An exact form for the magnetic field density of states for a dipole

Dependence of the frequency distribution around a sphere on the voxel orientation

Microscopically small magnetic field inhomogeneities within an external static magnetic field cause a free induction decay in magnetic resonance imaging that generally exhibits two transverse components that are usually summarized to a complex entity. The Fourier transform of the complex-valued free induction decay is the purely real and positive-valued frequency distribution which allows an easy interpretation of the underlying dephasing mechanism. Typically, the frequency distribution inside a cubic voxel as caused by a spherical magnetic field inhomogeneity is determined by a histogram technique in terms of subdivision of the whole voxel into smaller subvoxels. A faster and more accurate computation is achieved by analytical expressions for the frequency distribution that are derived in this work. In contrast to the usually assumed simplified case of a spherical voxel, we also consider the tilt angles of the cubic voxel to the external magnetic field. The typical asymmetric form of the frequency distribution is reproduced and analyzed for the more realistic case of a cubic voxel. We observe a splitting of frequency distribution peaks for increasing tilt of the cubic voxel against the direction of the external magnetic field in analogy to the case for dephasing around cylindrical, vessel-like objects inside cubic voxels. These results are of value, e.g., for the analysis of susceptibility-weighted images or in quantitative susceptibility imaging since the reconstruction of these images is performed in cubic-shaped voxels.

Pseudo-diffusion effects in lung MRI

Magnetic resonance imaging of lung tissue is strongly influenced by susceptibility effects between spin-bearing water molecules and air-filled alveoli. The measured lineshape, however, also depends on the interplay between susceptibility effects and blood-flow around alveoli that can be approximated as pseudo-diffusion. Both effects are quantitatively described by the Bloch-Torrey-equation, which was so far only solved for dephasing on the alveolar surface. In this work, we extend this model to the whole range of physiological relevant air volume fractions. The results agree very well with in vivo measurements in human lung tissue.

A quantitative study of susceptibility and additional frequency shift of three common materials in MRI

PURPOSE

This work quantifies magnetic susceptibilities and additional frequency shifts derived from different samples.

METHODS

Twenty samples inside long straws were imaged with a multiecho susceptibility weighted imaging and analyzed with two approaches for comparisons. One approach applied our complex image summation around a spherical or cylindrical object method to phase distributions outside straws. The other approach utilized phase values inside each straw from two orientations. Both methods quantified susceptibilities of each sample at each echo time. The R2* value of each sample was measured too. Uncertainty of each measurement was also estimated.

RESULTS

Quantified susceptibilities from complex image summation around a spherical or cylindrical object are consistent within uncertainties between different echo times. However, this is not the case for the other method. Nonetheless, most quantified susceptibilities are consistent between these two methods. Phase values due to additional frequency shifts in some of ferritin and nanoparticle samples have been identified. Only R2* values quantified from low concentration nanoparticle samples agree with the predictions from the static dephasing theory.

CONCLUSION

This work suggests that using the sample sizes and phase values only outside samples can correctly quantify the susceptibilities of those samples. With the presence of a possible additional frequency shift inside a material, it will not be suitable to obtain susceptibility maps without taking that into account. Magn Reson Med 76:1263-1269, 2016. © 2015 Wiley Periodicals, Inc.

Magnetic moment quantifications of small spherical objects in MRI

PURPOSE

The purpose of this work is to develop a method for accurately quantifying effective magnetic moments of spherical-like small objects from magnetic resonance imaging (MRI). A standard 3D gradient echo sequence with only one echo time is intended for our approach to measure the effective magnetic moment of a given object of interest.

METHODS

Our method sums over complex MR signals around the object and equates those sums to equations derived from the magnetostatic theory. With those equations, our method is able to determine the center of the object with subpixel precision. By rewriting those equations, the effective magnetic moment of the object becomes the only unknown to be solved. Each quantified effective magnetic moment has an uncertainty that is derived from the error propagation method. If the volume of the object can be measured from spin echo images, the susceptibility difference between the object and its surrounding can be further quantified from the effective magnetic moment. Numerical simulations, a variety of glass beads in phantom studies with different MR imaging parameters from a 1.5T machine, and measurements from a SQUID (superconducting quantum interference device) based magnetometer have been conducted to test the robustness of our method.

RESULTS

Quantified effective magnetic moments and susceptibility differences from different imaging parameters and methods all agree with each other within two standard deviations of estimated uncertainties.

