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British Journal of Radiology (2006) 79, 101-109
© 2006 British Institute of Radiology
doi: 10.1259/bjr/24908512
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Full Paper

Enhanced visualization and quantification of magnetic resonance diffusion tensor imaging using the p:q tensor decomposition

A Peña, PhD1,2,3, H A L Green, MBChB1,3, T A Carpenter, PhD3, S J Price, FRCS1,2,3, J D Pickard, MChir, FRCS, FMedSci1,3 and J H Gillard, BSc, MD, FRCR2,3

Departments of 1 Neurosurgery, 2 Radiology and the 3 Wolfson Brain Imaging Centre, Addenbrooke's Hospital and the University of Cambridge, Cambridge CB2 2QQ, UK

Correspondence: Dr Jonathan H Gillard, University Department of Radiology, Addenbrooke's Hospital, Cambridge CB2 2QQ, UK


   Abstract
Top
 Abstract
Introduction
 Materials and methods
 Results
 Discussion
 Conclusion
 References
 
Many scalar measures have been proposed to quantify magnetic resonance diffusion tensor imaging (MR DTI) data in the brain. However, only two parameters are commonly used in the literature: mean diffusion (D) and fractional anisotropy (FA). We introduce a visualization technique which permits the simultaneous analysis of an additional five scalar measures. This enhanced diversity is important, as it is not known a priori which of these measures best describes pathological changes for brain tissue. The proposed technique is based on a tensor transformation, which decomposes the diffusion tensor into its isotropic (p) and anisotropic (q) components. To illustrate the use of this technique, diffusion tensor imaging was performed on a healthy volunteer, a sequential study in a patient with recent stroke, a patient with hydrocephalus and a patient with an intracranial tumour. Our results demonstrate a clear distinction between different anatomical regions in the normal volunteer and the evolution of the pathology in the patients. In the normal volunteer, the brain parenchyma values for p and q fell into a narrow band with 0.976<p<1.063 x 10–3 mm2 s–1 and 0.15<q<1.08 x 10–3 mm2 s–1. The noise appeared as a compact cluster with (p,q) components (0.011, 0.141) x 10–3 mm2 s–1, while the cerebrospinal fluid was (3.320, 0.330) x 10–3 mm2 s–1. In the stroke patient, the ischaemic area demonstrated a trajectory composed of acute, sub-acute and chronic phases. The components of the lesion were (0.824, 0.420), (0.884, 0.254), (2.624, 0.325) at 37 h, 1 week and 1 month, respectively. The internal capsule of the hydrocephalus patient demonstrated a larger dispersion in the p:q plane suggesting disruption. Finally, there was clear white matter tissue destruction in the tumour patient. In summary, the p:q decomposition enhances the visualization and quantification of MR DTI data in both normal and pathological conditions.


   Introduction
Top
 Abstract
 Introduction
Materials and methods
 Results
 Discussion
 Conclusion
 References
 
Magnetic resonance (MR) diffusion tensor imaging (DTI) is a technique which allows the in vivo measurement of water diffusion in biological tissues from which tissue microstructure can be inferred [1–5]. It has been used successfully to investigate a number of neurological disorders that involve the disruption of white matter fibres including schizophrenia [6], head injury [7], multiple sclerosis [8] and stroke [9, 10]. In addition, DTI data can be used with a set of computational techniques called "tractography" [11] to reconstruct in vivo white matter tracts in the human brain, which is a very promising field, for example, to investigate their disruption due to an expanding tumour [12].

