High intensity focussed 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. The increasing interest in HIFU from biomedical researchers and clinicians has led to the development of a number of mathematical models to capture the effects of HIFU energy deposition in biological tissue. In this paper, we study the simplest such model that has been utilized by researchers to study temperature evolution under HIFU therapy. Although the model poses significant theoretical challenges, in earlier work, we were able to establish existence and uniqueness of solutions to this system of PDEs (see Efendiev et al. Adv Appl Math Sci 29(1):231–246, 2020). In the current work, we take the next natural step of studying the long-time dynamics of solutions to this model, in the case where the external forcing is quasi-periodic. 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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S. Sivaloganathan is grateful to NSERC for support of this research through an NSERC Discovery Grant; M.A. Efendiev is grateful to the University of Waterloo, Canada, for the award of the James D. Murray Distinguished Visiting Professorship and to the University of Marmara, Turkey, for the Rector’s Distinguished Visiting Professorship, during which time much of this research was carried out.
Dedicated to James D. Murray: Pioneer, Teacher, Colleague and Friend, on the occasion of his 90th birthday.
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Efendiev, M.A., Murley, J. & Sivaloganathan, S. Dimension Estimate of Uniform Attractor for a Model of High Intensity Focussed Ultrasound-Induced Thermotherapy. Bull Math Biol 83, 95 (2021). https://doi.org/10.1007/s11538-021-00928-x
- High intensity focussed ultrasound
- Uniform attractors
- Hausdorff dimension