ChE141 Heat and Mass Transfer

ChE141 Heat and Mass TransferStudy and design of cylindrical and spherical thermoseeds for cancer treatment using Comsol Multiphysics

Dunie Navarro3/14/2013Prof. Daniel Lepek

IntroductionThermal ablation consists of heat treatments of cancerous tumors delivered at temperatures above 50C for shorts periods of time. Its result is to terminate entire tumors, killing malignant cells and avoiding its effect on healthy tissue. The principle of operation of thermal ablation is to concentrate thermal energy to create a hyperthermic injury by using a needle-like applicator [1]. Thermal ablation is mainly used when surgery is not recommended on a patient. Its application is effective, low cost and can provide relief or cure completely from cancer. The project at hand focuses on the application of thermal ablation using small spheres made of ferromagnetic material. A magnetic field induces a volumetric generation of thermal energy in the thermoseeds, increasing the seed temperature and its surroundings. First, a spherical thermoseed will be modeled to analyze the heat transfer behavior to the surrounding of the ferromagnetic material, cancerous tissue. Properties of the spherical thermoseed are given in Table 1. Table 1 Thermosphere PropertiesParametersValues

Thermoseed thermal conductivity (W/m-K)10

Tissue thermal conductivity (W/m-K)0.5

Sphere radius (mm)1

Body temperature (C)37

Total energy generated (W)1.0

To model heat transfer behavior of the spherical thermoseed consider Poissons Equation [2]: (1) Considering heat transfer in the radial direction in spherical coordinates, (2)The equation above can be solved analytically or numerically by the finite difference method of the finite element method with the boundary conditions (B.C 1) (B.C 2)Using Comsol Multiphysics software package, heat transfer equations to model heat transfer behavior are built in and readily solved using algorithms with standard numerical methods implemented. Second, a new design for a thermoseed needed to be analyzed for similar heat transfer behavior to the spherical thermoseed. A cylindrical thermoseed is the proposed design to treat cancerous tissue. From the given radius of the sphere, volumetric and surface area constants are calculated. To determine the length and radius of the cylinder, the surface area and volume formulas are set equal to the spherical volume and surface area constants. A system of two equations for r and L can be solved; giving appropriates values of the radius and length of the cylinder. The thermocylinder properties are given in Table 2. Table 2 Thermocylinder PropertiesParametersValues

Thermoseed thermal conductivity (W/m-K)10

Tissue thermal conductivity (W/m-K)0.5 (Humans), 0.32 (Rats)

Cylinder radius (mm)1.225

Body temperature (C)37

Total energy generated (W)1.0

Cylinder length (mm)1.633

To model heat transfer behavior of the cylindrical thermoseed consider Poissons Equation: (1) Considering heat transfer in the radial direction in cylindrical coordinates, (2)The equation above can be solved analytically or numerically by the finite difference method of the finite element method with the boundary conditions (B.C 1) (B.C 2)

The boundary conditions are identical for both geometries since both possess axisymmetric properties, given at the center of the body. Cylindrical systems are readily solved in Comsol Multiphysics using the required 2-D axisymmetric space.

MethodsDesigns of a thermosphere and thermocylinder for treating cancerous tumors due to heat generation were modeled using Comsol Multiphysics. The thermosphere concept had been implemented as the current geometrical design of the medical device. The thermocylinder geometry needed to be modeled as a possible competitor to the spherical thermoseed. The spherical thermoseed was modeled using 3D space model, Heat Transfer in Solids Physics and Stationary Study Type. The chosen Geometry for the thermoseed-tissue system consisted of two concentric spheres, inner sphere being the thermoseed and outer sphere being the tissue. Two Materials were created to assign thermal properties to the geometries; mainly the materials thermal conductivities. The Heat Transfer in Solids tab was modified to include a heat source in the inner sphere and a temperature boundary condition in the surface of the outer sphere. To compute the set study, an Extra Fine mesh was selected to give precise results depending on memory. In the Study tab the computation was started to compute the required temperatures in the system. Using the methods to plot graphs from the Results tab, all required quantities were plotted or computed accordingly. To compute the maximum temperature in the tissue, the Derived Values tab was used and the extent of the lesion was computed by trial and error using temperature measurements at a point. To model the thermocylinder, all of the above steps were followed excepted for Geometry, where two concentric cylinders were built and thermal conductivities assigned to each medium. Two studies were set, Humans and Rats, to keep study results independent from each other.

