British Journal of Radiology (2004) 77, S176-S185
© 2004 British Institute of Radiology
doi: 10.1259/bjr/81090732
Analysis of MR diffusion weighted images
G J M Parker, BSc, PhD
Imaging Science and Biomedical Engineering, University of Manchester, Oxford Road, Manchester M13 9PT, UK
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Abstract
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Diffusion-weighted MR images provide information that is present in no other imaging modality. Whilst some of this information may be appreciated visually in diffusion weighted images, much of it may be extracted only with the aid of data post-processing. This review summarizes the methods available for interpreting diffusion weighted imaging (DWI) information using the diffusion tensor and other models of the DWI signal. This is followed by an overview of methods that allow the estimation of fibre tract orientation and that provide estimates of the routes and degree of anatomical cerebral white matter connectivity.
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Introduction
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Diffusion-weighted MRI (DWI) is an established technique in both clinical and research settings. The data produced provide the opportunity for varied and powerful data analysis, which is a major factor in its widespread, diverse and evolving application. Conventional radiological image interpretation of diffusion weighted images can yield important information; for example, a radiological judgement of brain parenchymal signal increase on DWI in hyperacute stroke provides a highly sensitive indicator of likely irreversible tissue damage due to cytotoxic oedema [1–3]. Highly complex computer-driven analysis of diffusion-weighted data is also possible, allowing extraction of the routes of cerebral fibre pathways [4]. Between these extremes lies the fertile field of diffusion tensor data analysis [5, 6], allowing quantitative assessment of tissue integrity, and the related field of q space imaging and its derivatives [7–9]. This article is designed to provide an overview of the application of diffusion-weighted data analysis in these settings.
The reason for the rapid expansion of the use of DWI lies in its unique sensitivity at the imaging voxel scale to microscopic tissue structure characteristics. The example of sensitivity to cytotoxic oedema in acute stroke is complemented by further examples of diffusion-related imaging parameters providing new insights into tissue damage and abnormalities in conditions such as multiple sclerosis [10–12], schizophrenia [13, 14], vascular dementia [15], and leukoaraiosis [16]. This sensitivity is coupled to the potential for identifying white matter fibre tracts and their routes and connections non-invasively. This provides the possibility for understanding human neuroanatomy in the normal and abnormal brain at a level of detail and accessibility that was previously impossible. As such, DWI may also prove to be an important tool in the neurosciences for understanding the key structure–function relationships of the brain.
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Diffusion and diffusion-weighted imaging
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Diffusion (or, more correctly for diffusion MRI, self-diffusion) is a passive process, driven by the ambient temperature of the substance of interest (a process also known as Brownian motion). MRI is sensitive to tissue water; the diffusion of water molecules, driven by body heat, is therefore what interests us. An analysis of diffusion describes the bulk properties observed due to the random thermal motion of millions of individual water molecules. As these molecules move within tissues, they encounter various restrictions and hindrances (for example cell membranes and macromolecules). We therefore do not observe "free" diffusion of water, and this is acknowledged by a general reference to an "apparent diffusion coefficient" (ADC), rather than a "diffusion coefficient". The fact that the observed diffusive process is affected in this way forms the basis of the utility of DWI. Subtle changes in the degree of restriction to diffusion (for example by an alteration in membrane permeability to water or a change in average intercellular spacing) are reflected in changes in the diffusion-weighted signal. The water molecules that influence the signal in a diffusion-weighted image acquisition can therefore be thought of as probes of tissue microstructure.
The signal observed in a diffusion-weighted image is determined by the apparent diffusion coefficient ADC and a weighting factor b (for convenience we will from now on mostly use the term "diffusion coefficient" and the symbol D in place of "apparent diffusion coefficient" and ADC):
where S is the observed signal and S0 is the signal intensity in the absence of any diffusion weighting. The diffusion weighting term, b, often referred to as the "b-factor", is determined by acquisition sequence parameters, and has units typically expressed as s mm–2. For the commonly used pulsed gradient spin echo diffusion sensitization scheme the weighting is calculated according to [17]:
where 
is the gyromagnetic ratio, 
and G are the duration and amplitude of the applied diffusion sensitization gradients, and 
is the time interval between these gradients (for a further description of diffusion sensitization and diffusion imaging pulse sequences see for example Le Bihan [18]).