CONCLUSION

An MRI method is developed to accurately quantify the effective magnetic moment of a given small object of interest. Most results are accurate within 10% of true values, and roughly half of the total results are accurate within 5% of true values using very reasonable imaging parameters. Our method is minimally affected by the partial volume, dephasing, and phase aliasing effects. Our next goal is to apply this method to in vivo studies.

Microfabricated high-moment micrometer-sized MRI contrast agents

While chemically synthesized superparamagnetic microparticles have enabled much new research based on MRI tracking of magnetically labeled cells, signal-to-noise levels still limit the potential range of applications. Here it is shown how, through top-down microfabrication, contrast agent relaxivity can be increased several-fold, which should extend the sensitivity of such cell-tracking studies. Microfabricated agents can benefit from both higher magnetic moments and higher uniformity than their chemically synthesized counterparts, implying increased label visibility and more quantitative image analyses. To assess the performance of microfabricated micrometer-sized contrast agent particles, analytic models and numerical simulations are developed and tested against new microfabricated agents described in this article, as well as against results of previous imaging studies of traditional chemically synthesized microparticle agents. Experimental data showing signal effects of 500-nm thick, 2-μm diameter, gold-coated iron and gold-coated nickel disks verify the simulations. Additionally, it is suggested that measures of location better than the pixel resolution can be obtained and that these are aided using well-defined contrast agent particles achievable through microfabrication techniques.

Susceptibility-weighted imaging: technical aspects and clinical applications, part 1

Susceptibility-weighted imaging (SWI) is a new neuroimaging technique, which uses tissue magnetic susceptibility differences to generate a unique contrast, different from that of spin density, T1, T2, and T2*. In this review (the first of 2 parts), we present the technical background for SWI. We discuss the concept of gradient-echo images and how we can measure local changes in susceptibility. Armed with this material, we introduce the steps required to transform the original magnitude and phase images into SWI data. The use of SWI filtered phase as a means to visualize and potentially quantify iron in the brain is presented. Advice for the correct interpretation of SWI data is discussed, and a set of recommended sequence parameters for different field strengths is given.

Quantifying magnetic nanoparticles in non-steady flow by MRI

OBJECTIVE

This work compares the measured R*2 of magnetic nanoparticles to their corresponding theoretical values in both gel phantoms and dynamic water flows on the basis of the static dephasing theory.

MATERIALS AND METHODS

The magnetic moment of a nanoparticle solution was measured by a magnetometer. The R*2 of the nanoparticle solution doped in a gel phantom was measured at both 1.5 and 4.7 T. A total of 12 non-steady state flow experiments with different nanoparticle concentrations were conducted. The R*2 at each time point was measured.

RESULTS

The theoretical R*2 on the basis of the magnetization of nanoparticles measured by the magnetometer agree within 11% of MRI measurements in the gel phantom study, a significant improvement from previous work. In dynamic flow experiments, the total R*2 calculated from each experiment agrees within 15% of the theoretical R*2 for 10 of the 12 cases. The MRI phase values are also reasonably predicted by the theory. The diffusion effect does not seem to contribute significantly.

CONCLUSIONS

Under certain situations with known R*2, the static dephasing theory can be used to quantify the susceptibility or concentration of nanoparticles in either a static or dynamic flow environment at a given time point. This approach may be applied to in vivo studies.

Signal decay due to susceptibility-induced intravoxel dephasing on multiple air-filled cylinders: MRI simulations and experiments

OBJECTIVE

Characterization of magnetic susceptibility artefacts with assessment of the gradient-echo signal decay function of echo time, pixel size, and object geometry in the case of air-filled cylinders embedded in water.

MATERIALS AND METHODS

Experiments were performed with a 0.2 T magnet on a network of small interacting air-filled cylinders along with Magnetic resonance imaging (MRI) simulations integrating intravoxel dephasing. Signal decay over echo time was assessed at different pixel sizes on real and simulated images. The effects of radius, distance between cylinders and main magnetic field were studied using simulation.

RESULTS

Signal loss was greater as echo time or pixel size increased. Voxel signal decay was not exponential but was weighted by sinus cardinalis functions integrating echo time, pixel size and field inhomogeneities which depended on main magnetic field strength and geometric configuration of the object. Simulation was able to model signal decay, even for a complex object constituted of several cylinders. The specific experimental signal modulation we observed was thus reproduced and explained by simulation.

