Accelerated regularized estimation of MR coil sensitivities using augmented Lagrangian methods

Several magnetic resonance parallel imaging techniques require explicit estimates of the receive coil sensitivity profiles. These estimates must be accurate over both the object and its surrounding regions to avoid generating artifacts in the reconstructed images. Regularized estimation methods that involve minimizing a cost function containing both a data-fit term and a regularization term provide robust sensitivity estimates. However, these methods can be computationally expensive when dealing with large problems. In this paper, we propose an iterative algorithm based on variable splitting and the augmented Lagrangian method that estimates the coil sensitivity profile by minimizing a quadratic cost function. Our method, ADMM-Circ, reformulates the finite differencing matrix in the regularization term to enable exact alternating minimization steps. We also present a faster variant of this algorithm using intermediate updating of the associated Lagrange multipliers. Numerical experiments with simulated and real data sets indicate that our proposed method converges approximately twice as fast as the preconditioned conjugate gradient method over the entire field-of-view. These concepts may accelerate other quadratic optimization problems.

Numerical Approximation of Solutions of a Nonlinear Inverse Problem Arising in Olfaction Experimentation

Identification of detailed features of neuronal systems is an important challenge in the biosciences today. Cilia are long thin structures that extend from the olfactory receptor neurons into the nasal mucus. Transduction of an odor into an electrical signal occurs in the membranes of the cilia. The cyclic-nucleotide-gated (CNG) channels which reside in the ciliary membrane and are activated by adenosine 3',5'-cyclic monophosphate (cAMP) allow a depolarizing influx of Ca(2+) and Na(+) and are thought to initiate the electrical signal.In this paper, a mathematical model consisting of two nonlinear differential equations and a constrained Fredholm integral equation of the first kind is developed to model experiments involving the diffusion of cAMP into cilia and the resulting electrical activity. The unknowns in the problem are the concentration of cAMP, the membrane potential and, the quantity of most interest in this work, the distribution of CNG channels along the length of a cilium. A simple numerical method is derived that can be used to obtain estimates of the spatial distribution of CNG ion channels along the length of a cilium. Certain computations indicate that this mathematical problem is ill-conditioned.

Director alignment at the nematic-isotropic interface: elastic anisotropy and active anchoring

Activity in nematics drives interfacial flows that lead to preferential alignment that is tangential or planar for extensile systems (pushers) and perpendicular or homeotropic for contractile ones (pullers). This alignment is known as active anchoring and has been reported for a number of systems and described using active nematic hydrodynamic theories. The latter are based on the one-elastic constant approximation, i.e. they assume elastic isotropy of the underlying passive nematic. Real nematics, however, have different elastic constants, which lead to interfacial anchoring. In this paper, we consider elastic anisotropy in multiphase and multicomponent hydrodynamic models of active nematics and investigate the competition between the interfacial alignment driven by the elastic anisotropy of the passive nematic and the active anchoring. We start by considering systems with translational invariance to analyse the alignment at flat interfaces and, then, consider two-dimensional systems and active nematic droplets. We investigate the competition of the two types of anchoring over a wide range of the other parameters that characterize the system. The results of the simulations reveal that the active anchoring dominates except at very low activities, when the interfaces are static. In addition, we found that the elastic anisotropy does not affect the dynamics but changes the active length that becomes anisotropic. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Numerical Study of Electro-Osmotic Fluid Flow and Vortex Formation

The phenomenon of electro-osmosis was studied by performing numerical simulations on the flow between parallel walls and at the nozzle microchannels. In this work, we propose a numerical approximation to perform simulations of vortex formation which occur after the passage of the fluid through an abrupt contraction at the microchannel. The motion of the charges in the solution is described by the Poisson-Nernst-Planck equations and used the generalized finite differences to solve the numerical problem. First, solutions for electro-osmotic flow were obtained for the Phan-Thien/Thanner model in a parallel walls channel. Later simulations for electro-osmotic flow were performed in a nozzle. The formation of vortices near the contraction within the nozzle was verified by taking into account a flow perturbation model.

Numerical approximation of the observed information matrix with Oakes' identity

An efficient and accurate numerical approximation methodology useful for obtaining the observed information matrix and subsequent asymptotic covariance matrix when fitting models with the EM algorithm is presented. The numerical approximation approach is compared to existing algorithms intended for the same purpose, and the computational benefits and accuracy of this new approach are highlighted. Instructive and real-world examples are included to demonstrate the methodology concretely, properties of the estimator are discussed in detail, and a Monte Carlo simulation study is included to investigate the behaviour of a multi-parameter item response theory model using three competing finite-difference algorithms.

A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper, we present a method based on Radial Basis Function (RBF)-generated Finite Differences (FD) for numerically solving diffusion and reaction-diffusion equations (PDEs) on closed surfaces embedded in ℝ d . Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.

