Cent. Eur. J. Phys. • 6(2) • 2008 • 211-222
DOI: 10.2478/s11534-008-0015-3
Central European Journal of Physics
Finite-element simulation of ultrasound brain
surgery: effects of frequency, focal pressure, and
scanning path in bone-heating reduction
Research Article
Sohrab Behnia1∗ , Amin Jafari2 , Farzan Ghalichi3 , Ashkan Bonabi1
1 Department of Physics, IAU, Urmia, Iran
2 Department of Physics, Urmia University, Urmia , Iran
3 Department of Biomedical Engineering, Sahand University, Sahand, Iran
Received 29 May 2007; accepted 17 December 2007
Abstract:
In this paper, the finite-element method (FEM) simulation of ultrasound brain surgery is presented.
The overheating problem of the post-target bone, which is one of the limiting factors for a successful
ultrasound brain surgery, is considered. In order to decrease bone heating, precise choices of frequency,
focal pressure, and scanning path are needed. The effect of variations in the mentioned scanning
parameters is studied by means of the FEM. The resulting pressure and temperature distributions of
a transdural ultrasound brain surgery are simulated by employing the FEM for solving the Helmholtz
and bioheat equations in the context of a two-dimensional MRI-based brain model. Our results show
that for a suitable value of the frequency, an increase in focal pressure leads to a decrease in the
required duration of the treatment and is associated with less heating of the surrounding normal tissue.
In addition, it is shown that at a threshold focal pressure, the target temperature reaches toxic levels
whereas the temperature rise in the bone is minimal. Wave reflections from sinus cavities, which result
in constructive interference with the incoming waves, are one of the reasons for overheating of the bone
and can be avoided by choosing a suitable scanning path.
PACS (2008): 87.50.yt, 43.35.Wa, 02.70.Dh
Keywords:
ultrasound brain surgery • bone overheating • finite-element method • Helmholtz equation • bioheat
equation
© Versita Warsaw and Springer-Verlag Berlin Heidelberg.
1.
Introduction
∗
E-mail: [email protected]
One of the most complicated tumor-removal surgeries is
that of brain tumors. Brain surgery requires precise and
sophisticated instruments and is associated with healthytissue damage. This can lead to quite complicated adverse
effects after treatment. The presence of the blood–brain
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Finite-element simulation of ultrasound brain surgery: effects of frequency, focal pressure, and scanning path in bone-heating reduction
barrier makes the delivery of drugs very difficult [1]. Also,
conventional methods have thus far failed to control tumor progression [2–4]. Because of the sensitivity of brain
tissue and its central role, minimally invasive and noninvasive procedures are desired. Compared to open surgeries, such approaches offer the advantage of reducing (a)
the surgery time, (b) the tissue damage associated with
surgery, and (c) transfusion requirements and their associated infection risks [5]. The results are a shorter recovery
time and hospital stay, a reduction in the cost of health
care, and a generally superior therapeutic outcome [5]. In
this respect, high-temperature thermal therapies are becoming increasingly acceptable. The methods applicable
in this context are high-intensity focused ultrasound, microwaves, radio-frequency currents, and lasers. Among
these, ultrasound has the advantage of accurately focusing energy into the body [6], a favorable range of energy
penetration [8], the ability to shape power depositions [7],
and the technical feasibility of constructing applicators of
any size and shape [8]. However, high acoustic absorption at bone interfaces and reflections from gas surfaces
may make certain clinical sites difficult to treat with ultrasound [7]. Ultrasound surgery of the brain has the advantage of being noninvasive and destroying just the tumor
while leaving the normal brain tissue intact. Therefore,
it can minimize the side effects that would be associated
with other treatments.
This method was first applied to target areas in the brains
of cats, dogs, and monkeys in the 1940’s [9, 10]. Due to
the large difference between the skull high acoustic velocity (about 3000 m s−1 ) and the brain acoustic velocity
(about 1500 m s−1 ), severe distortions in the beam shape
occured and prevented an effective focusing through an
intact skull [1]. Early brain treatments with focused ultrasound were therefore conducted through an acoustic window by removing a piece of the skull bone (craniectomy).
