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Bloch-Torrey partial differential equation

In diffusion MRI, a time-varying magnetic field gradient $\vec{g} : [0, T_\text{echo}] \to \mathbb{R}^3$ is applied to the tissue to encode water diffusion. The resulting complex transverse water proton magnetization in the rotating frame satisfies the Bloch-Torrey PDE:

\[\frac{\partial}{\partial t} M_i(\vec{x},t) = -\left(-\nabla \cdot \mathbf{D}_i \nabla + \frac{1}{T_{2,i}} + \underline{\mathrm{i}} \gamma \vec{g}(t) \cdot \vec{x}\right) M_i(\vec{x}, t),\]

where

  • $\vec{x} \in \Omega_i$,
  • $t \in [0, T_\text{echo}]$ where $T_\text{echo}$ is the echo time at which the signal is measured,
  • $\gamma = 2.67513\times 10^8 \, \mathrm{rad \, s^{-1} T^{-1}}$ is the gyromagnetic ratio of the water proton,
  • $\underline{\mathrm{i}}$ is the imaginary unit,
  • $\mathbf{D}_i$ is the intrinsic diffusion tensor in $\Omega_i$,
  • $T_{2,i}$ is the $T_2$-relaxation time in $\Omega_i$, and
  • $M_i$ is the magnetization in $\Omega_i$.

The initial conditions are given by

\[M_i(\vec{x}, 0) = \rho_i \in \mathbb{C}, \quad i \in \{1, \dots, N_\text{cmpt}\}.\]

where $\rho_i$ is the initial spin density in $\Omega_i$. The outer boundary conditions are given by

\[\mathbf{D}_i \nabla M_i(\vec{x}, t) \cdot \vec{n}_i(\vec{x}) = -\kappa_i M_i(\vec{x}, t), \quad \vec{x} \in \Gamma_i, \quad i \in \{1, \dots, N_\text{cmpt}\},\]

and the interface conditions by

\[\mathbf{D}_i \nabla M_i(\vec{x}, t) \cdot \vec{n}_i(\vec{x}) = -\mathbf{D}_j \nabla M_j(\vec{x}, t) \cdot \vec{n}_j(\vec{x}), \quad \vec{x} \in \Gamma_{i j}, \quad (i, j) \in \{1, \dots, N_\text{cmpt}\}^2,\]

\[\mathbf{D}_i \nabla M_i(\vec{x}, t) \cdot \vec{n}_i(\vec{x}) = \kappa_{i j} \left(c_{i j} M_j(\vec{x}, t) - c_{j i} M_i(\vec{x}, t)\right), \quad \vec{x} \in \Gamma_{i j}, \quad (i, j) \in \{1, \dots, N_\text{cmpt}\}^2.\]

where

  • $\vec{n}_i$ is the unit outward pointing normal vector of $\Omega_i$
  • $\kappa_i$ is a boundary relaxation coefficient for $\Omega_i$,
  • $\kappa_{i j} = \kappa_{j i}$ is the permeability coefficient on $\Gamma_{i j}$,
  • $c_{i j}$ and $c_{j i}$ account for the spin density equilibrium between the two compartments, with either
    • $c_{i j} = c_{j i} = 1$, in which case a uniform spin density across compartments is favored in the absence of a gradient, or
    • $c_{i j} = \frac{2 \rho_i}{\rho_i + \rho_j}$ and $c_{j i} = \frac{2 \rho_j}{\rho_i + \rho_j}$, which ensures that the non-uniform intitial spin density is preserved in the absence of a gradient Hong-Hsi Lee, Els Fieremans, Dmitry S. Novikov (2021).

The diffusion MRI signal is measured at echo time $t = T_\text{echo}$. This signal is computed as the spatial integral of the final magnetization $M(\cdot, T_\text{echo})$:

\[S(\vec{g}) = \int_\Omega M(\vec{x}, T_\text{echo}) \, \mathrm{d} \Omega(\vec{x}).\]