As can be seen from Equation 1
, diffusion sensitization leads to a decrease in signal intensity with increasing diffusion coefficient or increasing sensitization. As the application of a gradient is required to provide the sensitization (Equation 2), the direction along which the gradient is applied will affect the change in signal if diffusion is not uniform in all orientations. Figure 1 shows the effect of differently oriented diffusion sensitization gradients in an axial brain image.
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Figure 1. The effect of diffusion sensitization of b=1800 s mm–2 on an axial echo planar image acquisition at the level of the internal capsule and optic radiations. Sensitization direction varies from frame to frame. Note signal attenuation is dependent on applied gradient direction and on the orientation of white matter fibre bundles.
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As may be seen from Figure 1, the raw data quality of a typical DWI acquisition is relatively poor. Two main factors contribute to this problem. First, the signal attenuation process by which diffusion sensitization is applied (Equation 1) leads to diffusion weighted images being of inherently low signal to noise ratio (SNR). Second, the diffusion weighting process leads to a requirement for relatively long echo times (which in itself reduces SNR) in the imaging sequence (typically between 50 ms and 100 ms), and subsequently long repetition times, meaning that the only practical way to obtain whole brain coverage and to apply diffusion sensitization in multiple directions is to use an ultrafast imaging technique such as echo planar imaging (EPI) to sample the signal. This leads to subsequent problems associated with EPI acquisition, including coarse pixel resolution, image distortion and signal drop-out due to susceptibility problems (especially in the temporal and frontal lobes), and a range of image artefacts that may be present in imperfectly calibrated scanners.
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Diffusion anisotropy and the tensor model of diffusion
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As the diffusion that may be observed in vivo is directional or anisotropic (Figure 1), the opportunity arises to attempt to quantify this directionality. The diffusion tensor is a useful representation of diffusion anisotropy and allows estimation of the orientation of the dominant direction of diffusion within a tissue voxel [5, 6].
The diffusion anisotropy present in structures such as white matter tracts, and the fact that these structures are in general aligned at an arbitrary angle to the imaging gradients of the MR scanner, leads to the observation that the effects of diffusion-weighting gradients along different axes are in general correlated. This correlation necessitates the use of a tensor to describe these observations. Diffusive transport can be characterized by nine diffusion coefficients grouped in a symmetric second-rank tensor, D, which includes terms Dij accounting for these directional correlations:
The magnitudes of the tensor components depend on the subject orientation relative to the scanner frame of reference defined by the read, phase and slice gradient directions. Rotation of the tissue (for example by rotating the head) will change the individual components of the tensor at each point in the brain. The orientation of the structure at that point may be estimated by the process of tensor diagonalization, which extracts the dominant direction of diffusion, 
1, plus orthogonal minor directional information, 2 and 3 (1, 2 and 3 are the eigenvectors of the diffusion tensor). The diffusion coefficients (or diffusivities) along these three principal directions are provided by 
1, 2 and 3 (the eigenvalues of the tensor). In macroscopically anisotropic media such as white matter, with a single fibre bundle present, the principal eigenvector, 1, will reflect the orientation of the tissue microstructure [6, 19].
The diffusion tensor and its eigenvalues may be used to express the degree of diffusion anisotropy present in the tissue of interest. Anisotropy measurement is potentially of value, as it is generally high in white matter (Figure 2), whilst being low in grey matter and in damaged white matter, where the orientationally coherent diffusion restriction imposed by axonal bundles is disrupted. It has been demonstrated that anisotropy measurements can identify the effects of acute and chronic tissue damage in conditions such as stroke [20–22] and hemiparesis [23], and provide compelling evidence of axonal degeneration secondary to lesions or surgery [21, 22, 24].
