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Mapping Spatio-Temporal Diffusion inside the Human Brain Using a Numerical Solution of the Diffusion Equation
Abstract
Diffusion is an important mechanism for molecular transport in living biological tissues. Diffusion magnetic resonance imaging (dMRI) provides a unique probe to examine microscopic structures of the tissues in vivo, but current dMRI techniques usually ignore the spatiotemporal evolution process of the diffusive medium. In the present study, we demonstrate the feasibility to reveal the spatiotemporal diffusion process inside the human brain based on a numerical solution of the diffusion equation. Normal human subjects were scanned with a diffusion tensor imaging (DTI) technique on a 3-T MRI scanner, and the diffusion tensor in each voxel was calculated from the DTI data. The diffusion equation, a partial-derivative description of Fick's law for the diffusion process, was discretized into equivalent algebraic equations. A finite-difference method was employed to obtain the numerical solution of the diffusion equation with a Crank-Nicholson iteration scheme to enhance the numerical stability. By specifying boundary and initial conditions, the spatiotemporal evolution of the diffusion process inside the brain can be virtually reconstructed. Our results exhibit similar medium profiles and diffusion coefficients as those of light fluorescence dextrans measured in integrative optical imaging experiments. The proposed method highlights the feasibility to noninvasively estimate the macroscopic diffusive transport time for a molecule in a given region of the brain.
... In the many processes considered in biological, engineering or physical sciences normal diffusion or subdiffusion occurs in a system composed of two media, separated by a partially permeable thin membrane; in each part of the system different parameters characterizing diffusion may occur. As an example, we mention the transport of various substances (glucose, pyruvate, lactate, alanine) between blood and a cell [1], diffusion of various substances (medications, cosmetics) through the skin [2], diffusion of various substances in the brain [3], diffusion between extracellular brain space and cells [4], diffusion of large molecular drugs into the brain [5], nisin diffusion into agarose gel [6,7]. A list of similar examples can be significantly expanded, see also the problems discussed in [8,9,10]. ...
We consider subdiffusion in a system which consists of two media separated by a thin membrane. The subdiffusion parameters may be different in each of the medium. Using the new method presented in this paper we derive the probabilities (the Green's functions) describing a particle's random walk in the system. Within this method we firstly consider the particle's random walk in a system with both discrete time and space variables in which a particle can vanish due to reactions with constant probabilities and , defined separately for each medium. Then, we move from discrete to continuous variables. The reactions included in the model play a supporting role. We link the reaction probabilities with the other subdiffusion parameters which characterize the media by means of the formulae presented in this paper. Calculating the generating functions for the difference equations describing the random walk in the composite membrane system with reactions, which depend explicitly on and , we are able to correctly incorporate the subdiffusion parameters of both the media into the Green's functions. Finally, placing into the obtained functions we get the Green's functions for the composite membrane system without any reactions. From the obtained Green's functions, we derive the boundary conditions at the thin membrane. One of the boundary conditions contains the Riemann--Liouville fractional time derivative, which shows that the additional `memory effect' is created in the system. As is discussed in this paper, the `memory effect' can be created both by the membrane and by the discontinuity of the medium at the point at which the various media are joined.
... In the many processes considered in biological, engineering or physical sciences normal diffusion or subdiffusion occurs in a system composed of two media, separated by a partially permeable thin membrane; in each part of the system different parameters characterizing diffusion may occur [1,2,3,4,5,6,7,8,9,10]. A problem in the modelling of normal diffusion or subdiffusion in a system which consists of two different media is how to choose boundary condition at the border between the media. ...
We consider the subdiffusion--absorption process in a system which consists of two different media separated by a thin membrane. The process is described by subdiffusion--absorption equations with fractional Riemann--Liouville time derivative. We present the method of deriving the probabilities (the Green's functions) described particle's random walk in the system. Within the method we firstly consider the random walk of a particle in a system with both discrete time and space variables, and then we pass from discrete to continuous variables by means of the procedure presented in this paper. Using the Green's functions we derive boundary conditions at the membrane.
... [154] Magnetic resonance imaging (MRI) diffusion analyses in the brains of human subjects yielded matching results. [155] Therefore, the presence of brain ECM, as well as a 3D culture system, would significantly limit diffusion of particles with larger diameters. Viral infections illustrate the spread of nanoscale particles in the CNS. ...
Creating a cellular model of Alzheimer's disease (AD) that accurately recapitulates disease pathology has been a longstanding challenge. Recent studies showed that human AD neural cells, integrated into three‐dimensional (3D) hydrogel matrix, display key features of AD neuropathology. Like in the human brain, the extracellular matrix (ECM) plays a critical role in determining the rate of neuropathogenesis in hydrogel‐based 3D cellular models. Aging, the greatest risk factor for AD, significantly alters brain ECM properties. Therefore, it is important to understand how age‐associated changes in ECM affect accumulation of pathogenic molecules, neuroinflammation, and neurodegeneration in AD patients and in vitro models. In this review, mechanistic hypotheses is presented to address the impact of the ECM properties and their changes with aging on AD and AD‐related dementias. Altered ECM characteristics in aged brains, including matrix stiffness, pore size, and composition, will contribute to disease pathogenesis by modulating the accumulation, propagation, and spreading of pathogenic molecules of AD. Emerging hydrogel‐based disease models with differing ECM properties provide an exciting opportunity to study the impact of brain ECM aging on AD pathogenesis, providing novel mechanistic insights. Understanding the role of ECM aging in AD pathogenesis should also improve modeling AD in 3D hydrogel systems.
