Heat transfer analysis of hyperthermia treatment of the prostate D. Loyd," M. Karlsson,* B.-J. Erlandsson,' J.-G. Sjodin/ P. Ask*> "Departments of Mechanical Engineering and ^Biomedical Engineering, Linkoping University, S-581 83 Linkoping, Sweden ^Departments ofbiomedical Engineering and ^Urology, University Hospital of Northern Sweden, S-901 85 Umea, Sweden 1. INTRODUCTION This paper presents a method for analysis of heat transfer in patients undergoing hyperthermia treatment of the prostate. The application shown here is a simulation performed in order to evaluate a thermal injury in a patient that occured during a hyperthermia treatment session. Prostatic enlargement occurs in males over 50 years of age. The enlargement of the prostate leads to obstruction of the urinary flow and in worst case urinary retention. The standard method for surgial treatment of prostatic enlargement (benign prostatic hyperplasia, BPH) is transurethral resection, TUR-P. During the 1980's several alternative treatment modalities were introduced. The transurethral microwave thermotherapy method, TUMT, is one of these methods which is commonly used today; Rigatti et al [11]. TUMT means heating of the prostatic tissue by microwaves; Thuery [12]. The microwaves are transmitted from a water cooled antenna inserted in a urethral catheter. In clinical studies, TUMT is today a method demonstrating improvements both subjectively and objectively in patients with BPH; Baert et al [l]and [2]. 2. THERMAL ANALYSIS The heat generation in thetissuecaused by the microwaves depends on time and space. The mucosa in the prostatic urethra is protected from injurious heat by the cooling of the catheter. The catheter has an outer diameter of 6-8 mm. The domain of interest, which includes the prostatic gland and surrounding tissue, has a size of about 40 mm in diameter. The power supplied to the microwave antenna is generally up to 50 W. Several parameters in addition to the geometry will influence the maximum temperature and the temperature distribution in the prostate during microwave
618 Computer Simulations in Biomedicine hyperthermia treatment. Examples on such parameters are the equivalent heat transfer coefficient along the catheter, the power from the antenna, the absorption function of the microwave energy, the size of the prostate, the thickness of the addipose layer and the blood flow. Some of these parameters can vary strongly between individuals and between different transurethral microwave therapy units. The domain of interest is irregular in shape and volume, the boundary conditions are complex and different tissues of the domain have varying thermal properties. The finite element method, FEM, is suitable for analysis of such complex heat transfer problems; see e g Loyd et al [8]. 3. EQUATION AND BOUNDARY CONDITIONS The temperature distribution of the prostate and the surrounding tissue, as a function of time and space, can be obtained by solving the heat-conduction equation with an appropriate method, e g the finite element method. The equation in a cylindrical co-ordinate system (r,<j>,z) takes the form (see e g Carslaw and Jaeger [3] or Eckert and Drake [4]) 8T i 3 /' 3T\ i ^,^ ~~ +- kz- +Q (1) where the temperature T = T(r,<j>,z,t), i e the temperature depends on the space co-ordinates as well as the time t. The axes of the co-ordinate system used here coincide with the principal axes of the thermal conductivities of the anisotropic material. The density p, the specific heat capacity c, and the thermal conductivity in direction i, ki, depend all on the co-ordinates and the temperature. The domain contains distributed energy sources generating heat at a rate Q = Q(r,<}>,z,t) per unit time per unit volume. The term is used for the power from the microwave antenna. This term can also be used for phase transition; Loyd and Froier [8]. In such cases the heat generated depends on the temperature as well as on the space co-ordinates and the time; Q = Q(r,(t>,z,t,T). If some parameter of the equation or of the boundary conditions depends on the temperature, the problem is non-linear. Blood perfusion of the tissue can be included in the analysis by a perfusion term in the equation or can be estimated by an adjustment of the thermal conductivity. The boundary conditions for the equation (1) are in this case prescribed temperature, prescribed heat flux and convective heat transfer. The radiation can be included in the latter boundary condition; Hoi man [6]. The prescribed temperature can vary along the boundary (index B) and with time TB = TB (TB, #, ZB, t) (2) The prescribed heatflux,qg, can also vary along the boundary and with time which gives the boundary condition
