Mathematical model of the Bloch NMR flow equations for the analysis of fluid flow in restricted geometries using the Boubaker polynomials expansion scheme

In this study, the Bloch NMR flow equations are modelled into diffusion equation with constant transport coefficient in terms of the NMR transverse magnetization. Mathematical conditions are established for the diffusion coefficients to be constant or spatially varied with direction. When these conditions are met, the diffusion coefficients can then be easily evaluated in terms of Boubaker polynomials for the study of flow in restricted geometries.     

Generalized equation for describing the magnetization in spoiled gradient-echo imaging

The purpose of this study was to demonstrate a generalized equation for describing the magnetization in spoiled gradient-echo (SPGR) imaging in which the in-pulse relaxation and magnetization transfer (MT) effects are taken into account. First, the time-dependent Bloch equations for the two-pool exchange model with MT effect were reduced to an inhomogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation. Second, the equations describing the magnetization before and after the radiofrequency (RF) pulse were derived based on the above solution for the RF-pulse excitation and evolution phases. Finally, a generalized equation describing the steady-state magnetization was derived. The validity of this equation was investigated by comparing with the transverse magnetization obtained by the regular Ernst equation and analytical solution in which the in-pulse transverse relaxation is considered. When the same assumption was made in our method, there were good agreements between them, indicating the validity of our method. The in-pulse transverse and longitudinal relaxations decreased the transverse magnetization compared to the case in which these effects were neglected, whereas MT increased it. In conclusion, we derived a generalized equation for describing the magnetization in SPGR imaging. This equation will provide a suitable basis for understanding the signal intensity in SPGR imaging and/or T₁ measurement using an SPGR sequence in cases in which the effect of in-pulse relaxation and/or MT cannot be neglected.     

Computational modeling of MR flow imaging by the lattice Boltzmann method and Bloch equation

In this work, a computational model of magnetic resonance (MR) flow imaging is proposed. The first model component provides fluid dynamics maps by applying the lattice Boltzmann method. The second one uses the flow maps and couples MR imaging (MRI) modeling with a new magnetization transport algorithm based on the Eulerian coordinate approach. MRI modeling is based on the discrete time solution of the Bloch equation by analytical local magnetization transformations (exponential scaling and rotations).     

Generating spin turbulence through nonlinear excitation in liquid-state NMR

Chaotic dynamics of a water magnetization in a 600 MHz NMR spectrometer was generated by a radiation damping-based electronic feedback. Erratic induction signal was observed for several tens of seconds. The analysis of the data shows that this chaotic behaviour can be ascribed to spin turbulence in the sample and that a simpler model based on the three-dimensional Bloch equations modified to include a feedback field may not account for the experimental data.     

Efficient numerical methods for multiple surfactant-coated bubbles in a two-dimensional stokes flow

We present efficient and highly accurate numerical methods to compute the deformation of surfactant-coated, two-dimensional bubbles in a slow viscous flow. Surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion. In this paper, we restrict our attention to the case of a dilute insoluble surfactant. The convection–diffusion equation for the surfactant concentration on the interface is coupled with the Stokes equations in the fluid domain through a boundary condition based on the Laplace-Young condition. The Stokes equations are first recast as an integral equation and then solved using a fast-multipole accelerated iterative procedure. The computational cost per time-step is only O(N log N) operations, with N being the number of discretization points on the interface. The bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points. This equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient, implicit treatment of the stiffest terms in the dynamics. Several phenomenologically different examples are presented.     

Simulating magnetic nanoparticle behavior in low-field MRI under transverse rotating fields and imposed fluid flow