... For example, to take bidirectional BBB transport into account, one or two rate constants (s −1 ) can be included that specifically quantify the concentration-dependent exchange between the blood plasma and the brain ECF in one or two directions [128,133,135,180,181]. To account for the anisotropic diffusion within the white matter, a diffusion tensor can also be used instead of the effective diffusivity D* [182]. The modified diffusion equation has proven useful in predicting the local distribution profile of a drug after several invasive experimental measurements, such as microdialysis [128,135,152,181,[183][184][185][186][187][188]. ...
The blood brain barrier (BBB) is the main barrier that separates the blood from the brain. Because of the BBB, the drug concentration-time profile in the brain may be substantially different from that in the blood. Within the brain, the drug is subject to distributional and elimination processes: diffusion, bulk flow of the brain extracellular fluid (ECF), extra-intracellular exchange, bulk flow of the cerebrospinal fluid (CSF), binding and metabolism. Drug effects are driven by the concentration of a drug at the site of its target and by drug-target interactions. Therefore, a quantitative understanding is needed of the distribution of a drug within the brain in order to predict its effect. Mathematical models can help in the understanding of drug distribution within the brain. The aim of this review is to provide a comprehensive overview of system-specific and drug-specific properties that affect the local distribution of drugs in the brain and of currently existing mathematical models that describe local drug distribution within the brain. Furthermore, we provide an overview on which processes have been addressed in these models and which have not. Altogether, we conclude that there is a need for a more comprehensive and integrated model that fills the current gaps in predicting the local drug distribution within the brain.
... More studies exist that integrate more of the discussed properties into one model. For example, the distribution of a compound within the brain can be described by both compartmental exchange and transport through the brain ECF (Tan et al. 2003;Zhan et al. 2008;Calvetti et al. 2015;Ehlers and Wagner 2015;Jin et al. 2016). In Calvetti et al. (2015), a 3D model of brain cellular metabolism with spatial resolution of the location of the synapse relative to the brain capillaries demonstrates the importance of spatial distribution. ...
The development of drugs that target the brain is very challenging. A quantitative understanding is needed of the complex processes that govern the concentration–time profile of a drug (pharmacokinetics) within the brain. So far, there are no studies on predicting the drug concentration within the brain that focus not only on the transport of drugs to the brain through the blood–brain barrier (BBB), but also on drug transport and binding within the brain. Here, we develop a new model for a 2D square brain tissue unit, consisting of brain extracellular fluid (ECF) that is surrounded by the brain capillaries. We describe the change in free drug concentration within the brain ECF, by a partial differential equation (PDE). To include drug binding, we couple this PDE to two ordinary differential equations that describe the concentration–time profile of drug bound to specific as well as non-specific binding sites that we assume to be evenly distributed over the brain ECF. The model boundary conditions reflect how free drug enters and leaves the brain ECF by passing the BBB, located at the level of the brain capillaries. We study the influence of parameter values for BBB permeability, brain ECF bulk flow, drug diffusion through the brain ECF and drug binding kinetics, on the concentration–time profiles of free and bound drug.
... In the many processes considered in biological, engineering or physical sciences normal diffusion or subdiffusion occurs in a system composed of two media, separated by a partially permeable thin membrane; in each part of the system different parameters characterizing diffusion may occur [1,2,3,4,5,6,7,8,9,10]. A problem in the modelling of normal diffusion or subdiffusion in a system which consists of two different media is how to choose boundary condition at the border between the media. ...
We consider the subdiffusion--absorption process in a system which consists of two different media separated by a thin membrane. The process is described by subdiffusion--absorption equations with fractional Riemann--Liouville time derivative. We present the method of deriving the probabilities (the Green's functions) described particle's random walk in the system. Within the method we firstly consider the random walk of a particle in a system with both discrete time and space variables, and then we pass from discrete to continuous variables by means of the procedure presented in this paper. Using the Green's functions we derive boundary conditions at the membrane.
... In the many processes considered in biological, engineering or physical sciences normal diffusion or subdiffusion occurs in a system composed of two media, separated by a partially permeable thin membrane; in each part of the system different parameters characterizing diffusion may occur. As an example, we mention the transport of various substances (glucose, pyruvate, lactate, alanine) between blood and a cell [1], diffusion of various substances (medications, cosmetics) through the skin [2], diffusion of various substances in the brain [3], diffusion between extracellular brain space and cells [4], diffusion of large molecular drugs into the brain [5], nisin diffusion into agarose gel [6]. A list of similar examples can be significantly expanded, see also the problems discussed in [7][8][9]. ...
