Introduction

Diffusion-weighted MRI uses the motion of water molecules to generate contrast. Usually, gradient fields varying linearly in space encode the diffusion of water in the signal magnitude. For large spin ensembles, diffusion is symmetric in space, i.e. an equal number of spins move in positive and negative direction. For linear gradients, this results in zero bulk phase change. The phase does not contain information about the probed tissue. Instead, spins within a voxel de-phase and signal magnitude drops.This changes with a quadratic gradient field. The presented method replaces the linear gradient with a quadratic Z2 field $$$ B_z=z^2-0.5(x^2 + y^2))$$$. In anisotropic media, diffusion in this gradient field results in a phase change. For isotropic diffusion, no phase shift is expected due to concomitant terms that ensure $$$\vec{B_z}=0 $$$. Around saddle point of the field $$$(x=y=z=0)$$$, there are no linear components that de-phase the signal. Therefore, a quadratic gradient encodes diffusion in signal phase while preserving large portions of the magnitude, thus improving SNR. Here, the phase evolution in fiber phantoms is investigated in Monte-Carlo simulations and MR experiments.

Materilas and Methods

A theoretical concept was previously introduced [1] that allows predictions about the influence of a Z2 diffusion gradient on a spin ensemble. The expected change in net phase for 3D-diffusion is given by Equation(1)
$$\varphi=-2\gamma\beta T(T+\Delta t)(D_z-0.5(D_x+D_y))$$
This expression assumes two identical (rectangular shaped) quadratic diffusion gradient pulses with curvature β, duration T, mixing time Δt.γ is the gyromagnetic ratio of water. $$$D_{x,y,z}$$$ are the diffusion coefficients in the corresponding directions. Equation(1) predicts a phase change only for anisotropic diffusion. To validate the theory numerically and experimentally, a collection of densely packed fibers are chosen as anisotropic medium since high fractional anisotropy (FA) is possible. Also, its well-known structure can be studied in simulations[2]. A numerical phantom was created in Camino[3] by generating randomly packed, impermeable cylinders. FA of simulated phantom is determined combining Equation(1) in with the simulated phase accumulations for multiple packing-densities. Based on [2], a phantom was manufactured with bundles of parallel synthetic Dyneema fibers, which were soaked in water and tightly packed in shrink tube. To avoid phase errors due to bulk motion, the phantom was placed in a glass tube with fomblin. A six-direction DTI-measurement (b-value=[0,2000]) using the scanner's linear gradients was performed to determine diffusivity and FA of the sample. Quadratic-gradient diffusion experiments on this phantom were carried out on a 3T Philips Achieva scanner with a custom-made coil insert (Figure1). Schematic and description of the 1DFT-STEAM-sequence can be found in Figure2(a). This sequence allows long mixing times Δt, repeated phase measurements in one image acquisition and also enables to acquire data with/without diffusion-weighting within same acquisition (Figure2(b)).The non-weighted data serves as reference to correct sequence-induced phase errors. Diffusion experiments are performed for diffusion pulse duration T=30ms, Δt1=600ms and Δt2=250ms, TR=3ms, TE=80ms. Also, MR-experiments with RF coil only were performed to characterize uncertainty of phase measurements.

Results

Simulations in fiber phantoms confirm a phase change for anisotropic diffusion. In accordance with the theory, Figure3(a) shows phase change increasing with FA. Phase change also increases with longer Δt (see Figure3(b)). DTI-measurement of the fiber phantom yields FA=0.6. The acquired magnitude and phase images for diffusion experiments with two Δt can be found in Figure4. In Figure5(a), error-bars for the phase measurements of the spatial position with highest signal magnitude are plotted. This point in space corresponds to the area around the saddle point of the Z2 field. The spread of phase measurements suggest that they are affected by noise. Noise analysis of the measured phase shows an error of 0.056rad (Δt1) and 0.048rad (Δt2). The phase data acquired in the RF coil only experiments have a standard error of 0.0035rad (error-bar in Figure5(b)). SNR for magnitude (SNR_Δt1= 13.5, SNR_Δt2=25.0) and phase (SNR_Δt1= 2.12, SNR_Δt2=8.86) were determined.

Discussion and Conclusion

Diffusion MRI employing linear gradients to sensitize the signal magnitude to diffusion suffers from inherently poor SNR. The presented approach uses second order gradient fields to overcome these limitations by encoding (anisotropic) diffusion in the signal phase whilst preserving magnitude. Comparison of SNR shows a lower drop in SNR between Δt1 and Δt2 for phase than for magnitude data, although absolute value SNR is still higher for magnitude. This has to be addressed by reducing noise in phase acquisition (e.g. hardware improvements). Monte-Carlo simulations of fiber phantoms show a change in net phase dependent on the anisotropy of the sample as well as on experimental parameters, e.g. mixing time.MR data acquired with a custom made coil set-up show good signal for strong quadratic diffusion gradients and very long mixing times, both help, according to theory, to achieve a higher phase change. The measured phase difference of data acquired with two different Δt of 0.039 are in good agreement with the phase estimate provided by Equation(1) of 0.034rad. Higher uncertainty in diffusion measurements suggest additional noise is introduced when gradient coil is present and driven with current to generate gradient field. To reliably detect phase changes in this order of magnitude, the error on the phase measurements needs to be reduced, e.g. by acquiring more data points.

Acknowledgements

This work is funded by the King’s College London & Imperial College London EPSRC Centre for Doctoral Training in Medical Imaging (EP/L015226/1) andPhilips Healthcare.