Performance analysis of steady-state harmonic elastography

Stiffness reconstruction methods for MR elastography

Assessment of tissue stiffness is desirable for clinicians and researchers, as it is well established that pathophysiological mechanisms often alter the structural properties of tissue. Magnetic resonance elastography (MRE) provides an avenue for measuring tissue stiffness and has a long history of clinical application, including staging liver fibrosis and stratifying breast cancer malignancy. A vital component of MRE consists of the reconstruction algorithms used to derive stiffness from wave-motion images by solving inverse problems. A large range of reconstruction methods have been presented in the literature, with differing computational expense, required user input, underlying physical assumptions, and techniques for numerical evaluation. These differences, in turn, have led to varying accuracy, robustness, and ease of use. While most reconstruction techniques have been validated against in silico or in vitro phantoms, performance with real data is often more challenging, stressing the robustness and assumptions of these algorithms. This article reviews many current MRE reconstruction methods and discusses the aforementioned differences. The material assumptions underlying the methods are developed and various approaches for noise reduction, regularization, and numerical discretization are discussed. Reconstruction methods are categorized by inversion type, underlying assumptions, and their use in human and animal studies. Future directions, such as alternative material assumptions, are also discussed.

Estimation of Strain Elastography from Ultrasound Radio-Frequency Data by Utilizing Analytic Gradient of the Similarity Metric

Most strain imaging techniques follow a pipeline strategy: in the first step, tissue displacement is estimated from radio-frequency (RF) frames, and in the second step, a spatial derivative operation is applied. There are two main issues that arise from this framework. First, the gradient operation amplifies noise, and therefore, smoothing techniques have to be adopted. Second, strain estimation does not exploit the original RF data. It rather relies solely on the noisy displacement field. In this paper, a novel technique is proposed that utilizes both the displacement field and the RF frames to accurately obtain the strain estimates. The normalized cross correlation (NCC) metric between two corresponding windows around the samples of the pre- and post-compressed images is employed to generate a dissimilarity measurement. The derivative of NCC with respect to the strain is analytically derived using the chain rule. This allows an efficient minimization of the dissimilarity metric with respect to the strain using the gradient descent optimization technique. The effectiveness of the proposed method is investigated through simulation data, phantom experiments, and in vivo patient data. The experimental results show that exploiting the information in RF data significantly improves the strain estimates.

Model-based elastography: a survey of approaches to the inverse elasticity problem

Elastography is emerging as an imaging modality that can distinguish normal versus diseased tissues via their biomechanical properties. This paper reviews current approaches to elastography in three areas--quasi-static, harmonic and transient--and describes inversion schemes for each elastographic imaging approach. Approaches include first-order approximation methods; direct and iterative inversion schemes for linear elastic; isotropic materials and advanced reconstruction methods for recovering parameters that characterize complex mechanical behavior. The paper's objective is to document efforts to develop elastography within the framework of solving an inverse problem, so that elastography may provide reliable estimates of shear modulus and other mechanical parameters. We discuss issues that must be addressed if model-based elastography is to become the prevailing approach to quasi-static, harmonic and transient elastography: (1) developing practical techniques to transform the ill-posed problem with a well-posed one; (2) devising better forward models to capture the complex mechanical behavior of soft tissues and (3) developing better test procedures to evaluate the performance of modulus elastograms.

Viscoelastic properties of soft gels: comparison of magnetic resonance elastography and dynamic shear testing in the shear wave regime

Magnetic resonance elastography (MRE) is used to quantify the viscoelastic shear modulus, G*, of human and animal tissues. Previously, values of G* determined by MRE have been compared to values from mechanical tests performed at lower frequencies. In this study, a novel dynamic shear test (DST) was used to measure G* of a tissue-mimicking material at higher frequencies for direct comparison to MRE. A closed-form solution, including inertial effects, was used to extract G* values from DST data obtained between 20 and 200 Hz. MRE was performed using cylindrical 'phantoms' of the same material in an overlapping frequency range of 100-400 Hz. Axial vibrations of a central rod caused radially propagating shear waves in the phantom. Displacement fields were fit to a viscoelastic form of Navier's equation using a total least-squares approach to obtain local estimates of G*. DST estimates of the storage G' (Re[G*]) and loss modulus G″ (Im[G*]) for the tissue-mimicking material increased with frequency from 0.86 to 0.97 kPa (20-200 Hz, n = 16), while MRE estimates of G' increased from 1.06 to 1.15 kPa (100-400 Hz, n = 6). The loss factor (Im[G*]/Re[G*]) also increased with frequency for both test methods: 0.06-0.14 (20-200 Hz, DST) and 0.11-0.23 (100-400 Hz, MRE). Close agreement between MRE and DST results at overlapping frequencies indicates that G* can be locally estimated with MRE over a wide frequency range. Low signal-to-noise ratio, long shear wavelengths and boundary effects were found to increase residual fitting error, reinforcing the use of an error metric to assess confidence in local parameter estimates obtained by MRE.

