Diffusion magnetic resonance imaging is a noninvasively tool for investigating the microstructural layout of tissue in clinical settings. It can provide useful indices for investigating abnormalities of brain tissue^1-4. However, most commonly used methods, such as DTI¹ or DKI², are derived based on the estimated moments or the probability distribution of the displacements of water molecules. Though they may be sensitive to changes in tissue structure, these methods don’t provide any specific information about the changes. To overcome these limitations, we propose a novel approach for estimating the microstructural layout of tissue using in vivo dMRI measurements.

The evolution of the magnetization density $$$m({\bf r})$$$ of diffusing spins in biological tissue is described by the following modified version of the Block-Torrey equation $$ \partial_{t}{m({\bf r},t)}=-i{\bf g}(t)\cdot {\bf r} m({\bf r},t)+\nabla_{\bf r}\cdot D_0\nabla_{\bf r} m({\bf r},t)+u({\bf r},t,{\mathcal G}(t)), $$ where the novel term $$$u({\bf r},t,{\mathcal G}(t))$$$ denotes the disorder on the evolution process due to restrictions, hindrances and tissue heterogeneities. In particular, the disorder $$$u({\bf r},t,{\mathcal G}(t))$$$ is a function of $$${\bf r}, t$$$ and it also depends on the gradient sequence before the time point $$$t$$$. For example, if the disorder is caused by a spatially-varying diffusion coefficient $$$D({\bf r})=D_0+{\tilde D}({\bf r})$$$, then the disorder function can be represented as $$$u({\bf r},t,{\mathcal G}(t))= \nabla_{\bf r}\cdot {\tilde D}({\bf r})\nabla_{\bf r} m({\bf r},t)$$$. By using the spatial-Fourier transform of the modified Bloch-Torrey equation, we obtain the following equation $$ \partial_{t}{{\hat m}({\bf k},t)}= {\bf g}(t)\cdot \nabla_{\bf k} {\hat m}({\bf k},t)-\|{\bf k}\|_{D_0}^2 {\hat m}({\bf k},t)+{\hat u}({\bf k},t,{\mathcal G}(t)). $$ Thus the dMRI signal is given by $$$s(t)=\int m({\bf r},t) d {\bf r}={\hat m}(0,t)$$$. Assuming that the gradient sequence satisfies the echo condition, i.e. $$$\int_0^t {\bf g}(s) ds=0$$$, and the dMRI signal is normalized such as $$$s(0)=1$$$, then we can derive the following expression for diffusion signal by solving the above partial differential equation $$ s(t)=e^{-\int_0^t\|{\bf q}(s)\|^2_{D_0}ds}+\int_0^t {\hat u}({\bf q}(\tau),\tau,{\mathcal G}(\tau)) e^{-\int_{\tau}^t \|{\bf q}(s)\|^2_{D_0}ds}d\tau, $$ where $$${\bf q}(\tau):=\int_0^{\tau}{\bf g}(s)ds$$$. The integral term in the above equation implies that the diffusion signal is in general not an exponential function of the squared of gradient strength. Without this term, the diffusion signal corresponds to a Gaussian ensemble average propagator as in DTI.

In sPFG experiments, the gradient sequence $$${\mathcal G}(\tau)$$$ is linearly dependent on $$${\bf q}(\tau)$$$. Thus, for sPFG experiments with fixed diffusion time and pulse width, $$${\hat u}({\bf q}(t),t,{\mathcal G}(t))$$$ can be denoted by $$${\hat u}({\bf q}(t),t)$$$. In particular, the diffusion signal from narrow-pulsed sPFG experiments from isotropic structures is given by $$$s(q,t)=e^{-D_0q^2t}+\int_0^t {\hat u}(q,\tau) e^{-D_0q^2(t-\tau)}d\tau$$$. Using the expansion $$${\hat u}(q,t)=q^2 {\hat u}_2(t)+q^4{\hat u}_4(t)+\ldots$$$ and comparing the corresponding expansion of $$$s(q,t)$$$ with the result from the effective medium theory⁵, we can show that the instantaneous diffusion coefficient $$$D_{\rm inst}(t)=D_0-{\hat u}_2(t)$$$ and the long-time limit $$$\lim_{t\rightarrow \infty} {\hat u}_2(t)=\langle ({\tilde D}({\bf r}))^2\rangle/(D_0d)$$$ where $$$\langle ({\tilde D}({\bf r}))^2$$$ denotes the variance of diffusivities. For experiments with long diffusion time, the disorder can be assumed to be time-invariant. Then the diffusion signal is given by $$$s(q,t)=\frac{\hat{u}(q)}{D_0q^2}+(1-\frac{\hat{u}(q)}{D_0q^2})e^{-D_0q^2t}$$$.

We tested the proposed method on HCP data set. The experimental parameters are $$${\rm TR/TE}=5500/89\,{\rm ms}$$$, $$$\delta=10.6\,{\rm ms}$$$ and $$$\Delta=43.1\,{\rm ms}$$$. The data consists of 3 b-values with $$$b=1000, 2000, 3000\,{\rm s/mm^2}$$$. We applied the above model for characterizing the dMRI measurements by using different disorder functions and diffusion coefficients for dMRI signals in the isotropic cross-sectional plane and along the axonal direction, respectively. In particular, the disorder function is assumed to be given by $$${\bf q}^T U_2 {\bf q}$$$ where the disorder tensor $$$U_2$$$ has the same set of eigenvectors as the diffusion tensor $$$D_0$$$. The disorder coefficient $$${\hat u}_{2\perp}$$$ and the diffusion coefficient $$$d_{\perp}$$$ in the cross-sectional plane and the corresponding diffusion coefficient from DTI are shown in Figure (1). The yellow arrows in Figure (1a) point out some interesting patterns that are not shown in the diffusion coefficients. Figure (2) shows that the microstructural arrangement of axons varies from different segments of corpus callosum (CC). In particular, the axonal structure in the midbody has stronger disorder than the genu and the splenium areas.Since the estimated diffusivities in CC have similar values, the stronger disorder in the midbody implies that the corresponding $$$\langle ({\tilde D}({\bf r}))^2\rangle$$$ is higher, which is consistent with the observations from histology studies⁶.The PICASO model explains the relation between the structural organization and the dMRI signal. It provides novel information about the tissue microstructure, which can be used to investigate brain abnormalities in clinical settings.