Diffusion-Weighted Imaging of the Spinal Cord

3.1.1.1

Basic Concept of Diffusion Weighting

Water molecules in tissue undergo Brownian motion, which means that molecules are not perfectly static over time but experience random microscopic displacements due to thermal agitation. This effect can be characterized by a diffusion coefficient. Compared to free water, where this effect is equal whatever the direction of observation is (i.e., it is isotropic), in tissue the molecular displacement is restricted and hindered ^a

a Hindered diffusion refers to the diffusion of water with a Gaussian displacement pattern, such as in the extracellular space. Restricted diffusion refers to the diffusion of water in restricted geometries, yielding a non-Gaussian pattern of displacement, and is found in the intracellular space bounded for example by axonal or dendritic membranes.

by the tissue structures (i.e., it is anisotropic), where it is faster along certain directions and slower along others. This is a very important consideration for the discussion of water diffusion measurements in tissue using magnetic resonance imaging (MRI), as it will become clearer later on in this chapter.

MRI is inherently sensitive to the movement of water molecules in tissue because in MRI a spatial position is characterized by its own phase and frequency that are encoded during the experiment. On average, at a voxel size level of a few millimeter cube, the thermal motion of water molecules is negligible but can be encoded by sensitizing the MRI acquisition to the average displacement taking place during a certain time.

To aid in understanding diffusion-weighted MR images of the spinal cord, it may be helpful to explore the basic physical principles of the diffusion process in general terms. Traditionally, the net movement of a substance (e.g., spins or protons in the nuclear magnetic resonance (NMR) experiment, or generically “water protons” in the NMR experiment and throughout this chapter) can be characterized by a diffusion coefficient ( D ), a variable relating the concentration gradient to the rate of transfer of water molecules through a unit of area. The flux of water in one direction is related to the diffusion coefficient by Fick's first law of diffusion:

where F is the mass flux (mass/time), D is the diffusion coefficient (area/time), C is the concentration of water protons (mass/volume), and r is the length coordinate. If we assume the system is in steady state (i.e., no net water protons added or subtracted from the system), Fick's first and second law of diffusion can be combined to form the second-order wave equation for diffusion:

where C ( r , t ) denotes the concentration of water protons as a function of both distance, r , and time, t .

In the diffusion-weighted MRI experiment, no real or significant measurable “concentration gradients” of water protons exist (i.e., “zero-flux”). Thus, any “concentration gradients” in the diffusion MRI experiment must be thought of in terms of the concentration of “tagged” water protons, such as is performed in diffusion tracer experiments. In any sense, if no physical concentration gradients are assumed in this condition, we must describe the concentration of water protons in terms of the probability of their displacement across both space and time, $P\left(r_0|r\text{,}t\right)$ , where r ₀ is the initial position of the water protons. Assuming the probability displacement function follows a Gaussian distribution, which is the simple solution using random walks, Einstein's equation for mean displacement applies,

and it results in the solution:

The application of this Gaussian probability distribution of water protons to the Bloch equations is the basis of conventional diffusion-weighted MRI.

3.1.1.2

Bloch Equations with Diffusion Terms

The time-evolving magnetization density in an NMR experiment can be defined by the Bloch equations (including relaxation terms):

where $\overrightarrow{M}\left(t\right)$ is the magnetization density vector at time t , M _z (0) is the initial longitudinal magnetization, M _z ( t ) is the longitudinal magnetization at time t , M _x ( t ) is the transverse component of the magnetization density vector in the x -orientation at time t , M _y ( t ) is the transverse component of the magnetization density vector in the y -orientation at time t , γ is the gyromagnetic ratio for protons ( γ = 42.58 MHz/T), $\overrightarrow{B}$ is the external magnetic field vector, T ₁ is the spin–lattice relaxation rate, T ₂ is the spin–spin relaxation rate, j is the imaginary number $j=\sqrt{-1}$ , and $\hat{i}\text{,}\hat{j}\text{,}\text{and}\hat{k}$ denote unit vectors in the x , y , and z orientation.

