Studies of pediatric white matter development using diffusion-weighted MRI have generally focused on the examination of anisotropy measures within specific brain regions or on the analysis of connectivity measures derived from tractography. It is unclear, however, to what degree such methods are influenced by the specific diffusion models or tractography algorithms employed. In this work, we propose a new technique for quantifying global white matter development using the raw diffusion-weighted images directly. In particular, we establish a relationship between subject age and the efficiency with which a multidirectional diffusion-weighted data set can be sparsely represented using a learned dictionary.

We queried our institutional database for all outpatient brain MR studies performed over the 24 month period ending in September 2015 that included multidirectional diffusion-weighted imaging. Only studies performed on individuals younger than 21 years of age and reported as normal were further evaluated. Diffusion-weighted images were acquired at 1.5 T with b = 1200 s/mm² over 30 gradient directions. Images were corrected for head motion and eddy current effects, and fractional anisotropy maps were computed¹ and thresholded (FA > 0.2) to produce a white matter mask.

We decomposed each diffusion-weighted series using K-SVD², a coding method which simultaneously seeks both an optimal dictionary $$$\mathbf{D}$$$ and sparse coefficients $$$\mathbf{X}$$$ that approximate the observed data $$$\mathbf{Y}$$$ subject to a sparsity constraint $$$T_0$$$ on each coefficient vector $$$\mathbf{x}_p$$$:$$\DeclareMathOperator{\argmin}{argmin}\underset{\mathbf{D},\mathbf{X}}{\argmin} \lVert\mathbf{Y} - \mathbf{DX}\rVert^2_\mathrm{F}\quad\text{s.t.}\quad\forall p,\,\lVert\mathbf{x}_p\rVert_0\leq T_0\qquad\left(1\right)$$We summarize the K-SVD algorithm in Figure 1 and refer the reader to previous descriptions^3,4 for implementation details. A fixed dictionary size, $$$K=200$$$, and sparsity threshold, $$$T_0=10$$$, were used for this study.

We quantified the sparsity of the encoding by computing the Gini coefficient⁵, described by Equation 2, for each coding vector:$$\DeclareMathOperator{\gini}{Gini Coefficient}\gini\left(\mathbf{x}\right)=1-2\sum_{k=1}^K\frac{\mathbf{x}\left[k\right]}{\lVert\mathbf{x}\rVert_1}\left(\frac{K-k+\frac{1}{2}}{K}\right)\qquad\left(2\right)$$The Gini coefficient takes values over the interval $$$\left[0,1\right]$$$, with unity representing a coding vector with a single nonzero element. We selected this metric as it has an intuitive interpretation and several important advantages⁶ over the $$$\ell^0$$$ and $$$\ell^1$$$ norms, including invariance to both constant offset and scaling.

Finally, we postulated the model of Equation 3 to describe the distribution of mean white matter Gini coefficients across age:$$\text{Gini Coefficient}\sim\beta_1+\left(1-\beta_1\right)\left(1-e^{-\text{Age}/\beta_2}\right)\qquad\left(3\right)$$This hypothesis captures the mathematical bounds on the Gini coefficient as well as the physiologically expected asymptotic increase in fiber bundle organization. Model parameters were estimated using a nonlinear least-squares method⁷.

We identified 21 examinations satisfying the inclusion criteria. To illustrate the qualitative effects of K-SVD encoding, a representative diffusion-weighted volume from one subject is depicted in Figure 2 both in its original form and in its sparse representation. There is a clear reduction in image noise as a result of the sparse encoding. The mean white matter Gini coefficient for each subject’s sparse coded result is plotted in Figure 3 (open circles) as a function of age. A positive correlation is evident between the subject age and the sparsity of the encoded representation. The regression model of Equation 3 corresponding to these points is also shown in Figure 3 (dashed curve) with the estimated parameter values and summary statistics tabulated in Figure 4.The results reveal several important properties of the proposed technique. First, we have qualitatively verified prior reports⁴ indicating that sparse coding with a learned dictionary enables effective diffusion-weighted MRI denoising. In addition, we have shown that the K-SVD approach to this encoding produces a very efficient representation—the Gini coefficients in this study are all near unity. This efficiency is partially by design; we note that the minimum possible Gini coefficient given the chosen values for $$$K$$$ and $$$T_0$$$ is 0.95. Finally, by demonstrating that a model of the form described by Equation 3 fits our observed data, we have confirmed our hypothesis that the Gini coefficient increases throughout pediatric brain development, suggesting that more organized white matter is more efficiently encoded. Though further investigation will be needed, we expect the need for more complex (likely non-monotonic) models to describe the expected Gini coefficient through middle age and senescence.K-SVD enables the efficient decomposition of a diffusion-weighted MR data set into a sparse linear combination of elements from a trained dictionary. The Gini coefficient is a robust measure for quantifying the sparsity of the resulting encoding, and we have shown that it increases within the pediatric white matter as a predictable function of age—a relationship hypothesized to reflect the underlying increase in myelination and fiber bundle organization. This technique provides a means for studying global white matter development using the diffusion-weighted images directly and avoids the potential biases and assumptions inherent in diffusion model fitting and tractography.