1
Optimization of the Laser-Induced
Thermotherapy Procedure for Liver Tumors
12/21/2010
Intro to Biofluids CRN 42697
Alli Dickey and Kelli Martino
2
Executive Summary
Cancer is notorious for its destructive effect on the human body, capable of spreading
uncontrollably from one area to another. Several treatments are available to reduce or eliminate
tumor growth. Laser interstitial therapy (LITT) destroys tumor tissue by heating until the blood
vessels within the tumor coagulate or clot the vessels in the tumor [A], stopping blood supply.
Although laser interstitial thermotherapy effectively coagulates tumor tissue, it possesses
potential for improvement of precision. By creating a robust model of the LITT process using
COMSOL Multiphysics, the optimization of input parameters for the LITT procedure can be
performed without difficulty and expensive in vivo experiments. The model consists of a 2cm
spherical tumor [1] with a rectangular laser applicator assumed to be with a width of 0.05cm and
a length of 1cm [2]. This model uses a temperature range of 333.15K (60°C) to 363.15K (90°C)
[3, 4, 5] and a time domain of 120s (2 min) to 3300s (55min) [2]. Using these ranges, the optimal
value of coagulated tumor tissue to damaged healthy tissue is 1cm to 0.033cm from the center of
the tumor at 355K and 120s.
Introduction
Metastatic cancer, cancer that starts in one tissue then spreads to other areas of the body can
contribute to malignant tumor growth [1]. Despite the liver’s ability to regenerate, a liver
weakened by cancer poses a serious health risk to the patient because it is no longer able to
perform vital functions such as toxin removal. Careful ablation of the tumor is necessary to
maximize the destruction of tumor tissue while preserving as much of the healthy tissue as
possible. Surgical removal of malignant tissue is a favored option, but only approximately 20%
of patients are capable of receiving this treatment, leading to exploration of other methods [6].
LITT is an effective alternative in controlling tumor growth from liver metastatic cancer from the
breast and colon [7, 2].
The problem schematic, set up, and construction of the model can be found in the Design
Objectives. Following the description of the model the results of a parametric study are presented
in Results and Discussion. Finally, conclusions and design recommendations are made.
Design Objectives
Modeling LITT on a tumor allows optimization of the duration of the procedure and temperature
of the applicator edge that will maximize the ablated tumor tissue but minimize healthy tissue
damage. Time and temperature of the applicator are optimized through a parametric study.
Parametric studies allow the relationship between an output and input to be analyzed by isolating
and varying each input and evaluating the output. This optimization is quantified by minimizing
the amount of healthy tissue damage when the temperature at the edge of the tumor (r =
0.01000m) in the vertical direction is the temperature that coagulation occurs (323.15K or
higher). The coagulated tumor tissue is when the distance from the center of the tumor (r) is less
3
than 0.01000, and has a temperature greater than 323.15K (50°C). Damaged healthy tissue is
defined as the tissue with a temperature (T) between 323.15K (50°C) and 316.15K (43°C) when
the distance from the center of the tumor (r) is greater than 0.01000m. If healthy tissue (r >
0.01000m) is coagulated (T > 323.15K) the time/temperature combination is considered
undesirable. During optimization, the geometry, other boundary conditions, and other input
parameters remain constant. These input parameters consist of the initial temperature of the liver
and tumor at 310.15K, which is the average temperature of the body [8]. Also the heat flux is
zero on the outside boundaries of the liver and continuity is assumed for the internal boundaries
such as the tumor/tissue boundary. The applicator temperature is varied from 333 to 363K [3, 4,
𝑊
5], but the heat source is determined arbitrarily to be 5000𝑚3 . Time is another varied parameter
with a range of 120 (2min) to 3,300 (55min) seconds [2]. All boundary conditions and input
parameters can be found in Appendix A. These conditions were used for the 2-dimensional
model seen in Figure 1to simulate the LITT treatment.
Blood Vessels
Figure 1. A 2-dimensional Model Depicting the Subdomains Affected by the Heat from the Applicator
A vertical slice of the liver is used as the geometry for the model. The minor axis of the liver
slice is measured in the horizontal direction of the chest to back at 0.105m, while the major axis
is measured in the vertical direction from head to feet at 0.125m [M]. For simplicity, the tumor
was assumed to be in the approximate center of the liver with respect to the sagittal plane (as
seen in Figure 2). This assumption removes the need to consider the temperature distribution
4
outside of the liver. If the tumor is assumed to be near edge of the liver, surrounding organs,
arteries, and veins would have to be considered.
