology, Institute for Biological Research “Sinisa Stankovic”, National Institute of the Republic of Serbia, University of Belgrade, 11000 Belgrade, Serbia
4
Department of General Physiology and Biophysics, Institute of Physiology and Biochemistry, Faculty of Biology, University of Belgrade, 11000 Belgrade, Serbia
5
Institute of Medical and Clinical Biochemistry “Ljubisa Rakic”, Faculty of Medicine, University of Belgrade, 11000 Belgrade, Serbia
6
Clinical Centre of Serbia, Clinic of Neurosurgery, 11000 Belgrade, Serbia
7
Institute of Histology and Embryology “Aleksandar Ð. Kostić”, Faculty of Medicine, University of Belgrade, 11000 Belgrade, Serbia
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Fractal Fract. 2025, 9(9), 590; https://doi.org/10.3390/fractalfract9090590
Submission received: 2 July 2025 / Revised: 28 August 2025 / Accepted: 30 August 2025 / Published: 9 September 2025
(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)
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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.

1. Introduction

Every year, over 50 million people worldwide experience some form of traumatic brain injury (TBI) [1]. Patients’ symptoms can range from mild to fatal, and they may also experience post-traumatic amnesia and memory impairment, which are linked to hippocampal damage [2,3]. It is believed that injuries to the cortex can indirectly affect the morphology and number of neurons in the dentate gyrus (DG), a vulnerable part of the hippocampus [4,5]. Neuronal damage caused by brain injury can occur due to primary or secondary injury [6].
Adult neurogenesis occurs in specific brain regions, including the subgranular zone (SGZ) of the dentate gyrus (DG) in the hippocampus and the subventricular zone (SVZ) of the lateral ventricles [7]. Adult neural stem cells (NSCs) in the SGZ produce excitatory granule cells of the DG, which integrate into pre-existing neuronal circuits of the granular layer [8]. There is a large amount of research on the impact of cortical injury on adult hippocampal neurogenesis. The conclusions drawn from the results of those studies are complex and seemingly contradictory [9,10,11,12]. To better understand the effect of brain injury on creating new neurons and to study this process as one of the mechanisms of brain recovery after injury, it is necessary to study the effects of different models of traumatic brain damage on hippocampal neurogenesis. Neurons located around the lesion site, which are not directly affected by the primary mechanical damage, are at risk due to the cascade of events that occur during secondary brain damage [13]. All the processes that comprise the cascade of events are intertwined [14], representing potential targets for future therapeutic interventions.
Hyperbaric oxygenation (HBO) represents a therapeutic procedure where the patient intermittently breathes 100% oxygen (O2) at a pressure higher than the atmospheric pressure of 101.3 kPa [15]. HBO achieves therapeutic effects by creating a positive O2 diffusion gradient, increasing its concentration in the blood [16]. Additionally, intermittent exposure to hyperbaric oxygen triggers the activation of cellular mechanisms that also occur during hypoxia, a phenomenon known as the hyperoxia–hypoxia paradox [17,18]. It is known that the brain’s oxygen requirements increase during the acute phase of the injury, indicating the need to provide mitochondria with an additional amount of oxygen to re-establish aerobic metabolism in the brain [19]. HBO protocols used in experimental research exhibit variations in applied pressure, treatment duration, and treatment initiation [20,21]. Regardless, preclinical studies on various experimental models of TBI, spinal cord injury, and neurodegenerative diseases have shown that HBO exhibits a neuroprotective effect [15], as reflected in the reduction of lesion size, severity of damage, and amount of water in the brain [20].
Various morphometric approaches are employed to adequately study and quantify the histological changes associated with different brain diseases. Fractal analysis is one of the methods used with the primary goal of assessing different patterns of complexity in the nervous system [22]. Fractal analysis has proven especially useful in quantifying dendritic arborization complexity, which represents one of the most significant structural changes during aging, as well as in various brain diseases, including TBI [5,22,23,24,25,26]. Our previously published papers showed that SCA as a model of TBI leads to morphological changes of neurons in the SGZ and granular cells [5,26]. Pantic et al. used a grey-level co-occurrence matrix (GLCM) analysis to examine granule neurons in a TBI model. It found subtle changes in their morphology—such as texture uniformity, contrast, and variance—that conventional microscopy misses. Also, the findings in Jeremić et al. suggest that SCA reduces the number of developing neurons and impairs their structural integrity, which may contribute to functional deficits in the affected brain region. Thus, the primary objective of this study was to investigate whether different forms of fractal analysis can be utilized to examine dendritic arborization changes in newly formed neurons in the hippocampus and how these changes are reflected in the states of TBI and HBO therapy.

