EE591 Magnetic Resonance Imaging and Reconstruction Project Report

S S F P S i m u l a t i o n

Professor: Krishna Nayak

ID: 2486.9458.56, Name: Yongjin Cho

SSFP Simulation 1

1. Theory, simulation objective and method

A. Matrices and terms

i. Rx,y,z(θ) : rotation matrix about specified axis by θ

① Ry(α)= , Excitation about y-axis ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

− α

α

α

α

cos0

sin

0sin100cos

② Rz(φ)= , Rotation about z, free precession ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

1000cossin0sincos

φφφφ

ii. D : Dephaser or spoiler gradient matrix

① The spoiler gradient eliminates transverse component by dephasing all

transverse magnetizations.

② D = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100

000000

iii. Free precession matrix and vector

① A(t) : free precession matrix

- A(t)= , ⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−

−

1/

2/

2/

00

000

0

Tt

Tt

Tt

ee

e

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

1000cossin0sincos

φφφφ

φ=2π⋅off-resonance frequency⋅t

② B(t) : free precession vector

- B(t)= , Mo=1 assumed ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

− − 1/100

Tte

③ T1,T2 : Relaxation coefficients

④ TR : Repetition time, time interval between excitations

SSFP Simulation 2

⑤ TE : Echo time

- TE=0 for GRASS sequence and TE=TR/2 for SSFP sequence

B. Sequences and linear system diagram

i. GRASS (Gradient Recalled Acquisition in the Steady State)

① GRASS sequence is composed of excitation pulse train and dephaser /

crusher gradient at the end of 1 period.

α α α α

... ...

Crusher Gradient

TR

(a) GRASS sequence

RF(Ry) A(TR) +

B(TR)

DephaserM- M-M+

(b) Linear system diagram

Figure 1. GRASS sequence and linear system diagram.

ii. SSFP(Steady State Free Precession)

① SSFP sequence is the simplest sequence that is composed of excitation

pulse train.

② Since all gradients are refocused within a period, the net effect of

gradients is zero.

α α α α ... ...

TR

(a) SSFP sequence

SSFP Simulation 3

RF(Ry) A(TE) +

B(TR)

M- M-M+ M(TE)A(TR-TE) +

B(TR-TE)

(b) Linear system diagram

Figure 2. SSFP sequence and linear system diagram

C. Simulation objective

i. To characterize steady state and transient behaviors of magnetization for

GRASS and SSFP sequences over different values of parameters and

combinations of parameters.

ii. To compare GRASS and SSFP characteristics.

D. Method

i. Express each stage of sequence by matrix

ii. Relate magnetization of each stage using matrices

iii. Find proper expressions for transient magnetization and steady state

magnetization.

iv. After applying various values of function parameters (flip angle, T1, T2, TE,

TR, and off-resonance frequency) characterize behaviors of magnetization

over the different values.

E. Matlab functions

i. Rotation functions

① xrot.m, yrot.m, zrot.m

- xrot(θ), yrot(θ), zrot(θ)

- Rotation about a corresponding axis by θ

ii. Free precession

① freeprecess.m

- [Afp,Bfp]=freeprecess(T,T1,T2,df)

- Function simulates free precession and decay over a time interval T

- Afp : Free precession matrix

SSFP Simulation 4

- Bfp : Free precession vector

- T1, T2 : Relaxation coefficients

- df : off-resonance frequency

- Times are in ms and off-resonance is in Hz

iii. GRASS simulation functions

① grass.m

- [Msig, Mss] = grass(flip, T1, T2, TR, dfreq)

- Function computes the steady state magnetization of GRASS sequence.

- Msig : Steady state magnetization vector(Msig = [Mx My Mz])

- Mss : Steady state transverse magnetization in a phasor notation

(Mss = Mx +iMy)

- flip : flip angle in raidan

- T1, T2 : relaxation coefficients in ms

- TR : repetition time in ms

- dfreq : off-resonance frequency in Hz

② grasstr.m

- [Mte,Mtr] = grasstr(Mneg, flip, T1, T2, TR, dfreq)

- Function simulates one period of the GRASS sequence.

- Mte : Magnetization vector at the echo time (TE)

- Mtr : Magnetization vector at the end of one period, e.g., just before

the following excitation.

- Mneg : Magnetization just before the excitation or magnetization at TR

of the previous excitation period.

