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

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper evaluates fractal and multifractal analyses to quantify variations in dendritic complexity after a traumatic brain injury in a rodent model. In addition, the paper examines experimental and computational platforms towards the study of neuronal morphology with or without hyperbaric oxygen therapy. Before accepting the paper for publication, the following comments must be addressed:

As discussed, the introduction could be improved with a more comprehensive review of the literature on TBI and dendritic analysis, and by outlining the gaps that this research accomplished.
More rationale for the specific experimental parameters selected and the methods used in the analysis would improve the manuscript.
The discussion addresses important findings, but it would be good to expand the section and discuss the potential implications (for future research and clinical practices) of the current study's results, to a greater degree.
There is a number of typographical and grammatical errors that should be addressed for clarity, succinctness, and cohesiveness.
The abstract should be revised to more clearly summarize the contributions and findings of the current study.
Ensure that all mathematical and analytical methods are well-defined and any assumptions are assigned.
It might be worthwhile to provide a few more quantitative comparisons and statistical analyses to increase the strength of this work.
The conclusion should briefly summarize the important findings and provide scope for future research on novel interventions.
Content may benefit from more robust mechanistic rationale for HBO in context of neurogenesis and dendritic repair (e.g., hypoxia-inducible factors, reduction in oxidative stress).
Content could be streamlined by removing repetitive sections discussing TBI pathophysiology and focusing on a few key ideas important for dendritic complication and fractal analysis.
Clarify the selection of n=5/group. Include either power analysis or rationale for that sample size moved forward with to alleviate potential reviewer concerns about a lack of statistical power.
Provide specifications regarding the size and depth of cortical ablation and how you ensured consistency across animals.
Justification for the Q-range (-10 to +10) for box-counting and by grid position (n=12).
6. Clarification that non-parametric tests were selected (e.g., data distribution not normal).
Address multiplicity of the 255 multifractal parameters and consider that the intervention creates a false discovery rate (FDR) with your results or highlight explorative trends in the multifractal comparisons.
Report effect sizes (e.g., Cohen's d) and p-values to quantitate biological significance.
Discuss the potential reasons for roundness (Rd) showing no group differences when other parameters did.
Explicitly connect spectral regions to biological characteristics (e.g., negative Q = fine-scale dendrites, positive Q = coarse structure).
Highlight which parameters best discriminate groups (e.g., (e.g., f(α)min) and potential as biomarkers.

Overall, the manuscript provides a significant contribution to the analysis of dendritic complexity in the face of the impact from traumatic brain injury. With revisions to address the noted shortcomings, it can contribute importantly to the fields of neurobiology and related therapeutic studies.

Author Response

As discussed, the introduction could be improved with a more comprehensive review of the literature on TBI and dendritic analysis, and by outlining the gaps that this research accomplished.

Our comment and action: We have added the following text and reference regarding the TBI and dendritic analysis in the revised versions of the manuscript:

“Our previously published papers showed that SCA as a model of TBI leads to morphological changes of neurons in the SGZ and granular cells (Pantic et al. 2020; Jeremic et al. 2023). 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.”

More rationale for the specific experimental parameters selected and the methods used in the analysis would improve the manuscript.

Our comment and action: Thank you for your comment. The rationale behind the selection of both experimental parameters and analytical methods is rooted in their relevance to quantifying morphological complexity in 2D neuronal projections. Specifically, the employed methods (Euclidean, monofractal, and multifractal analyses) are well established in image-based morphometric studies for their ability to characterize space-filling properties and structural irregularity at various scales. To further clarify our approach, we have now added a brief introductory paragraph to Section 2.5 to clarify this rationale more explicitly in Section 2.5, where the analytical framework is introduced.

The discussion addresses important findings, but it would be good to expand the section and discuss the potential implications (for future research and clinical practices) of the current study's results to a greater degree.

Our comment and action: We thank the reviewer and have addressed this issue through numerous changes in the text that were suggested by the reviewer in the previous and following questions.

There is a number of typographical and grammatical errors that should be addressed for clarity, succinctness, and cohesiveness.