2. Materials and Methods

2.1. Experimental Groups

The current study was conducted on 15 adult Wistar albino rats, aged 10 weeks. All animals were classified into three groups (SHBO, L, and LHBO), with 5 animals in each group. Group SHBO consisted of animals without lesions on the sensorimotor cortex after surgical intervention (Figure 1: SHBO). Meanwhile, group L included animals where lesions were registered, after the right sensorimotor cortex was removed via suction ablation (SCA) (Figure 1: L). Finally, group LHBO consisted of animals that underwent the same surgical procedure as group L and were treated with oxygen (Figure 1: LHBO). This study was conducted following approval from the Ethical Committee at the University of Belgrade (Approval No. 61206-2915-20). All experimental procedures adhered to Directive 2010/63/EU on protecting animals used for scientific purposes. The following is a condensed protocol of the technical methods used in the present work; more detailed information can be found in our previous publication [5].

2.2. Surgical Procedure and Oxygen Treatment

Initially, the animals were anesthetized with an intraperitoneal injection of Zoletil®50 (Virbac, Carros, France) at 50 mg/kg body weight [27]. Subsequently, they were placed in a stereotaxic frame to perform a craniotomy using the following coordinates: 2 mm anterior to the bregma, 4 mm posterior to the bregma, and 4 mm lateral from the midline. The animals could recover for 5 h following the surgical procedure before commencing hyperbaric oxygen treatment (HBOT). This treatment was performed to the depth of the white matter to preserve the integrity of that layer. The animals in group LHBO were placed in an HBO chamber designed for animals (Holywell Neopren Commerce, Belgrade, Serbia). Following the previously described protocol, these animals were exposed to 100% oxygen for 60 min [5]. The hyperbaric oxygen therapy (HBOT) was administered once daily for ten consecutive days, using the same protocol routinely applied at the Centre for Hyperbaric Medicine in Belgrade, Serbia [28].

2.3. Tissue Preparation and Immunostaining

The animals were overdosed with CO2 after 10 days, followed by decapitation and brain dissection. The brains were then fixed overnight in 4% paraformaldehyde at +4 °C. Cryoprotection was followed by brain fixation at +4 °C and immersion in graded sucrose solutions. Subsequently, the brains were frozen in isopentane and cooled to −80 °C. The brains were then sectioned into 25-μm-thick coronal slices using a cryostat. Sections from 3.12 mm to 3.84 mm anteroposterior to the bregma were mounted on glass slides, air-dried at room temperature, and stored at −20 °C until further analysis.
The slides were tempered at room temperature for 20 min. The sections were activated by immersing the plates in 0.01 M physiological solution with phosphate buffer. After washing in PBS, endogenous peroxidase was blocked using hydrogen peroxide solution (Centrochem, Belgrade, Serbia) in methanol (Moss & Hemoss, Belgrade, Serbia), and the slides were incubated for 1 h in 5% normal donkey serum (NDS; Sigma, Munich, Germany). Then, the plates were incubated overnight at 4 °C in the primary antibody solution (doublecortin, DCX; goat, 1:200, Santa Cruz Biotechnology, Santa Cruz, CA, USA). The following day, after another wash in PBS, the slides were incubated for 2 h at room temperature with the appropriate horseradish peroxidase (HRP)-linked secondary antibody (HRP-antigoat, donkey, 1:200, Santa Cruz, CA, USA). After washing in PBS, 3,3′-diaminobenzidine (DAB; Dako, Glostrup, Denmark) was added to the preparations to detect the created immunocomplex. Subsequently, the slides were rinsed, dehydrated, and clarified in xylene (Moss & Hemoss, Belgrade, Serbia). Finally, the slides were mounted with coverslip medium (DPX, Sigma-Aldrich, Munich, Germany).

2.4. Recording Cells and Preparing the Image

The 80 DCX+ cells from the subgranular zone of the hippocampal dentate gyrus were reconstructed and captured using a Carl Zeiss Axiovert (ZEISS, Göttingen, Germany). The images were categorized into three groups, as previously explained in Section 2.1. Group SHBO contained 29 images, Group L contained 27 images, and Group LHBO contained 24 images. The original image of the recorded neuron was in RGB format and then converted to binary using the free image processing software ImageJ (v1.48, imagej.net/ij/). An example of converting a recorded cell from RGB to binary is shown in Figure 2A,B. To avoid errors in morphometric processing, all images were captured (and later processed) under the same conditions. The images had the same resolution (300 dpi), dimensions (2592 × 1944 pixels), and quality (8-bit depth). Finally, the cell body was removed from each image, leaving only the dendrites, which were then saved in TIFF format.

2.5. Morphometric Analysis

To quantitatively describe the morphological characteristics of the neuronal projections, we applied three complementary analytical approaches: Euclidean, monofractal, and multifractal analysis. These methods were selected due to their ability to quantify geometric complexity and space-filling capacity in two-dimensional images. While biological structures such as neurons often exhibit fractal-like behavior, this behavior is typically observed only within a limited range of spatial scales, constraining the direct application of ideal fractal models [29]. Nevertheless, within the spatial resolutions considered in this study, the analyzed structures are assumed to exhibit scale-invariant properties compatible with fractal and multifractal analysis.

