- Open Access
Influence of electrical and thermal properties on RF ablation of breast cancer: is the tumour preferentially heated?
© Ekstrand et al; licensee BioMed Central Ltd. 2005
- Received: 02 February 2005
- Accepted: 11 July 2005
- Published: 11 July 2005
Techniques based on radio frequency (RF) energy have many applications in medicine, in particular tumour ablation. Today, mammography screening detects many breast cancers at an early stage, facilitating treatment by minimally invasive techniques such as radio frequency ablation (RFA). The breast cancer is mostly surrounded by fat, which during RFA-treatment could result in preferential heating of the tumour due to the substantial differences in electrical parameters. The object of this study was to investigate if this preferential heating existed during experimental in vitro protocols and during computer simulations.
Excised breast material from four patients with morphologically diagnosed breast cancers were treated with our newly developed RFA equipment. Subsequently, two finite element method (FEM) models were developed; one with only fat and one with fat and an incorporated breast cancer of varying size. The FEM models were solved using temperature dependent electrical conductivity versus constant conductivity, and transient versus steady-state analyses.
Our experimental study performed on excised breast tissue showed a preferential heating of the tumour, even if associated with long tumour strands. The fat between these tumour strands was surprisingly unaffected. Furthermore, the computer simulations demonstrated that the difference in electrical and thermal parameters between fat and tumour tissue can cause preferential heating of the tumour. The specific absorption rate (SAR) distribution changed significantly when a tumour was present in fatty tissue. The degree of preferential heating depended on tissue properties, tumour shape, and placement relative to the electrode. Temperature dependent electrical conductivity increased the thermal lesion volume, but did not change the preferential heating. Transient solutions decreased the thermal lesion volume but increased the preferential heating of the tumour.
Both the computer model and the in vitro study confirmed that preferential heating of the tumour during RFA exists in breast tissue. However, the observed preferential heating in the in vitro studies were more pronounced, indicating that additional effects other than the difference in tissue parameters might be involved. The existing septa layers between the cancer tissue and the fatty tissue could have an additional electrical or thermal insulating effect, explaining the discrepancy between the in vitro study and the computer model.
- Electrical Impedance Tomography
- Radio Frequency Ablation
- Finite Element Method Simulation
- Specific Absorption Rate
- Bezier Curve
At least 10% of the women in the western world face the prospect of developing breast cancer. The tendency in modern treatment of these tumours is towards less invasive local treatment. Today breast conserving surgery (BCS) has become more common than mastectomy in many countries. BCS and mastectomy combined with radiation are associated with satisfactory long-term outcome. The survival rates after BCS of ductal carcinoma in situ is approximately 98%, whereas approximately 100% of these patents are cancer free after mastectomy [1, 2]. However, multiple treatments and additional adjuvant care are needed in up to 50% of the BCS cases, resulting in higher associated costs compared to mastectomy alone [3, 4]. As in all surgery for breast cancer, the goal is to remove all of the cancertogether with a sufficient margin of healthy tissue, to prevent local recurrence. Furthermore, today many breast cancers are detected at an early stage by mammography screening, raising a demand for new techniques that minimise alternation of breast configuration. Recently, approaches other than traditional surgery have been explored to satisfy these demands [5–7]. These techniques are minimally or totally non invasive, and include, cryosurgery, stereotactic excision, laser ablation, focused ultrasound, and radio frequency ablation (RFA). Potential benefits with these techniques are reduced morbidity rates, reduced treatment duration, and the ability to perform therapy for patients in poor medical condition on an outpatient basis. Of these new techniques, RFA is considered to be the most promising treatment for breast cancer because of its effective destruction of cancer cells and having a low complication rate [8, 9].
