Technical NoteAn objective method for regularization of fiber orientation distributions derived from diffusion-weighted MRI
Introduction
MRI can be used to map anisotropic diffusion of water in vivo (Le Bihan and van Zijl, 2002), providing opportunities for noninvasive mapping of fiber tracts in the brain (Lori et al., 2002, Mori and van Zijl, 2002). The tensor model of diffusion (Basser et al., 1994) has been the basis for most tracking studies. However, limitations of the tensor model, particularly in the case of partial volume of crossing fibers (Alexander et al., 2001), have motivated the development of a variety of more sophisticated approaches. Multiple-compartment fits (Tuch et al., 2002, Assaf et al., 2004) are a conceptually straightforward extension of the tensor model, but can be unstable for cases with more than two fibers (Tuch et al., 2002).
A number of model-independent approaches can describe diffusion in complex tissue. Diffusion spectrum imaging (Wedeen et al., 2005) can directly measure the water displacement probability distribution function (PDF) for all directions and displacement magnitudes, but is limited by prohibitively long scan times required for adequate q-space sampling. The functional form of the PDF can be integrated in the radial direction, leaving a function that describes the orientation dependence of diffusion. The integral has been approximated in q-ball imaging by the Funk-Radon transform (Tuch, 2004), or explicitly in the Diffusion Orientation Transform (Ozarslan et al., 2006). The orientation dependence can be estimated by sampling a subset of q-space that is small enough to be measured in a clinical setting.
Although the orientation dependence of diffusion can be used to indicate the direction of fibers, diffusion profiles and fiber directions are not necessarily the same. Model-dependent approaches such as persistent angular structure (Jansons and Alexander, 2003) and spherical deconvolution methods (Tournier et al., 2004, Anderson, 2005) attempt to describe the underlying arrangement of fiber tracts. It has been shown that persistent angular structure and spherical deconvolution are formally equivalent but can produce different results, depending on the details of the implementation (Alexander, 2005a). Linear methods (Tournier et al., 2004, Anderson, 2005) derive fiber orientation distribution functions (FODF) rapidly, but with high susceptibility to noise. Maximum entropy-based methods provide high resolution of fiber orientations, but require computationally costly nonlinear optimization (Alexander, 2005b, Alexander, 2005c).
Efforts at reliable evaluation of the FODF in the presence of noise have included low-pass filtering (Tournier et al., 2004), minimum entropy (Tournier et al., 2005), regularization by minimizing the intensity of negative peaks (Tournier et al., 2006), and the aforementioned maximum entropy based methods. The low-pass filter was optimized by simulations, therefore requiring significant user interaction. Entropy methods require nonlinear optimization, for which it can be hard to guarantee attainment of a global optimum in a reasonable amount of time. The negative peak minimization approach improves resolution with little cost, but is used in conjunction with a low-pass filter. In this paper, we examine the use of linear regularization as a practical method to derive the FODF. The method examined is fast enough to be optimized on a voxel-by-voxel basis in a reasonable amount of time. Furthermore, the methods are optimized with no user interaction and can be implemented with readily available software (Hansen, 1994). Through simulations and experiment, we compare the performance of linear regularization with an ad-hoc low-pass filter (AHLPF) previously suggested for spherical deconvolution (Tournier et al., 2004) and with minimum entropy (Tournier et al., 2005). We find that linear regularization provides more robust FODF estimation in regions of low anisotropy and low signal-to-noise ratio (SNR), while achieving similar results in the case of crossing fibers. Linear regularization with objective optimization is a general method that can be applied to the model-independent methods, as well.
Section snippets
Theory
Following the formalism of Tournier et al. (2004), one can determine the FODF by minimizing the chi-squared function:where is a matrix of spherical harmonic functions, is the matrix of coefficients of the rotational harmonic decomposition of the single-fiber response function, are coefficients of the spherical harmonic decomposition of the FODF and is the diffusion-weighted signal. Higher-order spherical harmonics are necessary to resolve fibers that are separated by
Results
Fig. 1 demonstrates the effects of the magnitude of regularization parameter on the FODF. The GCV function for a simulated FODF (θsep = 60°, partial volume fraction = 0.5, SNR0 = 30) is shown for a range of λ on a log–log scale. FODF surfaces and corresponding highlighted points on the GCV curve represent, from left to right, under-regularized, optimally regularized, and over-regularized cases. As the regularization parameter increases, one sees fewer spurious spikes, but at the cost of lower angular
Discussion
We have demonstrated the application of a standard regularization method to the calculation of fiber orientation distribution functions (FODFs). The method is linear, and hence can be applied rapidly on a voxel-by-voxel basis. Optimization is also performed automatically, based on the characteristics of the signal, and hence requires no user interaction. Due to the variability of noise from voxel to voxel in an image of a biological system, such optimization may improve the reliability of
Conclusions
Efforts to map white matter fiber tracks in regions of the brain with crossing fibers may well benefit from methods such as spherical deconvolution. As with most deconvolution methods, spherical deconvolution can be highly susceptible to noise. Standard linear regularization methods and algorithms for objective choice of regularization parameter improve the stability of the derivation of the fiber orientation distribution function. The approach examined here requires no user interaction and is
Acknowledgments
We gratefully thank the National Multiple Sclerosis Society for providing funding for this project. We also thank Jian Lin for developing and implementing the motion correction algorithm and Derrek Tew and Devyani Bedekar for performing the MR scans.
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