2.5.1. Euclidean and Monofractal Analysis

The reconstructed binary image of the neuron (Section 2.4) was analyzed by calculating parameters that quantify three features of the image: size, shape, and the tortuosity/complexity of dendrites. Each feature was quantified by two parameters: geometric (or computational) and monofractal. Neuron size was quantified by calculating the area (Ad) and fractal dimension (Dbin) of the binary image (Figure 2). Neuron shape was estimated by calculating the circularity, i.e., roundness (Rn). Also, the shape was quantified by the fractal dimensions of the boundary of a polygon (Dout) constructed by connecting the end points of dendrites into a convex polygon [29]. Finally, neuron complexity and tortuosity were assessed by calculating the dendrite’s maximum number of sections (Nm) and the circle whose centre is in the centre of the neuron’s body [30]. Additionally, this property was evaluated by calculating the fractal dimension of the skeletonized image (Dskel), in which the dendrites were compressed into a curved line with one pixel thickness. All three geometric parameters and the first monofractal parameter were calculated on the binary image of the neuron. The other two monofractal parameters were calculated when the binary image of the neuron was converted into an outline and skeletonized (Figure 2). All parameters were calculated using the ImageJ program, and the calculation methods are described in the literature [31,32,33].

2.5.2. Multifractal Analysis

Multifractal analysis offers a comprehensive framework for quantifying the morphological complexity of objects that exhibit multiple scaling laws [34]. This approach involves constructing multifractal spectra, which capture and characterize the intricate structural features of the object at various scales. In this study, we employed the non-overlapping box-counting method, implemented via the ImageJ plugin FracLac [31], to compute three commonly used multifractal spectra: the generalized dimension spectrum DQ(Q), the Hölder exponent spectrum α(Q), and the singularity spectrum f(α) vs. Q.
The generalized dimension spectrum DQ(Q) represents the fractal dimensions of a transformed version of the original pattern, where different structural features are selectively emphasized. As the parameter Q varies, the spectrum transitions from characterizing the fine-scale structures to capturing the dimensions of coarser features [30,33]. The Hölder exponent spectrum α(Q) provides information on the local regularity and irregularity of a point across scales, depending on the degree of distortion applied. This parameter enables the decomposition of a multifractal pattern into monofractal subsets, each exhibiting a single scaling law defined by a similar α value. These subsets possess distinct Hausdorff dimensions, which, depending on the level of distortion, are distributed across the f(α) vs. Q spectrum. The specific computational procedures for generating these spectra have been detailed in previous studies [31,35,36].
For this analysis, the parameter Q ranged from −10.0 to 10.0 in increments of 0.25, resulting in 81 distinct points per multifractal spectrum. This range was chosen to ensure sensitivity to both fine (Q < 0) and coarse (Q > 0) structures in the image, as is common in multifractal analyses of natural patterns. Each of these points was treated as an individual variable during statistical analysis. The box-counting method was applied using 12 different grid positions to mitigate quantization errors [31,37], as this number of grid positions provided an optimal balance between computational efficiency and error minimization [31].
Four additional parameters were extracted to characterize each multifractal spectrum further, reducing the dimensionality while preserving key spectral features. These parameters included the minimum and maximum values of the spectrum, the span (the difference between the maximum and minimum values), and the area under the spectrum (AUS), which was calculated using the trapezoidal rule for numerical integration over the equidistant Q-value points [35]. Specifically, the extracted parameters were DQmin, DQmax, DQspan, and AUS DQ(Q) for the generalized dimension spectrum; αmin, αmax, αspan, and AUS α(Q) for the Hölder exponent spectrum; and f(α)min, f(α)max, f(α)span, and AUS f(α) vs. Q for the singularity spectrum. Each of the three spectra comprised 81 distinct points and 4 additional extracted variables, for 255 multifractal parameters included in the analysis.

2.6. Statistical Analysis

Given that the number of images in each group was fewer than 30, and that preliminary tests indicated deviations from normality, all statistical comparisons were performed using non-parametric methods [38]. Results for the parameters in each group are presented with the median and range within that group. Euclidean and monofractal parameters are also shown with their standard deviations, which better illustrate the variability of values within the group. Differences between groups were tested using the median test for three independent samples [38]. If the p-value was less than 0.05, the difference in the median was considered statistically significant.