The absorbed power density at each point is:
where q = power density (W/m3), E = electric field (V/m), J = current density (A/m2) and σ = conductivity (1/(Ω·m)) (Vector variables have both a magnitude and a direction and are presented with an over-bar.). The total current through each closed surface enclosing one of the electrodes always equals the total current from the generator, independent of surface size. Hence, the heating will be concentrated in the region close to the active electrode, where the surface area is small and the current density is high. During experimental protocols with our electrode in in vitro muscle tissue, 87% of the total electrical energy is absorbed in an iso-potential volume with radius 2 cm . Between 43°C and 60°C the cell damage originates from denaturation of proteins. Over 60°C the tissue coagulates, because collagen is converted to glucose and the time frame of cell death is considered almost instantaneous. When the tissue temperature reaches approximately 90–110°C, phase transformation of the intra- and extra-cellular liquids occurs. Glucose develops an adhesive effect after dehydration. The gas bubbles created function as electrical insulation, which alters the effective current-path area and further increases the current density and tissue temperature. If the temperature reaches 200°C, tissue charring is initiated. This avalanche-like phenomenon ultimately insulates the electrode from the tissue, and heating often ceases non-reversibly. Thus, the output power during RFA is limited, ensuring that the maximum tissue temperature is below the initiation of the avalanche-like insulation phenomenon. The coagulation necrosis zone with one monopolar electrode will therefore be limited to approximately 13 mm in diameter during in-vivo protocols in muscle tissue . Several approaches have been proposed to increase the necrotic region. These are multiprobe electrodes , saline injected electrodes , and internally cooled electrodes . Ultrasonography, which is used to guide the electrode to the tumour, cannot adequately predict the thermal lesion margin. Instead, MRI and/or a core biopsy can be used to confirm adequate ablation . The success rate of the in-vivo studies performed before mastectomy varyies from 86% (19–22) to 100% (21–21) [14–17]. This variation might depend on different inclusion criteria, e.g. tumour size. The procedure was well tolerated, without complications and cosmesis was excellent. To date, no studies have used RFA as the only treatment method for breast cancer. Hence, the long-term oncologic results of this new method are yet to be evaluated.
The normal female breast is composed primarily of fat with varying concentration partly due to body habitus. The breast gland is composed of lobes, which empty into separate major ducts terminating in the nipple. Each lobe and its smaller subunits are separated by connective tissue. The amount of the latter usually increases with age. There are two types of breast cancers, which constitute more than 95% of the malignant tumours; ductal and lobular carcinomas. The former is derived from duct and the latter from terminal duct epithelium. They differ greatly in morphological as well as biological aspects. The classical "crablike" appearance of the ductal carcinoma, with solid central body from which strands stretch out in the surrounding tissue has, in fact, given name to all cancers (Krebs in German). The radiating branches consist of tumour cells and connective tissue. The latter partly considered to be induced by the tumour. The lobular carcinoma, on the other hand, has a diffuse growth pattern, is often multifocal, and is difficult to diagnose radiologically. Thus, RFA treatment of breast cancer comprises a unique situation, since the tumour is mostly embedded in fat. The difference in electrical parameters between the fat and the tumour tissues is substantial, which might result in preferential heating of the tumour. The goal of this study was to document if preferential heating existed during in vitro treatment, and if it could be induced in computer simulation by the dissimilarity of electrical and thermal parameters.
Several computer simulation studies have investigated RFA in a variety of locations such as the liver [18–21] or the heart . The main objective in most of these studies was to predict the thermal lesion size. Other relevant issues such as the effects of electrode cooling , vessel size , multiple probes , and temperature dependent conductivity  have also been addressed. Earlier FEM studies on breasts have primarily studied thermography for diagnostic purposes [23, 24].
Experimental In vitro study
The in vitro study, approved by the local ethics committee, was performed on excised breast material with a morphology diagnosis of breast cancer. After obtaining informed consent, four patients underwent modified radical mastectomy. After surgery the specimens, three ductal and one lobular carcinomas all over three centimetres in diameter, were sent for pathologic examination. Subsequently, an internally cooled steel electrode (VibraTech AB, Stockholm, Sweden, figure 1) was installed in the central part of the tumour. Temperature was measured at the needle tip with incorporated thermo-couples. The RF-generator was specially designed with a floating low-impedance output (0–950 W, 1.5 MHz). The maximum tissue temperature was maintained at approximately 100°C over 15 min. It is difficult to measure the true maximum temperature due to the circulating cooling media, i.e. the true maximum is located within the tissue. We have developed an algorithm that compensates for the deviation between the maximum temperature and the measured value at the electrode tip. Electrical impedance was measured to ensure that the initiation of the avalanche phenomenon did not occur. The temperature and the flow of the cooling media were 20°C and 12 ml/s, respectively. An electrode current (root mean squared) between 0.35 and 0.71 A was applied during the procedure. After the treatment, the tissue was immersed and fixed in a 4% formalin solution. The tumour, including the electrode canal and surrounding fat tissue, was cut out, trimmed, embedded and processed for large sections, which underwent histologic examination. Thermal lesion margin was defined as the zone with well established coagulation necrosis, i.e. condensation and loss of nuclear details and homogenisation of the cytoplasm.