Our comment and action: We thank the reviewer for his comments and have addressed the typographical and grammatical errors in the revised version of the manuscript.

The abstract should be revised to more clearly summarize the contributions and findings of the current study.

Our comment and action: We thank the reviewer for the valuable suggestion. The abstract has been revised to more clearly summarize the study’s contributions and key findings.
“Background: Traumatic brain injury (TBI) disrupts hippocampal neurogenesis and dendritic structure. Objective: 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.”

Ensure that all mathematical and analytical methods are well-defined and any assumptions are assigned.

Our comment and action: Thank you for this observation. The mathematical and analytical methods employed in our study are based on previously validated approaches, with all relevant calculations and theoretical frameworks detailed in the cited references. Moreover, the extraction of the morphological parameters was conducted using software that adheres to these established definitions and procedures, ensuring methodological consistency. We have revised the manuscript in Section 2.5 outlining the rationale and assumptions underlying the analytical methods used.

It might be worthwhile to provide a few more quantitative comparisons and statistical analyses to increase the strength of this work.

Our comment: We appreciate the reviewer’s suggestion; however, while working on the study design, our primary objective was to focus on a comprehensive fractal analysis paper. We believe that the analyses included in the current work have addressed the research question we set in the beginning of this study. Additional statistical testing beyond this scope would not add any substantial value and would extend the paper beyond the aims of the current work.

The conclusion should briefly summarize the important findings and provide scope for future research on novel interventions.

Our comment: We would like to note that the conclusion already provides a concise summary of the key findings and suggests some possible directions for future research. We have carefully reviewed the section again and made minor edits to ensure these points are clear.

Content may benefit from more robust mechanistic rationale for HBO in context of neurogenesis and dendritic repair (e.g., hypoxia-inducible factors, reduction in oxidative stress).

Our comment and action: Since this part was already mentioned in the original version of the manuscript, we have now rephrased and added additional information for better understanding, as suggested by the reviewer.

“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 investigated, there are a lot of ongoing studies trying to figure it out. Mu et al. suggested that activation of several signalling pathways and transcription factors, including Wnt, hypoxia-inducible factors (HIFs) and CREB (cAMP response element-binding), plays an important role in HBO-induced neurogenesis (Mu et al., 2011). In addition, Yang et al. proposed that activation of vascular endothelial growth factor may be another potential mechanism through which HBO affects neurogenesis (Yang et al., 2017). A recently published study (Ye et al., 2022) showed that HBO attenuates pyroptosis, which represents a new, yet insufficiently investigated mode of cell death of NSCs. HBO is thought to attenuate pyroptosis by blocking long non-coding RNA, which induces pyroptosis. In addition, the results of another study (Wang et al., 2022) showed that HBO enabled not only NSC migration but also differentiation into neurons and integration into neural circuits after brain injury. Moreover, they pointed out that factor 1, produced in stromal cells, is involved in regulating neurogenesis after brain injury.

As our understanding of the exact mechanisms through which HBOT produces its beneficial effects remains incomplete, we published a review (Pekovic et al. 2018) summarizing current findings on the possible cellular and molecular processes involved. We suggest that various signaling pathways and mechanisms may operate concurrently or in coordination, helping to create a favorable local environment that supports neurogenesis, tissue repair, and the recovery of compromised brain functions.

Content could be streamlined by removing repetitive sections discussing TBI pathophysiology and focusing on a few key ideas important for dendritic complication and fractal analysis.

Our comment and action: We thank the reviewer for his comments and have addressed this in the revised version of the manuscript.

Clarify the selection of n=5/group. Include either power analysis or rationale for that sample size moved forward with to alleviate potential reviewer concerns about a lack of statistical power.

Our comment and action: We thank the reviewer for raisin this point. In preclinical research studies, especially with ones focusing on histological and morphometric analysis of animal tissues, group sizes of n = 5 have been frequently used in similar studies. This number was also chosen due to ethical considerations for minimizing animal use, in line with the 3Rs principle (Replacement, Reduction, Refinement) and EU Directive 2010/63/EU on the protection of animals used for scientific purposes.

