Clearly, a division of SI(gj,r,i1) by SR(gj,r,�) will remove the diffusion term and leave the ratio

[3.73]

Taking the inverse of tangent function from both sides, the velocity can be

revealed

as

9 6

By

plotting tan- 1 (ratiovi) vs ('Y8gi�) for each q slice in which gi is the variable, the

velocity can

_be

obtained from the slope of the straight line fitted through these points.

There are two obvious disadvantages in the line fitting of

_Eq[3.74].

First, due to the discontinuity property of the tangent curve at ±mt/2, (n=

₁

,2, .. ), the velocity values at and near these regions would

_be

lost or contain large errors. This would be a problem if the sample has a wide flow field. The second disadvantage is, due to the periodic property of the tangent curve, there is a possibility of one velocity digital value representing two or more different velocities with phase angles n1t apart.

The self-diffusion coefficient can also be calculated through a similar least squares fit method, named Stejskal-Tanner method[50]. It is the classical method used widely in conventional

NMR

to measure the self-diffusion coefficient for liquid samples. This approach firstly takes the modulus of the complex data so that the phase information contained in the oscillatory term are lost while the decay information remains unaltered, as

[3.75]

Then all the pixels are normalized to the first pixel

_(q=O)

so that the amplitude

terms disappear. By taking the natural logarithm from both sides of

_Eq[3.75],

the self­

diffusion

can _be

revealed as

1

D = -

_{'Y 8 gi (1l.-8/3) 2 2 2}

In(modulusi)

[3.76]

By plotting the normalized

intensity against

_{['Y282gi2(�-8/3)]}

in

which

g

j₂ is

the

variable, the diffusion

coefficient

can be

calculated

from

the slope of the fitted line.

In

the

calculation of the pixel modulus using _Eq[3.75], the influence of the

noise power has to be taken into account.

By

denoting Re and

1m

as the true signals

and NR and NI

as

the real and imaginary part of the noise, the apparent image pixel

amplitudes

can be

written as

Re' = Re + NR

[3.77a]

9 7

Therefore the mcxiulus of the image pixels are

[3.78]

. The first term in

Eq[3.78]

is the true mcxiulus of the signal, the second term is

the contribution from the noise power which has to be subtracted, the last term is the

random noise term. Therefore in our software, instead of using

Eq[3.75]

directly, the

noise power for each pair of q images is calculated first using the first

128

pixels at the

edge of the images (for which Re and 1m will

be

zero) as

Noise(q)2 = �(Re'(q)

/

+ 1m' (q)j2) I

[3.79]

where the sum is over the i= 1 to

128

pixels. Then the mcxiulus becomes

mcxiulus(q) = (Re(q)2 + 1m(q)2 - Noise(q)2)

1/2

[3.80]

In the implementation of the least squares fit method, the scatter of the data makes the weighting important. For a set of q data, the first pixel is always the most reliable one because the signal is the highest. The data becomes much less reliable with the increase of the gradient g. There are many ways of weighting a set of discrete data in literatures. A double least squares fit approach is used in our software. That is, in the first time an unweighted least squares fit is performed, the slope b and the intercept a are calculated. Then a weighted fit is performed for the second time using the weighting factor of l/CJi2, in which the standard deviation CJ is calculated from the relationship Yi=a+bxi.

These two least squares fit methods have been used in our experimental data analysis, some results will be presented in Ch

6

to Ch 9. A short conclusion is given here first. The least squares fit methods can be used to analyze the data in Dynamic

NMR Microscopy experiments. If the signal-ta-noise ratio of the data images are good, these two methods should give more accurate results than the FFf method, especially in the case of the self-diffusion calculations where the least squares fit method avoids some artifacts associated with the conjugate domain. But if the signal-ta-noise ratio of

9 8

the data images is not very good, the FFf method is far better than these least squares

fit methods. This is due to the 'noise-resistant' algorithms used in the FFf method and

the procedure of zero filling in the conjugate space.

3 . 7

'One-shot' velocity m icroscopy

The ability to simultaneously construct velocity and self-diffusion images in Dynamic NMR Microscopy requires comprehensive information about the spin system

everywhere in both k space and _q space. Experiments are carried out by stepping

through a sequence of PGSE gradients and repeating the k-space imaging at each q value. Dynamic NMR Microscopy, in its simplest form, is essentially a four­ dimensional approach and is therefore time consuming. In situations where a velocity image is of principal interest, one can reconstruct velocity map from a single value of PGSE gradient[87. 941. This reduces the experimental time significantly.

The 'one-shot' velocity micro-imaging method used in this thesis utilizes the gradient phase cycling[94] which nulls the signals from stationary spins and produces

the sinusoidal dependency of the image intensity on the velocity of the moving spins.

_A

higher order suppression of stationary spin signal can be achieved by combining

gradient phase alternation with the use of a final rf 'z-storage' pulse[87] which rotates

the transverse magnetization of the stationary spins to the z-axis.

_A

four-quadrant

interpreting routine is also developed in our method to calculate the velocity from the

phase shift between

0

and 21t.

3 . 7 . 1 G rad ient p hase cycli n g and 'z-storage' rf p u lse

The stepping of PGSE gradient in Dynamic

NMR

Microscopy imposes

a

In document Dynamic NMR microscopy : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University (Page 116-119)