The Bloch-Torrey (BT) equations are an analytical approach that describes diffusion MRI (dMRI) experiments [1]. However, as increasingly more complex geometries are being considered and given the limitation of available image resolutions it becomes essential to update the analytical steps. We therefore propose a complete mathematical framework connecting between microscopic physical forces governing spin displacement in magnetic fields, geometric constraints on the diffusion process, and the BT equations describing the final MR signal. Bridging these different scales provides a microscopic interpretation to the BT equations. In addition, the framework allows us to generalize the BT equations to non-homogeneous specimens, thus inferring more accurate estimations of the MR signal for complex underlying geometries. Molecular displacement within an inhomogeneous magnetic field causes dephasing in dMRI experiments[2,3]. The displacement is governed by random forces (Brownian motion). However, it is also affected by deterministic microstructural boundaries, such as cellular membranes or the edges of pores. At the microscopic scale, it is not possible to predict which way a particular molecule will move at a given time, but the forces acting on each molecule and the geometry in which they displace can be modeled. We therefore propose the following stochastic equations to describe a microscopic model for molecular motion and precession: $$(1) \frac{d\mathbf{r}}{dt}=K(\mathbf{r},t)\mathbf{f}(t)+\mathbf{F}(\mathbf{r})\label{eq:random position}$$, $$ (2) \frac{dm_{l}}{dt}=\gamma(\mathbf{m}\times\mathbf{H})_{l}=i{\sum_{j=1}^d}\Lambda_{lj}m_{j}. $$ Eq. (1) is a Langevin equation that models a single molecule’s velocity in a position r and time t. The velocity changes as consequence of two forces. The stochastic force, $$$\mathbf{f}(t)$$$, which drives the Brownian motion where the factor $$$K(\mathbf{r},t)$$$ models the time-correlation of the process, which depends on the underlying deterministic geometry, allowing non-Gaussian dynamics. The second force is an external deterministic force, $$$\mathbf{F}(\mathbf{r})$$$, representing flow. Eq. (2) describes the magnetic moment change of each spin in the presence of a magnetic field [2-4], where $$$\Lambda$$$ is a matrix representation of the $$$H$$$. H can be space and time dependent. Since equations (1) and (2) describe a conventional stochastic Markov model, we derive the Stratonovich’s Fokker-Planck equation for the magnetic-diffusion propagator, which is the joint probability, $$$P$$$, of the magnetic moment and the position of a particle at a given time given the former position and magnetic moment: $$(3) \partial_{t}P=-\nabla_{\mathbf{m}}(i\Lambda\mathbf{m}P)-\nabla(\mathbf{F}P)+\nabla(D\nabla P)+\frac{1}{2}\nabla(P\nabla D).$$ Finally, using Eq. (3) we can derive the generalized BT equations: $$ (4) \frac{\partial}{\partial t}\langle\mathbf{m}\rangle=i\Lambda\langle\mathbf{m}\rangle+\nabla\left(D\nabla\langle\mathbf{m}\rangle\right)+\frac{1}{2}\nabla(\langle\mathbf{m}\rangle\nabla D)-\nabla\left(\mathbf{F}\langle\mathbf{m}\rangle\right).$$ Eq. (4) contains one additional term, $$$\frac{1}{2}\nabla(\langle\mathbf{m}\rangle\nabla D)$$$ with respect to the original BT equation [4]. When the $$$D=\frac{1}{2}K(\mathbf{r},t)^TK(\mathbf{r},t)$$$ diffusion coefficient is homogeneous, i.e., does not change across space, the additional term vanishes and Eq. (4) becomes identical to the original BT equation. This generalization suggests that the original BT equations are missing a term that might play an important role when considering complex structures with inhomogeneous diffusion coefficient.

We demonstrate the importance of the additional term by considering a simple inhomogeneous example of $$$D=D_0(x^2 + y^2)/a^2$$$ where is $$$a$$$ structural unit size, and with no flow ($$$\mathbf{F}(\mathbf{r})=0$$$). This example can be considered as a slow transition in diffusion coefficient between two linked environments, such as intra- and extracellular spaces. We simulate spin-echo experiments [5] with increasing gradient, $$$ b_{value}=20 - 1100[\frac{s}{mm^2}]$$$. Parameters were $$$D_0=2.3×10^{-6}[\frac{mm^2}{ms}]$$$, $$$ \Delta=50[ms]$$$ structural unit size $$$a=0.02mm$$$, and $$$\delta =5[ms]$$$. The BT equations were solved using the finite differences method with pseudo-periodic boundary conditions [6]. Fig. 1 compares the solutions for the generalized (circles) and the original (stars) BT equations. Notice that the difference between the two approaches is not linear. The linear fit shows that the apparent diffusion coefficient calculated using the generalized BT equations is much different than the one calculated with the conventional BT equations. Figure 2 simulates an homogeneous $$$D$$$, demonstrating that in this case the contribution of the additional term is negligible.

By proposing a simple microscopic model based on fundamental stochastic equations we derived the BT-equations, thus providing a novel analytical link between displacement, diffusion and the MR signal in the most general case where diffusion is not constant. Doing so we identify a missing term in the original BT equations. Our analysis suggests that the generalized BT equations can be used for more accurate simulations of MR signal. In addition, and importantly, the derivation includes a new closed form of the magnetic diffusion-propagator. This, for example, allows us to approximate the propagator and evaluate whether or not it is Gaussian.