We consider subdiffusion in a system which consists of two media separated by
a thin membrane. The subdiffusion parameters may be different in each of the
medium. Using the new method presented in this paper we derive the
probabilities (the Green's functions) describing a particle's random walk in
the system. Within this method we firstly consider the particle's random walk
in a system with both discrete time and space variables in which a particle can
vanish due to reactions with constant probabilities and , defined
separately for each medium. Then, we move from discrete to continuous
variables. The reactions included in the model play a supporting role. We link
the reaction probabilities with the other subdiffusion parameters which
characterize the media by means of the formulae presented in this paper.
Calculating the generating functions for the difference equations describing
the random walk in the composite membrane system with reactions, which depend
explicitly on and , we are able to correctly incorporate the
subdiffusion parameters of both the media into the Green's functions. Finally,
placing into the obtained functions we get the Green's functions
for the composite membrane system without any reactions. From the obtained
Green's functions, we derive the boundary conditions at the thin membrane. One
of the boundary conditions contains the Riemann--Liouville fractional time
derivative, which shows that the additional `memory effect' is created in the
system. As is discussed in this paper, the `memory effect' can be created both
by the membrane and by the discontinuity of the medium at the point at which
the various media are joined.
... The assessment of functional hemodynamic parameters on the basis of rapid-sequence tomography techniques (computed tomography as well as MRI) is increasing. [16][17][18][19] The indicator dilution technique was first used by Axel 20 for the quantification of cerebral blood flow. Microcirculatory changes via MRI can be sensitively detected in the brain, 19,21 the female breast, 10,22 and the lung. ...
Changes in liver microcirculation are considered essential in assessing ischemia-reperfusion injury, which in turn has an impact on liver graft function and outcome following liver transplantation (LTx). The aim of this study was to introduce dynamic magnetic resonance imaging (dMRI) as a new technique for overall quantification of hepatic microcirculation and compare it to perfusion measured by laser Doppler flowmetry (LDF; hepatic artery/portal vein) and thermal diffusion (TD). The study included 3 groups, measuring hepatic blood flow and microcirculation with the help of TD, LDF, and dMRI. In group I (9 landrace pigs; 26 +/- 5 kg), the native liver before and after partial portal occlusion was studied; in group II (6 landrace pigs; 25.5 +/- 4.4 kg), the liver 24 hours after LTx was studied; and in group III (14 patients), the liver on days 4 to 7 following LTx was studied. A close correlation was found between dMRI measurements and TD (r = 0.7-0.9, P < 0.01) in 4 defined regions of interest. Portal blood flow and partial occlusion of the portal vein were accurately detected by LDF flowmetry and correlated well with dMRI (r = 0.95, P < 0.01). In the clinical setting, representative TD measurements in segment 4b of the transplanted liver correlated well with dMRI analysis in other segments. Quantification of the portal blood flow and imaging of the whole liver could be performed simultaneously by dMRI. In conclusion, dMRI has been proved to be a sensitive modality for the quantification of liver microcirculation and hepatic blood flow in experimental and clinical LTx. It allows for a synchronous, noninvasive assessment of macrocirculation and microcirculation of the liver and could become a valuable diagnostic tool in advanced liver surgery and transplantation.
We show that some boundary conditions assumed at a thin membrane may result in normal diffusion not being the stochastic Markov process. We consider boundary conditions defined in terms of the Laplace transform in which there is a linear combination of probabilities and probability fluxes defined on both membrane surfaces. The coefficients of the combination may depend on the Laplace transform parameter. Such boundary conditions are most commonly used when considering diffusion in a membrane system unless collective or nonlocal processes in particles diffusion occur. We find Bachelier-Smoluchowski-Chapmann-Kolmogorov (BSCK) equation in terms of the Laplace transform and we derive the criterion to check whether the boundary conditions lead to fundamental solutions of diffusion equation satisfying this equation. If the BSCK equation is not met, then the Markov property is broken. When a probability flux is continuous at the membrane, the general forms of the boundary conditions for which the fundamental solutions meet the BSCK equation are derived. A measure of broken of semi-group property is also proposed. The relation of this measure to the non-Markovian property measure is discussed.
We present the model of a diffusion-absorption process in a system which consists of two media separated by a thin partially permeable membrane. The kind of diffusion as well as the parameters of the process may be different in both media. Based on a simple model of a particle's random walk in a membrane system we derive the Green's functions, then we find the boundary conditions at the membrane. One of the boundary conditions is rather complicated and takes a relatively simple form in terms of the Laplace transform. Assuming that particles diffuse independently of one another, the obtained boundary conditions can be used to solve differential or differential-integral equations describing the processes in multilayered systems for any initial condition. We consider normal diffusion, subdiffusion, and slow subdiffusion processes, and we also suggest how superdiffusion could be included in this model. The presented method provides the functions in terms of the Laplace transform and some useful methods of calculation of the inverse Laplace transform are shown.
Molecular (or tracer) diffusion coefficient data were obtained for 2-naphthol, benzoic acid, salicylic acid, camphor, and
cinnamic acid in water at temperatures that differ significantly from ambient value. Experimental values were determined for
the dissolution of 2-naphthol in water at 283 to 368K, of benzoic acid in water at 283 to 338K, of salicylic acid in water
at 283 to 343K, of camphor in water at 283 to 318K, and of cinnamic acid in water at 283 to 318K. Empirical correlations
are presented for the prediction of molecular diffusion coefficient over the entire range of temperatures studied, and they
are shown to predict the obtained data with very good accuracy.
