## Abstract

In this thesis we investigate a sequence of important inverse problems associated

with the bio-heat transient flow equation which models the heat transfer within

the human body. Given the physical importance of the blood perfusion coefficient

that appears in the bio-heat equation, attention is focused on the inverse problems concerning the accurate recovery of this information when exact and noisy measurements are considered in terms of the mass, flux, or temperature, which we sampled over the specific regions of the media under investigation.

Five different cases are considered for the retrieval of the perfusion coefficient,

namely when this parameter is assumed to be either constant, or dependent on

time, space, temperature, or on both space and time.

The analytical and numerical techniques that are used to investigate the existence

and uniqueness of the solution for this inverse coefficient identification are embedded in an extensive computational approach for the retrieval of the perfusion coefficient. Boundary integral methods, for the constant and the time-dependent cases, or Crank-Nicolson-type global schemes or local methods based on solutions of the first-kind integral equations, in the space, temperature, or space and time cases, are used in conjunction either with Gaussian mollification or with Tikhonov regularization methods, which are coupled with optimization techniques. Analytically, a number of uniqueness and existence criteria and structural results are formulated and proved.

with the bio-heat transient flow equation which models the heat transfer within

the human body. Given the physical importance of the blood perfusion coefficient

that appears in the bio-heat equation, attention is focused on the inverse problems concerning the accurate recovery of this information when exact and noisy measurements are considered in terms of the mass, flux, or temperature, which we sampled over the specific regions of the media under investigation.

Five different cases are considered for the retrieval of the perfusion coefficient,

namely when this parameter is assumed to be either constant, or dependent on

time, space, temperature, or on both space and time.

The analytical and numerical techniques that are used to investigate the existence

and uniqueness of the solution for this inverse coefficient identification are embedded in an extensive computational approach for the retrieval of the perfusion coefficient. Boundary integral methods, for the constant and the time-dependent cases, or Crank-Nicolson-type global schemes or local methods based on solutions of the first-kind integral equations, in the space, temperature, or space and time cases, are used in conjunction either with Gaussian mollification or with Tikhonov regularization methods, which are coupled with optimization techniques. Analytically, a number of uniqueness and existence criteria and structural results are formulated and proved.

Original language | English |
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Place of Publication | University of Leeds |

Publisher | PhD Thesis, University of Leeds |

Number of pages | 234 |

Publication status | Published - Jun 2009 |