Pamela Wochner^{1}, Jason Stockmann^{2}, Torben Schneider^{3}, Jack Lee^{1}, and Ralph Sinkus^{1}

Classical methods for Diffusion-Weighted MRI use linear gradients to obtain information about tissue microstructure encoded in the loss of MR magnitude. These linear gradients cause diffusing spins to dephase which prevents the use of phase information. When, however, gradient fields are applied that vary quadratically in space, diffusion in anisotropic media results in a shift of net phase with only a minimal loss in signal magnitude enabling the exploration of MR phase. Both the theory of this new concept and Monte-Carlo simulations predict net phase for diffusion experiments using quadratic gradients in anisotropic media in excellent agreement.

A theoretical concept was developed that allows predictions about the effect of a Z2 gradient field on the phase of a spin ensemble diffusing in three dimensions. The analytic expression

$$\varphi = -2 \gamma \beta T (T + \Delta t)(D_z - \frac{1}{2}D_x - \frac{1}{2}D_y)$$

predicts the net phase φ for a spin-echo sequence. Two identical gradient pulses of duration T are applied, one before and one after the refocusing pulse. Δt is the time between the pulses, β the curvature of the quadratic gradient and γ the gyromagnetic ratio of water. Since diffusion is generally anisotropic, a unique diffusion coefficient is assigned to each direction (Dx, Dy and Dz). Thus, in case of isotropic diffusion there in no net phase shift due to Maxwell’s law stating $$$\Delta \vec{B} = 0$$$. Monte-Carlo simulations were used to examine the correctness of the analytic calculus. An experimental set-up was developed for initial imaging on a 1.5T MR scanner. Core of the set-up is a coil insert shown in Figure 1(a) (schematic) and Figure 2 (picture). A Helmholtz pair serves as gradient coil to generate the quadratic gradient. A small-scale RF saddle coil was positioned in the separation between the individual coils and was used for transmit/receive. Away from the saddle point, the gradient field shows increasingly linear behavior. In these regions, diffusion effects are expected to destroy signal magnitude. To verify this, initial experiments on isotropic water (with 1% salt and 0.1% gadolinium, $$$D = 2.3\times10^{-9} m^2/s$$$) and sunflower oil ($$$D = 1.1\times10^{-12} m^2/s$$$) phantoms were performed.The equation predicts no net phase for isotropic diffusion ($$$D_x = D_y = D_z$$$). However, net phase is expected for anisotropic diffusion. The same findings were obtained in the simulations, see Figure 3(a). The plots for theoretical and simulated net phase are shown in Figure 3(b) for different degrees of anisotropy (varying ratio $$$D_x/D_z$$$) for symmetric diffusion in x- and y-direction ($$$D_x = D_y$$$). Comparison of theory and simulation results show excellent agreement.

Figure 4 shows the magnitude images obtained in the initial experiments for water and oil phantoms. The images clearly show that in presence of the quadratic gradient, signal away from the saddle point is dephased. The gradient has a stronger effect on water (higher diffusivity) than on oil. These effects are more pronounced for stronger gradient fields. These observations confirm that diffusion effects due to increasingly strong linear
components away from the saddle point dephase the MR signal. They also confirm that the MR signal in around the saddle point is preserved.

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2. Perez EE, Carelli AA, Crapiste GH. Temperature-dependent diffusion coefficient of oil from different sunflower seeds during extraction with hexane. Journal of Food Engineering. 2011 Jul 1;105(1):180-5.

3. Panagiotaki E, Walker-Samuel S, Siow B, Johnson SP, Rajkumar V, Pedley RB, Lythgoe MF, Alexander DC. Noninvasive quantification of solid tumor microstructure using VERDICT MRI. Cancer research. 2014 Feb 3.

4. Ianuş A, Shemesh N, Alexander DC, Drobnjak I. Double oscillating diffusion encoding and sensitivity to microscopic anisotropy. Magnetic resonance in medicine. 2017 Aug;78(2):550-64.

Figure 1:
(a) Schematic coil insert. The Helmholtz pair consists of two individual coils.
A RF
saddle coil (transmit/receive) is fitted into the separation between the
Helmholtz pair.
(b) Gradient field $$$B_z$$$ generated
by Helmholtz coil along z-axis.
In
between the maxima, the field has parabolic shape along z-axis. Due to
Maxwell’s equations the minimum
of the parabola
(circle at z=0) along z-direction
is in reality a saddle point. Around this point, the gradient field does not have linear
components. Hence the signal is not dephased. Away from the saddle point, field shows
increasingly linear behaviour that dephases the signal.

Figure 2:
Picture of coil insert used for initial experiments. Two coils (red) form the
Helmholtz pair to generate the gradient field. The individual coils have a length
of 15 mm and 50 windings of 24 AWG insulated copper wire each. Inner diameter
of the coil 10.5mm, outer diameter 12mm. The separation between coil 1 and coil
2 is 21mm.
For
the described coil design, $$$\beta =25 T/m^2$$$ is
obtained for 1A coil current.
A
small RF saddle coil was fit in the separation between. The RF saddle coil is
transmit/receive.

Figure 3: (a)
Net phase φ plotted against different curvatures β. The simulation results confirm that
approximately zero net phase is obtained for diffusion in isotropic material.
However, the simulation yields a net phase that is linearly dependent on β when diffusion is anisotropic ($$$D_z =
2.3\times10^{-9}m^2/s$$$, $$$D_x/D_z =
0.5$$$, $$$Dx = Dy$$$ ).
(b) Net phase φ for different degrees of anisotropy
given by ratio $$$D_x/D_z$$$ and $$$\beta = 150 T/m^2$$$
Monte-Carlo simulations were performed
for $$$10^6$$$ spins and two identical rectangular gradient
pulses. Pulse duration is T = 25ms, Δt=
20ms.

Figure 4: Magnitude images comparing effects of
quadratic gradient field on sunflower oil (a) and water sample (b). Images were
acquired for gradient pulses for two different coil currents. The images show
that with increasing gradient strength, more signal away from the gradient
centre is destroyed due to increasingly strong linear components. Moreover,
these effects are more pronounced for the medium with higher diffusivity.
Two
sine pulses were used as gradient pulses with T = 40ms and TE = 100ms.