CONCLUSION

The quantitative signal decay approach at 0.2 T can be used in characterization studies in the case of locally regular air/water interfaces as the signal depends on object size relative to pixel size and is relevant to the geometric configuration. Moreover, the good concordance between simulation and experiments should lead to further studies of magnetic susceptibility effects with other objects such as networks of spheres. MRI simulation is thus a potential tool for molecular and porous media imaging.

Local frequency density of states around field inhomogeneities in magnetic resonance imaging: effects of diffusion

A method describing NMR-signal formation in inhomogeneous tissue is presented which covers all diffusion regimes. For this purpose, the frequency distribution inside the voxel is described. Generalizing the results of the well-known static dephasing regime, we derive a formalism to describe the frequency distribution that is valid over the whole dynamic range. The expressions obtained are in agreement with the results obtained from Kubos line-shape theory. To examine the diffusion effects, we utilize a strong collision approximation, which replaces the original diffusion process by a simpler stochastic dynamics. We provide a generally valid relation between the frequency distribution and the local Larmor frequency inside the voxel. To demonstrate the formalism we give analytical expressions for the frequency distribution and the free induction decay in the case of cylindrical and spherical magnetic inhomogeneities. For experimental verification, we performed measurements using a single-voxel spectroscopy method. The data obtained for the frequency distribution, as well as the magnetization decay, are in good agreement with the analytic results, although experiments were limited by magnetic field gradients caused by an imperfect shim and low signal-to-noise ratio.

A complex sum method of quantifying susceptibilities in cylindrical objects: the first step toward quantitative diagnosis of small objects in MRI

A complex sum method of quantifying the magnetic susceptibility of a long, narrow cylinder embedded in a uniform medium has been developed. The radius of the cylinder can be as small as one pixel. The susceptibility inside the object is extracted from the magnetic resonance complex images, using two concentric circles around the axis of the cylinder. The numerical simulations of this complex sum method are in good agreement with the phantom studies. Specifically, the method was tested with a susceptibility difference of -9 ppm to mimic air/tissue interface in the human body at 1.5 T with an echo time of 5 ms. Phantom studies using an air-filled cylinder in a solidified gel have shown that the susceptibility of the gel cannot be determined by the usual least-squares-fit method but can be determined by the complex sum method to within 5-10% of the expected value.

Quantitative study of signal decay due to magnetic susceptibility interfaces: MRI simulations and experiments

Characterization of magnetic susceptibility artifacts is required in a wide range of MRI studies. Experiments with a 0.2T magnet and MRI simulations were used to assess the signal decay over echo time at different pixel sizes in the case of air-water interfaces. The specific experimental signal modulation which was reproduced in simulation was interpreted. Concordance between simulation and experiments should make possible further studies of magnetic susceptibility effects with more complex objects as in molecular imaging and in porous medium imaging.

Structure-specific magnetic field inhomogeneities and its effect on the correlation time

We describe the relationship between the correlation time and microscopic spatial inhomogeneities in the static magnetic field. The theory takes into account diffusion of nuclear spins in the inhomogeneous field created by magnetized objects. A simple general expression for the correlation time is obtained. It is shown that the correlation time is dependent on a characteristic length, the diffusion coefficient of surrounding medium, the permeability of the surface and the volume fraction of the magnetized objects. For specific geometries (spheres and cylinders), exact analytical expressions for the correlation time are given. The theory can be applied to contrast agents (magnetically labeled cells), capillary network, BOLD effect and so forth.

Feldverteilung magnetischer Dipole als Modell magnetisch markierter Zellen in der MRT

Due to the creation of intense local magnetic fields, iron oxide nanoparticles are used as a contrast agent to produce signal loss in Magnetic Resonance Imaging (MRI) in regions where labelled cells have migrated.

OBJECTIVE

To study effects of the intracellular distribution of magnetic moments on the extracellular magnetic field by means of numerical simulations.

METHODS

Various geometries of intracellular particle distributions were scrutinized and the extracellular field distortions were computed. The total magnetic moment of a labelled cell was assigned to various magnetic subcompartments. The implications on the intravoxel frequency distribution and the static MR signal decay were assessed.

RESULTS

The extracellular field perturbation was affected by the intracellular particle distribution only in close proximity to the labelled cell. With increasing distance from the labelled cell, the effects of the intracellular particle distribution were less pronounced. The intravoxel frequency distribution induced by a single labelled cell was non-lorentzian.

