Diffusion tensor imaging: Structural adaptive smoothing

Abstract

Diffusion Tensor Imaging (DTI) data is characterized by a high noise level. Thus, estimation errors of quantities like anisotropy indices or the main diffusion direction used for fiber tracking are relatively large and may significantly confound the accuracy of DTI in clinical or neuroscience applications. Besides pulse sequence optimization, noise reduction by smoothing the data can be pursued as a complementary approach to increase the accuracy of DTI. Here, we suggest an anisotropic structural adaptive smoothing procedure, which is based on the Propagation–Separation method and preserves the structures seen in DTI and their different sizes and shapes. It is applied to artificial phantom data and a brain scan. We show that this method significantly improves the quality of the estimate of the diffusion tensor, by means of both bias and variance reduction, and hence enables one either to reduce the number of scans or to enhance the input for subsequent analysis such as fiber tracking.

Introduction

Since the early times of nuclear magnetic resonance, it has been known that this phenomenon is sensitive to, and thus can be used to measure, diffusion of molecules in complex systems (Carr and Purcell, 1954). The basic principles of magnetic resonance diffusion weighted imaging (DWI) were introduced in the 1980s (LeBihan and Breton, 1985, Merboldt et al., 1985, Taylor and Bushell, 1985). Since then, DWI has evolved into a versatile tool for in-vivo examination of tissues in the human brain and spinal cord, leading to a plethora of clinical and neuroscience applications. The broad interest in this technique grows from the fact that DWI probes microscopic structures well beyond typical image resolutions through water molecule displacement, which can be used in particular to characterize the integrity of neuronal tissue in the central nervous system.

Diffusion in neuronal tissue is usually not isotropic but depends on the particular microscopic structure of the tissue. Different diffusion directions can be probed by application of corresponding bipolar magnetic field diffusion gradients (Stejskal and Tanner, 1965). In diffusion tensor imaging (DTI) (Basser et al., 1994a, Basser et al., 1994b), this direction dependence is utilized to reveal information not only about local diffusivity but also local diffusion anisotropy, and thus, fiber structure. The information contained in a voxel of a diffusion weighted image consists of the integral of the microscopic diffusion properties over the voxel volume. In DTI, this information is reduced to a three dimensional Gaussian distribution model for diffusion. Within this model, diffusion is completely characterized by the diffusion tensor, a symmetric positive definite 3 × 3 matrix with six independent components. This model describes diffusion completely if the microscopic diffusion properties within a voxel are homogeneous. In the presence of partial voluming effects, like crossing fibers, the Gaussian model is only an approximation. For these cases, more sophisticated models exist, which include higher order tensors or describe non-Gaussian diffusion distributions, as used in high angular resolution diffusion imaging (Tuch et al., 1999, Frank, 2001). In this paper, we restrict ourselves to the Gaussian diffusion tensor model for anisotropic diffusion, as used in DTI.

There are several clinical and basic neuroscience applications of DTI in the human brain. For reviews see Sundgren et al. (2004) and Mori and Zhang (2006), respectively. Diffusion in brain white matter is highly anisotropic due to its organization into fibers of neuronal axons. DTI is able to reveal information on this fiber structure and on connections of the different regions in the brain (Melhem et al., 2002). Fiber tracking has become an important field of interest, since it is possible to gain insight into the connections in the brain in vivo with applications such as surgical planning. In clinical or developmental studies, diffusion or anisotropy indices are usually compared by normalizing the brain volumes to a template before application of voxel-wise comparison tests (Schwartzman et al., 2005, Smith et al., 2006). In these studies, the accuracy of diffusion and anisotropy based measures is of crucial importance, since their variability directly determines the sensitivity to detect changes over time or differences between subjects. DTI has been used in longitudinal studies and comparison of patient groups, for example in cognitive studies (Liston et al., 2006, Khong et al., 2006), developmental studies (McKinstry et al., 2002, Schneider et al., 2004, Bengtsson et al., 2005, Mukherjee and McKinstry, 2006), assessment of brain injury (Liu et al., 1999, Nakayama et al., 2006) and cancer treatment (Leung et al., 2004), or genetic studies (Carbon et al., 2004, O'Sullivan et al., 2005). All these studies have in common that subtle changes in diffusion or anisotropy in white matter brain areas were used to indicate neuronal changes or differences.

DTI suffers from significant noise which may render subsequent analysis or medical decisions more difficult. This is especially important in low signal-to-noise applications, such as high-resolution DTI or DTI with high b-values (Clark and LeBihan, 2000; Yoshiura et al., 2003; Jones and Basser, 2004). It has been shown that noise may induce a systematically biased assessment of features. For example, a well-known phenomenon is the biased estimation of anisotropy indices in the presence of noise (Basser and Pajevic, 2000, Hahn et al., 2006). At high noise levels, in addition to the common random errors, the order of the diffusion eigenvectors is subject to a sorting bias. Noise reduction is therefore essential. Several approaches have been proposed for smoothing diffusion tensor data. They include common methods such as Gaussian smoothing (Westin et al., 1999), anisotropic kernel estimates (Lee et al., 2005), and methods based on non-linear diffusion (Perona and Malik, 1990, Weickert, 1998, Parker et al., 2000, Ding et al., 2005) or splines (Heim et al., 2007).

Smoothing of tensor data requires choosing a Riemannian (Fletcher, 2004, Pennec et al., 2006, Zhang and Hancock, 2006a, Zhang and Hancock, 2006b, Fletcher and Joshi, 2007) or log-Euclidian metric in the tensor space (Arsigny et al., 2006, Fillard et al., 2007). We see some conceptional advantages in smoothing the diffusion weighted images instead of the tensor estimates. Estimating the tensor from noisy data leads, with a certain probability, to results outside the tensor space. This requires some kind of regularization. Reducing the noise level instead in the diffusion weighted images allows for a reduction of this probability in case of an underlying non-degenerate tensor. In case of high noise level in the diffusion weighted images, the non-linearity of Eq. (1) leads to a bias in the tensor estimate. This bias can be reduced by smoothing the diffusion weighted images, but is not addressed if smoothing is performed in the tensor space itself. A correction for Rician bias (Gudbjartsson and Patz, 1995, Basu et al., 2006) can also be incorporated.

More specifically, we propose an alternative smoothing method based on the Propagation–Separation (PS) approach (Polzehl and Spokoiny, 2006). By naturally adapting to the structures of interest at different scales, the algorithm avoids loss of information on size and shape of structures, which is typically observed when using non-adaptive filters. More specific, by virtue of inspecting scale space in an iterative way, PS accumulates information on the spatial structure at small scales and uses this information to improve estimates at coarser scales. Like any other filter, the proposed method introduces a bias. However, this bias is controlled by the amount of achieved variance reduction (Polzehl and Spokoiny, 2006).

This article is organized as follows: In the Diffusion Tensor Imaging section we briefly review the basic notation of DTI and smoothing methods. The structural adaptive smoothing of DTI data section is dedicated to our structural adaptive smoothing algorithm based on the PS approach. We apply the method to artificial as well as experimental data in the Examples section. Summary and conclusions summarizes and discusses our results.