Imaging Restrictive Pore Geometry with Asymmetrical Chirped Pulses

Qutaibeh Katatbeh^{1}, Alexey Tonyshkin^{2}, and Andrew Kiruluta^{3}

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 scales^{2}.

1. F. B. Laun, T. A. Kuder, W. Semmler, and B. Stieltjes, "Determination of the Defining Boundary in Nuclear Magnetic Resonance Diffusion Experiments'', Phys. Rev. Lett. 107, 048012 (2011).

2. L. Stepinsik and P. T. Callaghan, "The long time-tail of molecular velocity correlation in a confined fluid: observation by modulated gradient spin echo NMR", Physica B, 292, 296-301 (2000).

Figure 1: For these simulations, dq=0.5 microns, FOV=12.5 microns, T=100 ms, L=10 microns, D=1 micron2/ms, for: (a) f_{max}=71.62 Hz, chirp rate=2.9 kHz/s and gradient rise time of 3.5 ms (b) f_{max}=79.58 Hz, 1st pulse chirp rate of 1.77 kHz/s, 2nd chirp rate of 1.6 kHz/s, and gradient rise time of 3.14 ms (c) f_{max}=397.89 Hz with first chirp rate of 8.8 kHz/s, and 2nd chirp rate of 1.6 kHz/s, with rise time of 0.63 ms (d) Laun et al.’s proposed asymmetric gradient sequence sequence.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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