PENNES bioheat equation, based on the heat diffusion

Transcription

1 1382 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 8, AUGUST 2007 Expanding the Bioheat Equation to Include Tissue Internal Water Evaporation During Heating Deshan Yang, Member, IEEE, Mark C. Converse*, Member, IEEE, David M. Mahvi, and John G. Webster, Life Fellow, IEEE Abstract We propose a new method to study high temperature tissue ablation using an expanded bioheat diffusion equation. An extra term added to the bioheat equation is combined with the specific heat into an effective (temperature dependent) specific heat. It replaces the normal specific heat term in the modified bioheat equation, which can then be used at temperatures where water evaporation is expected to occur. This new equation is used to numerically simulate the microwave ablation of bovine liver and is compared to experimental ex vivo results. Index Terms Ablation, bioheat equation, liver, microwave, specific heat. I. INTRODUCTION PENNES bioheat equation, based on the heat diffusion equation, is a much used approximation for heat transfer in biological tissue [1] [3]. Many publications have shown it is a valuable approximation [4], [5], especially at hyperthermia temperatures. However, at the higher temperatures seen during ablation, the Pennes bioheat equation does not incorporate all the physical processes affecting final tissue temperature. These processes include but are not limited to the effects of the movement and diffusion of tissue water due to temperature and changes in local water content due to heating, water evaporation at high temperatures, the diffusion of this generated water vapor, and its possible recondensation. To add to the complexity, the thermal and other physical properties are a function of temperature, water content and the changes in the mechanical stresses on the tissue. Attempting to model this complex physical system, which involves electromagnetic (EM), thermal and mass transfer modeling, mechanical stresses, etc, is challenging due to the interdependent nature of the physical properties. For example, in microwave ablation, temperature is partially dependent on the local power deposition, which is dependent on local microwave properties, which are a function of local tissue temperature, water content and physical structure of the tissue. In addition there appears to be a lack of data on all required physical parameters (i.e., dielectric properties of partially ablated tissue, diffusion rate Manuscript received November 15, 2005; revised November 8, This work was supported in part by the National Institutes of Health (NIH) under Grant DK Asterisk indicates corresponding author. D. Yang is with the Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI USA. *M. C. Converse is with the Department of Surgery, University of Wisconsin, 2151 Engineering Centers Building, 1550 Engineering Dr, Madison, WI USA. ( converse@cae.wisc.edu). D. M. Mahvi is with the Department of Surgery, University of Wisconsin, Madison, WI USA. J. G. Webster is with the Department of Biomedical Engineering, University of Wisconsin, Madison, WI USA. Digital Object Identifier /TBME of water in partially ablated tissue, a clear understanding of water evaporation and subsequent vapor diffusion within tissue at high temperatures, which would yield water vapor diffusion rates). When faced with such a complex modeling problem, one approach is to attempt to incrementally improve the model as increased understanding of the physical system and materials properties information becomes available. In this paper we attempt to improve the thermal model by incorporating simple water-related processes. These include tissue water evaporation, diffusion, water vapor diffusion and condensation. We feel these are very significant processes which, in fact, may become the dominant heat transfer processes in the system when tissue temperature approaches 100. Without considering these processes, results from the bioheat equation may significantly differ from experimental results at high temperatures. The study of tissue heating processes at high temperatures is relevant to therapeutic applications (such as RF, microwave, and laser ablation and hyperthermia) and food processing applications (such as baking and frying). Research on water evaporation related processes is sparse. Coupled heat transfer and liquid water transfer in porous material have been studied [6], [7]. Some work has been done for heat and mass transfer under laser radiation by considering tissue water evaporation on the heating surface [8] [11]. Other studies include food processing, such as baking and frying, using external heating sources [12], [13]. Most studies of ablative procedures do not consider tissue water related processes, or have mentioned tissue water effects but do not include them in models, or only consider surface evaporation [14] [16]. As not much study has been done, tissue water related processes are largely unquantified, and it is difficult to measure the physical effects or water movement and state changes. Our expectation during heating to high