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Diffusion MRI: From Quantitative Measurement to In vivo Neuroanatomy
Copyright © 2009 Elsevier Inc.
All right reserved.
Chapter One Introduction to Diffusion MR
Peter J. Basser and Evren Özarslan
I. What is Diffusion? 3 II. Magnetic Resonance and Diffusion 5 A. From the MR Signal Attenuation to the Average Propagator 8 III. Diffusion in Neural Tissue 8 IV. Concluding Remarks 9 Acknowledgments 9 References 9
Developments in the last century have led to a better understanding of diffusion, the perpetual mixing of molecules caused by thermal motion. In this chapter, the basic principles governing the diffusion phenomenon and its measurement using magnetic resonance (MR) are reviewed. The concepts of the apparent diffusion coefficient and of the diffusion propagator as well as their MR measurements are introduced from basic principles. Finally, the influence of neural tissue microstructure on the diffusion-weighted MR signal is briefly discussed.
Keywords: Diffusion, magnetic resonance, MRI, DWI, propagator, anisotropy, diffusion tensor, q-space, apparent diffusion coefficient
I. WHAT IS DIFFUSION?
Diffusion is a mass transport process arising in nature, which results in molecular or particle mixing without requiring bulk motion. Diffusion should not be confused with convection or dispersion — other transport mechanisms that require bulk motion to carry particles from one place to another.
The excellent book by Howard Berg (1983) Random Walks in Biology describes a helpful Gedanken experiment that illustrates the diffusion phenomenon. Imagine carefully introducing a drop of colored fluorescent dye into a jar of water. Initially, the dye appears to remain concentrated at the point of release, but over time it spreads radially, in a spherically symmetric profile. This mixing process takes place without stirring or other bulk fluid motion. The physical law that explains this phenomenon is called Fick's first law (Fick, 1855a, b), which relates the diffusive flux to any concentration difference through the relationship
J =∇C (1.1)
where J is the net particle flux (vector), ITLITL is the particle concentration, and the constant of proportionality, D, is called the "diffusion coefficient". As illustrated in Figure 1.1, Fick's first law embodies the notion that particles flow from regions of high concentration to low concentration (thus the minus sign in equation (1.1)) in an entirely analogous way that heat flows from regions of high temperature to low temperature, as described in the earlier Fourier's law of heating on which Fick's law was based. In the case of diffusion, the rate of the flux is proportional to the concentration gradient as well as to the diffusion coefficient. Unlike the flux vector or the concentration, the diffusion coefficient is an intrinsic property of the medium, and its value is determined by the size of the diffusing molecules and the temperature and microstructural features of the environment. The sensitivity of the diffusion coefficient on the local microstructure enables its use as a probe of physical properties of biological tissue.
On a molecular level diffusive mixing results solely from collisions between atoms or molecules in the liquid or gas state. Another interesting feature of diffusion is that it occurs even in thermodynamic equilibrium, for example in a jar of water kept at a constant temperature and pressure. This is quite remarkable because the classical picture of diffusion, as expressed above in Fick's first law, implies that when the temperature or concentration gradients vanish, there is no net flux. There were many who held that diffusive mixing or energy transfer stopped at this point. We now know that although the net flux vanishes, microscopic motions of molecule still persist; it is just that on average, there is no net molecular flux in equilibrium.
A framework that explains this phenomenon has its origins in the observations of Robert Brown, who is credited with being the first one to report the random motions of pollen grains while studying them under his microscope (Brown, 1828); his observation is illustrated in a cartoon in Figure 1.2. He reported that particles moved randomly without any apparent cause. Brown initially believed that there was some life force that was causing these motions, but disabused himself of this notion when he observed the same fluctuations when studying dust and other dead matter.
In the early part of the 20th century, Albert Einstein, who was unaware of Brown's observation and seeking evidence that would undoubtedly imply the existence of atoms, came to the conclusion that (Einstein, 1905; Fürth and Cowper, 1956) "... bodies of microscopically visible size suspended in a liquid will perform movements of such magnitude that they can be easily observed in a microscope". Einstein used a probabilistic framework to describe the motion of an ensemble of particles undergoing diffusion, which led to a coherent description of diffusion, reconciling the Fickian and Brownian pictures. He introduced the "displacement distribution" for this purpose, which quantifies the fraction of particles that will traverse a certain distance within a particular timeframe, or equivalently, the likelihood that a single given particle will undergo that displacement. For example, in free diffusion the displacement distribution is a Gaussian function whose width is determined by the diffusion coefficient as illustrated in Figure 1.3. Gaussian diffusion will be treated in more detail in Chapter 3, whereas the more general case of non-Gaussianity will be tackled in Chapters 4 and 7.
