Optimization of the Laser-Induced
Thermotherapy Procedure for Liver Tumors
11/30/2010
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; however laser interstitial thermotherapy (LITT) possesses potential for
refinement. By creating a robust model of the LITT process using COMSOL Multiphysics, the
optimization of parameters can be performed without the difficulty and expense of an
experiment. The model consists of a spherical tumor shape with a tumor size of 2cm [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], a heat source
𝑊
𝑊
range of 5,000𝑚3 to 9,000,000𝑚3 , and a time domain of 120s (2 min) to 3300s (55min) [2]. Using
these ranges, the optimal ratio of coagulated tumor tissue to damaged healthy tissue is 1cm to
𝑊
0.033cm from the center of the tumor at 355K, 120s, and 5000𝑚3 .
Introduction
A healthy liver performs many functions, the most well known being the removal of toxins.
Metastatic cancer in other areas of the body can contribute to malignant tumor growth in the
liver. Despite the liver’s ability to regenerate, a liver weakened by cancer poses a serious health
risk to the patient. Careful ablation of the tumor is necessary to preserve as much of the healthy
tissue as possible while maximizing the destruction of tumor tissue. Surgical removal of
malignant tissue is a favored option, but approximately 20% of patients are capable of receiving
this treatment, leading to exploration of other methods [6]. LITT has been shown to be effective
in controlling tumor growth from liver metastasis 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 the parametric study are
presented in Results and Discussion. Finally, conclusions and design recommendations are made.
Design Objectives
The intent of modeling LITT of a tumor is to optimize the strength of radiation that will
maximize the tumor tissue ablated but minimize the healthy tissue damage. Laser strength (heat
source), time and temperature will be optimized through a parametric study. The parameters will
be evaluated until the radius of coagulation is 0.0100m (1cm). The parameters that give the
smallest value of damaged healthy tissue with a radius of coagulation of 1cm will be considered
optimal. The geometry, other boundary conditions, and other input parameters will remain
constant. The ratio of coagulated tumor tissue to damaged healthy tissue will be analyzed by
evaluating the temperature distribution near the boundary between the tumor and the healthy
tissue. The threshold to coagulate tissue is 323.15K (50°C) or higher [8], and the threshold to
3
damage healthy tissue is 316.15K (43°C) or higher [8]. The LITT treatment was modeled in a 2dimensional domain seen in Figure 1.
Figure 1. A 2-dimensional model depicting the subdomains affected by the heat from the applicator.
The tumor and healthy tissue are symmetric about the z-axis. The tumor was assumed to be in
the approximate center of the liver for simplicity. The radius of the liver from the center of the
tumor was determined arbitrarily by using the average circumference of a Caucasian male’s
waist, which is 39in [9] or 0.991m. This circumference resulted in a waist diameter of 0.315m,
which was divided by 4 because the liver in Figure 2 appears to have its center a quarter of the
way into the abdomen. A male’s waist was used instead of a female because men are slightly
more likely to develop liver cancer [1].
Approximate
location of
tumor model
Figure 2. Size and location of liver in abdomen. [1]
4
A 2cm diameter was chosen because tumors less than 2cm are best for tumor ablations [1]. Table
1 shows that a 2cm tumor can be found in 3 stages of liver cancer [1]. The size of the laser
applicator (shown in Figure 1) was assumed to have a width of 0.05cm and a length of 1cm [2].
The length was chosen base on the size of the tumor and to be within the 1cm to 3cm range [2].
Table 1. Sizes of Tumors for Various Liver Cancer Stages [1]
Cancer Stage Number
Average Diameter
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
Initial conditions used for the model are the initial temperatures of each subdomain at 310.15K.
The boundary conditions are zero heat flux to the boundaries perpendicular to the axis of
symmetry, constant surrounding temperature of 37°C at the far right boundary, continuity
between the tumor and liver tissue, Qsource, temperature at the axis of symmetry. The Qsource term
was varied from 13,000 to 9,000,000 W/m2 and the temperature of the applicator varies from 333
to 363K [3, 4, 5]. The duration of the model was also varied from 120 to 3,300 seconds [2].
Results and Discussion
The model was determined to be robust through a sensitivity analysis and an accuracy
comparison with a published model. Inconsistency of density, thermal conductivity, and specific
heat values in different sources suggested the need to confirm insensitivity.