temperatures is that tissue loses water through an evaporation process. Generated water vapor increases the gas pressure within the area of water evaporation. This water vapor diffuses to lower pressure areas in which tissue temperature is also lower. In this lower temperature region, water vapor condenses to water liquid and releases its latent heat. Released latent heat energy heats the surrounding tissue and increases tissue temperature. Tissue in this region also gains water content during the condensation process. The entire process of water evaporation, water vapor diffusion and condensation is a process of water movement and energy movement and is as significant as direct thermal conduction. None of the procedures can be easily studied quantitatively. Evaporation or condensation could be analyzed alone with partial pressure rules if the phenomenon occurred at a free interface of water liquid and air [17]. Under such a condition, the maximal /$ IEEE

2 YANG et al.: EXPANDING THE BIOHEAT EQUATION TO INCLUDE TISSUE INTERNAL WATER EVAPORATION DURING HEATING 1383 amount of water vapor allowed in the air is equal to the water vapor saturated partial pressure at the current temperature and current air pressure. This rule does not apply directly to the situation of heating inside the tissue because the air pressure, water vapor diffusion rate and water liquid diffusion rate are largely unknown. In this paper we introduce a method to incorporate simple water-related processes into existing thermal models to improve ablation models at high temperatures. We first map tissue temperature to changes of tissue water content caused by heating and evaporation. With such a mapping from temperature to water content, we define a new term tissue effective specific heat and use this term in place of the normal tissue specific heat in the bioheat equation. This modified bioheat equation can be solved in the same way as the normal bioheat equation. The rest of this paper proceeds as follows. A new term is added to the bioheat equation and a new effective specific heat term is presented in Section III along with a description of the numeric simulation and experiment. Section IV presents the results of the comparison between simulation and experiment and is followed by the conclusion. The power density used for evaporation is related to the change in water content of tissue as a function of time where is the water latent heat constant, which is 2260 [kj/kg] and is the tissue water density [ ] which is assumed to be only a function of temperature. From the chain rule the derivative of with respect to time is Substituting this into (3), yields The modified bioheat equation then becomes (4) (5) (6) II. METHODS (7) A. Theoretical Solution of Tissue Water Evaporation With the Bioheat Equation Below is the Pennes bioheat diffusion equation where is density [ ], is specific heat [ ], is temperature [ ], is thermal conductivity [ ], is the microwave power density [ ], is a term which accounts for the effects of perfusion (see (2) [ ], and is the metabolic heat generation term [ ] which is considered insignificant with respect to the heating term and will be ignored for the purposes of this study where is the blood mass density ( ), is the blood specific heat [ ], is the blood perfusion rate [1/s], and is the ambient blood temperature [ ] before entering the ablation region. Note, all variables but are spatially dependent. For purposes of clarity the spatial dependence is left out of the equations and is to be implied. Evaporation requires energy, specifically termed the latent heat. To account for the energy needed to vaporize water we add a term to the bioheat equation, [ ], yielding a modified bioheat equation Note, here we have dropped the metabolic heat generation term. (1) (2) (3) Pulling the last term in the above equation to the left-handside Examining the above equation we can define an effective specific heat which yields a new modified Pennes bioheat equation (8) (9) (10) Equation (10) is in the same format as the original bioheat (1), with an effective specific heat used instead of the normal specific heat. Since is 0 when evaporation does not occur and is negative when evaporation occurs, effective specific heat is never less than normal specific heat value which is consistent with it requiring more energy to raise the temperature during a phase change. Tissue effective specific heat [ ] is the only new term in (10). It is similar to the normal specific heat as it is defined as the amount of energy required to increase the temperature of a unit mass of tissue by 1, and includes the water latent heat energy required if tissue water evaporation occurs. For this formulation we have assumed that the change in tissue water content and tissue effective specific heat are only dependent on tissue temperature. In actuality it is more complex than this. However as we stated earlier, we are implementing a model which, while still not complete, is more accurate than the existing thermal model. We will discuss the ramifications of this simplification in the results and discussion sections.