Diffusion is properly described by a high-dimensional mathematical quantity called a tensor. A tensor represents the generalization of scalars and vectors and as such, it contains more information than these. In three dimensions a scalar has one element, a vector three elements and a tensor nine elements. In order to quantify pathological changes in the diffusion tensor, a transformation is required which reduces the dimensionality of the tensor and to this end a number of tensor scalar measures have been proposed [1, 13]. From a theoretical point of view, tensor calculus establishes that many such measures exist. These include the lattice index (LI), relative anisotropy (RA), fractional anisotropy (FA), the volume ratio (VR) and ratios of the various eigenvalues ({lambda}i), the mean diffusivity (D), the Euclidean length of the tensor (L), its anisotropy angle ({phi}) and any algebraic combination of the first, second and third invariants of the tensor [14].

From a practical point of view, however, only a limited number of these measures are actually used in clinical studies. In the MR DTI literature, the most common of these measures are FA and D. Out of 30 recent studies on clinical applications of DTI, encompassing diseases such as schizophrenia, Alzheimer's disease, stroke, multiple sclerosis and head injury, 26 reported their results using both FA and D [7, 9, 12, 15–36], while only four reported D alone [37–40].

The caveat with exclusively using D and FA to characterize pathology in clinical applications is that it is not known a priori which tensor scalar measure is the most appropriate to quantify pathological changes in brain tissue. It is conceivable, for example, that a study might fail to show significant changes when the diffusion tensor is measured using FA but it may show differences when using RA or L or some other measure. We have previously shown this to be the case in acute stroke [34]. The identification of which is the "best" measure of the diffusion tensor is an empirical process, which will only be resolved after a large number of experiments are conducted and corroborated with external empirical information, such as histology. In these circumstances it seems reasonable to analyse as many scalar measures as possible, and not rely solely on D and FA.

This article explores the novel application of a mathematical technique to enhance the visualization and quantification of brain tissue in MR DTI, which we will term "p:q decomposition". The technique is based on a tensor transformation, which decomposes the diffusion tensor into its isotropic (p) and anisotropic (q) components. In contrast to the standard practice in the literature where only D and FA are analysed, this technique permits the visualization simultaneously of seven scalar measures. These are D, p, q, RA, FA, {phi}, and L. This technique is based on a classical decomposition used in tensor calculus, already observed by the major contributions of Basser et al [13] and Pierpaoli et al [5], but which has not been applied yet to visualize and quantify the diffusion tensor in MR DTI.

In the following sections we will describe the technique and apply it to data from a control volunteer and three clinical examples.


   Materials and methods
Top
 Abstract
 Introduction
 Materials and methods
Results
 Discussion
 Conclusion
 References
 
Theory
Diffusion in tissue can be mathematically represented as a second-order Cartesian tensor, which in matrix form is:


Formula 001

Given that the tensor is symmetric along its principal diagonal, i.e. Dyx = Dxy, it has only six independent components. From the tensor, the eigenvalues {lambda}i are calculated using a standard methodology, such as singular value decomposition [41], as {lambda}1, {lambda}2 and {lambda}3. According to tensor calculus, based on the eigenvalues many possible scalar measures of the diffusion tensor can be devised [42, 43].

The standard methodology in the DTI literature, however, is to calculate only two scalar measures of Dij. These are the mean diffusion (D) defined as:


Formula 002

where tr represents the trace of the tensor, and the fractional anisotropy (FA) or the relative anisotropy (RA) defined as:


Formula 003

The proposed technique is based on a classical tensor decomposition, already observed by Basser et al [13] and Pierpaoli et al [5]. Our contribution is to construct a graphical representation of the diffusion tensor based on this decomposition. This transformation has its conceptual roots in the mathematical theory of continuum mechanics [44, 45]. We will term the technique p:q decomposition.

Using this methodology, the first step is to decompose the diffusion tensor from Equation (1) according to the next equation:


Formula 004

into two tensors P and Q, i.e. Dij = Pij+Qij. Here Iij is the identity tensor Iij = diag(1,1,1). The first term on the right hand side of Equation (4) is the isotropic tensor, while the second term (in brackets) represents the deviatoric tensor. The magnitude of these tensors can be denoted by its isotropic (p) and anisotropic (q) components. The values of p and q can be computed as:


Formula 005

and


Formula 006

According to these definitions p is therefore a scaled measure of the mean diffusion in the tensor, while q is a measure of the variance or deviation of the eigenvalues with respect to the mean diffusion of the tensor.

