1 These changes were made 11/17/2004 to these notes: 1. Added an explanatory note (in pink!) in parentheses on page 2. 2. Corrected total derivatives to partial derivatives in Eq. (4) and preceding equation on page 4 and in Eqs. (32) and (33). 11. Distortion arising from static- field inhomogeneity for Spin- Echo images 11.1. In- plane effect Assume the usual (spin- warp) SE sequence, first with no static- field inhomogeneity (i.e., 0 B ∆ = ): ( = ( = ( = 2 , , , x f y p i x G t y G f y a s t G A x y z e d x d y p g t r - + = ∫ ∫ If we add inhomogeneity: ( = ( = ( = ( = 2 , , , , , x f y p a f f y i x G t y G B x y z t a s t G A x y z e d x d y p g t r - + + ∆ = ∫ ∫ (1) ( = ( = ( = ( = 2 , , , , , x f y p a f i x G t y G B x y z t f y a s t G A x y z e d x d y p g t r - + + ∆ = ∫ ∫ We note that there are two terms in the exponent that involve f t . We combine them: ( = ( = ( = ( = ( = 2 , , , , , x a f y p i x G B x y z t y G f y a s t G A x y z e d x d y p g t r - + ∆ + = ∫ ∫ Then we divide and multiply by G x to prepare for a change of integration variable: ( = ( = ( = , , 2 , , , a x f y p x B x y z i x G t y G G f y a s t G A x y z e d x d y p g t r ∆ - + + = ∫ ∫ The new integration variable is ( = ( = , , , , a a x B x y z x x y z x G ∆ ′ ≡ + (2) It’s meaning is that, because of a shift of ( = , , a B x y z g ∆ from the frequency, x xG g , at which they should have been precessing, the spins at ( = , , a x y z , act as if they are located at , , a x y z ′ . After the change of variables, we have ( = ( = ( = ( = 2 , , , , , , x f y p i x G t y G f y a a x x s t G A x x y z y z e d x d y p g t r ′ - + ∂ ′ ∂ ′ ′ = ∫ ∫ (3) This watermark does not appear in the registered version - http://www.clicktoconvert.com
Transcript
1
These changes were made 11/17/2004 to these notes:
1. Added an explanatory note (in pink!) in parentheses on page 2. 2. Corrected total derivatives to partial derivatives in Eq. (4) and preceding equation on page 4
and in Eqs. (32) and (33).
11. Distortion arising from static- field inhomogeneity for Spin-Echo images
11.1. In-plane effect
Assume the usual (spin-warp) SE sequence, first with no static- field inhomogeneity (i.e., 0BD = ):
( ) ( ) ( )2, , , x f y pi x G t y G
f y as t G A x y z e d x d yp g t
r- +
= ò ò
If we add inhomogeneity:
( )
( ) ( )( )2 , ,
,
, , x f y p a f
f y
i x G t y G B x y z t
a
s t G
A x y z e d x d yp g t
r- + + D
=
ò ò (1)
( ) ( ) ( )( )2 , ,, , , x f y p a fi x G t y G B x y z t
f y as t G A x y z e d x d yp g t
r- + + D
= ò ò
We note that there are two terms in the exponent that involve ft . We combine them:
( ) ( ) ( )( )( )2 , ,, , , x a f y pi x G B x y z t y G
f y as t G A x y z e d x d yp g t
r- + D +
= ò ò
Then we divide and multiply by Gx to prepare for a change of integration variable:
( ) ( )( ), ,
2
, , ,
ax f y p
x
B x y zi x G t y G
G
f y as t G A x y z e d x d yp g t
r
æ öæ öD- + +ç ÷ç ÷ç ÷
è øè ø= ò ò
The new integration variable is
( )( ), ,
, , aa
x
B x y zx x y z x
G
D¢ º + (2)
It’s meaning is that, because of a shift of ( ) , ,aB x y zg D from the frequency, xx Gg , at which
they should have been precessing, the spins at ( ), ,ax y z , act as if they are located at , ,ax y z¢ .
After the change of variables, we have
( ) ( )( ) ( )2, , , , , ,x f y pi x G t y G
f y a a
x
xs t G A x x y z y z e d x d y
p g tr
¢- + ¶
¢¶¢ ¢= ò ò (3)
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which looks surprisingly like Eq. (4), which ignored through-plane distortion.
