Method for fat fraction quantification in magnetic resonance imaging
View Patent ↗A method for determining a concentration of fat within a region of a subject includes acquiring at least three gradient echo images from at least two different flip angles. The first and third images occur at an in-phase time when resonance signals of fat and water are in-phase with one another. The second images occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another. T2* and T1 relaxation time maps are approximated based on the acquired images. Fat and water signals are distinguished based on observed relative signal phase. The amplitude of the fat and water signals are modulated using the T2* and T1 relaxation time maps. A ratio between fat and water signals for one or more voxels of the images is calculated based on the amplitude modulated fat and water signal images.
1. A method for determining a concentration of fat within a region of a subject, comprising:
acquiring at least three gradient echo images e 1 , e 2 , and e 3 of a region of a subject from at least two different flip angles α 1 and α 2 within a repetition time TR for a total of at least six gradient echo images e 1 α 1 , e 2 α 1 , e 3 α 1 , e 1 α 2 , e 2 α 2 , and e 3 α 2 , wherein gradient echo images e 1 α 1 , e 1 α 2 , e 3 α 1 , and e 3 α 2 occur at an in-phase time when resonance signals of fat and water are in-phase with one another and gradient echo images e 2 α 1 , and e 2 α 2 occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another;
approximating a T2* relaxation time map from the in-phase gradient echo images based on the α 1 flip angle or the α 2 flip angle;
approximating a T1 relaxation time map from the in-phase gradient echo images based on the α 1 flip angle and the α 2 flip angle;
distinguishing between fat and water signals within the gradient echo images based on observed relative signal phase;
modulating the amplitude of the fat and water signal images using the T2* relaxation time and T1 relaxation time maps;
determining a ratio between fat and water signals for one or more voxels of the gradient echo images based on the amplitude modulated fat and water signal images; and
displaying the determined ratio between fat and water for a desired voxel of the gradient echo images,
wherein the T2* relaxation time map is approximated from the in-phase gradient echo images according to the following equations:
T
2
α
1
*
=
(
TE
3
-
TE
1
)
ln
(
S
e
1
,
α
1
/
S
e
3
,
α
1
)
and
T
2
α
2
*
=
(
TE
3
-
TE
1
)
ln
(
S
e
1
,
α
2
/
S
e
3
,
α
2
)
where TE 1 is a time of the acquisition of the e 1 images, TE 3 is a time of the acquisition of the e 3 images, S e1,α1 is a signal of the e 1 α 1 gradient echo image, S e3,α1 is a signal of the e 3 α 1 gradient echo image, S e1,α2 is a signal of the e 1 α 2 gradient echo image, and S e3,α2 is a signal of the e 3 α 2 gradient echo image.
2. The method of claim 1 , wherein the gradient echo images are acquired in the following order: e 1 as an in-phase echo, then e 2 as an opposed-phase echo, and then e 3 as an in-phase echo.
3. The method of claim 1 , wherein the gradient echo images are acquired in the following order: e 1 as an opposed-phase echo, then e 2 as an in-phase echo, and then e 3 as an in-phase echo.
4. The method of claim 1 , wherein fat and water signals are distinguished within the gradient echo images using a two-point or three-point Dixon approach.
