Revisiting classical diffusion magnetic resonance methods as a means to measure time-dependent diffusion

The field of diffusion microstructural magnetic resonance (MR) aims to probe time-dependent diffusion, i.e., an ensemble-averaged mean-squared displacement ⟨<i>r</i> ²(<i>t</i>)⟩ that is not linear in time. This time-dependence contains rich information about the surrounding microenvironment. MR methods to measure time-dependent diffusion quantitatively, however, require either non-standard pulse sequences, such as oscillating gradients, or make non-physical assumptions, such as infinitely narrow gradient pulses. Here, we argue that standard spin echo and stimulated echo MR sequences can be used to probe ⟨<i>r</i> ²(<i>t</i>)⟩ directly. In particular, we propose a framework in which the log-signal ratio obtained from a pair of measurements with different inter-pulse spacing <i>Δ</i> is proportional to the MSD between these two <i>Δ</i> values along the gradient direction <i>x</i>: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>⟨</mml:mo> <mml:mrow> <mml:msubsup><mml:mrow><mml:mi>r</mml:mi></mml:mrow> <mml:mrow><mml:mi>x</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msubsup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:msub><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mo>⟩</mml:mo></mml:mrow> <mml:mo>-</mml:mo> <mml:mrow><mml:mo>⟨</mml:mo> <mml:mrow> <mml:msubsup><mml:mrow><mml:mi>r</mml:mi></mml:mrow> <mml:mrow><mml:mi>x</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msubsup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:msub><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow> <mml:mrow><mml:mn>1</mml:mn></mml:mrow> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mo>⟩</mml:mo></mml:mrow> </mml:math> . The framework is quantitative for short, finite-duration gradient pulses and under the Gaussian phase approximation (GPA). To validate the framework, we consider one-dimensional diffusion between impermeable, parallel planes, as well as periodically-spaced, permeable planes. Excellent agreement is obtained between the estimation and the ground truth in the regime where the GPA is expected to hold. Importantly, the GPA can be made to hold for any underlying microstructure, making the proposed framework widely applicable.