# Diffusion Magnetic Resonance Imaging in the Central Nervous System

*Kouhei Kamiya, Yuichi Suzuki, and Osamu Abe*

## Introduction

Diffusion magnetic resonance imaging (dMRI) measures the diffusion of water molecules along different dimensions The pattern of diffusion in biological tissues is determined by the geometry and organization of the sample at the micrometer scale - orders of magnitude below the size of the typical image voxel. The measurement was first introduced by Stejskal and Tanner in 1965 (Stejskal

& Tanner, 1965), and became available in clinical medicine approximately 20 years later (Le Bihan et al., 1986; Wesbey, Moseley, & Ehman, 1984). The unique contrast of diffusion-weighted images, and their sensitivity to tissue microstructure, had significant impact on clinical practice, as first discovered in acute stroke (Moseley et al., 1990). Anisotropy of diffusion in the nervous tissue led to the development of another clinically important tool called tractography, a non-invasive method for visualizing white matter bundles *in vivo* that is now indispensable for presurgical planning. This chapter is devoted to dMRI in the central nervous system, focusing on methods available in clinical settings and their theoretical backgrounds. First, we introduce the basic concepts of diffusion. Then, we move to applications, starting with the conventional diffusion-weighted image and diffusion tensor imaging (DTI). We next introduce methods going beyond DTI: diffusion kurtosis imaging (DKI) and biologically inspired compartment models. We also describe methods to estimate fiber orientation from dMRI, which forms the basis of tractography. Finally, we introduce emerging applications of non-conventional dMRI sequences and future perspectives. To keep the size of this chapter reasonable, we only touch on the basics. For more in-depth discussions on the theory and basic concepts of diffusion, the readers are referred to Price (2009), Jones et al. (2013), Kiselev (2017), and Novikov et al. (2019). Also, several recent reviews focus on recent topics such as microstructure models (Alexander et al., 2019; Jelescu & Budde, 2017), as well as tractography and its application in network analysis (Jeurissen et al., 2019; Sotiropoulos & Zalesky, 2019).

## Basics of dMRI

### Fick’s Laws

Adolf Fick (Fick, 1855) described the flux of particles in gases or liquids driven by a concentration gradient as:

where *F* is the rate of transfer of the diffusing particles passing through a unit area of the sample (i.e., flux), *C* is the concentration of the particles, and r is the position (a three-dimensional column vector) (Figure 6.1). *D* is referred to as the diffusion coefficient or diffusivity. Between 10°C and body temperature, £>_{waler} ranges from ~1 to 3 pm^{2}/ms. The negative sign indicates that the flow is in the opposite direction to that of increasing concentration. Equation (6.1) is called Fick’s first law of diffusion. Because the conservation of the total number of particles requires that the change of

FIGURE 6.1 Illustration of Fick’s laws, represented in one dimension with the concentration gradient along the x direction. The flux is driven by the concentration gradient (Fick’s first law of diffusion). The conservation of the total number of the particles requires that the change of concentration in the small volume *Ax* over short

*8C(x,t) 8F(x,l)*

time interval Д*t* is the net inflow: -^{5}-- =-----. Substituting this into the first law leads to Fick’s

*8t 8x*

second law of diffusion.

*дС ( Г t)*

concentration is related to the local flux divergence —*—=*** -VF(r,t), **Fick’s second law of diffusion can be derived: *^*

### Diffusion Propagator

In biological tissues, macroscopic concentration gradients are not the driving force behind diffusion. Rather, the process is driven by the local imbalance of exterior collisions with surrounding water molecules under random thermal motion (Brownian motion). Albert Einstein (Einstein, 1905) first showed the link between diffusion and Brownian motion. He showed that a probability density function (PDF) called the diffusion propagator P(r_{0}lr,,0, which represents the conditional probability that a particle starting at position **r _{0} **at time zero will move to

**r,**after a time

*t,*obeys Fick’s law in the same manner as the particle concentration. As we will see below, what we can actually measure is the ensemble average of net displacements,

**R**= r, -r

_{0}. The voxel-averaged PDF of net displacements

**R**over time

*t*is defined as

where **P(r _{0}) **is the probability of finding the spin at starting position

**r**Usually, we consider homogenous

_{0}.**P(**r

_{0}) = where

*V*is the voxel volume.