This paper describes the theory of an integrative optical imaging system and its application to the analysis of the diffusion of 3-, 10-, 40-, and 70-kDa fluorescent dextran molecules in agarose gel and brain extracellular microenvironment. The method uses a precisely defined source of fluorescent molecules pressure ejected from a micropipette, and a detailed theory of the intensity contributions from out-of-focus molecules in a three-dimensional medium to a two-dimensional image. Dextrans tagged with either tetramethylrhodamine or Texas Red were ejected into 0.3% agarose gel or rat cortical slices maintained in a perfused chamber at 34 degrees C and imaged using a compound epifluorescent microscope with a 10 x water-immersion objective. About 20 images were taken at 2-10-s intervals, recorded with a cooled CCD camera, then transferred to a 486 PC for quantitative analysis. The diffusion coefficient in agarose gel, D, and the apparent diffusion coefficient, D*, in brain tissue were determined by fitting an integral expression relating the measured two-dimensional image intensity to the theoretical three-dimensional dextran concentration. The measurements in dilute agarose gel provided a reference value of D and validated the method. Values of the tortuosity, lambda = (D/D*)1/2, for the 3- and 10-kDa dextrans were 1.70 and 1.63, respectively, which were consistent with previous values derived from tetramethylammonium measurements in cortex. Tortuosities for the 40- and 70-kDa dextrans had significantly larger values of 2.16 and 2.25, respectively. This suggests that the extracellular space may have local constrictions that hinder the diffusion of molecules above a critical size that lies in the range of many neurotrophic compounds.
The aim of this review is to describe two recent developments in the use of magnetic resonance imaging (MRI) in the study of biological systems: diffusion and perfusion MRI. Diffusion MRI measures the molecular mobility of water in tissue, while perfusion MRI measures the rate at which blood is delivered to tissue. Therefore, both these techniques measure quantities which have direct physiological relevance. It is shown that diffusion in biological systems is a complex phenomenon, influenced directly by tissue microstructure, and that its measurement can provide a large amount of information about the organization of this structure in normal and diseased tissue. Perfusion reflects the delivery of essential nutrients to tissue, and so is directly related to its status. The concepts behind the techniques are explained, and the theoretical models that are used to convert MRI data to quantitative physical parameters are outlined. Examples of current applications of diffusion and perfusion MRI are given. In particular, the use of the techniques to study the pathophysiology of cerebral ischaemia/stroke is described. It is hoped that the biophysical insights provided by this approach will help to define the mechanisms of cell damage and allow evaluation of therapies aimed at reducing this damage.
A method is presented for determining paths of anatomical connection between regions of the brain using magnetic resonance diffusion tensor information. Level set theory, applied using fast marching methods, is used to generate three-dimensional time of arrival maps, from which connection paths between brain regions may be identified. The method is demonstrated in the normal brain and it is shown that major white matter tracts may be elucidated and that multiple connections and tract branching are allowed. Maps of connectivity between brain regions are also determined. Four options are described for estimating the degree of connectivity between regions.
Assessment of the effectiveness of cancer therapy traditionally relies on comparison of tumor images acquired before and after therapeutic intervention by inspection of gross anatomical images to evaluate changes in tumor size. The potential for imaging to provide additional insights related to the therapeutic impact would be enhanced if a specific parameter or combination of parameters could be identified that reflect tissue changes at the cellular or physiological level. This information could also provide a more sensitive and earlier indicator of treatment response in an individual animal or patient. Diffusion magnetic resonance imaging can detect relatively small changes in tissue structure at the cellular level and thus provides an opportunity to quantitatively and serially follow therapeutic-induced changes in solid tumors. This article provides an overview of the use of diffusion magnetic resonance imaging as a surrogate marker for quantitating treatment responsiveness in both preclinical and clinical studies.
Complex valued data may be obtained by means of acquisition systems such as magnetic resonance imaging (MRI) or digital communication systems. If the signal amplitude is to be estimated from these (inevitably noise corrupted) complex data, one has two options. Either the signal amplitude is directly estimated from the complex valued data set, or, the complex data is first transformed into a magnitude data set after which the signal amplitude is estimated. Similarly, the noise variance can be estimated from both data sets.
The in situ assessment of axonal projections of the brain has been severely limited by the lack of noninvasive techniques to study this type of anatomy. We show here that in vivo three-dimensional (3D) reconstruction of axonal projections can be achieved using a rapid 3D high-resolution diffusion-weighted imaging technique combined with a recently designed fiber reconstruction algorithm. As a first example, neuronal pathways in the rat brain were probed. Eight well-known fiber projections; genu and splenium of corpus callosum, internal and external capsule, fimbria, anterior commissure, optic tract, and stria terminalis were tracked and shown to be in agreement with the location of these known axonal projections. The experiment took 2 hr and shorter times should be possible in the clinical situation. By combining anisotropy information with fiber tracking, the anisotropy of individual projections was also documented. (C) 1999 Wiley-Liss, Inc.
