State-of-the-art multi-compartment microstructure models of diffusion MRI in the human brain have limited capability to model multiple tissues at the same time. In particular, the available techniques that allow this multi-tissue modelling are based on multi-TE acquisitions. In (Frigo et al., 2020; Frigo et al., 2020; Frigo et al., 2021) we proposed a novel multi-tissue formulation of classical multi-compartment models that relies on more common single-TE acquisitions and can therefore be employed in the analysis of previously acquired datasets. We showed how modelling multiple tissues provides a new interpretation of the concepts of signal fraction and volume fraction in the context of multi-compartment modelling. The software that allows to inspect single-TE diffusion MRI data with multi-tissue multi-compartment models is included in the publicly available Dmipy Python package.

Recent studies have highlighted that all of the available MC models are transparent to the $$T_2$$ relaxation times of the modelled tissues. As a consequence, they implicitly assume that all the considered tissues have the same non-diffusion weighted signal $$S_0$$. While this is a reasonable assumption in some contexts, it is not true in general. In fact, each brain tissue is characterized by a specific relaxation time which makes $$T_2$$ imaging possible. Assuming that all the tissues have a single $$S_0$$ response simplifies the model at the cost of biophysical accuracy. Tissue fractions obtained with this assumption are called signal fractions, in contrast with the unbiased volume fractions which can be obtained with models that account for different $S_0$ responses of the modelled tissues. The former measures the linear relation between the signal generated by a single tissue compartment and the acquired signal, while the latter measures the volume of single tissue compartment that is present in the voxel.

Given the known interdependence between the $$T_2$$ times of tissues and the $$TE$$ of the acquisition, some attempts at addressing this issue have been formulated making use of multi-TE multi-shell dMRI acquisitions. Despite allowing to increase the signal-to-noise ratio (SNR), these techniques require a complete re-design of the experiments from acquisition to post-processing, posing severe limitations in terms of usability of already acquired data. This aspect is crucial in modern neuroimaging, where large studies like the Human Connectome Project (HCP), the UK Biobank and the Alzheimer Disease Neuroimaging Initiative (ADNI) invest significant amounts of time and financial resources to acquire data of large cohorts with standardised protocols that need to be carefully designed a priori.

In this work we will show that signal fractions are a biased estimation of volume fractions and that, under certain assumptions, the latter can still be retrieved from the first without acquiring new data or re-fitting the MC model. We call this new technique Multi-Tissue MC (MT-MC) model. To our knowledge, MT-MC is the only general framework that allows to estimate volume fractions from single-TE multi-shell dMRI data. This novel formulation is inspired by the technique of for the estimation of tissue-specific orientation distribution functions. The use of the MT-MC formulation solves some limitations of the previously mentioned multi-TE approaches and opens the door to the multi-tissue investigation of brain microstructure with data acquired with standard single-TE multi-shell dMRI protocols. Two algorithms for fitting the MT-MC model are proposed, one of which is designed to build on top of data already processed with standard MC models. Our new model is implemented and freely available in the Diffusion Microstructure Imaging in Python (Dmipy) framework, which is an open source tool designed for the abstraction, simulation, and fitting of MC models of dMRI. The ability of the MT-MC model to retrieve the unbiased volume fractions is tested on both synthetic data generated with Dmipy and real data obtained from the HCP database.