In the presence of alternating-sinusoidal or rotating magnetic fields, magnetic nanoparticles will act to realign their magnetic moment with the applied magnetic field. The realignment is characterized by the nanoparticle's time constant, τ. As the magnetic field frequency is increased, the nanoparticle's magnetic moment lags the applied magnetic field at a constant angle for a given frequency, Ω, in rad s⁻¹. Associated with this misalignment is a power dissipation that increases the bulk magnetic fluid's temperature which has been utilized as a method of magnetic nanoparticle hyperthermia, particularly suited for cancer in low-perfusion tissue (e.g., breast) where temperature increases of between 4 and 7 °C above the ambient in vivo temperature cause tumor hyperthermia. This work examines the rise in the magnetic fluid's temperature in the MRI environment which is characterized by a large DC field, B₀. Theoretical analysis and simulation is used to predict the effect of both alternating-sinusoidal and rotating magnetic fields transverse to B₀. Results are presented for the expected temperature increase in small tumors ( radius) over an appropriate range of magnetic fluid concentrations (0.002-0.01 solid volume fraction) and nanoparticle radii (1-10 nm). The results indicate that significant heating can take place, even in low-field MRI systems where magnetic fluid saturation is not significant, with careful the goal of this work is to examine, by means of analysis and simulation, the concept of interactive fluid magnetization using the dynamic behavior of superparamagnetic iron oxide nanoparticle suspensions in the MRI environment. In addition to the usual magnetic fields associated with MRI, a rotating magnetic field is applied transverse to the main B₀ field of the MRI. Additional or modified magnetic fields have been previously proposed for hyperthermia and targeted drug delivery within MRI. Analytical predictions and numerical simulations of the transverse rotating magnetic field in the presence of B₀ are investigated to demonstrate the effect of Ω, the rotating field frequency, and the magnetic field amplitude on the fluid suspension magnetization. The transverse magnetization due to the rotating transverse field shows strong dependence on the characteristic time constant of the fluid suspension, τ. The analysis shows that as the rotating field frequency increases so that Ωτ approaches unity, the transverse fluid magnetization vector is significantly non-aligned with the applied rotating field and the magnetization's magnitude is a strong function of the field frequency. In this frequency range, the fluid's transverse magnetization is controlled by the applied field which is determined by the operator. The phenomenon, which is due to the physical rotation of the magnetic nanoparticles in the suspension, is demonstrated analytically when the nanoparticles are present in high concentrations (1-3% solid volume fractions) more typical of hyperthermia rather than in clinical imaging applications, and in low MRI field strengths (such as open MRI systems), where the magnetic nanoparticles are not magnetically saturated. The effect of imposed Poiseuille flow in a planar channel geometry and changing nanoparticle concentration is examined. The work represents the first known attempt to analyze the dynamic behavior of magnetic nanoparticles in the MRI environment including the effects of the magnetic nanoparticle spin-velocity. It is shown that the magnitude of the transverse magnetization is a strong function of the rotating transverse field frequency. Interactive fluid magnetization effects are predicted due to non-uniform fluid magnetization in planar Poiseuille flow with high nanoparticle concentrations.     

Application of microwave amplitude modulation technique to measure spin-lattice and spin-spin relaxation times accurately with a continuous-wave EPR spectrometer: Solution of Bloch’s equations by a matrix technique and least-squares fitting

A matrix technique to calculate signals recorded using the microwave amplitude-modulation technique is described. The calculations are carried out for spin packets, on and off resonance, to take into account inhomogeneous broadening. Both, the transverse component of magnetization representing the continuous-wave signal in a resonator, such as a cross-looped resonator, as well as the signal (electromotive force) induced in a pickup coil oriented parallel to the external magnetic field, are calculated for an arbitrary value of the coefficient of modulation. This is accomplished by solving the relevant Bloch equations in the rotating frame for the case when the amplitude of the microwave field is modulated by a sinusoidal wave, using Fourier expansions of the longitudinal and transverse components of the magnetization in Bloch equations. This results in a series of coupled equations inM _α(n) (α=y,zz), the magnetic moments of vaarious orders, leading to a penta-diagonal matrix of infinite dimension. These equations are then truncated and solved by a fast matrix technique to calculateM _α(n), required to calculate the modulation signals as functions of the amplitudemodulation frequency Ω. It is outlined how to exploit the expressions for the modulation signals to estimate the spin-lattice relaxation timesT ₁ and spin-spin relaxation timesT ₂ accurately by the leastsquares procedure, fitting simultaneously all signals obtained for spin packets, on and off resonance, at various modulation frequencies. Illustrative examples are provided.     

1-D modeling of ion transport in rectangular nanofluidic channels

Based on the continuity hypothesis of fluid, 1-D mathematical models of ions’ transport in the rectangular nanofluidic channels are established by using the Poisson-Boltzmann (PB) equation and the modified Navier-Stokes (N-S) equations. The deduced equations are solved with MATLAB software. The results show that the distribution of the electric potential and the flow field could be predicted by the parameters, such as conductivity, surface charge density, solution concentration and channel height. The relationships between the parameters and the flow characteristics of the solution are also discussed. The research will help to the accurate manipulation of the solution in the nanofluidic channels.     

Spectral and temporal features of multiple spontaneous NMR-maser emissions

When a system composed of dissolved laser-polarized xenon with negative spin temperature is put inside a high field NMR magnet, a series of spontaneous maser emissions can be observed. We report here their spectral and temporal features using a processing model derived from the solution of the Bloch equations in the presence of radiation damping. We show, in particular, that by combining Fourier transformation and squared modulus, a parameter allowing the characterization of the burst of transverse magnetization can be determined. This parameter is shown to be correlated with the radiated energy. Moreover, this processing clearly reveals features which can probably be assigned to effects resulting from distant dipolar fields. Finally, the analysis of the experimental data reveals an unexpected behavior of the ¹²⁹Xe transverse self-relaxation.     

