Estimation of Gray Matter Fiber Direction Based on Anisotropy Diffusion

doi:10.21203/rs.3.rs-3022311/v1

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Estimation of Gray Matter Fiber Direction Based on Anisotropy Diffusion

https://doi.org/10.21203/rs.3.rs-3022311/v1

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In high-angle resolution diffusion imaging, the spherical deconvolution method can acquire information on fiber orientation in the white matter of the brain. However, this method views gray matter as isotropic diffusion, therefore SD has very limited use in gray matter. Related studies have shown that there are multiple cytosomes and dendrites in gray matter, and these structures result in the movement of water molecules in gray matter that are not simply free diffusion, and if viewed as completely isotropic, it will bring large uncertainties in the estimation of fiber orientation in superficial white matter regions. To solve this problem, this paper proposes a Multi-Anisotropy Response Function Constrained Spherical Deconvolution (MARF-CSD) method to solve the fiber orientation distribution functions of two anisotropic tissues in white matter and gray matter. In addition, in this paper, we use a modified damp Richardson-Lucy (dRL) method instead of the traditional non-negative least squares (NNLS) method for model solving. We use the diffusion kurtosis imaging (DKI) and neurite orientation dispersion and density imaging (NODDI) to model the white matter and gray matter signals, respectively, MARF-CSD method is compared with multi-shell constrained spherical deconvolution (MSCSD) by simulated data and brain data,. The simulated data results show that MARF-CSD can accurately solve the two anisotropic fiber direction distribution functions, and the dRL method greatly reduces the number of pseudo-peaks compared with the NNLS method. Results from brain data show that the MARF-CSD method has significant similarity to the fiber orientation estimated by the MSCSD method in white matter, but the MARF-CSD method shows anatomically plausible peaks in most gray matter voxels and the orientation is mostly perpendicular to the fiber tracts in the deep white matter region, obtaining a continuous fiber connection from the white matter to the gray matter of the brain.

high angular resolution diffusion imaging

spherical deconvolution

anisotropic diffusion

damp Richardson-Lucy method

Diffusion magnetic resonance imaging (dMRI) has been actively developed with the advantage of obtaining information on the connectivity and structural integrity of brain nerve fiber bundles, and has become a mature technique for modern medical imaging [1, 2]. The movement of water molecules in the brain is not simply free diffusion; the cellular structure causes its mobility to be affected [3, 4]. For example, the diffusion rate of water molecules in the cerebrospinal fluid is essentially the same in all directions, and this diffusion is called isotropic diffusion, its diffusion state is considered as homogeneous and the diffusion curve is mathematically normally distributed, so can used a large value of the average diffusion coefficient to describe this diffusion process. On the contrary, when water molecules are in the midst of inhomogeneous media, the inhomogeneity of these substances hinders the diffusive movement of water molecules, resulting in specific diffusion intensities in different directions, called anisotropic diffusion. Basser et al. [5] proposed diffusion tensor imaging (DTI) to describe the anisotropy of restricted or hindered diffusion processes and demonstrated that the orientation of fiber can be characterized by the principal eigenvector direction of the tensor. The DTI-based approach to fiber orientation estimation is effective, but again, it is subject to many limitations. For example, within each voxel, only one fiber direction can be tracked, but most of fibers in brain have a complex structure, and within these voxels, the DTI method will no longer be applicable [6, 7]. In addition, DTI-based fiber orientation estimation methods assume that the data are collected under Gaussian diffusion, the degree of anisotropic diffusion in the gray matter is low, using DTI methods leads to higher uncertainty in the main diffusion direction, so DTI is unlikely to be choice for studying the structure of gray matter fibers[8] .

