Biophysics of Computation

To the Editor: Neuroscientists measure the tree-like structures of neurons in order to better understand how neural circuits are constructed and how neural information is processed. In 1953, Donald Sholl published his well-known technique for quantitative analysis of the complex arbors of dendrites and axons1, but conventional methods still require reconstruction of arbors via time-consuming manual or semi-automated tracing from microscopy images. To bypass this reconstruction step and perform the Sholl technique directly on images instead, we developed Sholl Analysis (http://fiji.sc/Sholl), an open-source program for ImageJ/Fiji2 (Supplementary Fig. 1). The plug-in employs an improved algorithm to retrieve data from twoor three-dimensional (2D or 3D) bitmap images in any format supported by the Bio-Formats library (Supplementary Methods). It pairs this data retrieval with curve-fitting, regression analysis and statistical inference so that users can automatically extract a collection of Sholl-based metrics of arborization1,3 (Supplementary Note). Using individual cortical pyramidal neurons in 3D images, we found Sholl Analysis to be accurate when benchmarked against corresponding manual reconstructions (Supplementary Fig. 2). The method was also resilient to image degradation by simulated shot noise (Supplementary Fig. 3 and Supplementary Software). To further assess accuracy, and to explore the utility of Sholl Analysis in tackling neurons that are particularly slow to reconstruct manually, we studied cerebellar Purkinje cells in mice, which have large and intricate dendritic arbors. From tiled 3D image stacks of cerebellum (Fig. 1a), we selected seven Brainbow2.1-expressing Purkinje neurons and isolated their morphologies (Fig. 1b and Supplementary Note). We then used the Sholl Analysis software to retrieve ten metrics and found they were indistinguishable from those retrieved from manual reconstructions of the same 7 cells (Fig. 1c,d and Supplementary Methods). To probe the sensitivity of the Sholl Analysis software, we asked whether its metrics could be used to distinguish closelyrelated neocortical interneuron subtypes. Parvalbumin-positive (PV) interneurons in layer 5 of visual cortex can be morphologically classified into two subtypes on the basis of their axonal morphology: type 1 PV cells have ascending axons arborizing in layer 2/3, whereas axons of type 2 cells remain in layer 5 (ref. 4). Because their dendritic arbors are indistinguishable4, these two cell types otherwise appear highly similar (Fig. 1e,f). Using the Sholl Analysis software, we retrieved 18 metrics directly from 3D image stacks of 12 PV interneurons. We then used Ward’s hierarchical clustering based on these metrics to independently classify these cells (Fig. 1g and Supplementary Fig. 4). The 12 cells segregated into two groups: one group of five neurons and another of seven. We found that all the neurons but two were correctly classified, with one cell assigned incorrectly to each class (Fig. 1g). Thus, our use of the Sholl Analysis software to quantify arborization directly from bitmap images correctly identified 80–86% of cells. In agreement, linear Sholl plots of type 1 cells indicated more branching than was found for type 2 cells at a distance of 225–300 μm from the soma (Fig. 1h), which corresponds to check and inviting routine use. Second, the software can generate a summary report of the current system performance or a full report containing all individual PSF measurements and associated fitting parameters. Third, a table with the extracted resolution, planarity and colocalization data can be exported. This can be used for subsequent analysis, such as in an image processing or restoration pipeline. In addition, an average PSF from a user-selectable region of interest can be exported, for example, for image deconvolution. We used PSFj to quantify the performance of various high– numerical aperture (NA) objectives and to track day-to-day and system-to-system variation. The results showed substantial performance differences and allowed us to identify strengths and weaknesses of individual objectives as well as general shortcomings (Supplementary Figs. 1 and 2). In particular, we found that whereas lateral resolution performance generally fell short (~20–30%), axial resolution often met or exceeded expectations from the scalar approximation of the PSF commonly used in textbooks2 (Supplementary Note). Planarity was usually well corrected with variations over the FOV below the axial resolution and allowed for the detection of tilted slides caused, for example, by dust particles or misaligned slide holders or stages. Axial chromatic shifts were usually small, with little variation across the FOV (Supplementary Table 1). In contrast, chromatic shifts often showed circular symmetry and increased toward the edge of the FOV, which is a sign of lateral chromatic aberrations. Day-to-day performance variation of most objectives was relatively small (~2–6%) and comparable to single-measurement FOV variations (Supplementary Table 2). Furthermore, testing a limited number of identical objectives identified objective-toobjective and microscope-to-microscope variations of about 10% (Supplementary Tables 3 and 4). The PSFj software is open source and based on libraries from various sources, including ImageJ3 and μManager4, and it runs as a stand-alone application on the three major operating systems (using Java).

