Schematic showing tissue structure at three different spatial scales (tissue, cellular, and sub-cellular). The first panel displays an ensemble of mammalian cells, each bounded by an elastic membrane, suspended in a viscoelastic ECM. The second panel displays an individual cell at a higher level of magnification. The third panel displays the cell nucleus, consisting of a double membrane, a fluid-like nucleoplasm, and an elastic nucleolus in the interior, which contains chromatin (Ref. 43). Although the specific biological structures vary at each successive spatial scale, the essential features are the same: fluid substrates containing elastic compartments. This self-similar pattern forms the basis for the fractal structure shown in Fig. 2.
Layered fractal model for biological tissue based on the schematic shown in Fig. 1. The first panel displays an infinite number of thin elastic membranes with Young’s modulus alternating with viscous compartments that have coefficients of viscosity . The second panel zooms in on the first panel, thus showing the self-similar layered structure.
Fractal ladder model for tissue micro-structure. The continuum model depicted in Fig. 2 is described by an infinite fractal ladder consisting of springs with Young’s modulus and coefficients of viscosity .
Example of a recursive fractal ladder model where the dampers in the simple ladder shown in Fig. 3 are replaced with ladders. This particular fractal arrangement is denoted by and yields a fractional derivative with .
Example of a recursive fractal ladder model where the springs in the simple ladder shown in Fig. 3 are replaced with ladders. This particular fractal arrangement is denoted by and yields a fractional derivative with .
Equivalent generalized viscosity , Young’s modulus values , and ladder parameters calculated using density, speed of sound, and attenuation parameters from Ref. 4. The equivalent Young’s modulus is calculated assuming an equivalent bulk viscosity of water .
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