Because the tissue heating is induced by RF-energy, a quasi-static electrical model can be used to describe the electrical field and the current density. Under quasi-static conditions the electric potential can be solved using Laplace's equation:
▽·[σ(T)·▽V] = 0 (2)
where V is the electrical potential (V) and T is the temperature (°C). The thermal behaviour is governed by the bio-heat equation:
where ρ is the tissue density (kg/m3), c is the heat capacity (J/kg°C), k is the thermal conductivity (W/°C m), q is the electrical energy source, ρb is the density of blood, Cb is the heat capacity of blood, ω is the blood perfusion, Tblood is the basal temperature of the blood, and Qm is the metabolic heat source. In an in-vitro situation, the metabolic heat source and the blood perfusion are set to zero.
Description of cases
Tumour dimensions (mm)
Const / Temp dep. σ
Number of Elements
Steady state/Controlled Transient
Electrical and Thermal parameters
Electric Conductivity (1/(Ω·m))
Thermal Conductivity (W/(m·°C))
Specific heat (kJ/(kg·°C))
The electrical boundary condition for the outer boundary was set to V = 0, representing the ground electrode. The electrical boundary condition for the conducting electrode was set to a source potential of V = Vin. The non-conducting part had an insulating boundary condition, i.e. the current component orthogonal to the surface is zero. The thermal boundary condition of the outer surface was set to 20°C, i.e. the initial temperature of the tissue. With our equipment, the temperature at the electrode boundary does not diverge far from the cooling media temperature. Thus, the thermal boundary condition of the electrode was set to the cooling media temperature, T = 20°C, to incorporate the electrode cooling in the model.
Ten finite element method (FEM) [32, 33] models were developed (cases 1–10, table 1), and solved using FEMLAB 3.0 (Comsol, Stockholm, Sweden) on a 2.0 GHz AMD Athlon XP 2400 computer, with 1024 Mb RAM. When the electrical conductivity is independent of temperature, the electrical potential, V, can be found without solving the bio-heat equation. Thus, the calculated V, defined over the entire volume, is subsequently used to solve the bio-heat equation. When the electrical conductivity is temperature dependent, equations 2 and 3 must be solved by a coupled method. The required iterative computations for these non-linear problems are much more computer intensive. All models were solved without including blood perfusion and metabolic heating. Cases 1–8 were solved using a steady state approximation, where Vin was adjusted to obtain a final maximum temperature of 100°C. Additionally, two controlled 15 min transient simulations were performed, cases 9 and 10. During these simulations, the maximum temperature was maintained at approximately 100°C throughout the whole treatment, by successively adapting the boundary condition Vin.
A quadratic mesh consisting of Lagrange triangular elements was used for both the thermal and the electrical problems. The conservationof energy introduced by the source was checked and found highly reliable. This measurement serves as an indication of accuracy in the formulation. Furthermore, a common ad hoc procedure of successive mesh refinement was used, with the FEM solution considered converged when the difference in maximum temperature between successive calculations was less then 0.1% for a doubling of the number of elements. Thermal lesion size was determined using the 50°C margin. Even though the thermal lesion volume is dependent on both temperature and time of elevated temperature, similar lesion dimensions have been obtained using both thermal dose and threshold temperature .
