Mean apparent propagator (MAP) MRI: A novel diffusion imaging method for mapping tissue microstructure
Introduction
Non-invasive magnetic resonance imaging (MRI) has become paramount to the diagnosis and clinical management of many diseases of the central nervous system (CNS). MRI characterization of tissue water behavior also has contributed significantly to our fundamental understanding of CNS tissue microstructure. Diffusion-weighted MRI (DW-MRI), which is sensitized to the random motion of endogenous water molecules within the tissue environment, has proven particularly important to both clinical and basic science applications.
Conventional DW-MR utilizes two magnetic field gradients (Stejskal and Tanner, 1965) of equal magnitude and direction applied around the 180° radiofrequency (RF) pulse in a spin echo MR sequence. The magnetic moment of a hydrogen nucleus suffers a net phase shift if its locations during the application of the first and second gradients are different. A population of randomly moving nuclei exhibits phase incoherence, which leads to an attenuation in the overall signal. Mathematically, the signal is related to an important quantity, referred to as the propagator, through a Fourier relationship whose inversion yields the expression (Callaghan, 1991, Stejskal, 1965)where P(r) denotes the propagator indicating the likelihood for particles to undergo a net displacement r. The reciprocal space vector q = (2π)− 1γδG is an experimentally controlled parameter, where γ is the gyromagnetic ratio, and δ is the duration of the diffusion sensitizing gradients whose magnitude and orientation are determined by G. The normalized signal E(q) is just the ratio of the signal at q to its value at q = 0.
From a physical point of view, a propagator can be assigned to every location in space and within each voxel. If Pl(R, R + r) denotes the local propagator for a spin whose initial and final locations are R and R + r, respectively, the MR measurable propagator is given by (Callaghan, 1991, Stejskal, 1965)where ρ(R) is the likelihood of finding a spin at location R. Due to the averaging process inherent in the above equation and because the formulation assumes a simple pulse sequence that features a pair of infinitesimally short gradient pulses—conditions often violated to some extent in practice—while neglecting the influence of imaging gradients, we refer to P(r) that is computed through Eq. (1) as the mean apparent propagator (MAP).
From a practical point of view, Eq. (1) suggests that by collecting data at different q-vectors, e.g., by sampling a large Cartesian grid, one can reconstruct the MAP through Eq. (1) using the discrete Fourier transform. This scheme is referred to as q-space imaging (QSI) (Callaghan et al., 1990). The same analysis can be employed for spectroscopy or imaging data, where in the latter case, the transformation is repeated for every voxel of the image, and the results reveal an image of displacement profiles (Callaghan et al., 1988, Wedeen et al., 2005).
Eq. (1) establishes the fundamental relationship between the MR signal and the propagator. Both these functions are strongly influenced by the microscopic environment, which is impossible to resolve through direct MRI due to its limitations in sensitivity. However, if reliable models that link the diffusion process to either of these functions is available, voxel-averaged microscopic descriptors of the medium can be inferred from a collection of MR signals. Such an endeavor typically demands an accurate representation of the signal and/or propagator. For example, parameterizing the small |q| behavior of the signal profile through an oriented (multivariate) Gaussian function has lead to the introduction of diffusion tensor imaging (DTI) (Basser et al., 1994a). Since then numerous methods have been developed to unravel the complex tissue architecture within each voxel (Aganj et al., 2010, Alexander et al., 2002, Dell'Acqua et al., 2010, Frank, 2002, Jian et al., 2007, Kaden et al., 2007, Liu et al., 2003, Özarslan and Mareci, 2003, Özarslan et al., 2006b, Tournier et al., 2004, Tuch, 2004). Those methods that involve analytical representations of the signal were found to be most convenient as they provide compact representations of the signal as well as the estimated quantities and are inherently less susceptible to the effects of noise; for example, the advantage of an analytical representation was recognized (Anderson, 2005, Descoteaux et al., 2007, Hess et al., 2006) in q-ball imaging whose original realization (Tuch, 2004) lacked such a representation.
The above-mentioned techniques have focused almost exclusively on delineating the orientational features of the diffusion process even when there is more than one major fiber orientation within the voxel—a scenario, which DTI does not account for. However, orientational features, like the orientation distribution function (ODF), constitute only a part of the information that could be obtained from diffusion-attenuated signals. Of particular interest are the features that follow from the restricted character of the diffusion process, which are contained in the full displacement distribution and its dependence on the diffusion time. Such features contain information about cell size, shape, and transmembrane exchange, which are extremely important in biomedical applications of MR, and are obtainable from data acquired at large q-values. To infer such microstructural information and reconstruct the full MAP rather than its orientational features available in the ODF, acquisition of data with three-dimensional q-space coverage is beneficial (Callaghan et al., 1988, Wedeen et al., 2005, Wu and Alexander, 2007). Therefore, the development of a robust analytical model of the signal that could be used to describe data acquired over the entire three-dimensional q-space would be highly useful. To this end, several models have been introduced in recent years to represent the three-dimensional q-space signal (Assemlal et al., 2009, Assemlal et al., 2011, Cheng et al., 2010, Descoteaux et al., 2011, Hosseinbor et al., 2011, Özarslan et al., 2006b, Özarslan et al., 2009c, Ozcan, 2010, Ye et al., 2012, Yeh et al., 2011).
