An ultrasonic transducer focusing ultrasound into a thin waveguide by two elliptical reflectors

A high-power ultrasonic transducer for minimally invasive treatments is proposed, capable of treating areas inaccessible by high-intensity focused ultrasound treatments. This transducer employs two elliptical reflectors that efficiently focus ultrasound waves into a thin waveguide by utilizing mode conversion. Additionally, a slit structure is introduced to suppress wave diffraction, further enhancing the focusing capabilities of the transducer. Finite element analysis demonstrates that the proposed transducer achieves an impressive energy density magnification factor of 325, with an energy efficiency of 24.2 %. This efficiency is 3.4 times higher than that of a conventional transducer, Double Parabolic refLector wave-guided Ultrasonic transducer.

][3][4] In the HIFU method, powerful ultrasound waves are focused on lesions to destroy them from outside the body.While a significant advantage of HIFU is its noninvasive nature, HIFU has a few limitations.Since ultrasound propagation is significantly impeded by structures with an acoustic impedance, which is very different from soft tissues, such as bones and air-filled tissues, it is challenging to treat air-filled organs like the lungs and bowels 5,6) as well as areas blocked by bones like ribs. 7,8)][11][12] In MIT, a thin instrument is inserted into the body, delivering energy directly to the lesions.[19] The advantages of ultrasound are its relatively large penetration depth and precise treatment area control. 20,21)ltrasound for MIT can be excited with a piezoelectric element inside or outside the body.In the intracorporeal type, the size of the ultrasonic transducer is limited [17][18][19] and it is therefore difficult to achieve high output power with a short treatment time.
][24][25][26][27] As shown in Fig. 1(a), the DPLUS focuses ultrasonic waves generated from a piezoelectric element using two parabolic surface reflectors with the same focal point.These waves are then introduced into a thin waveguide to radiate high-power ultrasonic vibrations from the tip of the thin waveguide.Recent analyses have revealed that the transverse waves converted at the parabolic reflector affect the output ultrasound from the thin waveguide.This phenomenon is partly due to the long distance between the second reflector and the connection point of the thin waveguide.Detailed research is ongoing.
MIT using DPLUS has been proposed where a thin waveguide is inserted into the body through a catheter and an endoscope.Experiments have been conducted to produce thermal ablation of beef fat 23) and chicken breast tissue. 26)ese experimental results indicate that DPLUS is a feasible transducer for MIT.
To realize a more powerful ultrasonic transducer for MIT, two improvements are required.The first relates to mode conversion efficiency at the wave reflections.Reflectors are used to focus the ultrasound and direct it into the thin waveguide.Mode conversion splits the incident dilatational wave into dilatational and transverse waves.DPLUS uses parabolic reflectors to focus the incident wave from the piezoelectric ring.However, it has been found that a converted transverse reflector is a better candidate with respect to mode conversion efficiency.The second improvement is to reduce wave energy diffusion during focusing.Wave diffraction can be ignored only when the wavelength is short compared to the transducer's dimensions.In the several megahertz frequency range, the ultrasound wavelength in metal is on the order of millimeters, similar to the total size of the transducer, and hence wave diffusion during focusing cannot be ignored, resulting in energy loss.
Based on these considerations, we propose a high-power ultrasonic transducer named the Double ELliptical reflector transducer for hIgh-Power ultraSound (D-ELIPS).This transducer has two elliptical reflectors for high mode conversion efficiency.Moreover, a slit structure is introduced to reduce the distance between the second reflector and the waveguide to suppress wave diffusion.This improved transducer enables efficient medical treatment by MIT in a shorter time, reducing the burden on the patient.Furthermore, the lower mechanical stress of the operation inside the piezoelectric ceramics suppresses the piezoelectric nonlinear effect, allowing higher input voltage operation.
The D-ELIPS employs two elliptical reflectors to focus the ultrasound, as illustrated in Fig. 1(b).The incident dilatational wave is introduced from the piezoelectric ring, as in the DPLUS.However, the D-ELIPS utilizes the transverse wave, converted from the incident wave at the reflection.It has been clarified that the elliptical reflector enables wave focusing through mode conversion, as shown in Fig. 2(a).The elliptical reflector also converts focused transverse waves into dilatational plane waves, serving as the second reflector depicted in Fig. 2(b).In the D-ELIPS, the first elliptical reflector reflects the dilatational wave and focuses the transverse wave towards the focal point.The second elliptical reflector then reflects the transverse wave towards the focal point, transforming it into a dilatational plane wave and efficiently introducing the ultrasound into the thin waveguide.
Elliptical focusing offers two advantages over parabolic focusing, used in the DPLUS.Firstly, it has efficient energy conversion at the reflections.The energy conversion efficiency depends on the incidence angle, and the range of the effective incidence angle is wider for dilatational-to-transverse wave conversion than for dilatational-to-dilatational wave conversion.This advantageous characteristic provides more effective focusing of the D-ELIPS.The second advantage is that transverse waves have a shorter wavelength, about half that of dilatational waves, which reduces the wave diffraction loss.Using a focused transverse wave increases the amount of energy that reaches the second reflector.
As discussed, this research utilizes mode conversion for ultrasonic focusing.In general, when a dilatational plane wave is incident on the boundary between air and a solid, dilatational and transverse waves are generated, as shown in Fig. 3(a).When the incident wave is transverse, the reflected waves are composed of transverse and dilatational waves, as depicted in Fig. 3(b).This phenomenon is called mode conversion, and the reflection angle depends on the incidence angle and the wave propagation speed.This is because the wave propagation speed of dilatational waves c d and transverse waves c t and the angle of dilatational waves a d and transverse waves a t obey Snell's Law, as given in Eq. (1).

