Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverse problem in two dimensions

In vivo quantification of the anisotropic shear elasticity of soft tissue is an appealing objective of elastography techniques because elastic anisotropy can potentially provide specific information about structural alterations in diseased tissue. Here a method is introduced and applied to MR elastography (MRE) of skeletal muscle. With this method one can elucidate anisotropy by means of two shear moduli (one parallel and one perpendicular to the muscle fiber direction). The technique is based on group velocity inversion applied to bulk shear waves, which is achieved by an automatic analysis of wave-phase gradients on a spatiotemporal scale. The shear moduli are then accessed by analyzing the directional dependence of the shear wave speed using analytic expressions of group velocities in k-space, which are numerically mapped to real space. The method is demonstrated by MRE experiments on the biceps muscle of five volunteers, resulting in 5.5 ؎ 0.9 kPa and 29.3 ؎ 6.2 kPa (P < 0.05) for the medians of the perpendicular and parallel shear moduli, respectively. The proposed technique combines fast steadystate free precession (SSFP) MRE experiments and fully automated processing of anisotropic wave data, and is thus an interesting MRI modality for aiding clinical diagnosis. Magn Reson Med 56:489 -497, 2006.

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“…8. The obtained variations for v 1 and v 3 are reasonably low with regard to inverse problems in shear-wave-based elastography (39).…”

Section: Discussionmentioning

confidence: 69%

Shear wave group velocity inversion in MR elastography of human skeletal muscle

Papazoglou

Rump

Braun

et al. 2006

Magnetic Resonance in Med

108

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“…We consider both quasistatic strain fields and a time dependent strain field. The strain fields considered are similar to the examples given in [13], where they are shown to lead to non-unique modulus distributions under the assumption of incompressible material behavior.…”

Section: Analytical Examplesmentioning

confidence: 75%

“…Again by way of contrast, the same strain field in the incompressible model leads to the conclusion [13]:…”

Section: Two Dimensional Quasi-static Homogeneous Deformationmentioning

confidence: 90%

“…The exact solution contains only one arbitrary multiplicative constant, in stark contrast to the gross non-uniqueness apparent in the case the material is modeled as an incompressible solid [13]. One example in particular shows a large mismatch in the Lame' parameters where λ µ leads to a high sensitivity of the reconstructed shear modulus to the measured strains.…”

Section: Discussionmentioning

confidence: 99%

“…By contrast to above, under the assumption that the material is incompressible the same strain field leads to the conclusion (c.f. [13] for the derivation in 2D):…”

Section: Three Dimensional Quasi-static Homogeneous Deformationmentioning

confidence: 99%

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Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

Barbone¹,

Oberai²

2007

Phys. Med. Biol.

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Abstract. We consider several inverse problems motivated by elastography. Given the (possibly transient) displacement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density ρ, determine the Lamé parameters λ and µ. We consider several special cases of this problem: (a) For µ known apriori, λ is determined by a single deformation field up to a constant. (b) Conversely, for λ known apriori, µ is determined by a single deformation field up to a constant. This includes as a special case that for which the term λ∇ · u ≡ 0. (c) Finally, if neither λ nor µ is known apriori, but Poisson's ratio ν is known, then µ and λ are determined by a single deformation field up to a constant. This includes as a special case plane stress deformations of an incompressible material. Exact analytical solutions valid for 2D, 3D, and transient deformations are given for all cases in terms of quadratures. These are used to show that the inverse problem for µ based on the compressible elasticity equations is unstable in the limit λ → ∞. Finally, we use the exact solutions as a basis to compute nontrivial modulus distributions in a simulated example.

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Algorithms for quantitative quasi‐static elasticity imaging using force data

Tyagi

Goenezen

Barbone

et al. 2014

Numer Methods Biomed Eng

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Quasi-static elasticity imaging can improve diagnosis and detection of diseases that affect the mechanical behavior of tissue. In this methodology images of the shear modulus of the tissue are reconstructed from the measured displacement field. This is accomplished by seeking the spatial distribution of mechanical properties that minimizes the difference between the predicted and the measured displacement fields, where the former is required to satisfy a finite element approximation to the equations of equilibrium. In the absence of force data, the shear modulus is determined only up to a multiplicative constant. In this manuscript we address the problem of calibrating quantitative elastic modulus reconstructions created from measurements of quasi-static deformations. We present two methods that utilize the knowledge of the applied force on a portion of the boundary. The first involves rescaling the shear modulus of the original minimization problem to best match the measured force data. This approach is easily implemented but neglects the spatial distribution of tractions. The second involves adding a force-matching term to the original minimization problem and a change of variables, wherein we seek the log of the shear modulus. We present numerical results that demonstrate the usefulness of both methods.

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Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverse problem in two dimensions

Cited by 140 publications

References 37 publications

Shear wave group velocity inversion in MR elastography of human skeletal muscle

Shear wave group velocity inversion in MR elastography of human skeletal muscle

Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

Algorithms for quantitative quasi‐static elasticity imaging using force data

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