A subspace time-domain algorithm for automated NMR spectral normalization

Recently, two methods have been proposed for quantitatively comparing NMR spectra of control and treated samples, in order to examine the possible occurring variations in cell metabolism and/or structure in response to numerous physical, chemical, and biological agents. These methods are the maximum superposition normalization algorithm (MaSNAl) and the minimum rank normalization algorithm (MiRaNAl). In this paper a new subspace-based time-domain normalization algorithm, denoted by SuTdNAl (subspace time-domain normalization algorithm), is presented. By the determination of the intersection of the column spaces of two Hankel matrices, the common signal poles and further on the components having proportionally varying amplitudes are detected. The method has the advantage that it is computationally less intensive than the MaSNAl and the MiRaNAl. Furthermore, no approximate estimate of the normalization factor is required. The algorithm was tested by Monte Carlo simulations on a set of simulation signals. It was shown that the SuTdNAl has a statistical performance similar to that of the MiRaNAl, which itself is an improvement over the MaSNAl. Furthermore, two samples of known contents are compared with the MiRaNAl, the SuTdNAl, and an older method using a standard. Finally, the SuTdNAl is tested on a realistic simulation example derived from an in vitro measurement on cells.     

Optimization of residual water signal removal by HLSVD on simulated short echo time proton MR spectra of the human brain

Suppression of the residual water signal from proton magnetic resonance (MR) spectra recorded in human brain is a prerequisite to an accurate quantification of cerebral metabolites. Several postacquisition methods of residual water signal suppression have been reported but none of them provide a complete elimination of the residual water signal, thereby preventing reliable quantification of brain metabolites. In the present study, the elimination of the residual water signal by the Hankel Lanczos singular value decomposition method has been evaluated and optimized to provide fast automated processing of spectra. Model free induction decays, reproducing the proton signal acquired in human brain localized MR spectroscopy at short echo times (e.g., 20 ms), have been generated. The optimal parameters in terms of number of components and dimension of the Hankel data matrix allowing complete elimination of the residual water signal are reported.     

Application of high-resolution magic-angle spinning NMR spectroscopy to define the cell uptake of MRI contrast agents

A new method, based on proton high-resolution magic-angle spinning ((1)H HR-MAS) NMR spectroscopy, has been employed to study the cell uptake of magnetic resonance imaging contrast agents (MRI-CAs). The method was tested on human red blood cells (HRBC) and white blood cells (HWBC) by using three gadolinium complexes, widely used in diagnostics, Gd-BOPTA, Gd-DTPA, and Gd-DOTA, and the analogous complexes obtained by replacing Gd(III) with Dy(III), Nd(III), and Tb(III) (i.e., complexes isostructural to the ones of gadolinium but acting as shift agents). The method is based on the evaluation of the magnetic effects, line broadening, or induced lanthanide shift (LIS) caused by these complexes on NMR signals of intra- and extracellular water. Since magnetic effects are directly linked to permeability, this method is direct. In all the tests, these magnetic effects were detected for the extracellular water signal only, providing a direct proof that these complexes are not able to cross the cell membrane. Line broadening effects (i.e., the use of gadolinium complexes) only allow qualitative evaluations. On the contrary, LIS effects can be measured with high precision and they can be related to the concentration of the paramagnetic species in the cellular compartments. This is possible because the HR-MAS technique provides the complete elimination of bulk magnetic susceptibility (BMS) shift and the differentiation of extra- and intracellular water signals. Thus with this method, the rapid quantification of the MRI-CA amount inside and outside the cells is actually feasible.     

Improved Lanczos algorithms for blackbox MRS data quantitation

Magnetic resonance spectroscopy (MRS) has been shown to be a potentially important medical diagnostic tool. The success of MRS depends on the quantitative data analysis, i.e., the interpretation of the signal in terms of relevant physical parameters, such as frequencies, decay constants, and amplitudes. A variety of time-domain algorithms to extract parameters have been developed. On the one hand, there are so-called blackbox methods. Minimal user interaction and limited incorporation of prior knowledge are inherent to this type of method. On the other hand, interactive methods exist that are iterative, require user involvement, and allow inclusion of prior knowledge. We focus on blackbox methods. The computationally most intensive part of these blackbox methods is the computation of the singular value decomposition (SVD) of a Hankel matrix. Our goal is to reduce the needed computational time without affecting the accuracy of the parameters of interest. To this end, algorithms based on the Lanczos method are suitable because the main computation at each step, a matrix-vector product, can be efficiently performed by means of the fast Fourier transform exploiting the structure of the involved matrix. We compare the performance in terms of accuracy and efficiency of four algorithms: the classical SVD algorithm based on the QR decomposition, the Lanczos algorithm, the Lanczos algorithm with partial reorthogonalization, and the implicitly restarted Lanczos algorithm. Extensive simulation studies show that the latter two algorithms perform best.     