In 1956, Torrey described the Bloch equations in terms of self-diffusion of an NMR species (e.g., water protons) having a diffusion coefficient, D . By applying an arbitrary time-dependent linear magnetic field gradient, $\overrightarrow{g}\left(t\right)$ , we are able to describe the magnetization density vector as:

which has the solution:

This solution can be rewritten in generic terms as:

where b , generally referred to as the “ b -value”, represents the level of diffusion weighting (or attenuation) observed in the resulting magnetization as a function of diffusion “encoding” experimental parameters defined as:

For example, using a gradient recalled echo (GRE) experiment, the total magnetization at echo time TE at steady-state for a repetition time TR is:

Alternatively, using a spin-echo (SE) experiment, the total magnetization at echo time TE at steady-state for a repetition time TR is:

Note that while both of these experiments result in dramatically different dependencies of diffusion-related signal attenuation on the gradient waveforms, both sequences can be expressed in terms of $e^{-b·D}$ , differing only in their expression for b .

The simplest method of employing diffusion sensitivity is through the use of bipolar diffusion sensitizing gradients to the GRE experiment (in a single direction), as illustrated in Figure 3.1.1 (A) . Here, G reflects the gradient amplitude and δ reflects the duration of each bipolar gradient pulse. Using Eqn (3.1.9) , we see the expression for the b -value as a function of the diffusion-encoding scheme is:

Similarly, a monopolar pulse of similar amplitude and duration results in the same expression for b -value for the SE case ( Figure 3.1.1 (B)). This also can be extended to n -refocusing pulses in the SE experiment ( Figure 3.1.1 (C)), resulting in an expression for b -value of:

In 1965, Stejskal and Tanner introduced the most commonly employed diffusion-encoding scheme called pulsed gradient spin-echo (PGSE) encoding. In contrast to previous diffusion MR experiments, the PGSE experiment involves a diffusion time (also called mixing time), Δ, in addition to the encoding time, δ ( Figure 3.1.1 (D)).

Note that a stimulated echo acquisition mode (STEAM) , which replaces the π-refocusing pulse by two π/2-pulses separated by a length of time t , should be used to probe long diffusion times, since the phase encoded spins can be stored within the longitudinal orientation instead of transverse plane, making them subject to T ₁ decay and minimal T ₂ -related signal attenuation ( Figure 3.1.1 (E)). Taking into consideration limitations on gradient slew rates, the use of trapezoidal diffusion-encoding gradients ( Figure 3.1.1 (F)) allows for an accurate real-world approximation to the rectangular gradients proposed in the PGSE experiments, with a b -value of:

where G / ε is the gradient slew rate, which is typically assumed to be the maximum for a specific system. It is important to point out that the imaging gradients themselves, during readout of the image, contribute to diffusion weighting and should be included in the calculations of b for an exact solution of the equation.

3.1.1.3

The Apparent Diffusion Coefficient

Although the diffusion MR experiment can provide an estimate for the physical diffusion coefficient in an image voxel using the magnetic field gradients, the measured diffusion coefficient in MRI is often referred to as an apparent diffusion coefficient , or ADC , because it reflects the average diffusion coefficient in a voxel, affected by many biophysical factors and experimental setup.

