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This work details the observation of non-Gaussian apparent diffusion coefficient (ADC) profiles in multi-direction, diffusion-weighted MR data acquired with easily achievable imaging parameters (b ≈ 1000 s/mm2). A technique is described for modeling the profile of the ADC over the sphere, which can capture non-Gaussian effects that can occur at, for example, intersections of different tissue types or white matter fiber tracts. When these effects are significant, the common diffusion tensor model is inappropriate, since it is based on the assumption of a simple underlying diffusion process, which can be described by a Gaussian probability density function. A sequence of models of increasing complexity is obtained by truncating the spherical harmonic (SH) expansion of the ADC measurements at several orders. Further, a method is described for selection of the most appropriate of these models, in order to describe the data adequately but without overfitting. The combined procedure is used to classify the profile at each voxel as isotropic, anisotropic Gaussian, or non-Gaussian, each with reference to the underlying probability density function of displacement of water molecules. We use it to show that non-Gaussian profiles arise consistently in various regions of the human brain where complex tissue structure is known to exist, and can be observed in data typical of clinical scanners. The performance of the procedure developed is characterized using synthetic data in order to demonstrate that the observed effects are genuine. This characterization validates the use of our method as an indicator of pathology that affects tissue structure, which will tend to reduce the complexity of the selected model. Magn Reson Med 48:331–340, 2002. © 2002 Wiley-Liss, Inc.
Diffusion imaging, particularly diffusion tensor magnetic resonance imaging (DT-MRI) (1) has become popular because of the insight it provides into the structural connectivity of tissue (2, 3). Water is a large constituent of biological tissue, and water molecules in tissue constantly undergo random, Brownian motion. Tissue also contains rigid structures, such as the walls of cells, that form barriers to diffusion, and it is this hindrance to diffusion that allows tissue structure to be probed through measurements of water mobility due to diffusion processes. In some types of tissue, such as most gray matter in the brain, the structure has no preferred orientation and so causes approximately the same amount of hindrance to diffusion in all directions. The amount of diffusion or water mobility is thus approximately equal in all directions, i.e., isotropic. Other types of tissue, however, have ordered structure that hinders diffusion to different degrees in different directions, causing diffusion anisotropy. White matter in the brain, for example, consists of bundles of axon fibers, and water is free to diffuse along the axis of the fibers but is hindered in the perpendicular directions.
The diffusion of water molecules in tissue over some time interval, t, can be described by a probability density function, pt, on the displacement, x, of water molecules after time t. pt reflects the underlying tissue microstructure, because it is largest in the directions of least hindrance to diffusion and smaller in other directions. In white matter, for example, pt is largest in directions parallel to fibers, but is small in perpendicular directions and thus reveals fiber orientations. The goal of diffusion imaging is to obtain information about pt that leads to meaningful inferences about the microstructure of the material being imaged.
pt can be shown to relate to the NMR signal attenuation, S(q), measured through a pulsed gradient spin-echo experiment, via a Fourier transform (FT) with respect to (4, 5):
The spin-echo attenuation, S(q), is defined as the normalized diffusion-weighted signal, s(q)/s(0), where s(q) is the NMR signal in the presence of a diffusion-weighting gradient of magnitude G and direction k̂, and s(0) is the signal in the absence of any such gradient. δ is the length of the gradient pulse, and γ is the magneto-gyric ratio of protons in water. We note that Eq.  relies on the fact that δ is negligibly small compared to t. This assumption is rarely justified in practice, but the effect of non-negligible δ is merely to introduce a convolution over a range of diffusion times (6) into the measurements, which generally preserves the large-scale structure and orientation of the inferred pt.
Given enough measurements of S(q) spread over a suitable range of q, the FT can be inverted to obtain an estimate of pt. A more common approach, however, is to assume a simple model for pt the FT of which can be related directly to the spin-echo attenuation, which allows pt to be inferred from a much sparser set of measurements. The simplest model for pt that incorporates second-order statistics (anisotropy) is a multivariate, zero-mean Gaussian, which has covariance 2Dt at time t:
This is the distribution of an initial point concentration at x = 0, t = 0, diffusing according to the simple anisotropic diffusion equation (7):
The FT of pt is then also Gaussian, which gives rise to a simple relationship between the parameters of pt (the elements of D) and the spin-echo attenuation:
In Eq. , b is the diffusion-weighting factor given by b = t|q|2.