Although a formal a priori power analysis was not performed due to the novelty and complexity of the multifractal endpoints (with limited prior data available for accurate effect size estimation), our study design was informed by previous publications in which similar number of animals per group yielded sufficient statistical power to detect meaningful morphological differences between experimental groups using similar quantitative approaches (Ratliff et al., 2020, 2022; Jeremić 2023). Furthermore, the significant differences detected in multiple independent parameters in our results support that the chosen sample size was adequate for the study objectives.

Provide specifications regarding the size and depth of cortical ablation and how you ensured consistency across animals.

Our comment and action: It is already written in the manuscript (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 hours 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.)

To maximize consistency across animals, all procedures were performed by the same experienced surgeon using the same suction device and surgical microscope; the stereotaxic frame and coordinates were used for all animals. After surgery all brains were processed in parallel; lesion extent and location were confirmed on the coronal sections used for morphometry (sections from AP 3.12 mm to 3.84 mm relative to bregma).

Justification for the Q-range (-10 to +10) for box-counting and by grid position (n=12).
Clarification that non-parametric tests were selected (e.g., data distribution not normal).

Our comment and action: Thank you for your valuable suggestions. Regarding the Q-range (−10.0 to +10.0) and the step size (0.25), we selected this interval based on prior studies and practical considerations, as it offers a sufficiently wide spectrum to capture both fine (Q < 0) and coarse (Q > 0) structural features. The choice of 12 grid positions was guided by the recommendations in the literature and further tested for our image resolution and pattern size, as explained in Section 2.5.2. This number provided a good balance between scale resolution and computational robustness. More grid positions do not increase accuracy noticeably to justify the increase in computational time.

As for the statistical approach, we confirmed that the data did not meet the assumptions of normality and homogeneity of variance required for parametric tests. In addition, the sample size in each group was below 30, which further supported the decision to apply non-parametric statistical methods. We have clarified this in Section 2.6 (marked in yellow).

Address multiplicity of the 255 multifractal parameters and consider that the intervention creates a false discovery rate (FDR) with your results or highlight explorative trends in the multifractal comparisons.

Our comment and action: Thank you for raising this important point. We agree that performing multiple statistical tests across 255 multifractal parameters can increase the risk of false discoveries. However, our primary objective was exploratory, to investigate whether multifractal descriptors could differentiate between groups and identify trends that deserve further investigation. For this reason, we did not apply a formal FDR correction, as doing so in an exploratory context may increase the risk of Type II errors (false negatives) and mask potentially relevant trends [as pointed out in the publication: Rothman KJ. No adjustments are needed for multiple comparisons. Epidemiology. 1990].

Nevertheless, we recognize the importance of this issue and have added a sentence in the Discussion to reflect that the results (marked in yellow), particularly from individual spectrum points, should be interpreted with caution due to the potential for inflated Type I error. We also explicitly frame our findings as exploratory.

Report effect sizes (e.g., Cohen's d) and p-values to quantitate biological significance.

Our comment and action: Thank you for the suggestion. We fully agree that reporting effect sizes can enhance the interpretation of statistical findings, especially when considering biological relevance. Given that our analyses employed non-parametric methods due to small sample sizes and non-normal data distributions, the appropriate effect size metrics would be rank-based measures such as r (derived from the test statistic), rather than Cohen’s d.

While we initially considered including such metrics, the large number of multifractal parameters, which are displayed in detailed graphical formats, made comprehensive effect size reporting impractical without significantly increasing data presentation complexity. For clarity and readability, we opted to focus on p-values and graphical trends, particularly given the exploratory nature of the study. Nevertheless, we acknowledge the merit of this approach and will consider including summary effect size metrics in future confirmatory studies with a more targeted scope.

Discuss the potential reasons for roundness (Rd) showing no group differences when other parameters did.

Our comment and action: Thank you for your observation. We agree that the lack of statistically significant differences in the Roundness (R_d) parameter stands in contrast to the results of other morphological and complexity measures. One plausible explanation is that although the groups showed distinct differences in overall size and dendritic complexity, their global shape characteristics relative to a perfect circle remained relatively consistent. This suggests that roundness may not be sensitive enough to detect group-level morphological differences that manifest primarily through branching patterns and local irregularities rather than global outline geometry.