Estimation of the diffusion coefficient in fast-moving liquids will typically be underestimated in highly tortuous environments due to the relatively long diffusion times (milliseconds) used in conventional MR experiments. These relatively long diffusion times are a direct reflection of magnetic field gradient hardware limitations, primarily in terms of slew rate. For example, consider a single water molecule diffusing through a relatively tortuous environment with a fixed diffusion time, Δ ( Figure 3.1.2 , top row ). If this water molecule is diffusing relative slowly and the spacing between barriers is sufficiently large compared to the ensemble average distance traveled by the diffusing water molecule during this diffusion time $\left(\langle r-r_0\rangle <r_{\text{boundaries}}\right)$ , then the actual diffusion coefficient will be accurately represented by the measured, apparent, diffusion coefficient with MRI $(\mathrm{ADC}\cong D)$ ( Figure 3.1.2 (A)). Alternatively, if the water molecule is diffusing relatively quickly such that the spacing between barriers is smaller than the actual distance traveled by diffusion water molecule during the diffusion time of the experiment ( l _actual > r _boundaries ), then the diffusion coefficient will not be accurately represented by the measured, or apparent, diffusion coefficient detected by MRI (ADC < D ) ( Figure 3.1.2 (B)). In this case, the apparent diffusion coefficient, ADC, is related to the actual diffusion coefficient, D , by the square of the tortuosity of the environment, θ ² :

Tortuosity is a variable relating the actual total distance the water molecule travels during the diffusion time to the shortest distance between the start and end points of the water molecule's path.

The current example assumes a constant diffusion time; however, the same situation arises when different diffusion times are used within the same tissue architecture ( Figure 3.1.2 , bottom row ). Specifically, if short diffusion times are employed then minimal diffusion restriction is observed $(\mathrm{ADC}\cong D)$ ( Figure 3.1.2 (C)), whereas if longer diffusion times are used then the diffusion coefficient will again not be accurately represented by the apparent diffusion coefficient (ADC < D ) ( Figure 3.1.2 (D)).

The physical diffusion coefficient of water molecules can also be influenced by kinematic viscosity and temperature of the environment, as illustrated by the Einstein–Stokes equations :

Here, R is the universal gas constant, T is the absolute temperature, N is Avogadro's number, $\upsilon$ is the kinematic viscosity, and r is the approximate radius of the water molecule. As this equation suggests, the measured diffusion coefficient is proportional to temperature and inversely proportional to viscosity. This relationship is particularly important when interpreting spinal cord diffusion MR measurements during pathological conditions, where changes in tissue viscosity can occur as a result of axonal or myelin degeneration, necrosis, or inflammatory cell infiltration. Additionally, acute spinal cord injury above the T6 spinal level can result in changes in core body temperature from neurogenic shock, which may potentially confound diffusivity measurements.

Also, the ADC will depend on the direction of the applied diffusion gradients due to the complex tissue microstructure that can have different coherences. For example, measuring the ADC along a fiber bundle or across it, using the same diffusion time can give very different results.

3.1.2.1

Diffusion Tensor Imaging

A “diffusion tensor” is a 3 × 3 matrix useful in describing the diffusion characteristics of anisotropic media, where diffusion rates vary with respect to orientation. Diffusion tensor magnetic resonance images are calculated from data obtained from DWIs, but measured in six or more noncollinear diffusion-encoding directions. Specifically, the diffusion sensitizing gradients are applied using combinations of the three imaging gradients (slice select, phase encode, and frequency encode; or alternatively the z , y , and x directions), sampling the diffusion weighting along various orientations, one at a time. From these multiple DWI measurements, regression or nonlinear techniques can be used to estimate the 3 × 3 tensor field that describes the preferred magnitude and orientation of the self-diffusion properties of water ( Figure 3.1.4 (A) ). The diffusion tensor, D , is positive and symmetric, therefore it only has six unique values (i.e., D _xy = D _yx , D _xz = D _zx and D _yz = D _zy ). As a result, DWI data must be obtained in a minimum of six sensitizing directions to determine the tensor values. However, it is usually recommended to acquire at least 20 directions to obtain a robust and accurate estimate of the diffusion tensor.