3 1384 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 8, AUGUST 2007 TABLE I PARAMETERS USED IN THE MODEL Fig. 1. Axially symmetrical model geometry. A coaxial slot antenna is inserted through a plastic positioning template along the z-axis, into the liver tissue which is on a plastic cutting board. The radius of the coaxial antenna is 1.25 mm. It is inserted 20 mm deep into liver tissue. B. Numeric Simulation To examine the effectiveness of this new specific heat we attempt to model microwave ablation of ex vivo bovine liver. Fig. 1 shows the model geometry based upon the experimental setup (discussed subsequently). We used an axially symmetric model [15], which minimized computation time and allowed improved resolution while yielding a full 3-D solution. Both EM and thermal solutions were obtained. The EM solution was solved once and used as the heat source of the thermal model. While we expect the dielectric properties (and, therefore, the heating pattern) to change with temperature as well as loss of tissue water [18], for ease of computation, we assume that the initial heating pattern remains unchanged during the course of ablation. Normal thermal properties (i.e.,,, ) are assumed to be temperature and water content independent. Also, since we are simulating an ex vivo case, blood perfusion is not included in the computer model. Boundary conditions are set to be convective boundary conditions with the convective heat transfer coefficient set to 12 and the ambient air temperature set to 25. Microwave and thermal properties for the various tissues and materials are listed in Table I. To solve the modified bioheat equation we need the functional form of the temperature effective specific heat and, therefore, the temperature dependence of the water content. Based upon experiments that measured water content as a function of temperature, we have developed (11), shown at the bottom of the page, which defines the water content and thereby the effective specific heat as a function of temperature [19]. The equation and its derivative are plotted in Fig. 2(a) and (b). The dynamics of water vapor movement and the issue of condensation are not well understood at this point. So to incorporate (11)

4 YANG et al.: EXPANDING THE BIOHEAT EQUATION TO INCLUDE TISSUE INTERNAL WATER EVAPORATION DURING HEATING 1385 Fig. 3. Approximation of water vapor diffusion and condensation. Fig. 2. (a) Mapping from tissue temperature T to W [kg=m ], the mass of tissue water per unit volume of tissue. (b) Derivative of mapping function from tissue temperature T to W. Fig. 4. Schematic of solution procedure. Thermal simulation is performed in minor time steps. Time step 1t is adjustable during the simulation, from 0.2 to 1 s. It is controlled so that maximal tissue temperature change is less than 10 C and maximal tissue water content change is less than 1% of the liver tissue volume. the transfer of energy that occurs when water vapor is generated then condenses in a new location, we have developed a simple mechanism to account for condensation effects. In this method, we calculate the amount of tissue water vaporized in that time step after each time step. We assume that no water vapor escapes from the system and all water vapor diffuses to a tissue region of lower temperature and condenses uniformly in the tissue region of temperature from 60 to 80, shown in Fig. 3. During condensation, water vapor releases latent heat energy and heats the surrounding tissue where condensation occurs. We ignored the effects of thermal transfer between tissue and water vapor, assuming it was insignificant compared to the latent heat energy returned to the tissue during condensation. The amount of tissue water in the surrounding tissue increases after water vapor condenses back to water liquid. We assume that the amount of tissue water in the region at temperature from 60 to 80 does not change as a function of condensation. To simulate the model we used the multiphysics simulation tool FEMLAB in conjunction with MATLAB. The functional form of the effective specific heat term is too complex for FEMLAB to incorporate into its calculations, so a call to a MATLAB function is used in its place. Therefore, the solution proceeds as follows: is used to calculate and the effective specific heat. The bioheat equation is then solved in the solution region for. The entire computer simulation is illustrated in Fig. 4. For this model it took about 500 steps and 3.75 h to simulate a heating duration of 180 s. The average time step was about 0.4 s. C. Experiment Setup and Procedures We performed ex vivo experiments on bovine liver to validate the computer simulation results. Whole cow livers were obtained from a local slaughter house and were kept refrigerated

