在接近共振频率下使用气泡作为造影剂的声学成像模态的数学分析
Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies
摘要：
我们从数学上分析了使用气泡作为造影剂的声学成像方式。这些气泡的质量密度和体积模量享受对比尺度建模。这些对比度使它们能够在特定的入射频率下共振。我们考虑两种类型的这种对比。在第一种情况下，气泡是轻的，体积模量小，与背景相比，使它们产生Minnaert共振（对应于局部表面波）。在第二种情况下，气泡具有中等的质量密度，但仍然具有小的体积模量，使得它们产生一系列共振（对应于局部体波）。我们建议使用作为测量的远场之前和之后注入一个气泡，设置在一个给定的位置点在目标域中，在一个带的入射频率和在一个固定的单一的后向散射方向产生。然后，我们扫描的目标域与这样的气泡，并收集相应的远场。目标是重建目标区域中背景的可变质量密度和体积模量。1.我们表明，对于每个固定使用的气泡，对比的远场达到其最大值，入射，频率接近Minneart共振（或体波共振，取决于我们使用的气泡的类型）。因此，我们可以从我们的数据中重建这种共振。明纳特共振与背景质量密度的显式相关性使我们能够以直接的方式恢复它，即质量密度。2.此外，这种测量的对比远场允许我们恢复气泡位置点处的总场（即，不存在气泡时的总场）。例如，一个数值微分参数，使我们能够恢复目标区域的体积模量。
英文摘要：
We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves). We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixed single backscattering direction. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region. 1. We show that, for each fixed used bubble, the contrasted far-fields reach their maximum value at, incident, frequencies close to the Minneart resonance (or the body-wave resonances depending on the types of bubbles we use). Hence, we can reconstruct this resonance from our data. The explicit dependence of the Minnaert resonance in terms of the background mass density of the background allows us to recover it, i.e. the mass density, in a straightforward way. 2. In addition, this measured contrasted far-fields allow us to recover the total field at the location points of the bubbles (i.e. the total field in the absence of the bubbles). A numerical differentiation argument, for instance, allows us to recover the bulk modulus of the targeted region as well.
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