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

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

View larger version (14K): [in a new window]   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 , 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.

View larger version (50K): [in a new window]   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.

View larger version (31K): [in a new window]   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 , 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.

View larger version (42K): [in a new window]   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.

View larger version (37K): [in a new window]   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.