Computer Simulations in Biomedicine 619 -\rr, "\rr, l + kz U + qs = 0 where lp 1^ and 1^ are the appropriate direction cosines. The boundary condition for convective heat transfer is rtt rtt cjt k, The temperature of the fluid outside the analysed domain Too and the convective heat transfer coefficient h can vary along the boundary and with time. The heat transfer coefficient depends on geometry, flow situation and temperature. However, in forced convection problems, it is sometimes possible to assume the coefficient to be independent of the temperature; Holman [6]. 4. SOLUTION METHOD A variational method can be used for the time-dependent problem studied here; Zienkiewicz [14]. If the system at a particular instant is considered, the time derivative and the parameters can be treated as prescribed functions of the space co-ordinates. This is in principle the same situation as in a steady-state formulation. A standard finite element formulation is used in the analysis; Zienkiewicz [14]. The program used, THAFEM Thermal and Heat Analysis by Finite Element Method is a special-purpose program for heat transfer analysis; Loyd et al [7]. The program is intended for research and industrial applications but it is also frequently used for educational purposes. 5. APPLICATION OF THE METHOD - AN EXAMPLE The application of the method is exemplified by a simulation performed in order to evaluate a thermal injury in a patient undergoing TUMT-treatment because of symptomatic prostatic hyperplasia. Computer simulations were used in order to study the time dependant temperature distribution during the treatment. The treatment session was, according to all documents, performed without any abnormal findings or symptoms. The treatment took about 1 hour. Two hours post-treatment, when the patient had left the hospital, the patient felt pain in his penile shaft. He later developed an open wound in the peno-scrotal area. At cystoscopy, the urethra revealed a swollen area of 3 cm length on the left side of the urethral surface. No necrosis was found in the urethra. This wounded part of the urethra was situated in the area which corresponds to the skin wound on the surface. One of the investigations performed in order to find the cause of the injury was a computer simulation of the treatment session. The analysis was focused on the question if there was any possibility to achieve such a high temperature as
620 Computer Simulations in Biomedicine to induce a 3rd degree thermal injury 8 cm away from the transmitting centre of the microwave antenna properly placed in the prostatic urethra. The computational model shown in Figure 1 was used. The model is based on MR-pictures and X-ray investigations of the patient and on Gosling et al [5]. Axisymmetry is assumed and the model is extended in the radial direction in order to obtain well defined boundary conditions. The computational domain is divided into 4536 elements which represents 2373 nodes. The time step was 2 seconds for the first 100 seconds and then 10 seconds. (mm) H Figure 1: Computational domain. The prostate is assumed to have a density of 1050 kg/irp, specific heat 3720 J/kg K and heat conductivity 0.46 W/m K. The addipose tissue with fat has the density 815 kg/irp and specific heat 2300 J/kg K. The thermal conductivity of the addipose tissue is 0.19 W/m K; Martin et al [10]. The power from the microwave antenna is included in the heat generation term Q in Equation (1). The heat is calculated for each node from the power distribution shown in Figure 2. It can be seen that the main part of the heat generation takes place in a region close to the antenna; Loyd et al [9].
Computer Simulations in Biomedicine 621 Figure 2: Power distribution around the microwave antenna. The boundary condition between skin and air on the lower surface was convection and radiation with an equivalent heat transfer coefficient of 5 W/m^K and an air temperature of 25 C. The boundary condition to the surrounding tissue to therightwas prescribed temperature; normal body temperature 37 C. The upper boundary, the bladder, was insulated. The boundary condition along the catheter was convection with an equivalent heat transfer coefficient of 100 W/m2K and a fluid temperature of 5 C; Holman [6] and Wong [13]. The blood perfusion can be included in the analysis by an adjustment of the thermal conductivity. In the calcuation shown here it has been neglected. 6. RESULTS The temperature field and the heat flux were calculated as a function of time for different locations of the transmitting centre of the antenna and different temperatures of the cooling water. The influence from a reduced catheter cooling was studied as well as the influence from different power transmitted from the antenna. Figure 3 shows the temperature field 15 minutes after the start of the treatment session. In Figure 3 the location of the transmitting centre of the antenna is 3 cm from the centre of the prostate.