Experimental in vitro study
Ablation data from the computer simulation protocols
Comments on the FEM results
The largest thermal lesions were created using temperature dependent electrical conductivity and steady state solving. Furthermore, temperature dependent electrical conductivity mimics the impedance behaviour in the in vitro studies, which demonstrated a decreasing impedance of up to 50%. Hence, only temperature dependent electrical conductivity was used to document the effect of tumour spatial shape and the influence of time dependence. The baseline impedance in the experimental studies varied between 80 and 200 Ω. During FEM simulation the baseline impedance varied between 51 and 373 Ω, using only tumour tissue or only fat tissue, respectively. Consequently, the electrical parameters in the in vitro experiments and in the FEM model correspond well. In the models with only fat tissue, the heat was shifted towards the tip and away from the electrode because of the cooling along the electrode shaft and the increased SAR created by the curvature of the electrode tip. The shift in temperature between corresponding models with and without tumour is primarily explained by the shift in absorbed electrical power density (figures 7 and 9). In the models with tumour, the electrical absorption was relatively higher in the tumour tissue and relatively lower in the fat tissue compared to a non-tumour model. Even though the thermal lesion volume decreased during steady state simulation, preferential heating of the tumour increased. The thermal lesion volume decreased because of the shorter treatment time, i.e. less energy is transmitted by thermal conduction to heat the distant regions. Initially, the tumour shape could be detected in the isothermal lines. but with time this shape effect gradually decreased. Hence, as entropy increased and thermal conduction levelled out the differences in temperature with time, the temperature difference between the tumour and the fat gradually decreased to the steady state. The width of the tumour, compared to the length, has a more pronounced effect on the preferential heating. With increasing tumour width, both the location of the temperature maximum and the thermal lesion margin are pushed away from the electrode, along the symmetry axis, In case 8, the thermal lesion margin is closer to the electrode than in the original fat model. Yet the region near the electrode is relatively hotter in case 8. It seems that the heat dissipation from thin tumours decrease the lesion size. However, during transient analysis, this effect might decrease. The altering of the tumour length only changed the tumour in the region where the contribution to the impedance and resistive heating were small. Hence, almost no dependence on tumour length is detected. The tissue near the electrode is heated mainly by the absorbed electrical energy, while regions further away are mainly heated by thermal conduction. Thus, the preferential heating of the tumour is governed by the electrical parameters near the electrode, whereas thermal parameters become increasingly important further away.
The effects of temperature dependent electrical conductivity during RFA have been studied by Chang . Temperature dependent conductivity results in increased current density, whereas the electric field remains almost constant compared to constant conductivity. The maximum temperature increases by 5–8%, using constant voltage between the electrodes. Furthermore, one study has investigated the preferential heating of hepatic tumours . The study concludes that ablation at low frequencies, where the difference in electrical parameters are relatively higher, may preferentially target tumour tissue. This effect only arises if the active electrode is in contact with both liver and tumour tissue. The extent of the effect is also dependent on probe geometry and control algorithm. Moreover, Tungjitkusolmun, et al. have investigated the sensitivity of tissue parameters on thermal lesions for radio frequency cardiac ablation . Their study shows that the accuracy of tissue property values is critical to FEM modelling.
Limitations of the computer model
We have limited the computer model to include two tissue types: fat and tumour, giving a fair representation of the real situation. The electrical properties of glandular and connective tissue are similar to those of cancer. Thus, connective and glandular tissue are also preferentially heated by the RF-energy, which was confirmed in the in vitro studies. Hence, incorporation of glandular and connective tissue provides little additional information on the preferential heating of the tumour. To correctly mimic the SAR situation during RF ablation, a three dimensional model must be used, i.e. the iso-potential surface must increase with the distance squared from the electrode. This is for obvious reasons not the case in two dimensional or one dimensional Cartesian models. However, if the situation demonstrates rotational symmetry a two dimensional axially symmetric model can be used instead to significantly decrease calculation time and related computer resources. Non-linear models, transient analysis, and the use of different tissue types made rotational symmetry essential in our case. The tumour shapes considered are not claimed to be perfect representations of the true situation. A better representation would be a core tumour from which thin outgrowths extend. However, this configuration is impossible to achieve with rotational symmetry and with current limitations in our computational resources. Additionally, such a model would have too many variables to easily assess any conclusions. In this study the tumours introduced were shaped to present a similar situation as in the tumour strands extending from the core. We have, as a first step, developed a model with rotational symmetry and only two variables, thickness and length, which can demonstrate possible preferential heating of the tumour. Potential effects of temperature dependent thermal conductivity are not incorporated in this model. This phenomenon is usually not accounted for because of its low temperature coefficient, approximately 0.1%/°C for muscle tissue , and no breast specific data are available. The change in tissue electrical conductivity during heating has reversible and permanent effects . The permanent temperature effect increases with time and appears to be the result of structural changes of the tissue. Our model only accounts for the reversible part of the temperature dependence. However, the permanent temperature effect is less significant compared to the reversible effect. Furthermore, the 2%/°C coefficient used in this study is an over estimation of the measured reversible part, giving a fair approximation of the total change during our time interval of heating.
