## Peter Mansfield

Print publication date: 2013

Print ISBN-13: 9780199664542

Published to Oxford Scholarship Online: January 2013

DOI: 10.1093/acprof:oso/9780199664542.001.0001

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# (p.227) Appendix Elementary Principles of MRI

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There are four main variants of magnetic resonance imaging (MRI): (1) point scanning, (2) line scanning, (3) planar imaging, and (4) volumar imaging. All other MRI methods are variants or combinations of these four basic methods. In addition to these four there is the possibility of performing scans in either real space or reciprocal lattice space, often referred to as r-space or k-space. Many years ago Nobel Laureate Professor Richard Ernst showed that in nuclear magnetic resonance (NMR) it is better, from the signal/noise viewpoint, to work in the time domain rather than the frequency domain. In my own work on imaging, I have always worked in k-space.

The great French mathematician Joseph Fourier (1768–1830) invented a mathematical transformation, nowadays called the Fourier transform, that allows calculations performed in normal space, or r-space, to be transformed to an equivalent calculation in k-space, and vice versa.

In standard NMR, nuclear signals from an assembly of spins in a liquid, or liquid-like sample, may be observed as a function of angular frequency w, the absorption line shape, or as a function of time t, the transient signal. Again, signals in the two domains w,t, referred to as conjugate variables, are related by the Fourier transform. That is to say

$Display mathematics$
(1)

This is the standard Fourier transform used in NMR spectroscopy. However, in MRI we require a slightly modified (p.228) expression, because now our conjugate variables are position r and wave vector k. That is to say

$Display mathematics$
(2)

where

$Display mathematics$
(3)

in which ρ (r) is the spin density as a function of position r, G(t) is the time-dependent magnetic field gradient, and g is the magneto-gyric ratio of the particular spin species under observation. For example, protons have a g given by

$Display mathematics$
(4)

where g = 2 π × 42.577 MHz/T.

So far as imaging is concerned, it is always efficacious to use planar imaging over either point scanning or line scanning. Indeed, the most efficacious method of all is volumar imaging. However, to employ this technique requires an exceptionally fast computer with large memory in order to deal with the large data arrays involved. For this reason, echo volumar imaging (EVI) has so far been limited to data arrays describing eight contiguous planes, each plane being 1 cm thick and comprising 64 × 64 voxels. In this arrangement a Fourier transform of 32 k points was required taking approximately 90 msec. For the whole body images obtained using EVI, the typical voxel size was 5 × 5 × 10 mm3. However, an optimal voxel size to aim for would be 8 mm3, i.e. a factor of 32 smaller than currently achieved in 90 msecs. Ideally, the total imaging time should be kept at around 32 msec to circumvent T1 , T2 effects. These requirements are not practical with currently available equipment.

For example, with a voxel size of 8 mm3 and a total imaging time t = 32 msec scaling the currently achievable EVI (p.229) parameters, we have t = 32 msec = 3.2 × 104 μ sec = 1.28 × 105 t m sec, giving the imaging time per voxel element t = 0.25 m sec.

# Definitions

• T1 is the spin–lattice relaxation time. For human tissue at body temperature, this lies typically in the range 100 sec–1 sec.

• T2 is the spin–spin interaction time. In biological tissue at normal body temperature, T2T1.

• ρ(ω), ρ(r) are tissue densities lying typically in the range 0.8–1.0 gm cm–3.

• t, t are times

• ω is angular frequency

• i $= - 1$

• S(ω), S(k) are received nuclear signals

• k is the wave vector

• B is the magnetic field strength

• r is the position vector

• γ is the magneto gyric ratio = 2 π f

• f is the spin resonance frequency in a magnetic field of 1 Tesla. For protons this corresponds to a frequency f = 42.577 MHz.

• G is a magnetic gradient vector

• e = 2.8

• MHz is mega Hertz

• T is Tesla = 104 Gauss = magnetic field strength

(p.230)