A Mathematical Model of a Novel 3D Fractal-Inspired Piezoelectric Ultrasonic Transducer

Piezoelectric ultrasonic transducers have the potential to operate as both a sensor and as an actuator of ultrasonic waves. Currently, manufactured transducers operate effectively over narrow bandwidths as a result of their regular structures which incorporate a single length scale. To increase the operational bandwidth of these devices, consideration has been given in the literature to the implementation of designs which contain a range of length scales. In this paper, a mathematical model of a novel Sierpinski tetrix fractal-inspired transducer for sensor applications is presented. To accompany the growing body of research based on fractal-inspired transducers, this paper offers the first sensor design based on a three-dimensional fractal. The three-dimensional model reduces to an effective one-dimensional model by allowing for a number of assumptions of the propagating wave in the fractal lattice. The reception sensitivity of the sensor is investigated. Comparisons of reception force response (RFR) are performed between this novel design along with a previously investigated Sierpinski gasket-inspired device and standard Euclidean design. The results indicate that the proposed device surpasses traditional design sensors.

Modelling the Molecular Permeation through Mixed-Matrix Membranes Incorporating Tubular Fillers

Membrane-based processes are considered a promising separation method for many chemical and environmental applications such as pervaporation and gas separation. Numerous polymeric membranes have been used for these processes due to their good transport properties, ease of fabrication, and relatively low fabrication cost per unit membrane area. However, these types of membranes are suffering from the trade-off between permeability and selectivity. Mixed-matrix membranes, comprising a filler phase embedded into a polymer matrix, have emerged in an attempt to partly overcome some of the limitations of conventional polymer and inorganic membranes. Among them, membranes incorporating tubular fillers are new nanomaterials having the potential to transcend Robeson's upper bound. Aligning nanotubes in the host polymer matrix in the permeation direction could lead to a significant improvement in membrane permeability. However, although much effort has been devoted to experimentally evaluating nanotube mixed-matrix membranes, their modelling is mostly based on early theories for mass transport in composite membranes. In this study, the effective permeability of mixed-matrix membranes with tubular fillers was estimated from the steady-state concentration profile within the membrane, calculated by solving the Fick diffusion equation numerically. Using this approach, the effects of various structural parameters, including the tubular filler volume fraction, orientation, length-to-diameter aspect ratio, and permeability ratio were assessed. Enhanced relative permeability was obtained with vertically aligned nanotubes. The relative permeability increased with the filler-polymer permeability ratio, filler volume fraction, and the length-to-diameter aspect ratio. For water-butanol separation, mixed-matrix membranes using polydimethylsiloxane with nanotubes did not lead to performance enhancement in terms of permeability and selectivity. The results were then compared with analytical prediction models such as the Maxwell, Hamilton-Crosser and Kang-Jones-Nair (KJN) models. Overall, this work presents a useful tool for understanding and designing mixed-matrix membranes with tubular fillers.

Finite difference magnetoelastic simulator

We describe an extension of the micromagnetic finite difference simulation software MuMax3 to solve elasto-magneto-dynamical problems. The new module allows for numerical simulations of magnetization and displacement dynamics in magnetostrictive materials and structures, including both direct and inverse magnetostriction. The theoretical background is introduced, and the implementation of the extension is discussed. The magnetoelastic extension of MuMax3 is freely available under the GNU General Public License v3.

Compensated optimal grids for elliptic boundary-value problems

A method is proposed which allows to efficiently treat elliptic problems on unbounded domains in two and three spatial dimensions in which one is only interested in obtaining accurate solutions at the domain boundary. The method is an extension of the optimal grid approach for elliptic problems, based on optimal rational approximation of the associated Neumann-to-Dirichlet map in Fourier space. It is shown that, using certain types of boundary discretization, one can go from second-order accurate schemes to essentially spectrally accurate schemes in two-dimensional problems, and to fourth-order accurate schemes in three-dimensional problems without any increase in the computational complexity. The main idea of the method is to modify the impedance function being approximated to compensate for the numerical dispersion introduced by a small finite-difference stencil discretizing the differential operator on the boundary. We illustrate how the method can be efficiently applied to nonlinear problems arising in modeling of cell communication.

Electrical properties based B 1 + prediction for electrical properties tomography reconstruction evaluation

PURPOSE

In MR electrical properties tomography (EPT), conductivity and permittivity are reconstructed from MR measurements. However, depending on the reconstruction method, reconstructed electrical properties (EPs) show large variability in vivo, reducing confidence in the reconstructed values for clinical application in practice. To overcome this problem we present a method to evaluate the reconstructed EPs using a physics-basedB 1 + $$ {\mathrm{B}}_1^{+} $$ estimation model.

METHODS

A physics-based method using a finite difference based recurrent relation is used to estimate theB 1 + $$ {\mathrm{B}}_1^{+} $$ field from a set of given EPs and the boundary of the measuredB 1 + $$ {\mathrm{B}}_1^{+} $$ field. Reconstructed EPs can be evaluated by comparing the estimatedB 1 + $$ {\mathrm{B}}_1^{+} $$ field with the measuredB 1 + $$ {\mathrm{B}}_1^{+} $$ field. The method was first validated in simulations and afterward tested using MRI data from phantoms and in vivo.

