A dual spherical model for multi-shell diffusion imaging

Multi-shell diffusion imaging (MSDI) allows to characterize the subtle tissue properties of neurons along with providing valuable information about the ensemble average diffusion propagator. Several methods, both para- metric and non-parametric, have been proposed to analyze MSDI data. In this work, we propose a hybrid model, which is non-parametric in the angular domain but parametric in the radial domain. This has the advantage of allowing arbitrary number of fiber orientations in the angular domain, yet requiring as little as two b-value shells in the radial (q-space) domain. Thus, an extensive sampling of the q-space is not required to compute the diffusion propagator. This model, which we term as the dual-spherical" model, requires estimation of two functions on the sphere to completely (and continuously) model the entire q-space diffusion signal. Specifically, we formulate the cost function so that the diffusion signal is guaranteed to monotonically decrease with b-value for user-defined range of b-values. This is in contrast to other methods, which do not enforce such a constraint, resulting in in-accurate modeling of the diffusion signal (where the signal values could potentially increase with b-value). We also show the relation of our proposed method with that of diffusional kurtosis imaging and how our model extends the kurtosis model. We use the standard spherical harmonics to estimate these functions on the sphere and show its efficacy using synthetic and in-vivo experiments. In particular, on synthetic data, we computed the normalized mean squared error and the average angular error in the estimated orientation distribution function (ODF) and show that the proposed technique works better than the existing work which only uses a parametric model for estimating the radial decay of the diffusion signal with b-value.