The diffusion tensor may be visualized as an ellipsoid, with its orientation defined by the eigenvectors of the tensor (Figure 3). The magnitudes of the principal axes of the ellipsoid are defined as the mean effective diffusion distances for each eigenvalue at a given diffusion time [6]. Diffusion tensors may be grouped into three convenient classes: class I describes the case where 1
23. In this case diffusion is isotropic, and the ellipsoid equates to a sphere (Figure 3a). This is the form of tensor expected in free diffusion, and will be observed in cerebrospinal fluid and may closely be approximated in grey matter, where no dominant underlying structural orientation exists. Class II describes the case where 1>(23). In this case there is a well-defined dominant diffusion direction (the principal eigenvector, 1, with magnitude 1) and the ellipsoid takes on a cigar-shaped (prolate) form (Figure 3b, c). Class III describes the case where (12)>3 (Figure 3d). In this case the dominant direction of diffusion is less well-defined, and the underlying orientation of fibres is ambiguous. In this case the ellipsoid is disk-shaped (oblate).
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Figure 3. Diffusion tensor ellipsoids (above) and apparent diffusion coefficient (ADC) profiles (below). Eigenvalues ( x 10–6 mm2 s–1): (a) class I: 781, 670, 670; (b) class II: 1042, 529, 529; (c) class II: 1773, 164, 164; (d) class III: 931, 931, 283. Trace=2100 x 10–6 mm2 s–1. All ellipsoid axes in arbitrary spatial units. All ADC profile axes in 10–6 mm2 s–1.
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An alternative approach to visualizing the form of the diffusion tensor is to plot the orientation dependence of the ADC, giving an ADC profile [25, 26]. For the diffusion tensor, the diffusion profile along a direction 
is defined:
Figure 3 gives examples of class I, II, and III tensors using the ADC profile representation. This provides the same information as the ellipsoid, but presented in a different form. However, as we shall see later, the ADC profile is a more general visualization that is of use in tissues where the tensor is an inadequate representation of the observed diffusion weighted signal.
The anisotropy of the diffusion tensor is a scalar quantity that may be defined a number of ways. The most commonly used definition is fractional anisotropy (FA), which is defined [27]:
FA provides a quantitative rotationally invariant assessment of diffusion anisotropy, and is particularly effective for highlighting white matter fibre tracts (Figure 2).
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Visualizing fibre orientation using the diffusion tensor
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Presenting the information contained in the diffusion tensor throughout the brain is challenging due to the richness of information content. Potentially each of the eigenvectors and their associated eigenvalues, and/or an indication of diffusion anisotropy may be desirable for inclusion within each image voxel in such a display. In practice it is not possible to generate usable displays with individual graphical representations of the tensor such as diffusion ellipsoids on a voxel-by-voxel basis, as computer screen resolution and the capacity of the viewer to absorb information soon limits the ability to resolve such objects. More efficient modes of display concentrate on the diffusion eigenvectors, as shown in Figure 4, or use colour to represent the direction of fibre tracts [28].
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Figure 4. Principal eigenvector of the diffusion tensor (yellow needles) overlaid on orthogonal maps of fractional anisotropy. (a) Axial view through corpus callosum; (b) coronal view showing corticospinal tracts and corpus callosum; (c) sagittal view through corpus callosum and fornix.
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Complex fibre architecture and more accurate models of diffusion
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The diffusion tensor allows the reliable definition of a principal direction of diffusion, and thus the orientation of fibre bundles, only if tensors that approximate class II (Figure 3) are present. In the case of class I or III tensors the direction of any underlying fibre architecture is ambiguous. Unfortunately, such occurrences are common in the brain, within extensive regions where multiple non-colinear fibres are present within an imaging voxel. These situations occur when fibres cross (for example where the corticospinal tract, running superior–inferior, encounters callosal fibres running left–right), diverge (for example in the corona radiata) or display tight curvature (for example in the optic radiation). Figure 5 demonstrates the ADC profile that would be observed when combining two idealized class II tensors, representing fibres crossing at 90°, each with FA=0.9. In such cases the diffusion tensor is an inadequate description of the underlying phenomena (compare with Figure 3) and more sophisticated models of the observed diffusion characteristics are required. Additionally, data acquisitions with diffusion sensitization along many different orientations (often 50 or more) are required to sample such a distribution adequately (high angular resolution diffusion imaging (HARDI) see for example [29]).
