Università degli Studi di Genova
This thesis deals with two mathematical problems which play a very important role in medical applications: inverse scattering problem and cryosurgery planning problem. The first one is the problem of retrieving some physical information of an object from the knowledge of measurements of its scattered field. The second one is the problem of determining the best configuration of probes for freezing a tumoral tissue hosted in an organ, without damaging the surrounding healthy tissues. These problems are respectively at the basis of microwave tomography and cryosurgery operation. For what concerns inverse scattering, this thesis presents three distinct original contributions: 1) a hybrid approach to inverse scattering where a qualitative method is used preliminary to a quantitative method in order to reduce the complexity of the scattering scenario and to improve the quality of the reconstructions; 2) a finite element approach able to retrieve constant estimates on the refractive index of an object from the knowledge of its first transmission eigenvalue; 3) a physical interpretation of the linear sampling method, the most well-known (but still not completely mathematically justified) qualitative technique for inverse scattering by microwaves. The final part of the thesis is dedicated to introduce a cryosurgery planning technique based on stochastic optimization and able to define the position of the freezing probes together with other critical parameters with wide generality. All the proposed techniques are numerically validated on realistic computerized phantoms and both of the main problems and their medical applications are extensively introduced.
inverse problems, inverse scattering, cryosurgery planning, linear sampling method, transmission eigenvalues, contrast source inversion method, Pennes bioheat equation, Stefan problem, ant colony optimization method
15, 34, 45, 65, 78, 80
The work described in the thesis gave also rise to the following publications in refereed journals: The work described in the thesis gave also rise to the following publications in refereed journals: - R. Aramini, G. Caviglia, G. Giorgi “The role of point sources and their power fluxes in the Linear Sampling Method”, SIAM Journal on Applied Mathematics, Vol.71, No.4 - 2011. - G. Giorgi, H. Haddar “Computing estimates on material properties from transmission eigenvalues’, Inverse Problems, Vol.28, No.5 - 2012. - G. Giorgi, M. Brignone, R. Aramini, M. Piana “Application of the inhomogeneous Lippmann-Schwinger equation to inverse scattering problems”, SIAM Journal on Applied Mathematics, Vol. 73, No. 1 - 2013. - G. Giorgi, L. Avalle, M. Brignone, M. Piana, G. Caviglia “An optimization approach to multiprobe cryosurgery planning”, Computer Methods in Biomechanics and Biomedical Engineering- In Press. Thesis defended on April 18, 2012.