Skip to main content

Dimension Estimate of Uniform Attractor for a Model of High Intensity Focussed Ultrasound-Induced Thermotherapy


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.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. Bailey MR, Khokhlova VA, Sapozhnikov OA, Kargl SG, Crum LA (2003) Physical mechanisms of the therapeutic effect of ultrasound (a review). Acoust Phys 49(4):369–388

    Article  Google Scholar 

  2. Bhowmik A, Singh R, Repaka R, Mishra SC (2013) Conventional and newly developed bioheat transport models in vascularized tissues: a review. J Therm Biol 38(3):107–125

    Article  Google Scholar 

  3. Cavicchi T, OBrien WJ (1984) Heat generation by ultrasound in an absorbing medium. J Acoust Soc Am 76(4):1244–1245

    Article  Google Scholar 

  4. Chen MM, Holmes KR (1980) Microvascular contributions in tissue heat transfer. Ann N Y Acad Sci 335(1):137–150

    Article  Google Scholar 

  5. Chepyzhov VV, Vishik MI (1994) Attractors of nonautonomous dynamical systems and their dimension. J Math Pure Appl 73(3):279–333

    MathSciNet  MATH  Google Scholar 

  6. Chepyzhov VV, Efendiev MA (2000) Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: an example. Commun Pure Appl Math 53(5):647–665

    MathSciNet  Article  Google Scholar 

  7. Dobrakowski PP, Machowska-Majchrzak AK, Labuz-Roszak B, Majchrzak KG, Kluczewska E, Pierzchała KB (2014) MR guided focused ultrasound: a new generation treatment of Parkinsons disease, essential tremor and neuropathic pain. Interv Neuroradiol 20(3):275–282

    Article  Google Scholar 

  8. Efendiev MA (2009) Fredholm structures, topological invariants and applications. American Institute of Mathematical Sciences

  9. Efendiev MA, Murley J, Sivaloganathan S (2020) A coupled pde model of high intensity ultrasound heating of biological tissue. Adv Appl Math Sci 29(1):231–246

    MathSciNet  MATH  Google Scholar 

  10. Etehadtavakol M (2020) Survey of numerical bioheat transfer modelling for accurate skin surface measurements. Therm Sci Eng Prog 20:100681

    Article  Google Scholar 

  11. Hale JK (1988) Asymptotic behavior of dissipative systems. American Mathematical Soc

  12. Hariharan P, Myers MR, Banerjee RK (2007) HIFU procedures at moderate intensities - effect of large blood vessels. Phys Med Biol 52:3493–3513

    Article  Google Scholar 

  13. Haraux A (1991) Systèmes dynamiques dissipatifs et applications. Masson, Moulineaux

    MATH  Google Scholar 

  14. Hudson TJ, Looi T, Pichardo S, Amaral J, Temple M, Drake JM, Waspe AC (2018) Simulating thermal effects of MR-guided focused ultrasound in cortical bone and its surrounding tissue. Med Phys 45(2):506–519

    Article  Google Scholar 

  15. Kaltenbacher B (2015) Mathematics of nonlinear acoustics. Evol Equ Control Theory 4(4):447–491

    MathSciNet  Article  Google Scholar 

  16. Kreider W, Yuldashev PV, Sapozhnikov OA, Farr N, Partanen A, Bailey MR, Khokhlova VA (2013) Characterization of a multi-element clinical HIFU system using acoustic holography and nonlinear modeling. IEEE Trans Ultrason Ferroelectr Freq Control 60(8):1683–1698

    Article  Google Scholar 

  17. Ladyzhenskaya OA, Solonnikov VA, Ural’ceva NN (1968) Linear and quasi-linear equations of parabolic type. American Mathematical Soc

  18. Lesniewski P, Stepin B, Thomas JC (2010) Ultrasonic Streaming in incompressible fluids-modelling and measurements. Proceedings of 20th International Congress on Acoustics

  19. Makarov S, Ochmann M (1996) Nonlinear and thermoviscous phenomena in acoustics, part I. Acta Acust united Ac 82(4):579–606

    MATH  Google Scholar 

  20. Makarov S, Ochmann M (1997) Nonlinear and thermoviscous phenomena in acoustics, part II. Acta Acust United Ac 83(2):197–222

    MATH  Google Scholar 

  21. Modena D, Bassano D, Elevelt A, Baragona M, Hilbers PAJ, Westenberg MA (2019) HIFUpm: a visual environment to plan and monitor high intensity focused ultrasound treatments. In VCBM: 207-211

  22. Nirenberg L (1974) Topics in nonlinear functional analysis. American Mathematical Soc

  23. Nyborg WL (1981) Heat generation by ultrasound in a relaxing medium. J Acoust Soc Am 70(2):310–312

    Article  Google Scholar 

  24. Pennes HH (1948) Analysis of tissue and arterial blood temperatures in the resting human forearm. J Appl Physiol 1(2):93–122

    Article  Google Scholar 

  25. Renardy M, Rogers RC (2006) An introduction to partial differential equations. Springer Science & Business Media, Berlin

    MATH  Google Scholar 

  26. Roohi R, Baroumand S, Hosseinie R, Ahmadi G (2021) Numerical simulation of HIFU with dual transducers: The implementation of dual-phase lag bioheat and non-linear Westervelt equations. Int Commun Heat Mass Transf 120:105002

    Article  Google Scholar 

  27. Salgaonkar VA et al (2014) Model based feasibility assessment and evaluation of prostate hyperthermia with a commercial MR guited endorectal HIFU ablation array. Med Phys 41(3):033301

    Article  Google Scholar 

  28. Shen W, Zhang J, Yang F (2005) Modeling and numberical simulation of bioheat transfer and biomechanics in soft tissue. Math Comput Model 41:1251–1265

    Article  Google Scholar 

  29. Simon J (1987) Compact sets in the space Lp (0, T; B). Ann Mat Pure Appl 4(146):65–96

    MATH  Google Scholar 

  30. Temam R (2012) Infinite-dimensional dynamical systems in mechanics and physics, vol 68. Springer Science & Business Media, Berlin

    Google Scholar 

  31. Triebel H (1978) Interpolation theory, function space, differential operators. North-Holland, Amsterdam

    MATH  Google Scholar 

  32. Wissler EH (1987) Comments on Weinbaum and Jijis discussion of their proposed bioheat equation. J Biomech Eng 109(4):226–233

    Article  Google Scholar 

  33. Wu JR (2016) Handbook of contemporary acoustics and its applications. World scientific

  34. Wu J (2018) Acoustic streaming and its applications. Fluids 3(4):108

    Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to S. Sivaloganathan.

Additional information

Dedicated to James D. Murray: Pioneer, Teacher, Colleague and Friend, on the occasion of his 90th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation


  • High intensity focussed ultrasound
  • Uniform attractors
  • PDEs
  • Hausdorff dimension