In order to sensitise a pulse sequence to diffusion, gradient pulses are utilised. Spins m oving through a m agnetic field gradient experience a phase shift since the gradient defines the associated resonant frequency at each point. An expression for the phase shift, (j), is
( j ) = Y \r.G (t) dt'
[2.2]
w here y is the gyrom agnetic ratio, G(t) is the gradient at tim e t, and r is the distance m oved by the spin during this time. For an im aging voxel whose spins diffuse through such a gradient, the distribution o f phases will be of a gaussian form around a m ean of zero and a variance displacem ent given by Eq. [2.1]. In order to rephase these spins, a gradient can be applied with opposite polarity or alternatively, with the sam e polarity in com bination with a 7C pulse. For static spins, perfect rephasing then occurs. H ow ever, if the spin has moved during the application of these gradients, the phase shift accum ulated during the first gradient is not reversed by the phase dispersion caused by the second gradient pulse since the spin’s position and, therefore, phase will have altered. A net attenuation of the signal results due to incom plete rephasing of diffusing spins that can be described by
S = Sq e x p (-^ D )
[2.3] w here S and So are the signal levels before and after application o f the gradients respectively; D is the diffusion coefficient (usual units : [mm^/sec]) and the b-value is a factor that is dependent on the dephasing pow er o f the diffusion gradients (usual units : [sec/mm^]). Figure 2.1 depicts the effects of diffusion gradients on a spin in the rotating reference frame.
C h a p te r 2 C oncepts o f D iffusion and Perfu sio n 7z/2 p u ls e D i f f u s i o n g r a d i e n t I 71 p u ls e D i f f u s i o n g r a d i e n t I I S t a t ic s p i n s M o v i n g s p i n s
0
' R esultant signal Sq S o e x p ( - b D )(a)
( b) (c)(cl)
Fig 2.1 T h e e f f e c t o f d i f f u s i o n g r a d i e n t s on a s i m p l e 3 - s p i n s y s t e m in t he r o t a t i n g r e f e r e n c e f r a m e d u r i n g a s p i n - e c h o c o m b i n a t i o n o f p u l s e s . T h e d i f f u s i o n g r a d i e n t s are s e p a r a t e d b y a 7t p u l s e a n d t h e m a g n e t i s a t i o n state o f a set o f thr e e s p i n s is s h o w n at t he e n d o f t he t w o d i f f u s i o n g r a d i e n t s (b,c). T h e initial a n d final m a g n e t i s a t i o n s t at e s o f t h e s e s p i n s are d e p i c t e d (a,d). 43C h a p te r 2 C o n ce p ts o f D iffu sio n a n d P e rfu sio n
In this manner, a value for D can be obtained by m easuring the signal attenuation obtained at a number of different diffusion gradient strengths (i.e. at a num ber o f b- values) and fitting to Eq. [2.3]. M aps of the diffusion coefficient can be obtained by fitting each pixel in the series of differently diffusion-w eighted images. The classic pulse sequence for ADC m easurem ent was devised by Stejskal and Tanner (Stejskal, 1965) and is based upon a spin-echo 2D-FT sequence. It consists of a pair of diffusion gradients placed around the n pulse (see Fig. 2.2).
The b-value for this com bination of gradient pulses is approxim ately equal to
A --- 3
[2.4] w here Ô and A are as shown in Fig. 2.2 and the effects of the gradient ramps have not been taken into account. The b-value can be calculated for any com bination of gradients applied along any com bination of directions, with the generalised form ula (see ref. (Basser, 1992)) given by r 2 b =
J | / : ( 0 |
[2.5] t'= 0 with k(t) = y jG ( 0 ir/t' t'=0 RF nA
U GF ig 2.2 Stejskal-Tanner diffusion-w eighted sequence (Stejskal, 1965) show ing
C h a p te r 2 C o n ce p ts o f D iffu sio n and P e rfu sio n
D iffusion gradients can be incorporated into a large variety o f N M R pulse sequences such as EPI (Turner, 1990), steady state free precession (SSFP) (Le Bihan, 1989) and stim ulated echo sequences (M erboldt, 1992). M otion artefacts are a serious problem in diffusion im aging since the sequence is sensitised to sm all-scale m otion. As faster im aging sequences have becom e more w idespread on conventional M R I scanners, much w ork has been carried out to optim ise the application of diffusion sensitisation to these sequences. C hapter 6 discusses the suitability of the TurboFLA SH and F P I pulse sequences for rapid, quantitative diffusion imaging.
