Authors

Ghasem Abbasi, Alaeddin Malek

Abstract

In this paper, a novel scheme based on strongly continuous semigroup is proposed to find a pointwise optimal control function in a biological tissue. Here, mathematical model for hyperthermia therapy involves solution to the thermal wave equation as state while the control is given by the pointwise time dependent heat source. The target is the temperature at a given point within the tumor. Pointwise optimal control problem on and inside a tissue is solved subject to thermal wave model with Dirichlet and Rubin boundary conditions. The pointwise heating source induced by heating probe inserted at the tumor site as control at specific depth inside the biological body. Solutions for both thermal wave problem and its associated adjoint problem are proposed. Approximate controllability of the thermal wave problem is derived with the help of strongly continuous semigroups theory. We prove that the system is pseudo-port Hamiltonian. Pointwise time dependent optimal control problem is solved by using time discretization, conjugate gradient technique and strongly continuous semigroups theory. A set of numerical experiments concerning the design of optimal heating power strategy in cancer treatment by hyperthermia are presented.

Keywords

Hyperthermia; Semigroups theory; Thermal wave bioheat equation; Pseudo-port Hamiltonian; Controllability

Citation

BibTeX

@article{Abbasi_2020,
  title={{Pointwise optimal control for cancer treatment by hyperthermia with thermal wave bioheat transfer}},
  volume={111},
  ISSN={0005-1098},
  DOI={10.1016/j.automatica.2019.108579},
  journal={Automatica},
  publisher={Elsevier BV},
  author={Abbasi, Ghasem and Malek, Alaeddin},
  year={2020},
  pages={108579}
}

Download the bib file

References

  • Curtain, (1995)
  • Dehaye, J. R. & Winkin, J. J. LQ-optimal boundary control of infinite-dimensional systems with Yosida-type approximate boundary observation. Automatica vol. 67 94–106 (2016) – 10.1016/j.automatica.2015.12.033
  • Deng, Z.-S. & Liu, J. Analytical Study on Bioheat Transfer Problems with Spatial or Transient Heating on Skin Surface or Inside Biological Bodies. Journal of Biomechanical Engineering vol. 124 638–649 (2002) – 10.1115/1.1516810
  • Dhar, Problem on optimal distribution of induced microwave by heating probe at tumour site in hyperthermia. Advanced Modeling and Optimization (2011)
  • Dutta, J. & Kundu, B. Thermal wave propagation in blood perfused tissues under hyperthermia treatment for unique oscillatory heat flux at skin surface and appropriate initial condition. Heat and Mass Transfer vol. 54 3199–3217 (2018) – 10.1007/s00231-018-2360-0
  • Hinze, (2009)
  • Jacob, (2012)
  • kashcooli, M., Salimpour, M. R. & Shirani, E. Heat transfer analysis of skin during thermal therapy using thermal wave equation. Journal of Thermal Biology vol. 64 7–18 (2017) – 10.1016/j.jtherbio.2016.12.007
  • Lee, H.-L., Lai, T.-H., Chen, W.-L. & Yang, Y.-C. An inverse hyperbolic heat conduction problem in estimating surface heat flux of a living skin tissue. Applied Mathematical Modelling vol. 37 2630–2643 (2013) – 10.1016/j.apm.2012.06.025
  • Jing Liu, Xu Chen & Xu, L. X. New thermal wave aspects on burn evaluation of skin subjected to instantaneous heating. IEEE Transactions on Biomedical Engineering vol. 46 420–428 (1999) – 10.1109/10.752939
  • Liu, K.-C., Wang, Y.-N. & Chen, Y.-S. Investigation on the bio-heat transfer with the dual-phase-lag effect. International Journal of Thermal Sciences vol. 58 29–35 (2012) – 10.1016/j.ijthermalsci.2012.02.026
  • Loulou, T. & Scott, E. P. THERMAL DOSE OPTIMIZATION IN HYPERTHERMIA TREATMENTS BY USING THE CONJUGATE GRADIENT METHOD. Numerical Heat Transfer, Part A: Applications vol. 42 661–683 (2002) – 10.1080/10407780290059756
  • Malek, Optimal control solution for Pennes’ equation using strongly continuous semigroup. Kybernetika (2014)
  • Malek, A. & Abbasi, G. Heat treatment modelling using strongly continuous semigroups. Computers in Biology and Medicine vol. 62 65–75 (2015) – 10.1016/j.compbiomed.2015.03.030
  • Malek, A. & Abbasi, G. Optimal Control for Pennes’ Bioheat Equation. Asian Journal of Control vol. 18 674–685 (2014) – 10.1002/asjc.1059
  • Alizadeh Moghadam, A., Aksikas, I., Dubljevic, S. & Forbes, J. F. Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica vol. 49 526–533 (2013) – 10.1016/j.automatica.2012.11.016
  • Rabin, Y. A general model for the propagation of uncertainty in measurements into heat transfer simulations and its application to cryosurgery. Cryobiology vol. 46 109–120 (2003) – 10.1016/s0011-2240(03)00015-4
  • Strohbehn, J. W. & Douple, E. B. Hyperthermia and Cancer Therapy: A Review of Biomedical Engineering Contributions and Challenges. IEEE Transactions on Biomedical Engineering vol. BME-31 779–787 (1984) – 10.1109/tbme.1984.325238
  • Xu, F., Seffen, K. A. & Lu, T. J. Non-Fourier analysis of skin biothermomechanics. International Journal of Heat and Mass Transfer vol. 51 2237–2259 (2008) – 10.1016/j.ijheatmasstransfer.2007.10.024