A Continuous Tensor Field Approximation for DT-MRI Data - ismrm

A Continuous Tensor Field Approximation for DT-MRI Data Sinisa PAJEVIC1, Peter J. BASSER2, Akram ALDROUBI3

1National Institutes of Health, Bldg 12A, Rm 2007, Bethesda, MD USA; 2National Institutes of Health, Building 13, Room 3W16, Bethesda, MD USA; 3Department of Mathematics, Vanderbilt University, Nashville, TN USA; Introduction Here we (a) describe a general mathematical framework to create a continuous approximation of the discretely sampled DT-MRI data, (b) test its fidelity, (c) present new intrinsic MR "stains" depicting tissue structure and architecture, and (d) apply them to in vivo DT-MRI data. We treat DT-MRI data1 as discrete noisy samples of an underlying macroscopic diffusion tensor field, D(x), where, x = (x, y, z) are the spatial coordinates within imaging volume. This field is presumed to be continuous and smooth at a gross anatomical length scale within many soft fibrous tissues, including white matter, muscles, ligaments, etc. This work addresses critical but unmet needs in the analysis and representation of measured DT-MRI data. While the rigorous theoretical underpinnings of this approach are provided elsewhere2, here we apply this continuous tensor field representation to measured DT-MRI data. We demonstrate that the development of the continuous tensor approximation is useful for many new and important applications of DT-MRI in biology and medicine. Methods To construct a continuous approximation to a diffusion tensor field, we require that a set of continuous basis functions (approximants) defining an approximation space possess the following properties: (1) they must be sufficiently rich to represent the diffusion tensor field accurately; (2) certain higher derivatives (at least second) must be continuous (3) the mathematical description of the approximation space is computationally tractable and can be implemented using algorithms that are fast, robust, and accurate. To meet these requirements, we use atomic spaces2, in particular, we choose an atomic space which is a weighted sum of a finite number of tensor field generators, Bi(x/Δ). The scale parameter, Δ controls the mesh size of the grid in each direction. Specifically, Δj is proportional to the ratio of the number of measured data points and the number of unknown parameters in the jth direction. To approximate DT-MRI data by a continuous tensor field, we minimize the discrete energy norm, which is the magnitude of the difference between the tensor data measured at discrete points within the imaging volume and the approximated tensor field evaluated at those same points2. In our implementation, we choose B-spline functions to be the basis for our approximation. They have several advantages: 1) the generators have finite support (i.e., spatial extent), which speeds up and simplifies digital processing algorithms; 2) they can be expressed analytically at any point within the imaging volume; 3) by changing the polynomial order or degree of the B-spline functions, we can control the degree of smoothness and differentiability of our continuous approximation; 4) the derivatives of B-splines can be expressed recursively in terms of the original Bsplines; and 5) B-spline functions naturally generate multi-resolution structures that are useful in analyzing signals and images at different length scales. In our implementation we can choose between interpolation (i.e., fitting all data points exactly) and approximation (i.e., fitting data points approximately). We calculate new MRI stains from the diffusion tensor field, which characterize distinct, intrinsic structural or architectural features of the tissue. The evaluation of some of these requires spatial differentiation of components of the tensor field or of the fiber direction field. Since spatial differentiation amplifies noise, we expect the smooth continuous representation of the diffusion tensor to give more reliable estimates of spatial derivatives of these quantities. As an example, we calculate a new MRI stain which is a scalar function of the spatial rate of change of the tensor field with position, i.e., the gradient of the tensor field, D(x)ij,k, where i and j indicate the tensor component, and k indicates the coordinate direction along which partial derivatives are taken. A scalar that summarizes an intrinsic feature of the tensor field is the scalar contraction of this quantity with itself. Just as the magnitude of the gradient, |grad c(x)|2, detects intensity changes of a scalar field, c(x), the magnitude of the gradient of the tensor field detects changes in "intensity" of a spatially varying 2nd-order tensor field. Since it is a scalar contraction of a third order tensor, it is inherently a rotationally invariant quantity (i.e., independent of the choice of the coordinate system). We also obtain

the magnitude of the gradient of the isotropic portion of the tensor field (orientationally-averaged mean diffusivity at a given point x) and of the (normalized) anisotropic portion fo the diffusion tensor (deviatoric at x). To test the validity of our approach we applied it to a family of simulated continuous diffusion tensor fields that represent various structural or architectural motifs. Results DT-MRI was performed on normal subjects using methods described in (4). Figure 1a shows the gradient of the isotropic part of the diffusion tensor field. Note that the boundaries between regions in which there are differences in Trace(D(x)), i.e., between CSF and parenchyma in the gyri and sulci, and in the ventricles are clearly visible, but otherwise, the image has a relatively uniform intensity. Figure 1b shows an image of the magnitude of the gradient of the anisotropic part of the diffusion tensor field. Note that boundaries between white matter and gray matter are now highlighted, but no signal is seen in CSF or in gray matter per se. Other stains (not shown) can be developed characterizing the features of the curving and twisting of the triad of eigenvectors within the imaging volume, or equivalently, of the three level surfaces that lie perpendicular to each of the three eigenvectors of D(x), at any point.

Magnitude of the gradient of (a) isotropic and (b) anisotropic components of the approximated tensor field Discussion One application of this work is to improve statistical estimates of histological and physiological MRI "stains", including Trace(D(x)), the eigenvalues (principal diffusivities) and eigenvectors (principal directions) of D(x), as well as measures of diffusion anisotropy. Another is to compute intrinsic architectural or microstructural MRI "stains" based upon tissue fiber geometry. None of these quantities could be estimated accurately from measured diffusion tensor data, since their evaluation requires spatial differentiation of noisy tensor date. Our methodology provides more reliable estimates of such quantities but at the expense of lowered resolution. This approach has already been successfully used as the basis of elucidating the trajectory of individual fiber tracts3. Our ability to establish connectivity or continuity of neural pathways will benefit from this development. Finally, there are a number of generic image processing tasks to perform on high dimensional DT-MRI data that this framework facilitates, e.g., filtering noise, detecting edges and boundaries, etc. The strength of this approach is that it directly provides analytical representation of the diffusion tensor field and numerical evaluation at any point in the imaging volume in a very efficient way using Bsplines. References 1. P.J. Basser, et al. Biophys.J. 66(1), 259-67 (1994). 2. A. Aldroubi, et al, Contemporary Mathematics, p. 1-15, AMS, Providence, RI (1999). 3. P.J. Basser, et al. Magn.Reson. Med.44, 625-632, (2000). 4. A. Virta, et al.Magn.Reson. Imaging 17(8), 1121-33 (1999).

Proc. Intl. Soc. Mag. Reson. Med 9 (2001)

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Proc. Intl. Soc. Mag. Reson. Med 9 (2001)

1535