The aforementioned technique demonstrated the ability
of ultrasound to create deep lesions in the brain [11–13].
Following these studies, focused ultrasound was applied
through an intact dura after a craniectomy to treat patients with Parkinson’s disease [14]. Subsequently, this
method was clinically used to treat brain tumors [15, 16].
After the development of MRI-guided focused ultrasound,
which permitted the monitoring of temperature changes
during the treatment, precise image-guided targeting and
destruction of tumors became feasible. Taking advantage of this technique, craniectomy-based high-intensity
focused-ultrasound treatments were carried out in Rhesus
monkeys [17]. Focal heating was observed by MRI-derived
temperature imaging. Histological examination and MRI
images acquired immediately after ultrasound exposures
showed the focal lesions created by HIFU [17]. Recently,
the method of using MRI to guide and monitor focused ultrasound was employed to treat brain tumors in three patients [18]. These results have established the method as
a potentially effective procedure to destroy tumor tissue.
However, due to the required invasive surgery (e.g., the
creation of a bone window) there is an associated morbidity. A similar procedure was performed to treat one patient
with a malignant brain tumor [19]; the results have shown
some evidence of tumor coagulation. Precise coagulation
necrosis of specified targets has also been achieved after
a HIFU treatment of the brains of ten pigs [20]. All of the
brain treatments discussed above were performed through
an acoustic window after a craniectomy.
Recent studies have shown that the delivery of HIFU
through an intact skull is also feasible [1, 21–23]. The
method is based on using large-scale phased arrays
and correcting the phase abberations induced by the
skull bone by applying phase corrections to each element [1, 21–23]. The feasibility of this method has
been discussed in Ref. [1]. The required optimum sourceelement width was estimated for each frequency, and an
appropriate frequency range for trans-skull ultrasound
brain surgery was determined [21]. Based on the obtained results and on the numerical model in Ref. [21],
a large-scale phased array was manufactured and tested
both numerically and experimentally [22]. The results were
promising, and they demonstrated that practical and completely noninvasive ultrasound surgery is possible [22].
These results have therefore opened the way for the development of clinical phased-array systems; their application
has great potential in the near future. This technique has
now reached preclinical testing on primates [23]. A preclinical test was performed on three Rhesus monkeys using a 512-channel phased-array system under MRI guidance [23]. This research has demonstrated that adequate
acoustic intensity can be delivered to the brain. However,
the limiting factor in these studies has been the overheating of the brain surface.
The two aforementioned procedures for ultrasound surgery
have their specific benefits and drawbacks. The drawback
of the first method is that it requires bone removal from the
skull, which makes the treatment invasive. However, it is
possible to target a wider region inside the brain, to utilize
a wider frequency range for the treatment, and to shorten
the duration of the treatment as higher powers can be delivered to the target. The advantage of the second method
is its complete non-invasive nature. However, due to the
high acoustic absorption in the skull, larger transducers
must be used. In addition, skull heating limits the ultrasound power available for the focal-tissue coagulation and
may prevent coagulation of locations close to skull with
a single sonication [23]. Also, the frequency range of the
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treatment is limited to frequencies below 1 MHz.
During ultrasound surgery of the brain, overheating of the
post-target bone or the normal brain tissue can be problematic [17, 20, 24]. This issue can be ameliorated by
choosing a suitable frequency, focal pressure and scanning path. It is therefore necessary to study the effects of
these parameters on the treatment.
This work presents a theoretical investigation of this question employing the finite-element method (FEM). The reason for choosing the FEM as a tool for simulations is because of the inhomogeneity of the model and the ability
of the FEM to utilize nonuniform grids allowing for a high
rate of spatial sampling in specific areas of interest [25].
We simulate the transdural propagation of ultrasound from
an ultrasonic apparatus into an MRI-based brain model
and determine the resulting temperature by solving the
Helmholtz and bioheat equations with the aid of the FEM.
The effects of variations in frequency, scanning path, and
focal pressure on a sample target in the model are studied, and the choice of a suitable frequency, focal pressure,
and scanning path is discussed.
2.