622 Computer Simulations in Biomedicine Center of microwave antenna Air temperature 250C Figure 3: Temperature field 15 minutes from the beginning of the treatment session. Incorrect position of the microwave antenna. According to the calculations it was impossible to achieve such a high temperature to cause a 3rd degree thermal injury in the penile shaft with the antenna properly placed in the prostatic urethra. 7. DISCUSSION OF THE APPLICATION The application of the method shown here is the analysis of a treatment session in a patient with prostatic hyperplasia where a thermal injury had occure at a distance of 8 cm from the centre of the prostate. The question to be answered was: could an antenna properly placed in the prostate cause a thermal injury in the peno-scrotal area? The MR and X-ray pictures of the injured patient gave the geometry for the model. Different situations were implemented; the transmitting centre of the antenna properly placed within the prostatic urethra and the transmitting centre placed at different distances from the centre of the prostate. With the assumptions of the thermal properties of different tissues made in this paper, the fistant injury could not be explained if the antenna was properly placed in the prostate. It was also shown that an antenna dislocation will not necessarily be reveald by pain if the cooling system of the catheter was intact. In conclusion, the antenna must have been dislocated during the treatment session in order to explain the location of the thermal injury. In this case, the injury was an injury of the blood vessels supplying a specific skin area, leading to a 3rd degree thermal injury and necrosis of that area some hours posttreatment.
8. CONCLUSION Computer Simulations in Biomedicine 623 The paper presents a method for analysis of heat transfer in patients undergoing hyperthermia treatment of the prostate. The heat transfer problem is complex and require a finite element analysis. The domain to be analysed i e the prostate and the surrounding tissue is irregular in shape with complex boundary conditions. The bloodvessels have temperature-depending flow and the physical properties depend on the temperature. This type of treatment modalities will be futher enhanced as new methods will be introduced in medicine. Therefore, experimental work and computer simulations are of utmost importance to increase the knowledge. ACKNOWLEDGEMENTS We are very grateful to Mr Gunnar Andersson for valuable discussions and help with the computer simulations and to Ms Elisabeth Forslund for excellent secreterial assistance, both at Department of Mechanical Engineering, Linkoping University. REFERENCES 1. Baert, L. et al Tranurethral microwave hyperthermia for benign pro static hyperplasia: Preliminary clinical and pathological results, /. Urol., 1990, 144:1383-1387. 2. Baert, L. et al Treatment response with transurethral microwave hyperthermia in different forms of benign prostatic hyperplasia. A preliminary report, The Prostate, 1991, 18:315-320. 3. Carslaw, H.S. and Jaeger, J.C. Conduction of heat in Solids, 2 edn, Oxford University Press, U.K., 1959. 4. Eckert, E.R.G. and Drake, R.M. Analysis of Heat and Mass Transfer, McGraw-Hill, New York, 1972. 5. Gosling, J.A. et al Functional Anatomy of the Urinary Tract, Churchill Livingstone, London, 1983. 6. Holman, J.P. Heat Transfer, McGraw-Hill, Tokyo, 1989. 7. Loyd, D. et al THAFEM - a Finite Element Program for Heat Transfer Analysis, Finite Element Systems - A Handbook, ed C. A. Brebbia, 3 edn, pp. 721-732, Springer-Verlag, Berlin, 1985. 8. Loyd, D. and Froier, M. Heat Transfer Analysis of Structures containing Phase-Change Materials, Communications in Applied Numerical Methods, 1985, Vol. 4, 607-615.
624 Computer Simulations in Biomedicine 9. Loyd, D. et al Hyperthermia treatment of the prostate - a complex heat transfer problem, in Proc of Eighth International Conference on Numerical Methods in Thermal Problems, July 12-16, 1993 (ed R. W. Lewis), pp 1239-1250, Pineridge Press, Swansea, Wales, U. K., 1993. 10 Martin, G. et al Thermal model for the Local Microwave Hyperthermia Treatment of Benign Prostatic Hyperplasia, IEEE Trans, on Biomedical Engineering, 1992, Vol. 39, No. 8, pp. 836-844. 11. Rigatti et al Local deep microwave hyperthermia in the treatment of prostatic diseases, Arch. Urol, 1989, LXI: 179-181. 12. Thuery, J. Microwaves: Industrial, Scientific and Medical Applications, Ed E. H. Grant, Kings College London, Artech House, Boston, London, 1991. 13. Wong, H.Y. Heat Transfer for Engineers, Longman, London, 1977. 14. Zienkiewics, O.C. The Finite Element Method, McGraw-Hill, London, 1977.