The second step is to plot each tensor as a point in a Cartesian plane with p taken as the x-axis and q as the y-axis, as in Figure 1aGo. This plane will be denoted as the p:q plane. The effect of this transformation is to reduce the dimensionality of the tensor from six dimensions to two.


Figure 1
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  		Figure 1. (a) A point representing a sample of tissue in the p:q plane. The x axis corresponds to the isotropic component of diffusion (p) and the y axis the anisotropic component of diffusion (q). Any tensor can be decomposed into its p and q components po and qo, which will correspond to a point in the p:q plane. (b) Starting from a point in the p:q plane, we can deduce the standard anisotropy measures RA and FA using simple geometry. Both of these measures will be proportional to the angle {phi}, in fact RA is proportional to the tangent and FA proportional to the sine. (c) Two tissues will, in general, have different p and q components. Thus a tissue A with components pA and qA, will have a different location from a tissue B with components pB and qB. (d) A tissue A in general will have different p and q components at different times. By plotting these different components in the p:q plane we can obtain a trajectory that illustrates the evolution of tissue in time. In this example we see a trajectory demonstrating three time points for tissue A.

 
The third step is to use the p:q plane to deduce the five additional tensor scalar measures: D, RA, FA, L and {phi}. Four of these seven tensor measures (q, RA, FA, {phi}) are anisotropy measures, while D and p are measures of the magnitude of diffusion and L is a measure of the total diffusion of the tensor. D, p, q and L have units of 10–3 mm2 s–1, RA and FA are dimensionless, and {phi} has units of degrees.

These scalar measures can be deduced either analytically or graphically. The analytical method is to directly compute the measures based on the p,q components using the formulae:


Formula 007

However, the real advantage of using the p:q plane is that we can obtain these values directly from the graph as follows. Consider a tensor A from which we can compute its location in the p:q plane using Equations (5) and (6) as po and qo. Therefore, Lo is the distance between the origin of coordinates and the point (po,qo); {phi}o is the angle subtended between the p axis and a line originating in the centre of coordinates and passing through point (po,qo), i.e. the segment Lo; RAo is the ratio between qo and po; FAo is the ratio between qo and Lowith scale factor Formula ; D is Formula of the value in the p axis.

Figure 1bGo illustrates the geometrical relationship between these various quantities.

Data acquisition
In order to illustrate the use of the p:q decomposition with clinical data, four representative cases were selected. The first comprised five regions of interest (ROIs) in a healthy 27-year-old volunteer (to illustrate spatial variation in the tensor field, as in Figure 1cGo). The second reports the findings in a 76-year-old hypertensive woman who presented with a sudden onset of expressive dysphasia and right-sided hemiparesis. Imaging of her left middle cerebral artery territory stroke was undertaken at 37 h, 1 week and 3 months from stroke onset to illustrate the temporal variation in the tensor field (Figure 1dGo). The third case is a 85-year-old female hydrocephalus patient with a history of gait ataxia, falls and memory problems. And finally, the fourth case is a 46-year-old male patient with a WHO Grade II oligodendroglioma.