Sagittal Plane
Approximate
location of
tumor model
Figure 2. Location of Tumor in Abdomen. [1]
Since tumors with diameters of 2cm or less are best for LITT [1], the tumor is assumed to be
spherical with a diameter of 2cm. Table 1 shows that a 2cm tumor can be found in 3 stages of
liver cancer [1].
Table 1. Stages of Cancer that Have 2cm Tumors [1]
Cancer Stage Number
Diameter of Tumor(s)
1a, 1b
2
Single Tumor Any Size
Single Tumor any Size or Multiple Tumors
No Larger Than 5cm in Diameter
Multiple Tumors with at least One that is
Greater Than 5cm Across
3a
The tumor size is directly related to the size of the laser applicator (shown in Figure 1) [2]. Laser
applicators vary in length from 1cm to 3cm, [2] so a length of 1cm is assumed. The width is
arbitrarily chosen as 0.5 cm because the laser applicators are usually thin.
Results and Discussion
A sensitivity analysis and an accuracy comparison with a published model determined the model
to be credible. Inconsistency of values of input parameters such as density, thermal conductivity,
and specific heat from a literature search proved the need to confirm insensitivity. A sensitivity
analysis assessed the parameters shown in Tables 2 and 3.
5
Table 2. Properties and Their Ranges for a Sensitivity Analysis.
Material
Blood
Tumor
Liver
Applicator
ρ (kg/m^3 )
1000-1200
1040-1060
1040-1060
2100-2300
Properties [8]
k (W/mK)
N/A
0.47-0.58 [10]
0.5122-0.5737 [8]
1.25-1.45
C_p (J/kgK)
3200-3400
3500-3890
3500-3890
650-750
The given ranges for the density and specific heat for the tumor and liver were chosen based on
the densities and specific heats of other organs. The thermal conductivity values of the liver and
tumor were maintained from the stated sources. The values for blood and the laser applicator
properties were chosen arbitrarily. The ranges for the heat source and blood perfusion rate were
also chosen arbitrarily and can be seen in Table 3.
Table 3. Properties and Their Ranges for a Sensitivity Analysis
Parameter
Range
Laser Source Term(W/m^2)
5000-9000000
V_b (ml_blood/(ml_tissue*s))
0.0002-0.0006
After analyzing the minimum and maximum values for each input parameter, all of the
parameters except for the heat source and tumor density proved to have extremely little (0.5% or
less) or no effect on the solution. The maximum and minimum temperatures did not vary at all
when changing the specified variables, except for the 18K increase in initial temperature from
the increase in heat source. The temperature distribution only varied by a maximum of 5.92%
and 1.973% for the differences in the range of the heat source and tumor density respectively.
The laser heat source term proved to have a large effect on the maximum temperature, but the
laser source term only had a 2.714% or 6.18K difference in temperature distribution. This could
be the reason the COMSOL model is about 6K off of the published model because they do not
state what value of heat source they use. Since the temperature distribution was the main
concern, the differences of up to 5.92% proved the difference in parameters values do not
significantly change the solution. The x and y direction were also evaluated to determine the
direction that was most sensitive to the laser applicator. Since the edge of the tumor in the ydirection is closer the laser applicator, the edge of the tumor in the y-direction reached the
temperature of coagulation before the edge of the tumor in the x-direction. Because the
coagulation or damage of healthy tissue is undesirable, the y-direction was considered before the
x-direction for evaluation of the parametric study. These results can be seen in Appendix C.
A published model [3] was also used to verify the validity of this model. The COMSOL model
was altered to have the input parameters and geometry as the published model for a more
accurate comparison. The temperature distribution was compared using a contour plot of the
temperature distribution, which proved the COMSOL model to be accurate within +/- 6 degrees.
6
These results can be seen in Appendix C. This difference is reasonable because the COMSOL
model does not take radiation and varying paramters like the other model. After confidence was
gained in the COMSOL model, the parametric study was performed with the parameters and
ranges shown in Table 4.