Regarding the FEM simulations, a strict error analysis is not considered possible for this multi-field problem. An indication of accuracy in the formulation is the conservationof energy introduced by the source. This was checked and found highly reliable. This finding verified that the element approximation is able, in the limit, to reproduce the governing mathematical formulation. With respect to grading of the element mesh, a Cauchy convergence test was used to assess if the mesh had an appropriate size. The accuracy level in the two aspects of the simulations is thereby considerably higher than for the tissue material parameters used. The computer and experimental studies were performed in an in vitro situation, disregarding possible effects of blood perfusion. Thus, the lesion size is overestimated compared to the in vivo case. Furthermore, the preferential heating of the tumour is probably decreased due to the temperature dependent blood flow.
Comparison between experimental results and computer model
We have shown that RFA of breast cancer in vitro results in preferential heating of the tumour during both the experimental and the computer simulation studies. However, during computer simulation of thin tumours, the preferential heating at the thermal lesion margin vanishes due to increased heat conduction from the tumour. Thus, the preferential heating of the tumour observed in the experimental studies was more pronounced, especially in the long outgrowths extending from the core tumour, indicating that additional effects other than just differences in electrical and thermal parameters must be involved. During cancer growth, fibrous septa membranes are produced by the tumour and existing membranes are pushed in front of the tumour, creating numerous thin membrane layers at the tumour interface. These septa layers between the cancer tissue and the fatty tissue could have an additional electrical or thermal insulating effect. Thermal insulating membranes would probably increase the temperature in the tumour in distant regions where they dominate.
This work was supported by VINNOVA, Sweden
- Solin LJ, Fourquet A, Vicini FA, Taylor M, Olivotto IA, Haffty B, Strom EA, Pierce LJ, Marks LB, Bartelink H, McNeese MD, Jhingran A, Wai E, Bijker N, Campana F, Hwang WT: Long-term outcome after breast-conservation treatment with radiation for mammographically detected ductal carcinoma in situ of the breast. Cancer 2005, 103: 1137–4116. 10.1002/cncr.20886View ArticleGoogle Scholar
- Kricker A, Armstrong B: Surgery and outcomes of ductal carcinoma in situ of the breast: a population-based study in Australia. Eur J Cancer 2004, 40: 2396–402. 10.1016/j.ejca.2004.07.008View ArticleGoogle Scholar
- Bradley CJ, Given C, Baser O, Gardiner J: Influence of surgical and treatment choices on the cost of breast cancer care. Eur J Health Econ 2003, 4: 96–101. 10.1007/s10198-002-0150-5View ArticleGoogle Scholar
- Given C, Bradley C, Luca A, Given B, Osuch JR: Observation interval for evaluating the costs of surgical interventions for older women with a new diagnosis of breast cancer. Med Care 2001, 39: 1143–5. 10.1097/00005650-200111000-00002View ArticleGoogle Scholar
- Singletary S: Minimally invasive techniques in breast cancer treatment. Semin Surg Oncol 2001, 20: 246–250. 10.1002/ssu.1040View ArticleGoogle Scholar
