10007268

This study aimed at developing an inverse heat transfer approach for predicting the time-varying freezing front and the temperature distribution of tumors during cryosurgery. Using a temperature probe pressed against the layer of tumor, the inverse approach is able to predict simultaneously the metabolic heat generation and the blood perfusion rate of the tumor. Once these parameters are predicted, the temperature-field and time-varying freezing fronts are determined with the direct model. The direct model rests on one-dimensional Pennes bioheat equation. The phase change problem is handled with the enthalpy method. The Levenberg-Marquardt Method (LMM) combined to the Broyden Method (BM) is used to solve the inverse model. The effect (a) of the thermal properties of the diseased tissues; (b) of the initial guesses for the unknown thermal properties; (c) of the data capture frequency; and (d) of the noise on the recorded temperatures is examined. It is shown that the proposed inverse approach remains accurate for all the cases investigated.

[1] S. O. Pfleiderer, M. G. Freesmeyer, C. Marx, R. Kühne-Heid, A. Schneider and W. A. Kaiser, "Cryotherapy of breast cancer under ultrasound guidance: initial results and limitations," European radiology, vol. 12, no. 12, pp. 3009-3014, 2002.

[2] P. E. Huber, J. W. Jenne, R. Rastert, I. Simiantonakis, H. P. Sinn and H. J. Strittmatter, "A new noninvasive approach in breast cancer therapy using magnetic resonance imaging-guided focused ultrasound surgery," Cancer research, vol. 61, no. 23, pp. 8441-8447, 2001.

[3] M. Jaeger and M. Carin, "The front-tracking ALE method: application to a model of the freezing of cell suspensions," Journal of Computational Physics, vol. 179, no. 2, pp. 704-735, 2002.

[4] M. Carin and M. Jaeger, "Numerical simulation of the interaction of biological cells with an ice front during freezing," The European Physical Journal Applied Physics, vol. 16, no. 3, pp. 231-238, 2001.

[5] X. Zhao and K. J. Chua, "Studying the thermal effects of a clinically-extracted vascular tissue during cryo-freezing," Journal of Thermal Biology, vol. 37, no. 8, pp. 556-563, 2012.

[6] M. A. Khanday and F. Hussain, "Explicit formula of finite difference method to estimate human peripheral tissue temperatures during exposure to severe cold stress," Journal of thermal biology, vol. 48, pp. 51-55, 2015.

[7] G. Delhomme, A. Dittmar, W. H. Newman, H. F. Bowman and M. Jouvet, "Thermal diffusion probes for tissue blood flow measurements," Sensors and Actuators B: Chemical, vol. 6, no. 1-3, pp. 87-90, 1992.

[8] M. Burger and F. V. Breukelen, "Construction of a low cost and highly sensitive direct heat calorimeter suitable for estimating metabolic rate in small animals," Journal of Thermal Biology, vol. 38, no. 8, pp. 508-512, 2013.

[9] M. N. Ozisik and H. R. B. Orlande, Inverse Heat Transfer, New York: Taylor and Francis, 2000.

[10] H. L. Lee, T. H. Lai, W. L. Chen and Y. C. Yang, "An inverse hyperbolic heat conduction problem in estimating surface heat flux of a living skin tissue," Applied Mathematical Modelling, vol. 37, no. 5, pp. 2630-2643, 2013.

[11] H. L. Lee, T. H. Lai, W. L. Chen and Y. C. Yang, "An inverse hyperbolic heat conduction problem in estimating surface heat flux of a living skin tissue," Applied Mathematical Modelling, vol. 37, no. 5, pp. 2630-2643, 2013.

[12] K. Das, R. Singh and S. C. Mishra, "Numerical analysis for determination of the presence of a tumor and estimation of its size and location in a tissue," Journal of thermal biology, vol. 38, no. 1, pp. 32-40, 2013.

[13] R. Das, S. C. Mishra and R. Uppaluri, "Multiparameter estimation in a transient conduction-radiation problem using the lattice Boltzmann method and the finite-volume method coupled with the genetic algorithms," Numerical Heat Transfer, vol. 53, no. 12, pp. 1321-1338, 2008.

[14] P. W. Partridge and L. C. Wrobel, "An inverse geometry problem for the localisation of skin tumours by thermal analysis," Engineering Analysis with Boundary Elements, vol. 31, no. 10, pp. 803-811, 2007.

[15] K. Das and S. C. Mishra, "Non-invasive estimation of size and location of a tumor in a human breast using a curve fitting technique," International Communications in Heat and Mass Transfer, vol. 56, pp. 63-70, 2014.

[16] K. Das and S. C. Mishra, "Simultaneous estimation of size, radial and angular locations of a malignant tumor in a 3-D human breast–A numerical study," Journal of thermal biology, vol. 52, pp. 147-156, 2015.

[17] J. M. Luna, R. Romero-Mendez, A. Hernandez-Guerrero and F. Elizalde-Blancas, "Procedure to estimate thermophysical and geometrical parameters of embedded cancerous lesions using thermography," Journal of biomechanical engineering, vol. 134, no. 3, p. 031008, 2012.