At your suggestion, we have added the text regarding this in the Discussion section, the part related to Roundness (marked in yellow).

Explicitly connect spectral regions to biological characteristics (e.g., negative Q = fine-scale dendrites, positive Q = coarse structure).

Our comment and action: Thank you for this insightful suggestion. We fully agree that connecting specific spectral regions and distinct biological characteristics would greatly enhance the biological interpretability of multifractal spectra. We would love to be able to do so, and it is one of, if not the ultimate, aim of our research. However, based on the current state of knowledge and the practical limitations, we cannot make definitive claims regarding these associations. While such interpretations are theoretically plausible and intriguing, they remain speculative at this stage. We will nevertheless continue to strive towards the answer in our future studies designed to directly link multifractal spectral regions with quantifiable morphological elements of neuronal architecture.

Highlight which parameters best discriminate groups (e.g., (e.g., f(α)min) and potential as biomarkers.

Our comment and action: Thank you. We agree with the suggestion. Clearly identifying and emphasizing the most discriminative parameters is valuable, especially when considering their potential as biomarkers. We had already noted the particularly strong performance of f(α) min and D_Qmin, and have now revised the text slightly to make these findings and their potential significance more explicit in the context of group discrimination and possible future use as biomarkers. Added text is marked in yellow in the Discussion section regarding these parameters.

Our comment: We thank you for the kind words regarding our manuscript.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper by Jeremić et al., titled "Fractal and Multifractal Analysis as Methods of Quantifying Dendritic Complexity Changes in the Traumatic Brain Injury Model", presents the application of fractal and multifractal analysis to quantitatively describe changes in dendritic complexity of newly formed neurons in the dentate gyrus of rats subjected to traumatic brain injury (TBI) and treated with hyperbaric oxygen (HBO). The aim of the study was to determine whether geometric methods based on fractal theory can detect significant morphological differences between three experimental groups: control (SHBO), injured (L), and injured with HBO treatment (LHBO).

The authors employed a well-established approach: images of DCX+ neurons were processed and analyzed using ImageJ software and the FracLac plugin. In the monofractal analysis, three types of fractal dimensions were measured: for the binary image of the dendrites (Dbin), for their contour (Dout), and for the skeletonized version (Dskel). In the multifractal analysis, three spectra were computed: generalized dimensions Dq(Q)D_q(Q), Hölder exponents α(Q)\alpha(Q), and the singularity spectrum f(α)f(\alpha), using 81 values of Q in the range [−10,10][-10, 10]. Additionally, four summary parameters were extracted from each spectrum: minimum, maximum, span, and area under the curve (AUS), allowing for efficient statistical comparison between groups.

The results clearly show that dendrites in the L group exhibit significantly lower complexity—both geometrically and multifractally—compared to SHBO and LHBO. The LHBO group achieved values comparable to the control, suggesting a neuroprotective effect of HBO therapy. Particularly informative was the f(α)f(\alpha) spectrum, which not only distinguished the L group from the others but also revealed subtle differences between SHBO and LHBO, confirming the high sensitivity of the method.

Methodologically, the study is carefully designed. The authors ensured consistent image resolution, format, and processing (e.g., cell body removal, binarization, skeletonization) and used appropriate statistical methods (non-parametric tests for small samples). The tools used—ImageJ and FracLac—are well recognized in the field of biological image analysis and were applied correctly.

However, it should be noted that other modern approaches exist that could be considered in future studies or at least acknowledged in the discussion. First, dendritic structures are inherently anisotropic, while the current methods assume isotropy. It would therefore be worth considering the approach proposed by Rak et al. in the article "Quantifying multifractal anisotropy in two-dimensional objects" (Chaos 34, 103137, 2024), where the authors present a method for assessing directional multifractality in 2D images. This type of analysis allows for the detection of differences in structure that may go unnoticed under the assumption of symmetry. Second, while this study is based on 2D images, dendritic trees are intrinsically three-dimensional. Thus, the method of dynamic pyramidal volume correction described by Yin et al. in "Dynamic pyramidal volume correction method for calculating the three-dimensional fractal dimension of machined surfaces" (Applied Surface Science, Vol. 199, 116664, 2024) could be relevant. The authors of that paper proposed a geometric correction method that yields more realistic estimates of fractal dimensions in 3D structures. Adapting this methodology to dendritic morphologies reconstructed from confocal microscopy data could improve the accuracy of complexity assessment in future research. Mentioning such approaches does not diminish the value of the current study but rather points to directions for methodological expansion.

