Abstract
The goal of quantitative elastography is to identify biomechanical parameters from interior displacement data, which are provided by other modalities, such as ultrasound or magnetic resonance imaging. In this paper, we analyze the stability of several linearized problems in quantitative elastography. Our method is based on the theory of redundant systems of linear partial differential equations. We analyze the ellipticity properties of the corresponding PDE systems augmented with the interior displacement data; we explicitly characterize the kernel of the forward operators and show injectivity for particular linearizations. Stability criteria can then be deduced. While joint reconstruction of all parameters suffers from non-ellipticity even for more measurements, our results show stability of the separate reconstruction of shear modulus, pressure and density; they indicate that singular strain fields should be avoided, and show how additional measurements can help in ensuring stability of particular linearized problems.
Highlights
IntroductionElastography is a medical imaging technology; its current applications range from detection of cancer in the breast and in the prostate, liver cirrhosis and characterization of artherosclerotic plaque in hardened coronary vessels [19, 42, 54, 4, 53, 56, 15]
Elastography is a medical imaging technology; its current applications range from detection of cancer in the breast and in the prostate, liver cirrhosis and characterization of artherosclerotic plaque in hardened coronary vessels [19, 42, 54, 4, 53, 56, 15].Elastography is based on the fact that tissue has high contrast in biomechanical quantities and the health state of organs is reflected in the elastic properties of tissue [22, 2]
The most important of these is the shear modulus μ, which is the dominant factor in the propagation of shear waves in tissue; shear wave speed in tissue can change up to a factor of 4 with disease [47]
Summary
Elastography is a medical imaging technology; its current applications range from detection of cancer in the breast and in the prostate, liver cirrhosis and characterization of artherosclerotic plaque in hardened coronary vessels [19, 42, 54, 4, 53, 56, 15]. Elastography is based on the fact that tissue has high contrast in biomechanical quantities and the health state of organs is reflected in the elastic properties of tissue [22, 2]. The most important of these is the shear modulus μ, which is the dominant factor in the propagation of shear waves in tissue; shear wave speed in tissue can change up to a factor of 4 with disease [47]. SCHERZER specific excitation used, u can be space or space-time-dependent (both cases are treated in this work)
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