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Figure 5. Synthetic apparent diffusion coefficient (ADC) profile for two crossing fibres, generated with a two tensor model of diffusion. Fractional anisotropy=0.9 for both fibres, which are present in equal proportions and crossing at 90°. Eigenvalues and trace as in Figure 3.
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Perhaps the most comprehensive approach to describing this added complexity observed in HARDI is to describe the observations in the framework of q space. The use of the diffusion tensor requires a fundamental assumption regarding the nature of restricted or hindered diffusion in tissue to be made, namely that the probability of a water molecule experiencing a given displacement in a defined time period is governed by a Gaussian probability density function. This assumption allows a relatively straightforward derivation of estimates of fibre orientation using the diffusion tensor, which is in effect a description of this Gaussian distribution. However, the actual form of the probability distribution function is unlikely to be Gaussian in reality, and in some cases (such as when fibres cross within a voxel) such an approximation describes the observed diffusion profile poorly, meaning that estimates of fibre orientation are likely to be erroneous. The form of the probability density function may be described more fully with the aid of a parameter q, which describes the degree of diffusion sensitization due to a given gradient pulse, via the relationship q=G, where , , and G are defined as in Equation 2. The form of the probability density function may be inferred via a Fourier transform with respect to q of the signal intensity in a diffusion weighted experiment, in an analogous way to the formation of an image from MRI k space data. The net result is a function defining the orientation of any microscopic tissue structure.
Unfortunately, an accurate determination of a fibre orientation probability density function using q space is not possible using clinical MRI scanners, as it involves a requirement that the length of the diffusion sensitization pulse, , is infinitesimally small, which can only be approximated in experimental systems. However, good estimates of the most likely fibre orientations in a voxel may be obtained using appropriate approximations [7, 9, 25, 30, 31]. One such approximation of complex diffusion patterns may be obtained by the use of multiple tensor models. These model the form of the probability density function resulting from crossing fibres as a mixture of Gaussian densities, and as such form a compromise between classical diffusion tensor parameterizations and q space [8, 32]. Using this technique it can be shown that regions in the brain containing crossing fibres are relatively commonplace, as may be appreciated in Figure 6, where a comparison with the best estimate of fibre orientation from single tensor modelling is also presented. It is clear that modelling of complex fibre patterns is both possible and desirable, as single tensor modelling introduces the risk of interpreting artefactual apparent fibre orientations in single tensor data as true tissue structure.
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Figure 6. Axial, coronal, and sagittal views through a region of the brain centred on the superior longitudinal fasciculus. Top: best estimates of fibre orientations obtained from the diffusion tensor, overlaid on a fractional anisotropy map. Bottom: regions of crossing fibres highlighted in white, as defined using a mixture model of two tensors, with two resulting estimates of fibre orientation. Light grey regions indicate the presence of anisotropic single tensor tissue (single fibre bundle); dark grey indicates isotropic diffusion (mostly grey matter). Green ovals indicate regions where crossing fibres may be wrongly interpreted as representing coherent single fibre structure in single tensor analysis.
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Fibre tracking and investigation of cerebral connectivity
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The directional information concerning fibre bundle orientation that DWI provides may be exploited to analyse cerebral anatomical connectivity non-invasively. A number of methods have been developed to infer connection between tissues in neighbouring imaging voxels, or between brain regions [4, 33–43]. These methods fall into two broad groups: those that define a single route of connection for each start point (these will be referred to as "linear" approaches), and those that attempt to establish the spatially distributed degree of connection across the whole brain to a given start point or region (these will be referred to as "distributed" methods). Both linear and distributed methods can be used to perform "probabilistic" fibre tracking, although the type of probabilistic experiment for which each is suitable differs.