Using the displacement distribution concept, Einstein was able to derive an explicit relationship between the mean-squared displacement of the ensemble, characterizing its Brownian motion, and the classical diffusion coefficient, D, appearing in Fick's law (Einstein, 1905, 1926), given by
<[chi square]> = 2D Δ (1.2)
where >[chi square]> is the mean-squared displacement of particles during a diffusion time, Δ, and D is the same classical diffusion coefficient appearing in Fick's first law above.
At around the same time as Einstein, Smoluchowski (1906) was able to reach the same conclusions using different means. Langevin improved upon Einstein's description of diffusion for ultra-short timescales in which there are few molecular collisions, but we are not able to probe this regime using MR diffusion measurements in water. Since a particle experiences about 1021 collisions every second in typical proton-rich solvents like water (Chandrasekhar, 1943), we generally do not concern ourselves with this correction in diffusion MR.
II. MAGNETIC RESONANCE AND DIFFUSION
Magnetic resonance provides a unique opportunity to quantify the diffusional characteristics of a wide range of specimens. Because diffusional processes are influenced by the geometrical structure of the environment, MR can be used to probe the structural environment non-invasively. This is particularly important in studies that involve biological samples in which the characteristic length of the boundaries influencing diffusion are typically so small that they cannot be resolved by conventional magnetic resonance imaging (MRI) techniques.
A typical nuclear magnetic resonance (NMR) scan starts with the excitation of the nuclei with a 90 degree radiofrequency (rf) pulse that tilts the magnetization vector into the plane whose normal is along the main magnetic field. The spins subsequently start to precess around the magnetic field — a phenomenon called Larmor precession. The angular frequency of this precession is given by
ω = γB (1.3)
where B is the magnetic field that the spin is exposed to and γ is the gyromagnetic ratio — a constant specific to the nucleus under examination. In water, the hydrogen nucleus (i.e. the proton) has a gyromagnetic ratio value of approximately 2.68 x 108 rad/s/tesla. Spins that are initially coherent dephase due to factors such as magnetic field inhomogeneities and dipolar interactions (Abragam, 1961) leading to a decay of the voltage (signal) induced in the receiver.
As proposed by Edwin Hahn (Hahn, 1950), and illustrated in Figure 1.4, the dephasing due to magnetic field inhomogeneities can be reversed through a subsequent application of a 180 degree rf pulse, and the signal is reproduced. In this "spin-echo" experiment, the time between the first rf pulse and formation of the echo is called TE and it is twice the time between the two rf pulses, which is denoted by τ . The generated echo is detected by a receiver antenna (MR coil) and is used to produce spectra. Carefully devised sequences of rf pulses along with external magnetic field gradients linearly changing in space, enable the acquisition of magnetic resonance images. MR signal and image acquisition will be discussed in detail in Chapter 2.
The sensitivity of the spin-echo MR signal on molecular diffusion was recognized by Hahn. He reported a reduction of signal of the spin echo and explained it in terms of the dephasing of spins caused by translational diffusion within an inhomogeneous magnetic field (Hahn, 1950). While he proposed that one could measure the diffusion coefficient of a solution containing spin-labeled molecules, he did not propose a direct method for doing so.
A few years later, Carr and Purcell (1954) proposed a complete mathematical and physical framework for such a measurement using Hahn's NMR spin-echo sequence. They realized that the echo magnitude could be sensitized solely to the effects of random molecular spreading caused by diffusion in a way that permits a direct measurement. The idea they employed is not very different from what is utilized in most current studies of diffusion-weighted imaging. Because a spin's precession frequency is determined by the local magnetic field as implied by equation (1.3), if a "magnetic field gradient" is applied, spins that are at different locations experience different magnetic fields — hence they precess at different angular frequencies. After a certain time, the spins acquire different phase shifts depending on their location. Stronger gradients will lead to sharper phase changes across the specimen, yielding a higher sensitivity on diffusion. In most current clinical applications, a quantity called the "b-value", which is proportional to the square of the gradient strength, is used to characterize the level of the induced sensitivity on diffusion.
