Long-time behaviour of a coupled PDE model of high intensity focused ultrasound heating of biological tissue
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Date
2021-11-11
Authors
Murley, June
Journal Title
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Publisher
University of Waterloo
Abstract
High Intensity Focused Ultrasound (HIFU) has emerged as a novel therapeutic modality, for the treatment of various cancers, that is gaining significant traction in clinical oncology. It is a cancer therapy that avoids many of the associated negative side effects of other more well-established therapies (such as surgery, chemotherapy and radiotherapy), and does not lead to the longer recuperation times necessary in these cases. However, the mathematical modelling of this treatment for the sake of treatment planning and non-clinical study requires further development. In this thesis, popular models for the propagation of ultrasound and temperature, as well as the biological effects caused by these, are discussed. We introduce a coupled PDE model of the bioheat propagation caused by HIFU and establish that the solution exists and is unique. We further study the long-term dynamics of the solution under quasi-periodic external forcing, which corresponds to the periodic (or quasi-periodic) pulsing of the ultrasound under conditions where a patient is treated clinically with HIFU threapy. In this case, we are able to prove the existence of uniform attractors to the corresponding evolutionary processes generated by our model and to estimate the Hausdorff dimension of the attractors, in terms of the physical parameters of the system.
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Keywords
high intensity focused ultrasound, uniform attractors, partial differential equations, Hausdorff dimension, heating of biological tissue, oncology, maximal Lp-regularity, Leray-Schauder principle, pseudodifferential operator, a priori estimates