Linear methods
Linear methods for determining connectivity usually utilize the principal eigenvector (1) of the diffusion tensor to provide a propagation direction for each voxel along the path, as it is generally accepted that 1 is collinear with the principal orientation of fibre bundles when the tensor is an adequate model of tissue structure and noise is negligible [6]. Many of these approaches are analogous or identical to methods for determining streamlines (as developed for use in fluid dynamics, where instead of utilizing the principal direction of diffusion, the bulk flow direction propagates the path). Streamline methods as applied to diffusion tractography usually utilize interpolation of the tensor or eigenvector field to ensure smooth and possibly higher accuracy propagation [4, 35, 43, 44], although non-interpolated variants have also found applications [34, 45–47]. Figure 7 shows an example of streamlines used to isolate the superior longitudinal fasciculus (using the methods presented in Catani et al [44]). Such methods may be used to identify regions within the brain that are associated with abnormality or degeneration in specific white matter tracts [46–48].
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Figure 7. Medial sagittal view of a streamline reconstruction of the superior longitudinal fasciculus (left hemisphere), a long associative bundle composed of long and short fibres that connect the frontal lobe with the parietal, occipital, and temporal lobes (image courtesy Derek Jones).
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Other linear methods include the voxel linking approach introduced in Poupon et al [37], where voxels are linked together to form a chain of connection, using the degree of coincidence of neighbouring 1 to define the route of connection. Path integral approaches, aimed at minimizing the bending energy of a path defined using the 1 between any two given points have also been suggested [41, 49].
Distributed methods
A number of methods have been suggested that aim to generate distributed maps of "probability" of inter-region connection or of "connectivity" between brain regions [39, 42, 50–54]. For example, attempts have been made to assess cerebral connectivity by using models of the diffusion process, such as the diffusion ellipsoid, to propagate a simulated diffusion process, with the aim of establishing connectivity in a distributed manner using Monte Carlo methods [39, 40, 42]. This may be achieved using a grid-based simulated random walk process [42], in which a virtual particle is allowed to diffuse at a rate determined by a function of the magnitude of the diffusion coefficient along one of the possible intergrid point (intervoxel) directions chosen at random. This process is allowed to continue until the random walk has reached some stopping criterion (perhaps reaching some threshold in allowed tensor anisotropy) and each voxel in the brain encountered on the random walk is noted. This is then repeated a large number of times and the number of times over all randomizations that any voxel in the brain is encountered by the random walk provides an index of the "connectivity" of the start point to that voxel.
Other distributed tracking approaches include front evolution methods introduced in Parker et al [50, 54] and more recently in Tournier et al [55]. These methods aim to propagate a wave front through the directional information provided using DWI and differ from the Monte Carlo approaches in that the evolution of the front is entirely deterministic. However, in common with the results of the Monte Carlo methods discussed above, front evolution methods generate maps of a distributed "connectivity" index. An example of such an experiment, generated with the "fast marching tractography" (FMT) method [50, 54, 56, 57] is presented in Figure 8. These methods have the benefit over linear methods in that they are able to identify connectivity to more than one point in the brain from a given start point, if the data suggest such multiple connections exist. However, although they may provide some form of ranking of how much credence to give any identified connectivity, these methods are not truly probabilistic. Nevertheless, it has been shown that such methods are reproducible [58], and can provide information relating to altered cerebral connectivity in the presence of pathology [59, 60].
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Figure 8. Connectivity (hotwire colour scale) to a voxel in the splenium of the corpus callosum, as defined using the fast marching tractography method. Overlay onto fractional anisotropy map. Note branching pattern, indicating a distributed pattern of connectivity to start voxel.
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Another class of distributed method involves the use of Monte Carlo streamline approaches [61, 62]. These methods (closely related to the Monte Carlo simulated diffusion methods described above) define a distribution of possible streamline propagation directions at each point in the brain, based on the information in the DWI acquisition. Streamline propagation is then repeated (typically 1000–10 000 times from each start point [62]), with random sampling of these distributions on each iteration. Again, the number of times that each voxel in the brain encountered by a streamline is recorded, and this is used to assess a distributed pattern of connectivity.