The simplest form of diffusion-weighted (DW) MRI (8) models pt with a Gaussian in one dimension. A single spin-echo attenuation measurement allows the single parameter of pt—its scalar variance—to be estimated from Eq. :
The diffusion coefficient, d(k̂), is proportional to the variance of the Gaussian model, which describes the root mean squared displacement of water molecules in the direction of the applied DW gradient k̂. In DW-MRI, the measure of d(k̂) obtained from Eq.  is often called the apparent diffusion coefficient (ADC) (1), both because it represents a spatial average of the diffusion coefficient over an image voxel and because it is based on this Gaussian assumption, which may be unjustified.
DT-MRI extends this basic idea to 3D, where the diffusion coefficient is described by a diffusion tensor (DT), D, which is proportional to the covariance matrix of the trivariate Gaussian, as in Eq. . D relates to the 1D diffusion coefficient d(k̂) in any chosen direction k̂ as follows:
In 3D, D is a symmetric 3 × 3 matrix and thus has six free parameters. A minimum of six estimates of d(k̂) in independent directions is thus required to estimate D, which requires six measurements of the spin-echo attenuation (seven MR measurements in total) to be made with the DW gradient applied in these independent directions. The estimate of D obtained from such a set of measurements is referred to as the apparent diffusion tensor (ADT) (1), as in the case of the ADC.
We define the “ADC profile” to be the estimate of d(k̂) over the range of k̂, which is the unit sphere. When pt is Gaussian, the ADC profile is described by Eq. , and a standard approach (9) to the estimation of D is to acquire DW images in a large number of directions (many more than six) spread evenly over the unit hemisphere. This oversampling of the ADC profile provides a more robust estimate of D. Note that the antipodal symmetry of pt and hence d(k̂) is assumed, so that d(k̂) = d(−k̂) and only half of the sphere needs to be sampled. When pt is not Gaussian the ADC profile deviates from that described by Eq. , and this kind of multiple gradient direction scheme affords the possibility of observing significant deviations should they arise.
It has long been recognized (1, 10–14) that the Gaussian, DT model is inappropriate when complex tissue structure is found within a single image voxel. There are alternative models for pt that can capture certain non-Gaussian effects that occur in these circumstances. A simple alternative is the multi-Gaussian model (10, 11). This model is based on the assumption that a voxel contains n separate compartments, each containing a different tissue type in proportion ai (Σi ai = 1, i = 1, … n) and that the diffusion within each compartment can be described by a Gaussian pt with DT, Di. The model assumes further that there is no exchange of molecules between these separate compartments. pt then becomes a weighted sum of Gaussians and Eq.  becomes:
With this model for pt, the ADC profile can have shape very different from that described by Eq. , which is often modeled poorly by a single DT (10). Figure 1 shows ADC profiles simulated from prolate, oblate, and isotropic Gaussian pt's, together with profiles obtained from bi-Gaussian pt's that combine them. Note the characteristic peanut shape of the prolate Gaussian ADC profiles and the “filled bagel” or red blood cell shape of the oblate Gaussian profile, which are typical of Gaussian functions plotted over a sphere. The contours of the corresponding Gaussian functions in 3D have the more familiar ellipsoidal contours: cigar-shaped in the prolate case, and Frisbee-shaped in the oblate case.
