EE591 Magnetic Resonance Imaging and Reconstruction Project Report. S i m u l a t i o n. Professor: Krishna Nayak ID: , Name: Yongjin Cho

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1 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: , Name: Yongjin Cho

2 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 cosα 0 1 Ry(α)= 0 1 sinα 0 sinα 0, Excitation about y-axis cosα cosφ 2 Rz(φ)= sinφ 0 sinφ cosφ , Rotation about z, free precession D : Dephaser or spoiler gradient matrix 1 The spoiler gradient eliminates transverse component by dephasing all transverse magnetizations. 2 D = i Free precession matrix and vector 1 A(t) : free precession matrix t / T 2 e 0 0 cosφ sinφ 0 t / T 2 - A(t)= 0 e 0 sinφ cosφ 0, t / T1 0 0 e φ=2π off-resonance frequency t 2 B(t) : free precession vector 0 - B(t)= 0, Mo=1 assumed t 1 / T e 1 3 T1,T2 : Relaxation coefficients 4 TR : Repetition time, time interval between excitations

3 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 TR α α α Crusher Gradient (a) GRASS sequence RF(Ry) A(TR) + Dephaser M- M+ M- B(TR) (b) Linear system diagram Figure 1. GRASS sequence and linear system diagram. 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. TR α α α α (a) SSFP sequence

4 SSFP Simulation 3 RF(Ry) A(TE) + A(TR-TE) + M- M+ M(TE) M- B(TR) 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. To compare GRASS and SSFP characteristics. D. Method i. Express each stage of sequence by matrix i Relate magnetization of each stage using matrices 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 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

5 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 i 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)

6 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 M = AM B k +1 k +, 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

7 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) Transient Magnetization 1 M n = D*[A(TR)*Ry(α)*M n-1 +B(TR)] 2 M n (TE), Magnetization at echo time, is Ry(α)*M n-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 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.

8 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. 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

9 SSFP Simulation 8 i 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

10 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 E2 (1 E1)sinα Mss = Mo 1 ( E E )cosα E E , E1,2 = exp(-tr/t1,2) 3 When TR<<T1,T2 sin Mss = α Mo 1+ cosα + (1 cosα)( T1/ T 2) [1] 4 Mss is a function of flip angle, T1 and T2. Transient magnetization 1 M n (t)= A(t)*Ry(α)*M n-1 (TR)+B(t) 2 M n (TE) = A(TE)*Ry(α)*M n-1 (TR)+B(TE) 3 M n (TR) = A(TR)*Ry(α)*M n-1 (TR)+B(TR) or M n (TR) = A(TR-TE)*M n (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

11 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. Frequency response and flip angle 1 Mss shows different shape depending on flip angles over offresonance 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

12 SSFP Simulation 11 i 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

13 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.

14 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. 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

15 SSFP Simulation 14 i 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

16 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.

17 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 v 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

18 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

19 SSFP Simulation 18 B. SSFP characteristics i. Maximum SNR coming from the full refocus of magnetizations i Banding artifact due to the sensitivity in frequency response Long transition time not suitable for a fast sequence. C. Future work i. Analytical expressions of transient magnetizations [7] 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] i 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!!!

20 SSFP Simulation 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