INTRODUCTION

Diffusion magnetic resonance imaging (dMRI) is a non-invasive technique widely used to study brain microstructure. Biophysical modeling is often used to solve the inverse problem of inferring relevant tissue features (such as cell size, shape, and orientation) from the measured dMRI signal^1-3. However, when complex multi-compartment models are used, such an inverse problem can be highly ill-posed and give rise to degeneracies in the model-parameter estimation^4,5.
In this work, we focus on a recently proposed three-compartment model for non-invasive neural soma imaging called SANDI⁶, which assumes that it is possible to decompose the measured dMRI signal into three main sources: neurites, cell bodies (namely soma); and extra-cellular space where the diffusion in each of these compartments will have a time-dependence if there is any sensitivity to the size of that compartment.
Unfortunately, SANDI, like any other multi-compartment model⁴, may suffer from degeneracy of the model parameters. However, multidimensional diffusion encoding⁷ has been recently proposed to resolve such degeneracies^8-11. For example, Gyori et al.¹¹ used LTE+STE in a SANDI-like model. This study aims to investigate whether, and which combination of, multidimensional diffusion encodings can be used to effectively resolve the degeneracy of SANDI model parameter estimation.

METHODS

In the SANDI framework, the dMRI signal measured in brain tissue is assumed to arise from three main non-exchanging compartments: intra-neurite (modeled as diffusion in sticks), intra-soma (modeled as restricted diffusion in a sphere) and extra-cellular (modeled as isotropic Gaussian diffusion).
For a general B-matrix, the diffusion-weighted MR signal is:


$$S(\mathbf{B})/S_0=f_{ic}(f_{in}\int_{\mathbb{S}^2}W(\kappa,\mathbf{n})e^{-\mathbf{B}:\mathbf{D_i}(\mathbf{n})} d\mathbf{n}+f_{is}S_{is}(D_{is}(t),\mathbf{B}))+f_{ec}e^{-bD_{ec}}\quad(1)$$


where $$$f_{in}$$$, $$$f_{ec}$$$, $$$f_{is}$$$, $$$\mathbf{D_i}(\mathbf{n})=D_{in}\mathbf{n}\mathbf{n}^T$$$, $$$D_{ec}$$$ and $$$D_{is}$$$ are the intra-neurite, extra-cellular, and intra-soma signal fractions and diffusivities, respectively. $$$f_{ic}$$$ is the intracellular signal fraction where $$$f_{ic}+f_{ec}=1$$$ and $$$f_{in}+f_{is}=1$$$. $$$S_{is}$$$¹² is intra-soma diffusion signal, $$$W(\mathbf{n})$$$ is the Watson orientation distribution function (ODF) and $$$\kappa$$$ is the dispersion parameter. $$$B=b/3(1-b_\Delta)I_3+bb_\Delta\mathbf{g}\mathbf{g}^T$$$, where $$$\mathbf{g}$$$ is the diffusion gradient direction and the b-value, $$$b$$$, is the trace of the b-matrix. For linear (LTE), planar (PTE) and spherical (STE) tensor encoding, $$$b_\Delta$$$=1, -1/2, and 0 respectively¹³.

The powder-averaged signal is:


$$S/S_0=f_{ic}(f_{in}\frac{\sqrt{\pi}e^{-\frac{b}{3}D_{in}(1-b_\Delta)}erf(\sqrt{bb_\Delta D_{in}})}{2\sqrt{b b_\Delta D_{in}}}+f_{is}S_{is}(D_{is}(t),b_{\Delta}))+f_{ec}e^{-bD_{ec}}\quad(2)$$

Here, we use numerical simulations and experiments in-vivo in healthy volunteers to investigate the potential of combining efficient gradient waveforms for LTE, PTE and STE to improve the estimation of the five model parameters $$$f_{in}$$$, $$$f_{ec}$$$, $$$D_{in}$$$, $$$D_{ec}$$$ and $$$D_{is}$$$. As we analyze the direction-averaged signal, we don’t need to estimate dispersion. For these complex gradient waveforms, the diffusion time is ill-defined and therefore we consider the diffusion spectrum $$$D_{is}(\omega )$$$^14-16 in our analyses of compartment size.