Results and DiscussionPart ITwo geometries of ferromagnetic thermoseeds were characterized using heat transfer methods in Comsol Multiphysics. In Part I, heat transfer in a spherical thermoseed was analyzed given certain requirements of its geometry. The surface temperature profile and specified geometry of the thermoseed inside tissue are presented in Figure 1. Fig 1 Geometry and Surface Temperature of Spherical Thermoseed

The temperature scale on the right illustrates the profile starting in the top being the core temperature and moving outwards to the set surface temperature. As heat is generated in the thermoseed, it reaches the interface of the tissue and the thermoseed lowering temperature and dropping towards the surface. Symmetry in the geometry guarantees ease to resolve the model equations in order to provide a resultant temperature profile. Uniform distribution of the energy in the thermoseed is preserved as heat transfer to the tissue given the possibility of heat traveling in all possible directions uniformly. From the given surface profile it is inferred that temperature drops smoothly, avoiding sudden drops in temperature to satisfy the boundary condition of body temperature at the surface of the tissue. To confirm the proposed hypothesis, temperature profiles in the thermoseed and tissue regions were plotted in Comsol from the study results. The temperature within the thermoseed is presented in Figure 2. Fig 2 Temperature Profile in the Thermoseed

Temperature in the thermoseed drops steadily and slow due to its thermal conductivity and heat generation within its material. Temperature drops from 457.5K to 453.4K (184.5-180.25 C) as given above illustrating the low resistance to heat conduction within the ferromagnetic material. The shape of the temperature drop is parabolic indicating a polynomial drop. The temperature profile in tissue is presented in Figure 3. Given that the thermal conductivity of tissue is twenty times lower than the thermoseed, its temperature drops exponentially up to the surface of the tissue. The profile is characteristic of materials with similar thermal conductivities and boundary conditions. From the plot, a maximum temperature and minimum temperature in the tissue region can be calculated and considered as limits to the temperature profile. The value of maximum temperature at the interface of the thermoseed and tissue is highly relevant given that tumors are to be treated at a specific temperature. Fig 3 Temperature Profile in Tissue The heat flux through the tissue is provided in Figure 4 providing a measure of the transfer of thermal energy in the tissue surrounding the thermoseed. For each temperature difference in the system, heat flows from high to low temperature and its conduction is resisted by the thermal conductivity of the material. Heat flux drops exponentially 85000 W/m2 to zero after reaching the surface of the tissue in the model. The total heat flux decreases as it flows from the thermoseed-tissue interface to the outer surface of the system. From the given heat flux, temperature differences at different radius can be determined since for each flux interval, a temperature difference exists. Interest in temperature differences is considered when an intermediate temperature in the material needs to be determined. An intermediate temperature of interest in the system is the lethal temperature for tissue, Tlethal, the temperature at which cancer tissue is efficiently treated. Along with this temperature, the maximum temperature, Tmax, in the tissue needs to be determined as an upper bound for Tlethal. Fig 4 Heat Flux within Thermoseed

The maximum temperature, Tmax, and the extent of lesion rlesion, radial location of at where the tissue temperature reaches the lethal temperature, can be computed as functions of the thermal energy generation (in W). The thermal heat generation can be allowed to change in order to calculate the maximum temperature and rlesion. Maximum temperature and extent of the lesion in the tissue region are plotted in Figure 5 as functions of the rate of thermal energy generation, (W). As heat generation increases for values in the range of 0.1-1.5 W, the temperature increases linearly proportional and the extent of lesion logarithmically. It is logical that for higher heat generation, the thermoseed-tissue interface has a higher temperature than for lower heat generation values. The same increase in heat generation causes the lethal temperature for cancer tissue to shift farther away from the thermoseed giving a greater range of treatment for tissue according to energy input. In the specific case of energy generation of 1 W, the maximum computed temperature is 453.3K while the extent of lesion is located at rlesion = 5.5 mm from the interface. Fig 5 Maximum Temperature and Extent of Lesion (Spherical Thermoseed)

