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SPHERIOUSLY? The challenges of estimating spherical pore size non-invasively in the human brain from diffusion MRI1Cardiff University Brain Research Imaging Centre (CUBRIC), School of Psychology, Cardiff University, Cardiff, United Kingdom, 2Clinical Sciences Lund, Radiology, Lund University, Lund, Sweden, 3Centre for Medical Image Computing, Department of Computer Science, University College London, London, United Kingdom
Synopsis
Soma and Neurite Density Imaging (SANDI) was recently proposed to disentangle cylindrical and spherical geometries, attributed to neurite and soma compartments. In this work, using: (i) ultra-strong gradients; (ii) a combination of linear, planar, and spherical b-tensor encodings; and (iii) analysing the signal in the frequency domain, three main challenges were identified; First, the Rician noise floor biases estimation of soma properties. Second there is an empirical lower bound on the spherical signal fraction and pore-size. Third, if there is sensitivity to the transverse intra-cellular diffusivity in cylindrical structures, estimation of spherical pore-size is challenging.
Introduction
Diffusion magnetic resonance imaging (dMRI) is a non-invasive technique used to model brain microstructure1-5. Most dMRI models separate the tissue into intra- and extra- neurite non-exchanging compartments which is not a good representation of the signal in gray matter 6-12. Addressing this model insufficiency, Palombo et al.13 introduced a three-compartment model called Soma And Neurite Density Imaging (SANDI) which may suffer from degeneracy14-19.Q-space trajectory imaging was recently introduced20, where linear, planar, and spherical tensor encoding (LTE, PTE, and STE) are defined when the b-tensor has respectively, one, two, and three non-zero eigenvalues18,21-24.
In this study, we used b-tensor encoding and frequency-domain analysis in SANDI model and we found that fitting the spherical pore properties remains challenging for small signal fractions, i.e. <~10%.
F-statistic shows a lower bound on the detectable MRI signal fraction ($$$10\%$$$ for SNR=50).
Numerical simulations, yielded a lower limit of $$$3\,\mu m$$$ while the one from power-law is $$$7\,\mu m$$$.
The fitting becomes more challenging if we have cylinders instead of sticks.
Theory
For a general B-matrix, the diffusion-weighted MR signal of a three-compartment model is:$$S(\mathbf{B})/S_0=f_{\mathrm{cylinder}}\int_{\mathbb{S}^2} W(\kappa,\mathbf{n}) e^{-\mathbf{B}:\mathbf{D}_{\rm{cylinder}}(\mathbf{n},t)} \mathrm{d}\mathbf{n}+f_{\rm{sphere}}S_{\rm{sphere}}(D_{\rm{sphere}}(t),\mathbf{B})+f_{\rm{ball}} e^{-bD_{\rm{ball}}}\;\;\;(1)$$
where
$$$f_{\mathrm{cylinder}},\,f_{\mathrm{ball}},\,f_{\mathrm{sphere}},\,\mathbf{D}_{\mathrm{cylinder}}(\mathbf{n},t)=(D_{\mathrm{in}}^{\mid\mid}-D_{\mathrm{in}}^{\perp}(t))\mathbf{n}\mathbf{n}^T+D_{\mathrm{in}}^{\perp}(t)I,\,D_{\mathrm{ball}}\,\rm{and}\,D_{\mathrm{sphere}}$$$ are the cylinder, ball and sphere signal fractions and diffusivities, respectively 25. $$$W(\mathbf{n})$$$ is the Watson orientation distribution function with $$$\kappa$$$ as the dispersion parameter. $$$\mathbf{B}=\int_0^{TE}\mathbf{q}(t)\mathbf{q}^\intercal(t)\rm{d}t$$$ where $$$\mathbf{q}(t)=\gamma\int_0^t\mathbf{g}(t^\prime)\rm{d}t^\prime$$$ 20,26, and $$$\gamma$$$ is the gyromagnetic ratio. Axial and radial elements in the diagonal axisymmetric b-tensor are $$$b_{\mid\mid}$$$ and $$$b_\perp$$$ respectively, b-value, $$$b$$$ is the trace of $$$B$$$-matrix and $$$b_\Delta=(b_{\mid\mid}-b_\perp)/b$$$26.
The acquired signal is averaged over all diffusion directions for each shell27,28. Compartmental diffusion is represented with axisymmetric diffusion tensors with isotropic diffusivity, $$$D_I=1/3D_{\mid\mid}+2/3\,D_{\perp}$$$, and anisotropy, $$$D_\Delta=(D_{\mid\mid}-D_{\perp})/(D_{\mid \mid}+2\,D_{\perp})$$$ where $$$D_{\mid\mid}$$$ and $$$D_{\perp}$$$ are the axial and radial diffusivities, respectively. The signal attenuation from the $$$k$$$th compartment is given by19,26:
$$A_k(b,b_\Delta,D_{I;k},D_{\Delta;k})=\exp(-b\,D_{I;k}[1-b_\Delta\,D_{\Delta;k}])\cdot\,g(3b D_{I;k}\,b_\Delta\,D_{\Delta;k})\;\;\;(2)$$
where
$$g(\alpha)=\int_0^1\,\exp(-\alpha\,x^2)\mathrm{d}x=\sqrt{\frac{\pi}{4\alpha}}\rm{erf}(\sqrt{\alpha})\;\;\;(3)$$
and $$$\rm{erf}(.)$$$ is the error function 29. Therefore, the full signal equation is:
$$S/S_0=f_{\rm{cylinder}/\rm{stick}}A_{\rm{cylinder}/\rm{stick}}+f_{\rm{sphere}}A_{\rm{sphere}}+f_{\rm{ball}}A_{\rm{ball}}\;\;\;(4)$$
For complex gradient waveforms, the diffusion time is ill-defined. The analysis is taken in the frequency domain and therefore we consider the diffusion spectrum $$$D_{\rm{cylinder}}^\perp(\omega),\,D_{\rm{sphere}}(\omega)$$$30,31.
