Focused Ultrasound Induced Heating for Drug Delivery in
Cancer Therapies 
BEE/ENGRC 4530 Computer-Aided Engineering: Applications to Biomedical Processes
Amita Datta
Gabe Dreiman
Grace Livermore
Alice Wang
May 11, 2017
Keywords:
Targeted drug delivery; Cancer therapies; Focused Ultrasound; Induced heat generation;
COMSOL; Chemotherapeutic agents; Temperature-sensitive liposomes
Table of Contents
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
XIII.
XIV.
XV.
Executive Report Summary………………………………………………………….. 3
Introduction…………………………………………………………………………... 4
Code Verification…………………………………………………………………….. 5
Problem Statement and Design Objectives………………………………………...… 5
System Configuration of Our Model………………………………………………… 6
Governing Equations………………………………………………………………… 7
System Conditions…………………………………………………………………… 8
System Discretization…………………………………………………………..….… 9
Validation………..……………..…………………………………………………… 11
Results………………………………………………………………………………. 13
Limitations…………………………………………………………………...…...… 20
Conclusions…………………………………………………………………………..21
Appendix A…………………………………………………………………………..23
Appendix B……………………………………………………………………..……24
References……………………………………………………………………………26
2
I.
Executive Summary
Many treatment options are available in eliminating tumors including surgery, radiation, and
chemotherapy. These treatments, however, have heavy financial constraints and/or significant
systemic side effects. Focused ultrasound (FUS) is a new method that has been implemented for
targeted cancer treatments. Chemotherapeutic drugs are stored in temperature sensitive
liposomes and injected near the target tumor area. FUS is used to heat the target tissue to a
temperature range that causes the liposomes to burst and release their drugs. This methodology
allows targeted drug delivery to the tumor region while sparing the surrounding tissue,
preventing the systemic side effects of traditional chemotherapy.
We used COMSOL to optimize the duty cycle and pulse repetition frequency (PRF) of FUS to
maximize induced heating in a tissue analog, a bovine serum albumin phantom, from 21°C to a
specified temperature range, 23-28°C, while minimizing heating to the surrounding tissues. We
used a homologous tissue-mimicking phantom to have uniformity in parameters and to compare
our results with the experimental results from literature [1]. In tissue, body temperature is 37°C
and the liposomes are engineered to break at temperatures between 39°C and 44°C [1].
Therefore, we used a different initial temperature for the phantom but the same change in
temperature. In the model, as the pressure waves from the ultrasound transducer pass into the
phantom, some energy is lost. This attenuation of acoustic energy was assumed to be equal to the
heat energy gained by the phantom and thus was used as a heat source term. We were able to
determine the optimal duty cycle and PRF of the FUS to reach a high enough temperature that
would release chemotherapeutic agents from the liposomes into only the target region of the
phantom.
3
II.
Introduction
The most common types of cancers are skin, breast, prostate, colorectal, and lung cancer; all of
which are solid tumors of their respective cells [2]. Many treatment options are available for
eliminating tumors including surgery, radiation therapy, and chemotherapy. Surgical treatment is
only effective if there are clear margins after tumor removal [2]. Even still, some tumors are
inoperable due to their location and/or size. Radiotherapy uses ionizing radiation to induce DNA
damage and apoptosis within the tumor [2]. Radiation can be quite effective but sub-lethal
radiation to surrounding tissues can generate significant side effects, which may include
permanent nerve damage [3]. Chemotherapy involves systemic administration of cytotoxic drugs
that target rapidly proliferating cells [4]. Unfortunately, these drugs also kill healthy cells, are
carcinogenic themselves, and induce alopecia, diarrhea, anemia, and immunosuppression [4].
While the aforementioned treatment options are still clinically common, there exists a need for a
more effective and less invasive method for treating tumors.
