Review of the mathematical functions used to model the temperature dependence of electrical and thermal conductivities of biological tissue in radiofrequency ablation

Purpose: Although theoretical modelling is widely used to study different aspects of radiofrequency ablation (RFA), its utility is directly related to its realism. An important factor in this realism is the use of mathematical functions to model the temperature dependence of thermal (k) and electrical (sigma) conductivities of tissue. Our aim was to review the piecewise mathematical functions most commonly used for modelling the temperature dependence of k and sigma in RFA computational modelling. Materials and methods: We built a hepatic RFA theoretical model of a cooled electrode and compared lesion dimensions and impedance evolution with combinations of mathematical functions proposed in previous studies. We employed the thermal damage contour D63 to compute the lesion dimension contour, which corresponds to Omega = 1, Omega being local thermal damage assessed by the Arrhenius damage model. Results: The results were very similar in all cases in terms of impedance evolution and lesion size after 6 min of ablation. Although the relative differences between cases in terms of time to first roll-off (abrupt increase in impedance) were as much as 12%, the maximum relative differences in terms of the short lesion (transverse) diameter were below 3.5%. Conclusions: The findings suggest that the different methods of modelling temperature dependence of k and sigma reported in the literature do not significantly affect the computed lesion diameter.This work received financial support from the Spanish Plan Nacional de I þ D þ I del Ministerio de Ciencia e Innovacio´n, grant no. TEC2011-27133-C02-01, and from the PAID-06-11 UPV, grant ref. 1988. The authors alone are responsible for the content and writing of the paper.Trujillo Guillen, M.; Berjano, E. (2013). Review of the mathematical functions used to model the temperature dependence of electrical and thermal conductivities of biological tissue in radiofrequency ablation. International Journal of Hyperthermia. 29(6):590-597. https://doi.org/10.3109/02656736.2013.807438S590597296Radiofrequency ablation in liver tumours. (2004). Annals of Oncology, 15(suppl_4), iv313-iv317. doi:10.1093/annonc/mdh945McAchran, S. E., Lesani, O. A., & Resnick, M. I. (2005). Radiofrequency ablation of renal tumors: Past, present, and future. Urology, 66(5), 15-22. doi:10.1016/j.urology.2005.06.127Di Staso, M., Zugaro, L., Gravina, G. L., Bonfili, P., Marampon, F., Di Nicola, L., … Tombolini, V. (2011). A feasibility study of percutaneous radiofrequency ablation followed by radiotherapy in the management of painful osteolytic bone metastases. European Radiology, 21(9), 2004-2010. doi:10.1007/s00330-011-2133-3Sharma, R., Wagner, J. L., & Hwang, R. F. (2011). Ablative Therapies of the Breast. Surgical Oncology Clinics of North America, 20(2), 317-339. doi:10.1016/j.soc.2010.11.003Savoie, P.-H., Lopez, L., Simonin, O., Loubat, M., Bladou, F., Serment, G., & Karsenty, G. (2009). Résultat à deux ans de la thermothérapie prostatique par radiofréquence pour troubles mictionnels liés à l’HBP. Progrès en Urologie, 19(7), 501-506. doi:10.1016/j.purol.2009.03.004Akeboshi, M., Yamakado, K., Nakatsuka, A., Hataji, O., Taguchi, O., Takao, M., & Takeda, K. (2004). Percutaneous Radiofrequency Ablation of Lung Neoplasms: Initial Therapeutic Response. Journal of Vascular and Interventional Radiology, 15(5), 463-470. doi:10.1097/01.rvi.0000126812.12853.77Berjano, E. J. (2006). BioMedical Engineering OnLine, 5(1), 24. doi:10.1186/1475-925x-5-24Tungjitkusolmun, S., Woo, E. J., Cao, H., Tsai, J. Z., Vorperian, V. R., & Webster, J. G. (2000). Thermal—electrical finite element modelling for radio frequency cardiac ablation: Effects of changes in myocardial properties. Medical & Biological Engineering & Computing, 38(5), 562-568. doi:10.1007/bf02345754Shahidi, A. V., & Savard, P. (1994). A finite element model for radiofrequency ablation of the myocardium. IEEE Transactions on Biomedical Engineering, 41(10), 963-968. doi:10.1109/10.324528Solazzo, S. A., Liu, Z., Lobo, S. M., Ahmed, M., Hines-Peralta, A. U., Lenkinski, R. E., & Goldberg, S. N. (2005). Radiofrequency Ablation: Importance of Background Tissue Electrical Conductivity—An Agar Phantom and Computer Modeling Study. Radiology, 236(2), 495-502. doi:10.1148/radiol.2362040965Gabriel, C., Gabriel, S., & Corthout, E. (1996). The dielectric properties of biological tissues: I. Literature survey. Physics in Medicine and Biology, 41(11), 2231-2249. doi:10.1088/0031-9155/41/11/001Jo, B., & Aksan, A. (2010). Prediction of the extent of thermal damage in the cornea during conductive keratoplasty. Journal of Thermal Biology, 35(4), 167-174. doi:10.1016/j.jtherbio.2010.02.004Haemmerich, D., Chachati, L., Wright, A. S., Mahvi, D. M., Lee, F. T., & Webster, J. G. (2003). Hepatic radiofrequency ablation with internally cooled probes: effect of coolant temperature on lesion size. IEEE Transactions on Biomedical Engineering, 50(4), 493-500. doi:10.1109/tbme.2003.809488Jarrard, J., Wizeman, B., Brown, R. H., & Mitzner, W. (2010). A theoretical model of the application of RF energy to the airway wall and its experimental validation. BioMedical Engineering OnLine, 9(1), 81. doi:10.1186/1475-925x-9-81Dodde, R. E., Miller, S. F., Geiger, J. D., & Shih, A. J. (2008). Thermal-Electric Finite Element Analysis and Experimental Validation of Bipolar Electrosurgical Cautery. Journal of Manufacturing Science and Engineering, 130(2). doi:10.1115/1.2902858LAU, M., HU, B., WERNETH, R., SHERMAN, M., ORAL, H., MORADY, F., & KRYSL, P. (2010). A Theoretical and Experimental Analysis of Radiofrequency Ablation with a Multielectrode, Phased, Duty-Cycled System. Pacing and Clinical Electrophysiology, 33(9), 1089-1100. doi:10.1111/j.1540-8159.2010.02801.xBerjano, E. J., Alió, J. L., & Saiz, J. (2005). Modeling for radio-frequency conductive keratoplasty: implications for the maximum temperature reached in the cornea. Physiological Measurement, 26(3), 157-172. doi:10.1088/0967-3334/26/3/002Pätz, T., Kröger, T., & Preusser, T. (2009). Simulation of Radiofrequency Ablation Including Water Evaporation. World Congress on Medical Physics and Biomedical Engineering, September 7 - 12, 2009, Munich, Germany, 1287-1290. doi:10.1007/978-3-642-03882-2_341Jain, M. K., & Wolf, P. D. (2000). A Three-Dimensional Finite Element Model of Radiofrequency Ablation with Blood Flow and its Experimental Validation. Annals of Biomedical Engineering, 28(9), 1075-1084. doi:10.1114/1.1310219Chang, I. A., & Nguyen, U. D. (2004). BioMedical Engineering OnLine, 3(1), 27. doi:10.1186/1475-925x-3-27Yang, D., Converse, M. C., Mahvi, D. M., & Webster, J. G. (2007). Expanding the Bioheat Equation to Include Tissue Internal Water Evaporation During Heating. IEEE Transactions on Biomedical Engineering, 54(8), 1382-1388. doi:10.1109/tbme.2007.890740Bhavaraju, N. C., & Valvano, J. W. (1999). International Journal of Thermophysics, 20(2), 665-676. doi:10.1023/a:1022673524963Baldwin, S. A., Pelman, A., & Bert, J. L. (2001). A Heat Transfer Model of Thermal Balloon Endometrial Ablation. Annals of Biomedical Engineering, 29(11), 1009-1018. doi:10.1114/1.1415521Abraham, J. P., & Sparrow, E. M. (2007). A thermal-ablation bioheat model including liquid-to-vapor phase change, pressure- and necrosis-dependent perfusion, and moisture-dependent properties. International Journal of Heat and Mass Transfer, 50(13-14), 2537-2544. doi:10.1016/j.ijheatmasstransfer.2006.11.045Pennes, H. H. (1998). Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm. Journal of Applied Physiology, 85(1), 5-34. doi:10.1152/jappl.1998.85.1.5Pearce, J., Panescu, D., & Thomsen, S. (2005). Simulation of diopter changes in radio frequency conductive keratoplasty in the cornea. Modelling in Medicine and Biology VI. doi:10.2495/bio050451Zhao, G., Zhang, H.-F., Guo, X.-J., Luo, D.-W., & Gao, D.-Y. (2007). Effect of blood flow and metabolism on multidimensional heat transfer during cryosurgery. Medical Engineering & Physics, 29(2), 205-215. doi:10.1016/j.medengphy.2006.03.005Berjano, E. J., Burdío, F., Navarro, A. C., Burdío, J. M., Güemes, A., Aldana, O., … Gregorio, M. A. de. (2006). Improved perfusion system for bipolar radiofrequency ablation of liver: preliminary findings from a computer modeling study. Physiological Measurement, 27(10), N55-N66. doi:10.1088/0967-3334/27/10/n03Trujillo, M., Alba, J., & Berjano, E. (2012). Relationship between roll-off occurrence and spatial distribution of dehydrated tissue during RF ablation with cooled electrodes. International Journal of Hyperthermia, 28(1), 62-68. doi:10.3109/02656736.2011.631076Doss, J. D. (1982). Calculation of electric fields in conductive media. Medical Physics, 9(4), 566-573. doi:10.1118/1.595107Chang, S.-J., Yu, W.-J., Chang, C.-C., & Chen, Y.-H. (2010). 7 PROTEOMICS ANALYSIS OF MALE REPRODUCTIVE PHYSIOLOGY BY TOONA SINENSIS ROEM. Reproductive BioMedicine Online, 20, S3-S4. doi:10.1016/s1472-6483(10)62425-xBeop-Min Kim, Jacques, S. L., Rastegar, S., Thomsen, S., & Motamedi, M. (1996). Nonlinear finite-element analysis of the role of dynamic changes in blood perfusion and optical properties in laser coagulation of tissue. IEEE Journal of Selected Topics in Quantum Electronics, 2(4), 922-933. doi:10.1109/2944.577317Berjano, E. J., Saiz, J., & Ferrero, J. M. (2002). Radio-frequency heating of the cornea: theoretical model and in vitro experiments. IEEE Transactions on Biomedical Engineering, 49(3), 196-205. doi:10.1109/10.983453Barauskas, R., Gulbinas, A., & Barauskas, G. (2007). Investigation of radiofrequency ablation process in liver tissue by finite element modeling and experiment. Medicina, 43(4), 310. doi:10.3390/medicina43040039Ji, Z., & Brace, C. L. (2011). Expanded modeling of temperature-dependent dielectric properties for microwave thermal ablation. Physics in Medicine and Biology, 56(16), 5249-5264. doi:10.1088/0031-9155/56/16/011Labonte, S. (1994). Numerical model for radio-frequency ablation of the endocardium and its experimental validation. IEEE Transactions on Biomedical Engineering, 41(2), 108-115. doi:10.1109/10.28492

