Abstract

We derive and analyse a new way to calculate the shear modulus of an inhomogeneous elastic material from time-dependent magnetic resonance elastography (MRE) measurements of its interior displacement. Even with such a rich data source, this is a challenging inverse problem because the coefficient of the shear modulus in the governing equations can be small (or potentially zero). Our approach overcomes this by combining different data sets into an overdetermined matrix–vector equation. It uses finite differences to approximate space derivatives and a Fourier interpolant in time, and we do not need to assume that the inhomogeneous material is ‘locally homogeneous’. Crucially, our construction ensures that the computed value of the (real) shear modulus is real: approximation methods based on the frequency domain version of the problem often give a complex shear modulus for the elastic case and this can be hard to interpret, especially if its imaginary part dominates. We carry out careful numerical tests on a one (space) dimensional analogue of the problem and on experimental MRE data for an inhomogeneous gel phantom.

中文翻译：

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非均一的时变MRE的叠加频率方法：弹性剪切模量的反问题

摘要

我们推导并分析了一种新方法，该方法可以通过时变磁共振弹性成像（MRE）对其内部位移的测量来计算非均质弹性材料的剪切模量。即使具有如此丰富的数据源，这也是一个具有挑战性的反问题，因为控制方程式中的剪切模量系数可能很小（或可能为零）。我们的方法通过将不同的数据集组合到一个超定的矩阵-矢量方程中来克服这一问题。它使用有限的差异来近似逼近空间导数和时间上的傅立叶插值，而我们不必假设不均匀材料是“局部均匀”的。至关重要的是，我们的构造确保了（真实）剪切模量的计算值是真实的：基于问题的频域版本的近似方法通常会为弹性情况提供复杂的剪切模量，这可能很难解释，尤其是在其虚部占优势的情况下。我们对问题的一维（空间）模拟和非均质凝胶体模的实验MRE数据进行了仔细的数值测试。