Feedback control of temperature in specific geometry of porous media: application to hyperthermia
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Feedback control of temperature in specific geometry of porous media: application to hyperthermia Amir Rezvanian1 · Borhan Beigzadeh1 · Amir Hossein Davaei Markazi1 · Mahdi Halabian1 Received: 27 February 2020 / Accepted: 17 March 2020 © Akadémiai Kiadó, Budapest, Hungary 2020
Abstract One of the methods of treating cancer is hyperthermia which is performed in several different ways. One of the recent treatment modalities is with magnetic nanoparticles. By solving momentum and continuity equations in porous media numerically, the distribution of interstitial fluid pressure and velocity is calculated using which it is possible to numerically solve the bio-heat equation in the porous media considering magnetic nanoparticles could be treated as a heat source term. The purpose of this study is to heat up a certain part of the tissue considered as a cancerous tumor to the desired temperature of T = 316.15 K (43 ◦ C) in a controlled manner using a feedback control approach. To do that, first, the controller is designed and applied to a square region assuming that the temperature feedback is ideally available to the controller for the whole tissue; thus, the geometry is mapped onto a circular region while the same transformation is applied to the controller. Finally, a simple regressor is designed to estimate the temperature of the tissue by using the temperature of a single point on the tumor along with the information of the boundary conditions; the output of the regressor is fed back to the controller. Numerical simulation results show that the performance of this regressor controller in control of the temperature of the tissue and tumor is quite acceptable. Keywords Temperature control in porous media · Control of PDE systems · Distributed parameter control · Hyperthermia · Convection and diffusion List of symbols Ct Tissue-specific heat capacity (J kg−1 K−1) Ci Interstitial fluid specific heat capacity (J kg−1 K−1) CMNPs Distribution concentration of Fe3O4 magnetic nanoparticles (g mm−3) e Error function F The volume forces (N) G The permeability of the porous media (mm2) H The hydraulic conductivity of the vessel wall (mm Pa−1 s−1) Kt Tissue thermal conductivity (W mm−1 K−1) Ki Interstitial fluid thermal conductivity (W mm−1 K−1) Kd The derivative control gain Kp The proportional control gain LPL SL The lymphatic filtration coefficient (Pa−1 s−1) V * Borhan Beigzadeh [email protected] 1
School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
Pi Interstitial fluid pressure (Pa) PB Interstitial blood pressure (Pa) PL The hydrostatic pressure of the lymphatic (Pa) Qext The external heat source (W mm−3) S The surface area per unit volume of tissue for V transport in the interstitium (mm−1) SAR The specific absorption rate of Fe3O4 nanoparticles on the applied magnetic field (W g−1) T Temperature (K) Tt Tissue temperature (K) Tdes Desired temperature (K) X X-coordinate Y Y-coordinate x Non-dimensionalized X-coordinate y Non-dimensionalized Y-coor
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