## The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc. | |

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission
of the Department of Engineering Physics and Mathematics for public examination and debate in
Auditorium F1 at Helsinki University of Technology (Espoo, Finland) on the 20^{th} of May, 2005, at
12 noon.

Overview in PDF format (ISBN 951-22-7669-0) [528 KB]

Dissertation is also available in print (ISBN 951-22-7668-2)

In this thesis, models for light propagation and solution methods for the inverse problem in medical optical tomography (OT) in the presence of anisotropies are developed. Light propagation is modelled using the anisotropic diffusion equation (DE) in the frequency domain. The derivation of the diffusion equation in an anisotropic case is sketched, and the relevant boundary and source conditions presented. The numerical solution is obtained using the finite element (FE) method. The numerical work is done in two dimensional space in order to facilitate the testing of the novel methods for solving the inverse problem. To verify the light propagation model, the 2D FE solution is compared to the boundary element method solution to the DE and to a Monte Carlo simulation.

The main emphasis is on the solution of the inverse problem in OT in the presence of anisotropies. The anisotropic inverse problem is non-unique, and hence simultaneous reconstruction of both the anisotropic diffusion tensor and the absorption coefficient is not feasible without substantial prior knowledge. The goal in this work is to reconstruct the spatial distribution of the optical absorption coefficient and overcome the disturbing effect of the background anisotropies. At the same time, the prior knowledge available on the anisotropies is assumed to be rather limited. A few different approaches for estimating the absorption coefficient is presented. Firstly, an attempt to use a conventional isotropic reconstruction scheme is considered. In this case, the obtained estimates suffer from relative large artefacts. In the second approach the structure of the anisotropy is assumed known, and the spatially constant strength is reconstructed simultaneously with the absorption. The quality of the absorption estimate degrades quickly with the accuracy of the underlying anisotropy structure. To help this, statistical inversion methods are employed. Statistical methods provide means to model the prior information on unknown parameters through probability distributions. The anisotropy parameters are modelled using relatively loose Gaussian priors. By using Gaussian priors the posterior probability distribution of the absorption on condition of the measurements also assumes a Gaussian form. Hence the conditional mean estimates and covariances can be derived in a closed form without having to resort to numerical integration. The statistical approach is used to derive a modelling error approach, where the effect of anisotropies is treated as a modelling error and included into estimation. Implementing the statistical inversion methods enables recovering the main features in the absorption distribution. The statistical treatment of anisotropies is also applied to help the inverse problem when the boundary shape and source/measurement locations are modelled inaccurately.

This thesis consists of an overview and of the following 6 publications:

- Jenni Heino and Erkki Somersalo. 2002. Estimation of optical absorption in anisotropic background. Inverse Problems 18, pages 559-573. © 2002 Institute of Physics Publishing. By permission.
- Jenni Heino, Simon Arridge, Jan Sikora, and Erkki Somersalo. 2003. Anisotropic effects in highly scattering media. Physical Review E 68, 031908 (8 pages). © 2003 American Physical Society. By permission.
- Ilkka Nissilä, Tommi Noponen, Jenni Heino, Timo Kajava, and Toivo Katila. 2005. Diffuse optical imaging. In: James Lin (editor), Advances in Electromagnetic Fields in Living Systems 4, pages 77-130. Springer Science, in press.
- Jenni Heino, Erkki Somersalo, and Simon Arridge. 2002. Simultaneous estimation of optical anisotropy and absorption in medical optical tomography. Proceedings of IVth International Workshop "Computational Problems of Electrical Engineering". Zakopane, Poland, pages 191-194. © 2002 Warsaw University of Technology. By permission.
- Jenni Heino and Erkki Somersalo. 2004. A modelling error approach for the estimation of optical absorption in the presence of anisotropies. Physics in Medicine and Biology 49, pages 4785-4798. © 2004 Institute of Physics Publishing. By permission.
- Jenni Heino, Erkki Somersalo, and Jari Kaipio. 2005. Compensation for geometric mismodelling by anisotropies in optical tomography. Optics Express 13, pages 296-308. © 2005 Optical Society of America (OSA). By permission.

Errata of publications 1, 2, 3 and 4

**Keywords:**
optical tomography, medical imaging, anisotropies, inverse problems, finite elements

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© 2005 Helsinki University of Technology

Last update 2011-05-26