This book attempts to provide basic knowledge about the structure, properties, and processing of engineering materials and their applications. The first chapter gives an introduction to materials science and engineering and briefly describes the main types of engineering materials. Chapters 2 to 4 cover basic materials science subjects dealing primarily with structure. Chapters 5 and 6 relate structure to some basic electrical and mechanical properties. Polymeric materials are treated first in Chapter 7 since they do not require a background in phase-diagram analysis. Chapter 8 then discusses phase diagrams as preparation for the study of metallic and ceramic materials that are covered in Chapters 9 and 10, respectively. Chapter 11 deals with magnetic materials, while Chapter 12 discusses some aspects of corrosion. Finally, Chapter 13 presents some information on composite materials including some discussion of advanced composites, our latest-type materials used in engineering designs. The last part of Chapter 13 deals with the structure and properties of concrete and wood.
A two-dimensional finite element model of the prostrate region of the human body has been generated using the automatic mesh generation capabilities of the software package ANSYS. The tissue types included are skin, fat, muscle, bone, intestine, nerve, prostate and tumour tissue. The thermal response of the model undergoing hyperthermia treatment for cancer is obtained by solving the bioheat transfer equation using the method of finite elements. Variation of blood flow rates due to increasing temperatures during hyperthermia are included. Transient temperature profiles as well as constant temperature contours are displayed using the extensive post processing and graphics capabilities of ANSYS. The results show how selective heating can be obtained in the tumour region and the effects of varying blood flow rates. The limitations of using a commercial software package for modelling the thermal behaviour of the human body are discussed.
Perfusion of the ventriculo-cisternal system of the live dog with artificial spinal fluid containing radioactive inulin, sulfate or dextran resulted in the entry of these compounds into the brain tissue adjacent to the ventricles. The concentration of these compounds in the tissue decreased as a function of the distance from the ventricular surface in a manner consistent with diffusion. These data are consistent with an extracellular space of 7–14% in the areas of brain that were studied.
It has recently become apparent that a widespread misconception persists among computational fluid dynamicists concerning the von Neumann stability analysis of the explicit forward-time central-space (FTCS) model convective diffusion equation. The misconception appears to stem from a mathematical error in a paper by J. Fromm which was perpetuated a fortiori in the influential book by P. J. Roache. Although several professional workers have been aware of the error for some time, it is clear that many others are not, and this is particularly true of students (and teachers! ) who may be new to the subject and are using Roache's (generally excellent, although now of course somewhat dated) book as a text.
Diffusion plays a crucial role in brain function. The spaces between cells can be likened to the water phase of a foam and many substances move within this complicated region. Diffusion in this interstitial space can be accurately modelled with appropriate modifications of classical equations and quantified from measurements based on novel micro-techniques. Besides delivering glucose and oxygen from the vascular system to brain cells, diffusion also moves informational substances between cells, a process known as volume transmission. Deviations from expected results reveal how local uptake, degradation or bulk flow may modify the transport of molecules. Diffusion is also essential to many therapies that deliver drugs to the brain. The diffusion-generated concentration distributions of well-chosen molecules also reveal the structure of brain tissue. This structure is represented by the volume fraction (void space) and the tortuosity (hindrance to diffusion imposed by local boundaries or local viscosity). Analysis of these parameters also reveals how the local geometry of the brain changes with time or under pathological conditions. Theoretical and experimental approaches borrow from classical diffusion theory and from porous media concepts. Earlier studies were based on radiotracers but the recent methods use a point-source paradigm coupled with micro-sensors or optical imaging of macromolecules labelled with fluorescent tags. These concepts and methods are likely to be applicable elsewhere to measure diffusion properties in very small volumes of highly structured but delicate material.
The tensors produced by diffusion tensor magnetic resonance imaging (DT-MRI) represent the covariance in a Brownian motion model of water diffusion. Under this physical interpretation, diffusion tensors are required to be symmetric, positive-definite. However, current approaches to statistical analysis of diffusion tensor data, which treat the tensors as linear entities, do not take this positive-definite constraint into account. This difficulty is due to the fact that the space of diffusion tensors does not form a vector space. In this paper we show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. We show that these statistics preserve natural geometric properties of the tensors, including the constraint that their eigenvalues be positive. The symmetric space formulation also leads to a natural definition for interpolation of diffusion tensors and a new measure of anisotropy. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease. The framework presented in this paper should also be useful in other applications where symmetric, positive-definite tensors arise, such as mechanics and computer vision.
