**2. Materials and Methods**

In order to develop a design model, axicon lenses with φ angles between 80◦ and 170◦ in steps of 5◦ were simulated with increments of δ in steps of λ/4 with 8 grid points per wavelength (λ) in the stand-off medium. The final pressure field along with the RMS beam pattern was calculated. The normalized cross-section profile area at the focus F was determined, for each δ. The minimum area represents the greatest improvement with the optimum stand-off, which minimizes transmitted energy outside of the main beam and improves lateral spatial resolution with lower possible sidelobes.

The domain was discretized using a grid point spacing of 250 µm (giving a maximum supported frequency of 2.06 MHz), and a grid size of 512 × 512 grid points (corresponding to a domain size of 128 <sup>×</sup> 128 mm). Simulations were run on an NVIDIA® GTX 950 graphics processing unit (Santa Clara, CA, United States) using the MATLAB® Parallel Computing Toolbox (Natick, MA, United States). The simulation of all angles/stand-off combinations can be completed in approximately 60 h. By default, numbers in MATLAB® are stored in double precision. However, in almost all cases, k-Wave does not require this level of precision. In particular, the performance of the PML generally limits the accuracy to around 4 or 5 decimal places. We use a PML thickness of 20 grid points that gives a transmission coefficient of −100 dB. This corresponds to a reduction in the signal level of <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>5</sup> , which is significantly less than double precision. Further, there will also be uncertainties in the definition of the materials properties. A list of the main simulation inputs is given in Table A1. In this work, a heterogeneous medium was defined as a layered interface on both sides of the conical

cavity of the axicon lens as shown in Figure A1. The convective nonlinear effects from the convection of mass was considered. However, at low frequencies and amplitudes, nonlinearity will only have a small effect on the wave field. At higher frequencies and amplitudes, this effect become more important.

The accuracy of the implementation of ultrasound model with the k-Wave Toolbox was validated in our previous work [6] using experimental measurements of FUS made with the same axicon lens attached transducer and a needle hydrophone (Force Technology MH28) within a 6-L anechoic test tank. Our previous study has already shown that there is a good agreement between the simulated and experimental beam patterns. In this work, we also characterized the acoustic pressure amplitude of the beam pattern of the axicon lens when FUS was transmitted through a human skull phantom for experimental validation of simulated transcranial ultrasound propagation. There is negligible conversion to shear waves in the layers of skull when the incidence angle is within about 20◦ of normal [5]. The ability of bone to support shear waves can affect transcranial transmission, although the changes to the intracranial field are typically negligible for ultrasound applied at normal or near-normal incidence. Therefore, we will model only longitudinal waves. The phantom was created from the parietal portion of a mesh segmented from MRI head image data. Clear Med610 3D printing resin was used to create the skull bone phantom. The acoustic properties of Stratasys™ materials were recently reported in [15]; thus, these measurements were not repeated as part of the current work. The reported and measured property values, and estimated uncertainty in those measurements, are shown in Table 1.

To test the effects of a human skull on FUS fields, we inserted a 5 mm thick fragment of parietal bone phantom between the transducer and the hydrophone, as shown in Figure 2. The transducer described in our previous work [6] has an ultrasonic piezo-disc-type element of 28 mm diameter (SMD28T21F1000R, Steminc Steiner & Martins, Inc., Davenport, FL, USA) of PZT-4 mounted on stainless-steel housing operating in thickness mode vibration at 445 kHz. Epoxy (Resoltech 1040, Resoltech, Rousset, France) resin was used in order to build the conical cavity of the lens with an angle φ of 144◦ . Degassed PDMS (Sylgard 184, Dow Corning, Midland, MI, USA) was used to fill the conical cavity of the lens and for the lens-transducer interface with a stand-off δ of 30 mm. The ultrasonic lens is formed by the epoxy/PDMS interface. The specifications are summarized in Table 2.

**Table 1.** Compressional and shear speed, attenuation, and density of Clear Med610 material.


**Figure 2.** Photograph of the ultrasound test tank showing the axicon lens equipped transducer detail, parietal bone phantom, and hydrophone.


**Table 2.** Specifications of the axicon lens characterized.