- Hall-Craggs MA, Vaidya JS: Minimally invasive therapy for the treatment of breast tumours. Eur J Radiol 2002, 42: 52–57. 10.1016/S0720-048X(02)00019-0View ArticleGoogle Scholar
- Gazelle GS, Goldberg SN, Solbiati L, Livraghi T: Tumor ablation with radio-frequency energy. Radiology 2000, 217: 633–646.View ArticleGoogle Scholar
- Noguchi M: Minimally invasive surgery for small breast cancer. J Surg Oncol 2003, 84: 94–101. 10.1002/jso.10292View ArticleGoogle Scholar
- Singletary ES: Feasibility of radiofrequency ablation for primary breast cancer. Breast Cancer 2003, 10: 4–9.View ArticleGoogle Scholar
- Ekstrand V: Development and Clinical Evaluation of Therapeutic Techniques based on Acoustic and Electromagnetic Energy. In Licentiate thesis. Karolinska Institute, Department of surgical sciences; 2002.Google Scholar
- Izzo F, Thomas R, Delrio P, Rinaldo M, Vallone P, DeChiara A, Botti G, D'Aiuto G, Cortino P, Curley SA: Radiofrequency ablation in patients with primary breast carcinoma. Cancer 2001, 92: 2036–2044. 10.1002/1097-0142(20011015)92:8<2036::AID-CNCR1542>3.0.CO;2-WView ArticleGoogle Scholar
- Livraghi T, Goldberg N, Monti F, Bizzini A, Lazzaroni S, Meloni F, Pellicano S, Solbiati L, Gazelle S: Saline-enhanced Radio Frequency Tissue Ablation in the Treatment of Liver Metastases. Radiology 1997, 202: 205–210.View ArticleGoogle Scholar
- Goldberg S, Gazelle S, Solbiati L, Rittman W, Mueller P: Radiofrequency Tissue Ablation: Increased Lesion Diameter with a Perfusion Electrode. Acad Radiol 1996, 3: 636–644.View ArticleGoogle Scholar
- Burak WE Jr, Agnese DM, Povoski SP, Yanssens TL, Bloom KJ, Wakely PE, Spigos DG: Radiofrequency ablation of invasive breast carcinoma followed by delayed surgical excision. Cancer 2003, 98: 1369–76. 10.1002/cncr.11642View ArticleGoogle Scholar
- Izzo F, Thomas R, Delrio P, Rinaldo M, Vallone P, DeChiara A, Botti G, D'Aiuto G, Cortino P, Curley SA: Radiofrequency ablation in patients with primary breast carcinoma: a pilot study in 26 patients. Cancer 2001, 92: 2036–44. 10.1002/1097-0142(20011015)92:8<2036::AID-CNCR1542>3.0.CO;2-WView ArticleGoogle Scholar
- Hayashi AH, Silver SF, van der Westhuizen NG, Donald JC, Parker C, Fraser S, Ross AC, Olivotto IA: Treatment of invasive breast carcinoma with ultrasound-guided radiofrequency ablation. Am J Surg 2003, 185: 429–35. 10.1016/S0002-9610(03)00061-8View ArticleGoogle Scholar
- Fornage BD, Sneige N, Ross MI, Mirza AN, Kuerer HM, Edeiken BS, Ames FC, Newman LA, Babiera GV, Singletary SE: Small (< or = 2-cm) breast cancer treated with US-guided radiofrequency ablation: feasibility study. Radiology 2004, 231: 215–24.View ArticleGoogle Scholar
- Chang I: Finite Element Analysis of Hepatic Radiofrequency Ablation Probes using Temperature-Dependent Electrical Conductivity. Biomed Eng Online 2003, 2: 12. 10.1186/1475-925X-2-12View ArticleGoogle Scholar
- Haemmerich D, Tungjitkusolmun S, Staelin ST, Lee FT Jr, Mahvi DM, Webster JG: Finite-element analysis of hepatic multiple probe radio-frequency ablation. IEEE Trans Biomed Eng 2002, 49: 836–42. 10.1109/TBME.2002.800790View ArticleGoogle Scholar
- Tungjitkusolmun S, Staelin ST, Haemmerich D, Tsai JZ, Webster JG, Lee FT Jr, Mahvi DM, Vorperian VR: Three-Dimensional finite-element analyses for radio-frequency hepatic tumor ablation. IEEE Trans Biomed Eng 2002, 49: 3–9. 10.1109/10.972834View ArticleGoogle Scholar
