This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

The flow of homogeneous fluids through porous media: analogies with other physical problems

The variable projection algorithm of Golub and Pereyra (SIAM J. Numer. Anal. 10:413---432, 1973) has proven to be quite valuable in the solution of nonlinear least squares problems in which a substantial number of the parameters are linear. Its advantages are efficiency and, more importantly, a better likelihood of finding a global minimizer rather than a local one. The purpose of our work is to provide a more robust implementation of this algorithm, include constraints on the parameters, more clearly identify key ingredients so that improvements can be made, compute the Jacobian matrix more accurately, and make future implementations in other languages easy.

Variable Projection for Nonlinear Least Squares Problems

In this article, two novel deep learning methods are proposed for displacement estimation in ultrasound elastography (USE). Although convolutional neural networks (CNNs) have been very successful for displacement estimation in computer vision, they have been rarely used for USE. One of the main limitations is that the radio frequency (RF) ultrasound data, which is crucial for precise displacement estimation, has vastly different frequency characteristics compared with images in computer vision. Top-rank CNN methods used in computer vision applications are mostly based on a multilevel strategy, which estimates finer resolution based on coarser ones. This strategy does not work well for RF data due to its large high-frequency content. To mitigate the problem, we propose modified pyramid warping and cost volume network (MPWC-Net) and RFMPWC-Net, both based on PWC-Net, to exploit information in RF data by employing two different strategies. We obtained promising results using networks trained only on computer vision images. In the next step, we constructed a large ultrasound simulation database and proposed a new loss function to fine-tune the network to improve its performance. The proposed networks and well-known optical flow networks as well as state-of-the-art elastography methods are evaluated using simulation, phantom, and in vivo data. Our two proposed networks substantially outperform current deep learning methods in terms of contrast-to-noise ratio (CNR) and strain ratio (SR). Also, the proposed methods perform similar to the state-of-the-art elastography methods in terms of CNR and have better SR by substantially reducing the underestimation bias.

Displacement Estimation in Ultrasound Elastography Using Pyramidal Convolutional Neural Network

Due to its real-time, non-radiation based three-dimensional (3D) imaging capabilities, ultrasound (US) has been incorporated into various orthopedic procedures. However, imaging artifacts, low signal-to-noise ratio (SNR) and bone boundaries appearing several mm in thickness make the analysis of US data difficult. This paper provides a review about the state-of-the-art bone segmentation and enhancement methods developed for two-dimensional (2D) and 3D US data. First, an overview for the appearance of bone surface response in B-mode data is presented. Then, classification of the proposed techniques in terms of the image information being used is provided. Specifically, the focus is given on segmentation and enhancement of B-mode US data. The review is concluded by discussing future directions of research and additional challenges which need to be overcome in order to make this imaging modality more successful in orthopedics.

/pdf/ultrasound-imaging-and-segmentation-of-bone-surfaces-a-317xsn5yhg.pdf

Ultrasound imaging and segmentation of bone surfaces: A review.

Ultrasound poroelastography aims at assessing the poroelastic behavior of biological tissues via estimation of the local temporal axial strains and effective Poisson’s ratios (EPR). Currently, reliable estimation of EPR using ultrasound is a challenging task due to the limited quality of lateral strain estimation. In this paper, we propose a new two-step EPR estimation technique based on dynamic programming elastography (DPE) and Horn-Schunck (HS) optical flow estimation. In the proposed method, DPE is used to estimate the integer axial and lateral displacements while HS is used to obtain subsample axial and lateral displacements from the motion-compensated pre-compressed and post-compressed radio frequency data. Axial and lateral strains are then calculated using Kalman filter-based least square estimation. The proposed two-step technique was tested using finite-element simulations, controlled experiments and in vivo experiments, and its performance was statistically compared with that of analytic minimization (AM) and correlation-based method (CM). Our results indicate that our technique provides EPR elastograms of higher quality and accuracy than those produced by AM and CM. Regarding signal-to-noise ratio and elastographic contrast-to-noise ratio, in simulated data, the proposed method provides an average improvement of 30% and 75%, respectively, with respect to AM and of 100% and 169%, respectively, with respect to CM, whereas, in experiments, the proposed approach provides an average improvement of 30% and 67% with respect to AM and of 230% and 525% with respect to CM. Based on these results, the proposed method may be the preferred one in experimental poroelastography applications.

