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 timedependent 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 tissueOutcome/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 nonvanishing 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 (13). 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 socalled 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 (912), 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) (5,1518), 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 socalled 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 timedependent diffusivity nevertheless. For example, at
short diffusion times, the socalled 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 socalled
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 timedependent 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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