def plot(self,
type: str = 'Gradient',
time_range=(0, np.inf),
time_disp: str = 's',
save: bool = False):
"""
Plot `Sequence`.
Parameters
----------
type : str
Gradients display type, must be one of either 'Gradient' or 'Kspace'.
time_range : List
Time range (x-axis limits) for plotting the sequence. Default is 0 to infinity (entire sequence).
time_disp : str
Time display type, must be one of `s`, `ms` or `us`.
save_as : bool
Boolean flag indicating if plots should be saved. The two figures will be saved as JPG with numerical
suffixes to the filename 'seq_plot'.
"""
valid_plot_types = ['Gradient', 'Kspace']
valid_time_units = ['s', 'ms', 'us']
if type not in valid_plot_types:
raise Exception()
if time_disp not in valid_time_units:
raise Exception()
fig1, fig2 = plt.figure(1), plt.figure(2)
sp11 = fig1.add_subplot(311)
sp12, sp13 = fig1.add_subplot(312, sharex=sp11), fig1.add_subplot(
313, sharex=sp11)
fig2_sp_list = [
fig2.add_subplot(311, sharex=sp11),
fig2.add_subplot(312, sharex=sp11),
fig2.add_subplot(313, sharex=sp11)
]
t_factor_list = [1, 1e3, 1e6]
t_factor = t_factor_list[valid_time_units.index(time_disp)]
t0 = 0
for iB in range(1, len(self.block_events) + 1):
block = self.get_block(iB)
is_valid = time_range[0] <= t0 <= time_range[1]
if is_valid:
if hasattr(block, 'adc'):
adc = block.adc
t = adc.delay + [(x * adc.dwell)
for x in range(0, int(adc.num_samples))]
sp11.plot((t0 + t), np.zeros(len(t)), 'rx')
if hasattr(block, 'rf'):
rf = block.rf
tc, ic = calc_rf_center(rf)
t = rf.t + rf.delay
tc = tc + rf.delay
sp12.plot(t_factor * (t0 + t), abs(rf.signal))
sp13.plot(
t_factor * (t0 + t),
np.angle(
rf.signal * np.exp(1j * rf.phase_offset) *
np.exp(1j * 2 * math.pi * rf.t * rf.freq_offset)),
t_factor * (t0 + tc),
np.angle(rf.signal[ic] * np.exp(1j * rf.phase_offset) *
np.exp(1j * 2 * math.pi * rf.t[ic] *
rf.freq_offset)), 'xb')
grad_channels = ['gx', 'gy', 'gz']
for x in range(0, len(grad_channels)):
if hasattr(block, grad_channels[x]):
grad = getattr(block, grad_channels[x])
if grad.type == 'grad':
# In place unpacking of grad.t with the starred expression
t = grad.delay + [
0, *(grad.t + (grad.t[1] - grad.t[0]) / 2),
grad.t[-1] + grad.t[1] - grad.t[0]
]
waveform = np.array([grad.first, grad.last])
waveform = 1e-3 * np.insert(
waveform, 1, grad.waveform)
else:
t = np.cumsum([
0, grad.delay, grad.rise_time, grad.flat_time,
grad.fall_time
])
waveform = 1e-3 * grad.amplitude * np.array(
[0, 0, 1, 1, 0])
fig2_sp_list[x].plot(t_factor * (t0 + t), waveform)
t0 += calc_duration(block)
grad_plot_labels = ['x', 'y', 'z']
sp11.set_ylabel('ADC')
sp12.set_ylabel('RF mag (Hz)')
sp13.set_ylabel('RF phase (rad)')
[
fig2_sp_list[x].set_ylabel(f'G{grad_plot_labels[x]} (kHz/m)')
for x in range(3)
]
# Setting display limits
disp_range = t_factor * np.array(
[time_range[0], min(t0, time_range[1])])
sp11.set_xlim(disp_range)