1. The validity of the macroscopic laws of ion diffusion was critically examined within the microenvironment of the extracellular space in the rat cerebellum using ion-selective micropipettes and ionophoretic point sources. 2. The concepts of volume averaging, volume fraction (alpha) and tortuosity (lambda) were defined and shown to be theoretically appropriate for quantifying diffusion in a complex medium such as the brain. 3. Diffusion studies were made with the cations tetramethylammonium and tetraethylammonium and the anions alpha-naphthalene sulphonate and hexafluoro-arsenate, all of which remained essentially extracellular during the measurements. Diffusion parameters were measured for a period of 50s and over distances of the order of 0.1 mm. 4. Measurements of the diffusion coefficients of the ions in agar gel gave values that were very close to those derivable from the literature, thus confirming the validity of the method. 5. Measurements in the cerebellum did not reveal any systematic influences of ionophoretic current strength, electrode separation, anisotropy, inhomogeneity, charge discrimination or uptake, within the limits tested. 6. The pooled data from measurements with all the ions gave alpha = 0.21 +/- 0.02 (mean +/- S.E. of mean) and lambda = 1.55 +/- 0.05 (mean +/- S.E. of mean). 7. These results show that the extracellular space occupies about 20% of the rat cerebellum and that the diffusion coefficient for small monovalent extracellular ions is reduced by a factor of 2.4 (i.e. lambda 2) without regard to charge sign. The over-all effect of this is to increase the apparent strength of any ionic source in the cerebellum by a factor of lambda 2/alpha, about 12-fold in the present case, and to modify the time course of diffusion. 8. These conclusions confirm that the laws of macroscopic diffusion are closely obeyed in the cerebellum for small ions in the extracellular space, provided that volume fraction and tortuosity are explicitly taken into account. It is likely that these conclusions are generally applicable to other brain regions and other diffusing substances.
The diagonal and off-diagonal elements of the effective self-diffusion tensor, Deff, are related to the echo intensity in an NMR spin-echo experiment. This relationship is used to design experiments from which Deff is estimated. This estimate is validated using isotropic and anisotropic media, i.e., water and skeletal muscle. It is shown that significant errors are made in diffusion NMR spectroscopy and imaging of anisotropic skeletal muscle when off-diagonal elements of Deff are ignored, most notably the loss of information needed to determine fiber orientation. Estimation of Deff provides the theoretical basis for a new MRI modality, diffusion tensor imaging, which provides information about tissue microstructure and its physiologic state not contained in scalar quantities such as T1, T2, proton density, or the scalar apparent diffusion constant.
Quantitative-diffusion-tensor MRI consists of deriving and displaying parameters that resemble histological or physiological stains, i.e., that characterize intrinsic features of tissue microstructure and microdynamics. Specifically, these parameters are objective, and insensitive to the choice of laboratory coordinate system. Here, these two properties are used to derive intravoxel measures of diffusion isotropy and the degree of diffusion anisotropy, as well as intervoxel measures of structural similarity, and fiber-tract organization from the effective diffusion tensor, D, which is estimated in each voxel. First, D is decomposed into its isotropic and anisotropic parts, [D] I and D - [D] I, respectively (where [D] = Trace(D)/3 is the mean diffusivity, and I is the identity tensor). Then, the tensor (dot) product operator is used to generate a family of new rotationally and translationally invariant quantities. Finally, maps of these quantitative parameters are produced from high-resolution diffusion tensor images (in which D is estimated in each voxel from a series of 2D-FT spin-echo diffusion-weighted images) in living cat brain. Due to the high inherent sensitivity of these parameters to changes in tissue architecture (i.e., macromolecular, cellular, tissue, and organ structure) and in its physiologic state, their potential applications include monitoring structural changes in development, aging, and disease.
In diffusion tensor imaging (DTI) an effective diffusion tensor in each voxel is measured by using a set of diffusion-weighted images (DWIs) in which diffusion gradients are applied in a multiplicity of oblique directions. However, to estimate the diffusion tensor accurately, one must account for the effects of all imaging and diffusion gradient pulses on each signal echo, which are embodied in the b matrix. For DTI to be practical clinically, one must also acquire DWIs rapidly and free of motion artifacts, which is now possible with diffusion-weighted echo-planar imaging (DW-EPI). An analytical expression for the b matrix of a general DW-EPI pulse sequence is presented and then validated experimentally by measuring the diffusion tensor in an isotropic phantom whose diffusivity is already known. The b matrix is written in a convenient tabular form as a sum of individual pair-wise contributions arising from gradient pulses applied along parallel and perpendicular directions. While the contributions from readout and phase-encode gradient pulse trains are predicted to have a negligible effect on the echo, the contributions from other imaging and diffusion gradient pulses applied in both parallel and orthogonal directions are shown to be significant in our sequence. In general, one must understand and account for the multiplicity of interactions between gradient pulses and the echo signal to ensure that diffusion tensor imaging is quantitative.
The in situ assessment of axonal projections of the brain has been severely limited by the lack of noninvasive techniques to study this type of anatomy. We show here that in vivo three-dimensional (3D) reconstruction of axonal projections can be achieved using a rapid 3D high-resolution diffusion-weighted imaging technique combined with a recently designed fiber reconstruction algorithm. As a first example, neuronal pathways in the rat brain were probed. Eight well-known fiber projections; genu and splenium of corpus callosum, internal and external capsule, fimbria, anterior commissure, optic tract, and stria terminalis were tracked and shown to be in agreement with the location of these known axonal projections. The experiment took 2 hr and shorter times should be possible in the clinical situation. By combining anisotropy information with fiber tracking, the anisotropy of individual projections was also documented. Magn Reson Med 42:1123-1127, 1999.