In this article, we introduce a new method, referred to as MAP-MRI that subsumes DTI and extends it to generate a true and proper propagator or MAP in each voxel. By quantifying the non-Gaussian character of the diffusion process, this method more accurately characterizes diffusion anisotropy. This technique provides several new quantitative parameters, or MRI “stains,” derived from the entire displacement MAP that captures distinct novel features about nervous tissue microstructure. The technique is based on the idea of expressing the three-dimensional q-space MR signal in terms of the eigenfunctions of the quantum-mechanical simple harmonic oscillator (SHO) Hamiltonian, sometimes called the Hermite functions, which have also appeared in the reconstruction of the propagator from its cumulants (Liu et al., 2003, Liu et al., 2004). Estimation of probability distributions in a series of Hermite functions is well-studied in the statistics literature (Schwartz, 1967) and such expansions were shown to possess powerful properties, such as rapid convergence in both real and Fourier spaces (Walter, 1977) that make them ideally suited to problems of q-space signal analysis and mean propagator estimation. This representation is an extension of its one-dimensional (1D) counterpart (Özarslan et al., 2008a), which was shown to accurately represent the signal decay originating from very different environments (from free to restricted). In fact, the 1D version of the method was shown to estimate important microstructural properties such as the moments of the underlying compartment size distribution in a medium composed of isolated pores (Özarslan et al., 2011), and generating temporal scaling contrast (Özarslan et al., 2012) by employing a disordered media model for DW-MR (Özarslan et al., 2006a). We introduced an earlier version of the three-dimensional (3D) formulation in Özarslan et al. (2009c), which was instrumental in evaluating the robustness of sparse and optimal strategies for multiple-shell q-space MRI acquisitions (Koay et al., 2012). Here, we introduce a more refined, general, and comprehensive approach by incorporating an anisotropic scale parameter into the representation, which increases its adaptability to different diffusion profiles. The resulting representation reproduces DTI in its first term, and generalizes it to account for non-Gaussianity in the measured diffusion process.
Section snippets
One-dimensional SHO based reconstruction and estimation (1D-SHORE)
Before we introduce the formulation for representing three-dimensional q-space acquisitions, we examine a considerably simpler problem that involves q-space data obtained with different q-values while the gradient orientation, which defines the x-axis, is fixed. Such acquisitions have been utilized to address a number of important questions in biomedical research (Cohen and Assaf, 2002, Cory and Garroway, 1990). Although the focus of the paper is the modeling framework for three-dimensional
MAP-MRI for the 3D problem: a new generalization of DTI
Our formulation for 3D data follows very closely from the above treatment. There are two versions of the three-dimensional formulation. In the first approach, referred to as 3D-SHORE, the same scale parameter is used in all directions, intrinsically assuming that the spring constant is isotropic. This version can be formulated in Cartesian or spherical coordinates. The formulation with isotropic stiffness was introduced by Özarslan et al. (2009c), while an updated presentation with several
Scalar indices for 3D q-space imaging (QSI)
In this section, we shall introduce some indices that could be used to quantify various features of the three-dimensional diffusion process. In each subsection, one of the indices is discussed within the context of three-dimensional q-space imaging. Subsequently, an expression relating the index to the MAP-MRI coefficients is provided for easy estimation of the index.
Numerical implementation
The actual implementation of the MAP-MRI framework is relatively straightforward. Our implementation consists of the following steps:
- 1.
The MR data set was first used to fit the equation . It is important to impose a positive-definiteness constraint for the tensor . To this end, a scheme discussed in Koay et al. (2006) is used. This scheme employs the following steps: (i) By taking the logarithm of both sides of the above equation, one obtains a linear relationship. This
Resolution of orientational complexity
In Fig. 2, coronal grayscale PA maps of the excised marmoset brain located 0.7 mm posterior (top row), 1.2 mm anterior (middle row) and 4.9 mm anterior (bottom row) to bregma (Palazzi and Bordier, 2008) are shown with companion MAP-MRI-derived profiles visualized via color glyphs from selected regions. In the top row, there is a diagonal area of crossing between the inferior-located external medullary lamina and the superiorly-located internal capsule within the thalamic reticular nucleus. In
Orientation profiles in the anatomical reference frame
As described in the Theory section, the reconstruction is performed in the reference frame in which the diffusion tensor is diagonal. When the orientational features of the underlying propagator are to be visualized, the reconstructed MAP needs to be transformed back into the reference frame common to all voxels (the image reference frame). This was accomplished for the function by transforming its argument via the expression . This transformation was necessary to provide a correct
Influence and estimation of the tensorial scale parameter
The MAP-MRI framework requires the estimation of not only the coefficients but also the anisotropic scale parameter (hence A) as seen in Eq. (24). A novel aspect of our implementation, which proved to be very robust in practice, involves breaking this otherwise challenging computational problem into two very well-studied problems of (i) positive-definite tensor estimation to obtain , and (ii) convex quadratic programming to estimate the coefficients . Although the
Conclusion
MAP-MRI represents a new comprehensive analytical framework to model the three-dimensional q-space signal and transform it into apparent propagators. The key feature of the approach is the anisotropic spring constant or scale parameter. The anisotropically-scaled basis not only improves the ability of MAP-MRI to adapt to very different signal profiles, but can reduce the technique to the widely-employed DTI method if only the first of the basis functions is employed. Consequently, the MAP-MRI
Acknowledgments
The authors gratefully acknowledge Drs. John D. Newman and Afonso C. Silva for the marmoset brain specimen and Liz Salak for editing the manuscript. Special thanks to Jian Cheng for pointing out an error in (Özarslan et al., 2009c). Support for this work included funding from: (i) the Intramural Research Program of the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD), the National Institutes of Health (NIH), and (ii) the Department of Defense in the Center
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