Fractional Beer-Lambert law in laser heating of biological tissue

[EN] In this article we propose an alternative formulation to model a thermal-optical coupled problem involving laser heating. We show that by using the Fractional Beer-Lambert Law (FBLL) instead of the Beer-Lambert Law (BLL) as the governing equation of the optical problem, the formulation of the laser heat source changes, along with consequently, the distribution of temperatures. Our theoretical findings apply to laser thermal keratoplasty (LTK), used to reduce diopters of hyperopia. We show that the FBLL offers a new approach for heat conduction modeling of laser heating, which is more flexible and could better fit the data in cases where the BLL approach does not fit the data well. Our results can be extended to laser heating of other biological tissues and in other general applications. Our findings imply a new insight to improve the accuracy of thermal models, since they involve a new formulation of the external heat source rather than the heat equation itself.C. Lizama is partially supported by ANID Project FONDECYT 1220036. M. MurilloArcila is supported by MCIN/AEI/10.13039/501100011033, Project PID2019-105011GBI00, and by Generalitat Valenciana, Project PROMETEU/2021/070. M. Trujillo is supported by Grant RTI2018- 094357-B-C21 funded by MCIN/AEI/10.13039/501100011033 (Spanish Ministerio de Ciencia, Innovacion y Universidades ¿ / Agencia Estatal de Investigacion)Lizama, C.; Murillo Arcila, M.; Trujillo Guillen, M. (2022). Fractional Beer-Lambert law in laser heating of biological tissue. AIMS Mathematics. 14(4):14444-14459. https://doi.org/10.3934/math.2022796144441445914

Relationship between roll-off occurrence and spatial distribution of dehydrated tissue during RF ablation with cooled electrodes