Using focused ultrasound (FUS) for localized drug delivery to tumors is a new method of
treating cancer. Already, focused ultrasound is being used in ablation therapy, which involves
heating tumors to high temperatures, inducing cell death [5]. Ablation therapy serves as a
possible replacement for radiotherapy because the heating is localized to only the tumor, sparing
surrounding healthy tissue. Ultrasound is a non-invasive procedure, lacking the recovery period
and postoperative risks associated with surgical interventions and is much more patient
compliant. FUS for targeted drug delivery uses localized heating to break open temperature
sensitive liposomes that contain chemotherapeutic agents [6]. This allows targeted drug delivery
to the tumor regions while sparing the surrounding tissue, preventing the side effects of
traditional chemotherapy. The liposomes are engineered to break at temperatures between 39°C
and 44°C, below temperatures necessary to directly kill cells [7]. However, it is important to
only raise the temperature of the tumor because studies have shown ultrasound heating to
increase blood perfusion in surrounding tissues, which may cause unwanted drug diffusion away
from the tumor and into the systemic bloodstream [8].
III.
Code Verification
COMSOL code has been verified in general due to widespread usage. Our project
validation can be found in the validation section.
IV.
Problem Statement and Design Objectives
In this study, we address how to heat a targeted tissue area to a specific temperature range
such that the drug is delivered to that area without being released to the surrounding
tissues. The tissue will be modeled using a homogeneous tissue-mimicking phantom.
We will use COMSOL to optimize the following:
 Duty cycle and pulse repetition frequency (PRF) of the applied ultrasound.
4
Our design objectives are the following:
 To develop a model for ultrasound propagation and heat transfer within the
phantom.
 To optimize the ultrasound frequency to heat the focal zone of the phantom from
21°C to 23-28°C, a minimum two degree change similar to other studies that have
computationally modeled and experimentally validated this technique [1].
Table 1. Nomenclature of variables and their descriptions used throughout paper
Symbol Unit
Description
P
Pa
Acoustic pressure
r
m
Radial coordinate
z
m
Axial coordinate
3
kg/m
Density

rad/s
Angular frequency


c
T
t
k
I
cp
Cv
Q
𝑣0
κ
i
f
y
d
kw
dB/cm*MHz
m/s
˚C
s
W/(m*K)
W/m2
(J*kg)/K
J/m3K
J
m/s
m2/s
MHz
nm
Attenuation coefficient
Speed of sound
Temperature
Time
Thermal conductivity
Intensity
Specific heat capacity
Heat capacity per unit volume
Heat generation term from acoustic energy lost
Particle wave velocity
Thermal diffusivity constant
Complex number i
Frequency of the ultrasound
Attenuation factor constant
Displacement of the transducer
Wave number
5
V.
a)
System Configuration of Our Model
The target region was heated using an ultrasound transducer, which emitted acoustic
energy in the form of pressure waves. As the waves passed through the phantom to
the target the focal zone, energy was lost. We equated the acoustic energy lost from
the pressure wave to the thermal energy gained by the phantom. This multi-physics
problem solved for the temperature change in the phantom over time.
b)
6
Figure 1. a) 3D representation of the system. The geometry of the focal zone was
approximated to be an ellipsoid with dimension: 7mm x 1mm x 1mm [1]. b) 2D axisymmetric
representation of FUS heating of phantom in cylindrical coordinates. The transducer's
diameter of the curve was 80mm.
To reduce computation time, we used a 2D axisymmetric geometry instead of 3D. The
transducer was modeled as a truncated section of a sphere when implemented into
COMSOL with a diameter of 80 mm. The base of the transducer had a diameter of 50
mm across and top of the transducer had a diameter of 33.5 mm across [1]. The focal
region, which represents the tumor region, is highlighted in our model and is where we
hypothesize to see the greatest amount of temperature change. Overheating this region
and its surroundings could lead to cell damage or unwanted drug release.
7
VI.