Table 4. Parameters for Parametric Study with Ranges
Parameter
Ti of Applicator (K)
Time Domain
Range
333-363 [6,7,8]
2min-55min[1]
This time range was evaluated at the minimum and maximum temperatures to determine the
amount of healthy tissue damage at a radius of coagulation of 0.01000m. The radius of
coagulation was determined by using graphical data similar to Figure 3 to evaluate where the
temperature was 323.15K. When the temperature of coagulation reached 0.01000m, the optimal
value for the specific set of parameters was achieved. The healthy tissue damage was then
determined by finding the distance from the center in the y-direction where the temperature is
316.15K (the temperature where tissue damage begins). This evaluation was used to evaluate
both time and temperature and is shown in Figure 3 with the optimal results.
Healthy Tissue
Damaged
Above this line (T>323.15)
Tissue is Coagulated
Above this line (T>316.15)
Tissue is Damaged
Figure 3. Temperature Distribution from Center of Tumor in Y-direction Using the Optimal Parameters
The model was first tested at the minimum temperature for the minimum and maximum times.
If neither the minimum nor maximum time produced the desired radius of coagulation,
0.01000m, intuition of a time that would lead to that value was tested. If the estimated time did
not result in the desired amount of coagulated tissue, a revised estimation was made based on the
results. This process continued until a value of 0.01000m of coagulated tissue was determined.
7
This resulted in the radius of coagulation reaching 0.01000m (the edge of the tumor in the ydirection) after 2050 seconds for the minimum temperature. This was determined to be the new
maximum amount of time necessary to coagulate the tumor in the y-direction because a time of
above 2050s would result in coagulation of healthy tissue. This procedure was also used for
evaluation of the time domain at the maximum temperature, which resulted in the coagulation of
the tumor before the minimum time of 120seconds. Therefore, the temperature of 363.15K had
to be reduced to determine the temperature where coagulation reaches 0.01000m. To evaluate the
maximum temperature, temperature was then varied at 120s. This resulted in a maximum
temperature of 355K instead of 363K because anything larger than 355K would lead to
coagulation of healthy tissue. Temperature was further optimized using the same method for
time. After testing different time/temperature combinations, the optimal parameters were decided
by which minimized the amount of healthy tissue damage. The optimal settings were then
determined to be 355K at 120s, which only damaged 0.00330m of healthy tissue. With these
values, the amount of tumor coagulated was 0.0100m in the y-direction and 0.00868m in the xdirection. There was about 0.00330m of damaged healthy tissue in the y-direction and 0.0020m
in the x-direction. The temperature distribution for the optimal settings along the y-axis can be
seen in Figure 3, and the overall temperature distribution can be seen in Figure 4.
Max =355K
Y - Direction
Tumor
Laser Applicator
Arrows indicate
Temperature
Decrease decrease
Min =310.15K
Figure 4. Temperature Distribution Throughout the Tumor and Liver Tissue
Conclusions and Design Recommendations
A finite element analysis was performed with COMSOL to determine the optimal time and
temperature. The evaluated ranges consisted of 120s to 3300s [2] for time and 333K to 363K
[3,4,5] for temperature. The optimal coagulated tissue to damaged healthy tissue ratio was
8
approximately 1cm of tumor killed to 0.033cm of healthy tissue damaged at 355K (82°C) and
120s.
An important aspect to be evaluated is the transient heat transfer during the beginning and end of
the procedure. Since the laser cannot rapidly cool down to body temperature (315.15 or 37°C) or
rapidly heat to 355K, the transient heat transfer could affect the optimal parameters. This heat
transfer could be altered by changing the size and shape of the laser applicator.
Other complexities could be added to the model. Instead of assuming the tumor to be in the
center, the tumor could be assumed to be near the edge of the liver. The heat transfer analysis
would then require the assessment of the temperature distribution to surrounding organs, arteries,
and veins. Heat due to blood perfusion brings the model closer to a realistic situation. Despite
blood perfusion larger arteries exist near the liver such as the hepatic portal vein and hepatic
artery. A large artery or vein could introduce connective heat flux that could affect the
temperature distribution of the tumor.
9
Appendix A. Mathematical Model
Geometry
Laser interstitial thermotherapy utilizes heat transfer to coagulate cells. The geometry can be
simplified to a 2-dimensional axisymmetric domain.
Figure A1. Model Schematic.