3. Results

3.1. Euclidean and Monofractal Analysis

The results of six parameters examining the three morphological characteristics (size, shape, and complexity/tortuosity) of neurons are shown in Table 1. The first three parameters in the table (Ad, Rd, and Nm) are geometric, while the other three (Dbin, Dout, and Dskel) are monofractal. Additionally, data for each parameter are provided for each of the three examined groups. Given that the sample size for each group is less than 30, the results are presented as the median and range. The standard deviation is included to give readers a clearer picture of the variability within each group. The last column of the table shows the statistical evaluation of the differences in medians between the groups.
Based on the standard deviations of the parameters in the groups (Table 1), it is evident that the values within each group vary by more than 30% relative to the median. In some parameters, such as Ad and Rd, the variability exceeds 50%. Given these results, one might expect to find no significant differences between the groups, which is indeed the case for Rd. However, for Ad and Nm, group L is statistically different from the other two groups with a very high level of significance (p < 0.001). Similar differences between groups were observed for Dbin and Dskel. Specifically, for Dbin, a statistical difference was found between the medians of the groups SHBO and LHBO (p < 0.05). In the case of Dout, the SHBO group differs significantly from both groups L and LHBO at p < 0.01 and p < 0.001, respectively. Finally, the medians of Dout for groups SHBO and LHBO differ at the 0.01 level of significance.

3.2. Multifractal Analysis

Multifractal spectra contain a large number of variables; therefore, for clarity, the results are primarily presented graphically.

3.2.1. Generalized Dimension Spectra

Figure 3 presents the median value generalized dimension spectra DQ(Q) for the three groups. All groups exhibited a similarly shaped sigmoidal spectrum with statistically significant differences observed at 44 out of 81 points. Specifically, in the interval Q ∈ [−0.75, −0.25], significance was found at p < 0.05, while in the range Q ∈ [0, 10.0], significance reached p < 0.001. No significant differences were detected for Q < −0.75 (p > 0.05).
The LHBO group exhibited the highest median DQ values across the entire spectrum. In the negative Q-region, the L group displayed higher DQ values than the SHBO group until Q = −2, where an inversion occurred. In the statistically significant region, the LHBO group consistently had the highest DQ values, followed by the SHBO and L groups. Differences between the L and LHBO groups increased with Q, reaching their maximum in the statistically significant portion (positive Q-region), with the largest difference observed at Q = 3.25. Statistically significant differences between these two groups were found in the range Q ∈ [−1.5, 10.0]. Differences between the L and SHBO groups fluctuated, decreasing in the negative Q-region and peaking in the positive Q-region, where significant differences were observed (Q ∈ [0, 10.0]), with the maximum difference occurring at Q = 0.25. No statistically significant differences were found between the LHBO and SHBO groups across the entire DQ(Q) spectrum.
The median DQ values showed a general decreasing trend across the spectrum, except for slight increases around the central positive region. The most pronounced change in DQ values for all groups was observed around Q = −1 (i.e., Q = −1 for L and SHBO, and Q = −1.25 for LHBO).

3.2.2. Hölder Exponents Spectra

Figure 4 presents the median Hölder exponent spectra α(Q) for the three groups. The overall spectral shape was similar across groups, with statistically significant differences observed at 41 out of 81 points. The entire positive Q-region, including Q = 0, showed significant differences, with p < 0.05 at Q = 0 and p < 0.001 for Q ∈ [0.25, 10.0]. No significant differences were found in the negative Q-region (p > 0.05).
The relative differences in α values among groups closely mirrored those observed in the DQ(Q) spectra. The LHBO group exhibited the highest median α values across the entire spectrum. The L group had higher α values than the SHBO group in the negative Q-region until an inversion at Q = −0.25, occurring slightly later than in the DQ(Q) spectrum. In the statistically significant region, the LHBO group maintained the highest α values, followed by the SHBO and L groups. Differences between the L and LHBO groups increased with Q, reaching their maximum at Q = 2.25, with statistically significant differences in the range Q ∈ [−0.25, 10.0]. Differences between the L and SHBO groups remained relatively stable across the spectrum, except for fluctuations in the middle region, with significant differences observed in Q ∈ [0.25, 10.0]. The largest difference between these two groups occurred at Q = −10.0, although no statistically significant differences were found at this point. No significant differences were observed between the LHBO and SHBO groups across the entire α(Q) spectrum.
The median α values generally decreased across the spectrum, with slight increases in the central positive region for the L group and in the far negative Q-region for the LHBO and SHBO groups. The most pronounced change in α values occurred at Q = −0.75 for all three groups.

3.2.3. Singularity Spectra

Figure 5 presents the median singularity spectra f(α) vs. Q for the three groups. While all groups exhibited similar spectral shapes, noticeable shifts in values resulted in statistically significant differences across the entire spectrum.
The L group exhibited consistently lower f(α) values across the entire spectrum, while the LHBO and SHBO groups alternated in having the highest values. The SHBO group showed higher f(α) values in the negative Q-region until Q = −1.25, where the LHBO group assumed the highest values through the positive Q-region. Differences between the L and LHBO groups generally increased with Q, peaking at Q = 5.0, with statistically significant differences across the entire spectrum. Differences between the L and SHBO groups exhibited a slight decreasing trend with fluctuations, with the largest difference at Q = −6.5; statistically significant differences were observed for Q ∈ [−10.0, 4.75]. The LHBO and SHBO groups displayed fluctuating differences, peaking at Q = 6.25, with significant differences in Q regions of [−10.0, −3.0], [3.5, 3.75], and [4.25, 8.0]. The largest within-group spectral change occurred at Q = −1.0 for all three groups.

