A Langevin equation approach to diffusion magnetic resonance imaging

Abstract

Diffusion-weighted magnetic resonance imaging (DW MRI) is a noninvasive technique which is sensitive to the translational motion of water molecules because of their Brownian motion, allowing us to probe the microstructure of living cells and so achieve an understanding of tissue structure. The diffusive behaviour of water in cells alters in many disease states and during neuronal activation. Thus it has become a valuable and widely used diagnostic tool. The dramatic changes in water diffusivity in tissue following ischemia or observed in tumours, mean that the diffusion-weighted images can provide qualitative information for the purpose of diagnosis and treatment planning. As the degree of diffusion sensitivity is increased in a series of DWI acquisitions, decay of the image intensity due to dephasing, as a result of random modulation of the Larmor frequency caused by the translational Brownian motion of the spin containing liquid molecules is observed. The logarithm of the phase decay in general departs from that predicted by the normal Brownian motion due to the complex cellular environment experienced by the water molecules. For example, the Bloch-Torrey diffusion equation of the magnetization (based on Einstein's theory of the translational Brownian motion) describes accurately the normal phase diffusion behaviour of free water essentially predicting a logarithm of the decay, which is cubic in time. However, the phase decay for free water fails to describe the behaviour of water molecules in human tissue due to the complex cellular environment mentioned above [1]. Hence it has been adapted in a variety of ways (e.g. to stretched exponentials or Kohlrausch-Williams-Watts (KWW) functions i.e. a form of anomalous diffusion), to create empirical formulae which fit the decay curves, however there is much debate about the microscopic explanations which are used to justify these expressions. Thus, if a physically plausible model possessing a convincing microscopic explanation could be found to fit the decay observed in DWI imaging, it would have the potential to enhance the sensitivity of the technique for the observation of subtle changes in diffusion. As a step towards achieving this goal it is the purpose of this thesis to first outline the normal phase diffusion problem and its treatment without the use of probability density diffusion equations, by means of the inherently simpler Langevin equation for the random variables, describing the problem using only the properties of the characteristic function of Gaussian random variables [1]. Next, in order to provide a possible microscopic explanation for the stretched exponential behaviour referred to above, it will be shown how the Langevin treatment may be simply extended [1] to anomalous diffusion giving rise to stretched exponential decay using a fractional generalisation of the Langevin equation proposed by Lutz [2], which in the present instance describes the dependence of the phase decay on its past history. This equation describes the fractional Brownian motion of a free particle coupled to a fractal heat bath and so [2] represents Gaussian transport with the non-Markovian character being expressed via a memory function. In other words, in accordance with the pioneering work of Mandelbrot and van Ness [3] (See [4] for a comprehensive review), it allows one to include in the phase decay the fact observed in a host of natural time series, that the “span of interdependence” between samples of a random function may be infinite, unlike in Markov processes in which sufficiently distant samples of random functions are independent or nearly so. Indeed the concept of memory has been well put by Metzler and Klafter [4]; “unlike in a Markov process the now - state of the system depends on the entire history from its preparation”. The fractional Brownian motion, stemming from a fractional Langevin equation rather than the continuous time random walk (CTRW) often used to discuss anomalous diffusion is favoured here because it preserves many features associated with that walk, e.g. the stretched exponential decay while retaining the computational simplicity associated with Gaussian random variables. The theoretical results for the phase decay are then compared with the results of diffusion weighted experiments acquired from human subjects on a 3T MRI scanner, and animals on a 7T small bore MRI scanner.