Governing Equations
a. Ultrasound Equation
i. Details of Ultrasound Equation
In order to model ultrasound physics, the Helmholtz equation in cylindrical
coordinates was applied over the whole domain. Complex valued equations
for speed of sound and density can be found in the Appendix.
𝜕
𝑟
𝜕𝑝
𝜕
1
𝜔 2 𝑟𝑝
𝜕𝑝
[− 𝜌 (𝜕𝑟 )] + 𝑟 𝜕𝑧 [− 𝜌 ( 𝜕𝑧 )] − [(𝑐 ) ] 𝜌 = 0
𝜕𝑟
𝑐
𝑐
𝑐
𝑐
(Eqn. 1)
ii. Assumptions, simplifications and rationale
We assumed that the wave propagation from the ultrasound transducer was at
steady state. Additionally, we assumed that the wave propagation was linear
so that each successive wave was identical to the previous. We also assumed
the variables to be accurately defined by the equations in the Appendix based
of literature [1].
b. Heat Transfer Equation
i. Details of Heat Transfer Equation
The heat equation in cylindrical coordinates was solved to find the
temperature changes in the phantom. The heat source term was defined as the
ultrasound attenuation.
∂T
∂t
1 ∂
∂T
∂2 T
Q
= κ [ r ∂r (r ∂r ) + ∂z2 ] + C
v
(Eqn. 2)
Where:
k
ρCP
= κ [m2 /s] is the thermal diffusity constant
ρCP = Cv [J/m3 K] is the heat capacity per unit volume
Q [J ] is the heat gain from the ultrasound attenuation
𝑣0 [m/s]is the particle wave velocity
I [W/m2 ] is the intensity
Q = 2αI = αρc𝑣0 2
(Eqn. 3)
ii. Assumptions, simplifications and rationale
We eliminated the effects of perfusion in the target area because there is no
blood flow in the phantom. Additionally, we made the same assumption as
other papers that heat gain to the phantom is directly related to the acoustic
energy loss from the ultrasound [1].
8
VII.
System Conditions
a. Ultrasound Equation
i. Details of Ultrasound Boundary and Initial Conditions
The flux is 0 at side 1 because of the axisymmetric geometry. Flux is constant
at the edge of the transducer due to of an acceleration condition on the
transducer of −𝑑0 𝜔2 , which represents the surface intensity [5]. All
remaining sides of the domain were set to be absorbing boundaries, also
known as perfectly matched layers (PML). This ensures that flux in equals
flux out. The initial pressure over the whole domain was set to atmospheric
pressure, 101325 Pa, similar to existing literature experimental results [1].
ii. Assumptions, simplifications, and rationale
A PML boundary condition is important so that waves don’t bounce off the
edges of the domain and interfere with the physics of heating; this condition
makes the solution simpler.
b. Heat Transfer Equation
i. Details of Heat Transfer Boundary and Initial Conditions
Similar to the ultrasound conditions, the flux is 0 at side 1 because of the
axisymmetric geometry. All other sides of the heat transfer domain were
set to a constant temperature of 21C. The initial temperatures of the
phantom and water domains were 21C.
ii. Assumptions, simplifications, and rationale
We assumed that temperature outside the phantom in the water bath was at a
constant 21C. This is because the water domain is much larger when
compared to the size of the phantom.
VIII. System Discretization
a. Time Step and Solver Configuration
We used a direct solver to compute both physics at a time step of 0.25 seconds.
We chose these configurations in order to balance computation time and solution
accuracy.
b. Ultrasound Mesh
i. Details of Ultrasound Mesh
In order to obtain an accurate mesh for ultrasound properties, we defined the
size of mesh elements to depend on the frequency of the ultrasound transducer
and speed of sound in the medium [9]. The entire geometry was finely
meshed, excluding the boundary conditions, because that is where we
expected to see the largest amounts of change.