5. A method for determining a concentration of fat within a region of a subject, comprising:
acquiring at least three gradient echo images e 1 , e 2 , and e 3 of a region of a subject from at least two different flip angles α 1 and α 2 within a repetition time TR for a total of at least six gradient echo images e 1 α 1 , e 2 α 1 , e 3 α 1 , e 1 α 2 , e 2 α 2 , and e 3 α 2 , wherein gradient echo images e 1 α 1 , e 1 α 2 , e 3 α 1 , and e 3 α 2 occur at an in-phase time when resonance signals of fat and water are in-phase with one another and gradient echo images e 2 α 1 , and e 2 α 2 occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another;
approximating a T2* relaxation time map from the in-phase gradient echo images based on the α 1 flip angle or the α 2 flip angle;
approximating a T1 relaxation time map from the in-phase gradient echo images based on the α 1 flip angle and the α 2 flip angle;
distinguishing between fat and water signals within the gradient echo images based on observed relative signal phase;
modulating the amplitude of the fat and water signal images using the T2* relaxation time and T1 relaxation time maps;
determining a ratio between fat and water signals for one or more voxels of the gradient echo images based on the amplitude modulated fat and water signal images; and
displaying the determined ratio between fat and water for a desired voxel of the gradient echo images,
wherein the T1 relaxation time map is approximated from the in-phase gradient echo images according to the following equations:
T
1
e
1
=
-
TR
ln
[
S
e
1
,
α
1
sin
α
2
-
S
e
1
,
α
2
sin
α
1
(
S
e
1
,
α
1
sin
α
2
cos
α
1
)
-
(
S
e
1
,
α
2
sin
α
1
cos
α
2
)
]
and
T
1
e
3
=
-
TR
ln
[
S
e
3
,
α
1
sin
α
2
-
S
e
3
,
α
2
sin
α
1
(
S
e
3
,
α
1
sin
α
2
cos
α
1
)
-
(
S
e
3
,
α
2
sin
α
1
cos
α
2
)
]
wherein S e1,α1 is a signal of the e 1 α 1 gradient echo image, S e3,α1 is a signal of the e 3 α 1 gradient echo image, S e3,α2 is a signal of the e 1 α 2 gradient echo image, and S e3,α2 is a signal of the e 3 α 2 gradient echo image.
6. The method of claim 5 , wherein the gradient echo images are acquired in the following order: e 1 as an in-phase echo, then e 2 as an opposed-phase echo, and then e 3 as an in-phase echo.
7. The method of claim 5 , wherein the gradient echo images are acquired in the following order: e 1 as an opposed-phase echo, then e 2 as an in-phase echo, and then e 3 as an in-phase echo.
8. The method of claim 5 , wherein fat and water signals are distinguished within the gradient echo images using a two-point or three-point Dixon approach.
9. A method for determining a concentration of fat within a region of a subject, comprising:
acquiring at least three gradient echo images e 1 , e 2 , and e 3 of a region of a subject from at least two different flip angles α 1 and α 2 within a repetition time TR for a total of at least six gradient echo images e 1 α 1 , e 2 α 1 , e 3 α 1 , e 1 α 2 , e 2 α 2 , and e 3 α 2 , wherein gradient echo images e 1 α 1 , e 1 α 2 , e 3 α 1 , and e 3 α 2 occur at an in-phase time when resonance signals of fat and water are in phase with one another and gradient echo images e 2 α 1 , and e 2 α 2 occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another;
approximating a T2* relaxation time map from the in-phase gradient echo images based on the α 1 flip angle or the α 2 flip angle;
approximating a T1 relaxation time map from the in-phase gradient echo images based on the α 1 flip angle and the α 2 flip angle;
distinguishing between fat and water signals within the gradient echo images based on observed relative signal phase;
modulating the amplitude of the fat and water signal images using the T2* relaxation time and T1 relaxation time maps;
determining a ratio between fat and water signals for one or more voxels of the gradient echo images based on the amplitude modulated fat and water signal images; and displaying the determined ratio between fat and water for a desired voxel of the gradient echo images,
wherein fat and water signals are distinguished within the gradient echo images using a two-point or three-point Dixon approach,
wherein the gradient echo images are acquired in the following order: e 1 then e 2 then e 3 ;
the water signal W is determined as ½ (M 0,e1 +M 0,e2 ); and
the fat signal F is determined as ½ (M 0,e1 −M 0,e2 ),
wherein M 0,e1 is the initial magnetization of the e 1 image and M 0,e2 is the initial magnetization of the e 2 image, and
wherein the initial magnetization of the e 1 images M 0,e1 and the initial magnetization of the e 2 images M 0,e2 are calculated according to the following equations:
M
0
,
e
1
=
S
e
1
,
α
1
1
-
cos
(
α
1
)
ⅇ
-
TR
/
T
1
sin
(
α
1
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
1
/
T
2
*
and
M
0
,
e
2
=
S
e
2
,
α
1
1
-
cos
(
α
1
)
ⅇ
-
TR
/
T
1
sin
(
α
1
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
2
/
T
2
*
for
α
1
,
and
M
0
,
e
1
=
S
e
1
,
α
2
1
-
cos
(
α
2
)
ⅇ
-
TR
/
T
1
sin
(
α
2
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
1
/
T
2
*
and
M
0
,
e
2
=
S
e
2
,
α
2
1
-
cos
(
α
2
)
ⅇ
-
TR
/
T
1
sin
(
α
2
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
2
/
T
2
*
for
α
2
,
for α 2 ,
where S e1,α1 is a signal of the e 1 α 1 gradient echo image, S e2,α1 is a signal of the e 2 α 1 gradient echo image, S e1,α2 is a signal of the e 1 α 2 gradient echo image, and S e2,α2 is a signal of the e 2 α 2 gradient echo image, TE 1 is a time of the acquisition of the e 1 images, and TE 2 is a time of the acquisition of the e 2 images.