SD is one of the most popular methods for estimating fiber orientation distributions in white matte. The fiber orientation distribution function is obtained by deconvolution SD method, as opposed to estimating the directional distribution function, which quantifies the likelihood of tangential vectors of nerve fibers along a defined direction. SD method is based on the basic assumption that the dMRI signal obtained from a single voxel can be modeled as a convolution between the fiber orientation distribution function and the fiber response function, which can theoretically provide physical quantities similar to the real fiber orientation [9, 10]. SD method is highly susceptible to highly pathological solutions due to the discomfort of using NNLS method to solve the convolution problem. Later, some researchers combined the Richardson-Lucy (RL) method with an adaptive regularization to reduce pseudo-peaks and nonphysical fiber orientations in some brain regions which affected by partial volume effects, while also maintaining the angular resolution of the main fiber orientations, and this method was named dRL, dRL can solve the highly pathological problem of NNLS by solving the fiber the dynamic range of the directional distribution function [11, 12]. Studies have shown that the fiber orientation of superficial white matter regions is usually tangential to the cortical surface, which easily leads to early termination of the fiber bundle tracking; fiber orientation estimation in the cerebral cortex is highly susceptible lead to pseudo-peaks due to data noise, which eventually leads to uncertainty in fiber tracking; the presence of superficial white matter regions near the inner surface of the cortex can cause greater difficulties in fiber tracking from deep white matter to the cerebral gray matter [13, 14]. Considering the above issues together, the application of SD-based methods for fiber orientation estimation in studying the gray matter is limited. In 2014, Jeurissen et al. [15] proposed MSCSD method to further improve the accuracy of fiber orientation estimation in white matter by considering partial volume effects. For example, in some regions with non-single white matter tissues, this methodic has a greater advantage in fiber orientation estimation. Among many methods currently used for fiber bundle reconstruction, it is only assumed that the diffusion properties of white matter is anisotropic, and the gray matter is usually viewed as isotropic. However, due to the presence of multiple cytosomes and dendrites in gray matter, these structures result in the diffusion of water molecules not only having similarities to white matter, but also some of the water molecules are free to diffuse and move between different tissues. It was found that [16] that the diffusion behavior of water molecules in gray matter is intermediate between the white matter and cerebrospinal fluid, if the diffusion properties of gray matter are viewed as completely isotropic diffusion, it lead to greater uncertainty in estimating fiber orientation in superficial white matter regions.

To address the above problems, we considering the diffusion characteristics of white matter and gray matter simultaneously, proposing MARF-CSD method to model two anisotropic diffusion signals simultaneously. MARF-CSD method allows reconstructing multiple fiber orientation distribution functions simultaneously, aiming to improve the accuracy of gray matter fiber orientation estimation without changing the fiber orientation properties of white matter. Using MARF-CSD and MSCSD to estimate nerve fiber orientation in the brain by simulated data and real brain data The results show that the MARF-CSD method improves the accuracy of fiber direction estimation in gray matter without degrading the performance of the white matter region, demonstrating the advantage of the MARF-CSD method for fiber direction estimation in gray matter.

2.1 Multi-shell CSD (MSCSD)

Dell'acqua et al. [17] showed that the dMRI signal can be represented by the signal sum of regions with the same nature of diffusion direction and magnitude. The equation for the signal sum of three different diffusion tissues(white matter, gray matter, cerebrospinal fluid) is shown in Eq. (1):

$$\frac{S}{{{S_0}}}={f_{WM}}\left[ {\begin{array}{*{20}{c}} {R_{{WM,1}}^{1}}& \cdots &{R_{{WM,k}}^{1}} \\ \vdots & \ddots & \vdots \\ {R_{{WM,1}}^{n}}& \cdots &{R_{{WM,k}}^{n}} \end{array}} \right] \otimes FO{D_{WM}}+{f_{GM}}\left[ {\begin{array}{*{20}{c}} {R_{{GM}}^{1}} \\ \vdots \\ {R_{{GM}}^{1}} \end{array}} \right]+{f_{CSF}}\left[ {\begin{array}{*{20}{c}} {R_{{CSF}}^{1}} \\ \vdots \\ {R_{{CSF}}^{1}} \end{array}} \right]$$

where S is the diffusion-weighted signal, S₀ is the non-diffusion-weighted signal, n represents the number of different b values and k is the number of directions sampled uniformly on the unit sphere. R_WM, R_GM, and R_CSF matrices represent the response function matrices corresponding to white matter, gray matter, and cerebrospinal fluid, respectively, f_wm, f_gm, and f_csf are the signal fractions corresponding to white matter, gray matter, and cerebrospinal fluid, and FOD_WM is the fiber orientation distribution of white matter is a matrix of k*1. $\otimes$is spherical convolution operation. Only one anisotropic fiber orientation distribution function is included in Eq. (1), because both gray matter and cerebrospinal fluid are considered as isotropic diffusion.

