Quantitative Characterization of Brain Tissue Alterations in Brain Cancer Using Fractal, Multifractal, and IPR Metrics.

We studied the structural alterations between healthy and diseased brain tissues using a multiparametric framework combining fractal analysis, fractal functional transformation, multifractal analysis, and the Inverse Participation Ratio (IPR) analysis. Accurate characterization of brain tissue microstructure is crucial for early detection and diagnosis of cancer. By applying box-counting methods on brightfield microscopy images, we estimated the fractal dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mrow><mml:msub><mml:mi>D</mml:mi> <mml:mi>f</mml:mi></mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> and its logarithmic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced><mml:mrow><mml:mi>l</mml:mi> <mml:mi>n</mml:mi> <mml:mfenced> <mml:mrow><mml:msub><mml:mi>D</mml:mi> <mml:mi>f</mml:mi></mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> and functional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced><mml:mrow><mml:mi>l</mml:mi> <mml:mi>n</mml:mi> <mml:mfenced> <mml:mrow><mml:msub><mml:mi>D</mml:mi> <mml:mrow><mml:mi>t</mml:mi> <mml:mi>f</mml:mi></mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> forms to highlight spatial irregularities in the tissue architecture. While <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>D</mml:mi> <mml:mi>f</mml:mi></mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>l</mml:mi> <mml:mi>n</mml:mi> <mml:mfenced> <mml:mrow><mml:msub><mml:mi>D</mml:mi> <mml:mi>f</mml:mi></mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> exhibited long-tailed distributions distinguishing healthy from cancer tissues, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>l</mml:mi> <mml:mi>n</mml:mi> <mml:mfenced> <mml:mrow><mml:msub><mml:mi>D</mml:mi> <mml:mrow><mml:mi>t</mml:mi> <mml:mi>f</mml:mi></mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> provided significantly improved differentiation by emphasizing local structural variations. Additionally, multifractal analysis revealed broader <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> vs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> curves in cancerous samples, reflecting higher heterogeneity. IPR analysis based on light localization further demonstrated increased nanoscale variations in mass density, reflecting higher structural disorder in cancer tissues. Combining these complementary approaches creates a robust framework for measuring tissue complexity and holds great potential to improve microscopic diagnostic methods for brain cancer detection.