Frederik Bernd Laun^{1}, Kerstin Demberg^{2}, Michael Uder^{1}, Armin Michael Nagel^{1}, and Tristan Anselm Kuder^{2}

The short-time expansion of the
diffusion coefficient in powers of t^{1/2} is
universally connected to structural parameters of the boundaries restricting
the diffusive motion. The t^{1/2}-term
is proportional to the surface-to-volume ratio. The t-term is related to permeability and curvature. The short-time
expansion can be measured by application of oscillating gradients of long total duration. For
oscillating gradients, the inverse of the oscillation frequency becomes the
relevant time scale. The purpose of this work is to show that the
oscillating gradient approach is blind to the t-term. Thus, the t-term does not bias the determination of the t^{1/2}-term in experiments.

The short-time
expansion of the diffusion coefficient in powers of $$$t^{1/2}$$$, where $$$t$$$ is
the diffusion time, is universally connected to structural parameters of the
boundaries restricting the diffusive motion^{1}. The $$$t^{1/2}$$$-term is
proportional to the surface to volume ratio. The $$$t$$$-term is related to
permeability and curvature. The short-time expansion can be measured by the use
of diffusion encodings of short total duration or by application of
oscillating gradients of long total duration^{2}. For oscillating gradients, the
inverse of the oscillation frequency becomes the relevant time scale. The
purpose of this work is to show that the oscillating gradient approach is
blind to the $$$t$$$-term.

The effective diffusion coefficient in the short-time limit is

$$D(t)=\frac{<(x(t)-x(0))^2>}{2t}=\frac{<x(t)^2+x(0)^2-2x(t)x(0)>}{2t}=∑_{n=0}M_nc_nt^{n/2}$$with the $$$t^0$$$-term $$M_0=D_0,$$ the $$$t^{1/2}$$$-term$$M_1=-\frac{4}{3d\sqrt{π}}\frac{S}{V}\sqrt{D_0},$$and the $$$t$$$-term$$M_2=\frac{S}{V}(\frac{1}{d}\kappa-\frac{1}{12}D_0\overline{R^{-1}}).$$Here,
$$$D_0$$$ is the free diffusion coefficient, $$$S/V$$$ is the
surface-to-volume ratio, $$$d$$$ is the dimension, $$$\kappa$$$ is the
membrane permeability, and $$$\overline{R^{-1}}$$$ is a term related to
the membrane curvature. In diffusion MR experiments,
$$$<x(t)^2>$$$ and $$$<x(0)^2>$$$ do not contribute to the
signal attenuation due to the rephasing condition. Hence the diffusion
spectrum $$$\mathcal{D}(\omega)$$$ may be written as$$\frac{\mathcal{D}(\omega)}{\omega^2}=\frac{1}{2}\int_{-\infty}^\infty⟨x(t_2
)x(t_1 )⟩ e^{-i\omega t_{21}}
dt_{21}=-\frac{1}{2}\sum_{n=0}\int_{-\infty}^\infty M_n
|t_{21}|^{1+n/2}e^{-i\omega t_{21}}dt_{21}.$$with $$$t_{21}=t_2-t_1$$$. The last term is a Fourier transform: $$\frac{1}{2}\int_{-\infty}^\infty |t_{21}|^{1+n/2}e^{-i\omega t_{21}}dt_{21}=-\omega^{-2-n/2} cos(n\pi/4)\Gamma(2+n/2)$$ for $$$\omega>0$$$ and $$$n≥0$$$. Hence one finds:$$\mathcal{D}(\omega)=M_0+M_1 \sqrt{\frac{π}{2}}\frac{3}{4\omega^{1⁄2}}+0\cdot M_2+O(\omega^{-3⁄2} )$$and$$\mathcal{D}_{app}(\tau):=\mathcal{D}(2\pi/\tau)=D_0+M_1\frac{3}{8}\tau^{1⁄2}+0\cdot M_2+O(\tau^{3⁄2}).$$ Thus
the $$$t$$$-term is not visible in oscillating gradient diffusion MR diffusion experiments,
which build on the experimental detection of $$$D(\omega)$$$. Note that
the value of coefficient accompanying the $$$t^{1/2}$$$-term
was reported previously (e.g. in ^{3}).

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9. Laun FB, Demberg K, Nagel AM, et al.
On the
vanishing of the t-term in the
short-time expansion of the diffusion coefficient for oscillating gradients in
diffusion NMR. Front. Physics. Accepted.

Impact of the
t-term on the
short-time expansion of the diffusion coefficient. Markers indicate numerical simulations. Solid lines represent the short-time
expansion to order sqrt(t) and dotted lines to order
t. Red square markers represent bipolar gradients in the narrow pulse approximation and black circular
markers represent oscillating cosine gradients. a) Slab domain. The t-term is zero because
curvature and permeability are zero. b,c,d) Cylinder, sphere, and
bi-slab. For oscillating cosine gradients, the t-term is zero. Thus, the markers stay
close to the solid line in contrast to the markers indicating the bipolar
gradients, which stay close to the dotted line.