Fixed function and its application to Medical Science

TL;DR

This paper extends the concept of contraction in metric spaces to define fixed functions, providing theoretical results and an application to optimize radiation therapy treatment plans for tumor patients.

Contribution

It introduces a new notion of fixed functions in metric spaces and applies it to improve dose calculation in medical radiation therapy.

Findings

New fixed function theorems established with examples

Application improves accuracy of tumor treatment planning

Convergence of dose distribution sequences demonstrated

Abstract

In the present paper, the concept of contraction has been extended in a refined manner by introducing $\mathfrak{D}$-Contraction defined on a family $\mathfrak{F}$ of bounded functions. Also, a new notion of fixed function has been introduced for a metric space. Some fixed function theorems along with illustrative examples have also been given to verify the effectiveness of our results. In addition, an application to medical science has also been presented. This application is based on best approximation of treatment plan for tumor patients getting intensity modulated radiation therapy(IMRT). In this technique, a proper DDC matrix truncation has been used that significantly improves accuracy of results. In 2013, Z. Tian \textit{et al.} presented a fluence map optimization(FMO) model for dose calculation by splitting the DDC matrix into two components on the basis of a threshold intensity value. Following this concept, a sequence of functions can be constructed through the presented results which contains different dose distributions corresponding to different patients and finally it converges to a fixed function. The fixed function obtained in this application represents the suitable doses of a number of tumor patients at the same time. A nice explanation has also been given to check the authenticity of the application and the results.