Q -Space Imaging: A Model-Free Approach

In Chapter 3.1 , the diffusion of water molecules was described according to the diffusion tensor model, which assumes a Gaussian probability of displacement associated to the diffusion of water molecules. This assumption is true in free systems, but it is also applied for water in complex biological structures. In this chapter, the q -space theory is explored as a model free from assumptions, and we discuss its limitations and its translation to in vivo applications to the spinal cord. One of the main advantages of q -space imaging (QSI) is that it allows probing of tissue microstructures with higher fidelity than diffusion tensor imaging (DTI) because of its sensitivity to restriction and hindrance.

All concepts relative to DTI are discussed in Chapter 3.1 , and it is assumed that the user is familiar with magnetic resonance imaging (MRI) diffusion experiments.

3.2.1 Q -Space Theory

Nonparametric approaches (e.g., those based on q -space theory) do not model the displacement probability explicitly and therefore give a more accurate representation of the diffusion mechanism in complex biological tissue. The notion that structural information is obtainable through the study of q -space has been pioneered by Refs and . The following derivation forms the basis of q -space analysis, where $P (r | r^{'}, t)$ $P (r | r^{'}, t)$ is the conditional displacement probability of water (i.e., expressing the probability of a water molecule diffusing from position r to r ^′ over the time period t ). In a pulsed gradient spin echo (PGSE) diffusion experiment (see Chapter 3.1 for details), if the diffusion gradient pulse duration δ is negligible compared to the diffusion time Δ ( δ ≪ Δ ), any motion of water molecules during the diffusion gradient time can be neglected. In this short gradient pulse (SGP) regime, the normalized echo attenuation S for a specific PGSE experiment can be expressed as the integral over all water molecules of the net phase shift of each molecule caused by the diffusion gradient vector g and diffusion gradient duration δ , weighted by the probability of its movement from r to r ^′ during diffusion time Δ (also called the mixing time). Thus:

S (g, δ, Δ) = \iint P (r) P (r | r^{'}, Δ) \exp [- i \cdot γ δ g \cdot (r^{'} - r)] ⅆ r^{'} ⅆ r,

$S (g, δ, Δ) = \iint P (r) P (r | r^{'}, Δ) \exp [- i \cdot γ δ g \cdot (r^{'} - r)] ⅆ r^{'} ⅆ r,$

with γ being the gyromagnetic ratio for the nucleus under observation (e.g., the proton ¹ H).

In a typical diffusion MRI experiment, the spatial dimension of the prescribed voxel is several magnitudes bigger than the length scale of diffusion motion. Hence, it is useful to consider the average displacement probability density function (dPDF) $P^{'} (R, t)$ $P^{'} (R, t)$ (often referred to as the “average propagator” ), describing the ensemble average probability of a particle moving the distance R during diffusion time t independent of starting position r within the voxel, which is defined as:

P^{'} (R, t) = \int P (r) P (r | r + R, t) ⅆ r .

$P^{'} (R, t) = \int P (r) P (r | r + R, t) ⅆ r .$

Combining Eqns (3.2.1) and (3.2.2) , the signal attenuation can be expressed as:

S (g, δ, Δ) = \int P^{'} (R, Δ) \exp [- i \cdot γ δ g \cdot R] ⅆ R .

$S (g, δ, Δ) = \int P^{'} (R, Δ) \exp [- i \cdot γ δ g \cdot R] ⅆ R .$

Furthermore, by introducing the q -value q = (2π) ⁻¹ γδg and rewriting Eqn (3.2.3) as:

S (q, Δ) = \int P^{'} (R, Δ) \exp [i 2 π q \cdot R] ⅆ R,

$S (q, Δ) = \int P^{'} (R, Δ) \exp [i 2 π q \cdot R] ⅆ R,$

it is evident that there is a simple Fourier relationship between the observed signal S ( q , Δ ) and the average displacement probability P ^′ ( R , t ). This Fourier relationship can be exploited to infer the average displacement probability of diffusing water at a certain diffusion time within a sample without the need to impose any constraints on characteristics of the probability distribution itself.

The displacement PDF or diffusion propagator is the average probability of a particle moving a certain distance during a given diffusion time. The PDF is calculated by measuring the signal attenuation at various gradient amplitudes ( g ).

3.2.2 Free, Hindered, and Restricted Diffusion

Diffusion in nervous tissue can deviate significantly from simple Gaussian behavior in the presence of cell membranes and structures that hinder or restrict the diffusion of water molecules.

In the simplest case, the change in signal attenuation due to molecule–barrier interaction can be interpreted as a change in the apparent diffusion coefficient (ADC), while assuming that the displacement probability distribution remains Gaussian. This simple interpretation also forms the basis for DTI (see Chapter 3.1 for details) and has shown great sensitivity to pathological changes that affect this interaction. However, measuring ADC or the more popular rotationally invariant DTI indices lacks specificity. By employing the q -space analysis, instead, it is possible to infer the shape and size of the molecular surrounding environment.