5 1386 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 8, AUGUST 2007 Fig. 5. Ex vivo experiment setup. The beef liver sample was approximately cm. Through a plastic template, the antenna was inserted from the top to a position where the antenna slot was 2 cm deep. Four thermal sensors were all inserted 2 cm deep. Fig. 7. Tissue temperature at four thermal sensor positions. Curves without markers are measurement results from one of the ex vivo experiments. Curves with markers are corresponding results from computer simulation. Sensor positions were measured at 2.5, 4.5, 7, and 9.5 mm away from the antenna, respectively. All sensors were longitudinally aligned with the antenna slot. Fig. 8. (a) A lesion created in an ex vivo experiment at the end of a 180 s MWA procedure. Lesion size is about 3.9 cm22.7 cm. (b) Temperature contours from computer simulation at 180 s. Width of 2.7 cm roughly corresponds to 60 C boundary. two halves along the trace of the antenna to expose the lesion. The lesion was scanned with a HP scanner at a resolution of 300 DPI to measure lesion size and shape. Fig. 6. Plot of antenna heating pattern in decibel scale. It is normalized by the maximal value. over night. Initial tissue temperature was approximately 8. We selected the most uniform liver tissue with the fewest blood vessels for experiments. Liver tissue samples were a minimum of cm. Fig. 5 shows that the coaxial slot antenna was inserted straight down from the top, together with four optical fiber thermal sensors feeding a Luxtron 1000 fluoroptic thermometer. We connected the antenna to a CoberMuegge MG0300D 300 W, 2.45-GHz microwave generator through a 1-m-long flexible coaxial cable. The Luxtron thermometer was connected to a desktop PC via its serial port for data collection. Temperatures were measured four times per second for the entire duration of microwave heating. Data were stored on the PC for further processing. Liver tissue was heated using 75 W of power for 180 s. Antenna and thermal sensors were withdrawn after the procedure. Actual positions of thermal sensors to the antenna slot were then measured after ablation. Tissue was cut open longitudinally into III. RESULTS Fig. 6 shows the antenna heating pattern plot for this configuration. Fig. 7 compares computer simulation results of tissue temperature versus time, to ex vivo experimental measurement results. The results of the simulation compare reasonable well to the experimental results with similar trends in temperature profiles over the same approximate time range. A certain amount of mismatch between experiment and simulation is expected as the temperature dependence of the normal thermal properties is not considered nor is the change to the specific absorption rate (SAR) pattern from changes in the complex permittivity due to temperature and tissue water changes. Fig. 8 compares the lesion shape and size from experiments to temperature distribution from the computer simulation. The lesion in the experiment roughly conforms to the 60 contour line in the simulation. For the scanned lesion image from in vivo experiments, the lesion boundary can be easily identified as a clear dark ring, which is caused by increased blood perfusion at the boundaries. Lesion size is often associated with the 55 temperature contour of computer simulation. However

6 YANG et al.: EXPANDING THE BIOHEAT EQUATION TO INCLUDE TISSUE INTERNAL WATER EVAPORATION DURING HEATING 1387 Fig. 9. Comparison among the measurement results and two different computer simulation results. Simulation #1 uses the original Bioheat equation, which does not consider tissue water evaporation and condensation. Simulation #2 uses the modified heat equation to consider tissue water evaporation and includes condensation. The comparison is done for thermal sensors which were 4.5 mm and 9.5 mm away from the antenna. in the ex vivo case, it is usually very difficult to identify the lesion boundary because of the smooth tissue color changes. The relationship from the ex vivo lesion color change to tissue temperature is not well known so lesion color cannot be translated to tissue temperature directly. Fig. 9 compares results from two different computer simulations, to results from experiments. Temperature results from only two sensor positions are used for clarity. Simulation #1 uses the original bioheat equation, which does not consider tissue water evaporation or condensation. Evaporation and condensation are considered in simulation #2 with the modified bioheat equation. The results show that using the original bioheat equation (simulation #1), the temperature continues to grow, at 4.5 mm, even though experimentally and for simulation #2 it levels off. Also, at 9.5 mm the modified bioheat equation seems to match the experimental results better. This seems