# Capturing the voxel content in terms of a diffusion tensor distribution

## Signal description

The measured diffusion signal $$\mathcal{S}$$ probes diffusion processes over specific observational time-scales that depend on the choice of experimental time parameters. For given observational time-scales, a common description of the sub-voxel composition of heterogeneous tissues is obtained by considering a “snapshot” of the combined non-Gaussian diffusion effects of restriction and exchange, and by approximating the signal decay as a continuous weighted sum of exponential decays, giving the following multidimensional Laplace transform:

$\mathcal{S}(\mathbf{b}) = \mathcal{S}_0 \int_{\mathrm{Sym}^{+}(3)} \! \mathcal{P}(\mathbf{D})\, \exp(-\mathbf{b}:\mathbf{D}) \, \mathrm{d}\mathbf{D} \, ,$

where $$\mathbf{b}$$ is the diffusion-encoding tensor from tensor-valued diffusion encoding [@Westin:2016 ; @Topgaard:2017 ; @Reymbaut_book_chapter:2020], $$\mathcal{S}_0 = \mathcal{S}(\mathbf{b}=\mathbf{0})$$ is the non diffusion-weighted signal, and $$\mathcal{P}(\mathbf{D})$$ is the diffusion tensor distribution [@Jian:2007]. Here, $$\mathrm{Sym}^{+}(3)$$ denotes the space of symmetric positive-semidefinite $3\times3$ tensors and “:” is the Frobenius inner product.

## Time-dependent effects

Changing the observational time-scales may very well lead to a different set of exponential decays as a result of restricted diffusion [@Woessner:1963] and exchange [@Johnson:1993], which implies that the measured distribution $$\mathcal{P}(\mathbf{D})$$ may depend on the spectral content of the diffusion-encoding gradients [@Stepisnik:1981 ; @Stepisnik:1985 ; @Callaghan_Stepisnik:1995 ; @Topgaard_dim_rand_walks:2019 ; @Lundell_Lasic_book_chapter:2020 ; @Szczepankiewicz_arXiv:2020]. In other words, the retrieved diffusion tensors should be interpreted as apparent ones, including all potential time-dependent effects of restriction and exchange. Even though time-dependent effects have been measured in human-brain white matter [@Van:2014 ; @Baron_Beaulieu:2014 ; @Baron_Beaulieu:2015 ; @Fieremans:2016 ; @Veraart:2019 ; @Lundell:2019 ; @dellAcqua_ISMRM:2019], spinal cord [@Jespersen:2018 ; @Grussu:2019] and prostate [@Lemberskiy:2017 ; @Lemberskiy:2018] using pulse sequences specifically designed for varying the observational spectral content over extended ranges, the above $$\mathcal{P}(\mathbf{D})$$ description holds for the limited range of spectral contents probed by clinical dMRI experiments in the brain [@Clark:2001 ; @Ronen:2006 ; @Nilsson:2009 ; @Nilsson:2013a ; @Nilsson:2013b ; @deSantis_T1:2016 ; @Lampinen:2017 ; @Veraart:2018 ; @Grussu:2019 ; @Lampinen:2019 ; @Szczepankiewicz_ISMRM:2019].