Microstructure Modeling
Hong-Hsi Lee1,2
1Department of Radiology, A. A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, MA, United States, 2New York University School of Medicine, New York, NY, United States

Synopsis

Diffusion MRI enables to estimate microstructural length scale in cell dimension. Here we introduce three common examples in biophysical modeling of diffusion MRI: (1) Considering diffusion dephasing as transverse relaxation due to field inhomogeneity of applied diffusion gradient, intra-cellular signals mainly depend on gradient pulse duration in wide pulse limit. (2) The diffusion time-dependence in a heterogeneous medium is a process of spatial homogenization of restrictions, leading to power-law tails in diffusivity time-dependence. (3) Directional average of signals for each diffusion weighting b and the analysis of its deviations from 1/√b scaling provides a rotationally invariant axon size estimation.

Diffusion dephasing as the transverse relaxation

Diffusion dephasing can be considered as the transverse relaxation due to the field inhomogeneity induced by applied diffusion gradients [1]. Consider the pulsed-gradient sequence of gradient pulse width $$$\delta$$$ and time interval $$$\Delta$$$ between the gradient pairs. In the wide pulse limit, $$$\delta\gg t_c$$$, the problem maps onto that of transverse relaxation in the diffusion-narrowing regime (equivalent to the Gaussian phase approximation). In this limit, the time $$$t_c\sim r^2/D_0$$$ to diffuse across a cell of size $$$r$$$ (intrinsic diffusivity $$$D_0$$$) provides the correlation time, beyond which the contribution to the precession phase $$$\phi$$$ for each spin gets randomized.

It is then natural to split each Brownian path into $$$N=t/t_c\gg1$$$ steps of duration $$$t_c$$$, such that the total phase is $$$\phi\sim\sum\phi_n$$$, where each $$$\phi_n\sim\Omega\cdot t_c$$$ can be treated as an independent random variable with zero mean and variance $$$\langle\phi_n^2\rangle\sim(\Omega\cdot t_c)^2$$$. The $$$\Omega\sim g\cdot r$$$ is a typical value of the Larmor frequency inhomogeneity across a cell imposed by the applied Larmor gradient $$$g=\gamma G$$$, with $$$\gamma$$$ gyromagnetic ratio and $$$G$$$ diffusion gradient strength.

When the number $$$N$$$ of independent "steps" becomes large, according to the central limit theorem (CLT), the mean values and variances from the independent steps add up, i.e., $$$\langle\phi\rangle=0$$$ and $$$\langle\phi^2\rangle_c\equiv\langle\phi^2\rangle-\langle\phi\rangle^2\sim N\langle\phi_n^2\rangle\sim\Omega^2t_c\cdot t$$$, such that the intra-cellular diffusion-weighted (DW) signal $$$S_{\rm in}\sim\exp(-\langle\phi^2\rangle_c/2)\sim\exp(-R_2^*\cdot t)$$$, with effective $$$R_2^*\sim\Omega^2t_c$$$. In this case, it is the total pulse duration $$$t=2\delta$$$ that matters; the inter-pulse duration $$$\Delta$$$ $$$(>\delta)$$$ does not enter these considerations as long as $$$\delta\gg t_c$$$ (wide pulse), since no transverse relaxation occurs during the time when the gradient is off. Therefore, the DW signal inside a cell scales as
$$-\ln\,S_{\rm in}\sim\left(\frac{g^2r^4}{D_0}\right)\cdot\delta\,,$$
which agrees with Neuman's (1974) exact solution of intra-cellular DW signal transverse to a cylinder [2]
$$-\ln\,S_{\rm in}=\frac{7}{48}\cdot\frac{g^2r^4}{D_0}\cdot\delta + {\cal O}(g^4)\,.$$
Using the definition of apparent diffusivity, we find the radial diffusivity inside cylinder:
$$D_\perp\equiv-\tfrac{1}{b}\ln\,S_{\rm in}|_{b\to0}=\frac{7}{48}\frac{r^4}{D_0}\frac{1}{\delta(\Delta-\delta/3)}\,,\quad(1)$$
where $$$b=g^2\delta^2(\Delta-\delta/3)$$$ is the diffusion weighting.