9
Figure 2. Mesh for ultrasound physics. The yellow-boxed regions represent
regions of the PML boundary conditions, which are coarsely meshed as compared
to the rest of the phantom.
ii. Assumptions, simplifications, and rationale
Although we assume that the ultrasound is running at steady state, the
sinusoidal nature of the acoustics pressure changes would prevent an adaptive
meshing from working. Therefore, the mesh was created manually as per the
above description.
c. Heat Transfer Mesh
i. Details of Heat Transfer Mesh
The heat transfer mesh is less fine because the changes in temperature are not
as large as the changes in pressure. For the purposes of our project, we defined
a mesh that is finer around the focal zone of the phantom, which is the area
that we hypothesized the largest temperature gradients would occur.
10
Figure 3. Mesh for heat transfer physics. Note that only the phantom
region is meshed due to the assumption that the water surrounding the
phantom was maintained at a constant temperature.
ii. Assumptions, simplifications, and rationale
Since this is a transient problem, and adaptive meshing is limited to only
steady state problems, the mesh was created manually. Mesh elements are
smaller at the area near the focal zone, where a greater temperature change
was expected. Only the phantom region is meshed because we are only
focusing on heat transfer and temperature change within the phantom. The
water was considered to maintain a constant 21°C temperature, which was
also used as a boundary condition.
IX.
Validation
The temperature profile plots from experimental and simulated data values retrieved
from literature revealed a change of approximately 2.5˚C from the initial temperature
to the peak of the graph [1]. In our model we were able to achieve a very similar rise
in temperature of 2˚C. The curvature of our modeled results is very similar to that of
empirical data from the literature. These similarities make us confident in our model.
11
a)
b)
Figure 4. a) Temperature profile plots from both experimental and simulated values obtained
from literature [1]. The studies obtained around a maximum 2.4˚C change from the initial
temperature and reached a .5˚C total change after 50 seconds. b) Plot of temperature rise at
various distances from the focal region (z=.03, r=0) under 10s duration of continuous FUS
exposure. Our model obtained a similar ~2˚C temperature increase and overall 0.3˚C change
after 50 seconds, making us confident our simulation is functioning correctly.
12
X.
Results
a. Pressure Acoustics
This study revolves around focusing the ultrasound waves in the focal zone to
maximize its heating, while sparing other regions. We anticipated that the largest
acoustic pressures would occur within the focal zone due to the waves converging.
Our model confirmed this finding, with areas of high and low pressure confined
almost exclusively to the focal zone. By ensuring the ultrasound transducer is
focused on the correct area in our geometry, we were able later on optimize the
attenuation induced heating with confidence.
Figure 5. Total acoustic pressure field of the phantom and water domains at
steady state. The acoustic pressure waves are focused on the focal region with
minimal pressure field changes in the rest of the phantom and water.
b. Temperature
As mentioned previously, we are interested in only heating the focal region of the
phantom and wanted to minimize any temperature increase in the area surrounding
it. The temperature contour map shows the greatest amount of temperature change
in the focal region and minimal temperature increases in the non-focal region.
13
Figure 6. Temperature contour map of the phantom after 10s of FUS exposure.
Isotherms show that the max temperature occurs at the desired focal region and
radiates outward within the phantom in a relatively uniform fashion.
Figure 7. Temperature change over time with an input PRF of 2.5 Hz and 20% duty
cycle. Temperature changes are the greatest in the focal region (z=.03, r=0), implying
more heat generation, and heating decreases farther away from this region.
14
Note that the temperature increase is not constant because the ultrasound
transducer is creating pressure waves in pulses. Each peak in temperature is
related to a pulse of ultrasound emitted from the transducer wave propagation
pattern. Regardless of the pattern of the temperature profile, at the end of a 50s
non-continuous wave exposure, with non-optimized values for duty cycle and
PRF, there is about a 4˚C temperature change at 2mm away from the center, a
5.5˚C temperature change 1mm away from the center, and about 8˚C temperature
change at the center. These are promising results that show that our model has the
most temperature change within the focal zone.
c. Mesh Convergences
i. Ultrasound Mesh Convergence
The ultrasound mesh convergence results indicated that the mesh
converged after 50,000 elements. It is necessary that the ultrasound
physics has a dense mesh because of the complexity of wave propagation
through the phantom.
