Reševanje inverznih problemov prenosa toplote v tkivu z metodo robnih elementov - Iljaž, Jurij (visokošolska dela, 2012)
This work investigates inverse bioheat problem, which is a numerical estimation of space-dependent blood perfusion in non-homogeneous tissue under non-invasive temperature and heat flux measurement using Pennes' bioheat equation. Therefore the problem as well as the treated tissue can be describedmore realistically, which can lead to the estimation of blood perfusion in different types of tissue that is very important for medical diagnostic and also for some clinical applications. The treated problem has been solved numerically because of its complexity using newly developed numerical algorithm based on BEM (Boundary Element Method) and optimization. BEM has been used for solving direct bioheat transfer problem in non-homogeneous tissue under assumed function of blood perfusion to determine the objective function, which is then minimized using appropriate optimizationmethod. Therefore two optimization methods have been used; BFGS (Broyden-Fletcher-Goldfarb-Shanno) and LM (Levenberg-Marquardt), which have been upgraded with the first-order Tikhonov regularization and L-Curve method,because of the ill-posed problem. BEM based algorithm has been testedwith numerical examples using different blood perfusion test functions for homogeneous as well as non-homogeneous tissue, where boundary and initial values fulfils the uniqueness of the treated inverse problem. The analysis of numerical algorithm and solution stability has been done for exact as well as noisy measurement data. This way the complete analysis of the used optimization methods and regularization process for solving the treated inverse problem has been done, as well as the analysis of different affects like considering non-homogeneous tissue, initial value, data noise and blood perfusion function distribution that is reconstruction. The results show thatthe algorithm using LM method combined with the used regularization is more appropriate for this type of inverse problems than BFGS one, because of solution global stability better description of exact function and faster convergency. First-order regularization is appropriate only for smooth monotonic functions with low value of first derivative; otherwise the functioncan be retrieved only for the region near boundary measurement. It hasbeen showed that the non-homogeneous tissue has a strong affect on the function reconstruction, especially if the low conducting tissue is present. Function reconstruction is also possible for the noisy data under a low level of noise. If the noise level is high then the resolution is lost because of the high noise level and used regularization, which makes reconstruction more difficult. The work is focused only on the numerical part of the problem, which represents the base for further development of non-invasive methods of blood perfusion estimation in real non-homogeneous tissue.
Keywords: bioheat; blood perfusion; boundary element method; inverse problems; inverzni problemi; metoda robnih elementov; non-homogeneous tissue; optimizacija; optimization; perfuzijski pretok krvi; prenos toplote v tkivu