# The Pathologically Evolving Aggregation-State of Cells in Cancerous Tissues as Interpreted by Fractal and Multi-Fractal Dispersion Theory in Saturated Porous Formations

## TL;DR

This paper explores how fractal and multifractal theories can explain the structure of cancerous tissues and blood flow dynamics, suggesting a universal pattern in biological and non-biological systems.

## Contribution

The paper introduces a novel application of fractal dispersion theory to model blood perfusion in cancerous and healthy tissues.

## Key findings

- A critical fractal dimension (d ≅ 1.7) is linked to the aggregation state of cells in advanced cancer.
- The effective macrodispersion coefficient model shows potential as a descriptor of blood perfusion dynamics.
- A multifractal extension is proposed for enhanced diagnostic interpretation.

## Abstract

A recent author’s fractal fluid-dynamic dispersion theory in porous media has focused on the derivation of the associated nonergodic (or effective) macrodispersion coefficients by a 3-D stochastic Lagrangian approach. As shown by the present study, the Fickian (i.e., the asymptotic constant) component of a properly normalized version of these coefficients exhibits a clearly detectable minimum in correspondence with the same fractal dimension (d ≅ 1.7) that seems to characterize the diffusion-limited aggregation state of cells in advanced stages of cancerous lesion progression. That circumstance suggests that such a critical fractal dimension, which is also reminiscent of the colloidal state of solutions (and may therefore identify the microscale architecture of both living and non-living two-phase systems in state transition conditions) may actually represent a sort of universal nature imprint. Additionally, it suggests that the closed-form analytical solution that was provided for the effective macrodispersion coefficients in fractal porous media may be a reliable candidate as a physically-based descriptor of blood perfusion dynamics in healthy as well as cancerous tissues. In order to evaluate the biological meaningfulness of this specific fluid-dynamic parameter, a preliminary validation is performed by comparison with the results of imaging-based clinical surveys. Moreover, a multifractal extension of the theory is proposed and discussed in view of a perspective interpretative diagnostic utilization.