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15.2 Acquisition of DT Images

15.2.1 Fundamentals of DTI

DT images can be obtained using Nuclear Magnetic Resonance (NMR). NMR measures the frequency of precession of nuclear spins such as that of the proton (1H), which in a magnetic field B0, is given by the Larmor equation:

ω0 = γ B0.

(15.7)

The key to achieving spatial resolution in MRI is the application of time-dependent magnetic field gradients that are superimposed on the (ideally uniform) static magnetic field B0. In practice, the gradients are applied via a set of dedicated 3-axis gradient coils, each of which is capable of applying a gradient in one of the orthogonal directions (x, y, and z). Thus, in the presence of a magnetic field gradient g,

g =	∂ Bz	,	∂ Bz	,	∂ Bz	(15.8)
	∂ x		∂ y		∂ z

the magnetic field strength, and hence the precession frequency become position dependent. The strength of the magnetic field experienced by a spin at position r is given by:

B = B0 + g ·r

(15.9)

The corresponding Larmor precession frequency is changed by the contribution from the gradient:

∂ φ(r)

ω(r) = ∂ = γ (B0 + g ·r). (15.10) t

The precession frequency ω is the rate of change of the phase, φ, of a spin – that is, its precession angle in the transverse plane (Fig. 15.3). Therefore, the timedependent phase φ is the integral of the precession frequency over time. In MRI, we switch gradients on and off in different directions to provide spatial resolution, so the gradients are time dependent and the phase of a spin is given by:

t	t
φ(r, t) =	ω(r, t )dt = γB0t + γ g(t ) ·rdt .	(15.11)
0	0

We observe the phase relative to the reference frequency given by (15.7). For example if the gradient is applied in the x direction in the form of a rectangular pulse of amplitude gx and duration δ the additional phase produced by the gradients is

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Fig. 15.3 The effect of a magnetic field gradient on precession of spins. A constant magnetic field gradient g (illustrated by the blue ramp) applied in some arbitrary direction changes the magnetic field at position r from B0 to a new value B = B0 + g ·r. The gradient perturbs the precession of the spins, giving rise to an additional position-dependent phase φ , which may be positive or negative depending on whether the magnetic field produced by the gradient coils strengthens or weakens the static magnetic field B0

δ
φ (r, t) = γ gx(t )xdt = γδgxx = 2πkxx,	(15.12)
0

where the “spatial frequency” kx = γδgx/2π is also known as the “k value”. It plays an important role in the description of spatial encoding in MRI and can be thought of as the frequency of spatial harmonic functions used to encode the image.

In MRI to achieve spatial resolution in the plane of the selected slice (x, y), we apply gradients in both x and y directions sequentially. The NMR signal is then sampled for a range of values of the corresponding spatial frequencies kx and ky.

For one of these gradients (gx, say), this is achieved by keeping the amplitude fixed and incrementing the time t at which the signal is recorded (the process called ‘frequency encoding’).

In the case of the orthogonal gradient (gy), the amplitude of the gradient is stepped through an appropriate series of values. For this gradient, the appropriate spatial frequency can be written:

δ
ky = γ gy(t )dt = γ δ gy/2π.	(15.13)

0

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Fig. 15.4 Gradient pulse pairs used for diffusion attenuation. The first gradient sensitizes the magnetisation of the sample to diffusional displacement by winding a magnetization helix. The second gradient rewinds the helix and thus enables the measurement of the diffusion-attenuated signal. The two gradients must have the same amplitude if they are accompanied by the refocusing RF π pulse; otherwise, their amplitudes must be opposite

The MR image is then generated from the resulting two-dimensional data set S(kx, ky) by Fourier transformation:

S(x, y) = S(kx, ky)e−2πi(kx x+kyy)dkxdky. (15.14)

The Fourier transform relationship between an MR image and the raw NMR data is analogous to that between an object and its diffraction pattern.

15.2.2 The Pulsed Field Gradient Spin Echo (PFGSE) Method

Consider the effect of a gradient pair consisting of two consecutive gradient pulses of opposite sign shown in Fig. 15.4 (or alternatively two pulses of the same sign separated by the 180◦ RF pulse in a ‘spin echo’ sequence).

It is easy to show that spins moving with velocity v acquire a net phase shift (relative to stationary spins) that is independent of their starting location and given by:

φ(v) = −γ g ·vδ ,	(15.15)
where δ is the duration of each gradient in the pair and	is the separation of

the gradients. Random motion of the spins gives rise to a phase dispersion and attenuation of the spin echo NMR signal.

Stejskal and Tanner [3] showed in the 1960s that, for a spin echo sequence this

additional attenuation (Fig. 15.5) takes the form:
S( , g) = S0e−TE/T2 e−Dγ2g2δ2( −δ/3).	(15.16)

The first term is the normal echo attenuation due to transverse (spin-spin) relaxation. By stepping out the echo time TE, we can measure T2.

The second term is the diffusion term. By incrementing the amplitude of the magnetic field gradient pulses (g), we can measure the self-diffusion coefficient D.

For a fixed echo time TE, we write:

S = S0e−bD = S0e−TE/T2 e−bD,

(15.17)

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Fig. 15.5 A pulsed field gradient spin echo (PGSE) sequence showing the effects of diffusive attenuation on spin echo amplitude

	where			δ
	b = γ2g2δ2	−				(15.18)
					.
				3
	The ADC is then given by:
	ADC = −ln	S			b	(15.19)
		S
		0

For the case of anisotropic diffusion described by a diffusion tensor D, the expression for the echo attenuation in a PFG spin echo experiment becomes:

S( , g) = S0e−γ2g·D·gδ2( −δ/3),

(15.20)

where g = (gx, gy, gz) is the gradient vector, and the scalar product g ·D ·g is defined analogously to (15.5).

Overall, if diffusion is anisotropic, the echo attenuation will have an orientation dependence with respect to the measuring gradient g. Gradients along the x, y, and z directions sample, respectively, the diagonal elements Dxx, Dyy, and Dzz of the DT. In order to sample the off-diagonal elements, we must apply gradients in oblique directions – that is combinations of gx and gy or gy and gz, etc. Because the DT is symmetric, there are just 6 independent elements. To fully determine the DT therefore requires a minimum of 7 separate measurements – for example:

gx	0		g	0	0	1		g	1		g	1		0
gy	= 0		0		0						0
		,		, g ,		, √		g , √				, √		g .
							2			2			2
gz	0		0	0	g			0			g			g

(15.21)