10. A computer system comprising:
a processor; and
a non-transitory, tangible, program storage medium, readable by the computer system, embodying a program of instructions executable by the processor to perform method steps for determining a concentration of fat within a region of a subject, the method comprising:
acquiring at least three gradient echo images e 1 , e 2 , and e 3 of a region of a subject from at least two different flip angles α 1 and α 2 within a repetition time TR for a total of at least six gradient echo images e 1 α 1 , e 2 α 1 , e 3 α 1 , e 1 α 2 , e 2 α 2 , and e 3 α 2 , wherein gradient echo images e 1 α 1 , e 1 α 2 , e 3 α 1 , and e 3 α 2 occur at an in-phase time when resonance signals of fat and water are in-phase with one another and gradient echo images e 2 α 1 , and e 2 α 2 occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another;
approximating a T2* relaxation time map from the in-phase gradient echo images based on the α 1 flip angle or the α 2 flip angle;
approximating a T1 relaxation time map from the in-phase gradient echo images based on the α 1 flip angle and the α 2 flip angle;
distinguishing between fat and water signals within the gradient echo images based on observed relative signal phase;
modulating the amplitude of the fat and water signal images using the T2* relaxation time and T1 relaxation time maps;
determining a ratio between fat and water signals for one or more voxels of the gradient echo images based on the amplitude modulated fat and water signal images; and
displaying the determined ratio between fat and water for a desired voxel of the gradient echo images,
wherein the T2* relaxation time map is approximated from the in-phase gradient echo images according to the following equations:
T
2
α1
*
=
(
TE
3
-
TE
1
)
ln
(
S
e
1
,
α
1
/
S
e
3
,
α
1
)
and
T
2
α2
*
=
(
TE
3
-
TE
1
)
ln
(
S
e
1
,
α2
/
S
e
3
,
α2
)
where TE 1 is a time of the acquisition of the e 1 images, TE 3 is a time of the acquisition of the e 3 images, S e1,α 1 is a signal of the e 1 α 1 gradient echo image, S e3,α1 is a signal of the e 3 α 1 gradient echo image, S e1,α2 is a signal of the e 1 α 2 gradient echo image, and S e3,α2 is a signal of the e 3 α 2 gradient echo image.
11. The computer system of claim 10 , wherein the gradient echo images are acquired in the following order: e 1 as an in-phase echo, then e 2 as an opposed-phase echo, and then e 3 as an in-phase echo.
12. The computer system of claim 10 , wherein the gradient echo images are acquired in the following order: e 1 as an opposed-phase echo, then e 2 as an in-phase echo, and then e 3 as an in-phase echo.
13. The computer system of claim 10 , wherein fat and water signals are distinguished within the gradient echo images using a two-point or three-point Dixon approach.
