Assessing the tissue microstructure (in particular of brain white matter) using diffusion MRI (dMRI) is a great challenge. It requires solution of an ill-posed inverse problem. It is known that different models map well on the same data [1] and eventually the fitting quality cannot validate the underlying model even in the simplest case [2]. This deep problem is further exacerbated by the averaging of the cellular signal on the mesoscopic subvoxel level due to multiple orientations of neuronal fibers as described by the orientation distribution function (ODF), which makes the dMRI signal less specific to the microstructure.

The inverse problem is commonly (with a few exceptions, e.g. [3,4]) treated as finding the microstructural parameters $$$x$$$, that have the maximal a-posteriori probability (MAP) given the signal or its derived `features', $$$f$$$, aka fitting. Doing so, one has to introduce numerous constraints or priors, one has to account for the real noise statistics, and, last but not least, nonlinear fitting is very slow (lasting from a few hours to days). Here we propose a simultaneous solution to the above problems on the way to finding parameters of a given microstructural model. Not all of them are detectable, recognition of those is a constituent part of the proposed method.

We use derived features of the signal that are exclusively sensitive to the microstructure being invariant with respect to the ODF. Using the decomposition in spherical harmonics, $$$Y_{l,m}$$$, the signal takes the form of a product $$ S^b_{l,m}=M^b_l\,p_{l,m}\quad,$$where $$$M$$$ is the model, which is assumed to be axially symmetric, $$$p$$$ is the ODF and $$$b$$$ represents the dependence on a radial variable in q-space (e.g. the b-shells). A known ODF-invariant feature is given by $$$l=0$$$ [5,6]: $$ f_0^b:=S^b_{0,0}=M_0^b\,.$$We propose a larger set of ODF-independent features which are ratios of $$$l$$$-wise squared norms for $$$l>0$$$: $$ f^b_l := \frac{\sum_m |S^b_{l,m}|^2}{\sum_{m,b'} |S^{b'}_{l,m}|^2} = \frac{|M_l^b|^2}{\sum_{b'} |M_l^{b'}|^2} \label{eq:features}$$This infinite set is practically limited by the noise (Fig.1).

Instead of the commonly used fitting paradigm, we engage a Bayesian estimator to find the expectation values of the unknown parameters $$$\widetilde{x}_B(f)$$$. The fitting of the Bayesian estimator is closely related to machine learning. Therein, the full signal formation process is simulated for many parameter constellation of a specific tissue model (Fig.2) including the ODF and measurement noise. If the mapping $$$\widetilde x_B (f)$$$ is applied to the simulated data the fidelity of the parameter determination can be easily monitored (Fig.3).

We applied the method to data from a healthy subject from the HCP data set [7] and to two healthy subjects measured in-house using a two-shell scheme (2-shell60) and the most parsimonious sampling of q-space with only 28 points with $$$b>0$$$ (hex28).

A training of the Baesian estimator is performed on a regular desktop computer within minutes. The resulting estimator $$$\widetilde{x}_B(f)$$$ is applied to all data obtained with a given measurement scheme. The computation time for one brain is in the range of seconds. The number of acquired $$$b$$$-shells determines the number of features. We found the restriction $$$l\leq 2$$$ (Fig.1), which results in 5 features for HCP: $$$f=\{f^1_0,f^2_0,f^3_0,f^1_2,f^2_2\}$$$, and 3 for 2-shell60 and hex28: $$$f=\{f^1_0,f^2_0,f^1_2\}$$$. Figures 4 and 5 show maps of the obtained microstructural parameters. According to Fig.3, the volume fractions are reliable while the diffusivities are not.

The proposed method is 2--3 order of magnitude faster than the commonly used MAP (nonlinear fitting) approach. A further difference with MAP is how the two methods behave when data are insufficient for determination of all model parameters. MAP needs to be stabilized by a priori constraints, while the present approach can work with the unconstrained microstructural tissue models while monitoring the detectability of model parameters as illustrated in Fig.3. Effectively, constraints are replaced by the prior distributions of parameters.

Refraining from constraints in any form is not possible for the common acquisition schemes. According to our results shown in Fig.3 only two independent microstructural parameters, the volume fractions can be found reliably. In particular, this explains the surprisingly high quality of the diffusivity maps for the fastest hex28 measurement (Fig.5): The output values are determined by the prior distribution with some influence of gray or white matter is considered. One more detectable mesoscopic parameter is the width of the FOD, so the total number of detectable parameters is three.

The robustness and efficiency of the present method offers it as a working horse for development of better acquisition schemes, more adequate tissue models and clinically oriented research.