Purpose: To study the relationship between roll-off (sudden increase in impedance) and spatial distribution of dehydrated tissue during RF ablation using a cooled electrode (temperatures around 100°C). Methods: We used a double approach: (1) theoretical modelling based on the finite element method, and (2) 20 ablations using an experimental study on ex vivo excised bovine liver in which we measured impedance progress and temperature at three points close to the electrode surface: 0.5 (T1), 1.5 (T2) and 2.5 (T3) mm from the tip. T2 was located exactly at the centre of the 30 mm long electrode. Results: Temperatures at T1 and T3 quickly rose to 100°C (at ≈20 and 40 s, respectively), while at the rise at T2 was somewhat slower, stabilized around 50 s and reached a maximum value of 99°C at about 60 s. Impedance reached a minimum of 65 Ω (plateau), began increasing at 50 s and continued rising throughout the procedure, reaching a value equal to the initial value at 70 s. Likewise, computed impedance dropped to ≈73 Ω (plateau), began increasing at 50 s and reached an impedance value equal to the initial value at ≈78 s, which approximately coincided with the time when the entire zone surrounding the electrode was within the 100°C isotherm. Conclusion: There is a close relationship between the moment at which roll-off occurs and the time when the entire electrode is completely encircled by the dehydrated tissue. The mid-electrode zone is the last in which tissue desiccation occurs.This work received financial support from the Spanish Plan Nacional de I+D+I del Ministerio de Ciencia e Innovacion, grant no. TEC2008-01369/TEC and FEDER Project MTM2010-14909. The translation of this paper was funded by the Universitat Politecnica de Valencia, Spain. The authors alone are responsible for the content and writing of the paper.Trujillo Guillen, M.; Alba Martínez, J.; Berjano, E. (2012). Relationship between roll-off occurrence and spatial distribution of dehydrated tissue during RF ablation with cooled electrodes. International Journal of Hyperthermia. 28(1):62-68. https://doi.org/10.3109/02656736.2011.631076S6268281Poon, R. T.-P., Fan, S.-T., Tsang, F. H.-F., & Wong, J. (2002). Locoregional Therapies for Hepatocellular Carcinoma: A Critical Review From the Surgeon’s Perspective. Annals of Surgery, 235(4), 466-486. doi:10.1097/00000658-200204000-00004Solbiati, L., Livraghi, T., Goldberg, S. N., Ierace, T., Meloni, F., Dellanoce, M., … Gazelle, G. S. (2001). Percutaneous Radio-frequency Ablation of Hepatic Metastases from Colorectal Cancer: Long-term Results in 117 Patients. Radiology, 221(1), 159-166. doi:10.1148/radiol.2211001624Ahmed, M., Brace, C. L., Lee, F. T., & Goldberg, S. N. (2011). Principles of and Advances in Percutaneous Ablation. Radiology, 258(2), 351-369. doi:10.1148/radiol.10081634Pereira, P. L., Trübenbach, J., Schenk, M., Subke, J., Kroeber, S., Schaefer, I., … Claussen, C. D. (2004). Radiofrequency Ablation: In Vivo Comparison of Four Commercially Available Devices in Pig Livers. Radiology, 232(2), 482-490. doi:10.1148/radiol.2322030184Li, X., Zhang, L., Fan, W., Zhao, M., Wang, L., Tang, T., … Liu, Y. (2011). Comparison of microwave ablation and multipolar radiofrequency ablation, both using a pair of internally cooled interstitial applicators: Results inex vivoporcine livers. International Journal of Hyperthermia, 27(3), 240-248. doi:10.3109/02656736.2010.536967Burdío, F., Tobajas, P., Quesada-Diez, R., Berjano, E., Navarro, A., Poves, I., & Grande, L. (2011). Distant Infusion of Saline May Enlarge Coagulation Volume During Radiofrequency Ablation of Liver Tissue Using Cool-tip Electrodes Without Impairing Predictability. American Journal of Roentgenology, 196(6), W837-W843. doi:10.2214/ajr.10.5202Burdío, F., Navarro, A., Berjano, E. J., Burdío, J. M., Gonzalez, A., Güemes, A., … Grande, L. (2008). Radiofrequency hepatic ablation with internally cooled electrodes and hybrid applicators with distant saline infusion using an in vivo porcine model. European Journal of Surgical Oncology (EJSO), 34(7), 822-830. doi:10.1016/j.ejso.2007.09.029Burdío, F., Berjano, E. J., Navarro, A., Burdío, J. M., Güemes, A., Grande, L., … de Gregorio, M. A. (2007). RF tumor ablation with internally cooled electrodes and saline infusion: what is the optimal location of the saline infusion? BioMedical Engineering OnLine, 6(1), 30. doi:10.1186/1475-925x-6-30Haemmerich D, Mathematical modeling of impedance controlled radiofrequency tumor ablation and ex-vivo validation. Buenos Aires, Argentina: Proceedings of the 32nd Annual International Conference of the IEEE EMBS, 2010, pp. 1605–1608Arata, M. A., Nisenbaum, H. L., Clark, T. W. I., & Soulen, M. C. (2001). Percutaneous Radiofrequency Ablation of Liver Tumors with the LeVeen Probe: Is Roll-off Predictive of Response? Journal of Vascular and Interventional Radiology, 12(4), 455-458. doi:10.1016/s1051-0443(07)61884-3Haemmerich, D., Chachati, L., Wright, A. S., Mahvi, D. M., Lee, F. T., & Webster, J. G. (2003). Hepatic radiofrequency ablation with internally cooled probes: effect of coolant temperature on lesion size. IEEE Transactions on Biomedical Engineering, 50(4), 493-500. doi:10.1109/tbme.2003.809488McGahan, J. P., Loh, S., Boschini, F. J., Paoli, E. E., Brock, J. M., Monsky, W. L., & Li, C.-S. (2010). Maximizing Parameters for Tissue Ablation by Using an Internally Cooled Electrode. Radiology, 256(2), 397-405. doi:10.1148/radiol.09090662Berjano, E. J., Burdío, F., Navarro, A. C., Burdío, J. M., Güemes, A., Aldana, O., … Gregorio, M. A. de. (2006). Improved perfusion system for bipolar radiofrequency ablation of liver: preliminary findings from a computer modeling study. Physiological Measurement, 27(10), N55-N66. doi:10.1088/0967-3334/27/10/n03Pätz, T., Kröger, T., & Preusser, T. (2009). Simulation of Radiofrequency Ablation Including Water Evaporation. World Congress on Medical Physics and Biomedical Engineering, September 7 - 12, 2009, Munich, Germany, 1287-1290. doi:10.1007/978-3-642-03882-2_341Berjano, E. J. (2006). BioMedical Engineering OnLine, 5(1), 24. doi:10.1186/1475-925x-5-24Jo, B., & Aksan, A. (2010). Prediction of the extent of thermal damage in the cornea during conductive keratoplasty. Journal of Thermal Biology, 35(4), 167-174. doi:10.1016/j.jtherbio.2010.02.004Pearce, J., Panescu, D., & Thomsen, S. (2005). Simulation of diopter changes in radio frequency conductive keratoplasty in the cornea. Modelling in Medicine and Biology VI. doi:10.2495/bio050451Abraham, J. P., & Sparrow, E. M. (2007). A thermal-ablation bioheat model including liquid-to-vapor phase change, pressure- and necrosis-dependent perfusion, and moisture-dependent properties. International Journal of Heat and Mass Transfer, 50(13-14), 2537-2544. doi:10.1016/j.ijheatmasstransfer.2006.11.04