sp12.set_xlim(disp_range)
sp13.set_xlim(disp_range)
[x.set_xlim(disp_range) for x in fig2_sp_list]
fig1.tight_layout()
fig2.tight_layout()
if save:
fig1.savefig('seq_plot1.jpg')
fig2.savefig('seq_plot2.jpg')
plt.show()
TR_fill = (TR - n_slices * TE_train) / n_slices
TR_fill = system.grad_raster_time * round(TR_fill / system.grad_raster_time)
if TR_fill < 0:
TR_fill = 1e-3
warnings.warn(f'TR too short, adapted to include all slices to: {1000 * n_slices * (TE_train + TR_fill)} ms')
else:
print(f'TR fill: {1000 * TR_fill} ms')
delay_TR = make_delay(TR_fill)
delay_end = make_delay(5)
for k_ex in range(n_ex):
for s in range(n_slices):
rfex.freq_offset = GS_ex.amplitude * slice_thickness * (s - (n_slices - 1) / 2)
rfref.freq_offset = GS_ref.amplitude * slice_thickness * (s - (n_slices - 1) / 2)
rfex.phase_offset = rfex_phase - 2 * math.pi * rfex.freq_offset * calc_rf_center(rfex)[0]
rfref.phase_offset = rfref_phase - 2 * math.pi * rfref.freq_offset * calc_rf_center(rfref)[0]
seq.add_block(GS1)
seq.add_block(GS2, rfex)
seq.add_block(GS3, GR3)
for k_ech in range(n_echo):
if k_ex >= 0:
phase_area = phase_areas[k_ech]
else:
phase_area = 0
GP_pre = make_trapezoid(channel='y', system=system, area=phase_area, duration=t_sp, rise_time=dG)
GP_rew = make_trapezoid(channel='y', system=system, area=-phase_area, duration=t_sp, rise_time=dG)
if TR_fill < 0:
TR_fill = 1e-3
warnings.warn(
f'TR too short, adapted to include all slices to: {1000 * n_slices * (TE_train + TR_fill)} ms'
)
else:
print(f'TR fill: {1000 * TR_fill} ms')
delay_TR = make_delay(TR_fill)
for k_ex in range(n_ex + 1):
for s in range(n_slices):
rf_ex.freq_offset = gs_ex.amplitude * slice_thickness * (
s - (n_slices - 1) / 2)
rf_ref.freq_offset = gs_ref.amplitude * slice_thickness * (
s - (n_slices - 1) / 2)
rf_ex.phase_offset = rf_ex_phase - 2 * np.pi * rf_ex.freq_offset * calc_rf_center(
rf_ex)[0]
rf_ref.phase_offset = rf_ref_phase - 2 * np.pi * rf_ref.freq_offset * calc_rf_center(
rf_ref)[0]
seq.add_block(gs1)
seq.add_block(gs2, rf_ex)
seq.add_block(gs3, gr3)
for k_echo in range(n_echo):
if k_ex > 0:
phase_area = phase_areas[k_echo, k_ex - 1]
else:
phase_area = 0.0 # 0.0 and not 0 because -phase_area should successfully result in negative zero
gp_pre = make_trapezoid(channel='y',
system=system,
def calculate_kspace(self, trajectory_delay: int = 0):
"""
Calculates the k-space trajectory of the entire pulse sequence.
Parameters
----------
trajectory_delay : int
Compensation factor in millis to align ADC and gradients in the reconstruction.
Returns
-------
k_traj_adc : numpy.ndarray
K-space trajectory sampled at `t_adc` timepoints.
k_traj : numpy.ndarray
K-space trajectory of the entire pulse sequence.
t_excitation : numpy.ndarray
Excitation timepoints.
t_refocusing : numpy.ndarray
Refocusing timepoints.
t_adc : numpy.ndarray
Sampling timepoints.