RESULTS

The simulation experiments show that theB 1 + $$ {\mathrm{B}}_1^{+} $$ field can be accurately estimated, within 90 s for a typical brain at 1 mm3 isotropic resolution, when correct EPs are used as input. When incorrect EPs are used as input the estimatedB 1 + $$ {\mathrm{B}}_1^{+} $$ fields shows differences with the measuredB 1 + $$ {\mathrm{B}}_1^{+} $$ fields. These differences directly correspond to the errors in the underlying EPs, enabling detection of errors in the reconstructions. The results obtained in MRI experiments using phantoms and in vivo show the applicability of the method in practice.

CONCLUSION

With the proposed method,B 1 + $$ {\mathrm{B}}_1^{+} $$ fields can be accurately estimated from EPs. This approach can be used to evaluate EPT reconstructions and consequently gain more confidence in reconstructed EPs values in vivo.

A Nodal Immersed Finite Element-Finite Difference Method

The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. The IFED method uses a finite element (FE) method to approximate the stresses, forces, and structural deformations on a structural mesh and a finite difference (FD) method to approximate the momentum and enforce incompressibility of the entire fluid-structure system on a Cartesian grid. The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. With an FE structural mechanics framework, force spreading first requires that the force itself be projected onto the finite element space. Similarly, velocity interpolation requires projecting velocity data onto the FE basis functions. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. This paper provides both numerical and computational analyses of the effects of this replacement for evaluating the force projection and for the IFED coupling operators. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the IFED coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from standard FE methods, which require specialized treatments to accommodate mass lumping with higher-order shape functions. Our theoretical results are confirmed by numerical benchmarks, including standard solid mechanics tests and examination of a dynamic model of a bioprosthetic heart valve.

Analysis of periodic pulsating blood flow of fractional Maxwell power-law fluid in carotid artery with elastic vessel wall

Hemodynamic analysis reveals a highly significant effect on the prevention, diagnosis, and treatment of human vascular diseases. This article goes deeply into the periodic pulsatile blood flow in the carotid artery with an elastic vessel wall. In view of blood rheological experimental data, the constitutive equation of fractional Maxwell power-law fluid with yield stress, which can describe the four characteristics of yield stress, viscoelasticity, shear thinning, and thixotropy is established. Meanwhile, drawing support from the data of pulsatile flow, the finite Fourier series of pressure gradient with a period of 1 s has been proposed. Leading into Hooke's law can build the fluid-structure coupling boundary condition of blood flow and elastic vessel wall. The numerical solutions are got hold of finite difference method integrated with the newly developed L1-algorithm, and their convergence and stability of which are verified. The axial velocities of blood under different constitutive relationships are compared. The results throw light that other constitutive relationships underestimate the velocity of blood. Furthermore, the flow rate and wall shear stress on different fluid are calculated. It can be concluded that compared with Bingham fluid, the maximum and minimum flow rate/wall shear stress of fractional Maxwell power-law fluid with yield stress increases by 19% and 32%, respectively. The flow rate lags behind the pressure gradient and has time delay effect, on the contrary, the velocity of blood vessel wall is keeping pace with the pressure gradient. The effects of relevant physical parameters on velocity are discussed. In addition, the spatiotemporal distribution of blood flow in cerebral artery and femoral artery are analyzed.

Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models

In this work, an efficient and robust numerical scheme is proposed to solve the variable coefficients’ fourth-order partial differential equations (FOPDEs) that arise in Euler–Bernoulli beam models. When partial differential equations (PDEs) are of higher order and invoke variable coefficients, then the numerical solution is quite a tedious and challenging problem, which is our main concern in this paper. The current scheme is hybrid in nature in which the second-order finite difference is used for temporal discretization, while spatial derivatives and solutions are approximated via the Haar wavelet. Next, the integration and Haar matrices are used to convert partial differential equations (PDEs) to the system of linear equations, which can be handled easily. Besides this, we derive the theoretical result for stability via the Lax–Richtmyer criterion and verify it computationally. Moreover, we address the computational convergence rate, which is near order two. Several test problems are given to measure the accuracy of the suggested scheme. Computations validate that the present scheme works well for such problems. The calculated results are also compared with the earlier work and the exact solutions. The comparison shows that the outcomes are in good agreement with both the exact solutions and the available results in the literature.

Acoustic Forward Model for Guided Wave Propagation and Scattering in a Pipe Bend

The sections of pipe bends are hot spots for wall thinning due to accelerated corrosion by fluid flow. Conventionally, the thickness of a bend wall is evaluated by local point-by-point ultrasonic measurement, which is slow and costly. Guided wave tomography is an attractive method that enables the monitoring of a whole bend area by processing the waves excited and received by transducer arrays. The main challenge associated with the tomography of the bend is the development of an appropriate forward model, which should simply and efficiently handle the wave propagation in a complex bend model. In this study, we developed a two-dimensional (2D) acoustic forward model to replace the complex three-dimensional (3D) bend domain with a rectangular domain that is made artificially anisotropic by using Thomsen parameters. Thomsen parameters allow the consideration of the directional dependence of the velocity of the wave in the model. Good agreement was found between predictions and experiments performed on a 220 mm diameter (d) pipe with 1.5d bend radius, including the wave-field focusing effect and the steering effect of scattered wave-fields from defects.