3.2.4. Extracted Parameters

Four additional parameters were extracted for each spectrum type: minimum, maximum, span, and area under the spectrum (AUS). Median values and ranges for these parameters, along with the results of the Kruskal–Wallis test, are presented in Table 2. Eight out of twelve extracted parameters exhibited statistically significant differences. For the DQ(Q) and α(Q) spectra, the minimum and AUS values were important, whereas all extracted parameters for the f(α) vs. Q spectrum showed significant differences. These results align with the trends observed in Figure 3, Figure 4 and Figure 5, despite the different calculation methods. Notably, the extracted parameters were derived from individual spectra before median computation, rather than from the median spectra.

4. Discussion

Our study used fractal and morphometric analysis to investigate the effect of SCA on the morphological changes of DG granular neurons and the impact of HBO applied after TBI. Using both analyses, the results showed that the dendritic complexity of the granular cells decreases after the SCA. On the other hand, HBO increased complexity, and the neurons in this group resembled those in the SHBO group.
We have demonstrated previously that SCA, as a model of TBI, induces morphological changes in neurons in these results, showing that this model decreased TDL, NBP, and NDT, and reduced the fractal dimension of DCX+ neurons. Additionally, Ratliff et al. demonstrated that even the closed head injury model decreased the dendritic complexity of subgranular cells [2]. Further, they demonstrated that exposure to smoke led to a decrease in dendritic arbor complexity [4]. On the other hand, Saykally et al. pointed out that after repeated injury, the dendritic arbor of granular cells had significantly more branches, mainly in the distal part of the arbor [39]. Parenchymal models of brain injury result in the loss of immature neurons but also stimulate NSC proliferation in the SGZ and increase the synthesis of neurotrophic factors [40]. It should be noted that while each TBI model allows for the adjustment of disease severity, all models have limitations, and no model fully mimics human brain pathology [41]. Therefore, different experimental models of brain injury elicit distinct responses in the SGZ and SVZ.
A lot of work is being conducted to find an adequate treatment for brain injuries. Considering the complexity of TBI (ischemia, hypoxia, cerebral oedema), it is assumed that applying different therapeutic procedures would still give the best results [42]. Most therapeutic protocols applied immediately after TBI are based on calming the acute pathophysiology and aimed at preventing the development of secondary brain damage by maintaining adequate blood flow and preventing hypoxia [14]. This study demonstrates that HBO leads to the preservation of dendritic complexity, as evidenced by various mathematical models of fractal and multifractal analysis showing an increase in dendritic complexity.
The results of the monofractal analysis demonstrate a reduction in spatial complexity in the L group compared to both HBO-treated groups. This decrease is reflected in the lowest values across all analyzed monofractal and computational parameters. Except for roundness, all parameters showed statistically significant differences between group L and the two HBO groups. Neurons in the L group exhibited smaller surface area, reduced dendritic arborization, and diminished overall morphological complexity, including less pronounced dendritic tortuosity. Furthermore, the absence of statistically significant differences between the SHBO and LHBO groups for parameters, such as dendritic area, roundness, and maximum number of branches, supports the conclusion that HBO promotes recovery in neuronal size and branching, bringing these two groups to a similar morphological level. Although the SHBO and LHBO groups displayed closely aligned median values across most parameters, subtle distinctions in space-filling complexity and dendritic tortuosity persisted despite HBO treatment.
The lack of statistically significant differences in roundness, despite clear group-level variation in other computational and fractal measures, may indicate that this parameter is more reflective of the global shape contour rather than local morphological features such as branching or surface irregularity. As such, roundness may be less sensitive to the types of structural changes induced by HBO, particularly those affecting finer aspects of dendritic architecture. While multifractal analysis has found increasing application in neuroscience, its use has predominantly focused on characterizing biological signals [43,44,45]. In contrast, multifractal analysis in morphological assessment of neurons remains relatively uncommon, probably due to the tendency to treat neuronal structures as monofractal objects [46]. However, emerging evidence—including the findings of the present study—suggests that neuronal morphology may be governed by multiple scaling laws, supporting the notion that these structures are inherently multifractal in nature [35,47,48].
The sigmoidal shape of the spectrum of the generalized dimensions curve confirms the presence of multifractal characteristics in the two-dimensional neuronal projections. The most pronounced differences occur in the mid-to-high Q range, suggesting that the primary morphological distinctions between groups are driven by variations in coarser image features, i.e., regions composed of more extensive pixel clusters. In contrast, the finer structural elements (represented in the negative Q range) appear largely preserved across groups. Consequently, macrostructural differences dominate the observed morphological variability, while microstructural elements, such as tortuosity of the dendritic outlines, appear negligible. Notably, while group L differed significantly from both SHBO and LHBO, no significant differences were found between the latter two, reinforcing the proposed neuroprotective effect of HBO treatment.
The results derived from the spectrum of Hölder’s exponents are consistent with the spectrum of generalized dimensions. Statistically significant differences were again confined to the positive region of the spectrum, further confirming that the observed local morphological differences are predominantly attributable to the coarser structural features of the images. As in the DQ spectrum, group L exhibited the lowest α values and showed statistically significant differences compared to the SHBO and LHBO groups. In contrast, no significant differences were observed between SHBO and LHBO, once again implying a neuroprotective effect of HBO therapy.