To sum up, the DW signal of restricted diffusion inside cells solely depends on the gradient pulse duration in wide pulse limit. In other words, the sensitivity of axon diameter mapping (ADM) using wide pulse sequence is mainly tuned by the gradient pulse duration $$$\delta$$$, rather than the inter-pulse duration $$$\Delta$$$. Furthermore, the axon diameter estimation could be confounded by the features of realistic axonal shapes, such as axon caliber variations and undulations (Exercise 1) [3].

Power-law scaling of diffusivity time-dependence

Qualitatively, diffusion exemplifies the phenomenon of the gradual coarse-graining of medium details: The time-dependence of diffusion in a heterogeneous medium is a process of spatial homogenization of restrictions over a diffusion length scale $$$L(t)\propto\sqrt{t}$$$. At very short diffusion time $$$t$$$, the diffusion is roughly free (Gaussian) with an "intrinsic" diffusivity $$$D|_{t\to0}=D_0$$$. In contrast, in $$$t\to\infty$$$ limit, each water proton diffuses throughout the whole medium, and the particle loses all memories of structural details, leading to a Gaussian diffusion with a diffusivity $$$D|_{t\to\infty}=D_\infty$$$ lower than $$$D_0$$$ (Exercise 2).

When discussing diffusion time-dependence, it is convenient to focus on the change rate of the 2nd order displacement cumulant over time, i.e., instantaneous diffusion coefficient $$$D_{\rm inst}\equiv\frac{1}{2}\partial_t\langle x^2\rangle\,.$$$ At long times, local diffusivity fluctuation $$$\langle\left(\delta D(x)\right)^2\rangle|_{L(t)}$$$ is coarse-grained over $$$L(t)$$$ and decreases with time. This spatial homogenization of restriction information makes the diffusivity $$$D_{\rm inst}(t)$$$ decrease with time and gradually approach $$$D_\infty$$$. This self-averaging process yields the diffusivity time-dependence at long times. These qualitative considerations result in the exact statements [4,5]
$$D_{\rm inst}(t)-D_\infty\simeq \frac{1}{dD_\infty}\cdot\langle\left(\delta D(x)\right)^2\rangle|_{L(t)}\,.$$

The crucial observation is that the spatial fluctuations of $$$D(x)$$$ mimic those of the microstructure restriction density $$$n(x)$$$ at large $$$x$$$; in particular, the power spectrum of the diffusivity fluctuation in the spatial frequency $$$k$$$ domain [4,5]
$$\Gamma_D(k)\equiv\frac{\delta D(-k)\delta D(k)}{V}\propto k^p\,,k\to0$$
is characterized by the structural exponent $$$p$$$ describing long-range density fluctuations $$$n(x)$$$ via the power spectrum of the tissue microstructure in a voxel (of volume $$$V$$$), i.e., Fourier transform of the density autocorrelation function $$$\Gamma(k)\equiv{\rm FT}\{\langle n(x_0+x)n(x_0)\rangle_{x_0}\}\,,$$$ sharing the same power-law exponent $$$p$$$ at low $$$k$$$, i.e., $$$\Gamma_D(k)\sim\Gamma(k)\sim k^p$$$; this is because that, after coarse-graining, the small local fluctuations $$$\delta D(x)$$$ becomes asymptotically proportional to $$$\delta n(x)$$$.

In what follows, we relate the power-law tails in the time-dependent diffusivity to the first order in the power spectrum, based on the perturbative treatment up to the order $$${\cal O}\left((\delta D(x))^2\right)$$$ in $$$d$$$-dimension [4,5]:
$$D_{\rm inst}(t)-D_\infty\simeq\frac{1}{dD_\infty}\int\frac{{\rm d}^dk}{(2\pi)^d}\Gamma_D(k)e^{-D_\infty k^2t}=A\cdot t^{-\vartheta}\,,$$
where the dynamical exponent $$$\vartheta=(p+d)/2$$$ reveals the universality class of the microstructure. Using the relation of diffusivity and instantaneous diffusivity, $$$D(t)\equiv\frac{1}{2t}\langle x^2\rangle=\frac{1}{t}\int_0^t D_{\rm inst}(t'){\rm d}t'$$$, we find
$$D(t)\simeq D_\infty+c\cdot t^{-\vartheta}\,,$$
where $$$c=A/(1-\vartheta)$$$.