A New Method for Estimating the Effective Poisson’s Ratio in Ultrasound Poroelastography

The mechanical behavior of biological tissues has been studied using a number of mechanical models. Due to the relatively high fluid content and mobility, many biological tissues have been modeled as poroelastic materials. Diseases such as cancers are known to alter the poroelastic response of a tissue. Tissue poroelastic properties such as compressibility, interstitial permeability and fluid pressure also play a key role for the assessment of cancer treatments and for improved therapies. At the present time, however, a limited number of poroelastic models for soft tissues are retrievable in the literature, and the ones available are not directly applicable to tumors as they typically refer to uniform tissues. In this paper, we report the analytical poroelastic model for a non-uniform tissue under stress relaxation. Displacement, strain and fluid pressure fields in a cylindrical poroelastic sample containing a cylindrical inclusion during stress relaxation are computed. Finite element simulations are then used to validate the proposed theoretical model. Statistical analysis demonstrates that the proposed analytical model matches the finite element results with less than 0.5% error. The availability of the analytical model and solutions presented in this paper may be useful to estimate diagnostically relevant poroelastic parameters such as interstitial permeability and fluid pressure, and, in general, for a better interpretation of clinically-relevant ultrasound elastography results.

https://iopscience.iop.org/article/10.1088/1361-6560/aa9631/pdf

An analytical poroelastic model for ultrasound elastography imaging of tumors.

We report on the use of elastographic imaging techniques to assess the bone/soft tissue interface, a region that has not been previously investigated but may provide important information about fracture and bone healing. The performance of axial strain elastograms and axial shear strain elastograms at the bone/soft tissue interface was studied ex vivo on intact and fractured canine and ovine tibias. Selected ex vivo results were corroborated on intact sheep tibias in vivo. The elastography results were statistically analyzed using elastographic image quality tools. The results of this study demonstrate distinct patterns in the distribution of the normalized local axial strains and axial shear strains at the bone/soft tissue interface with respect to the background soft tissue. They also show that the relative strength and distribution of the elastographic parameters change in the presence of a fracture and depend on the degree of misalignment between the fracture fragments. Thus, elastographic imaging modalities might be used in the future to obtain information regarding the integrity of bones and to assess the severity of fractures, alignment of bone fragments as well as to follow bone healing.

https://iopscience.iop.org/article/10.1088/0031-9155/61/1/131/pdf

Ultrasound elastography assessment of bone/soft tissue interface

Effect of Boundary Conditions on Performance of Poroelastographic Imaging Techniques in Non Homogenous Poroelastic Media

Elastography is a well-established imaging modality. While a number of studies aimed at evaluating the performance of elastographic techniques are retrievable in the literature, very little information is available on the effects that the presence of an underlying permeability contrast in the tissue may have on the resulting elastograms. Permeability is a fundamental tissue parameter, which characterizes the ease with which fluid can move within a tissue. This parameter plays a central role both biomechanically in the description of the temporal behavior of fluid-filled tissues and clinically in the development of a number of diagnostic and therapeutic modalities. In this paper, we present a simulation study that investigates selected elastographic image quality factors in nonhomogeneous materials, modeled as poroelastic media with different geometries and permeability contrasts. The results of this study indicate that the presence of an underlying permeability contrast may create a new contrast mechanism in the spatial and temporal distributions of the axial strains and the effective Poisson's ratios experienced by the tissue and as imaged by the corresponding elastograms. The effect of permeability on the elastographic image quality factors analyzed in this study was found to be a nonsymmetric function of the underlying mechanical contrast between background and target, the geometry of the material and the boundary conditions.

Anuj Chaudhry

Papers

A New Method for Estimating the Effective Poisson’s Ratio in Ultrasound Poroelastography

An analytical poroelastic model for ultrasound elastography imaging of tumors.

Ultrasound elastography assessment of bone/soft tissue interface

Effect of Boundary Conditions on Performance of Poroelastographic Imaging Techniques in Non Homogenous Poroelastic Media

Effect of Permeability on the Performance of Elastographic Imaging Techniques