Synopsis

Diffusion-weighted imaging provides information to study the brain microstructure. Several studies in the literature have shown that there is degeneracy in the estimated parameters for a commonly used microstructural model. B-tensor encoding is one of the strategies that has been proposed to solve the degeneracy. The combination of linear-spherical tensor encoding (LTE+STE) and linear-planar (LTE+PTE) have been utilized in previous works. In this paper, we compare different combinations of b-tensor encoding, (LTE+STE), linear-planar (LTE+PTE), planar-spherical (PTE+STE) and linear-planar-spherical (LTE+PTE+STE). We also compare the results of fit using a nonlinear least square algorithm and microstructure imaging of crossing (MIX) method. The results show that the combination of tensor encodings with MIX fitting algorithm leads to lower bias and higher precision in the parameter estimates than single tensor encoding.

Introduction

Diffusion MRI (dMRI) is sensitive to the displacement of water molecules on the scale of the micrometer, and has, therefore, become one of the important tools for investigating brain microstructure ¹. Several models have been proposed that separate the tissue into intra and extra-neurite compartments ^2-4. Even with a high number of measurements, the fitting process suffers from degeneracy⁵. There are several studies trying to solve the degeneracy problem by adding extra measurements such as diffusion time, echo time and diffusion encoding beyond standard Stejskal-Tanner encoding (or linear tensor encoding, LTE) ^6-13. The literature has reported on improvements on some estimated parameters in the case of noise and a theoretical solution to the degeneracy in the noiseless case ^10-13. Here we extend this by comparing multiple combinations beyond what was previously proposed. We performed extensive simulations to address degeneracy, bias, and precision in realistic noise scenarios for 4 different combinations of tensor encoding; STE+LTE, PTE+LTE, PTE+STE, and PTE+LTE+STE.

Methods

We use the standard model of white matter ^14,15 with the modified extra axonal compartment and the fiber orientation distribution function (fODF) modeled with a Watson distribution. For a general B-matrix, the diffusion signal is:

$$ S(\mathbf{B})/S_0 = f\int_{\mathbb{S}^2} W(\mathbf{n}) e^{-\mathbf{B}: \mathbf{D_i}(\mathbf{n})} d\mathbf{n} + (1-f) \int_{\mathbb{S}^2} W(\mathbf{n}) e^{-\mathbf{B}:\mathbf{D_e}(\mathbf{n})} d\mathbf{n} \quad \quad \quad \quad \quad \quad \quad (1)$$

where $$$f$$$ is the intra-neurite water fraction, $$$\mathbf{D_i}(\mathbf{n})=D_a \mathbf{n}\mathbf{n}^T$$$ is the intra-neurite diffusivity, $$$\mathbf{D_e}(\mathbf{n})$$$ is the extra-neurite rank-2 diffusion tensor and the eigenvalues are given by the parallel and perpendicular diffusivities, $$$\mathbf{D_e}(\mathbf{n}) = (D_e ^{\mid\mid} - D_e ^\perp) \mathbf{n}\mathbf{n}^T + D_e ^\perp \mathbf{I}$$$. $$$W(\mathbf{n})$$$ is the Watson distribusion characterized by the mean orientation $$$\mu$$$ and the concentration parameter $$$\kappa$$$ around this orientation. The linear, planar and spherical tensors are defined as, $$$\mathbf{B_{lin}} = b\mathbf{g}\mathbf{g}^T$$$, $$$\mathbf{B_{pla}} = b(\mathbf{I}_3 − \mathbf{g} \mathbf{g}^T)/2$$$ and $$$\mathbf{B_{sph}} = b\mathbf{I}_3/3$$$ respectively, where $$$g$$$ is the diffusion gradient direction and the b-value $$$b$$$ is defined as the trace of the $$$\mathbf{B}$$$-matrix.

The fiber direction is estimated using nonlinear least square fitting of a single diffusion tensor. In order to find the other parameters of this model ($$$f$$$, $$$D_a$$$, $$$D_e ^{\mid\mid}$$$, $$$D_e^\perp$$$ and $$$ \kappa $$$), we use the microstructure imaging of crossing fibers (MIX) method ¹⁶. Synthetic dMRI signals were generated based on the model described in Equations (1), with ground truth parameter values defined by two sets of parameters. In set A, we fix all the parameters and create signal with SNR = 50 [$$$f$$$ = 0.38, $$$D_a$$$ = 0.5, $$$D_e^{\mid\mid}$$$ = 2.1, $$$D_e^\perp$$$ = 0.74 and $$$\kappa$$$ = 64]. Set B contains 8⁴ different physically plausible combinations of the five parameters that are spaced in 9 points in the intervals: 0.25 < $$$f$$$ < 0.75, 0.6 < $$$D_a$$$ < 2.5 $$$\mu m^2/ms$$$, 0.6 < $$$D_e^{\mid\mid}$$$ < 2.5 $$$\mu m^2/ms$$$, 0.1 < $$$D_e^\perp$$$ < 2 $$$\mu m^2/ms$$$and $$$\kappa\in$$$ [1, 2, 3, 4, 5, 6, 9, 16, 64]. The simulated protocol consisted of one b = 0 and two shells (b = 1 and b = 2 $$$ms/\mu m^2$$$) of 30 directions each. In order to make the number of samples equal for the different encoding combinations, we repeated each direction in the combination of (STE+LTE+PTE) twice, three times for the combination of (STE+LTE), (LTE+PTE), (PTE+STE) and six times for STE, LTE and PTE. Therefore, the number of measurements is (30+30+1)$$$\times$$$2$$$\times$$$3 = 366.

Results and discussion

Fig. 1 and 2 show the histograms of the fit results of set A using a nonlinear least square (NLLS) algorithm with random initial values and MIX framework respectively. The contour plots of different combinations of the parameters of Fig. 2 are illustrated in Fig. 3. In agreement with previous literature, none of the single encodings can solve the problem of degeneracy. The combination of encodings shows that the fitting is stable in all cases. The mean and the standard deviation (std) of the estimated parameters reflect the degree of degeneracy, bias, and precision. We quantify the total performance by fitting a linear trend to the plot of estimated parameter vs ground truth ( $$$R^2$$$, slope, $$$\beta_1$$$, and intercept, $$$\beta_0$$$). In Fig. 4 and Fig. 5, (set B) the combination of tensor encodings leads to better results while MIX outperforms NLLS.

CONCLUSION

Acknowledgements

The authors would like to thank Bibek Dhital, PhD, for the fruitful discussion about the combination of PTE and LTE. CMWT is supported by a Rubicon grant (680-50-1527) from the Netherlands Organisation for Scientific Research (NWO) and MA, CMWT and DKJ were all supported by a Wellcome Trust grant (096646/Z/11/Z).

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