CONCLUSION

The magnetic fields created by an iron loaded cell are sensitive to the intracellular distribution of nanoparticles only in close proximity to the cell. Far from the cell the field perturbation cannot be distinguished from the magnetic dipole field produced by a magnetic sphere with the same total magnetic moment.

Frequency distribution and signal formation around a vessel

We describe the NMR signal formation properties of a single vessel. Instead of assuming the frequency distribution to be a simple Lorentzian or Gaussian one, we take into account that the frequency distribution around the vessel is a complex function. Considering the static dephasing regime we find a relationship between signal formation and frequency distribution. Analytical expressions for the frequency distribution in a voxel and the magnetization decay are obtained. In the case of small volume fractions of blood and week magnetic fields the results can be used for describing signal formation processes in a vascular network. A relationship between the frequency distribution and the properties of the vascular network is derived. The magnetization decay in different time regimes is discussed. The result is relevant for describing signal formation processes around a vessel for arbitrary pulse sequences.

Imaging iron stores in the brain using magnetic resonance imaging

For the last century, there has been great physiological interest in brain iron and its role in brain function and disease. It is well known that iron accumulates in the brain for people with Huntington's disease, Parkinson's disease, Alzheimer's disease, multiple sclerosis, chronic hemorrhage, cerebral infarction, anemia, thalassemia, hemochromatosis, Hallervorden-Spatz, Down syndrome, AIDS and in the eye for people with macular degeneration. Measuring the amount of nonheme iron in the body may well lead to not only a better understanding of the disease progression but an ability to predict outcome. As there are many forms of iron in the brain, separating them and quantifying each type have been a major challenge. In this review, we present our understanding of attempts to measure brain iron and the potential of doing so with magnetic resonance imaging. Specifically, we examine the response of the magnetic resonance visible iron in tissue that produces signal changes in both magnitude and phase images. These images seem to correlate with brain iron content, perhaps ferritin specifically, but still have not been successfully exploited to accurately and precisely quantify brain iron. For future quantitative studies of iron content we propose four methods: correlating R2' and phase to iron content; applying a special filter to the phase to obtain a susceptibility map; using complex analysis to extract the product of susceptibility and volume content of the susceptibility source; and using early and late echo information to separately predict susceptibility and volume content.

Spectral characterization of local magnetic field inhomogeneities

The purpose of this study was the characterization of local magnetic susceptibility deviations by spectral analysis of their induced magnetic field inhomogeneities. Magnetic resonance spectra and related signal decay curves of local susceptibility deviations were simulated for different volume fractions and compositions of the object within the VOI. The size or composition of the object was varied at constant volume fraction, constant object size, or at constant 'magnetic strength' (defined as the product of the volume and the volume susceptibility of the object). Experimental spectra were acquired for individual metal spherical particles and a spherical air cavity. Where possible, spectra were used to characterize objects in terms of volume and composition. By simulations, a numerical relation was determined between the spectral broadening and the object's volume and composition. Comparison of spectra for various spherical objects showed the possibility of characterization with respect to size and composition. Experimental results confirmed the numerical results to a large extent, although the characterization was compromised by background signal decay, low volume fractions and limitations in signal-to-noise. In conclusion, spectral description of the field inhomogeneities related to small objects allows characterization of such objects with respect to size and composition. Practical applicability of the simulation results depends on background signal decay and volume fraction of the object.

On the artifact of a subvoxel susceptibility deviation in spoiled gradient-echo imaging

In MRI, susceptibility-based negative contrast amplifies the effect of objects that are too small to be detected by water displacement or intrinsic contrast properties. In this work, a simplified description of the susceptibility artifact of a subvoxel object in spoiled gradient-echo imaging is presented that focuses on the elimination of signal in its vicinity: the dephased-volume. The size and position of the dephased-volume are investigated using 3D time-domain simulations and in vitro experiments in which scan parameters and object magnetic moment are systematically varied. Overall signal loss is found to be linearly related to a dephasing parameter that contains the susceptibility difference with tissue, object volume, and echo time (TE), and thus allows the magnetic moment of the object to be assessed. Gradient strength, in-plane resolution, fractional echo, and slice orientation have limited influence. For the settings used, the center of mass of the artifact was always within 0.5 mm of the object's in-plane position.