"""
c_excitation = 0
c_refocusing = 0
c_adc_samples = 0
for i in range(len(self.block_events)):
block = self.get_block(i + 1)
if hasattr(block, 'rf'):
if not hasattr(block.rf,
'use') or block.rf.use != 'refocusing':
c_excitation += 1
else:
c_refocusing += 1
if hasattr(block, 'adc'):
c_adc_samples += int(block.adc.num_samples)
t_excitation = np.zeros(c_excitation)
t_refocusing = np.zeros(c_refocusing)
k_time = np.zeros(c_adc_samples)
current_dur = 0
c_excitation = 0
c_refocusing = 0
k_counter = 0
traj_recon_delay = trajectory_delay
for i in range(len(self.block_events)):
block = self.get_block(i + 1)
if hasattr(block, 'rf'):
rf = block.rf
rf_center, _ = calc_rf_center(rf)
t = rf.delay + rf_center
if not hasattr(block.rf,
'use') or block.rf.use != 'refocusing':
t_excitation[c_excitation] = current_dur + t
c_excitation += 1
else:
t_refocusing[c_refocusing] = current_dur + t
c_refocusing += 1
if hasattr(block, 'adc'):
k_time[k_counter:k_counter + block.adc.num_samples] = np.arange(
block.adc.num_samples
) * block.adc.dwell + block.adc.delay + current_dur + traj_recon_delay
k_counter += block.adc.num_samples
current_dur += calc_duration(block)
gw = self.gradient_waveforms()
i_excitation = np.round(t_excitation / self.grad_raster_time)
i_refocusing = np.round(t_refocusing / self.grad_raster_time)
k_traj = np.zeros(gw.shape)
k = [0, 0, 0]
for i in range(gw.shape[1]):
k += gw[:, i] * self.grad_raster_time
k_traj[:, i] = k
if len(np.where(i_excitation == i + 1)[0]) >= 1:
k = 0
k_traj[:, i] = np.nan
if len(np.where(i_refocusing == i + 1)[0]) >= 1:
k = -k
k_traj_adc = []
for i in range(k_traj.shape[0]):
k_traj_adc.append(
np.interp(
k_time,
np.arange(1, k_traj.shape[1] + 1) * self.grad_raster_time,
k_traj[i]))
k_traj_adc = np.asarray(k_traj_adc)
t_adc = k_time
return k_traj_adc, k_traj, t_excitation, t_refocusing, t_adc
system=system,
area=-(Ny / 2 - 0.5 - (Ny - Nyeff)) * delta_k,
duration=pre_time)
# Phase blip in shortest possible time
gy = make_trapezoid(channel='y', system=system, area=delta_k)
dur = math.ceil(calc_duration(gy) / seq.grad_raster_time) / i_raster_time
# Integer times needed for TE optimization
# The time(gy) refers to the number of blips, thus we substract 0.5 since the number of lines is always even.
# The time(gx) refers to the time needed to read each line of the k-space. Thus, if Ny is even, it would take half of the lines plus another half.
n_duration_center = math.ceil(
(calc_duration(gx) * (Ny / 2 + 0.5 - (Ny - Nyeff)) + dur *
(Ny / 2 - 0.5 -
(Ny - Nyeff)) + calc_duration(gx_pre, gy_pre)) / seq.grad_raster_time)
rf_center_with_delay = rf.delay + calc_rf_center(rf)[0]
n_rf90r = math.ceil((calc_duration(gz) - rf_center_with_delay + pre_time) /
seq.grad_raster_time)
n_rf180r = math.ceil((calc_duration(rf180) + 2 * calc_duration(gz_spoil)) / 2 /
seq.grad_raster_time)
n_rf180l = math.floor((calc_duration(rf180) + 2 * calc_duration(gz_spoil)) /
2 / seq.grad_raster_time)
# Group variables
seq_sys_Dict = {"seq": seq, "system": system, "i_raster_time": i_raster_time}
grads_times_Dict = {
"n_rf90r": n_rf90r,
"n_rf180r": n_rf180r,
"n_rf180l": n_rf180l,
def calculate_kspace(self, trajectory_delay=0):
c_excitation = 0
c_refocusing = 0
c_adc_samples = 0
for i in range(len(self.block_events)):
block = self.get_block(i + 1)
if hasattr(block, 'rf'):
if not hasattr(block.rf,
'use') or block.rf.use != 'refocusing':
c_excitation += 1
else:
c_refocusing += 1
if hasattr(block, 'adc'):
c_adc_samples += int(block.adc.num_samples)
t_excitation = np.zeros(c_excitation)
t_refocusing = np.zeros(c_refocusing)
k_time = np.zeros(c_adc_samples)
current_dur = 0
c_excitation = 0
c_refocusing = 0
k_counter = 0
traj_recon_delay = trajectory_delay
for i in range(len(self.block_events)):
block = self.get_block(i + 1)
if hasattr(block, 'rf'):