Moving Mesh Methods in Multiple Dimensions Based on Harmonic Maps

In practice, there are three types of adaptive methods using the finite element approach, namely the h-method, p-method, and r-method. In the h-method, the overall method contains two parts, a solution algorithm and a mesh selection algorithm. These two parts are independent of each other in the sense that the change of the PDEs will affect the first part only. However, in some of the existing versions of the r-method (also known as the moving mesh method), these two parts are strongly associated with each other and as a result any change of the PDEs will result in the rewriting of the whole code. In this work, we will propose a moving mesh method which also contains two parts, a solution algorithm and a mesh-redistribution algorithm. Our efforts are to keep the advantages of the r-method (e.g., keep the number of nodes unchanged) and of the h-method (e.g., the two parts in the code are independent). A framework for adaptive meshes based on the Hamilton–Schoen–Yau theory was proposed by Dvinsky. In this work, we will extend Dvinsky's method to provide an efficient solver for the mesh-redistribution algorithm. The key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure. Each iteration step is to move the mesh closer to the harmonic map. This procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration. The numerical schemes are applied to a number of test problems in two dimensions. It is observed that the mesh-redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi-dimensional computations.     

A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes

A new finite volume method is presented for discretizing general linear or nonlinear elliptic second-order partial-differential equations with mixed boundary conditions. The advantage of this method is that arbitrary distorted meshes can be used without the numerical results being altered. The resulting algorithm has more unknowns than standard methods like finite difference or finite element methods. However, the matrices that need to be inverted are positive definite, so the most powerful linear solvers can be applied. The method has been tested on a few elliptic and parabolic equations, either linear, as in the case of the standard heat diffusion equation, or nonlinear, as in the case of the radiation diffusion equation and the resistive diffusion equation with Hall term.     

Nonlinear Landau Damping in Spherically Symmetric Vlasov Poisson Systems

The Vlasov Poisson system is a partial differential equation widely used to describe collisionless plasma. It is formulated in a six-dimensional phase space, this prohibits a numerical solution on a complete phase space grid. In some applications, however, spherical symmetry is given, which introduces singularities into the Vlasov Poisson equation. We focus on such problems and propose a stable algorithm using accommodating boundaries. At first, the method is tested in the linear regime, where analytical solutions are available. Thereafter it is applied to large disturbances from equilibrium.     

A New Unconditionally Stable Algorithm for Steady-State Fluid Simulation of High Density Plasma Discharge

A new unconditionally stable algorithm for steady-state fluid simulation of high density plasma discharge is suggested. The physical origin of restriction on simulation time step is discussed and a new method to overcome it is explained. To compare the new method with previous other methods, a one-dimensional fluid simulation of inductively coupled plasma discharge is performed.     

Wave Simulation in Frozen Porous Media

We propose a numerical algorithm for simulation of wave propagation in frozen porous media, where the pore space is filled with ice and water. The model, based on a Biot-type three-phase theory, predicts three compressional waves and two shear waves and models the attenuation level observed in rocks. Attenuation is modeled with exponential relaxation functions which allow a differential formulation based on memory variables. The wavefield is obtained using a grid method based on the Fourier differential operator and a Runge–Kutta time-integration algorithm. Since the presence of slow quasistatic modes makes the differential equations stiff, a time-splitting integration algorithm is used to solve the stiff part analytically. The modeling is second-order accurate in the time discretization and has spectral accuracy in the calculation of the spatial derivatives.     

Positive Numerical Integration Methods for Chemical Kinetic Systems

Chemical kinetics conserves mass and renders nonnegative solutions; a good numerical simulation would ideally produce mass-balanced, positive concentration vectors. Many time-stepping methods are mass conservative; however, unconditional positivity restricts the order of a traditional method to one. The projection method presented in this paper ensures mass conservation and positivity. First, a numerical approximation is computed with one step of a mass-preserving traditional scheme. If there are negative components, the nearest vector in the reaction simplex is found by solving a quadratic optimization problem; this vector is shown to better approximate the true solution. A simpler version involves just one projection step and stabilizes the reaction simplex. This technique works best when the underlying time-stepping scheme favors positivity. Projected methods are more accurate than clipping and allow larger time steps for kinetic systems which are unstable outside the positive quadrant.     

