We study the following well-established model of reaction-diffusion type for brain tumour growth: <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cbegin%7Bequation%7D%0A%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Brcll%7D%0A%20%20%5Cpartial_%7Bt%7Du%20-%20div%20(D(x)%20%5Cnabla%20u)%20-%20f(u)%20&=&%200,&%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%5Cmbox%7Bin%20%7D%5COmega%5Ctimes(0,T)%5C%5C%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20u(0)%20&%20=%20&%5Cvarphi,&%5Cmbox%7Bin%20%7D%5COmega%5C%5C%0AD%5Cnabla%20u%5Ccdot%20n%20&=&0,&%20%5Cmbox%7Bon%20%7D%5Cpartial%5COmega%5Ctimes(0,T)%0A%20%5Cend%7Barray%7D%5Cright.%0A%20%5Cnonumber%0A%5Cend%7Bequation%7D" /> 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 <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctiny%5Cvarphi=u(0)" /> starting from a later state <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctiny%5Cpsi=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.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:liu-141982 |
| Date | January 2017 |
| Creators | Jaroudi, Rym |
| Publisher | Linköpings universitet, Tekniska fakulteten, Linköpings universitet, Kommunikations- och transportsystem, Linköping |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Licentiate thesis, comprehensive summary, info:eu-repo/semantics/masterThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | Linköping Studies in Science and Technology. Thesis, 0280-7971 ; 1787 |
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