= ( ) a c a c
sin sin 1 Due to the significant difference in acoustic impedance between the air and a solid, the energy of incident waves can be considered to be the same as that of reflected waves.
For focusing the transverse wave generated through mode conversion, the most appropriate reflector shape is an ellipse, as represented by Eq. (2). 28)The parameter C in Eq. ( 2) is a constant that indicates the size of the ellipse.
This equation shows that for an elliptical shape, the ratio of the major axis to the minor axis is expressed in terms of sound speed as c : which is demonstrated in Fig. 2(a).This ratio for the ellipse is dependent on the material's Poisson's ratio σ, as seen in the following equations.
Moreover, this elliptical reflector can convert the transverse focused wave into a dilatational plane wave, as shown in Fig. 2(b).This can be explained by considering it as shown in Fig. 2(c), where the solid lines represent waves in the solid and the dotted lines are supplementary lines from Fig. 2(a).The red line shows a transverse wave heading towards the focal point, showing that the solid and dotted lines are on a straight line.From the relationship of vertically opposite angles, the incidence angle equals a t and from Eq. ( 1), the reflection angle, ¢ a , d can be obtained from Eq. ( 5).
Since ¢ = a a , d d the relationship of vertically opposite angles holds, so the blue solid and dotted lines are on a straight line.In other words, the converging wave of the transverse wave is converted into a plane wave of the dilatational wave propagating in the vertical direction.
To perform the transformation shown in Fig. 2(b) at the second reflection in the D-ELIPS, the focal point of the converging wave must coincide with the focal point of the ellipse.Therefore, the first and second ellipses must share a focal point.As described later, stainless steel SUS304 (s = ) 0.29 was chosen for the waveguide material due to its high efficiency in mode conversion.From Eq. ( 2), the shapes of the first and second elliptical reflectors are designed as given in Eqs. ( 6) and (7), respectively.[The coordinate system is shown in Fig. 1 The diameter of the thin waveguide is 1.0 mm, and the distance between the second reflector and the thin waveguide tip is 22.1 mm.The piezoelectric ring used in this study is made of MT-18K PZT (Niterra Co., Ltd.) with a thickness of 1.3 mm, an inner radius of 8.0 mm, and an outer radius of 20.0 mm.In this study, the driving frequency is set to 1.581 MHz, which corresponds to the thickness resonance frequency of the piezoelectric ring.
The energy ratio of the reflected dilatational or transverse waves to the incident wave depends on / c c t d [or s, as discussed in Eq. ( 4)] and the incidence angle. 29)Figure 3(c) shows the calculated energy ratio for dilatational waves arising from incident dilatational waves versus the incidence angle.High conversion efficiency occurs near incidence angles of 0°or 90°.However, the range of angles that yield high efficiency around 90°is exceptionally narrow, making it difficult to use for focusing.While it is possible to utilize incidence angles near 0°for focusing, the range of angles that provide effective focusing remains limited.For example, in the case of duralumin A2017 (s = ) 0.33 , which is used for the DPLUS, incidence angles with over 80% efficiency are confined in the range 0°to 19°.
On the other hand, effective dilatational-to-transverse wave conversion can be realized for a broader range of incidence angles.Figure 3(d) shows an over 80% conversion efficiency from dilatational to transverse waves for incidence angles of 43°to 83°for s = 0.29.The D-ELIPS primarily utilizes this specific range for the first reflection to achieve efficient ultrasonic focusing.Figure 3(e) shows the case of dilatational waves reflected from transverse waves.From this figure, transverse-to-dilatational wave conversion efficiencies over 80% are achievable with s = 0.29 for incidence angles between 22°and 33°.Although this range seems narrow, the D-ELIPS confines the incident transverse wave angles in the second reflection, enabling efficient conversion of focused transverse waves into dilatational plane waves.Notably, the conversion efficiencies are equal in both reflections along the same pathway.