Real-time regularized ultrasound elastography

This paper introduces two real-time elastography techniques based on analytic minimization (AM) of regularized cost functions. The first method (1D AM) produces axial strain and integer lateral displacement, while the second method (2D AM) produces both axial and lateral strains. The cost functions incorporate similarity of radio-frequency (RF) data intensity and displacement continuity, making both AM methods robust to small decorrelations present throughout the image. We also exploit techniques from robust statistics to make the methods resistant to large local decorrelations. We further introduce Kalman filtering for calculating the strain field from the displacement field given by the AM methods. Simulation and phantom experiments show that both methods generate strain images with high SNR, CNR and resolution. Both methods work for strains as high as 10% and run in real-time. We also present in vivo patient trials of ablation monitoring. An implementation of the 2D AM method as well as phantom and clinical RF-data can be downloaded.

Effects of frequency- and direction-dependent elastic materials on linearly elastic MRE image reconstructions

The mechanical model commonly used in magnetic resonance elastography (MRE) is linear elasticity. However, soft tissue may exhibit frequency- and direction-dependent (FDD) shear moduli in response to an induced excitation causing a purely linear elastic model to provide an inaccurate image reconstruction of its mechanical properties. The goal of this study was to characterize the effects of reconstructing FDD data using a linear elastic inversion (LEI) algorithm. Linear and FDD phantoms were manufactured and LEI images were obtained from time-harmonic MRE acquisitions with variations in frequency and driving signal amplitude. LEI responses to artificially imposed uniform phase shifts in the displacement data from both purely linear elastic and FDD phantoms were also evaluated. Of the variety of FDD phantoms considered, LEI appeared to tolerate viscoelastic data-model mismatch better than deviations caused by poroelastic and anisotropic mechanical properties in terms of visual image contrast. However, the estimated shear modulus values were substantially incorrect relative to independent mechanical measurements even in the successful viscoelastic cases and the variations in mean values with changes in experimental conditions associated with uniform phase shifts, driving signal frequency and amplitude were unpredictable. Overall, use of LEI to reconstruct data acquired in phantoms with FDD material properties provided biased results under the best conditions and significant artifacts in the worst cases. These findings suggest that the success with which LEI is applied to MRE data in tissue will depend on the underlying mechanical characteristics of the tissues and/or organs systems of clinical interest.

Magnetic resonance poroelastography: an algorithm for estimating the mechanical properties of fluid-saturated soft tissues

Magnetic resonance poroelastography (MRPE) is introduced as an alternative to single-phase model-based elastographic reconstruction methods. A 3-D finite element poroelastic inversion algorithm was developed to recover the mechanical properties of fluid-saturated tissues. The performance of this algorithm was assessed through a variety of numerical experiments, using synthetic data to probe its stability and sensitivity to the relevant model parameters. Preliminary results suggest the algorithm is robust in the presence of noise and capable of producing accurate assessments of the underlying mechanical properties in simulated phantoms. Furthermore, a 3-D time-harmonic motion field was recorded for a poroelastic phantom containing a single cylindrical inclusion and used to assess the feasibility of MRPE image reconstruction from experimental data. The elastograms obtained from the proposed poroelastic algorithm demonstrate significant improvement over linearly elastic MRE images generated using the same data. In addition, MRPE offers the opportunity to estimate the time-harmonic pressure field resulting from tissue excitation, highlighting the potential for its application in the diagnosis and monitoring of disease processes associated with changes in interstitial pressure.

Optimized motion estimation for MRE data with reduced motion encodes

Motion estimation is an essential step common to all magnetic resonance elastography (MRE) methods. For dynamic techniques, the motion is obtained from a sinusoidal fit of the image phase at multiple, uniformly spaced relative phase offsets, phi, between the motion and the motion encoding gradients (MEGs). Generally, eight values of phi sampled at the Nyquist interval pi/4 over [0, 2pi). We introduce a method, termed reduced motion encoding (RME), that reduces the number of phi required, thereby reducing the imaging time for an MRE acquisition. A frequency-domain algorithm was implemented using the discrete Fourier transform (DFT) to derive the general least-squares solution for the motion amplitude and phase given an arbitrary number of phi. A closed form representation of the condition number of the transformation matrix which is used for estimating motion was introduced to determine the sensitivity to noise for different sampling patterns of phi. Simulation results confirmed the minimum error sampling patterns suggested from the condition number maps. The minimum noise in the motion estimate is obtained when the sampled phi are essentially evenly distributed over the range [0, pi) with an interval pi/n, where n is the number of phi sampled, or alternatively with an interval 2pi/n over the range [0, 2pi) which represents the Nyquist interval. Simulations also show that the noise level decreases as n increases as expected. The decrease in noise is the largest when n is small and it becomes less significant as n increases. The algorithm also makes it possible to estimate the motion from only two values of phi, which cannot be accomplished with traditional methods because sampling at the Nyquist interval is indeterminate. Finally, noise levels in motion estimated from phantom studies and in vivo results taken with different n agreed with that predicted by simulation and condition number calculations.