Nonlinear domain-wall dynamics in a system of two magnetic layers

The nonlinear dynamics of the magnetization in a spin-valve structure is investigated. Equations describing the dynamics of the magnetization in such a structure are obtained. The stability of the solution corresponding to a motionless flat domain wall is investigated. The nonlinear domain-wall dynamics are investigated in the approximation of a strong exchange interaction between the magnetic layers and in the approximation of a large magnetostatic energy. In the former case the nonlinear dynamical equations are shown to be similar to the equations describing the dynamics of the magnetization in a weak ferromagnet, and in the latter case they are similar to the equations of motion of a magnetic vortex (i.e., a vertical Bloch line) in a domain wall. Zh. éksp. Teor. Fiz. 116, 1365–1374 (October 1999)     

纵向弛豫对核磁共振线型的影响

根据经典 Bloch方程的解析解以及考虑辐射阻尼效应的Bloch方程的数值解,通过解析分析和数值模拟,从理论上研究了在射频场扰动下以及在辐射阻尼效应的作用下纵向弛豫对核磁共振线型的影响.结果表明:①射频场的扰动和辐射阻尼效应将导致纵向磁化与横向磁化的耦合,从而使纵向弛豫对线型产生了一定的影响.②在射频场的扰动下,峰强和线宽分别为2M0sin(θ)T1T2/(T1+T2)和(T1+T2)/(2πT1T2),即纵向弛豫将使谱线的峰强增大、线宽变窄,且影响程度随着比值T2/T1的减小而增大,峰强最大可增加1倍而线宽最多可减小1/2.③在强辐射阻尼效应的作用下,纵向弛豫会使谱线的峰强降低,降低的幅度与扳转角θ以及比值T2/T1密切相关.当θ从0到3π/4时,降低的幅度均较小,只有当θ＞3π/4时,降低的幅度才开始逐渐变大,且当 θ接近π时,降低的幅度急剧增大.谱线峰强降低的幅度与T2/T1呈较严格的正比关系,即T1越接近T2,峰强下降得越显著.     

Quantification of slow flow using FAIR

Phase contrast (PC)-based MRI methods are considered to be the most accurate approach for spatially resolved flow quantification, but the measurement of very slow velocities requires signal detection at long echo times and the application of strong field gradients. On the other hand, measurements based on time-of-flight or inflow effects can be conducted at short echo times and without flow-encoding gradients. A method for imaging flow at velocities of the order of 0.1 mm/s is presented and validated here. It consists of measuring the apparent spin-lattice relation rate (R₁*) of the flowing fluid using magnetization preparation by alternating slice-selective and nonselective inversion pulses (FAIR or flow-sensitive alternating inversion recovery) and a fast gradient-echo detection sequence. This method is appropriate for the quantitative imaging of slow flow at low Reynolds numbers in fluids where the T₂ values are too short to allow sensitive flow measurements by phase contrast-based methods.     

On MHD transient flow of a Maxwell fluid in a porous medium and rotating frame

Analytic solution for unsteady magnetohydrodynamic (MHD) flow is constructed in a rotating non-Newtonian fluid through a porous medium. Constitutive equations for a Maxwell fluid have been taken into consideration. The hydromagnetic flow in the uniformly rotating fluid is generated by a suddenly moved infinite plate in its own plane. Analytic solution of the governing flow problem is obtained by means of the Fourier sine transform. It is shown that the obtained solution satisfies both the associate partial differential equation and the initial and boundary conditions. The solution for a Navier-Stokes fluid is recovered if λ→0. The steady state solution is also obtained for t→∞.     

Time-of-flight flow imaging using NMR remote detection

A time-of-flight imaging technique is introduced to visualize fluid flow and dispersion through porous media using NMR. As the fluid flows through a sample, the nuclear spin magnetization is modulated by rf pulses and magnetic field gradients to encode the spatial coordinates of the fluid. When the fluid leaves the sample, its magnetization is recorded by a second rf coil. This scheme not only facilitates a time-dependent imaging of fluid flow, it also allows a separate optimization of encoding and detection subsystems to enhance overall sensitivity. The technique is demonstrated by imaging gas flow through a porous rock.     

