On an inverse problem from magnetic resonance elastic imaging

The imaging problem of elastography is an inverse problem. The nature of an inverse problem is that it is ill-conditioned. We consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by magnetic resonance imaging (MRI). This map is a nonlinear mapping, and our interest is in proving certain conditioning and regularity results for this operator which occurs implicitly in this problem of imaging in elastography. In this treatment we consider the tissue to be linearly elastic, isotropic, and spatially heterogeneous. We determine the conditioning of this problem of function reconstruction, in particular for the stiffness function. We further examine the conditioning when determining both stiffness and density. We examine the Frechet derivative of the nonlinear mapping, which enables us to describe the properties of how the field affects the individual maps to the stiffness and density functions. We illustrate how use of the implicit function theorem can considerably simplify the analysis of Frechet differentiability and regularity properties of this underlying operator. We present new results which show that the stiffness map is mildly ill-posed, whereas the density map suffers from medium ill-conditioning. Computational work has been done previously to study the sensitivity of these maps, but our work here is analytical. The validity of the Newton-Kantorovich and optimization methods for the computational solution of this inverse problem is directly linked to the Frechet differentiability of the appropriate nonlinear operator, which we justify

Recovering Young's moduli in heterogeneous stenosed carotid arteries: a numerical plane strain study

International audienceAssessing the vulnerability of atherosclerotic plaques requires an accurate knowledge of the mechanical properties of the plaque constituents. It is possible to measure displacements in vivo inside a plaque using ultrasounds or magnetic resonance imaging. The main issue is to solve the inverse problem that consists in estimating the elastic properties inside the plaque from measured displacements. This study focuses on the identifiability of elastic parameters. An idealised plane strain Finite Element (FE) model is used

Viscoelastic modulus reconstruction using time harmonic vibrations

This paper presents a new iterative reconstruction method to provide high-resolution images of shear modulus and viscosity via the internal measurement of displacement fields in tissues. To solve the inverse problem, we compute the Fr\'echet derivatives of the least-squares discrepancy functional with respect to the shear modulus and shear viscosity. The proposed iterative reconstruction method using this Fr\'echet derivative does not require any differentiation of the displacement data for the full isotropic linearly viscoelastic model, whereas the standard reconstruction methods require at least double differentiation. Because the minimization problem is ill-posed and highly nonlinear, this adjoint-based optimization method needs a very well-matched initial guess. We find a good initial guess. For a well-matched initial guess, numerical experiments show that the proposed method considerably improves the quality of the reconstructed viscoelastic images.Comment: 15 page

Analysis of DCE-MRI Data using a Nonnegative Elastic Net

We present a nonnegative Elastic Net approach for the analysis of Dynamic Contrast-Enhanced Magnetic Resonance Imaging data. A multi-compartment approach is considered, which is translated into a (restricted) least square model selection problem. This is done by using a set of basis functions for a given set of candidate rate constants. The form of the basis functions is derived from a kinetic model and thus describes the contribution of some compartment. Using the Elastic Net estimator, we chose clusters of basis functions, and hence, rate constants of compartments. As further challenge, the estimator has to be restricted to positive regression parameters, which correspond to transfer rates of the compartments. The proposed estimation method is applied to an in-vivo data set

Characterizing the elastic property distribution of soft materials nondestructively

