Pippa Storey^{1} and Dmitry S. Novikov^{1}

^{1}Department of Radiology, New York University School of Medicine, New York, NY, United States

R2* decay is nontrivial in the presence of magnetic microstructure; the logarithm of the signal is approximately quadratic over short times (the static dephasing regime) and asymptotically linear at long times (the diffusion narrowing regime). The transition can, in principle, provide an estimate of the characteristic length scales of the microstructure. Using Monte Carlo simulations of water molecules diffusing through a magnetic field perturbed by magnetic microstructure, we explore the regime of validity of the weak field approximation, in which the signal can be calculated analytically. We also illustrate how the signal behavior changes beyond the weak field limit.

In the weak field approximation$$$^{2-8}$$$ the signal decay is expressed as a cumulant expansion and truncated after the quadratic term $$S\left(t\right)=S_0e^{-R_2t}\left\langle e^{i\phi\left(t\right)}\right\rangle\approx{S_0}e^{-R_2t-\left\langle\phi^2\left(t\right)\right\rangle/2}\;\;\;\;\left[1\right]$$ where $$\phi\left(t\right)=\int_0^t\mathrm{d}t_1\,\Omega\left(t_1\right) \;\;\;\;\left[2\right]$$ is the cumulative phase of the spins and $$$\Omega\left(t\right)=\gamma{B_0}\left(t\right)$$$ is their Larmor frequency. The phase variance $$\left\langle\phi^2\left(t\right)\right\rangle=\int_0^t\mathrm{d}t_1\int_0^t\mathrm{d}t_2\,\left\langle\Omega\left(t_1\right) \Omega\left(t_2\right)\right\rangle\;\;\;\;\left[3\right]$$ is fully described by the temporal magnetic field correlation$$$^7$$$, which can be calculated analytically for certain geometries, notably a random distribution of permeable Gaussian sources of magnetic susceptibility, each with susceptibility profile $$$\chi\left(\mathbf{r}\right)=\chi_0e^{-\left|\mathbf{r}\right|^2/2l_c^2}$$$. The magnetic field correlation in this case equals $$\left\langle\Omega\left(0\right) \Omega\left(t\right)\right\rangle=\frac{\delta\Omega^2}{\left(1+t/t_c\right)^{3/2}}\;\;\;\;\left[4\right]$$ Similarly, for a random distribution of permeable spheres$$$^8$$$, each with susceptibility $$$\chi\left(\mathbf{r}\right)=\chi_0$$$ and radius $$$R=\left(6\sqrt{\pi}\right)^{1/3}l_c$$$, the magnetic field correlation equals $$\left\langle\Omega\left(0\right) \Omega\left(t\right)\right\rangle=\delta\Omega^2 \left[\mathrm{erf}\left(\sqrt{\frac{t_c}{t}}\right)+\frac{2}{\sqrt{\pi}}\left(\frac{t}{t_c}\right)^{3/2}\left(1-e^{-t_c/t}\right)+\frac{1}{\sqrt{\pi}}\left(\frac{t}{t_c}\right)^{1/2}\left(e^{-t_c/t}-3\right)\right]\;\;\;\;\left[5\right]$$ Here $$$\delta\Omega^2$$$ is the variance in Larmor frequency due to magnetic field inhomogeneity. The order of the cumulants is reflected in the power of the dimensionless parameter $$$\alpha=\delta\Omega\,t_c$$$, which represents the dephasing due to field inhomogeneity over the time interval $$$t_c$$$. Thus the weak field approximation is assumed to be valid in the limit $$$\alpha\ll1$$$.

In this work, we simulate the motion of water molecules among susceptibility sources of various geometries, and calculate the magnetic field correlation and signal decay over a range of $$$\alpha$$$. The results are validated against theory, and used to examine the regime of validity of the weak field approximation.

The cumulative phase $$$\phi\left(t_i\right)$$$ of each random walker at each time point $$$t_i$$$ was calculated and stored. Since the phases varied linearly with $$$\delta\Omega$$$, they could be arbitrarily scaled after the simulation, and the signal dephasing $$$\left\langle e^{i\phi\left(t_i\right)}\right\rangle$$$ reevaluated.

This is displayed in Figure 4. Note that the kurtosis of the cumulative phase approaches that of the underlying magnetic field as $$$t\rightarrow{0}$$$ since $$\lim_{\delta{t}\rightarrow{0}}\phi\left(\delta{t}\right)=\gamma{B_0}\left(0\right)\delta{t}\;\;\;\;\left[7\right]$$ It follows that $$$K\left(0\right)=0$$$ for spins diffusing through a magnetic field consisting of spatially correlated Gaussian-distributed noise.

The regime of validity of the weak field approximation is plotted as a function of $$$\alpha$$$ and $$$t/t_c$$$ in Figure 5. The weak field limit may be characterized by the condition that the fourth-order cumulant be much smaller than the second-order cumulant. For the geometries considered here, the kurtosis of the cumulative phase is $$$K\left(t\right)\sim{1}$$$, from which it follows that the weak field limit may most simply be characterized by the condition that the variance of the cumulative phase be small, $$$\left\langle\phi^2\left(t\right)\right\rangle\ll{1}$$$.

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Fig 1: Magnetic field correlation estimated from Monte Carlo
simulations and compared to theory for Gaussian and spherical sources of
magnetic susceptibility, and for a magnetic field consisting of spatially correlated
Gaussian-distributed noise.

Fig 2: Semilogarithmic plot of the free induction decay as a
function of normalized time $$$t/t_c$$$ for $$$\alpha$$$ = 0.05, 0.2 and 0.8. Monte
Carlo results (solid lines) are compared with the weak field approximation
(dashed lines) for a magnetic field perturbed by spherical and Gaussian sources
of magnetic susceptibility (top and bottom rows respectively) and for a
magnetic field consisting of spatially correlated Gaussian noise (bottom row). The
Monte Carlo results deviate from the weak field approximation for high $$$\alpha$$$
and/or long $$$t/t_c$$$.

Fig 3: Semilogarithmic plot of the signal as a function of $$$\alpha$$$
for fixed values of $$$t/t_c$$$. Monte Carlo results (solid lines) are compared
with the weak field approximation (dashed lines) for the same field geometries
as in Fig 2. The Monte Carlo estimates for $$$\ln{S}$$$ exhibit a quadratic
dependence on $$$\alpha$$$, in agreement with the weak field approximation, for
small $$$\alpha$$$ and short times $$$t/t_c$$$, but an approximately linear
dependence on $$$\alpha$$$ for large $$$\alpha$$$ or long times.

Fig 4: Excess kurtosis in the spins’ cumulative phase evaluated
from Monte Carlo simulations and plotted as a function of normalized time $$$t/t_c$$$
for a magnetic field perturbed by spherical sources of magnetic susceptibility and
for a magnetic field consisting of spatially correlated Gaussian noise. In the
latter case, the kurtosis was also calculated directly by numerical integration
of an analytic expression. The result (dashed line) agrees well with Monte
Carlo simulations.

Fig 5: The regime of validity of the weak field approximation as a
function of $$$\alpha$$$ and normalized time $$$t/t_c$$$. The lines represent
contours at which the percentage difference in $$$\ln{S}$$$ between the
weak field approximation (equation [1]) and Monte Carlo simulations equaled 5%. The weak
field approximation is valid over a larger region of parameter space for the
magnetic field consisting of spatially correlated Gaussian noise because the
kurtosis of the cumulative phase is lower for that geometry.