rf = block.rf
rf_center, _ = calc_rf_center(rf)
t = rf.delay + rf_center
if not hasattr(block.rf,
'use') or block.rf.use != 'refocusing':
t_excitation[c_excitation] = current_dur + t
c_excitation += 1
else:
t_refocusing[c_refocusing] = current_dur + t
c_refocusing += 1
if hasattr(block, 'adc'):
k_time[k_counter:k_counter + block.adc.num_samples] = np.arange(
block.adc.num_samples
) * block.adc.dwell + block.adc.delay + current_dur + traj_recon_delay
k_counter += block.adc.num_samples
current_dur += calc_duration(block)
gw = self.gradient_waveforms()
i_excitation = np.round(t_excitation / self.grad_raster_time)
i_refocusing = np.round(t_refocusing / self.grad_raster_time)
k_traj = np.zeros(gw.shape)
k = [0, 0, 0]
for i in range(gw.shape[1]):
k += gw[:, i] * self.grad_raster_time
k_traj[:, i] = k
if len(np.where(i_excitation == i + 1)[0]) >= 1:
k = 0
k_traj[:, i] = np.nan
if len(np.where(i_refocusing == i + 1)[0]) >= 1:
k = -k
k_traj_adc = []
for i in range(k_traj.shape[0]):
k_traj_adc.append(
np.interp(
k_time,
np.arange(1, k_traj.shape[1] + 1) * self.grad_raster_time,
k_traj[i]))
k_traj_adc = np.asarray(k_traj_adc)
t_adc = k_time
return k_traj_adc, k_traj, t_excitation, t_refocusing, t_adc
def write_tse(n=256,
fov=250e-3,
thk=5e-3,
fa_exc=90,
fa_ref=180,
te=50e-3,
tr=2000e-3,
slice_locations=[0],
turbo_factor=4,
enc='xyz'):
"""
2D TSE sequence with interleaved slices and user-defined turbo factor
Inputs
------
n : integer
Matrix size (isotropic)
fov : float
Field-of-View in [meters]
thk : float
Slice thickness in [meters]
fa_exc : float
Initial excitation flip angle in [degrees]
fa_ref : float
All following flip angles for spin echo refocusing in [degrees]
te : float
Echo Time in [seconds]
tr : float
Repetition Time in [seconds]
slice_locations : array_like
Array of slice locations from isocenter in [meters]
turbo_factor : integer
Number of echoes per TR
enc : str
Spatial encoding directions; 1st - readout; 2nd - phase encoding; 3rd - slice select
Use str with any permutation of x, y, and z to obtain orthogonal slices
e.g. The default 'xyz' means axial(z) slice with readout in x and phase encoding in y
Returns
-------
seq : pypulseq.Sequence.sequence Sequence object
Output sequence object. Can be saved with seq.write('file_name.seq')
pe_order : numpy.ndarray
(turbo_factor) x (number_of_excitations) matrix of phase encoding order
This is required for phase sorting before ifft2 reconstruction.
"""
# Set system limits
ramp_time = 250e-6 # Ramp up/down time for all gradients where this is specified
system = Opts(
max_grad=32,
grad_unit='mT/m',
max_slew=130,
slew_unit='T/m/s',
rf_ringdown_time=100e-6, # changed from 30e-6
rf_dead_time=100e-6,
adc_dead_time=20e-6)
# Initialize sequence
seq = Sequence(system)
# Spatial encoding directions
ch_ro = enc[0]
ch_pe = enc[1]
ch_ss = enc[2]
# Derived parameters
Nf, Np = (n, n)
delta_k = 1 / fov
k_width = Nf * delta_k
# Number of echoes per excitation (i.e. turbo factor)
n_echo = turbo_factor
# Readout duration
readout_time = 6.4e-3 + 2 * system.adc_dead_time
# Excitation pulse duration
t_ex = 2.5e-3
t_exwd = t_ex + system.rf_ringdown_time + system.rf_dead_time
# Refocusing pulse duration
t_ref = 2e-3
t_refwd = t_ref + system.rf_ringdown_time + system.rf_dead_time
# Time gaps for spoilers
t_sp = 0.5 * (te - readout_time - t_refwd
) # time gap between pi-pulse and readout
t_spex = 0.5 * (te - t_exwd - t_refwd
) # time gap between pi/2-pulse and pi-pulse
# Spoiling factors
fsp_r = 1 # in readout direction per refocusing
fsp_s = 0.5 # in slice direction per refocusing
# 3. Calculate sequence components
## Slice-selective RF pulses & gradient
#* RF pulses with zero frequency shift are created. The frequency is then changed before adding the pulses to sequence blocks for each slice.