2.2 Multi-anisotropy response function CSD (MARF-CSD)

Eq. (1) is a signal modeling of a single anisotropic diffusion tissue, while the diffusion behavior of water molecules in gray matter between the white matter and the cerebrospinal fluid, the gray matter can no longer be considered as isotropic diffusion, so we redesigned the deconvolution matrix to allow solve the fiber direction distribution function corresponding to multiple anisotropic diffusion tissues simultaneously, and based on Eq. (1), the improved signal model formula is shown in Eq. (2):

$$\frac{S}{{{S_0}}}=\left[ {\begin{array}{*{20}{c}} {R_{{WM,1}}^{1}}& \cdots &{R_{{WM,k}}^{1}} \\ \vdots & \ddots & \vdots \\ {R_{{WM,1}}^{n}}& \cdots &{R_{{WM,k}}^{n}} \end{array}\begin{array}{*{20}{c}} {R_{{GM,1}}^{1}}& \cdots &{R_{{GM,k}}^{1}} \\ \vdots & \ddots & \vdots \\ {R_{{GM,1}}^{n}}& \cdots &{R_{{GM,k}}^{n}} \end{array}} \right] \otimes \left[ {\begin{array}{*{20}{c}} {FO{D_{WM}}} \\ {FO{D_{GM}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{f_{WM}}} \\ {{f_{GM}}} \end{array}} \right]+{f_{CSF}}\left[ {\begin{array}{*{20}{c}} {R_{{CSF}}^{1}} \\ \vdots \\ {R_{{CSF}}^{1}} \end{array}} \right]$$

where S is the diffusion-weighted signal, S₀ is the non-diffusion-weighted signal, n represents the number of different b values and k is the number of directions sampled uniformly on the unit sphere. R_WM, R_GM, and R_CSF matrices represent the response function matrices corresponding to white matter, gray matter, and cerebrospinal fluid, respectively, f_wm, f_gm, and f_csf are the signal fractions corresponding to white matter, gray matter, and cerebrospinal fluid, $\otimes$is spherical convolution operation, FOD_WM, and FOD_GM are the fiber orientation distribution functions of white matter and gray matter, respectively, are convolved with the response functions of the corresponding tissues.

2.3 damp Richardson-Lucy (dRL)

RL is based on the traditional Bayesian method to solve the deconvolution problem and uses an iterative computational method to approximate the solution of the maximum likelihood problem in the Poisson noise case, which can directly derive the fiber direction distribution function from the signal domain, effectively avoiding the pseudo-direction caused by truncation and Gibbs effect when using the spherical harmonic function [18], which can reduce the pseudo-direction of fibers and make the fiber reconstruction results more smooth and stable compared with the NNLS method. Many previous works have been do to solve the deconvolution model using RL method, and achieved better results. In this paper, we use a modified RL method to solve the fiber direction distribution function. The RL method requires a pre-defined deconvolution matrix that can project the dMRI signal measured along each gradient direction onto the unit sphere, earlier related studies have used the signal collected from multiple gradient directions, iterating the expectation maximization method to solve the fiber orientation distribution function for each voxel [19] as shown in Eq. (3):

$${\left[ {FO{D^{\left( {k+1} \right)}}} \right]_i}={\left[ {FO{D^{\left( k \right)}}} \right]_i}\frac{{{{\left[ {{R^T}s} \right]}_i}}}{{{{\left[ {{R^T}R \cdot FO{D^{\left( k \right)}}} \right]}_i}}}$$

where FOD^(k) is the fiber orientation distribution function calculated for k iterations along the i-th sampling direction on the unit sphere, the tissue response function matrix is denoted by R, and R^T is the transpose of R. When solving Eq. (3), since the equation is semi-convergent, its solution actually requires an iteration stopping criterion, which is commonly defined as the maximum number of iterations or a threshold value is preset as the solution termination condition for the change between successive iterations of the computed FOD.