The Local Research Ethics Committee approved the study and informed consent was obtained. The diffusion tensor data sets were acquired using a 5 mm slice thickness. Imaging was performed on a 3 Tesla magnetic resonance machine (Bruker Medspec S300; Bruker Medical, Ettlingen, Germany). A single shot spin echo, echo planar imaging technique, with Stejskal-Tanner diffusion sensitizing pulses [46] was used. Imaging parameters were: repetition time (TR) = 5070 ms, echo time (TE) = 107 ms, {alpha} = 90°, {delta} = 21 ms and {Delta} = 66 ms. Eight interleaved supratentorial slices were acquired with a phase template in a near axial plane, using a 128 x 128 matrix, field of view of 25 cm x 25 cm. For each slice, images were collected from 12 non-collinear gradient directions [47]. For each gradient direction an unweighted bo image and five diffusion weighted images were collected at equally spaced b-values in the range bmin = 318 s mm–2 to bmax = 1541 s mm–2. Using a specially-written program in MATLAB (The MathWorks Inc., Natick, MA) the diffusion tensor was computed on a voxel by voxel basis, using a singular value decomposition algorithm to fit the signal intensities to the Stejskal-Tanner equation, following the method proposed by Basser et al [1, 2]. From the tensor, the p and q components were calculated based on Equations (5) and (6), and D, RA, FA, {phi}, and L using Equations (2) and (7).


   Results
Top
 Abstract
 Introduction
 Materials and methods
 Results
Discussion
 Conclusion
 References
 
Normal volunteer
D and FA maps of the volunteer were used to select square anatomical ROIs of 5 x 5 voxels, which were subsequently averaged to obtain a mean value for the ROI. These were placed in the corpus callosum (CC), occipital cortex (Cx), cerebrospinal fluid (CSF), internal capsule (IC) and noise regions, as illustrated in Figure 2a,bGo. When these ROIs were plotted in the p:q plane, they formed clearly segregated clusters (Figure 2cGo).


Figure 2
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  		Figure 2. (a) Map of the mean diffusion (D) for a horizontal slice of the normal volunteer investigated, demonstrating the regions of interest used in the study, which are (from top to bottom) noise (N), cerebrospinal fluid (CSF), internal capsule (IC), splenium corpus callosum (CC) and occipital cortex (Cx). The scale on the right indicates the magnitude of D. (b) Map of the fractional anisotropy (FA) for the same horizontal slice of the normal volunteer investigated, demonstrating the location of the same regions of interest. The scale on the right indicates the dimensionless magnitude of FA. (c) p:q plane illustrating the defined regions of interest (ROIs) in the normal volunteer. Three clusters are observed for the noise (N), with small components for both p and q. The three parenchyma ROIs (CC, IC, Cx) are located along a line with approximately the same value of p, but significantly different values of q. The CSF has a much larger dispersion and a larger value of mean diffusion.

 
The spherical diffusion and deviatoric diffusion fell within a narrow band with 0.976<p<1.063 x 10–3 mm2 s–1 for the three structures in brain parenchyma, i.e. CC, IC, Cx, D = 0.607, 0.563, 0.613 x 10–3 mm2 s–1 and p = 1.052, 0.976, 1.063 x 10–3 mm2 s–1, respectively. In contrast, all the other measures varied substantially, with a range bounded by maximum values for the CC (q = 1.086, L = 1.032) x 10–3 mm2 s–1 and RA = 0.879, 45.9°, FA = 1.512, and minimum values for the Cx (q = 0.150, L = 1.073) x 10–3 mm2 s–1 and RA = 0.141, 8.06°, FA = 0.171.

Noise appeared as a cluster close to the origin of the coordinates (D = 0.007, p = 0.011, q = 0.141) x 10–3 mm2 s–1, but whose additional scalar measures were amongst the highest (L = 1.073 x 10–3 mm2 s–1, RA = 12.145, 85.29°, FA = 1.220. CSF presented the opposite characteristics, being the most distant to the origin of coordinates (D = 1.917, p = 3.320, q = 0.330) x 10–3 mm2 s–1, and having small additional scalar measures (L = 1.073 x 10–3 mm2 s–1, RA = 0.099, 5.68°, FA = 0.121).