Fiber tract trajectories in coherently organized brain white matter pathways were computed from in vivo diffusion tensor magnetic resonance imaging (DT-MRI) data. First, a continuous diffusion tensor field is constructed from this discrete, noisy, measured DT-MRI data. Then a Frenet equation, describing the evolution of a fiber tract, was solved. This approach was validated using synthesized, noisy DT-MRI data. Corpus callosum and pyramidal tract trajectories were constructed and found to be consistent with known anatomy. The method's reliability, however, degrades where the distribution of fiber tract directions is nonuniform. Moreover, background noise in diffusion-weighted MRIs can cause a computed trajectory to hop from tract to tract. Still, this method can provide quantitative information with which to visualize and study connectivity and continuity of neural pathways in the central and peripheral nervous systems in vivo, and holds promise for elucidating architectural features in other fibrous tissues and ordered media.
The success of diffusion magnetic resonance imaging (MRI) is deeply rooted in the powerful concept that during their random, diffusion-driven displacements molecules probe tissue structure at a microscopic scale well beyond the usual image resolution. As diffusion is truly a three-dimensional process, molecular mobility in tissues may be anisotropic, as in brain white matter. With diffusion tensor imaging (DTI), diffusion anisotropy effects can be fully extracted, characterized, and exploited, providing even more exquisite details on tissue microstructure. The most advanced application is certainly that of fiber tracking in the brain, which, in combination with functional MRI, might open a window on the important issue of connectivity. DTI has also been used to demonstrate subtle abnormalities in a variety of diseases (including stroke, multiple sclerosis, dyslexia, and schizophrenia) and is currently becoming part of many routine clinical protocols. The aim of this article is to review the concepts behind DTI and to present potential applications.
The diffusion in voxels with multidirectional fibers can be quite complicated and not necessarily well characterized by the standard diffusion tensor model. High angular resolution diffusion-weighted acquisitions have recently been proposed as a method to investigate such voxels, but the reconstruction methods proposed require sophisticated estimation schemes. We present here a simple algorithm for the identification of diffusion anisotropy based upon the variance of the estimated apparent diffusion coefficient (ADC) as a function of measurement direction. The rationale for this method is discussed, and results in normal human subjects acquired with a novel diffusion-weighted stimulated-echo spiral acquisition are presented which distinguish areas of anisotropy that are not apparent in the relative anisotropy maps derived from the standard diffusion tensor model. Published 2001 Wiley-Liss, Inc.
The methods of group theory are applied to the problem of characterizing the diffusion measured in high angular resolution MR experiments. This leads to a natural representation of the local diffusion in terms of spherical harmonics. In this representation, it is shown that isotropic diffusion, anisotropic diffusion from a single fiber, and anisotropic diffusion from multiple fiber directions fall into distinct and separable channels. This decomposition can be determined for any voxel without any prior information by a spherical harmonic transform, and for special cases the magnitude and orientation of the local diffusion may be determined. Moreover, non-diffusion-related asymmetries produced by experimental artifacts fall into channels distinct from the fiber channels, thereby allowing their separation and a subsequent reduction in noise from the reconstructed fibers. In the case of a single fiber, the method reduces identically to the standard diffusion tensor method. The method is applied to normal volunteer brain data collected with a stimulated echo spiral high angular resolution diffusion-weighted (HARD) acquisition.
A method is presented for mapping intravoxel fiber structures using spectral decomposition onto a circular distribution of measured apparent diffusion coefficients (ADCs). The zeroth-, second-, and fourth-order harmonic components of the ADC distribution on the circle spanned by the major and median eigenvectors of the diffusion tensor can be used to provide quantitative indices for isotropic, linear, and fiber-crossing diffusion, respectively. A diffusion-weighted MRI technique with 90 encoding orientations was implemented to estimate the circular ADC distribution and calculate the circular spectrum. A digital phantom was used to simulate various diffusion patterns. Comparisons were made between the circular spectrum and regular DTI-based index maps. The results indicated that the zeroth- and second-order circular spectrum maps exhibited a strong consistency with the DTI-based mean diffusivity and linear indices, respectively, and the fourth-order circular spectrum map was able to identify the fiber crossings. MRI experiments were performed on seven healthy human brains using a 3T scanner. The in vivo fourth-order maps showed significantly higher densities in several brain regions, including the corpus callosum, cingulum bundle, superior longitudinal fasciculus, corticospinal tract, and middle cerebellar peduncle, which indicated the existence of fiber crossings in these regions.
While functional brain imaging methods can locate the cortical regions subserving particular cognitive functions, the connectivity between the functional areas of the human brain remains poorly understood. Recently, investigators have proposed a method to image neural connectivity noninvasively using a magnetic resonance imaging method called diffusion tensor imaging (DTI). DTI measures the molecular diffusion of water along neural pathways. Accurate reconstruction of neural connectivity patterns from DTI has been hindered, however, by the inability of DTI to resolve more than a single axon direction within each imaging voxel. Here, we present a novel magnetic resonance imaging technique that can resolve multiple axon directions within a single voxel. The technique, called q-ball imaging, can resolve intravoxel white matter fiber crossing as well as white matter insertions into cortex. The ability of q-ball imaging to resolve complex intravoxel fiber architecture eliminates a key obstacle to mapping neural connectivity in the human brain noninvasively.
