**1. Introduction**

Focused ultrasound (FUS) in pulsed mode is unique among transcranial brain stimulation methods in combining exceptional spatial resolution (on the millimeter scale) [1–3] with the potential to target subcortical structures (deeper than 10 cm) [4] through the intact skull. It also has potential for inducing neuronal excitation or suppression without evidence of tissue damage [5].

Recently, we demonstrated the advantages of focusing ultrasound (US) through polydimethylsiloxane (PDMS)-based axicon lenses to selectively drive brain activity [6]. The ultrasound axicon is shown in Figure 1. It has the shape of a cone. As the cone angle (φ) decreases, the focus moves closer to the lens. Developing low-intensity applications include the opening of the blood–brain barrier and ultrasonic neuromodulation. Both techniques have recently been extended to human subjects and are under active research. Spherically FUS transducer has been, until now, the most commonly used for transcranial focusing ultrasound, but it has a large focal length (several centimeters) that may hinder its coupling with the head, for example, when the focal plane is too close to skull inner ivory layer. With spherical segment ultrasound transducers, FUS is commonly delivered through a big plastic bag containing degassed water placed over the scalp. This is due to the fact that, for cortical stimulation,

the acoustic beam should be focused a few millimeters deep from the skull surface [3]. In this sense, the great advantage of the axicon lens is its ability to suppress the near-field and maintain a very near focus from the lens face outward. PDMS-based axicon lens affixed on the face of conventional transducers makes it possible to build more compact devices without liquids that can regasify or leak, with a greater spatial resolution.

However, for an axicon lens to offer high spatial resolution and depth, control is necessary to adjust the separation between the transducer and axicon lens, called stand-off (δ), for each particular case [7]. Given the geometry of the lens, if δ is not properly adjusted, internal reflections can occur, making the configuration useless as a focused transducer. The lens reduces the focal length (F) from the near-field distance (N) of the attached US transducer. The relation between axicon lens angle and the value of F/N produced was described in [8] for acrylic plastic/oil combination, but the effect of the stand-off was not modeled and the adequate value of δ was not described elsewhere. In [7], the stand-off of the different settings was adjusted experimentally to obtain the best signal-to-noise ratio (SNR) for ultrasonic contact inspection. In our approach, time domain ultrasound simulations, based on the k-space pseudo-spectral methods, play a key role in enabling the modeling and systematic design of ultrasonic axicon lenses with optimum stand-off for high-resolution ultrasonic applications, as transcranial stimulation, in high-resolution mapping, and in the evaluation of a wide variety of defects. Finally, the effects of human skull on axicon fields were tested.

**Figure 1.** (**a**) Exploded model of the axicon lens. The polydimethylsiloxane (PDMS) fills the conical cavity of the lens and the stand-off. An ultrasonic lens is formed by the PDMS/plastic interfaces. The window is a thin material layer between the conical cavity and the face of the lens; (**b**) schematic ultrasonic beam pattern for transducers with axicon lenses. These are described by its total included cone angle (φ). Depth of focus (DOF) is the focal region and F is the desired focal length. Both DOF and F depend on the sound velocity into the material used for the interface.

Skull is generally constituted of three relatively homogeneous layers: The outer and inner tables of solid ivory bone, and the central layer of diploe of cancellous bone, with a blood- and fat-filled porous structure. The dimensions of the blood- and fat-filled inclusions are random, and an average thickness in the direction normal to the surface is about 0.6 mm. [5], so the fundamental frequency used was chosen to help alleviate the concerns for acoustic energy absorption or refraction by the skull. For frequencies lower than about 0.5 MHz, reflection of sound is the principal cause of insertion loss. At frequencies between about 0.6 and 0.9 MHz, the absorption loss in the diploe layer begins to limit sound transmission, so the oscillations are damped out and the insertion loss increases linearly with frequency. At about 0.9 MHz, the scattering loss in the diploe layer begins to limit sound transmission, so the insertion loss begins to increase as the fourth power of frequency [5].

A full-wave nonlinear ultrasound model based on the k-space pseudo-spectral method was developed and released as part of the open-source k-Wave Acoustics Toolbox [9,10]. This model can account for the propagation of nonlinear ultrasound waves in homogeneous or heterogeneous media with power acoustic absorption and without restrictions on the directionality of the waves. The accuracy of the implementation of nonlinear ultrasound model in the k-Wave Toolbox was validated using experimental measurements of the ultrasound made with a linear diagnostic ultrasound probe and a membrane hydrophone [11].

A computational model for elastic wave propagation in heterogeneous media can be constructed based on the solution of coupled first-order acoustic equations given in Equations (1)–(3) using the Fourier pseudo-spectral method. This uses the Fourier collocation spectral method to compute spatial derivatives, and a leapfrog finite-difference scheme to integrate forwards in time. Using a temporally and spatially staggered grid, the field variables are updated in a time stepping [12,13]. To simulate free-field conditions, a perfectly matched layer (PML) is also applied to absorb the waves at the edge of the computational domain [14]. Without this boundary layer, the computation of the spatial derivate via the FFT causes waves leaving one side of the domain to reappear at the opposite side. The use of the PML thus facilitates infinite domain simulations without the need to increase the size of the computational grid.

$$\frac{\partial \mu}{\partial t} = -\frac{1}{\rho\_0} \nabla p - \alpha \cdot \mu \tag{1}$$

$$\frac{\partial \rho\_{\mathbf{x}}}{\partial t} = -\rho\_0 \frac{\partial u\_{\mathbf{x}}}{\partial \mathbf{x}} - \alpha\_{\mathbf{x}} \rho\_{\mathbf{x}} \tag{2}$$

$$p = c\_0^2 \sum \rho\_{\mathbf{x}, y, z} \tag{3}$$

where *u* is the acoustic particle velocity, ρ*<sup>0</sup>* is the medium density, ρ is the acoustic density, *c* is the thermodynamic sound speed, p is the acoustic pressure, and p<sup>0</sup> = p(*t* = 0) is the initial pressure distribution.

There are two main stages in this work: The definition of general design equations for PDMS-based axicon lenses with optimal stand-off, through comprehensive ultrasound simulations, for an optimal transcranial focusing; and the measurements and simulations performed to determine focusing performance of the proposed lenses, through a human skull phantom.