Multigrid Solution of the Incompressible Navier–Stokes Equations for Density-Stratified Flow past Three-Dimensional Obstacles

The steady incompressible Navier–Stokes equations in three dimensions are solved for neutral and stably stratified flow past three-dimensional obstacles of increasing spanwise width. The continuous equations are approximated using a finite volume discretisation on staggered grids with a flux-limited monotonic scheme for the advective terms. The discrete equations which arise are solved using a nonlinear multigrid algorithm with up to four grid levels using the SIMPLE pressure correction method as smoother. When at its most effective the multigrid algorithm is demonstrated to yield convergence rates which are independent of the grid density. However, it is found that the asymptotic convergence rate depends on the choice of the limiter used for the advective terms of the density equation, and some commonly used schemes are investigated. The variation with obstacle width of the influence of the stratification on the flow field is described and the results of the three-dimensional computations are compared with those of the corresponding computation of flow over a two-dimensional obstacle (of effectively infinite width). Also given are the results of time-dependent computations for three-dimensional flows under conditions of strong static stability when lee-wave propagation is present and the multigrid algorithm is used to compute the flow at each time step.     

Determination of membrane Peptide orientation by 1H-detected 2H NMR spectroscopy

We demonstrate the application of the proton inverse detected deuteron (PRIDE) NMR technique to the measurement of the orientation of membrane-bound peptides with enhanced sensitivity. Gramicidin D, a transmembrane peptide, and ovispirin, a surface-bound peptide, were used as model systems. The peptides were 2H-labeled by 1H/2H exchange and oriented uniaxially on glass plates. The directly detected 2H spectra of both peptides showed only a strong D(2)O signal and no large quadrupolar splittings. In contrast, the PRIDE spectrum of gramicidin exhibited quadrupolar splittings as large as 281 kHz, consistent with its transmembrane orientation. Moreover, the large D(2)O signal in the directly detected 2H spectra was cleanly suppressed in the PRIDE spectrum. For ovispirin, the 1H indirectly detected 2H spectrum revealed a 104 kHz splitting and a zero-frequency peak. The former reflects the in-plane orientation of most of the helix axis, while the latter results from residues with a magic-angle orientation of the N-D bonds. These are consistent with previous 15N NMR results on ovispirin. The combination of PRIDE and exchange labeling provides an economical and sensitive method of studying membrane peptide orientations in lipid bilayers without the influence of D(2)O and with the ability to detect N-D bonds at the magic angle from the bilayer normal.     

Imaging the long-range dipolar field in structured liquid state samples

We describe imaging experiments in which the pattern of the dipolar field generated by spatially modulated nuclear magnetization is directly visualized in simply structured phantoms. Two types of experiment have been carried out at 11.7 T using (1)H NMR signals. In the first, the field from a single spin species is imaged via its own NMR signal. In the second, the NMR signal from one spin species is used to image the field generated by a second species. The field patterns measured in these experiments correspond well with those calculated using simple theoretical expressions for the dipolar field. The results also directly demonstrate the spatial sensitivity of the signal generated using dipolar field effects, indicating that the range of the field depends upon the inverse of the spatial frequency with which the magnetization is modulated.     

Completely Conservative and Oscillationless Semi-Lagrangian Schemes for Advection Transportation

In this paper, we present a new type of semi-Lagrangian scheme for advection transportation equation. The interpolation function is based on a cubic polynomial and is constructed under the constraints of conservation of cell-integrated average and the slope modification. The cell-integrated average is defined via the spatial integration of the interpolation function over a single grid cell and is advanced using a flux form. Nonoscillatory interpolation is constructed by choosing proper approximation to the cell-center values of the first derivative of the interpolation function, which appears to be a free parameter in the present formulation. The resulting scheme is exactly conservative regarding the cell average of the advected quantity and does not produce any spurious oscillation. Oscillationless solutions to linear transportation problems were obtained. Incorporated with an entropy-enforcing numerical flux, the presented schemes can accurately compute shocks and sonic rarefaction waves when applied to nonlinear problems.     