Statistically significant differences between the various ROIs were investigated using unpaired Student's t-tests. The isotropic diffusion (p) between the noise and the parenchyma (CC, IC, Cx) was significantly different (p-value<0.01), and between the CSF and the parenchyma (p-value<0.01). It was significantly different between the CC and the IC (p-value<0.01), between the IC and the Cx (p-value<0.05), but not between the CC and the Cx (p-value = 0.5473). The deviatoric diffusion (q) was different for the three parenchyma ROIs (CC, IC, Cx). It was significantly different between the CC and IC (p-value<0.01), between the CC and Cx (p-value<0.01), and between the IC and Cx (p-value<0.01).

Results for the seven scalar measures are presented in Table 1Go.


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  		Table 1. Data from the MR diffusion tensor imaging(DTI) acquisitions for the normal volunteer (upper set) and the stroke patient (lower set)

 
Stroke patient
Lesion and contralateral control square anatomical ROIs of 5 x 5 voxels, were selected in the stroke patient using the FA and D maps at 37 h, 1 week and 1 month (Figure 3aGo). To clarify presentation and due to the multiple number of ROIs used, all variables analysed were averaged within each ROI to obtain a mean value and standard deviation. For each ROI the mean is in the crossing of the bars which represent the magnitude of standard deviation. The lesion (ischaemic region) in the stroke patient described a trajectory in the p:q plane composed of three phases, which occupied the regions: 0.824<p<2.624 x 10–3 mm2 s–1 and 0.25<q<0.42 x 10–3 mm2 s–1. The (p,q) components of the lesion ROI were (0.824, 0.420), (0.884, 0.254), (2.624, 0.325) at 37 h, 1 week and 1 month, respectively. All these results are shown in Figure 3bGo.


Figure 3
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  		Figure 3. (a) Mean diffusion (D) and fractional anisotropy maps (FA) for a stroke patient at three time points: 37 h (left column), 1 week (central column) and 3 months (right column). FA is lower row and D is upper row. These maps demonstrate the regions of interest (ROIs) used in this study. The lesion ROIs are presented in orange and the control ROIs in green. Each of these ROIs consisted of 5 x 5 x 1 voxels. (b) This figure illustrates the p:q plane for the stroke patient, with lesion and control ROIs at (a) 37 h, (b) at 1 week and (c) at 3 months. The control ROIs are denoted in blue and the lesion ROIs in red. The arrows demonstrate the trajectory followed by the lesion in this patient and show schematically how, while the control ROIs remain in roughly the same region in the p:q plane, the ischaemic lesion demonstrates a trajectory composed of acute (reduction in p, reduction in q), subacute (normalization of p while q remains low) and chronic (increased p while q remains low) phases. The inset shows schematically the location of the lesion ROI with respect to the control ROI and a line of constant fractional anisotropy. As FA is function of the angle {phi}, the figure indicates that at 37 h the lesion has a higher FA than the control, while at 1 week it has a lower FA than the control.

 
The corresponding contralateral control ROIs (Figure 3aGo, shown in blue), in contrast, demonstrated only a small degree of change, remaining in the region 1.154<p<1.288 x 10–3 mm2 s–1 and 0.47<q<0.77 x 10–3 mm2 s–1. The (p,q) components of the control ROI were (1.154, 0.475), (1.229, 0.777), (1.288, 0.586) at 37 h, 1 week and 1 month, respectively.

Results for the seven scalar measures are presented in Table 1Go.

Hydrocephalus patient
We have investigated microstructural changes in the internal capsule associated with the ventricular dilatation in this patient. For this analysis, four ROIs have been selected (two in the patient bilaterally and two in the control volunteer bilaterally) in one axial slice corresponding to the posterior limb of the internal capsule (IC) at the level of the foramen of Monro. Each ROI was composed of nine voxels. As a comparison, the same regions were selected in a control volunteer in the same manner. The results were: for the patient p = 1.29±0.466, q = 0.85±0.054 for the left IC and p = 0.96±0.195, q = 0.86±0.147 for the right IC; and for the control, p = 1.01±0.023, q = 0.80±0.084 for the left IC, and p = 0.96±0.034, q = 0.84±0.159 for the right IC. All the p, q units are in 10–3 mm2 and are illustrated in Figure 4Go. These results demonstrate that while the mean values of the four ROIs are roughly similar, there is a marked increase in the dispersion of the voxels in the IC of the patient.