2.1 .2 D iffusion anisotropy an d the diffusion trace
D iffusion is a three-dim ensional process and w ater m obility may be favoured along one axis o f a structure. The m easured ADC along this orientation will be larger than other directions. The directional dependence o f diffusion is known as diffusion anisotropy and is especially apparent in brain white m atter in w hich diffusion m obility is greater along the nerve axon fibres. A long other directions, diffusion is considered as being
restricted. Studies have also indicated that even grey m atter displays a certain degree of anisotropy (Lythgoe, 1997). The diffusion coefficient is therefore better represented
w ithin brain tissue by a rank-3, 2D tensor, D, that takes into account m olecular
displacem ents in the x, y and z directions (diagonal terms) and also their possible coupling terms (non-diagonal terms). The elements o f the tensor are denoted by the notation Djj where the elem ents i and j can take any o f the three gradient directions x, y and z (Basser, 1994).
C erebral ischaem ia is reflected in an acute reduction of the diffusive property o f tissue w ater. Im portant investigations of the pathophysiology o f ischaem ia have been undertaken using the contrast created by diffusion-w eighted im aging. The effects of diffusion anisotropy com plicate the interpretation of diffusion-w eighted images. V ariations in intensity across the image may reflect the geom etric relationship between the acting diffusion encoding gradients and the alignm ent of the structural organisation o f the tissue rather than the inherent m obility of w ater m olecules in an environm ent that has no directional-dependence. Contrast between ischaem ic and norm al tissue can hence be im paired. D irection-insensitive and, therefore, more generalised ADC m easurem ents are provided by scalar invariants of the tensor. One o f these param eters is provided by the m athem atical trace of the diffusion tensor that essentially represents the directional+y averaged ADC. The effects of adverse anisotropic contrast are hence
C h a p te r 2 C o n ce p ts o f D iffu sio n an d P e rfu sio n
avoided. The trace of a matrix is m athem atically invariant under rotations (M ori, 1995), and in the context of the diffusion tensor D, is denoted Trace(D ) = 3.Dav and is described by the relationship
Tmcg(D') = D,, +D,, '= D,, +
= Tmcg(D) =
[2 .6] where Dxx' is the A D C along the principal axis, x' (given by the eigenvector space of the diffusion tensor); D' is the diffusion matrix of the system within the space described by the principal axes; Dxx is the ADC along the laboratory x-axis; D is the diffusion matrix in the laboratory frame. The m atrices D and D are related by a rotation matrix (Basser, 1994).
A third of the trace of the diffusion tensor, l/3.T race(D ) ^ (also denoted Dav), can be m ost easily determ ined by obtaining separate diffusion-w eighted m easurem ents with the diffusion encoding gradients placed along orthogonal directions and by averaging the result (van G elderen, 1994). H owever, the effects of cross-talk from the im aging and background gradients (see Section 6.1) com plicate the acquisition of the purely unidirectional diffusion-w eighted im ages that represent the diagonal elem ents of the diffusion tensor (Basser, 1995). Cross-term s of the diffusion tensor result from interactions o f gradients in orthogonal directions. In addition, this m ethod does not represent a tim e efficient use of the gradient power. Single-shot sequences have therefore been developed that em ploy a set of gradient pulses applied along m ultiple directions in order to obtain the trace(D) within one scan. M ori et al. have proposed a gradients schem e in the three gradient axes to cancel the off-diagonal elem ents of the diffusion tensor (M ori, 1995). Cross-term s are avoided by the im m ediate refocusing of the diffusion gradients, which are arranged as a com bination of bipolar pairs. H owever, the gradient efficiency of these sequences is sub-optim al and numerical optim isation m ethods have been em ployed to im prove the relative efficiency o f such pulse sequences (W ong, 1995). This provides a schem e of tetrahedral diffusion encoding gradient w aveform s. This technique was im plem ented as part o f the investigations described in this thesis and is further described in Section 6.3.
C h a p te r 2 C o n ce p ts o f D iffu sio n an d P erfu sio n