to indicate an improvement in accuracy when incorporating the effect of water evaporation and condensation in the form of effective specific heat over the original bioheat equation. However, it is also possible that one of the temperature effects such as changing normal thermal properties or changing SAR pattern would also improve the simulation to more closely match experimental results if they were added in place of the evaporation effects. This question is currently under investigation as we incorporate these effects into our model. IV. DISCUSSION As we discussed in the introduction, a full simulation of ablation is extremely complex and not the goal of our paper. We have attempted to create a more accurate if not complete thermal model incorporating the effects of changing water content and evaporation. As this is not a complete model we discuss some of the limitations below. First of all, the antenna radiation pattern is assumed to be constant. Many of the tissue physical properties, including dielectric properties and thermal properties, are dependent on tissue temperature and tissue water composition. This means power deposition patterns will change during heating. Dielectric loss tends to increase with temperature while the water content tends to decrease. We expect this decrease of water content to lead to less attenuation of the wave propagating through the tissue. While these two opposing effects will not exactly cancel out, we expect it to reduce the level of simulation error due to the assumption of constant radiation pattern. However, this is something which should be investigated in more detail. We have also assumed that no water vapor escapes from the system. This means total water is conserved and all energy lost to evaporation is redeposited elsewhere in the tissue during condensation. This seems reasonable for percutaneous in vivo ablation where there is not any place for the water vapor to go. Although in the ex vivo case or during tissue surface laser ablation, this is less likely. As water vapor movement is largely unpredictable due to heterogeneous tissue mechanical structures and nonuniform mechanical stress on the tissue matrix we have utilized a simple method of accounting for water vapor dispersion and condensation. For ex vivo cases or tissue surface ablations, our simple method could slightly overestimate the temperature in the tissue, as the energy in the water vapor lost from the system would remain in the simulation and lead to a slight overestimation of temperature. The magnitude of this overestimation is dependent on the amount of loss of water vapor. We have also ignored tissue water diffusion by which tissue water diffuses from a low temperature wet region to a high temperature evaporation region. These could lead to relatively large errors especially for situations of slow tissue heating at lower powers. According to our preliminary computation modeling with normal tissue water diffusion coefficients, the effects of tissue water diffusion is minimal for high power and short duration tissue heating such as microwave ablation or laser ablation. Our assumption that tissue water content only depends on tissue temperature is also not exactly accurate because tissue water content depends on time, temperature and other factors. We expect this assumption is more accurate for high power thermal ablation than for slow thermal treatment. However, further research needs to be done in order to understand the exact dependencies. Finally, our water content vs. temperature function is based upon experimental measurements of ex vivo bovine liver. Thus extrapolation for other tissue types will only be approximate. While we have not quantitatively analyzed errors due to these limitations, we believe this modified bioheat equation accounting for energy lost during tissue water evaporation will be more accurate at the high temperatures seen during ablation than the original Pennes bioheat equation. Improvements on this model could incorporate other physical processes, such as tissue water condensation, diffusion, water vapor movement, etc. We have begun work designing such comprehensive computer models which will include tissue physical property dependencies on tissue water content, and model the additional physical processes. V. CONCLUSION We have presented a new modified bioheat equation, which incorporates the effect of water evaporation from the tissue in the form of an effective specific heat. We have performed experiments to validate a model of microwave ablation of bovine liver using this new bioheat equation. Comparing the simulation results to the experimental results, the new method has shown

