Of the British coastline and the interest of fractals in medicine

The Fractal Geometry of the Brain: AnOverview

The first chapter of this book introduces some history, philosophy, and basic concepts of fractal geometry and discusses how the neurosciences can benefit from applying computational fractal-based analysis. Further, it compares fractal with Euclidean approaches to analyzing and quantifying the brain in its entire physiopathological spectrum and presents an overview of the first section of this book as well.

Fractal dimension of the aortic annulus: a novel predictor of paravalvular leak after transcatheter aortic valve implantation

To evaluate the prognostic relevance of aortic annulus (AA) and left ventricular outflow tract (LVOT) Fractal dimension (FD). FD is a mathematical concept that describes geometric complexity of a structure and has been shown to predict adverse outcomes in several contexts. Computed tomography (CT) scans from the SOLVE-TAVI trial, which, in a 2 × 2 factorial design, randomized 447 patients to TAVI with the balloon-expandable Edwards Sapien 3 or the self-expanding Medtronic Evolut R, and conscious sedation or general anesthesia, were analyzed semi-automatically with a custom-built software to determine border of AA and LVOT. FD was measured by box counting using grid calibers between 0.8 and 6.75 mm and was compared between patients with none/trivial and mild/moderate paravalvular regurgitation (PVR). Overall, 122 patients had CT scans sufficient for semi-automatic PVR in 30-day echocardiography. PVR was none in 65(53.3%) patients, trace in 9(7.4%), mild in 46(37.7%), moderate in 2(1.6%) and severe in 0 patients. FD determined in diastolic images was significantly higher in patients with mild/moderate PVR (1.0558 ± 0.0289 vs. 1.0401 ± 0.0284, p = 0.017). Annulus eccentricity was the only conventional measure of AA and LVOT geometry significantly correlated to FD (R = 0.337, p < 0.01). Area under the curve (AUC) of diastolic annular FD for prediction of mild/moderate PVR in ROC analysis was 0.661 (0.542-0.779, p = 0.014). FD shows promise in prediction of PVR after TAVI. Further evaluation using larger patient numbers and refined algorithms to better understand its predictive performance is warranted.Trial Registration: www.clinicaltrials.gov , identifier: NCT02737150, date of registration: 13.04.2016.

Evaluation by fractal dimension of muscle regeneration after photobiomodulation

Abstract Introduction: Many treatment modalities are used for muscle tissue recovery. Photobiomodulation is a modality that can be employed to improve the quality of tissue repair. The use of fractal dimension (FD) is an innovative methodology in the quantitative evaluation of treatment efficacy. Objective: Use FD as a quantitative analysis method to evaluate the effect of photobiomodulation of 904 nanometers (nm) in the initial phase of the muscle regeneration process. Method: Thirty male Wistar rats were divided into three groups: Control Group (CG), Injured and Untreated Group (IUT), and Injured and Treated Group (IT). Muscle injury was induced by cryoinjury in the central region of the anterior tibial (AT) belly of the left posterior limb. This was performed by an iron rod that was previously immersed in liquid nitrogen. Applications started 24 hours after the injury and occurred daily for five days. They were performed at two points in the lesion area. The rats were euthanized on the seventh day. The AT muscles were removed and frozen in liquid nitrogen. Then, the histological sections were stained using the Hematoxylin-Eosin (HE) technique and submitted to FD analysis performed by the box-counting method using ImageJ software. The Kolmogorov-Smirnov test was used for data normality, and the Kruskall-Wallis test and Dunn's post-test were used for group comparison (p<0.05%). Results: Differences between IT and IUT groups were statistically significant, and it was possible to observe the reduction of fractability with p=0.0034. Conclusion: FD is a useful tool for the analysis of skeletal muscle disorganization in the initial phase of regeneration and confirms the potentially beneficial effects of photobiomodulation to this process.

Fractal dimension analysis reveals skeletal muscle disorganization in mdx mice

Duchenne Muscular Dystrophy (DMD) is characterized by muscle extracellular matrix disorganization due to the increased collagen deposition leading to fibrosis that significantly exacerbates disease progression. Fractal dimension analysis is a method that quantifies tissue/cellular disorganization and characterizes complex structures. The first objective of the present study was use fractal analysis to evaluate extracellular matrix disorganization in mdx mice soleus muscle. Next, we mimic a hyper-proliferation of fibrogenic cells by co-culturing NIH3T3 fibroblasts and C2C12 myoblasts to test whether fibroblasts induce disorganization in myoblast arrangement. Here, we show mdx presented high skeletal muscle disorganization as revealed by fractal analysis. Similarly, this method revealed that myoblasts co-cultured with fibroblast also presented cellular arrangement disorganization. We also reanalyzed skeletal muscle microarrays transcriptomic data from mdx and DMD patients that revealed transcripts related to extracellular matrix organization. This analysis also identified Osteoglycin, which was validated as a potential regulator of ECM organization in mdx dystrophic muscles. Our results demonstrate that fractal dimension is useful tool for the analysis of skeletal muscle disorganization in DMD and also reveal a fibroblast-myoblast cross-talk that contributes to "in vitro" myoblast disarrangement.