Notably, the spectra of generalized dimensions and Hölder’s exponents exhibit pronounced plateaus in their Q-positive regions. These plateaus suggest a limited diversity of scaling behaviours within this part of the spectrum, which coincides with the region where statistically significant differences between groups were observed. This implies that the coarser morphological features of the neuronal projections, represented by areas with a higher density of clustered pixels, demonstrate monofractal-like behavior. In contrast, the Q-negative portion of the DQ(Q) spectrum, which captures the finer details of the neuronal structure, displays greater variability in scaling laws, thus reflecting a higher degree of multifractality. However, despite this multifractal behavior in the fine-scale structures, the corresponding parameter distributions in this spectral region did not yield statistically significant group differences. A similar plateau is also present in the Q-negative α(Q) spectrum, reinforcing the partial suitability of monofractal analysis for characterizing these neuronal morphologies, as previously suggested in the literature [46]. Nonetheless, the multifractal characteristics observed in the DQ(Q) and f(α) vs. Q spectra highlight the need for further investigation on a larger sample to draw more definitive conclusions.
The third type of multifractal spectrum, the singularity spectrum, demonstrated the most excellent discriminatory power, successfully distinguishing the lesion (L) group from the two HBO-treated groups (SHBO and LHBO) and differentiating between the HBO groups. Statistically significant differences between the L and LHBO groups were observed across the entire Q range (−10.0 to 10.0). In contrast, differences between L and SHBO extended over a broad portion of the spectrum (Q from −10.0 to 4.75). These findings suggest that the morphological complexity of the monofractal subsets is reduced in the L group, as indicated by lower f(α) values.
In contrast to the other two multifractal spectra, the f(α) spectrum revealed statistically significant distinctions between SHBO and LHBO in several Q intervals, highlighting differences in fine and coarse structural features. Specifically, the SHBO group exhibited slightly higher complexity in fine-scale structures (Q-negative region), whereas the LHBO group displayed greater complexity in coarser morphological features (Q-positive region). These observations suggest that the lesion distinctly reduces the morphological complexity of the investigated neurons, while hyperbaric oxygen treatment appears to promote the restoration of complexity. Moreover, the differential effects observed between the SHBO and LHBO groups suggest that subtle structural variations are influenced by the injury context in which HBO is applied.
The results of the extracted multifractal parameters, presented in Table 2, are consistent with the findings obtained from the complete spectra, despite differences in the calculation method. Specifically, these parameters were computed individually for each binary image. In contrast, the values derived from the full spectra were obtained by calculating the group medians for each Q-value [35]. The minimum values of DQ min, α min, and f(α) min exhibited highly significant differences (p < 0.001), aligning with the regions of statistically significant differences observed in the respective spectra. Both DQ min and f(α) min successfully distinguished all three experimental groups, while α min differentiated the LHBO group from the other two. These results indicate that f(α) min and DQ min are particularly sensitive to group differences, as they successfully distinguished all three experimental groups. Their performance suggests they may be useful parameters for capturing subtle morphological changes in neuronal structure. While additional studies are needed to explore their generalizability, these measures could potentially support future efforts to identify quantitative markers of structural alterations.
In addition, all areas under the spectrum (AUS) parameters effectively separated the L and LHBO groups, while the AUS f(α) also differentiated the L and SHBO groups. These AUS measures offer a promising approach for summarising overall spectral complexity using a single, integrative parameter. The results suggest that this “compressed” or reduced form of multifractal analysis may serve as a practical alternative for situations requiring straightforward group differentiation, particularly when full-spectrum analysis is computationally intensive or unnecessary [35]. However, caution is warranted, as such simplification inevitably leads to the loss of detailed spectral information. Therefore, this approach should be reserved for cases where in-depth morphological characterization is not essential and where a rapid, statistically robust distinction between groups is the primary objective.
Furthermore, given the large number of multifractal parameters analyzed, the potential for inflated Type I error (false positives) due to multiple comparisons must be acknowledged. Although formal correction for false discovery rate (FDR) was not applied, reflecting the study’s exploratory nature, this limitation should be considered when interpreting the results, and future confirmatory research should incorporate appropriate statistical controls. After 10 HBO treatments, in this study, we showed that HBO led to preservation of dendritic complexity, as it increased FD, TDL, NBP, and NDT. As the mechanism of action of HBO is still insufficiently understood, numerous ongoing studies are attempting to elucidate it. Mu et al. suggested that the activation of several signaling pathways and transcription factors, including Wnt, hypoxia-inducible factors (HIFs), and CREB (cAMP response element-binding protein), plays an important role in HBO-induced neurogenesis [49]. In addition, Yang et al. proposed that activation of vascular endothelial growth factor may be another potential mechanism through which HBO affects neurogenesis [50]. A recently published study [51] demonstrated that HBO attenuates pyroptosis, representing a novel, yet underinvestigated, mode of cell death in NSCs. HBO is thought to attenuate pyroptosis by blocking the induction of long non-coding RNA, which otherwise triggers this form of cell death. In addition, the results of another study [52] demonstrated that HBO not only facilitated NSC migration but also promoted differentiation into neurons and integration into neural circuits following brain injury. Moreover, they pointed out that factor 1, produced in stromal cells, regulates neurogenesis after brain injury. Considering the above-mentioned results and facts, several conclusions can be drawn from this paper. Firstly, the present findings suggest that fractal and multifractal dimensions represent efficient methods for investigating and quantifying changes in dendritic complexity after sensorimotor cortex ablation and hyperbaric treatment. Additionally, this paper provides an in-depth analysis of how different modalities of multifractal analysis can be used to detect subtle changes in dendritic arborization patterns and thus provides a starting point for other research papers aiming to utilize fractal analysis for quantifying differences in neuron morphology under various experimental conditions. Finally, our results confirm the beneficial effects of hyperbaric treatment after brain injury and provide additional evidence for its usage as a neuroprotective agent.