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
Changing the density, thermal conductivity, and specific heat of blood, the applicator, liver
tissue, and tumor tissue confirmed the model is not sensitive to changes in these properties. The
temperature distribution was very similar between all variations as well as the maximum and
minimum temperatures within the subdomains.
5
Compared to a published mathematical model [3] the model was shown to produce realistic
results. Therefore a parametric study could be performed with reasonable confidence. The
𝑊
parametric study of variables proved a heat source of 5000𝑚3 , a time of 120s, and a temperature
of about 355K are the optimal values. With these values, the amount of tumor killed at 355K is
0.0100m in the y-direction and 0.00868m in the x-direction. This resulted in almost the entire
tumor killed. Healthy tissue is damaged is about 0.01330m in the y-direction and 0.0120m in the
x-direction. The tissue damage in the y-direction can be seen in Figure 3 and 4.
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
6
Max =355K
Y - Direction
Laser Applicator
Tumor
Arrows indicate
Temperature
Decrease decrease
Min =310.15K
Figure 4. Temperature distribution throughout the tumor and liver tissue
Testing the time limitations in the x and y directions determined the optimal time. The minimum
heat source and temperature were used to determine the maximum amount of time necessary
(within the 2 to 55 min range) to kill 0.0100m of the tumor in the radial direction. At 55min or
𝑊
3300s with the minimum heat source (5000𝑚3 ) and temperature (333.15K) of the applicator, the
coagulated tissue did not reach 0.0100m. However, in the y-direction, the tissue was coagulated
to 0.0100m at 2050s. This made clear that the y-direction was more sensitive as the proximity of
the applicator edge is closer to the boundary of the tumor than in the x-direction. If the applicator
was symmetrical about the x and y axis, the temperature distribution would be spherical, not
cylindrical. The non-spherical heat distribution limited the maximum time to be 2050s, so that no
healthy tissue is killed. The maximum time and heat source resulted in a coagulated tissue radius
of 0.0108m at the minimum temperature. Therefore the temperature was varied to determine the
𝑊
maximum temperature for 9,000,000𝑚3 and 3300s seen in Appendix C. The maximum
temperature at the maximum heat source and time in the y-direction was determined to be
355.06K also in Appendix C.
Using the determined maximum temperature and the minimum temperature, the temperature was
tested at different time intervals and values of Q. The test was performed with a heat source of
𝑊
𝑊
5000𝑚3 at 120s and 2050s, and a heat source of 9,000,000𝑚3 at 120s and 2050s. Although the
maximum temperature increased significantly with the heat source, the results for the two
different heat sources only varied about 0.05K in temperature that caused 0.100m of tumor tissue
to coagulate. This can be seen in Appendix C along with the remaining parametric study results.
The lower heat source was chosen to minimize the energy necessary for the procedure.
7
The temperature was further assessed at various times to determine the temperature that killed
0.0100m of the tumor from the center (in the y-direction) and damages the least amount of
healthy tissue. Temperature was optimized by first testing the maximum and minimum
temperatures to determine what temperature to simulate next. Coagulated tissue at 333.15K and
355K was assessed and an estimation of a temperature that would lead to 0.0100m of coagulated
tissue was tested. If the estimated temperature 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
0.0100m of coagulated tissue was determined. The optimal temperature was determined to be
𝑊
355K at 120s with a heat source of 5000𝑚3 , which only damages 0.00330m of healthy tissue.
The results reveal that an optimal LITT procedure exists within the researched ranges. The
increase in maximum temperature significantly increased by about 30K when the heat source
𝑊
𝑊
was increased from 5000𝑚3 to 9,000,000𝑚3 did not have a significant effect on the temperature
of the tissue at the boundary between the tumor and healthy tissue. The temperature at the
boundary remained around the same values because the temperature distributes faster with a
higher heat source. The temperature distributes at a faster rate with a higher heat source
providing a rationale for the temperature at the boundary changing little.