- flip, T1, T2, TR : same as in grass.m

- dfreq : off-resonance frequency in Hz

iv. SSFP simulation functions

① ssfp.m

- [Msig, Mss] = ssfp(flip, T1, T2, TE, TR, dfreq, phi)

SSFP Simulation 5

- Function computes the steady state magnetization of SSFP sequence

- Msig : Steady state magnetization vector

- Mss : Steady state transverse magnetization

- flip, T1, T2, TR, dfreq : same as in GRASS functions

- TE : echo time in ms set to TR/2

- phi : the amount of phase twist at the end of one period

② ab.m

- [A, B]= ab(flip, T1, T2, TE, TR, dfreq)

- Function for simulating transient behavior of SSFP sequence

- When the sequence of excitations are expressed as BAMM kk +=+1 ,

the function ab(•) gives out the corresponding transient matrix A and

vector B for each period.

- A : Transient matrix, B : transient vector

- flip, T1, T2, TE, TR, and dfreq : the same as in other functions

SSFP Simulation 6

2. GRASS Simulation

A. Magnetization expressions

i. Steady state magnetization

① Mss = Inv(I-Ry(α)*D*A(TR))*Ry(α)*B(TR)

ii. Transient Magnetization

① Mn = D*[A(TR)*Ry(α)*Mn-1+B(TR)]

② Mn(TE), Magnetization at echo time, is Ry(α)*Mn-1

B. Steady state signal characteristics

i. Frequency response and Mss over flip angle

① |Mss| does not vary over different off-resonance frequencies

② Given TR, |Mss| has its maximum strength when flip angle is at around

0.2 radian(11.46°)

(a) Frequency response. (b) Mss vs flip angle

Figure 3. Mss vs off resonance frequency and flip angle

ii. Mss vs T1, T2 and T2/T1

① |Mss| decreases as T1 gets longer.

② With varying T1 and fixed T2, When T1 varies, |Mss| is linearly

proportional to T2/T1

③ Since all transverse magnetizations are spoiled at the end of each

sequence, the net transverse magnetization at the moment of an

excitation is zero, which makes T2 have no effect on Mss.

SSFP Simulation 7

(a) Mss vs T1 (b) Mss vs T2/T1, T1=100 to 1000, T2=100

(a) Mss vs T2 (b) Mss vs T2/T1, T1=1000, T2=100 to 1000

Figure 4. Mss vs T1, T2 and T2/T1

C. Transient Response

i. Assume magnetization has reached its steady state once |Mxy| equals |Mss|

up to ten thousandth.

ii. T1=600ms, T2=100ms, TR=5ms, flip angle=π/3, off-resonance freq=0

① Tss : time before the steady state is reached

② Tss = 85ms Number of excitations = 85 / TR = 17 repetitions

Figure 5. Transient response

SSFP Simulation 8

iii. Transition time

① The transition time is constant over different resonant frequencies T1,

and T2.

② The transition time decreases exponentially as the flip angle increases.

③ Longer TR results in longer transition time.

(a) Off-resonance frequency and TR (b) Flip angle and TR

(c) T1 and TR (d) T2 and TR

Figure 6. Transient time

SSFP Simulation 9

3. SSFP Simulation

A. Magnetization expressions

i. Steady state magnetization

① Mss = inv(I-A(TE)*Ry(α)*A(TR-TE))*(A(TE)*Ry(α)*B(TR-TE)+B(TE))

② 2121

12

cos)(1sin)1(

EEEEEE

MoMss−−−

−=

αα

, E1,2 = exp(-TR/T1,2)

③ When TR<<T1,T2

)2/1)(cos1(cos1sin

TTMoMss

ααα

−++= [1]

④ Mss is a function of flip angle, T1 and T2.

ii. Transient magnetization

① Mn(t)= A(t)*Ry(α)*Mn-1(TR)+B(t)

② Mn(TE) = A(TE)*Ry(α)*Mn-1(TR)+B(TE)

③ Mn(TR) = A(TR)*Ry(α)*Mn-1(TR)+B(TR) or

Mn(TR) = A(TR-TE)*Mn(TE)+B(TR-TE)

④ In SSFP, TE is set to TR/2 when magnetizations with different resonant

frequencies are in phase.

B. Steady state characteristics

i. Frequency response

① A banding artifact occurs at on-resonance frequency.

② The frequency response of |Mss| repeats over every 2π, which is 1/TR,

and ∠Mss has a period of 4π

(a) Frequency response, |Mss| (b) Frequency response, ∠Mss

SSFP Simulation 10

(c) Shifted frequency response, |Mss| (d) Shifted frequency response. ∠Mss

Figure 7. Frequency response

③ The frequency response can be shifted by applying a phase shift of ϕ at

the end of each sequence.

