1. Diffusion MRI can provide non-invasive quantification of brain microstructure, but this requires models more accurate than diffusion tensor imaging.

2. Biophysical models of diffusion MRI describe the MR signal as originating from diffusion in distinct tissue components, such as the intra-axonal or extracellular space.

3. Comparment sizes, e.g., the average axon diameter, can be estimated using diffusion MRI, provided that the size is above the resolution limit of the acquisition protocol.

4. Orientation dispersion is essential to include in white matter diffusion models.

This talk aims to outline the models used in diffusion MRI and to provide an overview of

1. the diffusion tensor as a versatile model building block

2. the components required to successfully model diffusion in white matter

3. how to interpret axonal diameter estimates from diffusion MRI

4. the role of axonal orientation dispersion in white matter diffusion, and

5. how to compare diffusion MRI models

We refer to a model as equations that predict the diffusion-weighted MR signal intensity (dMRI data in short), given model parameters and parameters on how the experiment was performed, for example, b-values, amplitudes and directions of the magnetic field gradients, and diffusion times (Nilsson, 2011). Example of model parameters are the axon density, axon diameter, mean and variance of the axon diameter distribution, or the extracellular diffusion tensor. Estimating the model parameters from noisy dMRI data is a so-called inverse problem. By fitting the model to the data, we can infer microstructure parameters from the dMRI data.

The basic building block of most models employed for microstructure imaging is a diffusion tensor (Basser et al., 1994). However, models more capable than DTI does not assign a single diffusion tensor to the whole voxel. Separate diffusion tensors are rather assigned to different ensemble of water molecules within the voxel. Examples of such ensembles are water molecules in the extracellular space, the intra-axonal space, or the cerebrospinal fluid (CSF). Assuming negligible exchange between the components, the model can predict the total dMRI signal by simple summation of the dMRI signals from each component. The CSF component is modelled by a spherical tensor. Extra-cellular diffusion is often modelled by a cylinder-symmetric tensor assuming Gaussian diffusion (Assaf et al., 2004), however, the influence of the diffusion time on the effective extracellular diffusion tensor can also be considered (Novikov et al., 2014; Xu et al., 2014). Intra-axonal diffusion is commonly modelled by a cylinder-symmetric diffusion tensor where the radial diffusivity depends on the timing of the diffusion-encoding gradients (Alexander et al., 2010; Assaf et al., 2004). In the axial direction, along the fibres, the diffusivity is often represented by a single value. The diffusivity is often assumed to be equal in the intra-axonal and extracellular environments.

The simplest model of diffusion in white matter is composed of two tensors, representing extracellular and intra-axonal diffusion. The radial diffusivity of the intra-axonal space is calculated from two model parametes; the axonal radius, and the bulk diffusion coefficient. Examples of two-tensor models include the CHARMED model (Assaf et al., 2004). Similar but slightly more complex models also exist (Alexander et al., 2010). The AxCaliber model extends the CHARMED model by modelling intra-voxel variation in axon diameters by the Gamma distribution, parameterised by the mean and the variance of axon diameters (Assaf et al., 2008). Models such as CHARMED and AxCaliber may provide estimates of the axon diameter. When interpreting axon diameter estimates from dMRI, three aspects should be kept in mind.

(i) The parameter represents the volume-weighted axon diameter rather than the number average (Alexander et al., 2010).

(ii) Estimating small axon diameters from dMRI is intrinsically difficult. At a certain small diameter, which we refer to as the resolution limit, the apparent radial diffusivity of the intra-axonal component becomes inseparable from zero (Nilsson and Alexander, 2012; Nilsson et al., 2013). The protocol can be optimized to minimize the resolution limit (Alexander, 2008), but the limit is ultimately determined by MRI hardware specs such as the maximal gradient amplitude and slew rate. At present, the resolution limit of conventional MRI scanners is higher than the average axon diameters in most structures of the brain (Aboitiz et al., 1992; Dyrby et al., 2012; Liewald et al., 2014). Improved models and the use of oscillating gradients may help reduce the resolution limit (Xu et al., 2014).

(iii) The axon diameter is inferred from the variance of the diffusional displacements of water molecules, and thus the diameter will refer to the maximal distance between the restricting barriers (Nilsson et al., 2012). For straight cylinders, this distance agrees with the cylinder diameter, but axons do not run in straight paths (Nilsson et al., 2012; Ronen et al., 2013). A positive bias in the axon diameter may thus be expected when comparing results from dMRI with microscopy results. Orientation distribution of intra-axonal components has a large impact on microstructure estimates in the brain. White matter is not, as assumed in many models, composed of parallel cylinders. Instead, there is a large within-voxel orientation dispersion not only in regions of crossing fibres (Jeurissen et al., 2012), but also in the corpus callosum (Choe et al., 2012; Ronen et al., 2013).

Orientation dispersion of axons (or “neurites”) can be captured, for example, by the NODDI model (Zhang et al., 2012). In order to avoid overfitting, the effective axonal diameter is in this model set to zero, under the implicit assumption that the true axon diameter is below the resolution limit. The NODDI model also incorporates a spherical diffusion tensor that represent cerebrospinal fluid (CSF), to account for partial volume effects and “free water” (Pasternak et al., 2009). By performing powder averaging of the signal across diffusion encoding directions, the NODDI model can be simplified even further (Lampinen et al., ISMRM 2015).

Models for diffusion MRI can also be constructed to capture the within-voxel variance of diffusion tensors. Consider a voxel subdivided into regions. Assume that the diffusion in each region is well described by a diffusion tensor. DTI would yield the average diffusion tensor. Fourth-order tensors, found in the cumulant expansion of the MR signal (Jensen et al, 2005) can capture the diffusion tensor covariance, however, in order to fully capture the diffusion-tensor covariance, b-tensors of rank 2 or higher must be employed to acquire the data (Westin et al., 2014). The tensor covariance can then be analyzed to infer microstructure information such as the microscopic FA (µFA).

Models of the dMRI signal can be varied indefinitely. Some try to describe the data with few parameters. Others require more parameters, but may fit better to the data. Still others include parameters which cannot be estimated reliably and overfitting ensues. In order to assess the quality of a model, it has to be compared to other models in terms of how much of the variation in the data that it captures per model parameter. Adding another model parameter without obtaining a significantly better fit to the data leads to overfitting and reduce the trustworthiness of the fitted model parameters. Several tools have been employed for model comparisons in the context of diffusion MRI modelling, for example, the F-test or Bayesian Information Criterion (Nilsson and Alexander, 2012; Panagiotaki et al., 2012). Studies comparing a multitude of models have concluded that at least three tensors are typically required to describe white matter diffusion (Ferizi et al., 2013; Panagiotaki et al., 2012).

Biophysical models of the diffusion MRI signal allows estimation of microstructure-specific parameters from dMRI. Metrics related to axon density and axonal orientation dispersion can be reliably estimated from dMRI data. The axon diameter and its distribution are more challenging to estimate, although recent advancements in modelling and hardware design are promising (Huang et al., 2015; Xu et al., 2014). Apart from applications in white matter, these models also have applications in, for example, oncology (Panagiotaki et al, 2014).