Conclusions and Design Recommendations
A finite element analysis was performed with COMSOL to determine the optimal time,
temperature, and heat source. The evaluated ranges consisted of 120s to 3300s [2] for time,
𝑊
𝑊
333K to 363K [3,4,5] for temperature, and 5000𝑚3 to 9,000,000𝑚3 for the heat source. The
optimal coagulated tissue to damaged healthy tissue ratio was about 1cm of tumor killed to
𝑊
.033cm of healthy tissue damaged was determined at about 355K, 120s, and 5000𝑚3 .
This case study can be further assessed by determining the laser power required for the optimal
heat source and if it can be decreased to minimize the energy required. This study could be
altered to include the convention of blood flow and/or redrawn to include veins or arteries. The
size of the laser applicator could be optimized within the given range. Although the length is
specified in a range, the width and shape of the applicator can be modified for the desired results.
More testing can be done to determine the effect of the applicator after the 120s because
realistically the applicator has to cool down from 355K after 120s. The applicator cannot just
rapidly decrease from 355K to 310.15K for normal tissue at steady state [8]. The safety of the
rapid increase and decrease in temperature of the tissue should also be analyzed for safety. The
radiation dosage from the applicator should be assessed to assure the dosage is not poisonous to
the human body.
8
Appendix A. Mathematical Model
Geometry
Laser interstitial thermotherapy utilizes heat to kill tumor cells. The geometry can be simplified
to a 2-dimensional 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 direction of the chest to back at 0.105m, while the major axis is
measured in the direction of the head to feet at 0.125m. A spherical tumor is assumed with a
diameter of 0.02m. The applicator dimensions are 0.01m 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
where 𝜌 = density of the tissue
𝐶𝑝 = specific heat capacity of the tissue
T = temperature
t = time
𝑝
(1)
9
k = thermal conductivity of the tissue
Q = 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)
where 𝛼 = specific absorption rate of the laser
𝐼0 = laser power intensity
𝜌𝑏 = density of blood
𝐶𝑃𝑏 = specific heat of blood
𝑉𝑏 = dermal blood perfusion rate
𝑇𝑎 = arterial blood temperature
T = temperature of tissue
Initial Conditions
Human body temperature is self regulated and is normally at 310.15K. Initially, all tissues
including blood can be assumed to be this temperature.
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 is varied. The applicator boundary coincides with the axis of
symmetry.
10
Appendix B. Mesh
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
Degrees of Freedom
Figure B1. Convergence study
10000 12000
11
Appendix C Part 1: Sensitivity Analysis for X and Y Directions
Table C1. Damaged and Coagulated Tissue in the X-direction for Minimum Heat Source and Temperature with Varying
Time
Time (s)
120
3300
Time (Q=5000 W/m^3, Ti of Appl.=333.15K)
Distance from Center in x dir (m) for
Damaged Tissue
Coagulated
(316.15K)
Tissue(323.25K)
0.008973
0.00565
0.01835
0.00905
In the above table, the coagulated tissue never reaches 0.0100m from the center, therefore no
value for the minimum heat source and time can be evaluated for the optimal parameters.
Table C2. Damaged and Coagulated Tissue in the Y-direction for Minimum Heat Source and Temperature with Varying
Time
Time (s)
120
1982
1994
2000
2050
3300
Time (Q=5000 W/m^3, Ti of Appl.=333.15K)
Distance from Center in y dir (m) for
Coagulated
Damaged Tissue (316.15K)
Tissue(323.25K)
0.01045
0.00804
0.01420
0.00912
0.01750
0.00999
0.01835
0.00999
0.01840
0.01000
0.01915
0.01027
In Table C2 the coagulated tissue reaches 0.0100m at about 2050s for these conditions. This
means the maximum time for these conditions is 2050s.
12
Table C3. Damaged and Coagulated Tissue in the X-direction for Maximum Heat Source and Temperature with Varying
Time
Time (s)
120
182
193
208
264
547
1710
3300
Time (Q=9E6 W/m^3, Ti of
Appl.=363.15K)
Distance from Center in x dir (m) for
Damaged Tissue Coagulated Tissue
0.01190
0.00915
0.01345
0.00986
0.01375
0.01000
0.01400
0.01038
0.01480
0.01135
0.01890
0.01380
0.02580
0.01775
0.02900
0.01910
Table C3 proves that the maximum for the set parameters is 193s for the x-direction.
