Interpretation of Diffusion MR Imaging Data Using Non-negative Least Squares (NNLS)

Abstract

The interpretation of abdominal diffusion images remains difficult because different types of tissue coexist within each voxel. In this study, we estimated the distribution of the diffusion coefficient (D) within each voxel using non-negative least squares from diffusion-weighted images with multiple b-values. Additionally, a method for visualizing the results as a set of images representing the D ranges was developed. This method was tested on healthy volunteers.

Introduction

Although it has become clear that the primary contrast mechanism for pathological tissues is restricted diffusion, it is still difficult to interpret the diffusion MR images to understand underlying tissue structures. This difficulty is partly due to the many different tissue compartments within each MRI voxel. Existing interpretation methods include so-called intravoxel incoherent motion (IVIM)^1,2 and diffusional kurtosis imaging (DKI).³ These methods analyze the signal decay curves.

Another group of approaches, statistical models, utilizes models in which the diffusion coefficient D has a statistical distribution within each voxel.^4,5 In this approach, a D distribution function P(D) is converted into a signal decay function S(b) of b-value and the parameters of P(D) are estimated by curve fitting on S(b).

For example, in the gamma distribution model,⁴ the distribution of D in each voxel is modeled as

$$\rho \left(D\right)=\frac{m_0}{\text{Γ}\left(k\right){\theta }^k}{D}^{k-1}exp\left(-\frac{D}{\theta }\right)$$ [1]

where m₀ is proton density. θ and k are parameters that characterize the distribution. Γ is the gamma function, and the part [1/Γ(k)θ^k] is the normalization term to make the area under the curve to 1. The signal decay curve that corresponding to this distribution can be calculated as weighted sum of different exponential curves for all possible D values as

$$\begin{array}{cc}S(b)& =m_0{\displaystyle {\int }_0^{\infty }\rho (D)exp(-bD)dD}\\ & =\frac{m_0}{{(1+\theta b)}^k}\end{array}$$ [2]

which is a function of b.³ The signal decay curve is still described with the same parameters, θ and k, and can be estimated by curve fitting to the actually observed data. Once θ and k are estimated, the distribution function is obtained by inserting θ and k into eq[1]. While certain information on the distribution of D can be obtained with this process, it is still model-based approach and there is no guarantee of the estimated distribution represent the actual distribution of D.

In the current work, we propose a method to directly estimate the distribution of D from the signal decay curve using an approach called non-negative least squares (NNLS).⁶

Theory

The signal strength of multi-b i-th measurement S_i, obtained with b_i (i = 1, …, M), is expressed as below:

$$S_i=\sum _{j=1}^N{m}_j\mathrm{e}\mathrm{x}\mathrm{p}(-b_i{D}_j)$$ [3]

where D_j is the diffusion coefficient of j-th component (j = 1, …, N) and m_j is the amount of the component which has the diffusion coefficient D_j. Note that the sum of all components from 1 to N corresponds to the proton density. This can be written in matrix form

$$s=Am$$ [4]

where

$$A=\left[\begin{array}{cccc}exp\left(-b_1{D}_1\right)& exp\left(-b_1{D}_2\right)& \dots & exp\left(-b_1{D}_N\right)\\ exp\left(-b_2{D}_1\right)& exp\left(-b_2{D}_2\right)& \dots & exp\left(-b_2{D}_N\right)\\ ⋮& ⋮& \ddots & ⋮\\ exp\left(-b_M{D}_1\right)& exp\left(-b_M{D}_2\right)& \dots & exp\left(-b_M{D}_N\right)\end{array}\right]$$ [5]

$$m=\left[\begin{array}{c}{m}_1\\ m_2\\ ⋮\\ m_N\end{array}\right]$$ [6]

and

$$s=\left[\begin{array}{c}{S}_1\\ S_2\\ ⋮\\ S_M\end{array}\right]$$ [7]

This can be solved using NNLS approach

$$Min\text{}\parallel \text{}s-Am{\parallel }^{2}s.t.m_j\ge 0\forall j$$ [8]

Materials and Methods

Total of 5 volunteers were enrolled with informed consent, after approval of Institutional Review Board of Juntendo University School of Medicine (approval no. E23-0438).

Diffusion-weighted images were acquired using a single-shot diffusion-weighted EPI with multiple b-values on a 3T clinical scanner (Canon Medical Systems, Otawara, Japan). Typical imaging parameters were TR = 4692 msec (with respiratory gating), TE = 49 msec, b = 0, 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400 s/mm² (16 points), and FOV = 380 × 330 mm.

From multiple diffusion-weighted images, the distribution of the diffusion coefficient D within each voxel (D spectrum) was calculated. Instead of estimating the magnitude and D value of each component, only the magnitude of each component with a predetermined D value was calculated. By determining the D values, the system of equations becomes linear, and the problem is reduced to solving a linear set of equations. The unstable system can be regularized by forcing the estimated magnitude to be positive (non-negative). This process is called NNLS.⁶

The distribution of D was calculated using NNLS for D values of 0.10, 0.15, 0.23, 0.35, 0.54, 0.79, 1.20, 1.82, 2,75, 4.16, 6.30, 9.53, 14.4, 21.8, 33.0, 50.0 × 10⁻³ mm²/s (16 points, in exponentially equidistant scaling). The range of D values were determined to roughly cover the values for normal organs. The results obtained by the NNLS were 16 numbers per pixel or 16 images, corresponding to each D value.

A visualization process is necessary because these images are unsuitable for direct visual evaluation. Many different processing methods are possible; however, decomposition into 3 component images, Dlow, Dmid, and Dhigh, was used as the first attempt. These components roughly correspond to small cells, extracellular space, and flow, respectively. The proposed method is tested on healthy volunteers.

Also, a simple simulation was done to illustrate the difference between NNLS and ordinary LS (pseudo-inverse). The ranges of b-values and D are the same as the experiment. A test signal was generated for D = 1.0 × 10⁻³ mm²/s, which does not coincide with any of D values used for NNLS (between 0.79 and 1.20), and applied NNLS and LS (Fig. 1).

Fig. 1.

NNLS vs LS. A mono-exponential decay curve was generated for D = 1.0 × 10⁻³ mm²/s, and the result of NNLS and LS is shown in the figure. The ranges of b-values and D are the same as the experiment.D, diffusion coefficient; LS, least squares; NNLS, non-negative least squares.

Representative D spectra and their corresponding relative amounts for voxels in the hepatic parenchyma, gallbladder, and splenic parenchyma are shown in Fig. 2a. Based on these data, a conceptual weighting function model for separating the D spectrum into 3 components (Dlow, Dmid, and Dhigh) is illustrated in Fig. 2b. In this study, the threshold for the D spectrum was set at D = 1.7 × 10⁻³ mm²/s.

Fig. 2.

D spectrum and window function. Examples of D spectrum are shown in (a). The NNLS results were combined pixel-by-pixel using the window functions shown in (b).D, diffusion coefficient; GB, gallbladder; NNLS, non-negative least squares.

Results

The result of the simulation is shown in Fig. 2. A mono-exponential decay curve was generated for D = 1.0 × 10⁻³ mm²/s, and the result of NNLS and LS is shown in the figure. While NNLS gives a peak around the correct value (1.0), a large oscillation is seen in the result of LS.

The results of NNLS decomposition are shown in Fig. 3. The 16 images with different b-values were decomposed into 16 for each D value. In other words, 16 numbers representing the amount for 16 D values were calculated (D spectrum). These images were further grouped into 3 groups, Dhigh, Dmid, and Dlow, by multiplying with weighting functions as shown in Fig. 2 (weighting function & Dhigh/mid/low). Dlow showed higher values in regions with restricted diffusion such as the hepatic and splenic parenchyma. Dmid showed higher values in components with extracellular fluid or high viscosity, such as renal parenchyma and bile. Dhigh showed higher values in components suggesting vascular flow and cerebrospinal fluid. These findings indicated that the maps obtained were reasonable. Furthermore, in the color maps combining the 3 components, the respective contributions of Dlow, Dmid, and Dhigh were clearly visualized (Fig. 4).

Fig. 3.

Multi-b image and NNLS results. The multiple acquired images are shown in (a). The NNLS results are shown in (b). Images were obtained with 16 values of b (b = 0, 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, roughly equidistant spacing), and NNLS was applied for 16 D values (0.10, 0.15, 0.23, 0.35, 0.54, 0.79, 1.20, 1.82, 2,75, 4.16, 6.30, 9.53, 14.4, 21.8, 33.0, 50.0, exponential spacing).D, diffusion coefficient; NNLS, non-negative least squares.

Fig. 4.

Component images and color map. The resulting component images (Dlow, Dmid, and Dhigh) and a color map are shown. Colors assigned to each image are: Dlow: blue, Dmid: green, Dhigh: red (same as Fig. 2b).D, diffusion coefficient.

Discussion

Previously, estimating the D distribution within each voxel was considered difficult.⁴ However, it is possible to use the NNLS approach, which has been applied to body diffusion MR data. Since the primary goal of medical imaging is to acquire information about the target tissue/lesions, this approach of estimating the D distribution seems reasonable. It is known that certain changes in the tissue are reflected in changes in the diffusion characteristics. For example, small cells in the tumor tissue make apparent diffusion coefficient smaller due to restricted diffusion. Also, free water has larger D value, and flowing water has even larger D value. Using NNLS approach, it may become possible to get information on components with different diffusion characteristics in each voxel. Toward this goal, more clinically oriented volunteer study is being planned. However, whether the selection of 16 points for the D values is optimal for NNLS calculation remains an issue to be addressed in future studies.

Conclusion

A method for interpreting diffusion MR images involves decomposing multi-b images into the distribution of D values within each voxel. Although these images are currently further decomposed into 3 components representing different tissue characteristics, additional information within the D spectrum images is not fully represented in these 3 components. New approaches for interpreting these images might be developed in the future.