A large distance between the second reflector and the thin waveguide results in significant energy loss through wave diffusion.This is because the size of the second reflector of the D-ELIPS is small (1.0 mm in diameter) compared to the wavelength of the dilatational wave (3.66 mm, when the waveguide's material is SUS304 and the driving frequency is 1.581 MHz).Since the wavefront width of the plane wave after the second reflection is small, the plane wave does not travel straight but is diffused.To introduce the focused wave into the thin waveguide, a slit structure is added, eliminating the diffusion after the second reflection, as shown in Figs.1(b) and 5(a).
The appropriate distance between the second reflector and the thin waveguide depends on several factors.The shorter the distance, the more energy can be introduced into the waveguide before it diffuses after the second reflection.On the other hand, if the distance is too short, the focused transverse wave is blocked before arriving at the second reflector, and the introduced energy will decrease.To find the optimum distance, a finite element analysis is performed for various distances.
Transient analyses of the D-ELIPS and the DPLUS are performed with COMSOL Multiphysics 6.1 software (COMSOL, Inc.).The waveguide's material is SUS304 for the D-ELIPS and A2017 for the DPLUS.The material properties for the waveguides are listed in Table I.The shape of the DPLUS waveguide is the same as in a previous study [21].The thin waveguide tip is set as a low-reflecting boundary.The frequency is set to 1.581 MHz, corresponding to the thickness resonance frequency of the piezoelectric ring, and five burst sinusoidal waves are applied at 2.0 V pp .The transient analysis is performed for m 18.0 s. Figure 4 shows animation frames of the calculation results for the optimal structure in the D-ELIPS.In Fig. 4(a), a dilatational plane wave is introduced from the piezoelectric  Figure 5(b) shows the relationship between the position of the waveguide connecting part and the ratio of the energy introduced into the thin waveguide to the electrical energy input to the piezoelectric ring in the D-ELIPS.The horizontal axis shows the distance between the second reflector and the thin waveguide.This figure shows that the optimal distance for introducing energy into the thin waveguide is 0.27 mm. Figure 5(c) shows the energy flows in the thin waveguide as a function of time when a burst voltage is applied to the piezoelectric ring starting at = t 0 s.These energy flows are calculated at the thin waveguide cross-sections located at 15.0 mm for the DPLUS and at = z 25.0 mm for the optimal structure of the D-ELIPS.In the DPLUS, 7.1% of the electrical energy input to the piezoelectric element is subsequently introduced into the thin waveguide, while 24.2% is in the D-ELIPS.Therefore, the efficiency of the D-ELIPS is 3.4 times higher than that of the DPLUS.The energy that enters the waveguide from the piezoelectric ring is amplified by a factor of 325 in energy density at the thin waveguide.
In the period from m 10.6 s to m 14.5 s, the thin waveguide cross-section of the D-ELIPS is excited by waves propagating in the designed conversion sequence (dilatational to transverse to dilatational wave).Supporting the FEM simulation, this period can be calculated from the propagation speeds of dilatational and transverse waves in three-dimensional free space and the group velocity along the thin waveguide.The MATLAB package PCDISP 30) is used to calculate the group velocity for the wave in the thin waveguide.The energy flow is largely included in this time domain ( ) 99.4% , indicating that the wave is excited as designed.
In summary, we propose a powerful ultrasonic transducer that improves the mode conversion efficiency and suppresses the wave diffraction effect by introducing two elliptical reflectors and a slit structure.Finite element analysis confirms   064001-4 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd that the energy efficiency is increased by 3.4 times compared to the conventional device, DPLUS.In future work, we intend to manufacture a prototype of the D-ELIPS and experimentally validate its performance.

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ring, converted to a transverse wave by the first reflector, and then focused in Figs.4(b) and 4(c).The dilatational wave is successfully introduced into the thin waveguide in Fig. 4(d).

Fig. 5 .
Fig. 5. (a) Slit structure.(b) Relationship between the position of the waveguide and ratio of the energy introduced into thin waveguide to electrical energy input into the piezoelectric ring in the D-ELIPS.(c) Energy flows at thin waveguide cross-sections ( = z 15.0 mm for DPLUS, = z 25.0 mm for D-ELIPS).

Table I .
Material properties for waveguides.
©2024The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd