Tissue Segmentation Using Sparse Non-negative Matrix Factorization of Spherical Mean Diffusion MRI Data

Abstract

In this paper, we present a method based on sparse non-negative matrix factorization (NMF) for brain tissue segmentation using diffusion MRI (DMRI) data. Unlike existing NMF-based approaches, in our method NMF is applied to the spherical mean data, computed on a per-shell basis, instead of the original diffusion-weighted images. This is motivated by the fact that the spherical mean is independent of the fiber orientation distribution and is only dependent on tissue microstructure. Applying NMF to the spherical mean data will hence allow tissue signal separation based solely on the microstructural properties, unconfounded by factors such as fiber dispersion and crossing. We show results explaining why applying NMF directly on the diffusion-weighted images fails and why our method is able to yield the expected outcome, producing tissue segmentation with greater accuracy.

1. Introduction

In diffusion MRI (DMRI), tissue segmentation is a key step to tease apart signal contributions of white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF) for fine-grained analysis [1]. For example, it has been demonstrated in [2] that the overestimation of apparent WM density in voxels containing GM and CSF can be mitigated by explicitly considering the response functions associated with the individual tissue types. Unlike segmentation of anatomical (e.g., T1-weighted) images, which is typically intensity-based, DMRI segmentation is based on diffusion-attenuated signals stemming from the motion of water molecules.

Recently, nonnegative matrix factorization (NMF) has been proposed for tissue segmentation in DMRI [3]. NMF is a data-driven approach for blind source separation and can be viewed as a clustering algorithm [4–9]. The goal of NMF is to approximate a nonnegative data matrix by the product of two unknown matrices: A basis matrix and a weight matrix. The rank of the factorization is generally smaller than that of the data matrix and the product can hence be seen as a compressed representation of the data. The solution to the plain vanilla NMF [4] is not unique. However, this non-uniqueness problem can be mitigated by imposing sparsity or convexity constraints [3, 9, 10].

Jeurissen et al. [3] applied NMF directly to the diffusion-weighted (DW) images for tissue-type segmentation. The data matrix is formed with each row corresponding to a different DW image and each column corresponding to a different voxel. The factorization rank is set to 3 to recover the three main tissue types in the brain. Each column of the basis matrix returned by NMF is a basis vector that best represents a certain tissue type and the corresponding row of the weight matrix contains the weights of the voxels in relation to the tissue type.

We note that the method described above, while seemingly effective, suffers from a major limitation. Although it is straightforward to understand that CSF and GM can each be represented by a basis vector in the basis matrix, it is unclear how WM, which can come in a large variety of orientations and configurations, can be sufficiently captured using a single basis vector. In fact, we will show in this paper that this is an unreasonable representation and causes significant errors in segmentation.

We will address the above problem by applying NMF to spherical mean DMRI data. The spherical mean is computed for each shell associated with a diffusion weighting (i.e., b-value). The spherical mean is independent of the fiber orientation distribution and is only dependent on tissue microstructure [11]. Applying NMF on the spherical mean data will hence allow tissue classification based solely on the microstructural properties, unconfounded by factors such as fiber dispersion and crossing. We will show experimental results demonstrating why applying NMF directly on the DW images is not ideal and why our method is able to overcome the problem, producing tissue segmentation results with greater accuracy.

2. Approach

Our approach to tissue segmentation is inspired by the work of Jeurissen et al. [3]. However, instead of applying NMF directly to the DW images, we apply NMF on the spherical mean, which is computed on a per-shell basis.

2.1. The Spherical Mean

The spherical mean is not dependent on the fiber orientation distribution and is a function of per-axon diffusion characteristics [11]. This is based on the observation that for a fixed b-value the spherical mean of the diffusion-attenuated signal over the gradient directions, i.e.,

$${\overline{S}}_b=\frac{1}{4\pi }{\int }_{{\mathbb{S}}^2}{S}_b(g)dg$$ (1)

does not depend on the fiber orientation distribution. Assuming that the signal can be represented as the spherical convolution of a fiber orientation distribution p(ω) (p(ω) ≥ 0, ${\int }_{{\mathbb{S}}^2}p(\omega )d\omega =1$, $p(\omega )=p(-\omega ),\omega \in {\mathbb{S}}^2$) with an axial and antipodal symmetric kernel h_b(g, ω) ≡ h_b (|〈g, ω〉|), i.e.,

$$S_b(g)=S_0{\int }_{{\mathbb{S}}^2}{h}_b(g,\omega )p(\omega )d\omega ,$$ (2)

it can be shown that [11]

$${\overline{S}}_b=S_0{\overline{h}}_b.$$ (3)

This shows that the spherical mean is independent of the orientation distribution.

2.2. Tissue Segmentation via Sparse NMF

Brain tissue segmentation can be cast as a blind source separation problem, where the sources and the mixing weights are determined concurrently. NMF is an ideal tool for this problem owing to its ability to represent the data as an additive nonnegative combination of a set of non-negative basis vectors, resulting in a parts-based decomposition. NMF factorizes a nonnegative data matrix $V\in {ℝ}^{M\times N}$ into two unknown nonnegative matrices: The basis matrix $W\in {ℝ}^{M\times K}$ and the weight matrix $H\in {ℝ}^{K\times N}$:

$$V\approx W\times H.$$ (4)

In Jeurissen et al.’s [3] approach, each row of V corresponds to a different DW image and each column corresponds to a different voxel inside the brain. Instead of taking this approach, we show that a better approach is to perform segmentation based on the spherical mean. That is, each row of V should correspond to a shell associated with a b-value and each column to the voxels. Upon performing NMF, each column of W is a basis vector that best represents a certain tissue type and the corresponding row of the H contains the weights (also known as volume fractions when normalized to have a sum of 1) of the voxels in relation to the tissue type.

The parameter K of the factorization should be usually chosen to be smaller than M and N. Naturally, for brain tissue segmentation into WM, GM, and CSF, it should be set to 3. The value can be increased beyond 3 to account for pathology such as lesions. In general, NMF is undetermined and the factorization is obtained by solving the following problem

$$\underset{W,H}{\min }\Vert V-WH{\Vert }_F^2,w_{i,j}\ge 0,h_{i,j}\ge 0.$$ (5)

The non-uniqueness of NMF can be controlled by sparsity penalization [9]:

$$\underset{W,H}{\min }\Vert V-WH{\Vert }_F^2+{\lambda }_1\sum _j{\Vert h_j\Vert }_1+{\lambda }_2\sum _j{\Vert h_j\Vert }_2^2,w_{i,j}\ge 0,h_{i,j}\ge 0,$$ (6)

where h _j is j-th column vector of H. This form of penalization is reasonable since partial volume occurs mostly at tissue boundaries and hence most parts of the brain should be comprised of single tissue types [3].

The solution to (6) can be obtained by alternating between solving two subproblems, i.e., a non-negative least-squares problem:

$$\underset{W}{\min }\Vert V-WH{\Vert }_F^2,w_{i,j}\ge 0,$$ (7)

when H is fixed, and an elastic net problem

$$\underset{H}{\min }\Vert V-WH{\Vert }_F^2+{\lambda }_1\sum _j{\Vert h_j\Vert }_1+{\lambda }_2\sum _j{\Vert h_j\Vert }_2^2,h_{i,j}\ge 0,$$ (8)

when W is fixed. We use SPAMS [12–14] in our implementation.

Using the spherical mean instead of the original signal in V allows us to solve the problem inherent in Jeurissen et al.’s method [3]. This is because the spherical mean captures the per-axon diffusion characteristics independent of the orientation distribution. For example, voxels with very different fiber configuration, but with the same per-axon microstructural parameters, can be represented using exactly the same spherical mean vector.

Our formulation of the problem also means that a smaller matrix V needs to be decomposed since it has a significantly smaller number of rows. This implies that tissue segmentation can be performed much faster using our method.

3. Experiment

We evaluated our method with Jeurissen et al.’s [3] method as the comparison baseline. Both qualitative and quantitative results are reported.

3.1. Synthetic Data

Three different sets of synthetic datasets were generated for evaluation. The parameters used for generating the synthetic data were consistent with the real data described in the next section: b = 500, 1000, 1500, 2000, 2500, 3000 s/mm², with respectively 9, 12, 17, 24, 34, and 48 non-collinear gradient directions, 20 × 20 voxels. Rician noise with level comparable to the real data was added to the synthetic data. The three sets of synthetic datasets model three different scenarios: (1) Single: WM with single fiber orientations + GM + CSF; (2) Crossing: WM with two crossing fiber directions + GM + CSF; and (3) Partial Volume: WM with two crossing fiber directions with partial volume of varying degree with GM + CSF. The voxel number ratio is 9:9:2 for WM:GM:CSF, giving 45% WM voxels, 45% GM voxels, and 10% CSF voxels. The ground truth synthetic data with the fiber orientations (FOs) and color-coded tissue volume fractions are shown on the far left of Fig. 1.

Fig. 1.

Synthetic Datasets. Synthetic datasets with different configurations and with/without partial volume effects. Different fiber orientations are simulated. Volume fractions are colored coded with blue for WM, green for GM, and red for CSF. The ground truth volume fractions and those estimated by the baseline and the proposed method are shown

Comparing the estimation results obtained using the baseline and the proposed method, we can see that the baseline method is easily confused by the different WM orientations, resulting in incorrect volume fraction estimates. Note that the GM and CSF volume fraction estimates are also affected by the WM errors. On the other hand, the proposed method performs consistently for the different datasets, confirming the fact that it is not sensitive to variation in WM configurations.

We compute the mean absolute difference (MAD) between the estimated and the ground truth volume fractions. Table 1 shows the MAD values for the datasets, indicating that the proposed method yields estimates that are significantly closer to the ground truth. The baseline method, on the other, deviates significantly from the ground truth. Note that since the volume fraction is a normalized quantity, the estimation errors of one tissue type will affect others.

Table 1.

MAD between estimated volume fraction and ground truth

Method	Dataset	WM	GM	CSF
Baseline	Single	0.6206	0.5584	0.0167
	Crossing	0.4411	0.6512	0.0237
	Mixture	0.4775	0.2728	0.0219
Proposed	Single	0.0050	0.0043	0
	Crossing	0.0020	0.0028	0
	Mixture	0.0430	0.0403	0

3.2. Real Data

The DW images of an adult were acquired using a Siemens 3T Prisma MR scanner with the following protocol: 140 × 140 imaging matrix, 1.5mm × 1.5mm × 1.5mm resolution, TE = 88ms, TR = 2365ms, 32-channel receiver coil, and multi-band factor 9, 12, 17, 24, 34, and 48 non-collinear directions respectively for b = 500,1000,1500,2000,2500,3000s/mm². A non-DW image with b = 0s/mm² was collected for every 24 images, giving a total of 6.

Figure 2 shows the volume fraction maps of the baseline and the proposed method. One can appreciate that with the proposed method a clearer separation of the different types can be obtained. This is apparent from the better contrast of the volume fraction maps given by the proposed method. The WM map given by the proposed method is also more homogeneous in terms of intensity. This is an indicator that the proposed method is less affected by the WM orientation distribution.

Fig. 2.

Volume Fractions. Volume fraction maps given by the baseline and the proposed method

Figure 3 shows the plots for the basis vectors determined using the two methods. For the baseline method, the values (in W) are averaged over each shell to obtain the curves. The curves are expected to decay with diffusion weighting due to the increase in signal attenuation. It can be observed from the figure that the proposed method yields results that are closer to this expectation.

Fig. 3.

Basis. Plots of basis (normalized to 1 at b = 0s/mm²)

4. Conclusion

In this paper, we have shown that improved brain tissue segmentation can be improved by applying NMF on the spherical mean DMRI data. Specifically, we have demonstrated that, in contrast to Jeurissen et al.’s method, the proposed method is unconfounded by the differences in fiber orientation distributions. The proposed method will hence allow tissue segmentation to be performed more robustly using DMRI data.