A vertical slice of the liver is used as the geometry for the model. The minor axis of the liver
slice is measured in the horizontal direction of the chest to back at 0.105m, while the major axis
is measured in the vertical direction from head to feet at 0.125m. A spherical tumor is assumed
with a diameter of 0.02m. The applicator dimensions are 0.01000m in height and approximately
0.005m in width.
Governing Equations
Although the human body functions off of different types of physics, the model represents heat
transfer. The Cartesian form of the Energy Conservation Equation is used in the model.
𝜕𝑇
𝜕𝑡
𝜕2𝑇
𝑘
𝜕2 𝑇
1
= 𝜌𝐶 [𝜕𝑥 2 + 𝜕𝑦 2 ] + 𝜌𝐶 𝑄
𝑝
𝑝
𝑘𝑔
where 𝜌 [𝑚3 ] = density of the tissue
𝐽
𝐶𝑝 [𝑘𝑔∙𝐾] = specific heat capacity of the tissue
(1)
10
T [K] = temperature
t [s] = time
𝑊
k [𝑚∙𝐾] = thermal conductivity of the tissue
𝑊
Q [𝑚3 ] = heat generation
To make the model more realistic, blood perfusion is added to the physics. The heat generated by
blood perfusion added to the heat source equals the heat generation within the domain. Therefore
the heat source and blood perfusion must be defined.
𝑄ℎ𝑒𝑎𝑡 𝑠𝑜𝑢𝑟𝑐𝑒 = 𝛼𝐼0 𝑒 −𝛼𝑥
𝑄𝑏𝑙𝑜𝑜𝑑 𝑝𝑟𝑜𝑓𝑢𝑠𝑖𝑜𝑛 = 𝜌𝑏 𝐶𝑃𝑏 𝑉𝑏 (𝑇𝑎 − 𝑇)
1
where 𝛼 [𝑚] = specific absorption rate of the laser
𝑊
𝐼0 [𝑚2 ] = laser power intensity
𝑘𝑔
𝜌𝑏 [𝑚3 ] = density of blood
𝐽
𝐶𝑃𝑏 [𝑘𝑔∙𝐾] = specific heat of blood
𝐽
𝑉𝑏 [𝑠] = dermal blood perfusion rate
𝑇𝑎 [K] = arterial blood temperature
𝑇 [K] = temperature of tissue
Initial Conditions
Human body temperature self regulates and is normally at 310.15K. Initially, all tissues
including blood can be assumed to start at this temperature.
𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 310.15𝐾, t = 0s
Boundary Conditions
Surrounding the liver is human tissues which are assumed to be at normal body temperature.
Therefore, there is no heat flux at the outer edge of the liver. As a part of the optimization the
temperature of the applicator edge is varied.
𝑄̈ |𝑥=0.0525𝑚,𝑦=0.625𝑚 = 0
𝑊
𝑚2
𝑇𝑖 = Temperature of Applicator Edge
11
Input Parameters
The following parameters are needed for the COMSOL model.
Constants
𝐽
Cp_b = 3300 [𝑘𝑔∙𝐾]
T_a = 310.15 [K]
𝑘𝑔
Rho_b = 110 [𝑚3 ]
1
V_bp = 0.024/60 [𝑠 ]
Subdomain
Tumor
𝑊
k = 0.566 [𝑚∙𝐾]
𝑘𝑔
ρ = 1050 [𝑚3 ]
𝐽
Cp = 3800 [𝑘𝑔∙𝐾]
Applicator
𝑊
k = 1.38 [𝑚∙𝐾]
𝑘𝑔
ρ = 2203 [𝑚3 ]
𝐽
Cp = 703 [𝑘𝑔∙𝐾]
𝑊
Q = Heat Source [𝑚3 ]
Liver
𝑊
k = 0.566 [𝑚∙𝐾]
𝑘𝑔
ρ = 1050 [𝑚3 ]
𝐽
Cp = 3590 [𝑘𝑔∙𝐾]
𝑊
Q = Q_bp [𝑚3 ]
Boundary
Applicator Edge
Ti = Temperature of Applicator Edge [K]
Tedge_liver = 310.15 [K]
12
Global Expressions
𝑊
Q_h = 13000 [𝑚3 ]
𝑊
Q_bp = rho_b*Cp_b*V_bp*(T_a-T) [𝑚3 ]
Solver Parameters
Times
Range(120,10,3300)
13
Appendix B. Model Verification
Liver Tumor Mesh
The subdomains from smallest to largest are the applicator, the tumor, and the liver.