Applicator for RF Thermokeratoplasty: Feasibility Study Using Theoretical Modeling and Ex Vivo Experiments

Radiofrequency (RF) thermokeratoplasty uses RF currents to alter the curvature of the cornea by means of thermal lesions. An RF applicator which combined a microkeratome suction ring and a circular electrode was designed with the aim of creating circular thermal lesions in a predictable, uniform and safe way. An experimental study was conducted on ex vivo porcine eyes. A theoretical model was also designed. The experimental results showed a lesion depth of 34.2 ± 11.0% of corneal thickness at a constant voltage of 50 V up to roll-off (1000 X of impedance). With a voltage of 30 V for 30 s the mean depth was 36.8 ± 8.1%. The progress of electrical impedance throughout heating and lesion dimensions were used to compare the experimental and theoretical results. Both the impedance evolution and lesion dimensions obtained from the theoretical model showed good agreement with the experimental ¿ndings. The ¿ndings suggest that the new applicator could be a suitable option for creating uniform circular thermal lesions.This work received financial support from the Spanish "Plan Nacional de I+D+I del Ministerio de Ciencia e Innovacion" Grant No. TEC2008-01369/TEC and FEDER Project MTM2010-14909. The translation of this paper was partially funded by the Universitat Politecnica de Valencia, Spain.Trujillo Guillen, M.; Ribera, V.; Quesada, R.; Berjano, E. (2012). Applicator for RF Thermokeratoplasty: Feasibility Study Using Theoretical Modeling and Ex Vivo Experiments. Annals of Biomedical Engineering. 40(5):1182-1191. https://doi.org/10.1007/s10439-011-0492-1S11821191405Abraham, J. P., and E. M. Sparrow. A thermal-ablation bioheat model including liquid-to-vapor phase change, pressure- and necrosis-dependent perfusion, and moisture-dependent properties. Int. J. Heat Mass Transf. 50:2537–2544, 2007.Alió, J. L., M. I. Ramzy, A. Galal, and P. J. Claramonte. 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Feilmeier, and R. Kumar. Anterior chamber inflammation induced by conductive keratoplasty. J. Cataract Refract. Surg. 31:1676–1677, 2005.Ou, J. I., and E. E. Manche. Corneal perforation after conductive keratoplasty in a patient with previously undiagnosed Sjögren syndrome. Arch. Ophthalmol. 125:1131–1132, 2007.Pallikaris, I. G., T. L. Naoumidi, and N. I. Astyrakakis. Long-term results of conductive keratoplasty for low to moderate hyperopia. J. Cataract Refract. Surg. 31:1520–1529, 2005.Pearce, J., D. Panescu, and S. S. Thomsen. Simulation of diopter changes in radio frequency conductive keratoplasty in the cornea. WIT Trans. Biomed. Health 8:469–477, 2005.Stahl, J. E. Conductive keratoplasty for presbyopia: 3-year results. J. Refract. Surg. 23:905–910, 2007.Thomsen, S., J. A. Pearce, and W. F. Cheong. Changes in birefringence as markers of thermal damage in tissues. IEEE Trans. Biomed. Eng. 36:1174–1179, 1989.Trembly, B. S., N. Hashizume, K. L. Moodie, K. L. Cohen, N. K. 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Matemáticas urbanas

Rivera Herráez, RJ.; Trujillo Guillen, M. (2016). Matemáticas urbanas. Suma. 81:21-31. http://hdl.handle.net/10251/102335S21318

Computer modelling of RF ablation in cortical osteoid osteoma: Assessment of the insulating effect of the reactive zone

This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of Hyperthermia on 10 Feb 2016, available online: http://www.tandfonline.com/10.3109/02656736.2015.1135998Purpose: The aim was to study by computer simulations the insulating role of the reactive zone surrounding a cortical osteoid osteoma (OO) in terms of electrical and thermal performance during radiofrequency ablation (RFA). Material and methods: We modelled a cortical OO consisting of a nidus (10 mm diameter) enclosed by a reactive zone. The OO was near a layer of cortical bone 1.5 mm thick. Trabecular bone partially surrounds the OO and there was muscle around the cortical bone layer. We modelled RF ablations with a noncooled-tip 17-gauge needle electrode (300 s duration and 90 C target temperature). Sensitivity analyses were conducted assuming a reactive zone electrical conductivity value (rz) within the limits of the cortical and trabecular bone, i.e. 0.02 S/m and 0.087 S/m, respectively. In this way we were really modelling the different degrees of osteosclerosis associated with the reactive zone. Results: The presence of the reactive zone drastically reduced the maximum temperature reached outside it. The temperature drop was proportional to the thickness of the reactive zone: from 68 C when it was absent to 44 C when it is 7.5 mm thick. Higher nidus conductivity values (n) implied higher temperatures, while lower temperatures meant higher rz values. Changing rz from 0.02 S/m to 0.087 S/m reduced lesion diameters from 2.4 cm to 1.8 cm. Conclusions: The computer results suggest that the reactive zone plays the role of insulator in terms of reducing the temperature in the surrounding area. KeyworThis work was supported by a grant from the Agencia Nacional de Promocion Cientifica y Tecnologica de Argentina (Ref. PICT-2012-1201), and by the Spanish Programa Estatal de Investigacion, Desarrollo e Innovacion Orientada a los Retos de la Sociedad under grant number TEC2014-52383-C3-R (TEC2014-52383-C3-1-R). The authors alone are responsible for the content and writing of the paper.Irastorza, RM.; Trujillo Guillen, M.; Martel Villagran, J.; Berjano, E. (2016). Computer modelling of RF ablation in cortical osteoid osteoma: Assessment of the insulating effect of the reactive zone. International Journal of Hyperthermia. 32(3):221-230. https://doi.org/10.3109/02656736.2015.1135998S22123032

Two-compartment mathematical modeling in RF tumor ablation: New insight when irreversible changes in electrical conductivity are considered

