Cramer Rao Minimum Variance Bound

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The contribution of image noise to imprecision in the final parameter can be calculated. If a simple ratio of images is used (for example magnetization transfer ratio - see Chapter 8), then propagation of errors (Taylor, 1997) allows the effect of noise in each source image to be calculated. An analytical expression can be derived for the total noise, available online. The standard work is the Guide to the Expression of Uncertainty in Measurement (GUM), published by the International Standards Organization (ISO) in 1995. There is much commercial activity in this field, as organizations selling measurement services seek ISO accreditation. Many organizations produce guidance on 'the expression of uncertainty in measurement', and publish user-friendly versions of GUM. For example, the United Kingdom Accreditation Service publishes an excellent report 'The expression of uncertainty in testing', downloadable from (ref: LAB 12), and also the booklet 'M3003, 1997, The Expression of Uncertainty and Confidence in Measurement'. Eurochem offers a detailed guide with many examples in analytical chemistry ( European co-operation for Accreditation (EA) also has an excellent downloadable document 'Expression of the Uncertainty of Measurement in Calibration' (report EA-4/02; see The American Association for Laboratory Accreditation has a very helpful website, and the 'A2LA Guide for the Estimation of Measurement Uncertainty In Testing' can be downloaded ( See also 'A Beginner's Guide to Uncertainty of Measurement', Measurement Good Practice Guide no. 11, by Stephanie Bell, from

and this can be minimized as a function of imaging parameters such as TR and the number of averages, keeping the total imaging time fixed (see e.g. Tofts, 1996). If least-squares curve fitting is used to estimate a parameter from more than two images, simple noise propagation will not work, as the fitted parameter is not a simple function of the source images. However the Cramer-Rao minimum variance bound (van den Bos, 1982; Cavassila et al., 2001) is an analytical method making use of partial derivatives that does calculate the effect of image noise on the fitted parameters. The LC model for estimating spectral areas uses this method to estimate the minimum uncertainty in the metabolite concentration. Only uncertainty arising from data noise is included; other factors (both random and systematic) can make the uncertainty higher than this minimum variance bound. Another way to model noise propagation is to use numerical simulation, such as the Monte-Carlo technique, where known noise is added to the source data and the effect on the fitted parameter measured.


Phantoms can be made from a single component or mixtures. Geometric objects, used for size or volume standards, are often made of acrylic. Major manufacturers are Perspex in the UK and Plexiglas in North America. These are immersed in water (doped to reduce its T1 and T2 values). Objects with a specified T1, T2 or diffusion value can be made from a container filled with liquid or gel, often with various salts added to reduce the relaxation times. Chemical compounds are available from suppliers such as Sigma-Aldrich. Phantoms should ideally be stable with known properties.

3.4.1 Single Component Liquids

These may be water, oils or organic liquids such as alkanes. They all have the advantage of being readily available, either in the laboratory, from laboratory suppliers, or from the supermarket, at reasonable prices. No mixing, preparation, weighing or cookery is required. The only equipment needed is a supply of suitable containers. Handling the alkanes should be carried out in accordance with national health and safety regulations.9

Water has the advantage of being easily available, and of a standard composition. Its intrinsic Ti & 3.3 s, T2 & 2.5 s at room temperature (see Table 3.6 below), and in its pure form these long relaxation times usually cause problems. The long Ti can lead to incomplete relaxation with sequences that may allow full relaxation with normal brain tissue (Ti & 600 ms for normal white matter at 1.5T; see Chapter 5, Figure 5.3). The long T2 can cause transverse magnetization coherences that would be absent in normal brain tissue (T2 & 90-100 ms). Doped water overcomes these problems (see the next section). The low viscosity can also cause problems, with internal movement continuing for some time after a phantom has been moved, giving an artificial and variable loss of transverse magnetization in spin echo sequences used for T2 or diffusion.

Water has another particular disadvantage when used in large volumes. Its high dielectric constant (e = 80) leads to the presence of radiofrequency standing waves (dielectric resonance), where Bi is enhanced, giving an artificially high flip angle and signal (see Figure 3.8). The high dielectric constant reduces the wavelength of electromagnetic radiation, compared with its value in free space, by a factor ^/e; at 1.5 T the wavelength is 0.52 m, comparable with the dimensions of the subject (Glover et al., 1985; Tofts, 1994; Hoult, 2000). Standing waves are also present in the head, particularly at high field (see Figure 3.9), but to a much less extent, because electrical conductivity in the brain tissues damps the resonance. Even at 1.5T this effect is significant, and early attempts to measure head coil nonunifor-mity using large aqueous phantoms are now seen as fatally flawed (Table 3.4).

Oil has a low dielectric constant (e = 2 — 3), and has been used for nonuniformity phantoms (Tofts et al., 1997a). Several kinds are available, from various sources, and their properties have been described by Tofts et al. (i997a). It is stable and cheap; cooking oil is a convenient source.

9 In the UK this involves registering the project with a safety representative, using basic protective clothing and carrying out the pouring operation in a fume cupboard.

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