Figure 1. Simulated examples of ADC profiles. Top row: profiles corresponding to Gaussian diffusion processes: (i) prolate DT oriented along the x-axis (eigenvalues [1700, 200, 200] ×10−6 mm2/s), (ii) prolate DT oriented along the z-axis (eigenvalues [200, 200, 1700] ×10−6 mm2/s), (iii) oblate DT (eigenvalues [950, 950, 200] ×10−6 mm2/s), and (iv) isotropic DT (eigenvalues [700, 700, 700] ×10−6 mm2/s). Bottom row: ADC profiles corresponding to the bi-Gaussian model combining pairs of DTs from the top row in equal proportion with b set to 1000 s/mm2; (v) combines (i) and (ii), (vi) combines (i) and (iii), (vii) combines (ii) and (iii), and (viii) combines (i) and (iv).
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Alexander et al. (10) analyzed the behavior of the ADC profile within voxels containing multiple tissue compartments. They showed how the observability of higher-order profiles increases (their non-Gaussian shape becomes more pronounced) with the size of the diffusion-weighting factor, b. They used the instability of the DT and its derived scalar measures, such as the mean diffusivity and fractional anisotropy, to locate regions in the brain where the DT description of the ADC profile is poor. Several regions in data acquired with b = 3000 s/mm2 were highlighted by this approach, including the pons, corpus callosum, cingulum, internal capsule, and arcuate fasciculus.
Frank (11) showed that a 4th-order SH series provides a first approximation to the ADC profile obtained from a multi-Gaussian pt. Frank fitted a 4th-order SH series to ADC measurements acquired with b = 3000 s/mm2 and showed that significant 4th-order (i.e, non-Gaussian) components arise in locations of the human brain similar to those highlighted in Ref. 10 (see above).
In other related work, Wedeen and Tuch et al. (12, 13) used q-space techniques (4) to measure pt. This technique exploits the Fourier relationship between pt and the spin-echo attenuation directly by acquiring a large number of measurements of S(q) over a wide range of q in order to obtain enough samples of the FT of pt to perform a stable inversion. Distinctly non-Gaussian pt's have been observed in both the human brain and heart, particularly at locations where anisotropic fibers cross within a single voxel.
In this work, we investigate the observability of non-Gaussian ADC profiles in DW data acquired using acquisition parameters more typical of those used clinically. Our sequence consists of a multiple gradient direction scheme based on the work of Jones et al. (9) using a 1.5T scanner, with a maximum b of approximately 1000 s/mm2. We use the spherical harmonic (SH) series to provide a hierarchy of models for the ADC profile. In the Methods section, an efficient and robust method for fitting the SH series to DW-MRI data is described together with a method, based on the analysis of variance (ANOVA) test for addition/deletion of variables, for selecting the most appropriate level of truncation of the series. The combined fitting and model selection procedures developed are used to classify the profile in each voxel as arising from isotropic, anisotropic Gaussian, or non-Gaussian pt, and thus to produce maps of where these different types of behavior occur. Non-Gaussian behavior is most likely to arise and be observed in regions of high tissue complexity containing distributions of fiber orientations with multiple peaks, such as fiber intersections. The maps generated from the model selection procedure provide extra diagnostic information in pathologies involving neuronal loss, degeneration, or demyelination, since non-Gaussian behavior will tend to disappear in the affected areas because the complexity of the tissue structure is reduced. These maps can also be used to highlight regions in which the ADT and its derived indices are unreliable. We apply the method to both in vivo human brain data and to synthetic data for performance evaluation and validation.
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We have described methods for modeling and detection of non-Gaussian ADC profiles and shown that such profiles can be observed with scanning parameters typical of standard clinical DW-MRI data. The SH series up to order 8 was fit to samples of the ADC profile in each voxel, which provides a sequence of models of increasing complexity. A series of ANOVA tests was used to find the simplest of these models that adequately describes the data. This latter procedure classifies the profile at each voxel as isotropic, anisotropic Gaussian, or non-Gaussian, and allows maps of these different types of behavior to be produced, as shown in Fig. 4.
