# Expression of the peak time for time-domain boundary measurements in diffuse light

## TL;DR

This paper provides a mathematical analysis of the peak time in time-domain boundary measurements of diffusive media, establishing its properties and relation to object position, with implications for optical imaging.

## Contribution

It offers the first rigorous proof of existence, uniqueness, and explicit expression of the peak time in diffusive media, linking it to object location.

## Key findings

- Proved existence and uniqueness of the peak time.
- Derived explicit expression for the peak time.
- Established relationship between peak time and object position.

## Abstract

Light propagation through diffusive media can be described by the diffusion equation in a space-time domain. Further, fluorescence can be described by a system of coupled diffusion equations. This paper analyzes time-domain measurements, which measure the temporal point-spread function (TPSF), at a boundary of such diffusive media with a given source and detector. We focus on the temporal position of the TPSF maximum, which we refer to as the peak time. Although some unique properties of solutions of this system have been numerically studied, we give a mathematical analysis of peak time, providing proof of the existence, uniqueness, and the explicit expression of the peak time. We clearly show the relationship between the peak time and the object position in a medium.