Simulation methods
The simulations of this study are divided into three steps:
First, a two-dimensional model was constructed from a
brain MRI. Then a piece of the skull bone was removed to
create an acoustic window in order to allow ultrasound to
propagate inside the brain. Second, the pressure distribution was determined by solving the Helmholtz equation
with the FEM. Third, the temperature evolution resulting
from the associated ultrasound absorption was calculated
with the bioheat equation, which again involved the FEM.
It should be noted that using a two-dimensional model can
impose some limitations. The justification for this particular approach will be provided in other sections.
2.1.
Geometry of the model
Different boundaries of a sagittal brain MRI consisting of
the skull, brain tissue, cerebrospinal fluid, the frontal sinus, and the sphenoidal sinus were segmented, and the
FEM model for the simulations was created. Then, a
piece of the skull bone was removed, and an acoustic window for placing the transducer was created. A large-size
craniectomy was considered in order to study the effect
of variations in the scanning path. However, the actual
craniectomy to be performed in clinical treatments would
be smaller because a suitable scanning path can be chosen. Fig. 1 shows the model with a piece of the skull bone
removed.
Figure 1.
Model with craniectomy.
2.2. Wave-field calculation and the Helmholtz
equation
The calculation of the wave field is the crucial part of the
simulation, as it determines the spatial power distribution. Simple methods that are just capable of modeling
the propagation of waves in homogeneous media, such as
the Rayleigh integral [26], are insufficient in the case of the
brain. This arises because of the presence of tissues with
significantly different acoustical parameters (e.g., brain,
bone, and sinuses): these differences typically lead to
diverse patterns of absorptions, reflections, and possibly
interferences at the brain–bone interface, bone–gas interface, or inside the brain.
To appropriately model the wave field in inhomogeneous
media, full wave methods are necessary. These methods
are capable of determining the wave field in complex geometries, and they take into account wave reflections and
the generation of possible wave interferences. An alternative to full wave methods would be a modification of the
Rayleigh integral to account for variations in the acoustical parameters of multi-layer media [21]. This numerical model is able to calculate ultrasound-wave absorption,
diffraction, reflection, and refraction. It takes into account
the acoustic reflection and refraction at layer interfaces
but not the multiple reflections and reverberations in each
layer [21]. The advantage of this model is a reduction in
the large and intensive numerical loads associated with
the full wave method leading to rapid calculations and a
decrease in required memory. However, this method is
unable to simulate the standing-wave formation between
the incident and reflected beams. As will be shown in
this paper, standing-wave formation is one of the reasons
for severe bone overheating at the location of the sinus
interface.
The time-harmonic ultrasound field is the solution of the
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Finite-element simulation of ultrasound brain surgery: effects of frequency, focal pressure, and scanning path in bone-heating reduction
Figure 2.
(a)
(b)
(c)
(d)
Pressure distribution at frequencies of 300 kHz (a), 600 kHz (b), 800 kHz (c), and 1.0 MHz (d). The maximum focal pressure is
1.5 MPa.
Helmholtz equation in an inhomogeneous medium [26]:
1
k2 p
∇·
∇p +
=0,
(1)
ρ
ρ
where p is the pressure, ρ the density, and k the wave
number. In a dissipative medium, the wave number k is of
the form
f
k = 2π + iα ,
(2)
c
where α is the absorption coefficient, f is frequency and
c is the speed of sound [27]. The calculation of the wave
field is performed by applying the FEM to solve Helmholtz
equation (1).
2.3.
Thermal model and bioheat equation
The temperature elevation resulting from the absorption
of ultrasound in the tissue was simulated in our model by
solving the transient bioheat equation [28]:
ρ cT
∂T
= ∇ · (kT ∇T ) − ωB cB (T − TA ) + Q ,
∂t
(3)
where t is time, T is the temperature, cT the heat capacity of the tissue, kT the tissue’s thermal conductivity and
TA is the arterial blood temperature. The product ωB cB
denotes the blood-perfusion term and Q the heat source
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Sohrab Behnia, Amin Jafari, Farzan Ghalichi, Ashkan Bonabi
Figure 3.