[EN] The objective was to explore variations of temperature distribution and coagulation zone size computed by a two-compartment radiofrequency ablation (RFA) model when including simultaneously reversible changes in the tissue electrical conductivity (sigma) due to temperature and irreversible changes due to thermal coagulation. Two-compartment (tumor and healthy tissue) models were built and simulated. Reversible change of sigma was modeled by a piecewise function characterized by increments of +1.5%/degrees C up to 100 degrees C, and a 100 times smaller value from 100 degrees C onwards. Irreversible changes of sigma were modeled using an Arrhenius model. We assumed that both tumor and healthy tissue had a different initial sigma value (as suggested by the experimental data in the literature) and tended towards a common value as thermal damage progressed (necrotized tissue). We modeled a constant impedance protocol based on 90 V pulses voltage and three tumor diameters (2, 3 and 4 cm). Computer simulations showed that the differences between both models were only 0.1 and 0.2 cm for axial and transverse diameters, respectively, and this small difference was reflected in the similar temperature distributions computed by both models. In view of the available experimental data on changes of electrical conductivity in tumors and healthy tissue during heating, our results suggest that irreversible changes in electrical conductivity do not have a significant impact on coagulation zone size in two-compartment RFA models.This work was supported by the National Council of Science and Technology (CONACYT, Mexico) through a scholarship grant to Dora Luz Castro-Lopez, CVU registration No 446604; and by the Spanish Ministerio de Ciencia, Innovacion y Universidades under "Programma Estatal de I+D+i Orientada a los Retos de la Sociedad", Grant No "RTI2018-094357-B-C21".Castro-López, DL.; Trujillo Guillen, M.; Berjano, E.; Romero-Mendez, R. (2020). Two-compartment mathematical modeling in RF tumor ablation: New insight when irreversible changes in electrical conductivity are considered. Mathematical Biosciences and Engineering. 17(6):7980-7993. https://doi.org/10.3934/mbe.2020405S798079931762. D. Haemmerich, L. Chachati, A. S. Wright, D. M. Mahvi, F. T. Lee Jr, J. G. Webster, Hepatic radiofrequency ablation with internally cooled probes: Effect of coolant temperature on lesion size, IEEE Trans. Biomed. Eng., 50 (2003), 493-500.4. Z. Liu, S. M. Lobo, S. Humphries, C. Horkan, S. A. Solazzo, A. U. Hines-Peralta, et al., Radiofrequency tumor ablation: insight into improved efficacy using computer modeling, AJR Am. J. Roentgenol., 184 (2005), 1347-1352.5. S. M. Lobo, Z. J. Liu, N. C. Yu, S. Humphries, M. Ahmed, E. R. Cosman, et al., RF tumour ablation: computer simulation and mathematical modelling of the effects of electrical and thermal conductivity, Int. J. Hyperth., 21 (2005), 199-213.9. D. Haemmerich, D. J. Schutt, RF ablation at low frequencies for targeted tumor heating: In vitro and computational modeling results, IEEE Trans. Biomed. Eng., 58 (2011), 404-410.17. M. Pop, A. Molckovsky, L. Chin, M. C. Kolios, M. A. Jewett, M. D. Sherar, Changes in dielectric properties at 460 kHz of kidney and fat during heating: importance for radio-frequency thermal therapy, Phys. Med. Biol., 48 (2003), 2509-2525.18. U. Zurbuchen, C. Holmer, K. S. Lehmann, T. Stein, A. Roggan, C. Seifarth, et al., Determination of the temperature-dependent electric conductivity of liver tissue ex vivo and in vivo: Importance for therapy planning for the radiofrequency ablation of liver tumours, Int. J. Hyperth., 26 (2010), 26-33.19. E. G. Macchi, M. Gallati, G. Braschi, E. Persi, Dielectric properties of RF heated ex vivo porcine liver tissue at 480 kHz: measurements and simulations, J. Phys. D Appl. Phys., 47 (2014), 485401.21. E. Ewertowska, R. Quesada, A. Radosevic, A. Andaluz, X. Moll, F. G. Arnas, et al., A clinically oriented computer model for radiofrequency ablation of hepatic tissue with internally cooled wet electrode, Int. J. Hyperth., 35 (2019), 194-204.30. M. Qiu, A. Singh, D. Wang, J. Qu, M. Swihart, H. Zhang, P. N. Prasad, Biocompatible and biodegradable inorganic nanostructures for nanomedicine: Silicon and black phosphorus, Nano Today, 25 (2019), 135-155.33. A. Andreozzi, L. Brunese, M. Iasielllo, C. Tucci, G. P. Vanoli, Modeling heat transfer in tumors: A review of thermal therapies, Ann. Biomed. Eng., 47 (2019), 676-693

Can Fat Deposition After Myocardial Infarction Alter the Performance of RF Catheter Ablation of Scar-Related Ventricular Tachycardia?: Results from a Computer Modeling Study

