http://repository.iitr.ac.in/handle/123456789/10870| Title: | A study on DPL model of heat transfer in bi-layer tissues during MFH treatment |
| Authors: | Kumar D. Kumar P. Rai K.N. |
| Published in: | Computers in Biology and Medicine |
| Abstract: | In this paper, dual-phase-lag bioheat transfer model subjected to Fourier and non-Fourier boundary conditions for bi-layer tissues has been solved using finite element Legendre wavelet Galerkin method (FELWGM) during magnetic fluid hyperthermia. FELWGM localizes small scale variation of solution and fast switching of functional bases. It has been observed that moderate hyperthermia temperature range (41-46 °C) can be better achieved in spherical symmetric coordinate system and treatment method will be independent of the Fourier and non-Fourier boundary conditions used. The effect of phase-lag times has been observed only in tumor region. FCC FePt magnetic nano-particle produces more effective treatment with respect to other magnetic nano-particles. The effect of variability of magnetic heat source parameters (magnetic induction, frequency, diameter of magnetic nano-particles, volume fractional of magnetic nano-particles and ligand layer thickness) has been investigated. The physical property of these parameters has been described in detail during magnetic fluid hyperthermia (MFH) treatment and also discussed the clinical application of MFH in Oncology. © 2016 Elsevier Ltd. |
| Citation: | Computers in Biology and Medicine (2016), 75(): 160-172 |
| URI: | https://doi.org/10.1016/j.compbiomed.2016.06.002 http://repository.iitr.ac.in/handle/123456789/10870 |
| Issue Date: | 2016 |
| Publisher: | Elsevier Ltd |
| Keywords: | Bi-layer tissues Dual-phase-lag Finite element Legendre wavelet Galerkin method Magnetic fluid hyperthermia Magnetic nano-particles |
| ISSN: | 104825 |
| Author Scopus IDs: | 57202478211 57211464253 36442806900 |
| Author Affiliations: | Kumar, D., DST-CIMS, Faculty of Science, BHU, Varanasi, India Kumar, P., Department of Mathematical Sciences, IIT-BHU, Varanasi, India Rai, K.N., DST-CIMS, Faculty of Science, BHU, Varanasi, India, Department of Mathematical Sciences, IIT-BHU, Varanasi, India |
| Funding Details: | Authors would like to thanks for DST-Center for Interdisciplinary Mathematical Sciences, Banaras Hindu University Varanasi, India, for providing necessary facilities. Second author is also thankful to the UGC, New Delhi, India , for the financial support under the SRF (17- 06/2012/(i) EU-V) scheme. Appendix A 2.2.1. Non-Fourier BC of second kind . From Eq. (9) , we have found non-Fourier BC of second kind, i.e. (53) − k 2 ∂ ∂ r ( 1 + τ T 2 ∂ ∂ t ) T 2 ( r , t ) = q w at r = L , since q w is constant so that τ q 2 ∂ q w ∂ t = 0 . Taking the Laplace transform of Eq. (53) , we obtained (54) − k 2 ∂ ∂ r ( T 2 ˜ ( r , t ) + τ T 2 s T 2 ˜ ( r , s ) − τ T 2 T 2 ( r , 0 ) ) = q w s at r = L . Using initial condition T 2 ( r , 0 ) = T 02 in Eq. (54) , we have (55) − k 2 ∂ T 2 ˜ ( r , s ) ∂ r = q w s ( 1 + τ T 2 s ) at r = L . Inverse Laplace transform of Eq. (55) is (56) − k 2 ∂ T 2 ( r , t ) ∂ r = q w ( 1 − exp ( − t τ T 2 ) ) at r = L . 2.2.2. Non-Fourier BC of third kind . The non-Fourier constitutive relation at r = L is defined as (57) q 2 ( r , t ) + τ q 2 ∂ q 2 ( r , t ) ∂ t = − k 2 ( 1 + τ T 2 ∂ ∂ t ) ∂ T 2 ( r , t ) ∂ r , and taking into account that heat flux at the outer surface is (58) q 2 ( r , t ) = h [ T f − T 2 ( r , t ) ] at r = L . From Eqs. (57) and (58) , we obtained (59) − k 2 D τ T 2 ( ∂ T 2 ( r , t ) ∂ r ) = h [ T f − D τ q 2 T ( r , t ) ] at r = L , where A L = − k 2 , B L = h and f ( r , t ) = hT f when we assumed a constant temperature only at r = L . Taking Laplace transform and its inversion of Eq. (59) , we have found non-Fourier boundary condition at the outer surface, in case of constant temperature at r = L , i.e. (60) − k 2 ∂ T 2 ( r , t ) ∂ r = h ( T f − T 2 ( r , t ) ) ( 1 − exp ( − t τ T 2 ) ) at r = L . Theorem 2.2.3 Non-Fourier condition at interface ( r = L 1 ) ⇒ Fourier condition at that of the interface . Proof Non-Fourier condition at interface ( r = L 1 ) is defined in Eq. (13) and, taking the Laplace transform of this equation, we have (61) − k 1 ∂ ∂ r ( T ˜ ( L 1 , s ) + s τ T 1 T 1 ˜ ( L 1 , s ) − τ T 1 T 1 ( L 1 , 0 ) ) = − k 2 ∂ ∂ r ( T ˜ ( L 1 , s ) + s τ T 2 T 2 ˜ ( L 1 , s ) − τ T 2 T 2 ( L 1 , 0 ) ) . Using the initial condition T i ( r , 0 ) = T 0 i i = 1 , 2 in Eq. (61) and τ T 1 = τ T 2 , we obtained this equation in simplest form, i.e. (62) − k 1 ∂ T ˜ ( L 1 , s ) ∂ r = − k 2 ∂ T ˜ ( L 1 , s ) ∂ r . Inverse Laplace transform of Eq. (62) is (63) − k 1 ∂ T 1 ( L 1 , t ) ∂ r = − k 2 ∂ T 2 ( L 1 , t ) ∂ r , which is known as Fourier condition at interface ( r = L 1 ) .□ |
| Corresponding Author: | Kumar, D.; DST-CIMS, Faculty of Science, BHUIndia; email: dineshaukumar@gmail.com |
| Appears in Collections: | Journal Publications [ME] |
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