Table C4. Damaged and Coagulated Tissue in the Y-direction for Maximum Heat Source and Temperature with Varying
Time
Time (s)
120
3300
Time (Q=9E6 W/m^3, Ti of
Appl.=363.15K)
Distance from Center in y dir (m) for
Damaged Tissue Coagulated Tissue
0.01377
0.01080
0.02972
0.01985
The results in Table C4 show the maximum settings cannot be used because the minimum time
results in the destruction of healthy tissue.
13
Appendix C Part 2: Parametric Study Results
Lines highlighted in yellow denote the optimal values for that particular set of parameter values.
Blue denotes the overall optimal values.
Table C5. 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 C5 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 C6. 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
Table C7. 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
14
Table C8. Damaged and Coagulated Tissue in the Y-direction for Minimum Heat Source and Time with Varying
Temperature
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 C9. 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 C2 and C3 show 0.0100m of the tumor tissue is coagulated around 333.15K even though
different heat sources are used. Also Tables C4 and C5 show the same relationship with
0.0100m of the tumor tissue destroyed around 355K.
Table C10. 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
𝑊
the 0.00330m of damaged tissue when the time is 120s, the heat source is 5000𝑚3 , and the
temperature is 355K.
15
Table C11. 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 C11 shows that a smaller time of 200s still does not conclude in less than 0.00330 of
damaged healthy tissue.
Table C12. 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 C8 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.
16
Appendix D: References
[1] Time, By This. "Pancreatic Cancer." American Cancer Society :: Information and Resources
for Cancer: Breast, Colon, Prostate, Lung and Other Forms. Web. 05 Oct. 2010.
<http://www.cancer.org/Cancer/PancreaticCancer/DetailedGuide/pancreatic-cancer>.
[2] Mack, M. G., R. Straub, K. Eichler, O. Sollner, T. Lehnert, and T. J. Vogl. "Breast Cancer
Metastases in Liver: Laser-induced Interstitial Thermotherapy--Local Tumor Control Rate and
Survival Data." Radiology 233.2 (2004): 400-09. Print
[3] Fasano, Antonio, Dietmar Homberg, and Dmitri Naumov. "On a Mathematical Model for
Laser-induced Thermotherapy." Applied Mathematical Modeling 34 (2010): 3831-840. Print.
[4] Merkle, Elmar M., John R. Haaga, Jeffery L. Duerk, Gretta H. Jacobs, Hans-Juergen Brambs,
and Jonathan S. Lewin. "MR Imaging-Guided Radiofrequency Thermal Ablation of the Pancreas
in a Porcine Model with a Modified Clinical C-Arm System." Radiology (1999): 461-67. Print.
[5] Mueller-Lisse, Ullrich G., Martin Thoma, Sonja Faber, Andreas F. Heuck, Rolf Muschter,
Peter Schneede, Ernst Weninger, Alfons G. Hofstetter, and Maximilian F. Resier. "Coagulative
Interstitial Laser-induced Thermotherapy of Benign Prostatic Hyperplasia: Online Imaging with
a T2-weighted Fast Spin-Echo MR Sequence— Experience in Six Patients." Radiology (1999):
373-79. Print.
[6] Vogl, Thomas J., Katrin Eichler, Stefan Zangos, and Martin G. Mack. "Interstitial Laser
Therapy of Liver Tumors." Medical Laser Application 20.2 (2005): 115-18. Print.
[7] Vogl, T. J., R. Straub, K. Eichler, O. Sollner, and M. G. Mack. "Colorectal Carcinoma
Metastases in Liver: Laser-induced Interstitial Thermotherapy--Local Tumor Control Rate and
Survival Data." Radiology 230.2 (2004): 450-58. Print.
[8] Datta, Ashim K., and Vineet Rakesh. An Introduction to Modeling of Transport Processes:
Applications to Biomedical Systems. Cambridge, UK: Cambridge UP, 2010. Print.
[9] ONISHI, NORIMITSU. "Japan, Seeking Trim Waists, Measures Millions." The New York
Times. 13 June 2008. Web. 07 Oct. 2010.
<http://www.nytimes.com/2008/06/13/world/asia/13fat.html>.
[10] Müller, Gerhard J., and André Roggan. Laser-induced Interstitial Thermotherapy.
Bellingham, WA: SPIE Optical Engineering, 1995. Print.