Figure B1. Converged Mesh
To reduce discretization error, a convergence study shows the mesh converges around 11,000
degrees of freedom. Several other meshes with less degrees of freedom were plotted to show the
convergence shown in Figure B2.
Convergence Study
336
334
Temperature 332
(K)
330
328
326
0
2000
4000
6000
8000
10000 12000
Degrees of Freedom
Figure B2. Convergence Study Using Temperature at the Tumor Tissue and Healthy Tissue Boundary
14
Accuracy Check
To verify the accuracy of our model it was compared to a published model. Comparing Figures
B3 and B4 revealed that the model was fairly accurate in predicting the temperature distribution
throughout the model domains. Geometries, boundary and initial conditions were replicated as
well.
Figure B3. Temperature Distribution Through the Tumor and Liver in the Math Model [3]
Figure B4. Temperature Distribution Through the Tumor and Liver in LITT Model
15
Appendix C Part 1: Sensitivity Analysis
Table C1. Table of Contents for Appendix C Part 1 and Results
Material
Applicator
Blood
Properties
Liver
Tumor
Sensitivity analysis
Max Diff.
Property
(%)
ρ (kg/m^3 )
N/A
C_p (J/kgK)
N/A
k (W/mK)
N/A
Q (W/m^2)
5.92%
ρ (kg/m^3 )
N/A
C_p (J/kgK)
0.410%
V_b (ml_b/(ml_t*s))
N/A
T_a [K]
0.349%
ρ (kg/m^3 )
0.37%
C_p (J/kgK)
N/A
k (W/mK)
N/A
ρ (kg/m^3 )
1.973%
C_p (J/kgK)
0.41%
k (W/mK)
N/A
Max Diff.
(K)
N/A
N/A
N/A
18.555
N/A
1.304
N/A
1.101
1.169
N/A
N/A
6.187
1.311
N/A
Fig.
#
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
Page
#
15
16
16
17
17
18
18
19
19
20
20
21
21
22
*N/A is for no noticeable difference
**All Results achieved with Specified Input Parameters and Boundary Conditions for a Time of 120s, an
Applicator Ti of 333K, and Q=5000W/m^3 unless stated.
Figure C1. Sensitivity Analysis for Applicator Density.
335
Temperature [K]
330
325
320
Applicator Rho max
315
Applicator Rho min
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
Distance from Center in y-direction [m]
8.00E-02
16
Figure C2. Sensitivity Analysis for Specific Heat for Applicator.
335
Temperature [K]
330
325
320
Applicator Cp Max
315
Applicator Cp min
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
8.00E-02
Distance from Center in y-direction [m]
Figure C3. Sensitivity Analysis for Thermal Conductivity of Applicator.
335
330
325
320
Applicator k max
Applicator k min
315
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
8.00E-02
17
Figure C4. Sensitivity Analysis for Heat Source of Applicator.
355
350
Temperature [K]
345
340
335
330
325
Q min
320
Q min
315
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
8.00E-02
Distance from Center in y-direction [m]
The maximum heat source causes the initial temperature to increase by 18.55K, but the
temperature at the edge of the applicator remains the same. Although the heat source appears to
have a large effect, it only alters the values by a maximum of 5.92% or 6.813K around the edge
of the tumor. To verify the optimal results, values were tested with the optimal heat source and
still yielded the same optimal settings. Also the heat source is hard to very realistically, so it will
be dependent on the temperature of the laser applicator.
Figure C5. Sensitivity Analysis for Blood Density.
335
Temperature [K]
330
325
320
Blood Rho max
315
Blood Rho min
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
Displacement from Center in y-direction [m]
8.00E-02
18
Figure C6. Sensitivity Analysis for Specific Heat of Blood.
335
Temperature [K]
330
325
320
Blood Cp max
Blood Cp Min
315
310
305
0.00E+001.00E-022.00E-023.00E-024.00E-025.00E-026.00E-027.00E-02
Distance from Center in y-direction [m]
The differences in the specific heat of blood for blood perfusion still prove the specific heat does
not significantly alter the solution. The sensitivity analysis determined that the difference in
values for specific heat result in a 0.410% or 1.304K difference.