7 1388 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 8, AUGUST 2007 promise in generating a more accurate prediction of temperature profiles. Future studies will focus more basic research to understand how water changes state and moves in tissue during heating. This will allow a better understanding of the evaporation processes in tissue during heating to high temperatures as well as yielding a more complete understanding of the other processes which may be important during tissue heating such as water vapor diffusion, and condensation. REFERENCES [1] J. Lienhard, A Heat Transfer Textbook. Lexington, MA: Phlogiston, [2] E. H. Wissler, Pennes 1948 paper revisited, J. Appl. Phys., vol. 85, pp , [3] H. Arkin, Recent developments in modeling heat transfer in blood perfused tissues, IEEE Trans. Biomed. Eng., vol. 41, no. 2, pp , Feb [4] M. C. Kolios, Experimental evaluation of two simple thermal models using transient temperature analysis, Phys. Med. Biol., vol. 43, pp , [5] J. 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Eng., vol. 54, no. 1, pp , Jan Deshan Yang (S 04 M 05) received the B.S.E.E. degree from Tsinghua University, Beijing, China, in June He received the M.S. degree in computer science from Illinois Institute of Technology, Chicago, IL, in December He received Ph.D. degree in electrical engineering in the University of Wisconsin-Madison in December His Ph.D. research involved microwave hepatic ablation, measurement of tissue physical responses under thermal treatment, computer simulation of electromagnetic fields, and comprehensive modeling and computer simulation of heat transfer and tissue water transfer in tissue during heating. He worked in software and telecommunication engineering positions for Nokia Telecommunication, Motorola Inc, and Lucent Technologies from 1995 to He is as a Postdoctoral Research Associate with the Department of Radiation Oncology, School of Medicine, Washington University, St. Louis, MO. His research interests are computation and computer modeling in biomedical and medical applications, and medical image processing. Mark C. Converse received the B.S. degree in electrical engineering from the University of Wisconsin- Madison, in He received the M.S. and Ph.D. degrees from the same university in 1999 and 2003, respectively. During his graduate studies he was engaged in plasma processing research involving damage evaluation/analysis and mitigation during the etching process. After obtaining the M.S. degree in 1999, he began research in microwave vacuum electronics, investigating the impulse response of the helix traveling wave tube. In May of 2003, he began postdoctoral research examining the feasibility of using UWB microwave hyperthermia to treat breast cancer. Currently, he is an Assistant Scientist with the University of Wisconsin investigating microwave ablation of liver cancer. His research interests include electromagnetic interactions with materials, electrical/biological interfaces, and organic electronics. David M. Mahvi attended the University of Oklahoma, Norman, and subsequently received the M.D. degree in 1981 from the Medical University of South Carolina, Charleston. He then completed the following postgraduate medical clinical training programs at Duke University, Durham, NC: residency in surgery from ; fellowship in immunology ; residency in surgery In 1989, he joined the Section of Surgical Oncology, Department of Surgery at the University of Wisconsin-Madison where he is now Professor of Surgery and Chief of the Section. He is also a member, University of Wisconsin Comprehensive Cancer Center. John G. Webster (M 59 SM 69 F 86 LF 97) received the B.E.E. degree from Cornell University, Ithaca, NY, in 1953, and the M.S.E.E. and Ph.D. degrees from the University of Rochester, Rochester, NY, in 1965 and 1967, respectively. He is Professor Emeritus of Biomedical Engineering at the University of Wisconsin-Madison. In the field of medical instrumentation he teaches undergraduate and graduate courses in bioinstrumentation and design. He does research on improving electrodes for ablating liver to cure cancer. He does research on safety of electromuscular incapacitating devices. He does research on a miniature hot flash recorder. He is the editor of the most used text in biomedical engineering: Medical Instrumentation: Application and Design, Third Edition (Wiley, 1998) and has developed 22 other books including the Encyclopedia of Medical Devices and Instrumentation, 2nd edition (Wiley, 2006) and 190 research papers. Dr. Webster is a fellow of the Instrument Society of America, the American Institute of Medical and Biological Engineering, and the Institute of Physics. He has been a member of the IEEE-EMBS Administrative Committee and the National Institutes of Health (NIH) Surgery and Bioengineering Study Section. He is the recipient of the 2001 IEEE-EMBS Career Achievement Award.