Figure 4
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  		Figure 4. This is thep:q diagram for the internal capsule (IC) of a hydrocephalus patient. Regions of interest (ROIs) have been selected on the IC bilaterally at the level of the foramen of Monro. The same ROIs have been selected in a control subject. ROI location is shown in the insets (patient, above; control, below). The p:q diagram demonstrates an increased dispersion (disorganization) of the white matter tracts of the IC in the hydrocephalus patient as compared with the control.

 
Tumour patient
In this patient we investigated the hypothesis that there are significant differences between the microstructure in the tumour region and an equivalently-located contralateral control. For this analysis, two ROIs have been selected in the patient in one axial slice: one in the tumour region (the posterior pericallosal region) and another in the equivalent normal-appearing contralateral control region. For the tumour: p = 2.26±0.210, q = 0.37±0.049, while for the contralateral control region p = 1.331±0.128, q = 0.71±0.134. All the p, q units are in 10–3 mm2 and are illustrated in Figure 5Go. These results demonstrate both an increase in the isotropic component of diffusion and a decrease in the deviatoric component. We can interpret these changes as suggesting a loss in the microstructure of tissue, as a decrease in the density of fibres (more intercellular space) and a loss in fibre coherence. These changes are consistent with previous reports [48].


Figure 5
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  		Figure 5. This is thep:q diagram of a patient with a Grade II oligodendroglioma in the posterior pericallosal white matter. The location of the regions of interest (ROIs) is indicated in the inset (above), and the patient's MR fluid attenuation inversion recovery (FLAIR) image (below). Compared with the control region, the tumour region demonstrates both an increase in isotropic diffusion (p) and a decrease in deviatoric diffusion (q). This tissue signature is consistent with the destruction of white matter tracts in the tumour region.

 

   Discussion
Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
Conclusion
 References
 
We have presented a technique that permits the simultaneous visualization of multiple tensor scalar measures from MR DTI data. In particular, we have shown how, from a single graph (the p:q plane), it is possible to deduce seven scalar measures of the diffusion tensor, including D, p, q, RA, FA, {phi}, and L. This represents an improvement on the standard methodology in the MR DTI literature in which only two scalar measures (typically FA and D) are displayed.

There have already been a number of studies in the literature that have considered plotting simultaneously two tensor scalar measures, particularly FA and D. These include Pierpaoli et al [5] who distinguished various brain regions based on decomposing the tensor in terms of D and the volume ratio (VR). Werring et al [8] in an investigation of normal-appearing white matter lesions in multiple sclerosis, and Wieshmann et al [49] and Jones et al [50] have also demonstrated the potential of plotting FA and D simultaneously. Plotting D vs FA, however, does not allow visualization or quantitative analysis of the other scalar measures (e.g. p, q, RA, {phi}, L) from a single graph, while our study does.

To illustrate the use of our method for clinical data, we have applied it to a healthy volunteer, a sequential study in a patient with recent stroke, a patient with hydrocephalus and a patient with an intracranial tumour.

In all cases the p:q plane offers the analyst a concise and easy-to-use representation of the diffusion tensor. The first case illustrates the spatial variation in the tensor field and statistically significant differences between different tissue types (e.g. grey matter, white matter), while the second case illustrates the temporal variation in the tensor field and thus the evolution of the lesion (e.g. lesion, contralateral control).

We propose that the p:q decomposition is a powerful aid not only in the visualization of the data, but also in its analysis, by offering a unique opportunity to assess the additional non-standard tensor scalar measures (p, q, {phi}, L) and their relationship with the standard measures (D, RA, FA). In this context, several interesting observations from both the normal volunteer and the stroke patient have been possible by using the p:q technique.