Figure C7. Sensitivity Analysis for Blood Perfusion Rate.
335
Temperature [K]
330
325
320
V_b max
315
V_b min
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
Displacement from Center in y-direction [m]
8.00E-02
19
Figure C8. Sensitivity Analysis for Arterial Temperature.
335
Temperature [K]
330
325
320
T_a max
315
T_a min
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
8.00E-02
Distance from Center in y-direction [m]
Although the arterial temperature results in a change in temperature, the 0.349% or 1.101K
difference is not significant enough to rule the solution sensitive.
Figure C9. Sensitivity Analysis for Liver Density.
335
Temperature [K]
330
325
320
Liver rho max
Liver rho min
315
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
8.00E-02
Distance from Center in y-direction [m]
Much like the sensitivity analysis for the arterial temperature, the temperature difference of
0.37% or 1.169K did not produce enough evidence to consider the solution sensitive.
20
Figure C10. Sensitivity Analysis for Specific Heat of Liver.
335
Temperature [K]
330
325
320
Liver Cp max
Liver Cp min
315
310
305
0.00E+001.00E-022.00E-023.00E-024.00E-025.00E-026.00E-027.00E-02
Displacement from Center in y-direction [m]
Figure C11. Sensitivity Analysis for Thermal Conductivity of Liver.
335
Temperatre [K]
330
325
320
Liver k max
315
Liver k min
310
305
0.00E+00 1.00E-02 2.00E-02 3.00E-02 4.00E-02 5.00E-02 6.00E-02 7.00E-02
Displacement from Center in y-direction [m]
21
Figure C12. Sensitivity Analysis for Tumor Density.
335
Temperature [K]
330
325
320
Tumor rho max
315
Tumor rho min
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
8.00E-02
Distance from Center in the y-direction [m]
The tumor density was the second most influencing parameter with a difference of 1.973% of
6.187K, but this result is still not significant.
Figure C13. Sensitivity Analysis for Specific Heat of Tumor.
335
Tempertaure[K]
330
325
320
Tumor Cp max
315
Tumor Cp min
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
8.00E-02
Displacement from Center in y-direction [m]
Although the liver and tumor are assumed to the same specific heat, only the tumor displayed a
difference in temperature. This difference was 0.41% or 1.311K, which is not enough to
influence a solution. The specific heat of the tumor was a factor only because it the temperature
is much larger in the tumor than the healthy tissue.
22
Figure C14. Sensitivity Analysis for Thermal Conductivity of the Tumor .
335
Temperature [K]
330
325
320
Tumor k max
315
Tumor K min
310
305
0.00E+00
2.00E-02
4.00E-02
6.00E-02
Distance from Center in y-direction [m]
8.00E-02
23
Appendix C Part 2: Parametric Study Results
Lines highlighted in yellow denote the optimal values for that particular set of parameter values.
Green denotes the overall optimal values.
Table C2. Table of Contents for Appendix C Part 2
Q[W/m3]
C16
C17
C18
Varying
Parameter
Ti of Applicator
Ti of Applicator
Ti of Applicator
9000000
9000000
5000
Ti of Applicator
[K]
343.65-360.15
333.15-363.15
333.15-363.15
C19
Ti of Applicator
5000
333.15-363.15
120
C20
C21
C22
Ti of Applicator
Ti of Applicator
Ti of Applicator
9000000
5000
9000
333.15-355.06
333.15-355.15
333.15-355.15
120
1080
200
C23
Ti of Applicator
5000
333.15-355.15
125
Figure
Time [s]
Optimum Value
120
2050
2050
355.06[K]
333.15[K]
333.15[K]
355.00 &
355.15[K]
355.06[K]
333.15[K]
343.15[K]
354.15 &
355.15[K]
Table C16. Damaged and Coagulated Tissue in the Y-direction for Maximum Heat Source and Time with Varying
Temperature
Find temp that can be
used with Qmax at 120s
Temp of Appl. (K)
360.15
356.81
355.06
353.27
343.65
Laser Source (Q=9E6 W/m^3, Time =
120s)
Distance from Center in y dir (m) for
Damaged Tissue Coagulated Tissue
N/A
0.01048
N/A
0.01010
N/A
0.01000
N/A
0.00990
N/A
0.00929
Table C16 displays the maximum temperature that can be used for the maximum time and heat
source in order to have the coagulated tissue stay within 0.0100m from the center of the tumor.
