The Physics of Diffusion: What Are We Measuring?
Sune Jespersen1

1CFIN, University of Aarhus, Denmark

### Synopsis

This lecture covers the basic physics of diffusion. I cover the random walk as a conceptual model of diffusion as well as a tool for simulations. The relation for the mean square displacement is derived, and scenarios leading to time-dependent diffusivities are described, along with their universal short and long time regimes. A central quantity, the propagator, is introduced, and the diffusion equation describing its evolution derived. Examples of solutions are given, and the cumulant expansion as a general framework to describe diffusion in complex media is presented. The connection to the diffusion MR signal is outlined.

### Target audience

Scientists interested in the basic physics of diffusion in biological tissue

### Outcome/objectives

To gain a fundamental understanding of the physics of particles diffusing in complex media, supporting the ability of the participants to interpret and critically appraise diffusion MRI measurements.

### Highlights

- Due to thermal kinetic energy, water molecules move erratically on the micrometer scale – diffusion.
- The random walk is used for Monte Carlo simulation of diffusion
- The propagator encapsulates the statistics of diffusion and governs the diffusion MR signal
- The propagator is governed by the diffusion equation and is sensitive to microstructure
- The central limit theorem guarantees that the propagator is almost always Gaussian at long diffusion times
- The diffusion coefficient describes the width of the propagator (the mean square displacement), and depends in general on time

### Methods

Diffusion is the random motion of molecules, a kinetic manifestation of the thermal energy at any non-vanishing temperature. A basic understanding of its properties can be obtained by the picture of a ”random walk”, from which e.g. the relation $\left\langle \delta {{r}^{2}} \right\rangle =2dDt$ for the mean square displacement $\left\langle \delta {{r}^{2}} \right\rangle$ of molecules (in $d$ dimensions) during the time interval $t$ can be derived (1-3). The diffusion coefficient, $D$ in the equation, is a fundamental property of the medium, and is about 3 $\mu$m­2/ms for pure water at 37°C. This means that during e.g. 50 ms, a diffusion time relevant for MRI, molecules sample distances on the order of 10 $\mu$m of their environment, roughly the scale of individual cells. This property underlies the potential of diffusion to be used as a sensitive probe tissue microstructure (4,5). The random walk is also the conceptual basis of Monte Carlo simulations of diffusion. A central quantity for understanding diffusion, is the so-called propagator $P(r,{{r}_{0}};t)$, whose magnitude gives the probability for a molecule initially at ${{r}_{0}}$ to be found at $r$ after a time $t$ (3,6). As such, the propagator embodies the basic statistics of the diffusion process, and can be used to compute the MRI signal (7). Finding the diffusion propagator usually entails following the diffusion equation (also known as Fick’s 2nd law) (6) $$\frac{d}{dt}P(r,{{r}_{0}};t)=\nabla \cdot \text{D}\nabla P(r,{{r}_{0}};t)$$ which follows from the basic principle of mass conservation as well as Ficks first law, relating particle current to probability gradients. In this equation, we generalized from the diffusion constant $D$ to the 2nd rank diffusion tensor D, allowing for diffusion anisotropy. The solutions to the diffusion equation depend critically on boundary conditions, i.e. specification of the propagator or the current at e.g. tissue boundaries and interfaces. This is how the structure of the medium enters the formalism, and plays a crucial role in the appearance of the propagator (6,8). The most simple case of free diffusion, corresponding to a vanishing propagator at infinite distances, corresponds to the fundamental Gaussian expression (in $d=1$, $r\to x$) $$P(x,x_0;t)=\frac{1}{\sqrt{4\pi Dt}}{{e}^{-{{(x-{{x}_{0}})}^{2}}/(4Dt)}}$$ This solution is a component of many biophysical models of diffusion in tissue (9-12), specifically multiple gaussian compartments models, which can report on compartment volume fractions and diffusivities (4,13,14). The diffusion equation can be solved exactly only for a handful of other cases, such as diffusion between parallel plates, or inside spheres or cylinders, and these solutions also appear in biophysical diffusion models. Alternatively, the diffusion equation can be solved numerically using methods for partial differential equations such as finite elements. In general, the solution is not a Gaussian, although it can remain an arbitrarily good approximation on certain spatiotemporal scales. This explains the usefulness of the cumulant expansion (underlying e.g. diffusion kurtosis imaging) (5,15-18), which is essentially an expansion around a Gaussian distribution, each term accounting for finer and finer deviations from Gaussianity. Technically, the cumulant expansion is an expansion of the so-called characteristic function(15), the Fourier transform of the propagator, which is closely related to the MR diffusion signal(7). A full solution for the propagator in complex media such as biological tissue is not possible in general. A number of results exists for e.g. the time-dependent diffusivity nevertheless. For example, at short diffusion times, the so-called Mitra limit (19) applies $$D(t)={{D}_{0}}\left( 1-\frac{4}{3\sqrt{\pi }\,d}\frac{S}{V}\sqrt{{{D}_{0}}t}+\mathcal{O}(t) \right)$$ where $D_0$ is the microscopic diffusivity ($D(t\to 0$)) facilitating in principle a measurement of S/V, the surface to volume ratio of reflecting interfaces. In the opposite limit of long diffusion times $$D(t)\sim{\ }{{D}_{\infty }}+c{{t}^{-\tilde{\upsilon} }}$$ where $\tilde{\upsilon}$ is an exponent determined by the characteristics (correlation) of obstacles to the diffusing particles (20,21). In the tortuosity limit $t=\infty$, $D=D_\infty=D_0/\lambda^2$, where the tortuosity $\lambda$ reflects how much obstacles on average increase the distance of the shortest path between any two points. This equation implies a finite $D_\infty$, in agreement with measurements of diffusion in tissue. Note that this contradicts so-called anomalous diffusion and stretched exponentials, which imply that $D_\infty = 0$ or $D_\infty = \infty$ (even $D(t) = \infty$ for all $t$, for the latter case) (14). The finite $D_\infty$ is a consequence of the central limit theorem, which gives very broad conditions for the approach of sums of random variables (the random walk) to a Gaussian distribution (1). The time-dependent diffusivity can equivalently be studied in terms of frequency dependency of the velocity autocorrelation function, the Fourier transform of $\mathcal{D}(t)\equiv \theta (t)\left\langle v({{t}_{0}})v(t+{{t}_{0}}) \right\rangle$ (22,23).

### Acknowledgements

The author acknowledges support from the Dagmar Marshall foundation.

### References

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### Figures

The time evolution of the trajectory for a single particle is shown in top (blue lines) at 3 instances of time, $t_1$, $t_2$ and $t_3$. The spatial extent of the path is characterized by the root of the mean square displacement, which grows with time according to the top equation, defining the time-dependent diffusivity $D(t)$. The graph at the lower left illustrates a typical time dependence of the diffusivity, pointing out the short time regime and the long-term regime as mentioned in the text. The probability distribution of displacements, the propagator, is sketched on the lower right graph, along with a Gaussian approximation.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)