[18] K. Yue, X. Zhang and F. Yu, "Simultaneous estimation of thermal properties of living tissue using noninvasive method," International Journal of Thermophysics, vol. 28, no. 5, pp. 1470-1489, 2007.

[19] G. P. Flach and M. N. Özişik, "Inverse heat conduction problem of simultaneously estimating spatially varying thermal conductivity and heat capacity per unit volume," Numerical Heat Transfer, vol. 16, no. 2, pp. 249-266, 1989.

[20] C. H. Huang and C. Y. Huang, "An inverse biotechnology problem in estimating the optical diffusion and absorption coefficients of tissue," International journal of heat and mass transfer, vol. 47, no. 3, pp. 447-457, 2004.

[21] A. Jalali, M. B. Ayani and M. Baghban, "Simultaneous estimation of controllable parameters in a living tissue during thermal therapy," Journal of thermal biology, vol. 45, pp. 37-42, 2014.

[22] Y. T. Zhang and J. Liu, "Numerical study on three-region thawing problem during cryosurgical re-warming," Medical engineering & physics, vol. 24, no. 4, pp. 265-277, 2002.

[23] Y. Rabin and A. Shitzer, "Exact solution to the one-dimensional inverse-Stefan problem in non-ideal biological tissues," Journal of Heat Transfer, vol. 117, no. 2, pp. 425-431, 1995.

[24] K. J. Chua and S. K. Chou, "On the study of the freeze–thaw thermal process of a biological system," Applied Thermal Engineering, vol. 29, no. 17, pp. 3696-3709, 2009.

[25] H. Budman, A. Shitzer and J. Dayan, "Analysis of the inverse problem of freezing and thawing of a binary solution during cryosurgical processes," Journal of biomechanical engineering, vol. 117, no. 2, pp. 193-202, 1995.

[26] K. J. Chua, S. K. Chou and J. C. Ho, "An analytical study on the thermal effects of cryosurgery on selective cell destruction," Journal of biomechanics, vol. 40, no. 1, pp. 100-116, 2007.

[27] E. H. Wissler, "Pennes’ 1948 paper revisited," Journal of Applied Physiology, vol. 85, no. 1, pp. 35-41, 1998.

[28] T. L. Horng, T. L. Horng, W. L. Lin, C. T. Liauh and T. C. Shih, "Effects of pulsatile blood flow in large vessels on thermal dose distribution during thermal therapy," Medical physics, vol. 34, no. 4, pp. 1312-1320, 2007.

[29] H. H. Pennes, "Analysis of tissue and arterial blood temperatures in the resting human forearm," Journal of applied physiology, vol. 1, no. 2, pp. 93-122, 1948.

[30] S. Patankar, Numerical heat transfer and fluid flow, CRC press, 1980.

[31] C. H. Huang and C. Y. Huang, "An inverse problem in estimating simultaneously the effective thermal conductivity and volumetric heat capacity of biological tissue," Applied mathematical modelling, vol. 31, no. 9, pp. 1785-1797, 2007.

[32] J. Iljaž and L. Škerget, "Blood perfusion estimation in heterogeneous tissue using BEM based algorithm," Engineering Analysis with Boundary Elements, Vols. 75-87, p. 39, 2014.

[33] D. W. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," Journal of the society for Industrial and Applied Mathematics, vol. 11, no. 2, pp. 431-441, 1963.

[34] C. G. Broyden, "A class of methods for solving nonlinear simultaneous equations," Mathematics of computation, pp. 577-593, 1965.

[35] M. Hafid and M. Lacroix, "An inverse heat transfer method for predicting the thermal characteristics of a molten material reactor," Applied Thermal Engineering, vol. 108, p. 140–149, 2016.

[36] M. Hafid and M. Lacroix, "Inverse Heat Transfer Analysis of a Melting Furnace Using Levenberg-Marquardt Method," Int. J. of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, vol. 10, no. 7, pp. 1228-1235, 2016.

[37] M. Hafid and M. Lacroix, "Prediction of the Thermal Parameters of a High-Temperature Metallurgical Reactor Using Inverse Heat Transfer," Int. J. of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, vol. 10, no. 6, pp. 907-913, 2016.

[38] M. Hafid and M. Lacroix, "Inverse heat transfer prediction of the state of the brick wall of a melting furnace," Applied Thermal Engineering, Vols. 265-274, p. 110, 2017.

[39] S. Kumar and V. K. Katiyar, "Numerical study on phase change heat transfer during combined hyperthermia and cryosurgical treatment of lung cancer," International Journal of Applied Mathematics and Mechanics, vol. 3, no. 3, pp. 1-17, 2007.

[40] A. Zhang, L. X. Xu, G. A. Sandison and J. Zhang, "A microscale model for prediction of breast cancer cell damage during cryosurgery," Cryobiology, vol. 47, no. 2, pp. 143-154, 2003.

[41] F. J. González, "Thermal simulation of breast tumors," Revista mexicana de física, vol. 53, no. 4, pp. 323-326, 2007.