(a)
(b)
(c)
(d)
Temperature distribution at frequencies of 300 kHz (a), 600 kHz (b), 800 kHz (c), and 1.0 MHz (d). The maximum temperature of
the simulation reaches 56◦ C.
term. Since metabolic heat generation is negligible, Q
is determined by the absorbed ultrasound energy in the
tissue given by [26]:
Q=
α
|p|2 .
ρc
(4)
Here, α is the absorption coefficient, ρ the density, c the
speed of sound, and p the pressure.
3.
Simulation procedure
A sample target was considered at a depth of 3.5 cm inside the center of the brain. This location is suitable for
the study of different sonication parameters because it is
accessible from different angles. After the craniectomy, a
transducer with a 6 cm radius of curvature and a diameter
of 8.9 cm was placed at the top of the acoustic window.
The location of the transducer was adjusted such that its
focus was localized on the target. The simulations were
organized into the following three groups: first, the search
for a suitable frequency value for the sample target; sec215
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Finite-element simulation of ultrasound brain surgery: effects of frequency, focal pressure, and scanning path in bone-heating reduction
Figure 4.
(a)
(b)
(c)
(d)
Temperature distribution at focal pressures of 1.8 MPa (a), 2.0 MPa (b), 3.0 MPa (c), and 4.0 MPa (d). The frequency is 800 kHz.
ond, the study of effects on the temperature distribution
resulting from variations in the focal pressure; third, the
search for an appropriate scanning path associated with
a safe heating of the target.
The simulations were performed on a PC with a dual processor CPU (3.2 GHz), 4 GB of RAM, and a Linux operating system (SuSE 10). The calculations of the wave field
and the temperature elevation for a ten second sonication
lasted 382.574 s for the lowest frequency (which requires
the minimum mesh density) and 1254.821 s for the highest
frequency (which requires the maximum mesh density).
3.1. Suitable frequency value for the sample
target
The amount of beam absorption is proportional to the
frequency: higher beam frequencies result in increased
absorption in the medium. Thus, higher frequencies are
needed for safely heating targets located near a bone surface. By increasing the frequency we can increase the attenuation of the beams between the target and the bone
and therefore lower the amounts of energy that reach the
post-target bone. However, increasing the frequency is
not always the best choice because overheating of the
normal tissue between the transducer and the target can
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Sohrab Behnia, Amin Jafari, Farzan Ghalichi, Ashkan Bonabi
Table 2.
Acoustical parameters used in the simulations [34–36].
tissue c [m s−1 ] ρ [kg m−3 ]
brain 1545
Table 3.
1030
bone 2652
1796
water 1500
1000
air
1.16
383
Thermal parameters used in the simulations [35–39].
tissue cv [J kg−1 K−1 ] kT [W m−1 K−1 ] ωb [kg m−3 s−1 ]
Figure 5.
Table 1.
Temperature difference between the target and the posttarget bone versus pressure at a target temperature of
56◦ C.
Attenuation in bone versus frequency [32, 33].
Frequency
(MHz)
0.3 0.6 0.8 1.0
Attenuation
(N p m−1 )
20 52 86 124
occur [24].
We have examined the frequency range between 300 kHz
and 1 MHz in order to find a suitable frequency for heating the sample target. When using the FEM to solve the
Helmholtz equation the density of discretization points
should be ten points per wavelength in order to achieve a
reasonable accuracy (i.e., λ/h = 10, where λ is the wavelength and h the side length of a finite element). Also,
the density of discretization points per wavelength must
increase with the wave number to maintain a fixed level
of accuracy [29]. In this paper, the maximum element size
of meshing was chosen to satisfy λ/h = 12 in each case.
Moreover, the same mesh pattern was used for solving
the bioheat equation. For the study of suitable frequency
values, the pressure distribution was chosen to produce
a peak focal pressure of 1.5 MPa below the 300 kHz
brain 3640
0.528
10
bone 1300
0.630
0
water 4180
0.615
0
air
0.0263
0
1007
cavitation threshold for the lowest simulation frequency.
The measured cavitation-threshold pressure amplitude in
Ref. [30] was found to depend almost linearly on the frequency with a slope of 5.3 MPa MHz−1 . The calculation
of the wave fields was carried out by using the acoustical
parameters listed in Table 2. The attenuation coefficient of
brain tissue was chosen to be 7.5 N p m−1 MHz−1 [31, 32]
and those for bone are listed in Table 1 [31, 33]. The
resulting temperature distributions were calculated employing the properties listed in Table 3.