Effect of Fat Deposition on the Performance in RF Ablation IntroductionThe outcomes of catheter ablation of scar-mediated ventricular tachycardia (VT) remain far from perfect. The presence of fat as a component of the underlying substrate for scar-mediated VT could be relevant since this entity can seriously impede the passage of RF current due to its low electrical conductivity. Methods and ResultsComputer models of RF ablation were built in order to investigate the means by which the spatial heterogeneity of different tissues represented within the ventricular infarct zone, including the viable myocardium, fibrous tissue, and fat, could influence temperature distributions during RF ablation. The results demonstrated that spatial distributions of different tissue types significantly alter the density of electrical current largely as a result of fat impeding the passage of current. However, the thermal lesions appear minimally unaffected by this phenomenon, with variations in depth of approximate to 1 mm. ConclusionWhile during RF ablation of scar-related ventricular tachycardia differences in tissue characteristics may affect the density of electrical current on a small-scale, overall this does not appear to significantly impact the size of the created thermal lesions.This work was supported by the Spanish "Plan Estatal de Investigacion, Desarrollo e Innovacion Orientada a los Retos de la Sociedad" under Grant TEC2014-52383-C3 (TEC2014-52383-C3-1-R).Pérez, JJ.; D'avila, A.; Aryana, A.; Trujillo Guillen, M.; Berjano, E. (2016). Can Fat Deposition After Myocardial Infarction Alter the Performance of RF Catheter Ablation of Scar-Related Ventricular Tachycardia?: Results from a Computer Modeling Study. Journal of Cardiovascular Electrophysiology. 27(8):947-952. https://doi.org/10.1111/jce.13006S947952278Aryana, A., & d’ Avila, A. (2014). Contact Force During VT Ablation. Circulation: Arrhythmia and Electrophysiology, 7(6), 1009-1010. doi:10.1161/circep.114.002389Kottkamp, H., Hindricks, G., Horst, E., Baal, T., Fechtrup, C., Breithardt, G., & Borggrefe, M. (1997). Subendocardial and Intramural Temperature Response During Radiofrequency Catheter Ablation in Chronic Myocardial Infarction and Normal Myocardium. Circulation, 95(8), 2155-2161. doi:10.1161/01.cir.95.8.2155KOVOOR, P., DALY, M. P. J., POULIOPOULOS, J., BYTH, K., DEWSNAP, B. I., EIPPER, V. E., … ROSS, D. L. (2006). Comparison of Radiofrequency Ablation in Normal Versus Scarred Myocardium. Journal of Cardiovascular Electrophysiology, 17(1), 80-86. doi:10.1111/j.1540-8167.2005.00324.xBetensky, B. P., Jauregui, M., Campos, B., Michele, J., Marchlinski, F. E., Oley, L., … Gerstenfeld, E. P. (2012). Use of a Novel Endoscopic Catheter for Direct Visualization and Ablation in an Ovine Model of Chronic Myocardial Infarction. Circulation, 126(17), 2065-2072. doi:10.1161/circulationaha.112.112540Sasaki, T., Calkins, H., Miller, C. F., Zviman, M. M., Zipunnikov, V., Arai, T., … Zimmerman, S. L. (2015). New insight into scar-related ventricular tachycardia circuits in ischemic cardiomyopathy: Fat deposition after myocardial infarction on computed tomography--A pilot study. Heart Rhythm, 12(7), 1508-1518. doi:10.1016/j.hrthm.2015.03.041Goldfarb, J. W., Roth, M., & Han, J. (2009). Myocardial Fat Deposition after Left Ventricular Myocardial Infarction: Assessment by Using MR Water-Fat Separation Imaging. Radiology, 253(1), 65-73. doi:10.1148/radiol.2532082290Ichikawa, Y., Kitagawa, K., Chino, S., Ishida, M., Matsuoka, K., Tanigawa, T., … Sakuma, H. (2009). Adipose Tissue Detected by Multislice Computed Tomography in Patients After Myocardial Infarction. JACC: Cardiovascular Imaging, 2(5), 548-555. doi:10.1016/j.jcmg.2009.01.010Su, L., Siegel, J. E., & Fishbein, M. C. (2004). Adipose tissue in myocardial infarction. Cardiovascular Pathology, 13(2), 98-102. doi:10.1016/s1054-8807(03)00134-0Suárez, A. G., Hornero, F., & Berjano, E. J. (2010). Mathematical Modeling of Epicardial RF Ablation of Atrial Tissue with Overlying Epicardial Fat. The Open Biomedical Engineering Journal, 4(1), 47-55. doi:10.2174/1874120701004020047PÉREZ, J. J., D’AVILA, A., ARYANA, A., & BERJANO, E. (2015). Electrical and Thermal Effects of Esophageal Temperature Probes on Radiofrequency Catheter Ablation of Atrial Fibrillation: Results from a Computational Modeling Study. Journal of Cardiovascular Electrophysiology, 26(5), 556-564. doi:10.1111/jce.12630Berjano, E. J. (2006). BioMedical Engineering OnLine, 5(1), 24. doi:10.1186/1475-925x-5-24Hasgall PA Di Gennaro F Baumgartner C Neufeld E Gosselin MC Payne D Klingenböck A Kuster N 10.13099/VIP21000-03-0 www.itis.ethz.ch/databaseGonzalez-Suarez, A., & Berjano, E. (2016). Comparative Analysis of Different Methods of Modeling the Thermal Effect of Circulating Blood Flow During RF Cardiac Ablation. IEEE Transactions on Biomedical Engineering, 63(2), 250-259. doi:10.1109/tbme.2015.2451178Salazar, Y., Bragos, R., Casas, O., Cinca, J., & Rosell, J. (2004). Transmural Versus Nontransmural In Situ Electrical Impedance Spectrum for Healthy, Ischemic, and Healed Myocardium. IEEE Transactions on Biomedical Engineering, 51(8), 1421-1427. doi:10.1109/tbme.2004.82803

Could the heat sink effect of blood flow inside large vessels protect the vessel wall from thermal damage during RF-assisted surgical resection?