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Neuronal morphometry directly from bitmap images

https://escholarship.org/content/qt9xb8q5z6/qt9xb8q5z6.pdf?t=q7c1km

Psychedelics Promote Structural and Functional Neural Plasticity

The ever decreasing area of forests has lead to environmental and economical challenges and has brought with it a renewed interest in developing methodologies that quantify the extent of deforestation and reforestation. In this study we analyzed the deforested areas of the Apuseni Mountains, which has been under economic pressure in recent years and resulted in widespread deforestation as a means of income. Deforested surface dynamics modeling was based on images contained in the Global Forest Database, provided by the Department of Geographical Sciences at Maryland University between 2000 and 2014. The results of the image particle analysis and modelling were based on Total Area (ha), Count of patches and Average Size whereas deforested area distribution was based on the Local Connected Fractal Dimension, Fractal Fragmentation Index and Tug-of-War Lacunarity as indicators of forest fragmentation or heterogeneity. The major findings of the study indicated a reduction of the tree cover area by 3.8%, an increase in fragmentation of 17.7% and an increase in heterogeneity by 29%, while fractal connectivity decreased only by 0.1%. The fractal and particle analysis showed a clustering of forest loss areas with an average increase from 1.1 to 3.0 ha per loss site per year. In conclusion, the fractal and particle analysis provide a relevant methodological framework to further our understanding of the spatial effects of economic pressure on forestry.

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Dynamics of Forest Fragmentation and Connectivity Using Particle and Fractal Analysis

It is well established that microglial form and function are inextricably linked. In recent years, the traditional view that microglial form ranges between "ramified resting" and "activated amoeboid" has been emphasized through advancing imaging techniques that point to microglial form being highly dynamic even within the currently accepted morphological categories. Moreover, microglia adopt meaningful intermediate forms between categories, with considerable crossover in function and varying morphologies as they cycle, migrate, wave, phagocytose, and extend and retract fine and gross processes. From a quantitative perspective, it is problematic to measure such variability using traditional methods, but one way of quantitating such detail is through fractal analysis. The techniques of fractal analysis have been used for quantitating microglial morphology, to categorize gross differences but also to differentiate subtle differences (e.g., amongst ramified cells). Multifractal analysis in particular is one technique of fractal analysis that may be useful for identifying intermediate forms. Here we review current trends and methods of fractal analysis, focusing on box counting analysis, including lacunarity and multifractal analysis, as applied to microglial morphology.

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Quantitating the subtleties of microglial morphology with fractal analysis.

Breast cancer prognosis is a subject undergoing intense study due to its high clinical relevance for effective therapeutic management and a great patient interest in disease progression. Prognostic value of fractal and gray level co-occurrence matrix texture analysis algorithms has been previously established on tumour histology images, but without any direct performance comparison. Therefore, this study was designed to compare the prognostic power of the monofractal, multifractal and co-occurrence algorithms on the same set of images. The investigation was retrospective, with 51 patients selected on account of non-metastatic IBC diagnosis, stage IIIB. Image analysis was performed on digital images of primary tumour tissue sections stained with haematoxylin/eosin. Bootstrap-corrected Cox proportional hazards regression P-values indicated a significant association with metastasis outcome of at least one of the features within each group. AUC values were far better for co-occurrence (0.66–0.77) then for fractal features (0.60–0.64). Correction by the split-sample cross-validation likewise indicated the generalizability only for the co-occurrence features, with their classification accuracies ranging between 67 and 72 %, while accuracies of monofractal and multifractal features were reduced to nearly random 52–55 %. These findings indicate for the first time that the prognostic value of texture analysis of tumour histology is less dependent on the morphological complexity of the image as measured by fractal analysis, but predominantly on the spatial distribution of the gray pixel intensities as calculated by the co-occurrence features.