3.00E+07
Pressure (Pa)
2.50E+07
2.00E+07
1.50E+07
1.00E+07
5.00E+06
0.00E+00
0
20000
40000
60000
80000
Mesh Elements
Figure 8. Ultrasound mesh convergence for pressure acoustics. The solution
converges after 50,000 mesh elements.
15
ii. Heat Transfer Mesh Convergence
The heat transfer mesh’s solutions converge after 2,000 mesh elements. This
mesh is much coarser when compared to the ultrasound mesh because we
expected smaller temperature changes as compared to pressure changes.
While there is fine meshing at the focal region, the focal region is very small
when compared to the rest of the phantom, so this specificity did not add too
many mesh elements to the overall solution.
Figure 9. Temperature in the focal region as a function of time was obtained
from using a duty cycle of 10% for 50 seconds, as opposed to continuously
applying the ultrasound for 10 seconds. The solution converges after 2,000
mesh elements.
d. Sensitivity Analysis
i. Details of Sensitivity Analysis
To ensure our model is accurate, we analyzed the sensitivity of our results to
the following parameters in both the phantom and water: thermal
conductivity, specific heat capacity, density, speed of sound, transducer
displacement amplitude, and the attenuation coefficient.
16
1.2
Phantom
Change in Temperature (°C )
1
Water
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
+10%
-10%
Conductivity
[W/(mK)]
+10%
-10%
Specific Heat
[J/(kg*K)]
+10%
-10%
Density
[kg/m^3]
+10%
-10%
+10%
-10%
Speed of Sound Displacement
[m/s]
[nm]
+10%
-10%
Attenuation
[dB/cm]
% Change in Value
Figure 10. Resulting temperature change after a +/- 10% parameter difference in water
and phantom properties. The resulting temperature was taken at the center of the focal
zone at 9s of a 10s continuous wave exposure. Density, transducer displacement, and
attenuation have a larger effect on temperature change in the phantom, our domain of
interest.
Baseline values for the aforementioned parameters can be found in the
Appendix. In the phantom, changing the density, displacement, and
attenuation showed the most variability in results. Since the heat generation
term is determined from the acoustic energy lost from the water to the
phantom, changes in these mentioned parameters would affect the heat
generation the most, thus affecting the resulting temperature in the focal zone.
ii. Assumptions, simplifications, and rationale
In water, thermal conductivity, specific heat, and attenuation showed little to
no variation from the baseline when changed +/- 10%. The insensitivity may
have resulted because of too small a change in parameters or because heat
transfer was only modeled within the phantom, not the water. We applied a
range of +/- 10% change from our baseline parameters, not a Monte Carlo
distribution, which would have been more comprehensive but taken much
longer. A normal distribution of material properties is difficult to approximate
for human tissue properties. Within the phantom, the material properties are
17
homogenous, so for these simplified purposes we assumed that a range
sensitivity analysis would be acceptable.
e. Optimization
In order to optimize the delivery of drug to the focal zone, we developed an
optimization function that assigned positive (good) or negative (bad) scores to the
heating of each region. In the focal region, our goal was to maximize the area that
experienced a 2-7˚C temperature increase. This range represented the heating
required to break the liposomes and the release the drug. Drug delivery to the focal
zone is our primary goal so the score for this temperature range was weighted
heavily. Prior research demonstrated that the rate of drug delivery increases with
increasing temperature, thus we constructed a linear relationship between
temperature and optimization score in this region [10]. Above 28˚C the heating
induces vascular damage and decreases blood flow which limits drug delivery
[11]. In order to minimize the amount of focal zone that was overheated, we
assigned a score that became increasingly negative as the temperature exceeded
28˚C. Our second goal was to minimize the heating outside the focal region.
