The extracellular diffusion weighted signal predicts axon diameter distribution parameters

Introduction In this abstract we show that the diffusion-weighted (DW) signal from the extracellular space can be used to estimate white matter microstructure parameters, such as axon diameter distribution and volume fraction. A key challenge in Diffusion-Weighted Imaging (DWI) is to estimate more specific features of tissue microstructure than existing methods such as Diffusion Tensor Imaging provide. For example, model-based approaches [1, 2, 3] show some success in recovering axon density and diameter in white matter. White matter models typically separate the intra and extra-axonal water populations and assume that only the intra-axonal signal is sensitive to the distribution of axon diameters. In this abstract, we show that in fact the extracellular signal is highly sensitive to the axon diameter distribution for typical diffusion times. This finding motivates the future development of new models to exploit this sensitivity and thus provide better estimates of the axon diameter distribution and other microstructural parameters. Theory Biophysical models of human brain white matter generally consist of an intracellular and an extracellular compartment. Diffusion in the intracellular space is modeled as being restricted and is parameterized by the axon diameter, amongst other parameters. Diffusion in the extracellular compartments is hindered by the presence of the cells and the dispersion becomes Gaussian in the long diffusion time limit. This hindrance is characterized by the tortuosity factor λ and is defined as the ratio of the extracellular diffusion coefficient (ADC) and the free diffusion coefficient (Dfree): λ= Dfree/ADC. Both Szafer et al. [4] and Sen and Basser [5] derive models for the long-time limit where the tortuosity factor depends only on the intracellular volume fraction and is independent of the axon diameter distribution. However, in typical measurements from white matter, the long diffusion time limit does not hold. We use Monte Carlo simulations of diffusion to simulate the DW signal of the extracellular compartment in a large number of substrates that mimic human brain white matter. We use diffusion times that range from values typically used in DWI to values that will satisfy the long diffusion time limit. Because there are no analytical models available that models the extracellular DW signal for all diffusion times, we estimate a nonparametric model relating the DW signals of all substrates to their microstructure parameters using Gaussian Process Regression (GPR) [6]. Methods Monte Carlo Simulations White matter is modeled as a collection of non-abutting cylinders with radii which are distributed according to the gamma distribution, as in [3,7]. We generate 1000 substrates, each containing 200 cylinders with each combination of mean radius in {0.5, 1, 2, 5} μm and standard deviation in {0.1, 0.2, 0.5, 1, 2} μm. The volume fractions vary between 0.53 and 0.87. We allow no exchange between compartments. For each of these substrates we use Monte Carlo simulations [7] to synthesize diffusion-weighted signal. We simulate the signals using 2 DW directions, 1 in the parallel and 1 in the perpendicular direction to the cylinders. We use all possible combinations of the following set of DW parameters: δ = {8, 20} ms, G = {0.002, 0.005, 0.01, 0.02, 0.04, 0.06, 0.08, 0.10, 0.2, 0.4, 0.8, 1.0} T/m and Δ = {25, 100, 200, 1000} ms. The free diffusion coefficient of the extracellular compartment is set to 2 ×10 m/s. No noise is added, because the focus in this abstract is on whether there is any empirical relationship between extracellular diffusion and microstructure parameters. Gaussian Process Regression We perform GPR on the diffusion-weighted signals of all the substrates, using 80% of the substrates as training data, using publicly available GPR software [8] to perform model comparison on the neighborhood of interpolation and subsequent regression. The results show GPR-based prediction on the remaining 20% of the substrates (the test data).