  Acoustic holographic lenses, also known as acoustic holograms, can change the
phase of a transmitted wavefront in order to shape and construct complex
ultrasound pressure fields, often for focusing the acoustic energy on a target
region. These lenses have been proposed for transcranial focused ultrasound
(tFUS) to create diffraction-limited focal zones that target specific brain
regions while compensating for skull aberration. Holograms for tFUS are
currently designed using time-reversal approaches in full-wave time-domain
numerical simulations. However, such simulations need time-consuming
computations, which severely limits the adoption of iterative optimization
strategies. Furthermore, in the time-reversal method, the number and
distribution of virtual sources can significantly influence the final sound
field. Because of the computational constraints, predicting these effects and
determining the optimal arrangement is challenging. This study introduces an
efficient method for designing acoustic holograms using a volumetric
holographic technique to generate focused fields inside the skull. The proposed
method combines a modified mixed-domain method for ultrasonic propagation with
a gradient descent iterative optimization algorithm. This approach enables
substantially faster holographic computation than previously reported
techniques. The iterative process uses explicitly defined loss functions to
bias the ultrasound field's optimization parameters to specific desired
characteristics, such as axial resolution, transversal resolution, coverage,
and focal region uniformity, while eliminating the uncertainty associated with
virtual sources in time-reversal techniques. Numerical studies are conducted on
four brain structures: the anterior insula, hippocampus, caudate nucleus, and
amygdala. The findings are further validated in underwater experiments with a
3D-printed skull phantom.