Table C7. Damaged and Coagulated Tissue in the Y-direction for Maximum Heat Source and Maximum Time with
Varying Temperature
Ti of Applicator
(K)
333.15
363.15
Laser Source (Time=2050s, Q=9e6W/m^3)
Distance from Center in y dir (m) for
Damaged Tissue
Coagulated
(316.15K)
Tissue(323.25K)
0.01840
0.01000
0.02775
0.01900
24
Table C18. Damaged and Coagulated Tissue in the Y-direction for Minimum Heat Source and Maximum Time with
Varying Temperature
Ti of Applicator
(K)
333.15
363.15
Laser Source (Time=2050s, Q=5000W/m^3)
Distance from Center in y dir (m) for
Damaged Tissue
Coagulated
(316.15K)
Tissue(323.25K)
0.01850
0.01000
0.02800
0.01910
Table C19. Damaged and Coagulated Tissue in the Y-direction for Minimum Heat Source and Time with Varying
Temperature with Optimum Paraeters
Ti of Applicator
(K)
333.15
354.15
354.85
355.00
355.15
355.50
360.15
363.15
Laser Source (Time=120s, Q=5000W/m^3)
Distance from Center in y dir (m) for
Damaged Tissue
Coagulated
(316.15K)
Tissue(323.25K)
0.00776
0.00804
0.01320
0.00995
0.01328
0.00997
0.01329
0.00998
0.01330
0.01002
0.01332
0.01004
0.01360
0.01050
0.01377
0.01080
Table C20. Damaged and Coagulated Tissue in the Y-direction for Minimum Time and Maximum Heat Source with
Varying Temperature
Ti of Applicator
(K)
333.15
355.06
Laser Source (Time=120s, Q=9e6 W/m^3)
Distance from Center in y dir (m) for
Damaged Tissue
Coagulated
(316.15K)
Tissue(323.25K)
0.01042
0.00805
0.01326
0.01000
Table C13 and C14 show 0.0100m of the tumor tissue is coagulated around 333.15K even
though different heat sources are used. Also Tables C15 and C16 show the same relationship
with 0.0100m of the tumor tissue destroyed around 355K.
25
Table C21. Further Testing for Optimization of Damaged and Coagulated Tissue in the Y-direction
Ti of Applicator
(K)
333.15
355.15
Laser Source (Time=1080s, Q=5000W/m^3)
Distance from Center in y dir (m) for
Damaged Tissue
Coagulated
(316.15K)
Tissue(323.25K)
0.01668
0.00969
0.02277
0.01564
The time of 1080s was chosen and evaluated because it was the average between 120s and
2050s. This time was determined to be insignificant because the damaged tissue was larger than
the 0.00330m of damaged tissue when the time is 120s, the heat source is 5000
𝑊
𝑚3
, and the
temperature is 355K.
Table C22. Further Testing for Optimization of Damaged and Coagulated Tissue in the Y-direction
Try Time around
120s
Ti of Applicator
(K)
333.15
343.15
355.15
Laser Source (Time=200s, Q=9000W/m^3)
Distance from Center in y dir (m) for
Damaged Tissue
Coagulated
(316.15K)
Tissue(323.25K)
0.01200
0.00841
0.01358
0.00981
0.01501
0.01143
Table C22 shows that a smaller time of 200s still does not conclude in less than 0.00330 of
damaged healthy tissue.
Table C23. Further Testing for Optimization of Damaged and Coagulated Tissue in the Y-direction
Try Time closer
120s
Ti of Applicator
(K)
333.15
353.75
354.15
355.15
Laser Source (Time=125s, Q=5000W/m^3)
Distance from Center in y dir (m) for
Damaged Tissue
Coagulated
(316.15K)
Tissue(323.25K)
0.01063
0.00871
0.01333
0.00997
0.01335
0.00999
0.01343
0.01005
Table C19 shows that 120s is the optimal time resulting in 0.01330m of healthy tissue being
damaged, which is smaller than the 0.00335m or 0.00343m of healthy tissue damaged shown in
the table above.
26
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