Summary and recommendation
The article presents a solid and thoroughly executed scientific study. The methodology is well chosen and clearly described, and the results are biologically meaningful and well interpreted. Demonstrating the effect of HBO on the restoration of dendritic complexity after TBI using fractal metrics is a valuable contribution to the development of quantitative tools in experimental neurobiology. At the same time, the authors should acknowledge in the article the existence of alternative methods and consider, for future studies, approaches that account for the directionality and three-dimensionality of dendritic structures, in line with current methodological trends in complexity analysis.
Recommendation: publication after minor revisions

Author Response

The authors employed a well-established approach: images of DCX+ neurons were processed and analyzed using ImageJ software and the FracLac plugin. In the monofractal analysis, three types of fractal dimensions were measured: for the binary image of the dendrites (Dbin), for their contour (Dout), and for the skeletonized version (Dskel). In the multifractal analysis, three spectra were computed: generalized dimensions Dq(Q)D_q(Q)Dq(Q), Hölder exponents α(Q)\alpha(Q)α(Q), and the singularity spectrum f(α)f(\alpha)f(α), using 81 values of Q in the range [−10,10][-10, 10][−10,10]. Additionally, four summary parameters were extracted from each spectrum: minimum, maximum, span, and area under the curve (AUS), allowing for efficient statistical comparison between groups.

The results clearly show that dendrites in the L group exhibit significantly lower complexity—both geometrically and multifractally—compared to SHBO and LHBO. The LHBO group achieved values comparable to the control, suggesting a neuroprotective effect of HBO therapy. Particularly informative was the f(α)f(\alpha)f(α) spectrum, which not only distinguished the L group from the others but also revealed subtle differences between SHBO and LHBO, confirming the high sensitivity of the method.

Summary and recommendation

The article presents a solid and thoroughly executed scientific study. The methodology is well chosen and clearly described, and the results are biologically meaningful and well interpreted. Demonstrating the effect of HBO on the restoration of dendritic complexity after TBI using fractal metrics is a valuable contribution to the development of quantitative tools in experimental neurobiology. At the same time, the authors should acknowledge in the article the existence of alternative methods and consider, for future studies, approaches that account for the directionality and three-dimensionality of dendritic structures, in line with current methodological trends in complexity analysis.

Our action and comment: We thank you for the kind, thoughtful, and insightful words regarding our manuscript, and we will definitely take into consideration the comments and propositions suggested by the reviewer for our future work.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors present a study on the dendritic complexity of rodent neurons after traumatic brain injury based on both monofractal and multifractal analyses. The results of the article highlight the effectiveness of fractal and multifractal techniques for the evalution of neuronal modifications and their usefulness in appraising neurological treatments.

The article is technically sound and it is well written. It will be of interest not only for researchers working on traumatic brain injury but also for scientist in the broader fields of fractal and multifractal techniques.

Just a couple of minor corrections:

There is a spurious slash in Table 1.
In page 5, the format of the reference to the work of Salat et al. is not conformant to the style of the paper.

Author Response

Reviewer #3

The authors present a study on the dendritic complexity of rodent neurons after traumatic brain injury based on both monofractal and multifractal analyses. The results of the article highlight the effectiveness of fractal and multifractal techniques for the evaluation of neuronal modifications and their usefulness in appraising neurological treatments.

Just a couple of minor corrections:

There is a spurious slash in Table 1.
In page 5, the format of the reference to the work of Salat et al. is not conformant to the style of the paper.

Our comment and action: We thank you for the kind words regarding our manuscript. We have also corrected the errors highlighted by the reviewer in the revised version of the manuscript.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Accepted for publication in present form.