*F(R.t)*is termed the average or mean propagator.

Р(г_{о}1гь0 is the solution of the diffusion equation (Einstein, 1905),

Equation (6.4) is analogous to Fick’s second law of diffusion. In the absence of barriers (i.e., free diffusion), given the initial condition that all particles start from position **r _{0} **at time

*t=*0, the solution to Equation (6.4) is Gaussian

Equation (6.5) indicates that **P(r _{0}lr,,f) **does not depend on the initial position

**r**but only on the net displacement,

_{0}**R**=

**r, -r**The mean propagator

_{0}.

implies that the distribution of the particle mean displacement at any time *t* is Gaussian, and the width of distribution (mean squared displacement) increases linearly with *t:*

In one dimension, this is

Here, <•> denotes the ensemble average over the voxel. For a certain property w, it is defined as (tv) = j* *wp(w)dw,* where *p(w)* is the PDF of value tv. The range of diffusion time *t* explored in typical dMRI experiments of the brain is roughly from 20 to 100 ms. Assuming that the average diffusion coefficient in the brain is -1 pm^{2}/ms, dMRI probes a length scale on the order of 5 to 20 pm. This implies that dMRI is a powerful tool for probing subcellular structures of neural tissue, such as axons and cellular compartments, on the micrometer scale.

### Random Walk and Central Limit Theorem

The diffusion process can be best described as a random walk of particles (Kiselev, 2017). Here, we consider the case of free diffusion as the simplest example. The path of a particle can be split into very many small steps, taking place during a short time interval. In dMRI, we are interested in time intervals on the millisecond scale, which is much longer than the characteristic timescale of thermal motion, and therefore we can expect each step to be random and uncorrelated. This implies that the net displacement is the sum of very many short, statistically independent steps, thus satisfying the condition for the central limit theorem and indicting that the net displacement has a Gaussian distribution.

### Free, Hindered, and Restricted Diffusion

Diffusion in nervous tissue deviates from simple Gaussian distribution because of the presence of barriers such as cell membranes and myelin. In this setting, the diffusion propagator is influenced not only by the intrinsic diffusivity but also by the microstructural characteristics of the barriers. To express diffusion properties in biological tissues, it is often useful to distinguish between “restricted” and “hindered” diffusion (Jones et al., 2013). Restricted diffusion refers to particles residing inside small compartments, for example, inside impermeable membranes. Such particles cannot displace beyond the barriers. In hindered diffusion, the movement of the particles is impeded but not confined within a limited space. Typically, the extracellular space, the space between densely packed cells and axons, is regarded as hindered diffusion. Figure 6.2 illustrates the relationships between time and the mean square displacement for free, hindered, and restricted diffusion.

**FIGURE **6.2 Illustration of the mean squared displacement (MSD) and diffusion coefficient *(D)* as function of diffusion time *(t)* for the case of free, restricted, and hindered diffusion. For free diffusion (dark gray), MSD increases linearly with *t,* and *D* does not depend on *t.* Hindered diffusion (light gray) shows similar behavior to free diffusion for very small *t* where only a small portion of particles can reach a barrier. Increasing *t* deviates it from the case of free diffusion. For a very long *t,* it becomes indistinguishable from a uniform medium with the diffusion coefficient somewhat reduced as compared with the intrinsic diffusivity. Restricted diffusion (middle gray) refers to particles residing inside small compartments, which cannot displace beyond the barriers. In this case, when increasing *t,* MSD shows a plateau and *D* approaches zero.

### Bloch–Torrey Equation

Another approach to the dMRI signal is based on the Bloch-Torrey equation (Torrey, 1956). The transverse magnetization *m(r,t)* (a two-dimensional vector in the plane transverse to the *B _{0}* field, represented by a complex number) is represented by:

Here, r is the spin position, *T _{2}* is the transverse relaxation time, and g

*(t)*is the time-dependent diffusion-sensitizing gradient. For simplicity, we assumed a magnetically homogenous sample and homogenous relaxation rate in Equation (6.9). The above two approaches, the diffusion propagator and Bloch-Torrey equation, are tightly connected.