Part IIThe second geometry considered as a medium for analysis of the heat transfer behavior of ferromagnetic materials to treat cancer tissue was cylindrical. A thermocylinder, a cylindrical thermoseed, was designed to behave similarly to the spherical thermoseed previously studied. Two studies of the proposed geometry were carried according to parameters (thermal conductivity) for two tissue mediums (humans and rats). The cancer tissue of medium of rats is considered to model heat transfer behavior since clinical trials are carried on rats before human trials. The quantities previously considered in the spherical thermoseed are now analyzed for a cylindrical thermoseed. A surface temperature and geometry of the thermocylinder design for humans and rats is given in Figure 6 and 7. Heat generation within the thermocylinder is produced in the center and heat flux travels to the surface through human and rat tissue. The temperature scale on the right provides ranges of values for the different colors being hotter in the center and colder towards the surface.

Fig 6 Thermocylinder Surface Temperature (Humans)

Fig 7 Thermocylinder Surface Temperature (Rats)

Given the variations in thermal conductivity, being lower for rats tissue, the temperature in the thermocylinder and the tissue surrounding it is higher for rats. A difference of 70K is present between these two tissue mediums, a relevant difference that cannot be ignored in the design. Nonetheless, the difference is accounted for in the thermal conductivity of the tissue, where its value is constant. The temperature profile within the cylindrical thermoseed was computed for comparison to the spherical design. Temperature profiles for the cylindrical thermoseed in human and rat tissue were plotted as solution to the temperature variation in the ferromagnetic material in Figures 8 and 9. The temperature values for the thermocylinder surrounded by the rat tissue are higher than for human tissue. A low thermal conductivity in the surrounding medium of the thermocylinder causes a higher temperature at steady-state, but its heat transfer behavior is identical to higher thermal conductivity values. The heat transfer behavior in the human tissue is identical to the spherical thermoseed. Fig 8 Temperature Profile in Thermocylinder (Humans)

Fig 9 Temperature Profile in Thermocylinder (Rats)

Temperature results in the tissue of rats and humans differ in the higher temperature observed for surrounding cancerous tissue of rats. The behavior of the temperature through the tissue is exponentially dropping to its surface temperature, provided by the boundary condition. The temperature profile behavior is identical to the spherical thermoseed; accordingly if thermal conductivity changes, the temperature profile changes. The discussed profiles are provided in Figures 10 and 11. The temperature profiles in the cancerous tissue of humans and rats offers critical values in the treatment of the tissue. A maximum temperature and a lethal temperature for tissue need to be computed in order to assess the efficiency of the treatment. The maximum temperature is to be located in the tissue and ferromagnetic-material interface, where heat generation causes the highest temperature in the system. The lethal temperature at a desired radius value can be computed by locating the radius for the desired temperature of Tlethal. The location of this temperature, rlesion, is the needed value to evaluate the range of effectiveness of the thermocylinder. Fig 10 Temperature Profile in Human Tissue

Fig 11 Temperature Profile in Rats Tissue

The heat flux through the tissue surrounding the thermocylinder was computed to analyze the behavior of the rate of heat transfer per unit area. Heat flux drops as radius increases for the given geometry. Behavior in the cylindrical thermoseed matches the behavior in the spherical thermoseed since their surface areas for heat flux are nearly identical. Given that heat flux is dependent of thermal conductivity, the heat flux in rat tissue behaves identically to the flux in human tissue. Figures 12 and 13 provide evidence of the drop of the heat transfer flux through the thermocylinder. Heat flux near the interface of the materials is high but it drops quickly 1 mm away from the interface. Consequently, heat flux in both tissue mediums behaves in a similar fashion to the temperature profiles given above. Heat flux is controlled by temperature differences within the material. Thus, the heat flux profile can be predicted from a calculated temperature profile, if it werent readily available. The effectiveness of the treatment depends on how fast thermal energy can reach the cancerous tissue and effectively heat it to cause its failure.

