High intensity focussed ultrasound (HIFU) treatment planning with geometrical optics acoustics

聚焦超声越来越多地用于治疗应用，例如用于肿瘤的热消融和泌尿系结石的碎石破坏。精确的治疗计划需要声学建模来预测超声波将聚焦的位置。在这些治疗所需的高声压状态下，声波非线性传播-换句话说，它们使能量变陡并将能量推入更高的谐波，有时比基频高一个数量级。这可以显著改变热沉积的速率或声波的形状，从而对治疗产生影响。因此，有必要对非线性进行精确建模。声学传播的主要数值模型是基于网格或网格的，并且要求节点间隔比最高波长的一半更近（实际上更近），因此谐波的产生导致需要非常大的网格，并且计算变得不切实际地大。因此，HIFU模拟（前向问题）需要不同的方法。对于大多数这些应用，传播是线性的，直到接近焦点，因此焦点位置不会从线性状态改变，只是场的振幅。基于这一观察，我们建议设计一种方法，使用快速线性几何光学计算射线轨迹，然后通过求解非线性声学方程，例如，计算沿着这些射线的声压幅度。伯格斯方程或韦斯特丙酮方程，或它们的衍生物。然后，该方法可以在治疗计划的优化（逆问题）中用作正向求解器。

Focussed ultrasound is increasingly being exploited in therapeutic applications, for example for thermal ablation of tumours and lithotriptic destruction of urological stones. Accurate planning of the treatment requires acoustic modelling to predict where the ultrasound waves will focus. In the high acoustic pressure regimes required for these treatments, the acoustic waves propagate nonlinearly - in other words they steepen and push energy into higher harmonics, sometimes as much as an order of magnitude higher than the fundamental frequency. This can significantly alter the rate of heat deposition, or the shape of the acoustic wave, and thereby have an effect on the treatment. It is therefore necessary to model the nonlinearity accurately. The leading numerical models of acoustic propagation are grid or mesh-based, and require nodes spaced closer than half-the-highest-wavelength (in practice quite a bit closer) and so the generation of harmonics leads to a requirement for very large grids, and the computations become impractically large. HIFU simulations (forward problem) therefore call for a different approach. As for most of these applications, the propagation is linear until close to the focus, and so the focal position does not change from the linear regime, just the amplitude of the field. Based on this observation we propose to devise a method that calculates the ray trajectories using fast linear geometric optics, and then computes the acoustic pressure amplitude along these rays by solving nonlinear acoustic equations, eg. Burgers equation or Westervelt equation, or a derivative from them, along the rays. This method can then be used as a forward solver in the optimisation of the treatment plan (inverse problem).