3.2. Effects of focal-pressure variations on
the temperature distribution
The heat deposited by the ultrasound source is directly
proportional to the square of the focal pressure. For overcoming the effect of high blood perfusion, the value of the
focal pressure should therefore be high enough. In addition, cavitational effects should be avoided. To study
the effect of variations in the focal pressure, the simulations were carried out with a frequency of 800 kHz in the
focal-pressure range of 1.5–5.5 MPa.
3.3.
Study of the appropriate scanning path
The incident angle of the beams, the geometry of the posttarget bone, and the sinuses at the location of the inci217
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Finite-element simulation of ultrasound brain surgery: effects of frequency, focal pressure, and scanning path in bone-heating reduction
Figure 6.
(a)
(b)
(c)
(d)
Temperature distribution at scanning angles of −10◦ (a), −20◦ (b), +10◦ (c), +20◦ (d). The frequency is 800 kHz, and the focal
pressure is 1.8 MPa.
dence are the parameters that determine the patterns of
reflection, absorption, and possible interferences. For this
reason, the choice of an appropriate scanning path can
minimize unwanted effects. In this paper, the effects of
variations in the scanning path were studied by the rotation of the transducer from −20◦ to 20◦ around its focal
point. The simulations were performed using a frequency
of 800 kHz and a constant focal pressure of 1.8 MPa.
4.
Results and discussion
4.1.
Effects of frequency changes
Figs. 2a – 2d show the pressure field corresponding to
frequencies of 0.3, 0.6, 0.8, and 1.0 MHz, respectively.
They reveal that frequency changes can have significant
effects on the pressure distribution inside the brain. As
the frequency grows, the focal region becomes smaller
and the distribution of the pressure becomes more concentrated. The figures further show that there is no penetration through the surface of the sinus, and the waves
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Sohrab Behnia, Amin Jafari, Farzan Ghalichi, Ashkan Bonabi
undergo full reflection. The largest pressure amplitude at
the post-target bone is in the region between the surface of the sinus and the bone. The periodic pattern of
the pressure distribution in this region with distinct highamplitude maxima and minima, which is best seen in the
cases with 300 kHz and 600 kHz, results from the generated standing wave. However, the value of the pressure amplitude between the sinus surface and the bone is
lower for 800 kHz and 1.0 MHz. This is because higherfrequency ultrasound attenuates faster, and the energy
of the beams reaching the bone decreases. For lower
frequencies, when higher amounts of pressure reach the
brain–bone interface, the reflected beams have larger amplitudes, and when coupled to their slower dissipation,
the pattern of reflections becomes more pronounced. This
can increase the possibility of interference between the
reflected and the incident beams.
Figs. 3a – 3d show the temperature distributions obtained
from solving the bioheat equation, where the calculated
pressure fields corresponding to frequencies of 0.3, 0.6, 0.8,
and 1.0 MHz have been used as input. We have considered the case in which the peak temperature of the target
reaches 56◦ C. As shown in Ref. [40], in only one second
the tissue undergoes necrosis at this temperature. Figs.
3a – 3d also show that the highest temperature rise at
the post-target bone is between the sinus surface and the
bone. As is best seen in Figs. 3a and 3b, the created hot
spot is located close to the surface of the sinus. In other
parts of the post-target bone, where no sinus cavity exists,
the hot spots are created close to the surface of the bone
in the vicinity of the brain–bone interface. Our results reveal that in this model at frequencies of 300 and 600 kHz
the temperature of the post-target bone exceeds that of
the target and reaches toxic levels. Using the frequencies
of 800 kHz and 1.0 MHz results in a lower temperature
rise at the bone. The temperature of the post-target bone
reaches 49◦ C and 45◦ C for 800 kHz and 1 MHz, respectively. Although the temperature generated in the bone
at 800 kHz is non-toxic, it does exceeded the pain threshold. The threshold of thermally induced pain occurs at
about 45◦ C [41]. Our results also show that the beampath temperature elevation is faster for higher frequencies
because of the increased attenuation of ultrasound; this
is also stated in Ref. [24]. This may limit the duration of
the exposure for an effective treatment at higher frequencies [42]. This arises because tumor treatment typically
requires multiple sonications. But in the case of higher
frequencies, multiple sonications cannot be performed one
after another without any time delay. In this situation,
there should therefore be a delay between each sonication, in order to let the tissue cool down after each sonication. On the other hand, lower frequencies allow larger
treatable tumor sizes and smaller acoustic windows [24].