Fig 12 Thermocylinder Heat Flux (Humans)

Fig 13 Thermocylinder Heat Flux (Rats)

As considered in the spherical thermoseed design, two critical temperature values are needed to characterize the effectiveness of the design. Maximum temperature and the location of the lethal temperature provide an upper bound in tissue temperature and the reach of the medical device to give a lethal treatment to the cancerous tissue. The location of the maximum temperature is at the interface of the heat generation source and the heat receiver medium. An evaluation of temperature at the radius of the thermocylinder provided the desired temperature, being a maximum in the tissue. The lethal temperature was located by approaching the temperature using radius values. For the given heat generation of 1 W, the maximum temperature in human tissue was 442.6K (170C), and the extent of lesion was 5.1 mm which closely agrees with the thermosphere. In contrast, the maximum temperature in rat tissue was found to be 516.5K (243C) and the extent of lesion was 6.2 mm. Discrepancies in these values are inferred to be from differences in thermal conductivities within the tissue. Results are provided in Figures 14 and 15. Fig 14 Maximum Temperature and Extent of Lesion (Humans)

Fig 15 Maximum Temperature and Extent of Lesion (Rats)

ConclusionsHeat transfer behavior of a cylindrical thermoseed was analyzed using Comsol Multiphysics to determine its compatibility with the spherical thermoseed to treat cancerous tissue. The data and analysis collected on the thermocylinder design meets the required specifications in order to behave as the working design of the spherical thermoseed. Temperature profiles for the thermocylinder in the ferromagnetic material and tissue regions agree with the thermosphere device. Accordingly, heat transfer flux behavior of the cylindrical design matches the behavior of the thermosphere. The maximum temperature and extent of lesion in the thermocylinder agree fairly with the thermosphere values. Consequently, a cylindrical thermoseed can be implemented as replacement to the spherical thermoseed since manufacturing of the new design can be easier than a sphere and meets the required heat transfer behavior.

References

[1] Huang, Huang-Wen and Chihng-Tsung Liauh. "Review: Therapeutical Applications of Heat in Cancer Therapy." Journal of Medical and Biological Engineering (2011): 1-11.[2] Welty, James R., et al. Fundamentals of Momentum, Heat and Mass Transfer. New York: John Wiley & Sons, Inc, 2008.

Appendix Data Collection: Maximum Temperature and Extent of Lesion 1. Spherical Thermoseed

0273.150

0.1324.471.1

0.2338.791.9

0.3353.102.7

0.4367.423.3

0.5381.743.8

0.6396.064.2

0.7410.374.6

0.8424.694.9

0.9439.015.2

1.0453.335.5

1.1467.645.7

1.2481.965.9

1.3496.606.1

1.4510.596.3

1.5524.916.5

2. Cylindrical Thermoseed (Humans)

0273.150

0.1323.401.2

0.2336.651.9

0.3349.892.6

0.4363.153.1

0.5376.393.6

0.6389.644.0

0.7402.894.3

0.8416.144.6

0.9429.394.8

1.0442.645.1

1.1455.885.3

1.2469.145.5

1.3482.385.7

1.4495.635.9

1.5508.886.1

2. Cylindrical Thermoseed (Rats)

0273.150

0.1330.781.6

0.2351.412.6

0.3372.043.4

0.4392.674.0

0.5413.304.5

0.6433.934.9

0.7454.565.3

0.8475.195.6

0.9495.835.9

1.0516.456.1

1.1537.086.4

1.2557.716.6

1.3578.356.8

1.4598.986.9

1.5619.617.1