An Efficient Algorithm for Truncating Spatial Domain in Modeling Light Scattering by Finite-Difference Technique

The finite-difference time domain technique is one of the most robust and accurate numerical methods for the solution of light scattering by small particles with arbitrary composition and geometry. In practice, this method requires that the spatial domain for the computation of near-field be truncated. An absorbing boundary condition must be imposed in conjunction with this truncation. The performance of this boundary condition is essential to the stability of numerical computations and the reliability of results. In the present study, a new boundary condition, referred to as the mixed T algorithm, has been developed, which is a generalization of the transmitting boundary condition originally developed by Liao and co-workers. The present algorithm does not require spatial interpolation for wave values at interior grid points. In addition, it produces two minima of spurious reflections at small and large incident angles, allowing efficient absorption of the scattered waves at the boundary for large incident angles. When the third-order mixed T algorithm is used, the reflection coefficient of the boundary is less than 1% for incident angles from 0° to about 70°. We find that the numerical instability associated with the transmitting boundary condition is caused by the location-dependent amplitude of outgoing waves in the vicinity of the boundary. For this reason, the mixed T algorithm is stabilized by consistently introducing diffusive coefficients into the boundary equation. When the stabilized algorithm is applied, the near-field within the truncated domain can be computed by using single-precision arithmetic without overflows for more than 10⁵steps in the time-marching iteration. Finally, the new absorbing boundary condition is validated by carrying out numerical experiments involving the propagation of a TM wave excited by a sinusoidal point source, simultaneous simulation of the wave propagation in small and large domains, and the scattering of a TM wave by an infinite circular cylinder.     

A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows

We propose a new model and a solution method for two-phase compressible flows. The model involves six equations obtained from conservation principles applied to each phase, completed by a seventh equation for the evolution of the volume fraction. This equation is necessary to close the overall system. The model is valid for fluid mixtures, as well as for pure fluids. The system of partial differential equations is hyperbolic. Hyperbolicity is obtained because each phase is considered to be compressible. Two difficulties arise for the solution: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum and energy equations. We propose robust and accurate discretisation of these terms. The method solves the same system at each mesh point with the same algorithm. It allows the simulation of interface problems between pure fluids as well as multiphase mixtures. Several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible. The method provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.     

Tagging of the Magnetization with the Transition Zones of 360° Rotations Generated by a Tandem of Two Adiabatic DANTE Inversion Sequences

A new technique is presented for generating myocardial tagging using the signal intensity minima of the transition zones between the bands of 0° and 360° rotations, induced by a tandem of two adiabatic delays alternating with nutations for tailored excitation (DANTE) inversion sequences. With this approach, the underlying matrix corresponds to magnetization that has experienced 0° or 360° rotations. The DANTE sequences were implemented from adiabatic parent pulses for insensitivity of the underlying matrix to B₁ inhomogeneity. The performance of the proposed tagging technique is demonstrated theoretically with computer simulations and experimentally on phantom and on the canine heart, using a surface coil for both RF transmission and signal reception. The simulations and the experimental data demonstrated uniform grid contrast and sharp tagging profiles over a twofold variation of the B₁ field magnitude.     

The Adjoint Method for an Inverse Design Problem in the Directional Solidification of Binary Alloys

This paper presents a systematic procedure based on the adjoint method for solving a class of inverse directional alloy solidification design problems in which a desired growth velocityv_fis achieved under stable growth conditions. To the best of our knowledge, this is the first time that a continuum adjoint formulation is proposed for the solution of an inverse problem with simultaneous heat and mass transfer, thermo-solutal convection, and phase change. In this paper, the interfacial stability is considered to imply a sharp solid–liquid freezing interface. This condition is enforced using the constitutional undercooling criterion in the form of an inequality constraint between the thermal and solute concentration gradients,GandG_c, respectively, at the freezing front. The main unknowns of the design problem are the heating and/or cooling boundary conditions on the mold walls. The inverse design problem is formulated as a functional optimization problem. The cost functional is defined by the square of theL₂norm of the deviation of the freezing interface temperature from the temperature corresponding to thermodynamic equilibrium. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, and velocity fields such that the gradient of the cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the finite element method solutions of the continuum direct, sensitivity, and adjoint problems. The developed formulation is demonstrated with an example of designing the directional solidification of a binary aqueous solution in a rectangular mold such that a stable vertical interface advances from left to right with a desired growth velocity.     

Physics-Based Preconditioning and the Newton–Krylov Method for Non-equilibrium Radiation Diffusion

An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical methods, Jacobian-free Newton–Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, we employ a Jacobian–free implementation of Newton's method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov method used to solve the linear systems that come from the iterations of Newton's method. The preconditioner in this solution method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid methods. The preconditioner also addresses the strong coupling between equations with local 2×2 block solves. The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurring minimal memory requirements.