④ The constant phase shows an echo at the pass band.

ii. Frequency response and flip angle

① |Mss| shows different shape depending on flip angles over off-

resonance frequency.

② When flip angle is π/4, the frequency response is the most flat over the

pass band interval.

③ The signal response of near on-resonance signal over flip angle shows a

potential artifact cause. [5]

④ Since on-resonance signal yields strong signal for low flip angle, it may

produce some signal in unwanted location in case of a selective

excitation.

(a) vs Off-resonance frequency (b) vs Flip angle

Figure 8. Frequency response with different flip angles

SSFP Simulation 11

iii. Frequency response and T1 and T2

① Tissues with different T1 and T2 show different frequency response

② Longer T1 yields smaller |Mss| and longer T2 gives bigger |Mss|

③ The pass band plot of short T1 and long T2 has the similar shape to low

flip angle frequency response

④ The frequency response is the most flat when T1 is about 300 ms and

T2 is about 400 ms.

⑤ The frequency response of |Mss| with different T2/T1 will be exactly the

same as the figure 9. (b).

(a) T1 frequency response, T2=100ms (b) T2 frequency response, T1=1000ms

Figure 9. Frequency response with different T1 and T2

iv. Mss over T1 and T2

① Mss in SSFP is known to be proportional to T2/T1, which makes T2/T1-

ish contrast possible.

② Mss shows a linear proportionality to square root of T2/T1 for low T2/T1.

③ Mss seems not to vary over different TR values.

(a) Mss vs T1 (b) Mss vs T2

SSFP Simulation 12

(c) Mss vs T2/T1 (d) Mss vs sqrt(T2/T1)

Figure 10. Mss vs T1, T2, T2/T1 and sqrt(T2/T1)

④ The lines in the gray scale image gives the result that |Mss| is constant

for fixed T2/T1 but when T1 and T2 are small some different

dependency is observed.

⑤ When T1 and T2 are small, the magnitude of steady state signal is

dependent on T1/TR and T2/TR, which shows a dependency of |Mss|

on TR.

(a) Gray scale image (b) Contour over T2/TR and T1/TR, and TR/T2 and TR/T1

Figure 11. Gray scale image and Contour of |Mss|

⑥ Magnetization plot on xz-plane over flip angles and off-resonance

frequencies forms an elliptical shape.

⑦ Each ellipse represents different off-resonant frequency. Since the

interval between off-resonance frequencies is 10Hz and the periodicity is

100Hz, 11 different ellipses can be observed.

⑧ For each off-resonance frequency, whenever a line with an angle α from

the vertical axis passes the middle point of the corresponding ellipse,

the maximum transverse magnetization can be acquired with the flip

angle α.

SSFP Simulation 13

(a) Mxz vs flip angle and T2/T1 (b) Mxz vs flip angle and off-resonance frequency

Figure 12 Mxy over flip angle, off-resonance frequency and T2/T1

C. Transient Response

i. Assumption

① The difference between the magnitudes of consecutive magnetizations

decrease.

② The steady state satisfies the condition that the difference between

consecutive magnetizations is less than 0.001, i.e., |Mxy(k) -Mxy(k-1)| ≤

0.001, which indicates a smooth variation.

③ Even though the magnitude of the transverse magnetization is not quite

close to the reference steady state magnetization, if the oscillation, or

the variation is smooth the signal can be acquired and used.

ii. Transient Oscillation

① Before the steady state is reached, magnetizations show an oscillatory

behavior.

(a) Oscillation in 3D space (b) Transverse plane oscillation

Figure 13. Transient Oscillation

SSFP Simulation 14

iii. Transient response of |Mxy|

① T1=1000ms and T2=100ms. TR=10ms, flip angle = 70°

② As off-resonance frequency gets closer to 0, where the banding artifact

occurs, the transition time gets longer and the signal gets weaker.

Figure 14. |Mxy| over repetitions and Off-resonance frequency

iv. Transient behavior over T1 and T2

① Lower T1, higher T2 or higher T2/T1 produces stronger steady state

signal.

② A separate plot of odd and even excitations shows no oscillation. But

this non-oscillatory behavior is only for the middle frequency of the pass

band.