Method
The numerical simulations were performed using the model in Eq (1), with $$$f_{\rm{sphere}}=0.01:0.01:0.1,\,0.15,\,0.2:0.1:1,\,f_{\rm{ball}}=f_{\rm{stick}}=(1-f_{\rm{sphere}})/2,\,D_{\rm{in}}^{\mid\mid}=2\;\mu m^2/ms,\,D_{\rm{ball}}=0.6\;\mu m^2/ms,\,R_{\rm{sphere}}=1:1:10\;\mu m$$$, and $$$R_{\rm{cylinder}}=4\;\mu m$$$. The simulated protocol comprised 10 $$$b=0$$$ and 8 non-zero shells ($$$b=1,\,2,\,3,\,4.5,\,6,\,7.5,\,9,\,10.5\;ms/\mu m^2$$$) in ($$$10,\,31,\,31,\,31,\,31,\,61,\,61,\,61,\,61$$$) directions for LTE and 5 shells ($$$b=1,\,2,\,3,\,4.5,\,6\;ms/\mu m^2$$$) in ($$$31,\,31,\,31,\,31,\,61$$$) directions 32 for PTE and 5 shells for STE ($$$b=0.2,\,1,\,2,\,3,\,4.5\;ms/\mu m^2$$$) in ($$$6,\,9,\,9,\,12,\,15$$$) directions and SNR=50.To identify the empirical lower bound on detectable spherical pore size, we fitted a power-law8,9,17 to the direction-averaged signal from the LTE measurements for $$$b=6,\,7.5,\,9,\,10.5\,ms/\mu m^2$$$ ($$$S/S_0=\beta\,b^{-\alpha}$$$) and then compared the values of $$$\alpha$$$, with values observed in vivo.
Diffusion-weighted images (DWIs) were acquired from two healthy participants with the protocol detailed in the simulation on a 3T Connectom MR imaging system. Forty-two axial slices with $$$3\,mm$$$ isotropic voxel size and a 78$$$\times$$$78 matrix size, $$$TE=88\,ms$$$, $$$TR=3000\,ms$$$ were obtained.
DWIs were corrected for Gibbs ringing33, motion, eddy currents 34, and gradient nonlinearity 35. A 3D Gaussian filter with a standard deviation of 0.5 was applied to smooth the images.
Results
Fig. 1 shows the gradient waveforms and the signal decay inside the spherical and cylindrical compartments.Fig. 2 (a) shows the results of fitting the sphere radius under different noise simulations. The results of F-test (p-value) show when the sphere radius or the signal fraction of the sphere is small ($$$R_{\rm{sphere}}<2\,\mu m$$$ and $$$f_{\rm{sphere}}<0.05$$$) the simplified model is preferred.
Fig. 2 (b) shows the estimated sphere size when there is a non-zero diameter cylinder, (SNR = 200).
Fig. 3 shows the results of fitting the model to the signal from five different ROIs in the brain.
Fig. 4 illustrates the estimated parameters on in vivo data including the noise floor as an extra parameter.
Fig. 5 (a) shows the effect of sphere signal fraction and size on the estimated exponent, $$$\alpha$$$, in the power-law fit ($$$S/S_0=\beta\,b^{-\alpha}$$$).
In Fig. 5 (b) we do not observe values of $\alpha$ less than 0.5 in the white matter. This places a lower bound on the size of spherical pores of around $$$7\,\mu m$$$.
Discussion and Conclusion
In this work, we have demonstrated key challenges and limitations in estimating spherical pore size non-invasively in the human brain from dMRI.Our simulations show the effect of Rician bias and identified the lower bound limit of the sphere signal fraction and size that can be detected from the diffusion-weighted signal from both an SNR and empirical perspective.
We showed that for small signal fraction of soma, ($$$<10\%$$$) this is a problem. However, we know from detailed microscopy of brain cortex 36,37, that in GM the soma signal fraction is on average $$$>20\%$$$. Therefore, reliable estimation of soma properties in GM is possible, while in WM it presents several challenges.
Using the ultra-strong gradients of the Connectom scanner, the diffusion signal in the white matter can be made sensitive to the axon diameter, and therefore we have the cylinder+ball+sphere model which has two time-dependencies. Disentangling these two by only changing the frequency content of the encoding waveform is challenging.
Studying all these challenges prevents misinterpretation of the biased estimated parameters as the potential biomarkers in clinical studies.
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. We thank Lars Mueller for setting up the protocol for b-tensor encoding. MP is supported by UK EPSRC EP/N018702/1 and UKRI Future Leaders Fellowship MR/T020296/1.References
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Figures
Fig . 5 (a) The effect of sphere size and signal fraction on exponent α (similar to Fig. 2 in 10). ($$$f_{\rm{sphere}} = 0.01:0.01:0.1,\,0.2:0.1:0.5$$$, $$$f_{\rm{ball}} = f_{\rm{stick}} = (1 - f_{\rm{sphere}})/2$$$, $$$D_{\rm{in}}^{\mid\mid} = 2 \; \mu m^2/ms$$$, $$$D_{\rm{ball}} = 2 \; \mu m^2/ms$$$, $$$R_{\rm{sphere}} = 1:1:10 \; \mu m$$$, δ = 29.65 ms, and Δ = 37.05 ms). (b) Estimated FA, parameter β and α of the power-law fit ($$$S/S_0 = \beta b^{-\alpha}$$$) from axial, coronal, and sagittal views of the brain image.