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.

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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