Our procedure was applied to human brain data collected with parameters typical of those used in clinical scans, and appeared to classify isotropic (GM) and anisotropic (WM) regions correctly as order 0 and order 2, respectively. On average, 5% of profiles in voxels within the brain were classified as order 4 or above (non-Gaussian). Several regions—in particular, the pons, optic radiation, and corona radiation—were found consistently to contain dense clusters of order 4 models. Validation of our method was performed by characterizing its performance using synthetic data with realistic noise properties, as well as applying it to DT models of data in regions of the brain that were consistently classified as non-Gaussian. Although results from only one data set are shown here, our method was applied to four data sets and other results can be found in Ref. 23. Acquisition of a larger ensemble of data sets is currently underway, which will enable parametric mapping to be performed in order to allow comparisons to be made between the occurrence of non-Gaussian diffusion in different population groups.
The behavior maps produced by our method provide new insights into the complexity of tissue structure in the brain. These maps have a number of practical applications. As mentioned in the Introduction, one aim is to use these maps as a stain for diagnosis of structure-reducing pathology. Furthermore, these maps can be used in postprocessing algorithms, such as tractography algorithms, which need to identify when diffusion is anisotropic and when the principal direction of the DT can be relied upon to describe the orientation of the underlying tissue. Finally, these maps indicate when a more complex model (for example, a bi-Gaussian model) than the DT needs to be used to describe pt adequately. Typically, such models are more difficult to fit to data than the DT and nonlinear fitting algorithms must be employed (see for example Ref. 21). Such procedures are computationally expensive, so it is advantageous to be able to identify only when they need to be performed.
It seems likely that most of the non-Gaussian behavior we observe is due to the intersection of WM fibers with different orientation. This kind of tissue structure gives rise to profiles similar to those that are derived from multi-Gaussian pt's (22), although it is also likely that some exchange of particles between the tissue compartments corresponding to each fiber occurs in the timescale of the diffusion measurement, which will cause pt to deviate from the multi-Gaussian (22). Other types of non-Gaussian processes almost certainly occur in the brain, caused by restriction due to impermeable barriers (22). However, the deviation from Gaussian that is caused by these effects is less marked than those due to multicompartmentation, so such behavior may not be observed reliably—particularly at low b-values such as those used here.
The advantages of the SHs as models for ADC profiles lie both in their generality and in the simplicity and robustness of the fitting procedure. Linearity of the fitting procedure is a significant advantage in terms of computational effort, but also ensures that the fitting procedure is well posed and is not prone to spurious erroneous results. There is little physical justification for the use of the SHs in this context, and there may be more appropriate sets of basis functions that better reflect the kind of ADC profiles that are likely to arise given the underlying physical processes. An advantage of this series, however, is its generality: any profile can be represented, and we do not limit ourselves to particular models of the underlying processes. We note for clarity that when pt is non-Gaussian the shape of the ADC profile does not relate to pt in a straightforward way, and thus the SH shape does not provide any direct information about the underlying tissue structure. Although it is possible to extract some information of this type from non-Gaussian profiles on the sphere (21), this issue is beyond the scope of this work.
There are many techniques for model selection in the literature. The use of ANOVA and the F-test for deletion of variables has proved more successful than most in our application, but there may be others that improve performance to some extent. The test we used is based on an assumption of Gaussian errors in the measurements. Although this is a reasonable first approximation for our data (23), the full analytic form of the errors in ADC measurements is not Gaussian. There are tests that incorporate noise models for the data, which may improve classification performance, particularly in extreme cases such as very anisotropic diffusion. We note that the same technique could be used to produce a finer classification of diffusion profiles; for example, we could distinguish between axisymmetric (two eigenvalues equal) and nonsymmetric (all eigenvalues unequal), anisotropic Gaussian diffusion.
In the present study we used only data acquired with common clinical imaging parameter settings, and one of our goals was to demonstrate that significant non-Gaussian behavior can be observed at b-values as low as 1000 s/mm2. Recently there has been a trend to move toward higher b-values, which can produce profiles richer in information; in particular, non-Gaussian behavior becomes more pronounced (10). Our methods are equally applicable to data acquired with higher b-values, and we would expect to observe a higher proportion of non-Gaussian profiles in such data, which may highlight other regions of the brain in which non-Gaussian behavior occurs.