Soft materials have the advantage that their subsurface structure may be imaged utilizing imaging modalities such as ultrasound, magnetic resonance imaging, computed tomography (CT) scans, and optical coherence tomography. In recent decades subsurface displacement fields were successfully measured using these imaging modalities. These displacement fields are of time-harmonic, transient, or quasi-static nature and can be used to solve an inverse problem in elasticity to determine the elastic material property distribution within the region of interest of the soft material. One important application area is in the detection and diagnosis of diseased tissues, such as breast tumors. This can be done as tumors are often stiffer than their background tissue, and based on the stiffness contrast they can be visualized and distinguished from their surrounding healthy tissues. We will present the solution of the inverse problem in finite elasticity for quasi-static displacement fields. Here, the quasi-static displacement fields may be determined by taking a sequence of ultrasound images while slowly compressing the tissue with the ultrasound transducer. From this sequence of ultrasound images one may determine the displacement fields utilizing well-developed cross-correlation techniques. We solve the inverse problem iteratively by posing it as a constrained optimization problem, where the difference between a measured and computed displacement field is minimized in the L-2 norm and in the presence of Tikhonov regularization. The computed displacement field satisfies the constraint, which are the equations of equilibrium and solved using finite element methods for the current estimate of the elastic property distribution. We model the material to be hyperelastic with a strain energy density function of exponential form and two elastic property parameters: the shear modulus describes the linear elastic behavior, whereas the nonlinear elastic property describes the rate at which the material is stiffening at large strains [1]. In addition, we assume that the material is isotropic and incompressible. This method has been tested on hypothetical data and breast tumor patients [2] and proofed to be robust in the presence of noisy displacement data to recover the tumors. However, this method appears to be sensitive to the applied boundary conditions, i.e., the solution of the inverse problem for uniform boundary compression appears to be different than applying a linearly changing boundary compression. Obviously, the solution of the inverse problem lacks uniqueness with respect to changing boundary conditions. We realize that the uniqueness issue here is primarily because of the formulation of the displacement correlation term in the objective function. We provide a new formulation and show that this leads to a “more unique” solution of the inverse problem and improves the overall contrast. REFERENCES [1] Goenezen, S., Barbone, P., and Oberai, A.A., Solution of the nonlinear elasticity imaging inverse problem: the incompressible case. Computer Methods in Applied Mechanics and Engineering. 2011, 200(13–16), 1406–1420. [2] Goenezen, S., Dord, J.F., Sink, Z., Barbone, P., Jiang, J., Hall, T.J., and Oberai, A.A., Linear and nonlinear elastic modulus imaging: an application to breast cancer diagnosis. IEEE Trans Med Imaging. 2012, 31(8), 1628–1637

Application of the Finite Element Method in a Quantitative Imaging technique

We present the Finite Element Method (FEM) for the numerical solution of the multidimensional coefficient inverse problem (MCIP) in two dimensions. This method is used for explicit reconstruction of the coefficient in the hyperbolic equation using data resulted from a single measurement. To solve our MCIP we use approximate globally convergent method and then apply FEM for the resulted equation. Our numerical examples show quantitative reconstruction of the sound speed in small tumor-like inclusions

Frequency-splitting Dynamic MRI Reconstruction using Multi-scale 3D Convolutional Sparse Coding and Automatic Parameter Selection

Department of Computer Science and EngineeringIn this thesis, we propose a novel image reconstruction algorithm using multi-scale 3D con- volutional sparse coding and a spectral decomposition technique for highly undersampled dy- namic Magnetic Resonance Imaging (MRI) data. The proposed method recovers high-frequency information using a shared 3D convolution-based dictionary built progressively during the re- construction process in an unsupervised manner, while low-frequency information is recovered using a total variation-based energy minimization method that leverages temporal coherence in dynamic MRI. Additionally, the proposed 3D dictionary is built across three different scales to more efficiently adapt to various feature sizes, and elastic net regularization is employed to promote a better approximation to the sparse input data. Furthermore, the computational com- plexity of each component in our iterative method is analyzed. We also propose an automatic parameter selection technique based on a genetic algorithm to find optimal parameters for our numerical solver which is a variant of the alternating direction method of multipliers (ADMM). We demonstrate the performance of our method by comparing it with state-of-the-art methods on 15 single-coil cardiac, 7 single-coil DCE, and a multi-coil brain MRI datasets at different sampling rates (12.5%, 25% and 50%). The results show that our method significantly outper- forms the other state-of-the-art methods in reconstruction quality with a comparable running time and is resilient to noise.ope