Described herein are exemplary methods for estimating a nonparametric joint radius-length (R-L) distribution of an ensemble of porous elements represented generally by finite cylinders. Some described methods comprise estimating an eccentricity distribution of a group of anisotropic porous elements. For example, disclosed methods can be applied to estimate a nonparametric joint R-L distribution of injured axons in nervous tissue, muscle tissue, plant elements, or other porous materials. Employing a novel three dimensional (3-D) double pulsed-field gradient (d-PFG) magnetic resonance (MR) acquisition scheme, both the marginal radius and length distributions of a population of generally cylindrical porous elements can be obtained. The marginal radius and length distributions can then b e used to constrain and stabilize the estimate of the joint radius-length distribution. Using the marginal distributions as constraints allows the joint R-L distribution to be reconstructed from an underdetermined system (i.e., more variables than equations).
Determination of a joint probability distribution of radius and length of anisotropic pores from double pulsed field gradient MRI data