Normal volunteer
The first observation is that the p:q plane provides a graphical means to understand the complex equations that describe RA and FA. From Figure 1bGo and Equation (7), one can easily observe that both RA and FA are composite measures of other more basic tensor quantities. In particular, RA is simply the ratio between q and p, and FA the ratio between q and L scaled by a factor ofFormula .

The second observation is that the p:q plane explains some anomalies when using FA and RA as measures of anisotropy. For example, if we take the value of FA for noise from Table 1Go, we obtain the theoretical maximum FA value of 1.22. This is confusing, as one would not expect empty space to have a large degree of organization (anisotropy). This paradoxical result is in fact a methodological artefact in using FA as an anisotropy measure, and is due to the presence of L in the denominator of FA. Given that the diffusion of empty space should be zero (or close to zero due to experimental error), a very small L will imply a very large FA. Thus FA is a measure of tissue anisotropy, but weighted by its total diffusion. The same argument applied to the value of RA, which gives the enormous value of 12 (while the corpus callosum, for instance, is 1.032).

The third observation is the insight that q might offer as a measure for the background noise in the data. Equation (6) can be interpreted in statistical terms such that q is a measure of the variance of the eigenvalues of the tensor with respect to the mean diffusion D. Therefore, isotropic elements in the data (such as the empty space, Cx and the CSF) should theoretically have all the eigenvalues equal and thus a q equal to zero. However, due to experimental error, background noise and other MR acquisition influences, there is a small discrepancy and the eigenvalues are not exactly the same. Our results show that Cx and the noise have approximately the same q values (0.141 and 0.15), while the CSF presented a larger dispersion (q = 0.330), which might be attributed to the contribution of diffusion and bulk flow during the acquisition time.

Stroke patient
The first observation is the ability of the p:q planes to visually convey simultaneous changes in the isotropic and anisotropic components of the diffusion tensor as they change in time, in other words the "trajectory" of the tensor. In addition to the qualitative nature of the trajectory, the magnitude of tensor changes can be read directly from the p and q axes of the plots. In our example, the trajectory describing the lesion evolution is composed of three segments or phases (Figure 3bGo), which can be interpreted in biological terms as the acute (reduction in p, reduction in q), sub-acute (pseudonormalization of p, while q remains reduced) and chronic (increase in p, while q remains reduced) phases that have been well-documented in association with stroke [9].

A second observation demonstrates another methodological artefact or anomaly of FA and RA. Close inspection of Figure 3bGo demonstrates that the lesion clusters (shown in red) with respect to the control clusters (shown in blue) are displaced first above a line of constant FA in the acute phase (37 h) and subsequently below this line in the sub-acute phase (1 week), as shown in the inset. As both RA and FA are functions of the angle {phi}, this would imply that they are increased in the lesion as compared with the control, which is absurd. This paradox of increased tissue anisotropy (as measured in terms of RA or FA) was reported by Nusbaum and colleagues [51] in normal ageing. As we have explored in more detail in the case of acute stroke [34], the p:q technique provides a graphical explanation of why this can be the case, and that this apparent increase in anisotropy (as measured in terms of RA or FA) can be purely a graphical consequence of the manner in which FA and RA are calculated and thus a methodological artefact. Quantitatively, the anisotropy measured with FA and RA of the lesion's acute phase (37 h): RA increased from 0.412 to 0.510 (or +24%) and FA increased from 0.466 to 0.556 (or +19%). In contrast, q decreased from 0.475 to 0.420 (or –12%). This behaviour suggests, albeit tentatively, that theoretically q may detect early changes in tissue anisotropy that are misrepresented by RA and FA.

