Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound

Elena A. Kaye*, Yoni Hertzberg, Michael Marx, Beat Werner, Gil Navon, Marc Levoy, Kim Butts Pauly

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Purpose: To study the phase aberrations produced by human skulls during transcranial magnetic resonance imaging guided focused ultrasound surgery (MRgFUS), to demonstrate the potential of Zernike polynomials (ZPs) to accelerate the adaptive focusing process, and to investigate the benefits of using phase corrections obtained in previous studies to provide the initial guess for correction of a new data set. Methods: The five phase aberration data sets, analyzed here, were calculated based on preoperative computerized tomography (CT) images of the head obtained during previous transcranial MRgFUS treatments performed using a clinical prototype hemispherical transducer. The noniterative adaptive focusing algorithm [Larrat, MR-guided adaptive focusing of ultrasound, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57(8), 1734-1747 (2010)]10.1109/TUFFC.2010.1612 was modified by replacing Hadamard encoding with Zernike encoding. The algorithm was tested in simulations to correct the patients' phase aberrations. MR acoustic radiation force imaging (MR-ARFI) was used to visualize the effect of the phase aberration correction on the focusing of a hemispherical transducer. In addition, two methods for constructing initial phase correction estimate based on previous patients data were investigated. The benefits of the initial estimates in the Zernike-based algorithm were analyzed by measuring their effect on the ultrasound intensity at the focus and on the number of ZP modes necessary to achieve 90 of the intensity of the nonaberrated case. Results: Covariance of the pairs of the phase aberrations data sets showed high correlation between aberration data of several patients and suggested that subgroups can be based on level of correlation. Simulation of the Zernike-based algorithm demonstrated the overall greater correction effectiveness of the low modes of ZPs. The focal intensity achieves 90 of nonaberrated intensity using fewer than 170 modes of ZPs. The initial estimates based on using the average of the phase aberration data from the individual subgroups of subjects was shown to increase the intensity at the focal spot for the five subjects. Conclusions: The application of ZPs to phase aberration correction was shown to be beneficial for adaptive focusing of transcranial ultrasound. The skull-based phase aberrations were found to be well approximated by the number of ZP modes representing only a fraction of the number of elements in the hemispherical transducer. Implementing the initial phase aberration estimate together with Zernike-based algorithm can be used to improve the robustness and can potentially greatly increase the viability of MR-ARFI-based focusing for a clinical transcranial MRgFUS therapy.

Original languageEnglish
Pages (from-to)6254-6263
Number of pages10
JournalMedical Physics
Issue number10
StatePublished - Oct 2012


FundersFunder number
National Institutes of HealthR21 EB011559
National Cancer InstituteP01CA159992


    • MR-ARFI
    • MRgFUS
    • Zernike polynomials
    • adaptive focusing
    • transcranial


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