Again, the intensity can change, but this time it is a function of the derivative in a direction between the x and z axes. 11.3. Echo-Planar Imaging (EPI) We ignore the effects of slice selection for now and again alter Eq. (1), this time to correspond to Spin-Echo EPI. Note that we are assuming a sequence with only forward-going readouts:
( ) ( )( )( )( )
( )( )( )
, ,
1 1
2 , ,
, ,
, , ,
z
x x b p a x y
B x y z
z G
i x G m t y n G B x y z m t n t
a
s m n z A p G z z
x y z e d x d y d zp g t
g
r
D
- D + + D D + D
= + - ´ò
ò ò (21)
and the integral over x and y can be transformed as follows:
where yGD is the “blip” gradient, which is used to provide the phase encoding between
readouts, and ( )e f f
y y p yG G ttº D D is the “effective” gradient for the purposes of assessing the
effects of inhomogeneity. It is the gradient that would required to produce the same change in
phase by being turned on at a constant rate during ytD as yGD would by being turned on for
only only pt . Since it has so much more time to act, it is much smaller than yGD .
It is important to note here that the time dependence of the blip gradient need not be a square wave. The important thing is the effect caused by the gradient over the time that it is applied. Its time dependence may be of any shape without affecting the image, so long as the integral
( )0
p
y b bG t d tt
Dò
has the desired value. For this more general case,
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(This independence of time shape is true of all phase encoding gradients, including those used for the non-EPI sequences.) As usual, we define spatial frequencies for x and y. For EPI they are as follows:
( )e f f
x x x
y y y
k m G t
k n G t
g
g
= D
= D
Similarly, as before for spin-warp imaging, we can deduce from these expressions that
( )
( )( )
1
1e f f
x x
y y
X G t
Y G t
g
g
-
-
º D
º D
With these definitions, the x,y integral becomes
( )
( ) ( )( )e f f
, , , ,2
, , .
a ax y
x y
B x y z B x y zi x k y k
G Gx y z e d x d y
p
r
æ öæ öæ öD Dç ÷ç ÷- + + +ç ÷ç ÷ç ÷è ø è øè øò ò (22)
Now we transform the integration using
( )( )
( )( )
( )
( )( )
e f f
, ,, ,
, ,, ,
, ,, ,
x
y
z
B x y zx x y z x
G
B x y zy x y z y
G
B x y zz x y z z
G
D¢ º +
D¢ º +
D¢ º +
(23)
The Jacobian for this transformation can be shown to be equal to
1 x y z
x y z B B BJ G G G
x y z x y z
¢ ¢ ¢æ ö ¶ D ¶ D ¶ D= + + +ç ÷ç ÷ ¶ ¶ ¶è ø
(24)
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Using Eqs. (23) and (23) in Eq. (22) and inserting that into the z integration that we last saw in Eq. (21), yields
( )
( )( )( ) ( ) ( ) ( )
1
2
1
, , , , , ' , , , '
1
, ,
x y
x y
i x k y k
zx y z
x x y z y x y z z x y z
B B BG G Gx y z
s k k z
A p G z z d z e d x d yp g
rg
æ öç ÷ ¢ ¢- +è ø
¢ ¢ ¢ ¢
¶ D ¶ D ¶ D+ + +¶ ¶ ¶
=
æ öç ÷¢ ¢ ¢-ç ÷è ø
ò ò ò
Taking the inverse Fourier transform gives
( ) ( ){ } ( ) ( )
( )( )( ) ( ) ( )( )
1 1211 1 1 1 1 1
1 1 1 1 1 1
1
, , , , , , ,
, , , , , , , ,,
1
x yi x k y k
x y x y x y
z
x y z
i x y z s k k z x y s k k e d k d k
x x y z y x y z z x y zB p G z z d z
B B BG G G
x y z
p g
rg
+ +-= =
¢ ¢ ¢¢ ¢= -
¶ D ¶ D ¶ D+ + +
¶ ¶ ¶
ò ò
ò
F
(25)
which is very similar to Eq. (7). Making the same assumptions that we made after Eq. (7) but including changes with respect to y as well as x and z yields
( ) ( )( )( )( ) ( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( )( )
1 1 1 1 1 1 1 1 1
1 1 1 1
1 1 1 1 1 1 1 1 1
0 1 1 1 1 1 1 1 1 1
, , , , , , , ,, ,
1
, , , , , , , ,
1
, , , , , , , ,,
1
z
x y z
x y z
x y z
x x y z y x y z z x y zi x y z B p G z z d z
B B BG G G
x y z
x x y z y x y z z x y zC
B B BG G G
x y z
i x x y z y x y z z x y z
B B BG G G
x y z
rg
r
¢ ¢= -¶ D ¶ D ¶ D
+ + +¶ ¶ ¶