The preferred diffusion magnitude and direction are found from decomposition of the eigenvalues and eigenvectors for the diffusion tensor ( D ) in each image voxel. The eigenvalues ( λ ) are represented by a diagonal matrix, as shown in Figure 3.1.4 (B). To each one of the three eigenvalues is associated an eigenvector ( ε ), i.e., a unitary vector indicating the direction of the diffusion process. The eigenvalues are also ordered so that λ ₁ is the principal eigenvalue representing the diffusivity along ε ₁ . Popular DTI metrics are derived from these eigenvalues, such as the fractional anisotropy ( FA ), axial diffusivity (AD) and radial diffusivity (RD), defined in the “Diffusion Tensor indices” box. Because diffusion of water in spinal cord white matter is more anisotropic than gray matter, anisotropy indices are often useful for identifying spinal gray and white matter. The mean diffusivity (MD) is defined as the average of the three eigenvalues.

3.1.2.2

Ball-and-Stick Model

Introduced in Ref. , the ball-and-stick model is a partial volume model, where the diffusion-weighted MR signal is split into several anisotropic components (each one representing a fiber orientation) and a single isotropic component. This model can be fitted to DWI data (similar sampling scheme as for DTI). This model is notably implemented in BedPostX available in FSL ( http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT ) and in camino ( http://www.camino.org.uk ).

3.1.2.3

High Angular Resolution Diffusion Imaging

Diffusion tensors quantify mean diffusion within the space of a voxel, on the order of millimeters, providing only an average quantification of the microscopic diffusion process. The result is integration of diffusion characteristics for every axon localized in each voxel. If axons were homogeneously aligned within an image voxel, the first eigenvector of the tensor would accurately approximate their direction. However, the tensor model is not capable of resolving multiple fiber orientations within the same voxel. Although the use of the second eigenvector has been proposed as a means of resolving crossing fibers in spinal cord, there are major limitations imposed by the tensor model. For example, the three eigenvectors are, by definition, orthogonal. Thus, when the primary direction is defined by the first eigenvector—longitudinal fibers in the case of the spinal cord—the second eigenvector is limited in terms of degrees of freedom since its direction is necessarily on the plane orthogonal to longitudinal fibers. In the presence of nonorthogonal fiber crossings, the typical process of orthogonal decomposition of the tensor becomes less efficient.

To overcome this issue, model-free approaches have been proposed to measure the microscopic diffusion properties without constraining its representation. These methods are collectively known as diffusion spectrum imaging and have already demonstrated benefits for imaging the brain. However, long acquisition times are required to adequately sample q -space and retrieve the three-dimensional diffusion profile. To reduce acquisition times, sampling of q -space in a single direction has been proposed, allowing the distinction of diffusion properties of various types of axons. Q-space techniques for the spinal cord will be discussed in Chapter 3.2 . Another approach is to sample the q -space restricted to a single sphere. This method is known as high angular resolution diffusion imaging (HARDI). Some popular HARDI reconstructions methods include QBI, deconvolution techniques, diffusion orientation transform and persistent angular structure. Note that HARDI data can always be used to fit the diffusion tensor model to it too.

HARDI was successfully applied in the ex vivo spinal cord of rats, cats, and monkey. Results demonstrated that HARDI is able to retrieve crossing fiber information, where the DTI approach was constrained to a unique diffusion direction. The retrieval of longitudinal, commissural, and dorsoventral fibers was notably demonstrated. Berens et al. demonstrated particular diffusion properties in the dorsal horn of spinal cord injured rats, such as increased diffusivity in the plane orthogonal to dorsoventral tracts ( Figure 3.1.5 ).

3.1.2.4

Diffusion Kurtosis Imaging

The diffusional kurtosis is a quantitative measure of the degree to which the diffusion displacement probability distribution deviates from a Gaussian form. As such, diffusion kurtosis imaging (DKI) may provide new markers of pathological processes in the white matter. A spinal cord study showed that kurtosis imaging is sensitive to axonal and myelin damage in a rat model of axotomy. Acquisition of kurtosis imaging can be done by sampling q -space with multishells (e.g., 5 shells) varying from 200 to 3000 s/mm ² . For a comprehensive explanation of the diffusion kurtosis, see Chapter 3.2 .