Multi-scale modelling of diffusion magnetic resonance imaging in heterogeneous media
Keywords: multi-scale modelling
diffusion MRI
magnetic resonance imaging

Diffusion magnetic resonance imaging (diffusion MRI) is an imaging technique which is sensitive to the micrometre scale movement of water molecules within a medium. It is routinely used in clinical practice, as well as increasingly in biomedical research as a quantitative tool. However, mathematical relations describing the image signal in a heterogeneous medium are poorly established. There is therefore a need for a better mathematical foundation for diffusion MRI which accounts for the heterogeneous nature of biological tissue, both in terms of molecular transport and magnetic properties.
By using a novel distribution function formulation describing diffusion-weighted spin-echo (DW-SE), which is the most common form of diffusion MRI, it is shown that the local modulation wave-vector, known as the q-vector, and the time scale define the operating regime of a DW-SE pulse sequence. Different locations in the q-t parameter space therefore correspond to respective asymptotic models describing DW-SE.
The effects of micro-scale magnetic heterogeneities are then analysed using multiple scales. It is found that in the long-time regime of DW-SE, local induced variations in the q-vector are typically of comparable magnitude to the macro-scale applied q-vector, whereas they are negligible in the short-time regime. Therefore, in the subsequent analysis of the long-time regime, the spatial q-vector variation is considered simultaneously with transport heterogeneities. Using multiple scales homogenisation, a multi-compartmental effective medium model has been derived. The effective diffusion tensors can be calculated by solving a cell problem over a periodic cell of the microstructure, however it is shown that the leading order effect of the spatial q-vector variation integrates exactly to zero.
Meanwhile, the short-time regime of DW-SE is analysed using a boundary layer model. It is first considered using an unphysical assumption of instantaneous modulation of spins, with results agreeing with literature. Taking advantage of the reduced problem complexity, the effects of realistic gradient pulses are then numerically computed. The image signal is found to vary approximately linearly with the pulse duration of a rectangular pulse, thus leading to a proposed two-point extrapolation method for correction. Meanwhile, for more general irregular pulse shapes, it is found that as long as they are symmetric, they correspond to an equivalent rectangular pulse with easily computable parameters.
Finally, for the intermediate-time regime, numerical solutions to the full problem are sought, using simple model geometries and a microscopy-derived realistic microstructure. The results agree with the two derived models at the respective asymptotes, with a transitional region of about a decade in the q-value. Additionally, the transitional region occurs at smaller q-values for isolated intra-cellular spaces compared to the connected extra-cellular space. This finding can inform future experiment design and modelling, particularly in relation to separating and analysing the intra-cellular signal component.