White et al. [20] use a new function to modified the likelihood function that gives smoother solution when the difference between the estimated model and the data is small, this method known as dRL method. The negative fiber orientation component is reduced by adding an additional canonical term to attenuate the overfitting of the FOD to noise or isotropic tissue (e.g., cerebrospinal fluid). dRL method as shown in Eq. (4):

$${\left[ {FO{D^{\left( {k+1} \right)}}} \right]_i}={\left[ {FO{D^{\left( k \right)}}} \right]_i}\left( {1+{{\left[ {{u^{\left( k \right)}}} \right]}_i}\left( {\frac{{{{\left[ {{R^T}s - {R^T}R \cdot FO{D^{\left( k \right)}}} \right]}_i}}}{{{{\left[ {{R^T}R \cdot FO{D^{\left( k \right)}}} \right]}_i}}}} \right)} \right)$$

where the regular term is represented by a vector u, which performs a constraint operation on each element of the FOD. u(k) is updated at each iteration according to the standard deviation std(s) of the FOD amplitude and dMRI signal obtained from the k-th calculation, calculated as shown in Eq. (5):

$$\begin{gathered} {\left[ {{r^{\left( k \right)}}} \right]_i}=1 - \frac{{{{\left[ {{{\left( {FO{D^k}} \right)}^v}} \right]}_i}}}{{{{\left[ {{{\left( {FO{D^{\left( k \right)}}} \right)}^v}+{\eta ^v}} \right]}_i}}} \hfill \\ {u^{\left( k \right)}}=1 - \hbox{max} \left( {0.1 - 4 \times std\left( s \right)} \right){r^{\left( k \right)}} \hfill \\ \end{gathered}$$

Where r is the level of constraint that controls each element in FOD, v is a geometric parameter that describes the speed of the constraint from the beginning to the end. The smaller of v, the larger the range of FOD amplitudes required to be calculated when r transitions from 0 to 1. Therefore, a suitable value of v needs to be set so that the effect of suppressing the FOD amplitude is dynamically balanced; otherwise, a larger value of the fiber directional distribution coefficient is suppressed, while a smaller coefficient does not have a suppressive effect, which defeats the original purpose of the study. According to Dell'acqua et al. [12], the solution of the fiber orientation distribution coefficient is more likely to converge when the constant v is set to 8. ƞ is taken to be twice the maximum FOD coefficient calculated from the current iterations, and such a value of ƞ is chosen to minimize the contribution of the gray matter in solving for the fiber orientation distribution coefficient in white matter. u was used to adjust the suppression effect between different voxels and can be used as an anisotropy index in dMRI data, for the initial classification of brain tissues with different diffusion properties. This parameter is less affected by partial volume effects in cross-fiber region compared to the anisotropy fraction. In this paper, the value range of u is set from 0.2 to 1 according to the diffusion properties of brain tissue, indicating the span of voxels from fully anisotropic to fully isotropic diffusion.

In this paper, we based on MARF-CSD, used dRL method to solve the deconvolution model, obtained two anisotropic fiber direction distribution functions for white matter and gray matter, respectively, which are calculated as shown in Eq. (6) and Eq. (7):

$${\left[ {FO{D_{WM}}^{{\left( {k+1} \right)}}} \right]_i}={\left[ {FO{D_{WM}}^{{\left( k \right)}}} \right]_i}\left( {1+{{\left[ {{u^{\left( k \right)}}} \right]}_i}\left( {\frac{{{{\left[ {{R_{WM}}^{T}s - {R_{WM}}^{T}{R_{WM}} \cdot FO{D_{WM}}^{{\left( k \right)}}} \right]}_i}}}{{{{\left[ {{R_{WM}}^{T}{R_{WM}} \cdot FO{D_{WM}}^{{\left( k \right)}}} \right]}_i}}}} \right)} \right)$$

