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optimal control; Pennes' bioheat equation; semigroup theory; thermal therapy; hyperthermia
A distributed optimal control problem on and inside a homogeneous skin tissue is solved subject to Pennes' equation with Dirichlet boundary condition at one end and Rubin condition at the other end. The point heating power induced by conducting heating probe inserted at the tumour site as an unknown control function at specific depth inside biological body is preassigned. Corresponding pseudo-port Hamiltonian system is proposed. Moreover, it is proved that bioheat transfer equation forms a contraction and dissipative system. Mild solution for bioheat transfer equation and its adjoint problem are proposed. Controllability and exponentially stability for the related system is proved. The optimal control problem is solved using strongly continuous semigroup solution and time discretization. Mathematical simulations for a thermal therapy in the presence of point heating power are presented to investigate efficiency of the proposed technique.
[1] Aghayan, S. A., Sardari, D., Mahdavi, S. R. M., Zahmatkesh, M. H.: An inverse problem of temperature optimization in hyperthermia by controlling the overall heat transfer coefficient. Hindawi Publishing Corporation J. Appl. Math. 2013 (2013), 1-9. MR 3090615
[2] Curtain, R. F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag 21 of Text in Applied Mathematics, 1995. MR 1351248 | Zbl 0839.93001
[3] Cheng, K. S., Stakhursky, V., Craciunescu, O. I., Stauffer, P., Dewhirst, M., Das, S. K.: Fast temperature optimization of multi-source hyperthermia applicators with reduced-order modelling of 'virtual sources'. Physics in Medicine and Biology 53 (2008), 6, 1619-1635. DOI 10.1088/0031-9155/53/6/008
[4] Deng, Z. S., Liu, J.: Analytical Solutions to 3D Bioheat Transfer Problems with or without Phase Change. In: Heat Transfer Phenomena and Applications (S. N. Kazi, ed.), Chapter 8, InTech, 2012.
[5] Deng, Z. S., Liu, J.: Analytical study on bioheat transfer problems with spatial or transient heating on skin surface or inside biological bodies. J. Biomech. Eng. 124 (2002), 638-649. DOI 10.1115/1.1516810
[6] Dhar, R., Dhar, P., Dhar, R.: Problem on optimal distribution of induced microwave by heating probe at tumour site in hyperthermia. Adv. Model. Optim. 13 (2011), 1, 39-48. MR 2889921
[7] Dhar, P., Dhar, R., Dhar, R.: An optimal control problem on temperature distribution in tissue by induced microwave. Adv. Appl. Math. Biosciences 2 (2011), 1, 27-38.
[8] Dhar, P., Dhar, R.: Optimal control for bio-heat equation due to induced microwave. Springer J. Appl. Math. Mech. 31 (2010), 4, 529-534. DOI 10.1007/s10483-010-0413-x | MR 2647997 | Zbl 1205.49004
[9] Gomberoff, A., Hojman, S. A.: Non-standard construction of Hamiltonian structures. J. Phys. A: Math. Gen. 30 (1997), 14, 5077-5084. DOI 10.1088/0305-4470/30/14/018 | MR 1478610 | Zbl 0939.70020
[10] Heidari, H., Malek, A.: Optimal boundary control for hyperdiffusion equation. Kybernetika 46 (2010), 5, 907-925. MR 2778921 | Zbl 1206.35138
[11] Heidari, H., Zwart, H., Malek, A.: Controllability and Stability of 3D Heat Conduction Equation in a Submicroscale Thin Film. Department of Applied Mathematics, University of Twente, Enschede 2010, pp. 1-21.
[12] Karaa, S., Zhang, J., Yang, F.: A numerical study of a 3D bioheat transfer problem with different spatial heating. Math. Comput. Simul. 68 (2005), 4, 375-388. DOI 10.1016/j.matcom.2005.02.032 | MR 2141455 | Zbl 1062.92018
[13] Loulou, T., Scott, E. P.: Thermal dose optimization in hyperthermia treatments by using the conjugate gradient method. Numer. Heat Transfer, Part A 42 (2002), 7, 661-683. DOI 10.1080/10407780290059756
[14] Malek, A., Bojdi, Z., Golbarg, P.: Solving fully 3D microscale dual phase lag problem using mixed-collocation, finite difference discretization. J. Heat Transfer 134 (2012), 9, 094501-094506. DOI 10.1115/1.4006271
[15] Malek, A., Nataj, R. Ebrahim, Yazdanpanah, M. J.: Efficient algorithm to solve optimal boundary control problem for Burgers' equation. Kybernetika 48 (2012), 6, 1250-1265. MR 3052884
[16] Malek, A., Momeni-Masuleh, S. H.: A mixed collocation-finite difference method for 3D microscopic heat transport problems. J. Comput. Appl. Math. 217 (2008), 1, 137-147. DOI 10.1016/ | MR 2427436 | Zbl 1148.65082
[17] Momeni-Masuleh, S. H., Malek, A.: Hybrid pseudo spectral-finite difference method for solving a 3D heat conduction equation in a submicroscale thin film. Numer. Methods Partial Differential Equations 23 (2007), 5, 1139-1148. DOI 10.1002/num.20214 | MR 2340665
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