The Solution of Inverse Non‐Linear Elasticity Problems That Arise When Locating Breast Tumours

Jonathan P Whiteley

doi:10.1080/10273660500148606

Abstract

Non‐linear elasticity theory may be used to calculate the coordinates of a deformed body when the coordinates of the undeformed, stress‐free body are known. In some situations, such as one of the steps in the location of tumours in a breast, the coordinates of the deformed body are known and the coordinates of the undeformed body are to be calculated, i.e. we require the solution of the inverse problem. Other than for situations where classical linear elasticity theory may be applied, the simple approach for solving the inverse problem of reversing the direction of gravity and modelling the deformed body as an undeformed body does not give the correct solution. In this study, we derive equations that may be used to solve inverse problems. The solution of these equations may be used for a wide range of inverse problems in non‐linear elasticity.

Highlights

The accurate location of cancerous tumours in patients undergoing surgery is of considerable clinical importance
We demonstrate that the forwards and the inverse problem are identical in the special case, where quadratic and higher terms in the strain may be neglected, i.e. under the conditions where classical linear elasticity theory may be applied
We plot, the boundary of the deformed body; the boundary of the region that is the solution of the inverse problem; and the boundary of the region that is calculated by reversing the direction of gravity

Summary

Introduction

The accurate location of cancerous tumours in patients undergoing surgery is of considerable clinical importance. Tumours in women’s breasts are often located using magnetic resonance (MR) imaging. When using this technique, the woman lies in the prone position with her breasts hanging downwards into the machine used to detect the signal. Imaging techniques are used to locate the position of the tumour. Should the woman require surgery, during the surgery she will usually lie in the supine position, and so her breast will occupy a different shape and position. The location of the tumour when the woman lies in this different position is not obvious, and so a technique for locating this tumour is, of much clinical interest

Methods

Results

Conclusion