Probabilistic methods
A number of linear and distributed methods have been developed that attempt to quantify the probability of anatomical connection between brain regions using DWI data. Although some of the methods discussed in the previous section are sometimes described as "probabilistic", the definition of probability in each case must be considered carefully. In all cases a model of diffusion is required (implicitly or otherwise) that tells us how to infer fibre orientation information from a DWI acquisition. A probability distribution function (PDF) describing the likely distribution of possible fibre orientations at any given point in the brain is subsequently generated using this model. A measure of probability that two brain regions are connected may be generated if: (1) the model describing the relationship between the observed DWI signal and fibre orientation is accurate; and (2) the connection probability is defined as the path integral of the local fibre orientation PDFs. In practice the relationship between DWI observations and underlying fibre orientation distributions within an imaging voxel is unknown, with the exception of identifying the dominant orientation, if one is present; therefore condition 1 is not in general met. However, even with this significant caveat in place, a large amount of progress has been made in deriving probabilistic maps of brain connectivity that attempt to answer the question, "how confident can I be that a route of connection, as represented by my model of the relationship between fibre orientation and my DWI data, exists between point A and point B in the brain?"
The distributed methods introduced in the previous section generate indices of connectivity that may be interpreted as probabilities of connection, given the methodology and models of diffusion employed. Another major class of probabilistic methods is that which attempts to define a probability of connection in the presence of data noise. Many linear and distributed tracking approaches are susceptible to errors in estimating fibre orientation due to noise [63] meaning that it is not possible to assign confidence to any connection that is established. Thus, an erroneous pathway, caused by perhaps noise or partial volume effects, is assigned as much significance as "true" pathways. The probabilistic tracking work in [32, 64] overcomes this limitation by explicitly estimating the confidence with which the orientation of fibres within the brain may be estimated in the presence of data noise. These methods are able to generate maps of the probability of connection between imaging voxels based on the data noise and (as always) a given model of the relationship between fibre orientation and the diffusion signal. Other work has suggested the use of q space-derived measures of the likely orientation of tissue microstructure to generate probabilities of connection between voxels [7, 38]. However, even with these methods it is unclear how precisely it is possible to infer the true orientational distribution of fibre bundles within a voxel.
A related area of probabilistic tracking uses the results of multiple experiments in different individuals to generate group maps of connected volumes that represent the probability of finding a position in the brain with connection to a given location. These "commonality" maps may be generated using linear [46, 47] or distributed [60, 62, 65, 66] methods, and give an indication of how reproducibly a given connection is found across a population of subjects. These maps are of use in determining the fibre bundles in the normal brain that are positionally invariant and in comparison with situations where tracts may be displaced or obliterated due to pathology such as tumours [46, 60] or in developmental abnormalities.
Tracking with complex fibre architecture
All above methods may be used either with tensor data or with more sophisticated models of complex fibre orientation [32, 67]. For example, it has been shown that it is possible to perform streamline tracking through multitensor data [67], to perform group mapping analyses using such multitensor streamlining [68], and to perform probabilistic analyses with multifibre datasets [32]. Figure 9 shows an example of a probabilistic fibre tracking experiment in the normal visual system, using crossing fibre information [32]. This clearly shows the potential for identifying complex "wiring patterns" in the human brain using probabilistic fibre tracking and crossing fibre analysis.
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Figure 9. Probability of connection to a region of interest in the left lateral geniculate nucleus (LGN), as defined using the multifibre probabilistic index of connectivity (PICo) method. Overlay onto axial T2 weighted images. Note multiple routes of connection identified from the LGN. OR, optic radiation.
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Conclusions
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DWI provides powerful non-invasive indications of tissue structure and cerebral connectivity. Assessments of tissue structure alterations range from the identification of signal intensity changes on diffusion weighted images, through to quantitative interpretations of the diffusion tensor and other more sophisticated models of the DWI signal. Cerebral anatomical white matter connectivity is assessed using software approaches designed to interpret the fibre tract orientation information that DWI provides. These may generate linear or distributed results that may be derived in a deterministic or probabilistic manner. These methods are beginning to fulfil their early promise as key tools in the understanding of brain pathology and abnormality and in understanding the interconnections that mediate cerebral function in both the healthy and diseased or damaged brain.
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Acknowledgments
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The author is grateful to Dr Derek Jones for the provision of Figure 7.
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L. French and P. Pavlidis
Informatics in neuroscience
Brief Bioinform,
November 1, 2007;
8(6):
446 - 456.
[Abstract]
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Abstract
Introduction
