A first-order adjoint and a second-order hybrid method for an energy output least-squares elastography inverse problem of identifying tumor location

Nathan D Cahill,Akhtar A Khan,Brian C Winkler,Miguel Sama,Baasansuren Jadamba

doi:10.1186/1687-2770-2013-263

Abstract

Abstract In this paper we investigate the elastography inverse problem of identifying cancerous tumors within the human body. From a mathematical standpoint, the elastography inverse problem consists of identifying the variable Lamé parameter μ in a system of linear elasticity where the underlying object exhibits nearly incompressible behavior. This problem is subsequently posed as an optimization problem using an energy output least-squares (EOLS) functional, but the nonlinearity that arises makes the computation of the EOLS functional’s derivatives challenging. We employ an adjoint method for the computation of the gradient, something shown to be an efficient method in recent studies, and also give a parallelizable hybrid method for the computation of the EOLS functional’s second derivative. Detailed discrete formulas and nontrivial computational examples are provided to show the feasibility of both the adjoint and hybrid approaches. Furthermore, all results are given in the framework of a general saddle point problem allowing easy adaptation to numerous other inverse problems. MSC:35R30, 65N30.

Highlights

1 Introduction Consider the following system of partial differential equations describing the response of an isotropic elastic object to certain body forces and traction applied to its boundary:
In this work our objective is to investigate the elastography inverse problem of locating cancerous tumors within the human body
6 Concluding remarks In this work we have presented a detailed application of the adjoint method for efficiently computing the gradient of the energy output least-squares functional as well as a hybrid method for calculating the functional’s second derivative

Summary

Introduction

Consider the following system of partial differential equations describing the response of an isotropic elastic object to certain body forces and traction applied to its boundary:. The primary objective of this work is to develop an efficient computational framework for the elastography inverse problem For this we employ an adjoint approach for the derivative computation of a recently proposed energy output least-squares (EOLS) functional [ ]. Saddle point problem ( a)-( b) connected to the elastography inverse problem of identifying a variable parameter μ in the system of incompressible linear elasticity can be deduced by setting: a(μ, u , v) = μ (u ) · (v), b(u , q) = (div u )q, c(p, q) = pq,. In [ ], the following objective functional was proposed to solve the inverse problem of identifying the variable parameter ∈ A in saddle point problem ( a)-( b): J( ) = a.

Discretization formulas for the adjoint and the hybrid method

Computation of the gradient by using the adjoint method