Specifically, we wanted to minimize the portion of the non-focal zone that had a
temperature increase greater than 2˚C. Exceeding 2˚C would cause liposomes in
the non-focal area to burst and delivery drug to the healthy tissue in a real patient.
Therefore, the function score is 0 below 23˚C and -1 above it.
12
Optimization value
10
8
6
Focal Zone
4
Non-focal Zone
2
0
-2
-4
-6
20
22
24
26
Temperature (°C)
28
30
32
Figure 11. Plot of our optimization function in both the focal and non-focal
regions. These two functions were later summed over their respective nodes for
optimization.
18
The objective functions are defined below:
Jfocal
0 for T < 23°C
T − 23
) for 23°C ≤ T ≤ 28°C
= {5 (1 +
5
−1(T − 28) for T > 28°C
0 for T ≤ 23°C
Jnon−focal = {
−1 for T > 23°C
Each of these functions was applied to their respective nodes in the domain and
then summed using the following equation:
Jtotal = ∑Focal nodes Jfocal + ∑Non−focal nodes Jnon−focal
(Eqn. 4)
A parametric sweep was conducted, calculating the value of Jtotal for combinations
of 10-30% for duty cycle and 1-3Hz for PRF.
-125
PRF=1
PRF=1.5
PRF=2
PRF=2.5
-175
Jtotal
-225
-275
-325
-375
-425
0.1
0.15
0.2
0.25
0.3
Duty Cycle (%)
Figure 12. Plot of variability in optimization score with different PRF and duty cycle in our
model. The maximum occurred at a duty cycle of 20% and a PRF of 2.5 Hz
After varying duty cycle from 10-30% and PRF from 1-3 Hz, we found the
maximum of our optimization function to occur at a duty cycle of 20% and PRF of
2.5 Hz.
19
XI.
Limitations
Our model had many constraints both because of our initial geometry and our use of a
tissue-mimicking phantom. We used a simple 2D axisymmetric geometry of a
rectangle rotated about an axis to make our phantom a cylinder. This shape is not
common among organs or tissues. Because we modeled FUS heating in a phantom
and not in a tissue, many of our parameters were not realistic. For example, tissue is
not by any extent homogenous. Such properties as specific heat capacity, density,
attenuation coefficient, and thermal conductivity would be different in different tissue
types and would differ throughout the tissue domain. Developing a model to represent
these variations in tissue properties would be very complex.
We were also constrained by some of the assumptions we made during our modeling.
We assumed that the initial pressure of the domain was atmospheric while actual
pressures within the body can vary depending on the location of the tumor. We also
did not include a perfusion term in the heat transfer equation, which would account
for heat loss or gain from blood flow in the tumor. Perfusion could have a large effect
on the heating of the tumor region, allowing for more heat transfer from the focal
zone to the surrounding tissue. Additionally, this could introduce vascular system
damage to the surrounding tissues of the focal region, leading to more systemic
problems. Finally, we assumed that the transducer was operating at steady state while
it is likely that there were transient effects, particularly when the transducer was
turned on and off to generate pulses.
XII.
Conclusions/Discussion
a. FUS Heating
Our model proves that implementing FUS heating for targeted drug delivery is a
promising avenue for treating cancer. While more complex models are necessary
before pre-clinical and clinical trials can be done safely, our results confirm that a
COMSOL implementation of FUS can effectively heat a focal zone within a larger
domain. We found that a duty cycle of 20% and PRF of 2.5 Hz optimized drug
delivery in our model, and these should be kept constant in subsequent models
with consistent geometries and other parameters. Due to variations in geometry
and tissue composition the optimized parameters would not necessarily be
applicable for all other models. However, our optimized variables could be useful
for designing an algorithm for moving the focal zone within a larger tumor to
achieve even heating.
b. Future Improvements
Future models should use more realistic tumor geometries than our cylinder
geometry. These geometries can be retrieved from medical imaging such as CT
scans of solid tumors in different locations of the body.