Due to the increased attenuation, higher-frequency beams
should be directed to the target from a larger area. This
will spread the deposited energy in a larger region and
therefore decrease the temperature elevation in the beam
path. For this purpose, a larger acoustic window should
be created, and a bigger transducer should be used. By
choosing an appropriate frequency, one can necrose larger
tumor volumes with each single exposure, and the tissue
temperature that exceeds pain threshold can be minimized
[42]. By considering the location, depth, and size of the tumor numerical calculations can yield a frequency suitable
for reducing off-target heating, and they can also determine the smallest possible acoustic-window size ensuring
only minimal skull damage.
4.2. Variations of focal pressure and temperature distribution
Figs. 4a – 4d show the temperature distribution corresponding to focal pressures of 1.8, 2.0, 3.0, and 4.0 MPa,
respectively; the peak temperature of the target is 56◦ C.
These figures reveal that using higher focal pressures increases the sharpness of the region that reaches toxic temperatures and decreases the temperature generated in the
post-target bone and the surrounding normal tissue. Using higher focal pressures decreases the time required for
the target to gain enough energy. Therefore, the effect
of conduction and diffusion decreases. This, in turn, increases the precision of the treatment and restricts the
area of normal tissue that reaches high temperatures. The
reason for the lower temperature generated at higher intensities in the post-target bone is the following: The high
blood perfusion in the brain (≈ 10 kg m−3 s−1 ) compared to
the extremely low value in bone (≈ 0 kg m−3 s−1 ) generates
a stronger heat sink in the target than in the bone. This
heat sink grows linearly with increasing temperature, so it
reduces the total heat source at the target. This, in turn,
will decrease the rate of temperature growth. Considering the fact that the rate of temperature growth in bone
is constant and not influenced by any heat sink, the temperature evolution in bone is near that of the target. By
increasing the focal pressure, the effect of blood perfusion
becomes negligible in comparison with the external heat
source for therapeutic temperature levels (≈ 56◦ C) and
cannot affect the temperature evolution. In addition, the
decrease in treatment time prevents the bone from gaining
enough energy.
Fig. 5 shows the temperature difference between the target and the bone as a function of focal pressure when
the temperature of the target reaches 56◦ C. It is apparent that the temperature difference grows with increas219
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Finite-element simulation of ultrasound brain surgery: effects of frequency, focal pressure, and scanning path in bone-heating reduction
ing pressure. However, the rate of growth decreases with
increasing pressure and saturates at a threshold. The
generated temperature at the post-target bone is 43.9◦ C,
which is much lower than in the case of applying 1.5 MPa
of focal pressure at a frequency of 800 kHz. Therefore,
not only increasing the frequency, but also using a higher
focal pressure enhances the safety of the treatment by reducing the overheating of the post-target bone and the
surrounding tissue.
4.3.
Variations in the scanning path
Considering a maximum focal pressure of 1.8 MPa in all
of the models, we let the peak temperature of the target
reach 56◦ C. Figs. 6a – 6d show the temperature distribution for the respective scanning angles of −20◦ , −10◦ ,
10◦ , and 20◦ around the focal point. These figures show
that by changing the scanning path one can significantly
reduce the temperature of the post-target bone. The highest temperature rise at the post-target bone occurs when
parts of the beams end up at the sinus interface. Choosing
a suitable scanning path that moves the beams away from
this region, decreases overheating of the bone. This is
because the external heat source depends on the square
of the acoustic pressure, and a small degree of constructive interference between the incident and reflected beams
leads to very high acoustic heating. In our model, the best
scenario is attained by applying the scanning path of 20◦
when the overheating of the post-target bone decreases
to 42◦ C.