Fractal Dimension in Quantifying Experimental-Pulmonary-Hypertension-Induced Cardiac Dysfunction in Rats

BACKGROUND

Right-sided heart failure has high morbidity and mortality, and may be caused by pulmonary arterial hypertension. Fractal dimension is a differentiated and innovative method used in histological evaluations that allows the characterization of irregular and complex structures and the quantification of structural tissue changes.

OBJECTIVE

To assess the use of fractal dimension in cardiomyocytes of rats with monocrotaline-induced pulmonary arterial hypertension, in addition to providing histological and functional analysis.

METHODS

Male Wistar rats were divided into 2 groups: control (C; n = 8) and monocrotaline-induced pulmonary arterial hypertension (M; n = 8). Five weeks after pulmonary arterial hypertension induction with monocrotaline, echocardiography was performed and the animals were euthanized. The heart was dissected, the ventricles weighed to assess anatomical parameters, and histological slides were prepared and stained with hematoxylin/eosin for fractal dimension analysis, performed using box-counting method. Data normality was tested (Shapiro-Wilk test), and the groups were compared with non-paired Student t test or Mann Whitney test (p < 0.05).

RESULTS

Higher fractal dimension values were observed in group M as compared to group C (1.39 ± 0.05 vs. 1.37 ± 0.04; p < 0.05). Echocardiography showed lower pulmonary artery flow velocity, pulmonary acceleration time and ejection time values in group M, suggesting function worsening in those animals.

CONCLUSION

The changes observed confirm pulmonary-arterial-hypertension-induced cardiac dysfunction, and point to fractal dimension as an effective method to evaluate cardiac morphological changes induced by ventricular dysfunction.

Comparison of fractal dimension and Shannon entropy in myocytes from rats treated with histidine-tryptophan-glutamate and histidine-tryptophan cetoglutarate

INTRODUCTION

Solutions that cause elective cardiac arrest are constantly evolving, but the ideal compound has not yet been found. The authors compare a new cardioplegic solution with histidine-tryptophan-glutamate (Group 2) and other one with histidine-tryptophan-cetoglutarate (Group 1) in a model of isolated rat heart.

OBJECTIVE

To quantify the fractal dimension and Shannon entropy in rat myocytes subjected to cardioplegia solution using histidine-tryptophan with glutamate in an experimental model, considering the caspase markers, IL-8 and KI-67.

METHODS

Twenty male Wistar rats were anesthetized and heparinized. The chest was opened, the heart was withdrawn and 40 ml/kg of cardioplegia (with histidine-tryptophan-cetoglutarate or histidine-tryptophan-glutamate solution) was infused. The hearts were kept for 2 hours at 4ºC in the same solution, and thereafter placed in the Langendorff apparatus for 30 min with Ringer-Locke solution. Analyzes were performed for immunohistochemical caspase, IL-8 and KI-67.

RESULTS

The fractal dimension and Shannon entropy were not different between groups histidine-tryptophan-glutamate and histidine-tryptophan-acetoglutarate.

CONCLUSION

The amount of information measured by Shannon entropy and the distribution thereof (given by fractal dimension) of the slices treated with histidine-tryptophan-cetoglutarate and histidine-tryptophan-glutamate were not different, showing that the histidine-tryptophan-glutamate solution is as good as histidine-tryptophan-acetoglutarate to preserve myocytes in isolated rat heart.

Quantification of fractal dimension and Shannon's entropy in histological diagnosis of prostate cancer

Background

Prostate cancer is a serious public health problem that affects quality of life and has a significant mortality rate. The aim of the present study was to quantify the fractal dimension and Shannon’s entropy in the histological diagnosis of prostate cancer.

Methods

Thirty-four patients with prostate cancer aged 50 to 75 years having been submitted to radical prostatectomy participated in the study. Histological slides of normal (N), hyperplastic (H) and tumor (T) areas of the prostate were digitally photographed with three different magnifications (40x, 100x and 400x) and analyzed. The fractal dimension (FD), Shannon’s entropy (SE) and number of cell nuclei (NCN) in these areas were compared.

Results

FD analysis demonstrated the following significant differences between groups: T vs. N and H vs. N groups (p < 0.05) at a magnification of 40x; T vs. N (p < 0.01) at 100x and H vs. N (p < 0.01) at 400x. SE analysis revealed the following significant differences groups: T vs. H and T vs. N (p < 0.05) at 100x; and T vs. H and T vs. N (p < 0.001) at 400x. NCN analysis demonstrated the following significant differences between groups: T vs. H and T vs. N (p < 0.05) at 40x; T vs. H and T vs. N (p < 0.0001) at 100x; and T vs. H and T vs. N (p < 0.01) at 400x.

Conclusions

The quantification of the FD and SE, together with the number of cell nuclei, has potential clinical applications in the histological diagnosis of prostate cancer.

Method for computing the three-dimensional capacity dimension from two-dimensional projections of fractal aggregates

The current theory of projections of fractals is considered in this paper with application to fractal aggregates. In particular, this theory does not accurately enable the computation of the capacity dimension of three-dimensional aggregates from the capacity dimension of their two-dimensional projections. Herein we propose to compute the three-dimensional capacity dimension from the perimeter-based fractal dimension, using a semiempirical equation, an approach not applied earlier.