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Inverse Mathematical Models for Brain Tumour Growth
Linköping University, Faculty of Science & Engineering. Linköping University, Department of Science and Technology, Communications and Transport Systems.
2017 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

We study the following well-established model of reaction-diffusion type for brain tumour growth:

$\begin{equation} \left\{\begin{array}{rcll} \partial_{t}u - div (D(x) \nabla u) - f(u) &=& 0,& \mbox{in }\Omega\times(0,T)\\ u(0) & = &\varphi,&\mbox{in }\Omega\\ D\nabla u\cdot n &=&0,& \mbox{on }\partial\Omega\times(0,T) \end{array}\right. \nonumber \end{equation}$

This equation describes the change over time of the normalised tumour cell density u as a consequence of two biological phenomena: proliferation and diffusion.

We discuss a mathematical method for the inverse problem of locating the brain tumour source (origin) based on the reaction-diffusion model. Our approach consists in recovering the initial spatial distribution of the tumour cells $\tiny\varphi=u(0)$ starting from a later state $\tiny\psi=u(T)$, which can be given by a medical image. We use the nonlinear Landweber regularization method to solve the inverse problem as a sequence of well-posed forward problems.

We give full 3-dimensional simulations of the tumour in time on two types of data, the 3d Shepp-Logan phantom and an MRI T1-weighted brain scan from the Internet Brain Segmentation Repository (IBSR). These simulations are obtained using standard finite difference discretisation of the space and time-derivatives, generating a simplistic approach that performs well. We also give a variational formulation for the model to open the possibility of alternative derivations and modifications of the model. Simulations with synthetic images show the accuracy of our approach for locating brain tumour sources.

##### Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1787
##### National Category
Computational Mathematics
##### Identifiers
ISBN: 9789176854402 (print)OAI: oai:DiVA.org:liu-141982DiVA, id: diva2:1149624
##### Presentation
2017-10-20, TP1, Täppan, Campus Norrköping, Norrköping, 10:15 (English)
##### Supervisors
Available from: 2017-10-16 Created: 2017-10-16 Last updated: 2019-10-12Bibliographically approved
##### List of papers
1. Source Localization of Reaction-Diffusion Models for Brain Tumors
Open this publication in new window or tab >>Source Localization of Reaction-Diffusion Models for Brain Tumors
2016 (English)In: PATTERN RECOGNITION, GCPR 2016 / [ed] Rosenhahn, Bodo, Andres, Bjoern, Springer Publishing Company, 2016, Vol. 9796, p. 414-425Conference paper, Published paper (Refereed)
##### Abstract [en]

We propose a mathematically well-founded approach for locating the source (initial state) of density functions evolved within a nonlinear reaction-diffusion model. The reconstruction of the initial source is an ill-posed inverse problem since the solution is highly unstable with respect to measurement noise. To address this instability problem, we introduce a regularization procedure based on the nonlinear Landweber method for the stable determination of the source location. This amounts to solving a sequence of well-posed forward reaction-diffusion problems. The developed framework is general, and as a special instance we consider the problem of source localization of brain tumors. We show numerically that the source of the initial densities of tumor cells are reconstructed well on both imaging data consisting of simple and complex geometric structures.

##### Place, publisher, year, edition, pages
Springer Publishing Company, 2016
##### Series
Lecture Notes in Computer Science, ISSN 0302-9743, E-ISSN 1611-3349
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:liu:diva-133404 (URN)10.1007/978-3-319-45886-1_34 (DOI)000389019900034 ()978-3-319-45886-1 (ISBN)978-3-319-45885-4 (ISBN)
##### Conference
38th German Conference on Pattern Recognition (GCPR)
Available from: 2016-12-27 Created: 2016-12-22 Last updated: 2018-02-20

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