Beyond the linear domain - The way forward in MFEIT image reconstruction of the human head L Horesh a1 , R H Bayford a, b , R J Yerworth a , A Tizzard b , G M Ahadzi b and D S Holder a a Department of Medical Physics and Bioengineering, and Clinical Neurophysiology, UCL, London, UK b Middlesex University, Archway Campus, Furnival Building, Highgate, London N19 3UA, UK ABSTRACT: Multi-Frequency Electrical Impedance Tomography (MFEIT) is a promising technique for generating pseudo-absolute images of the head in conditions such as acute stroke, when a time referenced dynamic image is not possible. Unfortunately, the forward solution of EIT problem is non-linear and regional changes in admittivity of more than 50% with frequency exceed the linear domain. Linear solvers fail to compensate for these complex changes. A non-linear reconstruction could provide a better solution, but this introduces computational difficulties due to the large mesh size required for modelling the human head. We have investigated the suitability of six non- linear algorithms, three line-search methods, and two regularization schemes, with respect to computational requirements, robustness and accuracy. Despite a close competition between Variable-Metric and Conjugate- Gradients, both with Brent line search and with regularized search direction, the latter demonstrated faster convergence, with lower run-time and memory demand. However, more efficient methods ought to be developed to tackle properly the scalability problem. Keywords: EIT, Non-linear, Multi-frequency, Conjugate Gradients, Variable Metric, Levenberg-Marquardt 1. INTRODUCTION Our group at University College London is primarily interested in developing EIT for imaging brain function [1]. At present, we are focusing on two clinical applications: epilepsy [2] and stoke [3]. The first application employs a simple linear Truncated Singular Value Decomposition (TSVD) time-difference algorithm [2]. However, a different approach is required for stoke, as there is no reference data. Admittivity changes vary quite drastically with frequency by more than 50% [4]. This is due to frequency dependent permeability of cell membranes. Hence, the changes are beyond the linear reconstruction domain. This suggests that a non-linear approach could provide an accurate solution to this problem, and a comparison study of the most appropriate method is required. This is the second of two papers presented here, the first focuses on whether a non-linear approach could cope better than linear with pseudo-absolute imaging [5]. This paper addresses which non-linear method is most suitable for large-scale pseudo-absolute stroke imaging. 2. METHODOLOGY A set of non-linear reconstruction algorithms was chosen and evaluated with respect to convergence, robustness, image quality, and computational requirements. A data set was created to model a haemorrhage or a tumour in the brain by simulating a 16mm radius spherical perturbation with a 100% or 400% change with respect to brain conductivity. The forward problem was modelled with Finite Elements Models (FEM) of homogenous spheres of 9,670, and 30,270 elements, shelled sphere of 9,998 elements, and shelled human head of 31,111 elements. Thirty-one electrodes were positioned according to the extended 10-20 EEG convention. Injection-measurement protocol included 258 entries, 21 independent injection pairs and 51 independent measurements pairs. Electrical Impedance and Diffuse Optical Reconstruction Software (EIDORS) [6] FEM generation functions have been used to produce the forward solution system matrix. Two preconditioners were used for the linear Precondioned Conjugate Gradients (PCG) forward solver: incomplete Cholesky and Algebraic Multi-Grid (AMG) [7]. Non-linear inverse solvers consisted of two types, Newton-type: (a) Gauss-Newton [8], (b) Levenberg-Marquardt [9], (c) compact form of Levenberg-Marquardt [9], (d) Variable Metric Broyden-Fletcher-Goldfarb-Shanno (BFGS) [8] and Krylov-space: (e) Conjugate Gradients Polak-Ribiere (PR) [9], (f) Conjugate Gradients PR with adjoint field gradients [6], [9], [10]. 1 l.horesh@ucl.ac.uk ; tel 442073879300x(3045); fax 442073879002; http://madeira.physiol.ucl.ac.uk/midx-group/