form. Two examples of fully automatic approaches are atlas-based segmentation and statistical shape models. No automatic methods are perfect, but given the laborious nature of manual segmentation and the pressures on clinicians’ time, segmentation is a very active research field.

11.6 Diffusion Imaging: An Illustration of a Full Pipeline

Although the physics of the 3D imaging modalities can be considered quite complex, involving different wavelengths, different fields, radioactivity and so on, it could be argued that the main output is fairly simple in terms of the techniques described in this chapter. Essentially, we get grayscale images. But this is misleading, medical imaging can also provide dynamic frame images from ultrasound, dynamic MRI or CT of the heart, contrast enhanced images showing the evolution of a contrast agent (MRI, CT, PET), flow images, and even tensor matrix images. All of this requires sophisticated post-processing and data analysis. Clearly, it is not possible to describe every technique, rather, in this section, we concentrate on tensor imaging. This relates to the most complex objects (tensors) and illustrates issues with other techniques, thus we use it as a case study of data processing in medical imaging. In particular, we use this to illustrate that nice images do not come for free, but only after processing steps that all require care. Specifically, we see the steps:

1.From raw images to the object which is a tensor field; this requires spatial alignment of different images, the input is a set of scalar images; the output consists of tensor images;

2.Data processing from the tensor field to a field of glyphs: input is a tensor image; the output is a visualization (glyphs), or vector images (showing principal direction of diffusion);

3.Information extraction from the glyph shapes, first pointwise: the input is a tensor image; the output consists of different quantitative scalar images;

4.We can also extract global information by connectivity: the input is a vector image, the output consists of a collection of 3D streamlines.

Processing requires different steps mentioned in previous sections, such as segmentation to delineate white matter, smoothing and thresholding to stop lines with unrealistic shapes and so on.

11.6.1 From Scalar Images to Tensors

This entire chapter tries to convey the message that the underlying physics of imaging cannot be ignored. Understanding where the images come from, and why they look like how they do is important. In some cases one might waste months of work trying to perform an imaging operation that would have just required a radiographer

Fig. 11.19 Diffusion weighted images with corresponding gradient directions below. Note that differences are subtle—for each gradient direction (red line in lower row), the image has darker voxels when it contains structures forcing diffusion in that direction. For this reason, such images are rarely used directly, but replaced by the FA, MD, etc. below, which are orientation independent

to change scan settings or inject a different contrast agent. In diffusion MRI, we can and do manipulate magnetic field gradients in different directions, g. As a second order diffusion tensor can be represented by a symmetric 3 × 3 matrix, it contains six independent components. Thus, provided we have a model of the physics of how these gradients affect images, we should be able to reconstruct the matrix. The images in Fig. 11.19 show the diffusion weighted images corresponding to the gradient directions below.

One model, the default for diffusion imaging, is that each of these 6 images obeys the equation S(g) = S0 exp(−bgt Dg) where g = [x, y, z]T is a unit vector in the gradient direction, the scalar b comprises several acquisition parameters, such as gradient strength and duration, and D is the unknown tensor matrix with unknowns [Dxx , Dyy , Dzz, Dxy , Dxz, Dyz]. A more compact way to describe them is via a matrix, or tensor D, which is a table of numbers called diffusion coefficients:

Fig. 11.20 Components of the diffusion tensor image. In other words, this is a matrix of images, or the image of a matrix (tensor). This provides very rich information, but MD shown in Fig. 11.21 and FA shown in Fig. 11.22 are easier to interpret directly, although they have lost information

By solving the equations using least squares, we can get tensor images as output, as shown in Fig. 11.20.

11.6.2 From Tensor Image to Information

Although, in theory, the diffusion tensor image in Fig. 11.20 contains all available information on diffusion, it is hard to interpret. For this reason, we construct images which display the information in a more intuitive way. The main tool for this is that the 3 × 3 diffusion tensor can be decomposed in orientation and anisotropy information. Shape information is encoded by three numbers describing the specific diffusivities in three orthogonal directions.

The Mean Diffusivity (MD) describes how ‘strong’ the diffusion is, or how unhindered, or how far in average water molecules diffuse. It is a mean, over all directions, and thus contains no directional information. To interpret it, higher MD, i.e. with a light to white contrast, correspond to regions where there are few barriers. For example, a region with higher MD than normal could describe a degradation of the

Fig. 11.21 An image of Mean Diffusivity (MD). Bright areas correspond to voxels containing water able to diffuse relatively freely.

Dark regions contain more restricted water molecules

cellular structure. On the other hand, dark regions indicate that the water molecules find it harder to move around. Macroscopically it would be as if the medium has become more viscous, or microscopically cells could have swollen. In units, MD has the same units as the diffusion tensor, mm2/s, see Fig. 11.21.

The Fractional Anisotropy (FA) measures, as its name suggests, how anisotropic the diffusion is. It is an index, thus has no units, and ranges between 0–1, where 0 indicates fully isotropic diffusion and 1 indicates that diffusion happens in a single direction. The definition of FA is the standard deviation of the specific diffusivities, suitably normalized. Typically, cerebro-spinal fluid (CSF) has a low FA as it is essentially a liquid, so should not have a preferred direction. FA is high in white matter, in particular in the Corpus Callosum, or in any fibre like structure such as muscle fibres. (See Fig. 11.22a.)

Both MD and FA indicate nothing about direction. A useful way to do this is to use colors. The three components in RGB are matched to spatial directions. Note the blue large balls in the ‘smarties’ picture above a FA region. They indicate strong diffusion, but not very directional. The smaller ellipsoids show that in regions where diffusion is directional it is also much smaller (see Fig. 11.22b).

Fig. 11.22 An image of Fractional Anisotropy (FA). Bright regions indicate high anisotropy; note that this information complements the MD, also note that ventricles, for example, contain fully isotropic voxels (dark in FA) as they contain Cerebro-Spinal Fluid (CSF), but the corresponding voxels are bright in MD, as the water molecules are essentially free. Regions of the Corpus Callosum, on the other hand, contain lots of white matter fibres pointing in a single direction thus they appear dark in MD, as molecules cannot move much, but bright in FA as, when they move, they do so preferably in a single direction. (b) Colored FA, with colors indicating direction