14. A computer system comprising:
a processor; and
a non-transitory, tangible, program storage medium, readable by the computer system, embodying a program of instructions executable by the processor to perform method steps for determining a concentration of fat within a region of a subject, the method comprising:
acquiring at least three gradient echo images e 1 , e 2 , and e 3 of a region of a subject from at least two different flip angles α 1 and α 2 within a repetition time TR for a total of at least six gradient echo images e 1 α 1 , e 2 α 1 , e 3 α 1 , e 1 α 2 , e 2 α 2 , and e 3 α 2 , wherein gradient echo images e 1 α 1 , e 1 α 2 , e 3 α 1 , and e 3 α 2 occur at an in-phase time when resonance signals of fat and water are in-phase with one another and gradient echo images e 2 α 1 , and e 2 α 2 occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another;
approximating a T2* relaxation time map from the in-phase gradient echo images based on the α 1 flip angle or the α 2 flip angle;
approximating a T1 relaxation time map from the in-phase gradient echo images based on the α 1 flip angle and the α 2 flip angle;
distinguishing between fat and water signals within the gradient echo images based on observed relative signal phase;
modulating the amplitude of the fat and water signal images using the T2* relaxation time and T1 relaxation time maps;
determining a ratio between fat and water signals for one or more voxels of the gradient echo images based on the amplitude modulated fat and water signal images; and
displaying the determined ratio between fat and water for a desired voxel of the gradient echo images,
wherein the T1 relaxation time map is approximated from the in-phase gradient echo images according to the following equations:
T
1
e
1
=
-
TR
ln
[
S
e
1
,
α1
sin
α
2
-
S
e
1
,
α2
sin
α
1
(
S
e
1
,
α1
sin
α
2
cos
α
1
)
-
(
S
e
1
,
α2
sin
α
1
cos
α
2
)
]
and
T
1
e
3
=
-
TR
ln
[
S
e
1
,
α1
sin
α
2
-
S
e
1
,
α2
sin
α
1
(
S
e
1
,
α1
sin
α
2
cos
α
1
)
-
(
S
e
3
,
α2
sin
α
1
cos
α
2
)
]
wherein S e1,α1 is a signal of the e 1 α 1 gradient echo image, S e3,α1 is a signal of the e 3 α 1 gradient echo image, S e1,α2 is a signal of the e 1 α 2 gradient echo image, and S e3,α2 is a signal of the e 3 α 2 gradient echo image.
15. A computer system comprising:
a processor; and
a non-transitory, tangible, program storage medium, readable by the computer system, embodying a program of instructions executable by the processor to perform method steps for determining a concentration of fat within a region of a subject, the method comprising:
acquiring at least three gradient echo images e 1 , e 2 , and e 3 of a region of a subject from at least two different flip angles α 1 and α 2 within a repetition time TR for a total of at least six gradient echo images e 1 α 1 , e 2 α 1 , e 3 α 1 , e 1 α 2 , e 2 α 2 , and e 3 α 2 , wherein gradient echo images e 1 α 1 , e 1 α 2 , e 3 α 1 , and e 3 α 2 occur at an in-phase time when resonance signals of fat and water are in-phase with one another and gradient echo images e 2 α 1 , and e 2 α 2 occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another;
approximating a T2* relaxation time map from the in-phase gradient echo images based on the α 1 flip angle or the α 2 flip angle;
approximating a T1 relaxation time map from the in-phase gradient echo images based on the α 1 flip angle and the α 2 flip angle;
distinguishing between fat and water signals within the gradient echo images based on observed relative signal phase;
modulating the amplitude of the fat and water signal images using the T2* relaxation time and T1 relaxation time maps;
determining a ratio between fat and water signals for one or more voxels of the gradient echo images based on the amplitude modulated fat and water signal images; and
displaying the determined ratio between fat and water for a desired voxel of the gradient echo images,
wherein fat and water signals are distinguished within the gradient echo images using a two-point or three-point Dixon approach,
wherein the gradient echo images are acquired in the following order: e 1 then e 2 then e 3 ;
the water signal W is determined as ½ (M 0, e1 +M 0,e2 ); and
the fat signal F is determined as ½ (M 0,e1 −M 0,e2 ),
wherein M 0,e1 is the initial magnetization of the e 1 image and M 0,e2 is the initial magnetization of the e 2 image, and
wherein the initial magnetization of the e 1 images M 0,e1 and the initial magnetization of the e 2 images M 0,e2 are calculated according to the following equations:
M
0
,
e
1
=
S
e
1
,
α
1
1
-
cos
(
α
1
)
ⅇ
-
TR
/
T
1
sin
(
α
1
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
1
/
T
2
*
and
M
0
,
e
2
=
S
e
2
,
α
1
1
-
cos
(
α
1
)
ⅇ
-
TR
/
T
1
sin
(
α
1
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
2
/
T
2
*
and
α1
,
and
M
0
,
e
1
=
S
e
1
,
α
2
1
-
cos
(
α
2
)
ⅇ
-
TR
/
T
1
sin
(
α
2
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
1
/
T
2
*
and
M
0
,
e
2
=
S
e
2
,
α
2
1
-
cos
(
α
2
)
ⅇ
-
TR
/
T
1
sin
(
α
2
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
2
/
T
2
*
for
α2
,
where S e1,α1 is a signal of the e 1 α 1 gradient echo image, S e2,α1 is a signal of the e 2 α 1 gradient echo image, S e1,α2 is a signal of the e 1 α 2 gradient echo image, and S e2,α2 is a signal of the e 2 α 2 gradient echo image, TE 1 is a time of the acquisition of the e 1 images, and TE 2 is a time of the acquisition of the e 2 images.