Two-dimensional array of room-temperature nanophotonic logic gates using InAs quantum dots in mesa structures

We present a detailed study of the dynamics of light in passive nonlinear resonators with shallow and deep intracavity periodic modulation of the refractive index in both longitudinal and transverse directions of the resonator. We investigate solutions localized in the transverse direction (so-called Bloch cavity solitons) by means of envelope equations for underlying linear Bloch modes and solving Maxwell’s equations directly. Using a round-trip model for forward and backward propagating waves we review different types of Bloch cavity solitons supported by both focusing (at normal diffraction) and defocussing (at anomalous diffraction) nonlinearities in a cavity with a weak-contrast modulation of the refractive index. Moreover, we identify Bloch cavity solitons in a Kerr-nonlinear all-photonic crystal resonator solving Maxwell’s equations directly. In order to analyze the properties of Bloch cavity solitons and to obtain analytical access we develop a modified mean-field model and prove its validity. In particular, we demonstrate a substantial narrowing of Bloch cavity solitons near the zero-diffraction regime. Adjusting the quality factor and resonance frequencies of the resonator optimal Bloch cavity solitons in terms of width and pump energy are identified.     

Exact travelling solutions for some nonlinear physical models by ( <Emphasis Type="Italic">G</Emphasis>′/ <Emphasis Type="Italic">G</Emphasis>)-expansion method

In this paper, we establish exact solutions for some special nonlinear partial differential equations. The (G′/G)-expansion method is used to construct travelling wave solutions of the two-dimensional sine-Gordon equation, Dodd–Bullough–Mikhailov and Schrödinger–KdV equations, which appear in many fields such as, solid-state physics, nonlinear optics, fluid dynamics, fluid flow, quantum field theory, electromagnetic waves and so on. In this method we take the advantage of general solutions of second-order linear ordinary differential equation (LODE) to solve many nonlinear evolution equations effectively. The (G′/G)-expansion method is direct, concise and elementary and can be used with a wider applicability for handling many nonlinear wave equations.     

Nonlinear and multiquantum effects in dual-frequency EPR experiment: the intensities of sideband resonances

Two microwave frequencies ω₁ and ω₂ simultaneously exciting a paramagnetic spin probe near the resonance condition ω₁ = γB ₀ effect a pattern of resonances occurring at multiples of the difference frequency δ = ω₂ ? ω₁. Their intensities, measured in the absorption or dispersion mode in the magnetic field-sweep experiment, decrease rapidly as the distance from the centerband increases. Numerically solved Bloch equations are used to discuss the intensities of the transverse components of magnetization up to seven, harmonics of the frequency difference δ. In conclusion, it is suggested that the experimental investigation of these patterns can be used for the purpose of a continuous-wave monitoring the relaxation rates of spin probes.     

Design of self-refocused pulses under short relaxation times

The effect of using self-refocused RF pulses of comparable duration to relaxation times is studied in detail using numerical simulation. Transverse magnetization decay caused by short T2 and longitudinal component distortion due to short T1 are consistent with other studies. In order to design new pulses to combat short T1 and T2 the relaxation terms are directly inserted into the Bloch equations. These equations are inverted by searching the RF solution space using simulated annealing global optimization technique. A new T2-decay efficient excitation pulse is created (SDETR: single delayed excursion T2 resistive) which is also energy efficient. Inversion pulses which improve the inverted magnetization profile and achieve better suppression of the remaining transverse magnetization are also created even when both T1 and T2 are short. This is achieved, however, on the expense of a more complex B1 shape of larger energy content.     

A classical theory of superparamagnetic relaxation

A Fokker-Planck equation for the relaxation of a classical ferromagnetic particle coupled to a classical heat bath is derived from the Nakajima-Zwanzig equation. The equation of motion for the mean magnetization of an ensemble of particles is found to be closed only under special circumstances. In the strong motional narrowing limit the equation of motion reduces to the Bloch equations in the limit MH ? k_BT, i.e. for small particles, and to the Landau-Lifshitz equation in the opposite limit. For the motional narrowing region in toto the particular case of uniaxial anisotropy is analysed, giving an equation of motion which for large particles reduces to a modified Landau-Lifshitz equation with g-shift and a reduced damping constant. This equation cannot be meaningfully identified with the Gilbert equation.Approximate expressions for superparamagnetic relaxation rates by Kramers' method are obtained for the case of (i) triaxial (i.e. orthorhombic) and (ii) cubic (K +ve and ?ve) anisotropy, assuming large energy barriers. The results supplement Brown's expression for uniaxial anisotropy and show a more complicated dependence on the Landau-Lifshitz parameter λ than the linear dependence found by Brown. For small λ the rates tend to constant values compatible with the transition.