$${\left[ {FO{D_{GM}}^{{\left( {k+1} \right)}}} \right]_i}={\left[ {FO{D_{GM}}^{{\left( k \right)}}} \right]_i}\left( {1+{{\left[ {{u^{\left( k \right)}}} \right]}_i}\left( {\frac{{{{\left[ {{R_{GM}}^{T}s - {R_{GM}}^{T}{R_{GM}} \cdot FO{D_{GM}}^{{\left( k \right)}}} \right]}_i}}}{{{{\left[ {{R_{GM}}^{T}{R_{GM}} \cdot FO{D_{GM}}^{{\left( k \right)}}} \right]}_i}}}} \right)} \right)$$

where FOD_WM, FOD_GM are the fiber orientation distribution functions of white matter and gray matter, respectively, and R_WM, R_GM denote the convolutional response function matrices of white matter and gray matter, respectively. i is the i-th direction sampled on the unit sphere. The data used in this paper were collected under multiple sampling directions based on multiple b-values, so the anisotropic response function matrix consists of each submatrix. n represents the number of b-values, and k is the number of directions sampled uniformly on the unit sphere, as shown in Eq. (8) and Eq. (9):

$${R_{WM}}=\left[ {\begin{array}{*{20}{c}} {R_{{WM,1}}^{1}}& \cdots &{R_{{WM,k}}^{1}} \\ \vdots & \ddots & \vdots \\ {R_{{WM,1}}^{n}}& \cdots &{R_{{WM,k}}^{n}} \end{array}} \right]$$

$${R_{GM}}=\left[ {\begin{array}{*{20}{c}} {R_{{GM,1}}^{1}}& \cdots &{R_{{GM,k}}^{1}} \\ \vdots & \ddots & \vdots \\ {R_{{GM,1}}^{n}}& \cdots &{R_{{GM,k}}^{n}} \end{array}} \right]$$

2.4 Response function

To emphasize that the MARF-CSD method is independent, three different diffusion models are used for the response function matrices R of white matter, gray matter and cerebrospinal fluid, respectively. The columns of the response function matrix are generated using a specific signal model, such as a signal generated using a predefined set of parameters, and then projected in multiple spatial directions defined along the unit sphere. The cerebrospinal fluid response function matrix R_CSF was used the apparent diffusion coefficient model with a diffusion parameter at 0.003 mm² /s, which is typical for the free diffusion of water molecules at 37°. Since the DKI model takes into account non-Gaussian diffusion, this model is more suitable for the construction of white matter response function matrix R_WM [21]. The gray matter response function matrix R_GM uses the NODDI model, and related studies have shown that the NODDI model can better characterize the microstructure of gray matter compared to the common tensor model [22]. The graphical representation of the relevant response function matrix is shown in Fig. 1, the white matter response function matrix constructed with DKI model exhibits a pie shape with a high degree of anisotropy; NODDI model indicates that the gray matter signal is also clearly anisotropic and shows a blimp-like shape, and the cerebrospinal fluid shows a spherical shape with an isotropic diffusion.

2.5 Simulation

The feasibility of simultaneously calculating multiple anisotropic fiber orientation distribution functions was experimentally assessed by simulated data. The DKI model was used to generate R_WM, and the eigenvalues of white matter diffusion tensor were set to [0.0017, 0.0002, 0.0002] mm² /s. The gray matter response function matrix R_GM was generated using the NODDI model, according to the diffusion properties of the real human brain, the parameter set was set as follows: the diffusivity from intracellular to extracellular was set to 0.4, the diffusivity parallel to the diffusion direction of water molecules was set to 0.0017 mm² /s. The average diffusion coefficient of cerebrospinal fluid was set to 0.003 mm² /s. Simulated data I was designed to evaluate the feasibility of spherical deconvolution for solving multiple anisotropic response functions, while simulated data II was used to analyze the dRL method compared with the NNLS method.