(a) T1=100ms to 1000ms, T2=100ms, off-resonance frequency=π

(b) T2=100ms to 1000ms, T1=1000ms, off-resonance frequency=π

SSFP Simulation 15

(c) Transient behavior in odd and even excitations, Off-resonance frequency=10 to 50 Hz

Figure 15. Transient behavior over T1 and T2

v. Transition time and Off-resonance frequency

① Again the transition time is periodic over 2π.

② As the off-resonance frequency gets closer to the middle of the pass

band, the transition time tends to decrease.

③ A small peak is observed right at the center of the pass band.

④ When TR=6ms, the result does not show the expected periodicity.

When TR=6ms, the period is supposed to be 167Hz but it seems to be

500Hz. But, if high peaks are used for the reference of periodicity, it

shows an about 167Hz periodicity.

⑤ The variation of TR seems not to affect the transition time.

(a) TR=2ms (b) TR=4ms (c) TR=6ms

Figure 16. Transient time vs Off-resonance frequency

⑥ The shapes of the graphs are quite similar to the graph of the frequency

response of the steady state signal if the vertical axis is flipped over.

SSFP Simulation 16

⑦ The similarity indicates that it takes more time for signals with low

steady-state intensity, e.g., signals near the banding frequency, to reach

their steady state.

vi. Transient time and flip angle

① The transient time is proportional to flip angle and TR does not affect

much on the transient time.

Figure 17. Transient time vs Flip angle

vii. Transition time over T1 and T2

① For fixed T2, the transition time increases along with the increase of T1.

② As the figure 17 (a) shows, the transition time shows a linear decrease

with T2/T1 when T2 is fixed and T1 is varying.

③ When T1 is fixed and T2 varies from 100 to 1000ms, the transition time

is proportional to the increase of T2 or T2/T1. In this case T2/T1 is

directly proportional to T2, and thus the graphs look exactly the same.

(a) T1 = 100 to 1000ms, T2=100ms

SSFP Simulation 17

(b) T2 = 100 to 1000ms, T1=1000ms

Figure 18. Transient time vs T1, T2 and T2/T1

④ From figure 17 (a) and (b), it is found that the transition time is not a

function of T2/T1. Moreover, it can be inferred that the transition time is

more dependent on the variation of T2 rather than the variation of T1.

⑤ Both simulations, in which T1 and T2 varies respectively, show that the

transition time does not have any strong dependence on TR.

4. Discussion

A. Comparison between GRASS and SSFP

i. The SSFP sequence gives the maximum possible SNR since in SSFP

sequence every magnetization is refocused such that there is no waste of

magnetization. On the other hand, in GRASS sequence, transverse

magnetizations are de-phased over repetitions, and thus it results in weak

signal intensity.

GRASS SSFP

Signal Strength Weak Strong

Frequency response Uniform over frequency Banding Artifact

Contrast T1 T2/T1

Transition Time Short Long

Table 1. GRASS and SSFP

SSFP Simulation 18

B. SSFP characteristics

i. Maximum SNR coming from the full refocus of magnetizations

ii. Banding artifact due to the sensitivity in frequency response

iii. Long transition time not suitable for a fast sequence.

C. Future work

i. Analytical expressions of transient magnetizations [7]

ii. Search literatures for current solutions and improvements for SSFP

shortcomings.

① Banding artifact

Ex) Multiple acquisition method [4], Fat suppression

② Transition time

Ex) Catalyzing steady state [3], T1-map acquisition with transient-state

magnetization [6]

iii. Effective RF design for desirable signal / slice profile [8],[5]

① Improvement of frequency response

- To make pass band wide and flat

② Improvement of slice profile

- To eliminate signals from out-of slice.

- To make in-slice profile flat.

iv. Improvement and enhancement of SSFP performance via trials of new ideas

and a through research and study of literatures !!!

SSFP Simulation 19

5. Reference

[1] Scheffler K, Lehnhardt S, Principles and applications of balanced SSFP techniques

[2] Perrin LR et al., Comparative study of fast gradient echo MRI sequences : Phantom

study

[3] Hargreaves BA et al., Characterization and reduction of the transient response in

steady-state MR imaging

[4] Bangerter NK et al., Analysis of multiple-acquisition SSFP

[5] Sung K, Excitation pulse design improving the spatial-spectral profile in SSFP imaging

[6] Hsu J et al., Spin-lattice relaxation and a fast T1-map acquisition method in MRI with

transient-state magnetization

[7] Nazarova I et al., Analytical analysis of multi-pulse NMR

[8] Pauly J et al., Parameter Relations for the SLR selective excitation pulse design

algorithm