Effective diffusion tensor computed by homogenization

Introduction Diffusion MRI can give useful information on cellular structure and structural change (for a review see [1]). We show that the effective diffusion tensor obtained by mathematical homogenization theory (see e.g. [2,3]) is a good approximation to the long time apparent diffusion tensor under realistic DMR scanning conditions for both isotropic and anisotropic diffusion and general geometries. The homogenized diffusion tensor is obtained by solving three steady-state Laplace equations, which is a more computationally efficient approach than long time simulation in the time domain, either via Monte-Carlo simulation or numerical solution of the time-dependent Bloch-Torrey PDE. Theory In the two-compartment model, we consider the two compartments, i Ω and e Ω , to be the ensemble of cells and the extra-cellular compartment, respectively. The two compartments have the same intrinsic diffusion coefficient D. The cell membrane is modeled by an infinitely thin permeable interface characterized by permeability κ. Given the diffusion gradient with profile f(t) and gradient strength γ / : q g r r = , where γ is the gyro-magnetic ratio, the DMRI signal attenuation is ) , ( t q r Ψ , from which we define the apparent diffusion tensor D from the Taylor expansion in q r :