2.5.1 Simulation I

The following three assumptions were made for the simulated data I used in the experiment:

Simulated data 1: simulated voxels containing only two anisotropic diffusion tissues, white matter and gray matter, with f_CSF = 0 and f_WM, f_GM increasing from 0 to 1 in steps of 0.1;

Simulated data 2: simulated voxels containing only two tissues with different diffusion properties, white matter and cerebrospinal fluid, with f_GM = 0 and f_WM, f_CSF increasing from 0 to 1 in steps of 0.1;

Simulated data 3: The simulated voxels contain only two tissues with different diffusion properties, gray matter and cerebrospinal fluid, with f_WM = 0 and f_GM, f_CSF increasing from 0 to 1 in steps of 0.1.

Rice noise with SNR of 80 was added to the simulated data during the experiments to study the performance of the MARF-CSD method in distinguishing one isotropic diffusion signals and two anisotropic diffusion signals.

2.5.2 Simulation Ⅱ

The simulated data II was used from the phantom data simulated data in ExploreDTI (http://www.ExploreDTI.com/exampledataset.htm). This data containing 6 unweighted images and 60 weighted images with a voxel size of 1 × 1 × 1.

2.6 In-vivo data

2.6.1 In-vivo data Ⅰ

The real brain data Ⅰ used in the experiment came from the public multicenter dMRI dataset (multicenter dataset of multi-shell diffusion MRI in healthy traveling adults with identical setting) from the CBIST team of the magnetic resonance platform of Zhejiang University (https://doi.org/10.6084/m9.figshare.8851955), which was acquired using a spin-echo planar imaging sequence with the following parameters: voxel volume of 146 × 146 × 92, voxel size of 1.5 × 1.5 × 1.5, and selected b-values of {0, 1000. 2000, 3000}, the number of diffusion gradient vectors as {6, 30, 30, 30}.

2.6.2 In-vivo data Ⅱ

Real brain data II from the center of functionally integrative neuroscience (CFIN) dataset publicly available at DIPY database (https://github.com/dipy), which was acquired at multiple diffusion-sensitive gradient intensities, with the following parameters: voxel volume of 96 × 96 × 19, voxel size of 2.5 × 2.5 × 2.5, and image data selected with b-values of {0, 1000, 3000}, corresponding to the number of diffusion gradient vectors of {1, 33, 33}.

3.1 Simulation

Figure 2 shows the linearly varying simulated white matter, gray matter, and cerebrospinal fluid signal fractions and the results calculated by the MARF-CSD method. Each row show the white matter volume fraction f_WM, gray matter volume fraction f_GM, and cerebrospinal fluid volume fraction f_CSF, respectively, and each column shows the experimental results for simulated data 1, simulated data 2, and simulated data 3, respectively. In all three cases, the estimated signal fractions agree well with the linearly varying simulated values, and the linear variation is in the correct direction. The maximum errors were 8%, 20% and 2% for white matter, gray matter and cerebrospinal fluid, respectively. The pearson correlation coefficient between the simulated values and the estimated values ranged between 0.99 and 1.

Figure 3 shows the fiber reconstruction results of the dRL method with the NNLS method on the phantom data. Each row shows the experimental results for different regions, where the yellow rectangular box and red rectangular box regions are cross-fiber regions and the green rectangular box regions are single-fiber regions; the first column shows the experimental results of NNLS method on three fiber regions, and the second column shows the experimental results of dRL method on three fiber regions. The results show that using dRL method can improve the angular resolution of the cross-fibers, and in the single-fiber region (green rectangular box), the fiber direction estimation is more stable, which greatly reduces the number of pseudo-peaks.

3.2 In-vivo data

Figure 4 shows the fiber reconstruction results of dRL method with NNLS method on real brain data I. The first row shows the fiber reconstruction results for the brain region of interest, and the second row selects several voxels of the region selected in the first row to be displayed in magnification, marked with red rectangular boxes and yellow rectangular boxes in the whole-brain fiber orientation estimation images, respectively. Each column represents different method, the first column being the NNLS method and the second column being dRL method. The experimental results are consistent with the previous simulation experiments, indicating that dRL method can improve the stability of fiber direction estimation and reduce the number of pseudo-peaks to some extent.