Purpose: To assess by means of computer simulations whether the heat sink effect inside a large vessel (portal vein) could protect the vessel wall from thermal damage close to an internally cooled electrode during radiofrequency (RF)-assisted resection. Methods: First, in vivo experiments were conducted to validate the computational model by comparing the experimental and computational thermal lesion shapes created around the vessels. Computer simulations were then carried out to study the effect of different factors such as device-tissue contact, vessel position, and vessel-device distance on temperature distributions and thermal lesion shapes near a large vessel, specifically the portal vein. Results: The geometries of thermal lesions around the vessels in thein vivo experiments were in agreement with the computer results. The thermal lesion shape created around the portal vein was significantly modified by the heat sink effect in all the cases considered. Thermal damage to the portal vein wall was inversely related to the vessel-device distance. It was also more pronounced when the device-tissue contact surface was reduced or when the vessel was parallel to the device or perpendicular to its distal end (blade zone), the vessel wall being damaged at distances less than 4.25 mm. Conclusions: The computational findings suggest that the heat sink effect could protect the portal vein wall for distances equal to or greater than 5 mm, regardless of its position and distance with respect to the RF-based device. © 2014 American Association of Physicists in Medicine.This work received financial support from the Spanish "Plan Nacional de I+D+I del Ministerio de Ciencia e Innovacion" Grant No. TEC2011-27133-C02-01 and -02, also from the Universitat Politecnica de Valencia (INNOVA11-01-5502; and PAID-06-11 Ref. 1988). A. Gonzalez-Suarez is the recipient of a Grant VALi+d (ACIF/2011/194) from the Generalitat Valenciana. E.B and F.B. declare a stock ownership in Apeiron Medical S.L. This company has a license for Patent U.S. 8.303.584, on which the device considered in this study is based. The remaining authors have no conflict of interest or financial ties to disclose.González Suárez, A.; Trujillo Guillen, M.; Burdío, F.; Andaluz, A.; Berjano, E. (2014). Could the heat sink effect of blood flow inside large vessels protect the vessel wall from thermal damage during RF-assisted surgical resection?. Medical Physics. 41(8):083301-1-83301-13. https://doi.org/10.1118/1.4890103S083301-183301-13418Poon, R. T., Fan, S. T., & Wong, J. (2005). Liver resection using a saline-linked radiofrequency dissecting sealer for transection of the liver. Journal of the American College of Surgeons, 200(2), 308-313. doi:10.1016/j.jamcollsurg.2004.10.008Burdío, F., Grande, L., Berjano, E., Martinez-Serrano, M., Poves, I., Burdío, J. M., … Güemes, A. (2010). A new single-instrument technique for parenchyma division and hemostasis in liver resection: a clinical feasibility study. The American Journal of Surgery, 200(6), e75-e80. doi:10.1016/j.amjsurg.2010.02.020Topp, S. A., McClurken, M., Lipson, D., Upadhya, G. A., Ritter, J. H., Linehan, D., & Strasberg, S. M. (2004). Saline-Linked Surface Radiofrequency Ablation. Annals of Surgery, 239(4), 518-527. doi:10.1097/01.sla.0000118927.83650.a4Tepetes, K. (2008). Risks of the radiofrequency-assisted liver resection. Journal of Surgical Oncology, 97(2), 193-193. doi:10.1002/jso.20900Marchal, F., Elias, D., Rauch, P., Zarnegar, R., Leroux, A., Stines, J., … Villemot, J. P. (2006). Prevention of Biliary Lesions That May Occur During Radiofrequency Ablation of the Liver. Annals of Surgery, 243(1), 82-88. doi:10.1097/01.sla.0000193831.39362.07Sutton, P. A., Awad, S., Perkins, A. C., & Lobo, D. N. (2010). Comparison of lateral thermal spread using monopolar and bipolar diathermy, the Harmonic Scalpel™and the Ligasure™. British Journal of Surgery, 97(3), 428-433. doi:10.1002/bjs.6901Lee, J. M., Han, J. K., Chang, J. M., Chung, S. Y., Kim, S. H., Lee, J. Y., … Choi, B. I. (2006). Radiofrequency Ablation of the Porcine Liver In Vivo: Increased Coagulation with an Internally Cooled Perfusion Electrode. Academic Radiology, 13(3), 343-352. doi:10.1016/j.acra.2005.10.020Goldberg, S. N., Grassi, C. J., Cardella, J. F., Charboneau, J. W., Dodd, G. D., Dupuy, D. E., … Silverman, S. G. (2005). Image-guided Tumor Ablation: Standardization of Terminology and Reporting Criteria. Radiology, 235(3), 728-739. doi:10.1148/radiol.2353042205Pennes, H. H. (1948). Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm. Journal of Applied Physiology, 1(2), 93-122. doi:10.1152/jappl.1948.1.2.93Abraham, J. P., & Sparrow, E. M. (2007). A thermal-ablation bioheat model including liquid-to-vapor phase change, pressure- and necrosis-dependent perfusion, and moisture-dependent properties. International Journal of Heat and Mass Transfer, 50(13-14), 2537-2544. doi:10.1016/j.ijheatmasstransfer.2006.11.045Jo, B., & Aksan, A. (2010). Prediction of the extent of thermal damage in the cornea during conductive keratoplasty. Journal of Thermal Biology, 35(4), 167-174. doi:10.1016/j.jtherbio.2010.02.004Berjano, E. J. (2006). BioMedical Engineering OnLine, 5(1), 24. doi:10.1186/1475-925x-5-24Zhao, G., Zhang, H.-F., Guo, X.-J., Luo, D.-W., & Gao, D.-Y. (2007). Effect of blood flow and metabolism on multidimensional heat transfer during cryosurgery. Medical Engineering & Physics, 29(2), 205-215. doi:10.1016/j.medengphy.2006.03.005T. Pätz T. Körger T. Preusser Simulation of radiofrequency ablation including water evaporation 2009Chang, I. A., & Nguyen, U. D. (2004). BioMedical Engineering OnLine, 3(1), 27. doi:10.1186/1475-925x-3-27Chang, I. A. (2010). Considerations for Thermal Injury Analysis for RF Ablation Devices~!2009-09-09~!2009-12-19~!2010-02-04~! The Open Biomedical Engineering Journal, 4(2), 3-12. doi:10.2174/1874120701004020003Tungjitkusolmun, S., Staelin, S. T., Haemmerich, D., Jang-Zern Tsai, Hong Cao, Webster, J. G., … Vorperian, V. R. (2002). Three-dimensional finite-element analyses for radio-frequency hepatic tumor ablation. IEEE Transactions on Biomedical Engineering, 49(1), 3-9. doi:10.1109/10.972834Beop-Min Kim, Jacques, S. L., Rastegar, S., Thomsen, S., & Motamedi, M. (1996). Nonlinear finite-element analysis of the role of dynamic changes in blood perfusion and optical properties in laser coagulation of tissue. IEEE Journal of Selected Topics in Quantum Electronics, 2(4), 922-933. doi:10.1109/2944.577317Antunes, C. L., Almeida, T. R. O., & Raposeiro, N. (2012). Saline‐enhanced RF ablation on a cholangiocarcinoma: a numerical simulation. COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, 31(4), 1055-1066. doi:10.1108/03321641211227302Doss, J. D. (1982). Calculation of electric fields in conductive media. Medical Physics, 9(4), 566-573. doi:10.1118/1.595107Haemmerich, D., Wright, A. W., Mahvi, D. M., Lee, F. T., & Webster, J. G. (2003). Hepatic bipolar radiofrequency ablation creates coagulation zones close to blood vessels: A finite element study. Medical & Biological Engineering & Computing, 41(3), 317-323. doi:10.1007/bf02348437Berjano, E. J., Burdío, F., Navarro, A. C., Burdío, J. M., Güemes, A., Aldana, O., … Gregorio, M. A. de. (2006). Improved perfusion system for bipolar radiofrequency ablation of liver: preliminary findings from a computer modeling study. Physiological Measurement, 27(10), N55-N66. doi:10.1088/0967-3334/27/10/n03Burdío, F., Berjano, E. J., Navarro, A., Burdío, J. M., Grande, L., Gonzalez, A., … Lequerica, J. L. (2009). Research and development of a new RF-assisted device for bloodless rapid transection of the liver: Computational modeling and in vivo experiments. BioMedical Engineering OnLine, 8(1), 6. doi:10.1186/1475-925x-8-6Modelling in Medicine and Biology VI. (2005). doi:10.2495/bio05Haemmerich, D., Chachati, L., Wright, A. S., Mahvi, D. M., Lee, F. T., & Webster, J. G. (2003). Hepatic radiofrequency ablation with internally cooled probes: effect of coolant temperature on lesion size. IEEE Transactions on Biomedical Engineering, 50(4), 493-500. doi:10.1109/tbme.2003.809488Berjano, E., & d’ Avila, A. (2013). Lumped Element Electrical Model based on Three Resistors for Electrical Impedance in Radiofrequency Cardiac Ablation: Estimations from Analytical Calculations and Clinical Data. The Open Biomedical Engineering Journal, 7(1), 62-70. doi:10.2174/1874120720130603001Hannesson, P., Stridbeck, H., Lundstedt, C., Andren-Sandberg, Å., & Ihse, I. (1995). Intravascular Ultrasound of the Portal Vein — Normal Anatomy. Acta Radiologica, 36(4), 388-392. doi:10.3109/02841859509173394Chen, X., & Saidel, G. M. (2008). Mathematical Modeling of Thermal Ablation in Tissue Surrounding a Large Vessel. Journal of Biomechanical Engineering, 131(1). doi:10.1115/1.2965374T. Peng D. O'Neill S. Payne Mathematical study of the effects of different intrahepatic cooling on thermal ablation zones 2011CIONI, G., D’ALIMONTE, P., CRISTANI, A., VENTURA, P., ABBATI, G., TINCANI, E., … VENTURA, E. (1992). Duplex-Doppler assessment of cirrhosis in patients with chronic compensated liver disease. Journal of Gastroenterology and Hepatology, 7(4), 382-384. doi:10.1111/j.1440-1746.1992.tb01003.xRíos, J. S., Zalabardo, J. M. S., Burdio, F., Berjano, E., Moros, M., Gonzalez, A., … Güemes, A. (2011). Single Instrument for Hemostatic Control in Laparoscopic Partial Nephrectomy in a Porcine Model Without Renal Vascular Clamping. Journal of Endourology, 25(6), 1005-1011. doi:10.1089/end.2010.0557Dos Santos, I., Haemmerich, D., Pinheiro, C., & da Rocha, A. (2008). Effect of variable heat transfer coefficient on tissue temperature next to a large vessel during radiofrequency tumor ablation. BioMedical Engineering OnLine, 7(1), 21. doi:10.1186/1475-925x-7-21Consiglieri, L., Santos, I. dos, & Haemmerich, D. (2003). Theoretical analysis of the heat convection coefficient in large vessels and the significance for thermal ablative therapies. Physics in Medicine and Biology, 48(24), 4125-4134. doi:10.1088/0031-9155/48/24/010Consiglieri, L. (2012). Continuum Models for the Cooling Effect of Blood Flow on Thermal Ablation Techniques. International Journal of Thermophysics, 33(5), 864-884. doi:10.1007/s10765-012-1194-0Huang, H.-W. (2013). Influence of blood vessel on the thermal lesion formation during radiofrequency ablation for liver tumors. Medical Physics, 40(7), 073303. doi:10.1118/1.4811135Welp, C., Siebers, S., Ermert, H., & Werner, J. (2006). Investigation of the influence of blood flow rate on large vessel cooling in hepatic radiofrequency ablation / Untersuchung des Einflusses der Blutflussgeschwindigkeit auf die Gefäßkühlung bei der Radiofrequenzablation von Lebertumoren. Biomedizinische Technik/Biomedical Engineering, 51(5_6), 337-346. doi:10.1515/bmt.2006.067Lehmann, K. S., Ritz, J. P., Valdeig, S., Knappe, V., Schenk, A., Weihusen, A., … Frericks, B. B. (2009). Ex situ quantification of the cooling effect of liver vessels on radiofrequency ablation. Langenbeck’s Archives of Surgery, 394(3), 475-481. doi:10.1007/s00423-009-0480-1Ng, K. K. C., Lam, C. M., Poon, R. T. P., Shek, T. W. H., Fan, S. T., & Wong, J. (2004). Delayed portal vein thrombosis after experimental radiofrequency ablation near the main portal vein. British Journal of Surgery, 91(5), 632-639. doi:10.1002/bjs.4500Metcalfe, M. S., Mullin, E. J., Texler, M., Berry, D. P., Dennison, A. R., & Maddern, G. J. (2007). The safety and efficacy of radiofrequency and electrolytic ablation created adjacent to large hepatic veins in a porcine model. European Journal of Surgical Oncology (EJSO), 33(5), 662-667. doi:10.1016/j.ejso.2007.02.011Bangard, C., Gossmann, A., Kasper, H. U., Hellmich, M., Fischer, J. H., Hölscher, A., … Stippel, D. L. (2006). Experimental Radiofrequency Ablation Near the Portal and the Hepatic Veins in Pigs: Differences in Efficacy of a Monopolar Ablation System. Journal of Surgical Research, 135(1), 113-119. doi:10.1016/j.jss.2006.02.026T. Kröger T. Preusser H. O. Peitgen Blood flow induced cooling effect in radio frequency ablation for hepatic carcinoma 200