=¶ D ¶ D ¶ D
+ + +¶ ¶ ¶
=¶ D ¶ D ¶ D
+ + +¶ ¶ ¶
ò
(26)
where 0i is, as before, the image that would be acquired if there were no inhomogeneity, and
( )( )
( )( )
( )( )
1
1
1
, ,, ,
, ,, ,
, ,, ,
x
y
z
B x y zx x y z x
G
B x y zy x y z y
G
B x y zz x y z z
G
Dº +
Dº +
Dº +
(27)
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Where these equations are similar to Eqs. (8) and (9). As before, we insist that the Jacobian for our transformation, Eq. (with primes replaced by subscript ones) (24), is nonzero
Since the transformation in Eqs. (9) is one-to-one, we can rewrite the last of Eqs. (8) as
( ) ( ) ( )( )( )
( ) ( ) ( )
1 1 1
0
, , , , , , , ,
, , ,
, , , , , ,1 x y z
i x x y z y x y z z x y z
i x y z
B x y z B x y z B x y zG G G
x y z
=
¶ D ¶ D ¶ D+ + +
¶ ¶ ¶
(28)
As before, we can rotate our coordinate system to make things simpler. By defining a new coordinate system rotated so that
( )
( ) ( ) ( )
( )
( ) ( ) ( )
22 2
1 1 1
1 22 2
1 1 1
1 1 1
x y z
x y z
x y z
x y z
x G y G z Gx
G G G
x G y G z Gx
G G G
+ +=
+ +
+ +=
+ +
%
%
(There are an infinite number of such coordinate systems!), we can show that
( )
( ) ( ) ( )
1
1
1
1
, ,,
,
,
, , , , , ,1 .x y z
B x y zx x
G
y y
z z
B x y z B x y z B x y z xG G G
x y z x
D= +
=
=
¶ D ¶ D ¶ D ¶+ + + =
¶ ¶ ¶ ¶
% %%% %
%
% %
% %
%
%
(29)
where
( ) ( ) ( )
2 2 2
2 2 21 1 1 1 x z z
x y z
x y y z z x
G G GG G G G
G G G G G G= + + =
+ +
% (30)
With these relationships Eq. (28) becomes
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11.5 EPI for back-and-forth readout EPI rarely has forward-going readouts only. Typically the readouts go back and forth. The x,y integral in that case has the same form for even values of n, but for odd values, it looks like this:
We alter the signal in this case by reversing the sign of kx for odd values of n, where
( )e f f
y y yk n G tg= D .
As result we have
( ) ( )
( ) ( )( )e f f
, , , ,2
1, , , , ,
a ax y
x y
B x y z B x y zi x k y k
G G
x ys k k z x y z e d x d yp
r
æ öæ öæ öD Dç ÷ç ÷- ± + +ç ÷ç ÷ç ÷è ø è øè ø= ò ò
where the ± is a plus for even n and minus for odd n. This change of sign means that our change of variables for x no longer works. The effect on the image is not at all obvious. It causes a “ghost” image to appear above and below the true image. The effect is called “N-over-2 ghosting”, and it is called that because the ghost appears to be a faint, rough facsimile of the “unghosted” image shifted up N/2 pixels and another version shifted down N/2 pixels. (For those who are interested in the reason for this effect, it is a consequence of the fact that even lines are slightly different from odd lines. The
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mathematical description of this effect involves multiplying in k-space by a function that changes from one to zero and back as we move upward (y-direction) in k-space by one discrete unit. Multiplying by a modulation function in k-space is equivalent to convolving with a kernel in image space, where the kernel is the Fourier transform of the modulation function. The Fourier transform for the modulation function in this space is a kernel that equals zero everywhere except at the origin and at positions that are above or below the origin at a distance of / 2q N± , where q is any integer.
away. The value of the kernel at these non-zero positions is 1 for the even lines; for the odd lines it is one for even values of q and -1 for odd values.) However, since Gx is very large, the effect is negligible, so we can neglect this term. As a result, we can use the approximations given by Eqs. , as if the readout were always forward going.
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