16. A system for determining a concentration of fat within a region of a subject, comprising:
a magnetic resonance imaging scanner for acquiring at least three gradient echo images e 1 , e 2 , and e 3 of a region of a subject from at least two different flip angles α 1 and α 2 within a repetition time TR for a total of at least six gradient echo images e 1 α 1 , e 2 α 1 , e 3 α 1 , e 1 α 2 , e 2 α 2 , and e 3 α 2 , wherein gradient echo images e 1 α 1 , e 1 α 2 , e 3 α 1 , and e 3 α 2 occur at an in-phase time when resonance signals of fat and water are in-phase with one another and gradient echo images e 2 α 1 , and e 2 α 2 occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another; and
a computing device for approximating a T2* relaxation time map from the in-phase gradient echo images based on the α 1 flip angle or the α 2 flip angle and approximating a T1 relaxation time map from the in-phase gradient echo images based on the α 1 flip angle and the α 2 flip angle, distinguishing between fat and water signals within the gradient echo images based on observed relative signal phase, modulating the amplitude of the fat and water signal images using the T2* relaxation time and T1 relaxation time maps, determining a ratio between fat and water signals for one or more voxels of the gradient echo images based on the amplitude modulated fat and water signal images, and displaying the determined ratio between fat and water for a desired voxel of the gradient echo images,
wherein the T2* relaxation time map is approximated from the in-phase gradient echo images according to the following equations:
T
2
α1
*
=
(
TE
3
-
TE
1
)
ln
(
S
e
1
,
α1
/
S
e
3
,
α1
)
and
T
2
α2
*
=
(
TE
3
-
TE
1
)
ln
(
S
e
1
,
α2
/
S
e
3
,
α
2
)
where TE 1 is a time of the acquisition of the e 1 images, TE 3 is a time of the acquisition of the e 3 images, S e1,α 1 is a signal of the e 1 α 1 gradient echo image, S e3,α1 is a signal of the e 3 α 1 gradient echo image, S e1,α2 is a signal of the e 1 α 2 gradient echo image, and S e3,α2 is a signal of the e 3 α 2 gradient echo image.
17. The system of claim 16 , wherein the gradient echo images are acquired in the following order: e 1 as an in-phase echo, then e 2 as an opposed-phase echo, and then e 3 as an in-phase echo.
18. The system of claim 16 , wherein the gradient echo images are acquired in the following order: e 1 as an opposed-phase echo, then e 2 as an in-phase echo, and then e 3 as an in-phase echo.
19. The system of claim 16 , wherein fat and water signals are distinguished within the gradient echo images using a two-point or three-point Dixon approach.
20. A system for determining a concentration of fat within a region of a subject, comprising:
a magnetic resonance imaging scanner for acquiring at least three gradient echo images e 1 , e 2 , and e 3 of a region of a subject from at least two different flip angles α 1 and α 2 within a repetition time TR for a total of at least six gradient echo images e 1 α 1 , e 2 α 1 , e 3 α 1 , e 1 α 2 , e 2 α 2 , and e 3 α 2 , wherein gradient echo images e 1 α 1 , e 1 α 2 , e 3 α 1 , and e 3 α 2 occur at an in-phase time when resonance signals of fat and water are in-phase with one another and gradient echo images e 2 α 1 , and e 2 α 2 occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another; and
a computing device for approximating a T2* relaxation time map from the in-phase gradient echo images based on the α 1 flip angle or the α 2 flip angle, approximating a T1 relaxation time map from the in-phase gradient echo images based on the α 1 flip angle and the α 2 flip angle, distinguishing between fat and water signals within the gradient echo images based on observed relative signal phase, modulating the amplitude of the fat and water signal images using the T2* relaxation time and T1 relaxation time maps, determining a ratio between fat and water signals for one or more voxels of the gradient echo images based on the amplitude modulated fat and water signal images, and displaying the determined ratio between fat and water for a desired voxel of the gradient echo images,
wherein the T1 relaxation time map is approximated from the in-phase gradient echo images according to the following equations:
T
1
e
1
=
-
TR
ln
[
S
e
1
,
α1
sin
α
2
-
S
e
1
,
α
2
sin
α
1
(
S
e
1
,
α1
sin
α
2
cos
α
1
)
-
(
S
e
1
,
α2
sin
α
1
cos
α
2
)
]
and
T
1
e
3
=
-
TR
ln
[
S
e
3
,
α1
sin
α
2
-
S
e
3
,
α
2
sin
α
1
(
S
e
3
,
α1
sin
α
2
cos
α
1
)
-
(
S
e
3
,
α2
sin
α
1
cos
α
2
)
]
wherein S e1,α1 is a signal of the e 1 α 1 gradient echo image, S e3,α1 is a signal of the e 3 α 1 gradient echo image, S e1,α2 is a signal of the e 1 α 2 gradient echo image, and S e3,α2 is a signal of the e 3 α 2 gradient echo image.