Example 1: Diffusion outside highly aligned, densely packed axons.
White matter axons are commonly modeled as highly aligned, randomly packed fibers. The hindered diffusion transverse to axons in extra-axonal space, thus, corresponds to the short-range disorder in 2d ($$$p=0$$$, $$$d=2$$$, and $$$\vartheta=1$$$), yielding the $$$\log(\Delta/\delta)/\Delta$$$ diffusivity time-dependence transverse to axons for wide pulse sequence [6-8]. The translation of narrow pulse diffusivity to wide pulse diffusivity is based on the Gaussian phase approximation, where the effect of finite gradient pulse duration $$$\delta$$$ serves as an additional low-pass filter.

Example 2: Diffusion along realistic axons with caliber variation.
In the brain white matter, axon beading are randomly positioned along axons and correspond to the short-range disorder in 1-dimension ($$$p=0$$$, $$$d=1$$$, and $$$\vartheta=1/2$$$), leading to a diffusivity power-law tail of $$$1/\sqrt{t}$$$-dependence at long times of narrow-pulse sequence [7-9].

Power-law scaling of diffusion signal at strong diffusion weighting

In the widely accepted picture of white matter (Standard Model), the DW signal in the unit gradient direction $$$\hat{g}$$$ is given by [10-11]
$$S(\hat{g},b)=f\int {\rm d}\hat{n} {\cal P}(\hat{n}) \psi_\hat{n}(\hat{g},b)+S_{\rm ex}(\hat{g},b)\,,$$
where the first term comes from the collection of narrow cylinders representing axons (with intra-axonal volume fraction $$$f$$$), parametrized by the orientational distribution function (ODF) $$${\cal P}(\hat{n})$$$, and the second term comes from the hindered diffusion in extra-axonal space. The intra-axonal response function, i.e., the signal from water inside an axon pointing in the direction $$$\hat{n}$$$
$$\psi_\hat{n}(\hat{g},b)=e^{-bD_\perp(1-\cos^2\theta)}\cdot e^{-bD_a\cos^2\theta}\,,\quad \cos\theta\equiv\hat{g}\cdot\hat{n}\,,$$
factorizes into a non-Gaussian restricted diffusion propagator in the direction transverse to the axon, and a simple Gaussian diffusion propagator along the axon in $$$t\to\infty$$$ limit. The $$$D_\perp$$$ and $$$D_a$$$ is the intra-axonal diffusivity transverse and parallel to the axon segment of length $$$\sim L(t)$$$. For large $$$b$$$, such that $$$bD_{\rm ex}\gg1$$$, the extra-axonal signal $$$S_{\rm ex}(\hat{g},b)\sim e^{-bD_{\rm ex}(\hat{g})}$$$ decays exponentially and can be negligible. Averaging the signal over diffusion directions for each $$$b$$$-shell cancels the shape of orientation distribution and yields the spherical mean signal $$$\bar{S}(b)$$$ (Exercise 3) [10-11]:
$$\bar{S}(b)\simeq\beta e^{-bD_\perp+{\cal O}(b^2)} b^{-1/2}\,, \quad(2)$$
where $$$\beta=f\sqrt{\pi/(4D_a)}$$$. The asymptotic power-law scaling $$$\bar{S}\sim1/\sqrt{b}$$$ can only originates from intra-axonal water in the limit of zero stick radius, $$$bD_\perp\ll1$$$.