- Haemmerich D, Chachati L, Wright AS, Mahvi DM, Lee FT Jr, Webster JG: Hepatic radiofrequency ablation with internally cooled probes: effect of coolant temperature on lesion size. IEEE Trans Biomed Eng 2003, 50(4):493–500. IEEE Trans Biomed Eng 2003, 50:493–500. 10.1109/TBME.2003.809488View ArticleGoogle Scholar
- Tungjitkusolmun S, Woo EJ, Cao H, Tsai JZ, Vorperian VR, Webster JG: Thermal – electrical finite element modelling for radio frequency cardiac ablation: effects of changes in myocardial properties. Med Biol Eng Comput 2000, 38: 562–8.View ArticleGoogle Scholar
- Osman MM, Afify EM: Thermal modeling of the malignant woman's breast. J Biomech Eng 1988, 110: 269–76.View ArticleGoogle Scholar
- Sudharsan NM, Ng EY, Teh SL: Surface Temperature Distribution of a Breast With and Without Tumour. Comput Methods Biomech Biomed Engin 1999, 2: 187–199.View ArticleGoogle Scholar
- Gautherie M, Qenneville Y, Gros CH: Thermogenesis of mammary epitheliomas. III. Study, by means of fluvography, of the thermal conductivity of mammary tissue and of the influence of tumor vascularization. Biomedicine 1975, 22: 237–245.Google Scholar
- Erdmann B, Lang J, Seebass M: Optimization of temperature distributions for regional hyperthermia based on a nonlinear heat transfer model. Ann N Y Acad Sci 1998, 858: 36–46.View ArticleGoogle Scholar
- Werner J, Buse M: Temperature profiles with respect to inhomogeneity and geometry of the human body. J Appl Physiol 1988, 65: 110–1118.Google Scholar
- Gautherie M: Thermopathology of breast cancer: measurement and analysis of in vivo temperature and blood flow. Ann N Y Acad Sci 1980, 335: 383–415.View ArticleGoogle Scholar
- Jossinet J: Variability of impedivity in normal and pathological breast tissue. Med Biol Eng Comput 1996, 34: 346–350.View ArticleGoogle Scholar
- Grimnes S, Martinsen OG: Bioimpedance & Bioelectricity. Academic press; 2000.Google Scholar
- Foster KR, Schwan HP: Dielectric properties of tissues and biological materials: A critical review. Crit Rev Biomed Eng 1989, 17: 25–104.Google Scholar
- Cook RD, Malkus DS, Plesha ME, Witt RJ: Concepts and Applications of Finite Element Analysis. 4th edition. Wiley; 2002.Google Scholar
- Zienkiewicz OC, Taylor RL: The Finite Element Method. In The basis. Volume 1. 5th edition. Butterworth-Heinemann; 2000.Google Scholar
- Graham SJ, Chen L, Leitch M, Peters RD, Bronskill MJ, Foster FS, Henkelman RM, Plewes DB: Quantifying tissue damage due to focused ultrasound heating observed by MRI. Magn Reson Med 1999, 41: 321–328. 10.1002/(SICI)1522-2594(199902)41:2<321::AID-MRM16>3.0.CO;2-9View ArticleGoogle Scholar
- Haemmerich D, Mahvi DM, Lee FT jr, Webster JG: RF ablation at audio frequencies preferentially targets tumor – A Finite Element Study. Proceedings EMBS-BMES Houston 2002.Google Scholar
- Tungjitkusolmun S, Cao H, Tsai JZ, Webster JG: Using ANSYS for three-dimensional electrical-thermal models for radio-frequency catheter ablation. Proceedings – 19th International Conference IEEE/EMBS: 30 October – 2 November 1997; Chicago 1997, 161–164.Google Scholar
- Pop M, Molckovsky A, Chin L, Kolios MC, Jewett MA, Sherar MD: Changes in dielectric properties at 460 kHz of kidney and fat during heating: importance for radio-frequency thermal therapy. Phys Med Biol 2003, 48: 2509–25. 10.1088/0031-9155/48/15/317View ArticleGoogle Scholar
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