A third observation is that during the sub-acute phase (1 week after the stroke), the best sensitivity to the pathology is offered by q rather than by FA or RA. q was reduced from 0.777 to 0.254 (–67%), In contrast, FA and RA decreased by the smaller amounts of 0.632 to 0.287 (–55%), and 0.654 to 0.338 (–48%), respectively.

Based on the previous observations, we can conclude that, at least in some circumstances, some non-standard anisotropy measures (such as q) can provide a higher sensitivity to detect pathological conditions than standard measures such as RA and FA. We have also shown that RA and FA have the potential to give "paradoxical" results and thus must be used with caution. However, this analysis does not resolve what is perhaps the most important question in MR DTI: from all the various tensor measures, which one is the best one to characterize damage to brain tissue? As has been recently noted by Pierpaoli et al [23], the fact remains that we do not know a priori which is the best measure because this is not a theoretical question but an empirical one. It is equivalent to asking which statistical measure, e.g. the mean or the variance for example, will better describe a population. They describe different aspects of a population and therefore will be useful in answering different questions. Tensor calculus can only help by defining which measures can be used in our analysis. Which one best describes some aspect of the brain (be it a tissue type or a pathological condition, such as oedema or necrosis) can only be answered empirically, by relating the observed tensor measures with independent biological data such as histology, other imaging modalities and/or cognitive tests.

Hydrocephalus patient
The disruption observed in IC using the p:q diagram from the hydrocephalus patient is encouraging. In patients with hydrocephalus it is common to observe clinical symptoms that are thought to be associated with the disruption of deep white matter tracts. Similar findings were observed in this patient at the level of the internal capsule (IC). It has been suggested that during ventricular dilatation, these tracts are being stretched and thus become mechanically compromised. Our results support this notion by demonstrating that the MR DTI diffusion signature of the IC is altered. In particular, this disruption is not due to changes in the anisotropy of tissue but to changes in its mean diffusion. This suggests that the white matter tracts have been disrupted.

Tumour patient
In the case of the tumour patient, the p:q decomposition was useful to illustrate simultaneously changes in both the isotropic and the anisotropic components of the diffusion tensor. There was a decrease in anisotropy (q) and an increase in mean diffusion (p). These changes are thought to be associated with white matter destruction.

There are a number of limitations in using the p:q decomposition. Just like D, RA and FA, the location of tissue in the p:q plane does not give information about the directionality of diffusion. Also, there are other important tensor scalar measures that are not directly conveyed by the p:q plane, such as the eigenvalues and the tensor invariants. Finally, from a practical point of view, the p:q decomposition must be applied to other brain pathologies in order to establish how beneficial it might be in those situations.


   Conclusion
Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 Conclusion
References
 
The p:q tensor decomposition enhances the visualization and quantification of MR DTI data in both normal and pathological conditions. In particular it is an aid to visualize simultaneously seven scalar tensor measures. We have also shown the pitfalls of using FA and RA exclusively, and the potential of using other tensor measures, particularly q. However, it is important to note that, despite the enhanced visualization and quantification provided by our technique, the choice of which tensor scalar measure best describes brain tissue and its changes remains an empirical matter. We hope that the enhanced repertoire of analysis tools that we propose might enable improved categorization of tensor abnormalities in pathology.


   Acknowledgments
 
AP is in receipt of a Wellcome Trust Fellowship in Mathematical Biology. The Cambridge Commonwealth Trust supports HALG. The Medical Research Council Technology Foresight grant and the Wolfson Foundation support the Wolfson Brain Imaging Centre. We acknowledge the help of radiographers Tim Donovan, Victoria Lupson and Ruth Bisbrown-Chippendale, the many useful discussions with Dr Neil G Harris, Dr Brian K Owler, Dr Luzius A Steiner and Dr Shahan Momjian, as well as the excellent computing support of Mr Julian Evans.

Received for publication April 22, 2003. Revision received May 24, 2005. Accepted for publication June 1, 2005.


   References
Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 Conclusion
 References
 

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