The deviation of spherical mean signal from the $$$1/\sqrt{b}$$$ scaling in Eq. (2) yields an estimated intra-axonal radial diffusivity $$$D_\perp$$$ in Eq. (1), effectively providing an axon size estimation in wide pulse limit [10-11]:
$$r_{\rm MR}=\left(\frac{48}{7}\delta(\Delta-\delta/3)D_0 D_\perp\right)^{1/4}\,.$$
The $$$r_{\rm MR}$$$ estimates the effective axon radius $$$r_{\rm eff}=\left(\langle r^6\rangle/\langle r^2\rangle\right)^{1/4}$$$, where the $$$r^4$$$ dependence in Eq. (1) is weighted by each cross section's volume $$$\propto r^2$$$. In other words, the MR-estimated axon size is heavily skewed by thick axons.

Exercises

  1. For diffusion inside an undulating thin fiber of undulation wavelength $$$\lambda$$$ and amplitude $$$w$$$ (radius $$$\ll \lambda, w$$$), please estimate the radial diffusivity inside the fiber by using the idea of transverse relaxation due to the diffusion gradient of wide pulse sequence (gradient pulse duration $$$\delta$$$, inter-pulse duration $$$\Delta$$$). The intrinsic diffusivity is $$$D_0$$$.
    Hint: The estimation of radial diffusivity in an undulating fiber couples the longitudinal and transverse diffusion. According to the CLT, the variance of diffusional phase (transverse to the fiber) from the independent steps add up, and the number $$$N$$$ of of independent steps is counted along the fiber. The time scale of diffusion along fiber is $$$t_u\sim\lambda^2/D_0\ll\delta$$$ (wide pulse), yielding an $$$N\sim\delta/t_u$$$ during the pulse duration $$$\delta$$$.

  2. (a) In a 1-dimensional medium of $$$N$$$ layers, of which the intrinsic diffusivity and layer thickness is $$$D_i$$$ and $$$l_i$$$ in the $$$i$$$-th layer, please calculate the overall diffusivity $$$D_\infty$$$ across the $$$N$$$ layers at long times.
    Hint: Fick's law of diffusion (diffusivity $$$D=J/(-\partial_x\varphi)$$$) is analogous to the Ohm's law of electricity (electrical conductance $$$G = I/V$$$), sharing the same rule of summing up the total diffusivity/conductance in a series circuit.

    (b) Given that the myelin sheath could be approximated by $$$N\gg1$$$ layers of myelin lamellae (intrinsic diffusivity $$$D_m$$$, thickness $$$l_m$$$) interleaved by $$$(N-1)$$$ layers of myelin water compartment (intrinsic diffusivity $$$D_0$$$, thickness $$$l_0$$$), please calculate the overall diffusivity $$$D_\infty$$$ across the myelin sheath at long times. In the limit of slow diffusion in thin myelin lamellae, i.e., both $$$D_m$$$ and $$$l_m$$$ vanish, the ratio $$$\kappa\equiv D_m/l_m$$$ is by definition the permeability of each myelin lamellae [12]. What is the overall diffusivity $$$D_\infty$$$ in this limit?

  3. (a) Please estimate the spherical mean signal $$$\bar{S}(b)$$$ for the diffusion inside a thin cylinder (stick) at strong diffusion weighting $$$b$$$, i.e., $$$bD_0\gg1$$$ (Intrinsic diffusivity $$$D_0$$$). How does the signal scale with $$$b$$$?
    Hint: $$$\int_0^a\exp(-x^2){\rm d}x=\sqrt{\pi}/2\cdot{\rm erf}(a)\simeq\sqrt{\pi}/2\,,a\gg1$$$.

    (b) Please estimate the spherical mean signal $$$\bar{S}(b)$$$ for the diffusion between parallel planes of very small distance in-between at strong diffusion weighting $$$b$$$, i.e., $$$bD_0\gg1$$$. How does the signal scale with $$$b$$$? [13]
    Hint: $$$\int_0^a\exp(x^2){\rm d}x=\sqrt{\pi}/2\cdot{\rm erfi}(a)\simeq\exp(a^2)/(2a)\,,a\gg1$$$.

Acknowledgements

I would like to thank Els Fieremans, Dmitry S Novikov, and Marco Palombo for the discussion.

References

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Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)