# Topological complexity of photons' paths in biological tissues

## TL;DR

This paper introduces methods to measure the topological complexity of photon paths in biological tissues using Monte Carlo simulations, revealing increased complexity with larger source/detector spacing and the formation of knotted paths.

## Contribution

It proposes three novel measures of geometrical and topological complexity for photon paths and applies them to Monte Carlo simulations in biomedical optics.

## Key findings

- Complexity increases with source/detector spacing.
- Photon paths often form knotted structures.
- Results have implications for biomedical optics and mathematical topology.

## Abstract

In the present contribution three means of measuring the geometrical and topological complexity of photons' paths in random media are proposed. This is realized by investigating the behavior of the average crossing number, the mean writhe, and the minimal crossing number of photons' paths generated by Monte Carlo (MC) simulations, for different sets of optical parameters. It is observed that the complexity of the photons' paths increases for increasing light source/detector spacing, and that highly "knotted" paths are formed. Due to the particular rules utilized to generate the MC photons' paths, the present results may have an interest not only for the biomedical optics community, but also from a pure mathematical point of view.