In the absence of any interacting barriers, free diffusion (or unrestricted diffusion) describes the pure Brownian motion of water (i.e., molecules diffusing freely in all directions). As described in Chapter 3.1 , in the case of free diffusion in all three dimensions, the average displacement probability takes the form of a simple Gaussian distribution and Einstein's equation for the mean displacement R applies:

R = \sqrt{6 D Δ} .

$R = \sqrt{6 D Δ} .$

In other words, in a free diffusing medium, the mean displacement has a simple dependency on the diffusion time Δ and the diffusivity D of the medium.

In reality, free diffusion is rarely encountered in a biological tissue sample. Instead, the presence of restricting barriers, such as cell walls, membranes, or myelin, impedes the motion of the water molecules and alters the displacement probability. In this case, the diffusion displacement probability not only is influenced by the diffusivity of the medium but also, more importantly, informs about the characteristics of the surrounding environment on the scale of the mean displacement R .

The observed effects on the diffusion MR signal can be quite diverse, depending on the type and location of barriers within the sample. It is helpful to further distinguish between restricted and hindered diffusion (see Figure 3.2.1 for illustration). Restricted diffusion is observed if the movement of water molecules is confined to closed spaces, e.g., inside impermeable cell walls. Those molecules experience restricted diffusion in that the molecules cannot displace farther than the confines of the cell. In hindered diffusion, the movement of molecules is impeded, although it is not confined within a limited space. Hindered diffusion best describes water motion in the space that is between densely packed cells or axons with semipermeable membranes and surrounded by myelin sheaths.

Free diffusion Water molecules move freely in all directions of space, following a Gaussian distribution
Restricted diffusion Water molecules are confined in closed spaces, such as the intra-axonal space
Hindered diffusion Water molecules are constrained within extra-axonal spaces

FIGURE 3.2.1, Cartoon of diffusion paths in the presence of different tissue environments. Restricted diffusion (shown in green): the molecule's motion is constrained within a confined space (e.g., within in the intra-axonal spaces). Hindered diffusion (shown in red): the free motion of water molecules is hindered by the dense packing of axons (e.g., within the extra-axonal space). Free diffusion (shown in blue): pure Brownian motion in the absence of barriers.

3.2.2.1 Restricted Diffusion and the Long Diffusion Time Limit

Figure 3.2.2 illustrates the expected relationship between diffusion time and the root mean square displacement $(RMSD = \sqrt{〈 R^{2} 〉})$ $(RMSD = \sqrt{〈 R^{2} 〉})$ in hindered and restricted diffusion. According to Einstein's equation (Eqn 3.2.4) , free diffusion shows a linear relationship between diffusion time and RMSD. In the hindered-diffusion scenario, the RMSD is reduced compared to free diffusion, but it is still approximately linear with distance. In the restricted diffusion scenario, the RMSD is limited by the confinement of the restricting compartment. From Figure 3.2.2 , it also becomes clear that the observed motion pattern of restricted diffusion has a strong dependency on the diffusion time. Consider the simple case of water molecules exclusively diffusing within an impermeable compartment of size a . For short diffusion times Δ ≪ a ² / D , the motion pattern is very similar to free diffusion, as most molecules have not yet explored the confines of the compartment. With increasing diffusion time, more diffusing molecules will exhibit a behavior influenced by the boundaries of the compartment. Finally, at long diffusion times Δ ≫ a ² / D , most molecules explore the extent of the confining geometry, and the mean displacement converges to the upper limit of the confining dimension a .

FIGURE 3.2.2, Illustration of the behavior of molecule displacement in the long diffusion time limit. For free diffusion (blue), the linear relationship between the squared mean molecule displacement and diffusion time is governed by Einstein's equation. The squared mean displacement in hindered diffusion (red) also increases monotonously with diffusion time, albeit at a lower rate than in free diffusion. In restricted diffusion (green), the possible displacement of each molecule is limited by the size of the confining space and therefore ultimately reaches a plateau for long diffusion times.

In the long diffusion time limit, all molecules have fully experienced the confining space; the conditional displacement probability (Eqn (3.2.1) ) becomes independent of starting position and therefore is equivalent to the molecular density function ρ ( r ) of the confining space. Furthermore, in the long diffusion time limit, the average displacement probability P ^′ (Eqn (3.2.2) ) also reduces to the autocorrelation function of ρ ( r ):

P^{'} (R, t \to \infty) = \int ρ (r) ρ (r + R) ⅆ r

$P^{'} (R, t \to \infty) = \int ρ (r) ρ (r + R) ⅆ r$

In simple geometries, this autocorrelation relationship can be directly exploited to estimate the size of the confining space.

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3.2.1

Q -Space Theory

3.2.2

Free, Hindered, and Restricted Diffusion

3.2.2.1

Restricted Diffusion and the Long Diffusion Time Limit

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