The restricted diffusion signal inside the sphere is, $$$S_{is}=\exp{(-\beta)}$$$ where $$$\beta$$$ is^{15, 16}:

$$\beta=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathbf{F}^T(\omega)D_{is}(\omega)\mathbf{F}(-\omega)d\omega\quad(3)$$

where $$$\mathbf{F}(\omega)=\int_{0}^{\tau}\mathbf{F}(t)e^{-i\omega t}dt$$$, $$$\mathbf{F}(t)=\gamma\int_{0}^{t}\mathbf{g}(t')dt'$$$, $$$\mathbf{g}(t)$$$ is the gradient waveform, and $$$\gamma$$$ is the gyromagnetic ratio. $$$D_{is}(\omega)$$$ is given by¹⁵:

$$D_{is}(\omega)=\sum_iB_i\frac{a_iD_0\omega^2}{a_i^2D_0^2+\omega^2}\quad(4)$$

where $$$a_i=(\mu_i/R_s)^2$$$, $$$B_i=2(R_s/\mu_i)^2/(\mu_i^2-2)$$$, $$$R_s$$$ is the sphere radius and $$$\mu_i s$$$ are the roots of the derivatives of the first order spherical Bessel function $$$j_1'(\mu_i)= 0$$$. $$$D_{0}$$$=2 $$$\mu m^2/ms$$$, as proposed in⁶.

Simulations. The numerical simulations were performed using the model in (1), with 100 different physically plausible combinations of the five parameters randomly distributed in the intervals: 0.25<$$$f_{in}$$$<0.75, 0.25<$$$f_{ec}$$$<0.75, 0.6<$$$D_{in}$$$ and $$$D_{ec}$$$<2.5$$$\mu m^2 /ms$$$, 3<$$$R_s$$$<10 $$$\mu m$$$ and $$$\kappa$$$=11. The simulated protocol comprised one b=0 and 15 shells (b=1-15$$$ms/ \mu m^2$$$ with step size of 1$$$ms/ \mu m^2$$$) for LTE and 12 shells ($$$b_{max}$$$=12 $$$ms/ \mu m^2$$$) for PTE of 60 directions each and 5 shells for STE ($$$b_{max}$$$=5 $$$ms/ \mu m^2$$$) of 3 directions each and SNR = 50 with Rician noise. The diffusion signal was direction-averaged over the same b-value and b-tensor shapes.

Experiments. In vivo data were collected using a 3T Connectom scanner with a maximum gradient strength of 300 $$$mT/m$$$. The imaging protocol was the same as the simulation. One healthy participant was scanned with TE=88ms and TR=3000ms. The imaging matrix was $$$78×78×22$$$ and the voxel size 3mm isotropic.

RESULTS

Fig. 1 shows the free gradient waveforms of the linear, planar, spherical tensor encoding and the corresponding spectrum. Fig. 2 illustrates the signal decay inside the spherical and cylindrical compartments using different encoding schemes. Fig. 3 shows the results of the simulation. Different combinations of tensor encoding have been compared and the combination of LTE+PTE+STE gives the best estimation of the signal fractions. For $$$D_{in}$$$, LTE+PTE is the best and finally, LTE provides the best estimation of $$$D_{ec}$$$ and $$$R_s$$$. Fig. 4 shows the estimated parameters of the three-compartment model for the in vivo brain image. Fig. 5 illustrates the mean and std of the parameters in the corpus callosum and cerebellum. Note with 3 mm isotropic resolution, partial volume effects from CSF can be substantial, which may be addressed in future work by including a CSF compartment in the model (and additional low b-value measurements to aid its fitting).

DISCUSSION AND CONCLUSION

In this study, we used the frequency domain approach to identify the combination of b-tensor encodings that maximizes the accuracy of estimates of the SANDI model parameters. Our results extend and complement existing SANDI work, showing (in vivo and in silico) that different combinations of b-tensor encodings provide optimal estimates for different SANDI parameters, and provide the first ever in vivo estimates of soma size using the frequency domain.

Acknowledgements

The data were acquired at the UK National Facility for In Vivo MR Imaging of Human Tissue Microstructure funded by the EPSRC (grant EP/M029778/1), and The Wolfson Foundation. MA and DKJ are supported by a Wellcome Trust Investigator Award (096646/Z/11/Z) and a Wellcome Trust Strategic Award (104943/Z/14/Z). The authors would like to thank Filip Szczepankiewicz for providing the pulse sequences for b-tensor encoding. DCA is supported by UK EPSRC (EP/M020533/1, EP/N018702/1, EP/R006032/1), EU Horizon 2020 (634541-2); GHZ and MP are supported by UK EPSRC EP/N018702/1.