Anna Bofill s Use of Mathematics in Her Architecture

[EN] The work of Anna Bofill Levi, architect and composer, is an example of using mathematics to envision or conceive a project in both architecture and music. In this paper we focus on the architectural aspect and try to highlight the impact of mathematics on her work by analyzing several of her projects. The point of reference is the 1975 thesis on form generation written by Anna Bofill for the Universitat PolitScnica de Barcelona. We conclude with excerpts from an interview with Anna Bofill herself.This work has been partially supported by the Department of Applied Mathematics of Universitat Politecnica de Valencia (PID-DMA 2014 y PID-DMA-2016).Gómez Collado, MDC.; Rivera Herráez, RJ.; Trujillo Guillen, M. (2017). Anna Bofill s Use of Mathematics in Her Architecture. Nexus Network Journal. 19(2):239-254. https://doi.org/10.1007/s00004-016-0322-8S239254192Bofill Levi, Anna. 1975. Contribución al estudio de la generación de formas arquitectónicas y urbanas, Ph.D. thesis, Universitat Politècnica de Barcelona.Bofill Levi, Anna. 2008. Guía para el planeamiento urbanístico y la ordenación urbana con la incorporación de criterios de género. Generalitat Catalana.Bofill Levi, Anna. 2010. Generations of forms: space to inhabit, time to think, The Schelling Lectures, Art Stock Books Ltd.Bofill Levi, Anna, Rosa M. Dumenjó Mertí, Isabel Segura Soriano and Carmen Martinez Garrote. 1998. Las mujeres y la ciudad: libro blanco para una concepción del entorno habitado desde el punto de vista del género. Consejo de la mujer de la Comunidad de Madrid.Borja, Jordi and Zaida Muxi. 2000. El espacio público. Ciudad y ciudadanía. Ed. Electa, BarcelonaGarcía Hernández, Pedro. 2013. La agregación modular como mecanismo proyectual residencial en España: el Taller de Arquitectura, Ph. D. thesis. Universitat Ramón Llul. http://hdl.handle.net/10803/108286 .Geddes, Patrick. 1915. Cities in evolution. Williams and Nortgate.González Virós, Itziar. 2006. Anna Bofill Levi (Entrevista). Quaderns d’arquitectura i urbanisme 250: 112-19. http://www.raco.cat/index.php/QuadernsArquitecturaUrbanisme/article/view/235108/349829 .Martín Nieva, Helena. 2002. Número y género de dos términos: arquitectura y música, en Anna Bofill Levi. DC PAPERS: revista de crítica y teoría de la arquitectura 23 (Juliol 2012): 57–68.Sert, Josep Lluis. 1942. Can Our Cities Survive? An ABC of Urban Problems, their Analysis, their Solutions, Cambridge, Harvard University Press