In this talk, I will review and systematize brain diffusion models and their assumptions, by putting them into an overarching physical context of the coarse-graining over an increasing diffusion length scale. In the Overview part, I go over basic definitions and place diffusion models into the $$$q-t$$$ space (Figure 1).

A. The mesoscopic Bloch-Torrey equation

The main equation that governs the evolution of the observed transverse magnetization $$$M(t,x)$$$:

$$ \partial_t M(t,x) = \partial_x \left[D(x)\partial_x M(t,x)\right] - \left[R_2(x) + i\Omega(t,x)\right] M(t,x) \qquad (1) $$

Here, $$$D(x)$$$ is the spatially varying local diffusivity (including membranes is equivalent to either boundary conditions on $$$M(t,x)$$$ or to sharply varying $$$D(x)$$$); $$$R_2(x)$$$ is the locally varying transverse relaxation rate; and $$$\Omega(t,x) \equiv \Omega(x) + g(t)\cdot x$$$, where $$$\Omega(x)$$$ is the local Larmor frequency offset. The diffusion encoding $$$q$$$ is created by the t-dependent gradient $$$g(t) = dq/dt$$$.

The microstructural tissue properties are embodied in $$$D(x)$$$, $$$R_2(x)$$$ and $$$\Omega(x)$$$ spatially varying on the micrometer scale. The measured signal in a given voxel $$$ S(t,q) \propto \int {\rm d}x \, M(t,x) $$$ summed over all spins in the voxel. This averaging washes out most of the cellular-level tissue properties. The purpose of biophysical modeling and advanced diffusion acquisition is to identify and quantify the relevant degrees of freedom -- the features in $$$D(x)$$$, $$$R_2(x)$$$ and $$$\Omega(x)$$$ that survive the voxel-averaging inherent in the measured signal $$$S(t,q)$$$.

B. Hierarchy of diffusion models based on coarse-graining: Depending on the temporal scale we are able to resolve, the corresponding diffusion length sets the lower bound to the size of the tissue features the diffusion signal is sensitive to. This naturally orders the diffusion models in terms of the length scales they are probing -- from the Mitra limit (Mitra et al. Phys Rev Lett 1992), to probing the disorder correlation length (Novikov et al. PNAS 2014; Section Time-dependent diffusion as a non-Gaussian effect below), to the long-time (tortuosity) limit (Section Multiple Gaussian compartments below).

C. Models versus representations: One has to draw the distinction between genuine biophysical models (that have concrete assumptions in terms of tissue parameters) and convenient basis sets which merely represent the data well. One can only validate the models, and should use representations for storing the data, as an intermediate step.

D. The cumulant expansion as a default representation. DTI (Basser et al. Biophys J 1994), DKI (Jensen et al. MRM 2005), and beyond. Applicability of DTI, DKI; convergence of the series.

E. The $$$q-t$$$ space of diffusion MRI See Figure 1

In this Section, multi compartmental brain diffusion models are considered in the infinitely long-time limit, when diffusion in each of the compartments is assumed to be Gaussian. The parameters of these compartments (e.g. intra- and extra-neurite space) are then deemed to be effective parameters, i.e. the diffusivities can be very different from those of pure water or axoplasm, due to the presence of different kinds of hindrances.

A. The concept of sticks (Behrens et al., MRM 2003; Kroenke et al. MRM 2004; Assaf et al. MRM 2004; Jespersen et al., NI 2007): Early on, it has been hypothesized (and universally adopted), that the distinctive building block of brain diffusion models is a so-called stick --- a picture of a neurite as a one-dimensional channel with effectively zero radius and impermeable walls, so that its diffusivity along the stick is finite, and the diffusivity transverse to it is zero. The rest of the water resides in the extra-neurite space, for which diffusivity does not vanish in any direction.

B. The Standard Model (SM) of diffusion in the neural tissue

$$ S_{{\bf g}}(b) = \int \! {\rm d}{\bf n} \, {\cal P}({\bf n}) \, {\cal K}({\bf g},{\bf n}) \qquad (2) $$

Here $$${\cal K}({\bf g},{\bf n})$$$ is the response of an individual fiber segment pointing in direction n when measured with gradient direction g, and $$${\cal P}({\bf n})$$$ is the fiber ODF.