5.
Limitations of this study
There are some limitations to this study. The simulations were carried out employing a two-dimensional brain
model. As discussed, the mesh density required to reach
a reliable accuracy in the results is already fairly high.
The mesh density needed for higher frequencies or in the
three-dimensional case is even higher and will lead to the
problem of computational complexities that are unfeasible
to carry out with a single PC.
For thermal studies, the primary difference between the
two- and three-dimensional simulations is that in three
dimensions the heat transfer due to conduction and perfusion is larger leading to a faster cooling effect. As is
discussed in Ref. [44, 45], perfusion is the dominant factor
affecting the temperature distribution. Using shorter sonication times has shown to produce perfusion-independent
temperature elevation and, as indicated in Refs. [46, 47],
produces a thermal damage independent of the tissue’s
thermal parameters and perfusion. Adequate temperatures
in simulations were obtained for durations below 10 s, so
the cooling effects of perfusion and conduction between
the two- and three-dimensional cases become negligible.
Since these are the main sources for errors, we expect
no significant differences between the results in two- and
three-dimensional thermal simulations when the proposed
higher focal pressure and shorter durations are used.
For simulations of acoustic fields in three dimensions, the
boundary surface of the transducer is a two-dimensional
area instead of a one-dimensional line. In three dimensions, there is more versatility in the placement of the
elements, so that achieving the desired focus in the tissue
is actually easier [43]. This discussion shows that if the
treatment is predicted to be safe in two dimensions, the
actual three-dimensional treatment is likely to be safe as
well. Also, the suitable parameters determined by using
two-dimensional simulations can be applied and tested
in three dimensions, which will lead to a substantial decrease in computing time. However, in order to get quantitative results before clinical treatments, the simulations
should also be run in three dimensions, especially when
the actual treatment time is long.
6. Geometrical parameters affecting the results
Other model parameters affecting the achieved temperature elevations are the geometry of the transducer and the
depth of the target. The quantities that should be considered as parameters in this context are the geometrical dimensions of the transducer, i.e., its radius of curvature and
its diameter. These parameters determine the beam divergence past the focal point. A suitable transducer should
diverge the beams into a larger area on the post-target
bone. This can decrease the sharp temperature elevations in the bone by reducing the power that reaches a
specific point. A future study could for example address
the effects of variations in these parameters. First, variations in the diameter with the curvature radius held fixed
should be studied with the goal to characterize the effects
of diameter variations on the reduction of heating in the
post target bone. Second, variations in the curvature radius should be studied while holding the diameter fixed.
Finally, the effect of simultaneous variations in both of
these parameters should be considered; it may be possible to characterize the obtained results with the F -number
(i.e., the ratio of curvature radius to diameter F = R/d).
However, in order to study accurately the effects of variations in these parameters, homogeneous models should
be employed. In the case of using inhomogeneous models, the ending location of the beams would be different
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with different transducer geometries. Even for a transducer with optimal geometrical parameters, one may not
get the right result (as one would with homogeneous models) if the beams end at the sinus interface location. In
order not to encounter such an inconsistency, we propose
to find the best geometrical parameters using homogeneous models, and then applying them to the real case.
By choosing a suitable scanning path, we can then further reduce overheating of the post-target bone. Similar
considerations apply for studying the effect of variations
in depth for a given transducer because in this case the
beam endings move with variations in the target depth.
We also remark that due to the inhomogeneity in the geometry of the post-target bone and the presence of the
sinuses, the distance of the target to the bone cannot be
measured accurately. The stated facts suggest using homogeneous models to study the geometrical parameters of
the transducer.
7.
Conclusion
The effects of variation in frequency, focal pressure, and
scanning path on the temperature distribution have been
studied. The primary findings of this study are threefold:
First, not just variations in frequency, but also variations
in focal pressure and scanning path can lead to safer treatments by decreasing the overheating of the post-target
bone and the surrounding normal tissue. Second, when
the target temperature reaches toxic levels, a threshold
focal pressure can be found at which the temperature rise
in the bone is minimal. Third, by choosing a scanning
path such that the beam endings move away from the sinus location, one can strongly decrease bone overheating.
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