Uniqueness of the elastography inverse problem for incompressible nonlinear planar hyperelasticity

We reconstruct the in vivo spatial distribution of linear and nonlinear elastic parameters in ten patients with benign (five) and malignant (five) tumors. The mechanical behavior of breast tissue is represented by a modified Veronda–Westmann model with one linear and one nonlinear elastic parameter. The spatial distribution of these elastic parameters is determined by solving an inverse problem within the region of interest (ROI). This inverse problem solution requires the knowledge of the displacement fields at small and large strains. The displacement fields are measured using a free-hand ultrasound strain imaging technique wherein, a linear array ultrasound transducer is positioned on the breast and radio frequency echo signals are recorded within the ROI while the tissue is slowly deformed with the transducer. Incremental displacement fields are determined from successive radio-frequency frames by employing cross-correlation techniques. The rectangular regions of interest were subjectively selected to obtain low noise displacement estimates and therefore were variables that ranged from 346 to 849.6 mm2. It is observed that malignant tumors stiffen at a faster rate than benign tumors and based on this criterion nine out of ten tumors were correctly classified as being either benign or malignant.

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“…This behavior is consistent with the uniqueness of the inversion, which may be expected from the results of [47]. …”

Section: Methodssupporting

confidence: 91%

“…This is because we do not impose any nonzero traction boundary condition in our boundary value problem. The nonlinear parameter γ is reconstructed uniquely if its value is specified at some location in the domain, or if a lower bound is specified [47]. We adopt the latter approach and select the lower bound to be one.…”

Section: Methodsmentioning

confidence: 99%

Linear and Nonlinear Elastic Modulus Imaging: An Application to Breast Cancer Diagnosis

Goenezen

Dord

Sink

et al. 2012

IEEE Trans. Med. Imaging

114

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“…However, for a large class of hyperelastic constitutive models with two parameters it has been shown that given two deformation fields: one at small strain, and another at large strain, we are guaranteed a unique solution for the two parameters up to two undetermined constants [Ferreira et al, 2012]. These constants can be determined by knowing these parameters at one point within the domain.…”

Section: The Inverse Problemmentioning

confidence: 99%

Inferring spatial variations of microstructural properties from macroscopic mechanical response

Liu

Hall

Barbone

et al. 2016

Biomech Model Mechanobiol

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Disease alters tissue microstructure, which in turn affects the macroscopic mechanical properties of tissue. In elasticity imaging, the macroscopic response is measured and is used to infer the spatial distribution of the elastic constitutive parameters. When an empirical constitutive model is used these parameters cannot be linked to the microstructure. However, when the constitutive model is derived from a microstructural representation of the material, it allows for the possibility of inferring the local averages of the spatial distribution of the microstructural parameters. This idea forms the basis of this study. In particular, we first derive a constitutive model by homogenizing the mechanical response of a network of elastic, tortuous fibers. Thereafter, we use this model in an inverse problem to determine the spatial distribution of the microstructural parameters. We solve the inverse problem as a constrained minimization problem, and develop efficient methods for solving it. We apply these methods to displacement fields obtained by deforming gelatin-agar co-gels, and determine the spatial distribution of agar concentration and fiber tortuosity, thereby demonstrating that it is possible to image local averages of microstructural parameters from macroscopic measurements of deformation.

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“…2. In order to solve for both the shear modulus and the nonlinear parameter , it is necessary to measure a displacement field at small strains .jruj 1/ and another at a finite value of strain [17]. The first displacement field is then used to determine the shear modulus while fixing to a small constant (say 1) throughout the domain.…”

Section: Inverse Problemmentioning

confidence: 99%

“…In quasi-static elasticity imaging, one typically has at most a measured quasi-static displacement field. From such data, then even in the most optimistic modeling case, one can determine the shear modulus of the tissue only up to a multiplicative constant [15][16][17][18]. This is because in the equations of motion, there is no term on the right-hand-side in order to calibrate the material parameters (see Equation (1) in the following section).…”

mentioning

confidence: 99%

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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Uniqueness of the elastography inverse problem for incompressible nonlinear planar hyperelasticity

Cited by 17 publications

References 29 publications

Linear and Nonlinear Elastic Modulus Imaging: An Application to Breast Cancer Diagnosis

Linear and Nonlinear Elastic Modulus Imaging: An Application to Breast Cancer Diagnosis

Inferring spatial variations of microstructural properties from macroscopic mechanical response

Algorithms for quantitative quasi‐static elasticity imaging using force data

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