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Transcranial-focused ultrasound (tFUS) has potential for both neuromodulation and neuroimaging. Due to the influence of head tissue, especially the skull, its attenuation is a key issue affecting precise focusing. The objective of the present study was to construct a mathematical model of ultrasound attenuation inclusive of skull thickness. First, combined with real skull phantom experiments and simulation experiments, tFUS attenuation of different head tissues was investigated. Furthermore, based on the system identification method, a mathematical model of ultrasound attenuation was constructed taking skull thickness into account. Finally, the performance of the mathematical model was tested, and its potential applications were investigated. For different head tissues, including scalp, skull, and brain tissue, the skull was found to be the biggest influencing factor for ultrasound attenuation, the attenuation caused by it being 4.70 times and 7.06 times that of attenuation caused by the brain and scalp, respectively. Consistent with the results of both the simulation and phantom experiments, the attenuation of the mathematical model increased as the skull thickness increased. The average error of the mathematical model was 1.87% in the phantom experiment. In addition, the experimental results show that the devised mathematical model is suitable for different initial pressures and different skulls with correlation coefficients higher than 0.99. Both simulation and phantom experiments validated the effectiveness of the proposed mathematical model. It can be concluded from this experiment that the proposed mathematical model can accurately calculate the tFUS attenuation and can significantly contribute to further research and application of tFUS.

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