The loss of phase information in MR diffusion encoding techniques was recently shown by Laun et al.¹ as a direct consequence of the use of anti-symmetric diffusion encoding gradients. They proposed lifting this restriction through the use of an asymmetrical gradient pair such that the first gradient is applied for a sufficiently long time for the random walkers to acquire a phase as that of a particle located at the center of mass of the imaging domain. A second very short diffusion encoding gradient is then applied but it is too short to record the diffusion dynamics in the spin system. It simply reverses the phase of all non-diffusing spins imposed by the first gradient such that the diffusion weighted signal now bears a phase as in conventional MR imaging. However, their proposed method recovers the diffusion walkers phases at the expense of practical implementation due to the requirement of a very narrow second gradient with unrealistic gradient slew rates. Here we propose a novel modification to the above sequence that uses null zeroth order moment chirped gradients to provide an optimal probe for small pore sizes in diffusion MR while the asymmetric diffusion encoding allows recovery of the diffusion walkers phases. The approach renders itself to alternative implementations that use a single chirped gradient lobe to encode diffusion but with the requirement for zero total gradient area.

We propose an approach that makes the practical implementation of the asymmetrical diffusion gradient encoding method practical on MR scanners. The proposed sequence uses an asymmetrical chirped gradient pair to encode diffusion dynamics. We will restrict ourselves to a linear chirp encoding but non-linear chirps could also be employed in the analysis. Thus, the displacement vector becomes:

$$q(t)=g \int_0^t \cos(\kappa \pi t^2)dt=g \frac{C\left( t \sqrt{ 2\kappa }\right)}{ \sqrt{2\kappa }}\,\,\, \mbox{where}\,\,\, C(t)=\int_0^t\cos(t^2)dt=FresnelC(t)$$

The diffusion weighted signal in the long time limit for an asymmetric chirped diffusion encoding gradient is modified from the asymmetrical diffusion encoding method (ADE) proposed in [1] as:

$$s(q)=<\exp(iq\cdot(x_{c.m.}-x_2))>=\exp[iq\cdot x_{c.m.}]\int_\Omega dx_2\exp(-iq(t)\cdot x_2)\,\,\,\mbox{where}\,\,\, q(t)=\gamma\int_0^{t^\prime}\tilde Fresnel(t\sqrt{2\kappa})/\sqrt{2\kappa}dt$$

is the displacement vector which is now temporally varying and xc.m. is the center of mass of the pore space function of a closed domain. When switching from pure sinusoid to chirps, the stationarity attached to the linear phase is replaced by a time-varying one which connects time and frequency by means of a one dimensional curve. The frequency structure of a chirp thus appears as that of a distorted monochromatic signal. Hence, the use of chirp-based substitutes to the ordinary Fourier analysis naturally takes into account possible time evolution of spectral properties of varying diffusion time scales².

We can thus break the pulse symmetry either by changing the chirp rate, bandwidth, duration or amplitude, giving us more degrees of freedom in optimizing the diffusion encoding gradient to preserve the phase of the diffusing spins. We adopt the restrictive equilateral triangular pore domain1 of length L as our imaging volume and inverse Fourier transform the diffusion-weighted signal derived from each of the encoding methods as shown in Figure 1. We see that the triangular restrictive geometry becomes discernible once we start to progressively break the symmetry between the gradient encoding chirped pulses from (a) to (c) while preserving the necessary condition that the zeroth order moments of the two gradients remains equal. In this case, symmetry is broken by changing the chirp rate of the second encoding gradient to be six times less than the first gradient pulse leading to a visualization of the structural information from the underlying triangular restriction. As a reference, the asymptotic case in which the second gradient is a delta function is shown to image the entire restrictive geometry while that in (c) is within the hardware specifications of most MR scanners. The proposed chirped asymmetric encoding method asymptotically approaches the result in (d) with longer chirp duration pulses.The ADE method for imaging the underlying restrictive geometry using an asymmetrical gradient encoding scheme to preserve the phases of the diffusing spins suffers from the challenge of implementation due to the requirement for a very short second gradient lobe. By extending this method to use an asymmetrical chirped gradient diffusion encoding pair, we showed that the degrees of freedom for breaking the pair symmetry were extended to the chirp rate, chirp bandwidth, duration and amplitude of the encoding gradients making the method much more versatile and compatible with typical gradient specifications (slew rate and amplitude) of conventional clinical MR system. The proposed method allows for the recovery of arbitrary shaped restrictive pores while also enabling sweeping the spectral range of the diffusion encoding gradient to probe even smaller pore sizes limited only by the chirp duration of the encoding gradient. Experimental implementation of this proposed approach and validation are currently underway.