Figure 5 and Fig. 6 show the fiber orientation reconstruction results of two regions of interest estimated by MSCSD and MARF-CSD methods, respectively. Two anisotropic fiber orientation distribution functions for white matter and gray matter are obtained by MARF-CSD method, named MARF-CSD₁ and MARF-CSD₂, respectively. The last column shows a few voxels of the region of interest enlarged to highlight more details, which are marked with red rectangles in the whole brain fiber direction estimation map; each row represents MSCSD, MARF-CSD₁, MARF CSD₂, MARF-CSD four method’s results, where MARF-CSD combines the information content of MARF-CSD₁ in white matter with that of MARF-CSD₂ in gray matter. The fiber orientation of white matter estimated by MARF-CSD method (MARF-CSD₁) has significant similarity with MSCSD. The fiber orientations estimated by both methods had similar orientations in the regions adjacent to the white matter and smaller orientation coefficients along the deeper cortical gray matter orientation. MSCSD compared with MARF-CSD, MARF-CSD₂ was non-zero only in gray matter and its adjacent regions.The gray matter fiber orientations of the MARF-CSD method (MARF-CSD₂ ) showed anatomically plausible peaks, and the orientation was mostly perpendicular to the fiber tracts in the deep white matter region. When MARF-CSD₁ and MARF-CSD₂ are linearly combined (MARF-CSD), a continuous fiber connection from white matter to gray matter of the brain is obtained.

Figure 7 shows the fiber reconstruction results of MARF-CSD and MSCSD in the brain region of interest. The first two columns show the results of MARF-CSD and MSCSD, respectively, and the rightmost column zooms in on a 3 × 3 area, the red rectangular box showing the results of the MARF-CSD method and the yellow rectangular box showing the results of the MSCSD method. The fiber orientations obtained with these two methods look similar in general, but the fiber orientations determined with the MARF-CSD method look clearer. The rightmost column clearly shows that the fiber orientation calculated by the MARF-CSD method has a lower angular uncertainty than the MSCSD method and a better angular resolution of fiber crossing can be observed (blue arrows).

Since the diffusion of water molecules in gray matter is intermediate between the white matter and the cerebrospinal fluid, and the traditional SD method treats the gray matter as isotropic diffusion, and the method can accommodate only one anisotropic diffusion of tissue [23, 24]. Therefore, in this paper, a MARF-CSD method is proposed to simultaneous model two anisotropic diffusion signals. Simulation experiments results show that the signal fractions estimated by the MARF-CSD method are in high agreement with the linearly varying simulated values, indicating that MARF-CSD method can distinguish one isotropic diffusion signal and two anisotropic diffusion signal from the mixed signal. Further, experiments were conducted with real human brain data from publicly available databases, the results shows that the white matter fiber orientation estimated by the MARF-CSD method (MARF-CSD₁) was significantly similar to MSCSD, but the fiber orientation distribution coefficients along the gray matter direction of the MSCSD method became progressively smaller, and the MARF-CSD method (MARF-CSD₂) for gray matter fiber orientation showed more reliable information in most gray matter voxels. The fiber orientation calculated by the MARF-CSD method has lower angular uncertainty and higher angular resolution of cross-fibers than MSCSD method. In conclusion, the MARF-CSD method can reconstruct the fiber orientation distribution of multiple anisotropic diffusion tissues simultaneously, extracting more information of gray matter tissue while preserving the superior performance of white matter.

