Reševanje inverznih problemov prenosa toplote v tkivu z metodo robnih elementov

doktorska disertacija

Jurij Iljaž (Author), Leopold Škerget (Mentor)

Abstract

Delo obravnava inverzni problem prenosa toplote v tkivu, tj. numerično določitev krajevno odvisnega perfuzijskega pretoka v nehomogenem tkivu na osnovi neinvazivnih meritev temperature in toplotnega toka. Pri tem obravnavani problem temelji na Pennesovem matematičnem modelu. Tako pridemo dorealnejšega opisa problema oziroma stanja tkiva ter možnosti določitve perfuzijskega pretoka v posameznem tkivu, ki pa je še kako pomemben tako za diagnostiko kot tudi nekatere klinične aplikacije. Problem je bil zaradi svoje zahtevnosti obravnavan numerično, pri čemer je bil postavljen nov numerični algoritem na osnovi MRE (Metode Robnih Elementov) in optimizacije. Pri tem je bila MRE uporabljena za reševanje direktnega problema prenosa toplote v nehomogenem tkivu ob predpostavljeni neznani spremenljivki ter s temdoločitve namenske funkcije, ki jo želimo minimizirati s primerno optimizacijsko metodo. Izbrani sta bili dve optimizacijski metodi; BFGS (Broyden-Fletcher-Goldfarb-Shanno) in LM (Levenberg-Marquardt), ki smo ju zaradi nestabilnosti obravnavanega inverznega problema nadgradili z uporabo Tihonove regularizacije prvega reda, pri tem pa uporabili metodo L-krivulje zadoločitev optimalnega regularizacijskega parametra. Numerični algoritem je bil pri tem testiran za različne testne funkcije perfuzijskega koeficienta tako homogenega kot nehomogenega tkiva, pri čemer robni in začetni pogoji zadostijo enoličnosti inverznega problema. S tem je bil analiziran numerični algoritem kot tudi reševanje inverznega problema za primer eksaktnih meritev in meritev s šumom. Tako smo prišli do celovite analize primernosti optimizacijske metode, uporabljene regularizacije za reševanje tovrstnih primerov, kot tudi vpliva nehomogenosti, začetne vrednosti, šuma v meritvah inporazdelitev neznane funkcije, ki jo želimo rekonstruirati. Rezultati pri tem kažejo, da je numerični algoritem z uporabo LM metode ter uporabljene regularizacije primernejši od BFGS metode za podani inverzni problem, saj je rešitev globalno stabilna, natančneje določi neznano funkcijo in tudi hitreje konvergira. Regularizacija prvega reda je primerna le za gladke monotone funkcije z nizko vrednostjo prvega odvoda, drugače je rekonstrukcija funkcije možna le v območju blizu meritev, pri čemer ima nehomogenost snovnih lastnostivelik vpliv, še zlasti ob prisotnosti tkiva z nižjo toplotno prevodnostjo. Rekonstrukcija je možna tudi ob prisotnosti nižje stopnje šuma, medtem ko je resolucija pri višji stopnji na račun regularizacije izgubljena, kar oteži reševanje problema. Delo je usmerjeno le v numerični del problema in tako predstavlja osnovo za nadaljnji razvoj neinvazivnih metod določitve perfuzijskega pretoka krvi v realnem nehomogenem tkivu.

Keywords

inverzni problemi;prenos toplote v tkivu;metoda robnih elementov;optimizacija;perfuzijski pretok krvi;

Data

Language:	Slovenian
Year of publishing:	2012
Source:	[Maribor
Typology:	2.08 - Doctoral Dissertation
Organization:	UM FS - Faculty of Mechanical Engineering
Publisher:	J. Iljaž]
UDC:	536.24:519.61/.64(043.3)
COBISS:	16157718
Views:	2372
Downloads:	248
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Other data

Secondary language:	English
Secondary title:	Boundary element method for inverse bioheat problems
Secondary abstract:	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.
Secondary keywords:	inverse problems;bioheat;boundary element method;optimization;non-homogeneous tissue;blood perfusion;
URN:	URN:SI:UM:
Type (COBISS):	Dissertation
Thesis comment:	Univ. Maribor, Fak. za strojništvo
Pages:	VIII, 118 str.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;physics;fizika;heat;thermodynamics;statistical physics;termodinamika;toplota;statistična fizika;heat conduction;heat transfer;mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;computational mathematics;numerical analysis;računska matematika;numerična analiza;
ID:	1026123