Synopsis

This lecture introduces key concepts behind the physics of diffusion MRI (dMRI) signal contrast, and motivate why these concepts are relevant in the context of quantifying tissue microstructure. Following this lecture, researchers and clinicians who are interested in understanding the basics of molecular diffusion, will gain intuition about the diffusion process as conceptualised by random-walks of particles, familiarise with representing the diffusion process by the diffusion propagator, understand the regimes in which the diffusion can and cannot be considered Gaussian and understand how these concepts are relevant in the context of tissue microstructure. Hands-on exercises will give intuition into the concepts discussed.

Target Audience

Outcome/Objectives

Following this lecture, the audience will:

• gain intuition on the diffusion process as conceptualised by random-walks of particles
• familiarise with representing the diffusion process by the diffusion propagator
• understand the regimes in which the diffusion can and cannot be considered Gaussian
• for Gaussian diffusion, understand the definitions of diffusion coefficient and diffusion tensor
• understand how these concepts are relevant in the context of tissue microstructure

Purpose

Methods

This lecture will cover the following topics:

Introduction to the diffusion process

The lecture introduces the molecular self-diffusion as (stochastic) random thermal motion of molecules by following the random-walk theory of Brownian motion. The simple case of mono-dimensional random-walk will be used as educational example to derive the diffusion equation. We show how this description is coherent with the macroscopic description of the diffusion process as a flux of particles arising from a gradient in concentration, and the resulting two Fick’s laws of diffusion [1,2].

Diffusion propagator and Central Limit Theorem

Solving the diffusion equation provides the conditional probability P(r0,r1,t) of finding a particle initially at a position r0, at a position r1 after a time t. This probability density function is the so called diffusion propagator (or the Green function of the diffusion equation), and it is very important for analyzing MRI diffusion measurements. The ensemble averaged (over all of the starting positions, r0) probability that an arbitrarily selected particle will displace by R = r1 - r0, during the period t is the so called average or mean propagator P(R,t). Following the example of a mono-dimensional random-walk, we will show that whenever the second moment of the displacement of a step in a symmetric random walk exists, then the probability density function of the displacement of the walker after n steps (i.e. P(r0,r1,t), and it ensemble average P(R,t)) tends, for n large enough, to a Gaussian distribution [2,3]. This statement provides the content of the Central Limit Theorem (CLT).

Quantifying the Gaussian diffusion process

The value of the CLT and random-walk theory will be discussed in the case of free isotropic diffusion, and anisotropic Gaussian diffusion [1-4].

Diffusion coefficient

For mono-dimensional free diffusion in homogeneous medium, the self-diffusion coefficient will be defined in terms of diffusion time and particles mean squared displacement (MSD). From its definition, we will introduce the concept of diffusion length as characteristic length scale associated to the diffusion process, and provide intuitive meaning with relation to biological tissue.

Diffusion tensor

For three-dimensional Gaussian diffusion in anisotropic medium, the concept of scalar self-diffusion coefficient will be extended to a rank two tensor, the diffusion tensor. Because symmetrical and real, the diffusion tensor can be diagonalized, and the meaning of the diffusion tensor eigenvalues/eigenvectors will be discussed, together with derived scalar metrics. These concepts are of particular importance since they are at the basis of the popular diffusion-weighted MRI technique known as Diffusion Tensor Imaging (DTI) [5].

Counter-examples to Gaussian diffusion

Restricted diffusion

Diffusion in restricted environment will be provided as example of CLT break down and non-Gaussian diffusion, by solving the diffusion equation in the case of diffusion restricted between parallel planar boundaries, a toy model for modelling cell membranes in biological tissue [1-3].

Different regimes: from Gaussian to non-Gaussian diffusion

Making use of the concept of diffusion length and diffusion time, we will define different diffusion regimes for restricted diffusion and characterize them in terms of apparent or time-dependent diffusion coefficient and the diffusion propagator features. Intuitively, diffusion measurements in restricted geometries can be split into three main regimes according to the size of the diffusion length compared to the size of the restricting geometry [3,4]:

1. the ‘short time’ or ‘free diffusion’ limit: in this limit the particle does not diffuse far enough to ‘feel’ the effects of restriction and measurements performed within this timescale lead to the true diffusion coefficient
2. the ‘crossover regime’ from the short- to long-time diffusion: in this regime some of the particles ‘feel’ the effects of restriction and the diffusion coefficient measured within this timescale will be a function of time. The fraction of particles that ‘feel’ the effects of the boundary will be dependent on the surface-to-volume ratio of the restricting geometry
3. the ‘long-time’ limit: in this regime all of the diffusing particles ‘feel’ the effects of restriction and now the displacement of the particle is independent of Δ and depends only on a.

Time dependence

The interesting case of apparent or time-dependent diffusion coefficient will conclude the lecture. Examples of time-dependent diffusion coefficient for diffusion restricted in planes, cylinders and spheres will be derived [1-3], and the results will be linked to cases of interest for biological tissues, with particular focus on brain tissue.

Acknowledgements

References

[1] Callaghan, Paul T. Principles of nuclear magnetic resonance microscopy. Clarendon Press, 1991.

[2] Callaghan, Paul T. Translational dynamics and magnetic resonance: principles of pulsed gradient spin echo NMR. Oxford University Press, 2011.

[3] Price, William S. NMR studies of translational motion: principles and applications. Cambridge University Press, 2009.

[4] Sen, Pabitra N. "Time‐dependent diffusion coefficient as a probe of geometry." Concepts in Magnetic Resonance Part A: An Educational Journal 23, no. 1 (2004): 1-21.

[5] Basser, Peter J., James Mattiello, and Denis LeBihan. "MR diffusion tensor spectroscopy and imaging." Biophysical journal 66, no. 1 (1994): 259-267.