The process of calculating the fiber orientation distribution function for multiple anisotropic diffusion tissues is more complicated, and the solving process requires a matrix rank much lower than the number of possible solutions (fiber orientation distribution coefficients calculated on the unit sphere), which poses a challenge to the solution of deconvolution [8]. MARF-CSD method accommodates the fiber orientation distribution functions of multiple anisotropic tissues and NNLS method has highly uncomfortable, we introduced the dRL method in the model solving scheme. Figure 2 shows that the MARF-CSD method can effectively distinguish one isotropic diffusion signal and two anisotropic diffusion signal in mixed signals containing both isotropic and anisotropic diffusion signals. Among them, the mixed signals from white matter and cerebrospinal fluid were the easiest to achieve accurate signal separation with the least angular error. In contrast, separation of gray matter from white matter and cerebrospinal fluid is a more difficult task, which may be due to the lower degree of anisotropic diffusion of gray matter.

Pietsch et al. [25] in 2019 proposed an method for solving multiple white matter fiber orientation distribution functions, aiming to address the shortcomings of SD method for white matter fiber reconstruction in infant’s brain.The MARF-CSD method can reconstruct multiple fiber orientation distribution functions, and the fiber orientation distribution functions are specific to different tissue types (e.g., white matter and gray matter), as shown in Fig. 5 and Fig. 6. These two figures show that MARF-CSD (MARF-CSD₁) and MSCSD estimate very similar fiber orientations in white matter, both have small fiber orientation distribution coefficients among gray matter. In contrast, the gray matter fibers estimated by MARF-CSD₂ have a clear diffusion direction. In this work, we use the NODDI model to describe the gray matter signal, and related studies have shown that the NODDI model can better characterize the gray matter microstructure compared with the common tensor model. In Fig. 7, we investigate the performance of MARF-CSD for brain nerve fiber orientation estimation. The experimental results show that the MARF-CSD method is more beneficial in fiber orientation estimation for tracking to deeper brain gray matter regions, as the fibers reconstructed by this method are sharper and have higher resolution compared to the MSCSD method.

The response function models describing the diffusion properties of white matter and gray matter in this paper were chosen from the commonly used methods currently, but the combination of DKI and NODDI may not be the best combination for estimating multiple fiber orientation distribution functions. Therefore, the combination of anisotropic diffusion models can be further adjusted or even replaced with other more specific tissue models to further improve the MARF-CSD method.

This study have proposed the MARF-CSD method to implement tissue-specific SD solutions, which can reconstruct the fiber orientation distribution functions of multiple anisotropic diffusion tissues simultaneously. By combining specific white matter and gray matter diffusion models, we have demonstrated that the MARF-CSD method can improve the performance of SD in gray matter while preserving the superior performance of SD method for nerve fiber orientation estimation in white matter.

Competing interests

All authors of this paper declare that there is no conflict of interest.

Authors' contributions

Yingyu Pan was the experimental designer and executor of the experimental study, completing the collection of experimental data and writing the first draft of the paper; Yuanjun Wang was the leader of the study, directing the experimental process, writing and revising the paper. Both authors read and agreed to the final text.

Funding

Supported by the National Natural Science Foundation of China (61201067) and Natural Science Foundation of Shanghai (18ZR1426900).

Data availability

We used open-access data from the public multicenter dMRI dataset (multicenter dataset of multi-shell diffusion MRI in healthy traveling adults with identical setting) from the CBIST team of the magnetic resonance platform of Zhejiang University (https://doi.org/10.6084/m9.figshare.8851955) and the center of functionally integrative neuroscience (CFIN) dataset publicly available at DIPY database (https://github.com/dipy). The review board approved all study procedures with regards to collecting and sharing data, and consent was obtained for each participant.

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No competing interests reported.

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Estimation of Gray Matter Fiber Direction Based on Anisotropy Diffusion

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Abstract

Figures

1. Introduction

2. Theory and Methods

2.1 Multi-shell CSD (MSCSD)

2.2 Multi-anisotropy response function CSD (MARF-CSD)

2.3 damp Richardson-Lucy (dRL)

2.4 Response function

2.5 Simulation

2.5.1 Simulation I

2.5.2 Simulation Ⅱ

2.6 In-vivo data

2.6.1 In-vivo data Ⅰ

2.6.2 In-vivo data Ⅱ

3. Results

3.1 Simulation

3.2 In-vivo data

4. Discussion

5. Conclusion

Declarations

References

Additional Declarations

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