### Stejskal–Tanner Pulsed Gradient Spin–Echo Sequence

Stejskal and Tanner (1965) developed the theory and methodology to measure diffusion with MRI. Figure 6.3 illustrates the Stejskal-Tanner pulsed gradient spin-echo (PGSE) sequence, the most commonly used dMRI sequence. A 90° radiofrequency (RF) pulse is applied at *t=0* to excite

FIGURE 6.3 The effect of diffusion weighting in a Stejskal and Tanner pulsed gradient spin-echo (PGSE) sequence. (A) and (B) show two ensembles of spins, static (A) and diffusing (B). Different spins are represented by different grays. For the static spins, the dephasing induced by the first and the second lobes of the diffusion gradient are completely balanced, which means no signal loss occurs other than that caused by T2 relaxation. In contrast, the diffusing spins will experience different magnetic fields depending on the positions at each moment, and hence the dephasing induced by the first lobe cannot be reversed by the second one, leading to extra loss of signal amplitude as compared with the static spins.

the sample, and a 180° RF pulse is applied at *t=*TE/2 to reverse the phase of the spins. Diffusion- sensitizing gradients are applied on both sides of the 180° pulse. For static spins, dephasing before and after the 180° pulse is exactly reversed and therefore cancels because each spin experiences a constant magnetic field. In contrast, for diffusing molecules, each spin experiences a different magnetic field depending on its position. The dephasing cannot be completely reversed, resulting in a reduced signal amplitude (i.e., extra signal loss compared with the signal obtained without the diffusion-sensitizing gradient). In typical dMRI experiments, we are concerned with the effect of a spatially homogenous gradient g, defined as the gradient of the magnetic field in the direction of B_{()}

For the spin-echo sequence, we define g*(t)* as the “effective” gradient, accounting for the phase reversals due to the 180° pulse. Under the presence of such gradient, the magnetic field at point r is

and the phase shift at time *t* of a spin following a path r(/) is given by the temporal integration of the Larmor frequency

where у is the gyromagnetic ratio. To characterize the gradient imposed in the pulse sequence, the following notations are introduced:

Equation (6.14) defines the b-value, or the й-factor (Le Bihan et al., 1986), an index commonly used to represent the strength of diffusion weighting. For the example in Figure 6.3, Equation (6.14) *i (* <5 ^

yields *b = у G SI* A— , where *G* is the strength of the diffusion-sensitizing gradient (see also V 3,

Figure 6.4). Diffusion weighting is often expressed by combination of the 6-value and a unit vector indicating the direction of the diffusion-sensitizing gradient

### q-Space

To appreciate the relationship between the signal and the diffusion propagator, it is useful to consider the short gradient pulse (SGP) limit («5 <к A) (Figure 6.4). The transverse magnetization of a

FIGURE 6.4 The effective gradient *g(t)* (top row) and the corresponding *q(t)* (bottom row). (A) Stejskal and Tanner PGSE sequence and (B) short gradient pulses.

spin can be described with a complex number *e~ ^{u}‘* where

is the phase acquired in the magnetic field. According to Equation (6.12), the net phase shift of a spin which was at position r_{0} during the first gradient pulse and at r, during the second is q (r, -r_{0}). Thus, the normalized signal intensity (Stejskal & Tanner, 1965) is

where *S _{0}* is the reference signal that would be obtained without the diffusion-sensitizing gradient. Note that q here is different from the time-dependent function q(f) (Figure 6.4).

Using the mean propagator, Equation (6.16) can be rewritten as

Equation (6.17) implies that the dMRI signal is the Fourier transform of the mean propagator (Figure 6.5). This leads to the formalism of g-space imaging (QSI) (Assaf et al., 2000), which aims to recover the mean propagator by measurements varying q.

In clinical scanners equipped with a maximal gradient strength of 40 to 80 mT/m, the high q-values necessary for QSI cannot be achieved without increasing 6; hence the SGP condition is inevitably violated. In addition, in biological tissues, because QSI measures the displacement probability averaged over the voxel, the contributions from multiple compartments (e.g., intra- and extracellular compartments) are entangled.