Parameters of the kernel $$${\cal K}({\bf g},{\bf n})$$$, and of the ODF $$${\cal P}({\bf n})$$$, are qualitatively different. While the achievable ODF shape depends on the maximum angular resolution, equivalent to the maximum order L_max in the spherical harmonics basis, the scalar parameters of the kernel $$${\cal K}({\bf g}, {\bf n})$$$ describe the microstructural properties of the architecture of the fiber at the scale of the diffusion length, and are independent of the angular resolution.

C. SM parameter estimation with high-quality data (Jespersen et al. NI 2010)

D. SM parameter estimation using realistic data: Fixing parameters and/or the functional shape of ODF $$${\cal P}({\bf n})$$$ for increased precision (Zhang et al. NI 2012)

E. SM parameter estimation challenge: Multiple minima and flatness of parameter landscape As it has been shown recently (Jelescu et al., NBM 2016; Novikov et al., ISMRM 2015), a general feature of multi-compartmental fiber response $$${\cal K}({\bf g},{\bf n})$$$ is the presence of distinct minima in the optimization landscape for the fitting to any model of the form (2). In particular, for the 2-compartmental $$${\cal K}({\bf g},{\bf n})$$$ (intra- and extra-axonal space), the two minima are characterized by different sets of biologically plausible diffusivities and the axonal water fraction. At the time of writing this syllabus, there is no definitive answer as to which minimum is the "correct" one; both describe the realistic noisy data quite well, yet they represent very different biophysical realities.

F. A toy model of bi-modality: Parallel fibers (Fieremans et al, NBM 2010, NI 2011) One can trace the existence of 2 sets of biologically plausible parameters to a model of parallel fibers, where this "duality", or "degeneracy" persists up to the order $$$b^2$$$. It has been also proven (Novikov et al, ISMRM 2015) that this duality persists for any fiber ODF, so the toy model is representative.

G. Factorization of the ODF and of the fiber response. Isotropic averaging (Kaden et al., MRM 2015; Jespersen et al., ISMRM 2015); an infinite family of rotational invariants (Reisert et al.; Novikov et al.)

H. Validation of SM. Validation of the stick response: Observation of $$$b^{-1/2}$$$ scaling (Veraart, Fieremans, Novikov) for the isotropically averaged diffusion signal in human white matter. Ways to resolve of the multiple minima for parameter estimation challenge: Isotropic diffusion weighting (Dhital et al., ISMRM 2015). Results from DKI-based parameter estimation (Fieremans et al., 2011-2016).

In this Section, we lift the assumption of Gaussian diffusion in every brain compartment. Indeed, at finite (not infinitely long diffusion times), the diffusion coefficient exceeds its tortuosity limit.

A. The origin of the time-dependence: Structural correlations and gradual coarse-graining (Novikov et al. PNAS 2014). Approaching the long time limit: Dependence on disorder class.

B. Time-dependent diffusion due to disorder in dimension d = 1 relevant for $$$D_\parallel(t)$$$ in WM (along fibers: Fieremans et al., NI 2016) and for GM (inside neurites: Does et al., MRM 2003; Novikov et al., PNAS 2014). Time dependence signifying that neurites are not hollow cylinders; ideas about what kind of microstructure could cause this time dependence. Quantification of the disorder correlation length along the neurites.

C. Time-dependent diffusion due to disorder in dimension d = 2: $$$D_\perp(t)$$$ transverse to WM tracts. Contributions of intra- and extra-axonal water to overall $$$D(t)$$$; predominance of the extra-axonal contribution (Burcaw et al. NI 2015). Determination of the correlation length of axonal packing in human WM (Fieremans et al. NI 2016).

The field of microstructural mapping is experiencing a revolution as the quality of experiments reaches the level where we can observe important microstructural effects. Revealing these effects requires sharpening our analytical and parameter-estimation tools, drawing on exciting connections with mesoscopic condensed matter physics and other modern quantitative methods, as well as puts an ever-increasing emphasis on the model validation. There remains a lot to be understood; here is my (incomplete) list of unresolved issues:

Parameter estimation challenge for the "Standard Model" of Gaussian compartments still remains.

Permeability/exchange time? how well we can approximate compartments as non-exchanging?

Modeling and validation in gray matter (especially in the view of possible exchange).

Confounding effects of mesoscopically varying $$$R_2(x)$$$ and $$$\Omega(x)$$$ on the diffusion metrics.

What tissue length scales can we realistically probe in brain? axonal diameters, packing correlation length for axons, mean distance between axonal beads, anything else? Which length scales are dominant?

Probing and interpreting non-Gaussian effects, separately in intra- and extra-cellular spaces, through particular t and q dependencies.