Scientists with physics or engineering background interested in modeling diffusion and understanding its effects on the magnetic resonance signal.

· The goal is to provide the basic tools and a brief description of the building blocks for developing a comprehensive model of diffusion taking place in complex media.

· The audience will gain an overview of the essential concepts and mathematical techniques employed.

Within the timescale of signal formation via a conventional magnetic resonance (MR) technique, spin bearing molecules undergo random movements, which lead to variations in the detected signal. Because such motion occurs on the order of several micrometers, it is the microscale characteristics of the host medium that influence the MR signal. Thus, diffusion MR is often interpreted as a means to overcome the resolution limitation of conventional MR imaging. In principle, one can obtain features of the underlying microstructure from the diffusion-attenuated MR signal if adequate models of diffusion are employed. Because microscale features are quite sensitive to changes associated with development, aging, and numerous diseases, diffusion MR has been quite a powerful probe not only for basic science applications but also as a diagnostic tool.

Most tissues feature extremely complex architecture, making it difficult to come up with models featuring an acceptable number of parameters that could be estimated from the acquired signal. However, some features of the MR signal intensity, such as its anisotropy, seem to be influenced primarily by cellular membranes, presumably due to their restricting character [1]. This observation is important also from a practical point of view as the assumption of restricted diffusion makes it possible to parameterize the microstructure in terms of quantities such as cell size, cell size distribution, membrane permeability, etc., which are extremely relevant for the diagnosis and management of many diseases. In a commonly employed approach, one describes the cells as cylinders and spheres. Though such a description amounts to a substantial simplification of the host medium, it leads to two very desirable consequences: tractable solutions and characterization of the compartments via one or two parameters.

There are several experimental regimes for which analytical solutions for the MR signal intensity have been obtained. For Stejskal and Tanner’s conventional pulsed field gradient sensitization that employs a pair of gradients [2], analytical solutions have been obtained for the case in which the gradients are infinitesimally short [3]. However, this “narrow pulse” regime cannot be realized in clinical acquisitions. Perhaps a more feasible regime for the case of small attenuation (large signal), employs what is referred to as the Gaussian phase approximation [4].

Studies in recent years have demonstrated the potential of sophisticated gradient waveforms to provide novel information inaccessible by traditional measurements. In fact, in the small attenuation regime, the signal has been derived in terms of a straightforward integral involving the time-dependent gradient waveform [5].

For the case of larger diffusion sensitivity (lower signal values), the derivations tend to be considerably more cumbersome, and one typically has to resort to semi-analytical or numerical techniques. Among these, two approaches need to be mentioned. In the first case, an arbitrary gradient profile is represented by a train of impulses [6] leading to the multiple propagator technique. A matrix product formulation of this technique has been introduced [7], which can be considered a numerical evaluation of a path integral [5]. The second approach also discretizes the time axis, representing an arbitrary gradient profile via a piecewise constant function [8,9]. With the evaluation of some sophisticated integrals, this approach also leads to a matrix product representation [10], which has been generalized to the case of gradient waveforms that employ gradients along multiple directions [11]. The relationships between the multiple propagator and multiple correlation function techniques have been investigated recently [12]. Despite their resemblance, the latter approach leads to computationally more efficient implementations in the case of general pulsed field gradients.

Accurate inferences of microscale information from specimens under examination demand theoretical developments that typically employ sophisticated mathematical techniques. Significant advances have been made in the past most notably in porous media research. Some of these developments have been adopted to biomedical MR. With the ever increasing demand for accurate diagnostic tools and technical developments that improve the quality and speed of data acquisition, determination of microstructure via diffusion MR is expected to be an active area with potentially significant outcomes.