Comparison of Monofractal, Multifractal and gray level Co-occurrence matrix algorithms in analysis of Breast tumor microscopic images for prognosis of distant metastasis risk.

Fractal geometry is being used in diverse research areas and it is proving to be an increasingly useful tool. Since there is a growing interest in the applications of fractal geometry in many branches of science and art, questions about its methodology, underlying principles and meaningful use, become more and more current. The present paper deals with the conceptual and methodological aspects of fractal geometry. By means of the fractal analysis and calculus we discuss some basic concepts of fractal geometry using as an example the triadic Koch curve. We present a system of parametric equations for that fractal, and derive its capacity dimension and two main inverse-power laws. Since little evidence is available on the properties of the Koch surface, our main endeavour is directed toward investigating its characteristics. Starting from an experimentally stated hypothesis that interior of some natural objects is solid we support this hypothesis theoretically on the Koch surface. We show that the areas of that surface converge to a finite number whose value we calculated. Using the capacity dimension we demonstrate that the surface area of the limit Koch curve is structureless so that it does not belong to the “first category” of fractals.

Fractal and nonfractal properties of triadic Koch curve

The prediction of induction chemotherapy response at the time of diagnosis may improve outcomes in osteosarcoma by allowing for personalized tailoring of therapy. The aim of this study was thus to investigate the predictive potential of the so far unexploited computational analysis of osteosarcoma magnetic resonance (MR) images. Fractal and gray level cooccurrence matrix (GLCM) algorithms were employed in retrospective analysis of MR images of primary osteosarcoma localized in distal femur prior to the OsteoSa induction chemotherapy. The predicted and actual chemotherapy response outcomes were then compared by means of receiver operating characteristic (ROC) analysis and accuracy calculation. Dbin, Λ, and SCN were the standard fractal and GLCM features which significantly associated with the chemotherapy outcome, but only in one of the analyzed planes. Our newly developed normalized fractal dimension, called the space-filling ratio (SFR) exerted an independent and much better predictive value with the prediction significance accomplished in two of the three imaging planes, with accuracy of 82% and area under the ROC curve of 0.20 (95% confidence interval 0-0.41). In conclusion, SFR as the newly designed fractal coefficient provided superior predictive performance in comparison to standard image analysis features, presumably by compensating for the tumor size variation in MR images.

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Fractal and Gray Level Cooccurrence Matrix Computational Analysis of Primary Osteosarcoma Magnetic Resonance Images Predicts the Chemotherapy Response.

This study calls attention to the difference between traditional box-counting method and its modification. The appropriate scaling factor, influence on image size and resolution, and image rotation, as well as different image presentation, are showed on the sample of asymmetrical neurons from the monkey dentate nucleus. The standard BC method and its modification were evaluated on the sample of 2D neuronal images from the human neostriatum. In addition, three box dimensions (which estimate the space-filling property, the shape, complexity, and the irregularity of dendritic tree) were used to evaluate differences in the morphology of type III aspiny neurons between two parts of the neostriatum.

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Box-Counting Method of 2D Neuronal Image: Method Modification and Quantitative Analysis Demonstrated on Images from the Monkey and Human Brain

Fractal analysis in the neurosciences has advanced over the past twenty years. The fractal dimension, besides its ability to discriminate among different cell types, can work as a reliable parameter in cell classification. A qualitative analysis of the morphology of neurons and glia cell types involves a detailed description of the structure and features of cells, and accordingly, their classification into defined classes and types. This paper outlines how fractal analysis can be used for further quantitative classification of these cell types using box-counting and multifractal analysis.

Nebojša T. Milošević

Papers

Comparison of Monofractal, Multifractal and gray level Co-occurrence matrix algorithms in analysis of Breast tumor microscopic images for prognosis of distant metastasis risk.

Fractal and nonfractal properties of triadic Koch curve

Fractal and Gray Level Cooccurrence Matrix Computational Analysis of Primary Osteosarcoma Magnetic Resonance Images Predicts the Chemotherapy Response.

Box-Counting Method of 2D Neuronal Image: Method Modification and Quantitative Analysis Demonstrated on Images from the Monkey and Human Brain

Box-Counting and Multifractal Analysis in Neuronal and Glial Classification