Author Contributions

Conceptualization, R.J., N.R. and I.Z.; methodology, R.J., S.P., S.D., N.R., N.T.M. and I.Z.; software, N.R., N.T.M. and I.Z.; validation, N.R., N.T.M. and I.Z.; formal analysis, R.J., N.R., S.D., S.P., N.T.M., I.Z., M.J.; investigation, R.J., S.D., S.P., P.B., N.R., N.T.M., I.Z.; resources, S.P., P.B.; data curation, N.R., N.T.M., I.Z.; writing—original draft preparation, R.J., N.R., N.T.M. and I.Z.; writing—review and editing, S.D., S.P., I.B., I.L., V.B., P.B.; visualization, R.J., N.R., I.Z.; supervision, P.B., S.D., S.P., N.T.M., I.Z.; funding acquisition, S.P., P.B., I.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by grants from the Ministry of Science, Technological Development, and Innovations of the Republic of Serbia (No. 200110, No. 451-03-47/2025-03/200007, and No. 451-03-66/2024-03/200110).

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflict of interest.

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Fractalfract 09 00590 g001
Figure 1. Representative neurons from all three groups were selected after the necessary graphical processing and image adjustments steps (removal of cell body, adequate orientation, and positioning of dendritic tree).
Figure 1. Representative neurons from all three groups were selected after the necessary graphical processing and image adjustments steps (removal of cell body, adequate orientation, and positioning of dendritic tree).
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Figure 2. An example of a DCX+ neuron from the subgranular zone of the hippocampus was captured from the immunostained slices obtained from the SHBO group (A) and the same reconstructed neuron after image binarization (B).
Figure 2. An example of a DCX+ neuron from the subgranular zone of the hippocampus was captured from the immunostained slices obtained from the SHBO group (A) and the same reconstructed neuron after image binarization (B).
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Figure 3. Median generalized dimension spectra DQ(Q) for the three groups. Statistically significant differences (p < 0.05) were observed in the Q intervals from −0.75 to 10.0, with p < 0.05 in the range −0.75 to −0.25 and p < 0.001 from 0 to 10.0, highlighted in green.
Figure 3. Median generalized dimension spectra DQ(Q) for the three groups. Statistically significant differences (p < 0.05) were observed in the Q intervals from −0.75 to 10.0, with p < 0.05 in the range −0.75 to −0.25 and p < 0.001 from 0 to 10.0, highlighted in green.
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Figure 4. Median Hölder exponent spectra α(Q) for the three groups. Statistically significant differences were observed for Q values from 0 to 10.0 (p < 0.05 at Q = 0 and p < 0.001 for Q ∈ [0.25, 10.0]), highlighted in green.
Figure 4. Median Hölder exponent spectra α(Q) for the three groups. Statistically significant differences were observed for Q values from 0 to 10.0 (p < 0.05 at Q = 0 and p < 0.001 for Q ∈ [0.25, 10.0]), highlighted in green.
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Figure 5. Median singularity spectra f(α) vs. Q for the three groups. Statistically significant differences were observed across the entire spectrum (Q ∈ [−10.0, 5.0] with p < 0.001 and Q ∈ [5.25, 10.0] with p < 0.05), highlighted in green.
Figure 5. Median singularity spectra f(α) vs. Q for the three groups. Statistically significant differences were observed across the entire spectrum (Q ∈ [−10.0, 5.0] with p < 0.001 and Q ∈ [5.25, 10.0] with p < 0.05), highlighted in green.