21. A system for determining a concentration of fat within a region of a subject, comprising:
a magnetic resonance imaging scanner for acquiring at least three gradient echo images e 1 , e 2 , and e 3 of a region of a subject from at least two different flip angles α 1 and α 2 within a repetition time TR for a total of at least six gradient echo images e 1 α 1 , e 2 α 1 , e 3 α 1 , e 1 α 2 , e 2 α 2 , and e 3 α 2 , wherein gradient echo images e 1 α 1 , e 1 α 2 , e 3 α 1 , and e 3 α 2 occur at an in-phase time when resonance signals of fat and water are in-phase with one another and gradient echo images e 2 α 1 , and e 2 α 2 occur at an opposed-phase time when the resonance signals of fat and water are 180° out of phase with one another; and
a computing device for approximating approximating a T2* relaxation time map from the in-phase gradient echo images based on the α 1 flip angle or the α 2 flip angle, approximating a T1 relaxation time map from the in-phase gradient echo images based on the α 1 flip angle and the α 2 flip angle, distinguishing between fat and water signals within the gradient echo images based on observed relative signal phase, modulating the amplitude of the fat and water signal images using the T2* relaxation time and T1 relaxation time maps, determining a ratio between fat and water signals for one or more voxels of the gradient echo images based on the amplitude modulated fat and water signal images, and displaying the determined ratio between fat and water for a desired voxel of the gradient echo images,
wherein fat and water signals are distinguished within the gradient echo images using a two-point or three-point Dixon approach,
wherein the gradient echo images are acquired in the following order: e 1 then e 2 then e 3 ;
the water signal W is determined as ½ (M 0,e1 +M 0,e 2 ); and
the fat signal F is determined as ½ (M 0,e1 −M 0,e2),
wherein M 0,e1 is the initial magnetization of the e 1 image and M 0,e1 −M 0,e2 is the initial magnetization of the e 2 image, and
wherein the initial magnetization of the e 1 images M 0,e1 and the initial magnetization of the e 2 images M 0,e2 are calculated according to the following equations:
M
0
,
e
1
=
S
e
1
,
α
1
1
-
cos
(
α
1
)
ⅇ
-
TR
/
T
1
sin
(
α
1
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
1
/
T
2
*
and
M
0
,
e
2
=
S
e
2
,
α
1
1
-
cos
(
α
1
)
ⅇ
-
TR
/
T
1
sin
(
α
1
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
2
/
T
2
*
and
α1
,
and
M
0
,
e
1
=
S
e
1
,
α
2
1
-
cos
(
α
2
)
ⅇ
-
TR
/
T
1
sin
(
α
2
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
1
/
T
2
*
and
M
0
,
e
2
=
S
e
2
,
α
2
1
-
cos
(
α
2
)
ⅇ
-
TR
/
T
1
sin
(
α
2
)
(
1
-
ⅇ
-
TR
/
T
1
)
ⅇ
+
TE
2
/
T
2
*
for
α2
,
where S e1,α1 is a signal of the e 1 α 1 gradient echo image, S e2,α1 is a signal of the e 2 α 1 gradient echo image, S e1,α2 is a signal of the e 1 α 2 gradient echo image, and S e2,α2 is a signal of the e 2 α 2 gradient echo image, TE 1 is a time of the acquisition of the e 1 images, and TE 2 is a time of the acquisition of the e 2 images.