### The Effect of Diffusion Time

The mean propagator **P(R,t) **is a function of **R **and *t,* therefore implying that at least two parameters

(q and *t)* are needed to characterize dMRI experiments (Kiselev, 2017; Novikov et al., 2019). The 6-value alone does not characterize a measurement, because two measurements with the same *b* but different *q* and *t* will result in different signals. The exception to this is the case of free diffusion. In the above example of the SGP limit, substituting Equation (6.6) into Equation (6.17) results in

FIGURE 6.5 The diffusion MRI signal is the Fourier transform (FT) of the mean propagator. The top row shows the signal obtained from t-butanol solution (free diffusion), and the bottom row shows the signal obtained from the bovine optic nerve (restricted diffusion). From left to right: (a) the normalized signal attenuation as a function of q for different diffusion times (Д), (b) the mean displacement obtained by FT of the data shown in (a), and (c) the root mean square displacement as a function of *t ^{m}.* (Reprinted with permission from Assaf et al. 2000.)

where *b=q ^{2}t.* As we will see in the following sections, the b-value is indeed useful for experimental conditions that can be implemented on clinical scanners, and hence is widely used. Typically, we estimate the diffusivity using Equation (6.18) in a low g-value regime. Here, we emphasize that estimated diffusivity is a function of the diffusion time

*t*(Figure 6.2), although it does not explicitly appear in Equation (6.18). We will return to the relationship between diffusivity and the diffusion propagator in Section 6.4.1.

Along the *t* axis, at least three characteristic regimes have been described in the dMRI literature (Kiselev, 2017; Novikov et al., 2019; Sen, 2004). These regimes are defined relative to the correlation length, *l _{c},* the typical distance over which the environment of each local microstructure changes (cell size, cell packing, axon undulation, caliber change, etc.). The correlation length sets the timescale

*t*/

_{c}= l_{c}^{2}*D,*which is the typical time for molecules to diffuse over the distance

*l*

_{c}.*1) Short time limit (t t_{c})*

Within a very short diffusion time, only a small portion of particles can reach a barrier, and the other particles do not experience the effect of the barrier. The relevant parameter here is the net surface-to-volume ratio (S/V) of the barriers. The diffusion coefficient in this regime is described (Mitra et al., 1992) as

where *D _{0}* is the intrinsic diffusivity and

*d*is the number of dimensions along which molecules can diffuse. For biological tissues, meeting this short time limit by using PGSE is practically impossible. Oscillating gradient spin echo (OGSE) is better suited to probe diffusion times shorter than PGSE (Does et al., 2003; Novikov & Kiselev, 2011). So far, the short time limit has been probed mainly in preclinical studies using animal models of tumors that have relatively large cell sizes (Reynaud, 2017).

2) Approaching the long time limit *(t* -> oo)

*D(t)* decreases with *t,* approaching the limit *I), = D(t* -> oo) in a medium-specific manner

- (de Swiet & Sen, 1996; Novikov et al., 2014). Novikov et al. (2014) demonstrated that the behavior approaching the long time limit depends on the distinct types of long-range spatial correlations of the structure, for example, regular lattices, or short-range disorder where the barriers are uncorrelated or exhibit correlation over a finite length. This theory has been shown to give a good explanation for the diffusion time dependence observed in the human brain within the range of diffusion time relevant for clinical studies (Fieremans et al., 2016).
- 3) Long time limit
*(t ->*oo)

For sufficiently long diffusion times, Gaussian distribution becomes the effective description for connected pores (hindered diffusion) (Kiselev, 2017) (Figure 6.2). In biological tissues, the premise for the central limit theorem (Section 6.2.3) does not always hold because the presence of barriers results in correlations between adjacent steps of particle motion. However, such correlation is only effective over the timescale *t _{c}* =

*l*

_{c}

^{2}/

*D.*For sufficiently long times, the steps become uncorrelated, justifying the use of the central limit theorem. Thus, at long times, the effective diffusion propagator becomes Gaussian. In other words, with increasing diffusion time, the effects of small local environments are “gradually forgotten” (Novikov et al., 2019), and their effects on measured signal finally become indistinguishable from that of a uniform medium, with the diffusion coefficient

*D*somewhat reduced as compared with the intrinsic diffusivity

_{x}*D*If the sample consists of multiple non-exchanging compartments, the model for the long time limit becomes the sum of Gaussians. We will see such models in Section 6.4.2.

_{0}.