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Table 1. The calculated values of six parameters in each of the three groups of dentate nucleus neurons are presented. Results are expressed as the group median and the range of values. The last column shows the statistical evaluation of differences between groups using the Kruskal–Wallis test.
Table 1. The calculated values of six parameters in each of the three groups of dentate nucleus neurons are presented. Results are expressed as the group median and the range of values. The last column shows the statistical evaluation of differences between groups using the Kruskal–Wallis test.
ParameterGroupsMedian (Range)SDp
Ad (μm2)
SHBO254 (918)220*** L
L109 (128)37
LHBO241 (478)140*** L
Rd
SHBO0.241 (0.467)0.15-
L0.170 (0.746)0.20
LHBO0.261 (0.649)0.15
Nm
SHBO3.30 (3.8)1.1*** L
L1.90 (2.20)0.7
LHBO2.95 (4.50)1.0*** L
Dbin
SHBO1.359 (0.253)0.06* LHBO
L1.325 (0.178)0.05*** SHBO
LHBO1.427 (0.250)0.06*** L
Dout
SHBO1.193 (0.106)0.03** L
L1.163 (0.059)0.02*** SHBO
LHBO1.164 (0.064)0.02** SHBO
Dskel
SHBO1.062 (0.218)0.05*** L
L1.017 (0.123)0.04
LHBO1.074 (0.124)0.04*** L
* p < 0.05, ** p < 0.01, and *** p < 0.001.
Table 2. Extracted parameters for the three groups. Statistically significant p-values are bolded.
Table 2. Extracted parameters for the three groups. Statistically significant p-values are bolded.
ParameterMedian Value (Range)Kruskal–Wallis Hdfp
Group LGroup LHBOGroup SHBO
DQ min1.133 (0.542)1.276 (0.381)1.182 (1.031)19.2502<0.001
DQ max2.142 (1.372)2.168 (0.656)2.069 (1.384)2.56520.277
DQ span1.022 (0.904)0.936 (0.651)0.875 (1.065)5.06420.079
α min1.106 (1.076)1.243 (0.384)1.131 (1.696)16.1422<0.001
α max2.329 (1.591)2.338 (0.741)2.239 (1.598)2.73120.255
α span1.220 (0.723)1.143 (0.690)1.093 (0.875)3.07220.215
f(α) min0.356 (0.360)0.426 (0.332)0.482 (0.426)25.9262<0.001
f(α) max1.177 (0.286)1.340 (0.374)1.279 (0.326)21.0962<0.001
f(α) span0.872 (0.395)0.888 (0.350)0.809 (0.478)10.25520.006
AUS DQ(Q)30.230 (14.947)32.318 (7.524)30.548 (21.287)8.39820.015
AUS α(Q)33.393 (19.219)35.284 (8.346)33.610 (26.228)6.65620.036
AUS f(α)14.952 (3.404)16.965 (6.772)17.558 (6.525)26.2332<0.001
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Jeremić, R.; Rajković, N.; Peković, S.; Dacić, S.; Lavrnja, I.; Bjelobaba, I.; Jeremić, M.; Baščarević, V.; Brkić, P.; Milošević, N.T.; et al. Fractal and Multifractal Analysis as Methods of Quantifying Dendritic Complexity Changes in the Traumatic Brain Injury Model. Fractal Fract. 2025, 9, 590. https://doi.org/10.3390/fractalfract9090590

AMA Style

Jeremić R, Rajković N, Peković S, Dacić S, Lavrnja I, Bjelobaba I, Jeremić M, Baščarević V, Brkić P, Milošević NT, et al. Fractal and Multifractal Analysis as Methods of Quantifying Dendritic Complexity Changes in the Traumatic Brain Injury Model. Fractal and Fractional. 2025; 9(9):590. https://doi.org/10.3390/fractalfract9090590

Chicago/Turabian Style

Jeremić, Rada, Nemanja Rajković, Sanja Peković, Sanja Dacić, Irena Lavrnja, Ivana Bjelobaba, Marija Jeremić, Vladimir Baščarević, Predrag Brkić, Nebojša T. Milošević, and et al. 2025. "Fractal and Multifractal Analysis as Methods of Quantifying Dendritic Complexity Changes in the Traumatic Brain Injury Model" Fractal and Fractional 9, no. 9: 590. https://doi.org/10.3390/fractalfract9090590

APA Style

Jeremić, R., Rajković, N., Peković, S., Dacić, S., Lavrnja, I., Bjelobaba, I., Jeremić, M., Baščarević, V., Brkić, P., Milošević, N. T., & Zaletel, I. (2025). Fractal and Multifractal Analysis as Methods of Quantifying Dendritic Complexity Changes in the Traumatic Brain Injury Model. Fractal and Fractional, 9(9), 590. https://doi.org/10.3390/fractalfract9090590

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