In the scheme considered by Carr and Purcell, a constant magnetic field gradient is applied throughout the entire Hahn spin-echo experiment as shown in Figure 1.5. Such an acquisition can be performed either in a spatially linear main field, or using another coil that is capable of creating a linear magnetic field on top of the homogeneous field of the scanner (B0). In their description, at a particular time t, a particle situated at position x experiences a magnetic field of B0 G x(t). If the particle is assumed to spend a short time, τ', at this point before moving to another location, it suffers a phase shift given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
as a result of the Larmor precession at the field modified by the constant gradient. Here, the minus sign is necessary for protons whose precession is in the clockwise direction on the plane perpendicular to the main magnetic field. Therefore, the net phase shift that influences the MR signal at t = 2τ is related to the motional history of the particles in the ensemble. By exploiting this phenomenon Carr and Purcell proposed MR sequences to sensitize the MR spin echo to the effects of diffusion, and developed a rigorous mathematical framework to measure the diffusion coefficient from such sequences. This elevated NMR as a "gold standard" for measuring molecular diffusion. An alternative mathematical formulation of the problem was introduced by Torrey (1956) who generalized the phenomenological Bloch equations (Bloch, 1946) to include the effects of diffusion.
After about a decade, Stejskal and Tanner (1965) introduced many innovations that made modern diffusion measurements by NMR and MRI possible. First, they introduced the pulsed gradient spin-echo (PGSE) sequence, which replaced Carr and Purcell's constant magnetic field with short duration gradient pulses as illustrated in Figure 1.6. This allowed a clear distinction between the encoding time (pulse duration, δ) and the diffusion time (separation of the two pulses, Δ). A particularly interesting case of this pulse sequence that makes the problem considerably simpler — from a theoretical point of view is obtained when the diffusion gradients are so short that diffusion taking place during the application of these pulses can be neglected. In this "narrow pulse" regime, the net phase change induced by the first gradient pulse is given simply by
φ1 = -q x1 (1.5)
where x1 is the position of the particle during the application of the first pulse and we ignore the phase change due to the B0 field, which is constant for all spins in the ensemble. In the above expression all experimental parameters were combined in the quantity q = γδG, where δ and G are the duration and the magnitude of the gradient pulses, respectively. Similarly, if the particle is situated at position x2 during the application of the second pulse, the net phase change due to the second pulse is given by
φ2 = -q x2 (1.6)
The 180 degree rf pulse applied in between the two gradient pulses reverses the phase change that occurs prior to it, i.e. that induced by the first gradient pulse. Therefore, the aggregate phase change that the particle suffers is given by
φ2 - φ1 = -q (x2 - (x1 (1.7)
Clearly, if particles remained stationary, i.e. (x1 - (x2, the net phase shift would vanish. In this case, and in the case in which all spins are displaced by the same constant amount, the magnitude of the echo will be unchanged (except for the T1 and T2 decay that is occurring throughout the sequence). However, if particles diffuse randomly throughout the excited volume, the phase increment they gain during the first period does not generally cancel the phase decrement they accrue during the second period. This incomplete cancellation results in phase dispersion or a spreading of phases among the randomly moving population of spins. Therefore, the overall signal, given by the sum of the magnetic moments of all spins, is attenuated due to the incoherence in the orientations of individual magnetic moments.
It is more convenient to introduce a new quantity, E(q), called MR signal attenuation, than to deal with the MR signal itself. E(q) is obtained by dividing the diffusion-attenuated signal, S(q), by the signal in the absence of any gradients, S00 = S(0), i.e. E(q) = S(q)/ S0. Since relaxation-related signal attenuation is approximately independent of the applied diffusion gradients, dividing S(q) by S0 eliminates the effects of relaxation, and the q-dependence of E(q) can be attributed solely to diffusion. The MR signal attenuation is then given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
where we employed two new quantities. The first of these, ρ(x1), is the spin density at the time of application of the first pulse quantifying the likelihood of finding a spin at location x1. In most applications, this function can be taken to be a constant throughout the water-filled region, where the value of the constant is determined by setting the integral of ρ(x1) equal to unity. The second function that we used, P(x1, x2, Δ), is the diffusion propagator (Green's function) that denotes the likelihood that a particle initially located at position x1 will have ended up at x2 after a time Δ — the separation of the two gradients. These two functions are related through the expression
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)
Excerpted from Diffusion MRI: From Quantitative Measurement to In vivo Neuroanatomy Copyright © 2009 by Elsevier Inc.. Excerpted by permission of Academic Press. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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