20
Figure 13. Theoretical model of the application of FUS induced heating for
subcutaneous xenograft tumor model. A more realistic tissue model for FUS
targeted drug delivery in cancer therapies.
Another way to improve FUS heating modeling is to model more realistic tissue
rather than a phantom. By modeling tissue, the results would take into account
tissue heterogeneity such as different heat conduction, attenuation, and density
values in the tissue, skin, and muscle. Moreover, real tissues include perfusion
terms, which account for heat being brought into and out of the system by blood
flow.
In closing, while modeling FUS induced heating for the purpose of targeted drug
delivery has been proven to be effective, further models and studies are required
before implementation of this technique in preclinical trials.
21
XIII. Appendix A.
Table 2. Input parameter values retrieved from literature
Parameter
𝑇0
𝑓0
𝑑0
Value
294 K
1 MHz
14 nm
Description
Initial Temperature
Source Frequency
Displacement Amplitude of Transducer
Source
[1]
[1]
[5]
Table 3. Material property values retrieved from experiments in literature
Property
Value
Description
0.0022 dB/(cm*MHz)
Attenuation Coefficient of Water
𝛼𝑤𝑎𝑡𝑒𝑟
0.104 dB/(cm*MHz)
Attenuation Coefficient of Phantom
𝛼𝑝ℎ𝑎𝑛𝑡𝑜𝑚
1500 m/s
Speed of Sound in Water
𝑐𝑤𝑎𝑡𝑒𝑟
1544 m/s
Speed of Sound in Phantom
𝑐𝑝ℎ𝑎𝑛𝑡𝑜𝑚
4200 J/(kgK)
Specific Heat in Water
𝐶𝑤𝑎𝑡𝑒𝑟
4000 J/(kgK)
Specific Heat in Phantom
𝐶𝑝ℎ𝑎𝑛𝑡𝑜𝑚
0.61 W/(mK)
Thermal Conductivity in Water
𝑘𝑤𝑎𝑡𝑒𝑟
0.55 W/(mK)
Thermal Conductivity in Phantom
𝑘𝑝ℎ𝑎𝑛𝑡𝑜𝑚
Source
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
Table 4. Equations for complex variables used in governing equations
Equation
Value Description
Source
𝜔
Angular frequency
𝜔 = 2𝜋𝑓
𝑘𝑤
Wave number
𝑘𝑤 =
𝛼
𝜌𝑐
Attenuation Coefficient [dB/cm/MHz]
Complex-Valued Density [kg/𝑚3 ]
𝛼 = 𝛼0 𝑓 𝑦
𝑐𝑐
Complex-Valued Speed of Sound [m/s]
𝜌𝑐 =
𝜔
𝑐
𝜌𝑐 2
𝑐𝑐 2
𝜔
𝑐𝑐 = 𝑘
− 𝑖𝛼
[5]
[5]
[8]
[5]
[5]
𝑤
Figure 14. Visualization of the Pulse Repetition Frequency (PRF) pattern range. We
varied the PRF in our model to optimize heating in the focal region of our phantom.
22
Figure 15. Visualization of a continuous
ultrasound wave pulse for 1s. This wave pulse was used when obtaining general pressure
acoustics and temperature contour plots.
XIV. Appendix B.
Figure 16. CPU time taken and memory being used by heat transfer validation run on
COMSOL. The heat transfer physics took a significant amount of time to run because the
ultrasound physics was required to be solved simultaneously at every time point in order
to incorporate the heat generation term.
23
Figure 17. CPU time taken and memory being used by ultrasound validation run on COMSOL.
The ultrasound physics did not take a significant time to run because it was at steady state.
24
XV.
References
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