Methods are presented to map complex fiber architectures in tissues by imaging the 3D spectra of tissue water diffusion with MR. First, theoretical considerations show why and under what conditions diffusion contrast is positive. Using this result, spin displacement spectra that are conventionally phase-encoded can be accurately reconstructed by a Fourier transform of the measured signal's modulus. Second, studies of in vitro and in vivo samples demonstrate correspondence between the orientational maxima of the diffusion spectrum and those of the fiber orientation density at each location. In specimens with complex muscular tissue, such as the tongue, diffusion spectrum images show characteristic local heterogeneities of fiber architectures, including angular dispersion and intersection. Cerebral diffusion spectra acquired in normal human subjects resolve known white matter tracts and tract intersections. Finally, the relation between the presented model-free imaging technique and other available diffusion MRI schemes is discussed.
The diffusion tensor of N-acetyl aspartate (NAA), creatine and phosphocreatine (tCr), and choline (Cho) was measured at 3T using a diffusion weighted STEAM (1)H-MRS sequence in the healthy human brain in 6 distinct regions (4 white matter and 2 cortical gray matter). The Trace/3 apparent diffusion coefficient (ADC) of each metabolite was significantly greater in white matter than gray matter. The Trace/3 ADC values of tCr and Cho were found to be significantly greater than NAA in white matter, whereas all 3 metabolites had similar Trace/3 ADC in cortical gray matter. Fractional anisotropy (FA) values for all 3 metabolites were consistent with water FA values in the 4 white matter regions; however, metabolite FA values were found to be higher than expected in the cortical gray matter. The principal diffusion direction derived for NAA was in good agreement with expected anatomic tract directions in the white matter.
Diffusion within the extracellular space (ECS) of the brain is necessary for chemical signaling and for neurons and glia to access nutrients and therapeutics; however, the width of the ECS in living tissue remains unknown. We used integrative optical imaging to show that dextrans and water-soluble quantum dots with Stokes–Einstein diameters as large as 35 nm diffuse within the ECS of adult rat neocortex in vivo. Modeling the ECS as fluid-filled “pores” predicts a normal width of 38–64 nm, at least 2-fold greater than estimates from EM of fixed tissue. ECS width falls below 10 nm after terminal ischemia, a likely explanation for the small ECS visualized in electron micrographs. Our results will improve modeling of neurotransmitter spread after spillover and ectopic release and establish size limits for diffusion of drug delivery vectors such as viruses, liposomes, and nanoparticles in brain ECS.
• drug delivery
• integrative optical imaging
• nanoparticles
• restricted diffusion
• somatosensory cortex
An improved finite difference (FD) method has been developed in order to calculate the behaviour of the nuclear magnetic resonance signal variations caused by water diffusion in biological tissues more accurately and efficiently. The algorithm converts the conventional image-based finite difference method into a convenient matrix-based approach and includes a revised periodic boundary condition which eliminates the edge effects caused by artificial boundaries in conventional FD methods. Simulated results for some modelled tissues are consistent with analytical solutions for commonly used diffusion-weighted pulse sequences, whereas the improved FD method shows improved efficiency and accuracy. A tightly coupled parallel computing approach was also developed to implement the FD methods to enable large-scale simulations of realistic biological tissues. The potential applications of the improved FD method for understanding diffusion in tissues are also discussed.
In MRI, the raw data, which are acquired in spatial frequency space, are intrinsically complex valued and corrupted by Gaussian-distributed noise. After applying an inverse Fourier transform, the data remain complex valued and Gaussian distributed. If the signal amplitude is to be estimated, one has two options. It can be estimated directly from the complex valued data set, or one can first perform a magnitude operation on this data set, which changes the distribution of the data from Gaussian to Rician, and estimate the signal amplitude from the obtained magnitude image. Similarly, the noise variance can be estimated from both the complex and magnitude data sets. This article addresses the question whether it is better to use complex valued data or magnitude data for the estimation of these parameters using the maximum likelihood method. As a performance criterion, the mean-squared error (MSE) is used.
The iterated Crank-Nicholson method has become a popular algorithm in numerical relativity. We show that one should carry out exactly two iterations and no more. While the limit of an infinite number of iterations is the standard Crank-Nicholson method, it can in fact be worse to do more than two iterations, and it never helps. We explain how this paradoxical result arises. Comment: 2 pages
In vivo diffusion analysis with quantum dots and detrains predicts the width of brain extracellular space
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A finite element model of molecular diffusion in brain incorporation in vivo diffusion tensor MRI data
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A finite element model of molecular diffusion in brain incorporation in vivo diffusion tensor MRI data
- McQueen
In vivo fiber tractography using DT-MRI data
- Basser
Maximum likelihood estimation of signal amplitude and noise variance from MR data
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Evaluation of cancer therapy using diffusion magnetic resonance imaging
- Ross