# 90 deg pulse (+y')
rf_ex_phase = np.pi / 2
flip_ex = fa_exc * np.pi / 180
rf_ex, g_ss, _ = make_sinc_pulse(flip_angle=flip_ex,
system=system,
duration=t_ex,
slice_thickness=thk,
apodization=0.5,
time_bw_product=4,
phase_offset=rf_ex_phase,
return_gz=True)
gs_ex = make_trapezoid(channel=ch_ss,
system=system,
amplitude=g_ss.amplitude,
flat_time=t_exwd,
rise_time=ramp_time)
# 180 deg pulse (+x')
rf_ref_phase = 0
flip_ref = fa_ref * np.pi / 180
rf_ref, gz, _ = make_sinc_pulse(flip_angle=flip_ref,
system=system,
duration=t_ref,
slice_thickness=thk,
apodization=0.5,
time_bw_product=4,
phase_offset=rf_ref_phase,
use='refocusing',
return_gz=True)
gs_ref = make_trapezoid(channel=ch_ss,
system=system,
amplitude=gs_ex.amplitude,
flat_time=t_refwd,
rise_time=ramp_time)
rf_ex, g_ss, _ = make_sinc_pulse(flip_angle=flip_ex,
system=system,
duration=t_ex,
slice_thickness=thk,
apodization=0.5,
time_bw_product=4,
phase_offset=rf_ex_phase,
return_gz=True)
## Make gradients and ADC
# gs_spex : slice direction spoiler between initial excitation and 1st 180 pulse
# gs_spr : slice direction spoiler between 180 pulses
# gr_spr : readout direction spoiler; area is (fsp_r) x (full readout area)
# SS spoiling
ags_ex = gs_ex.area / 2
gs_spr = make_trapezoid(channel=ch_ss,
system=system,
area=ags_ex * (1 + fsp_s),
duration=t_sp,
rise_time=ramp_time)
gs_spex = make_trapezoid(channel=ch_ss,
system=system,
area=ags_ex * fsp_s,
duration=t_spex,
rise_time=ramp_time)
# Readout gradient and ADC
gr_acq = make_trapezoid(channel=ch_ro,
system=system,
flat_area=k_width,
flat_time=readout_time,
rise_time=ramp_time)
# No need for risetime delay since it is set at beginning of flattime; delay is ADC deadtime
adc = make_adc(num_samples=Nf,
duration=gr_acq.flat_time - 40e-6,
delay=20e-6)
# RO spoiling
gr_spr = make_trapezoid(channel=ch_ro,
system=system,
area=gr_acq.area * fsp_r,
duration=t_sp,
rise_time=ramp_time)
# Following is not used anywhere
# gr_spex = make_trapezoid(channel=ch_ro, system=system, area=gr_acq.area * (1 + fsp_r), duration=t_spex, rise_time=ramp_time)
# Prephasing gradient in RO direction
agr_preph = gr_acq.area / 2 + gr_spr.area
gr_preph = make_trapezoid(channel=ch_ro,
system=system,
area=agr_preph,
duration=t_spex,
rise_time=ramp_time)
# Phase encoding areas
# Need to export the pe_order for reconsturuction
# Number of readouts/echoes to be produced per TR
n_ex = math.floor(Np / n_echo)
pe_steps = np.arange(1, n_echo * n_ex + 1) - 0.5 * n_echo * n_ex - 1
if divmod(n_echo, 2)[1] == 0: # If there is an even number of echoes
pe_steps = np.roll(pe_steps, -round(n_ex / 2))
pe_order = pe_steps.reshape((n_ex, n_echo), order='F').T
savemat('pe_info.mat', {'order': pe_order, 'dims': ['n_echo', 'n_ex']})
phase_areas = pe_order * delta_k
# Split gradients and recombine into blocks
# gs1 : ramp up of gs_ex
gs1_times = [0, gs_ex.rise_time]
gs1_amp = [0, gs_ex.amplitude]
gs1 = make_extended_trapezoid(channel=ch_ss,
times=gs1_times,
amplitudes=gs1_amp)
# gs2 : flat part of gs_ex
gs2_times = [0, gs_ex.flat_time]
gs2_amp = [gs_ex.amplitude, gs_ex.amplitude]
gs2 = make_extended_trapezoid(channel=ch_ss,
times=gs2_times,
amplitudes=gs2_amp)
# gs3 : Bridged slice pre-spoiler
gs3_times = [
0, gs_spex.rise_time, gs_spex.rise_time + gs_spex.flat_time,
gs_spex.rise_time + gs_spex.flat_time + gs_spex.fall_time
]
gs3_amp = [
gs_ex.amplitude, gs_spex.amplitude, gs_spex.amplitude, gs_ref.amplitude
]
gs3 = make_extended_trapezoid(channel=ch_ss,
times=gs3_times,
amplitudes=gs3_amp)
# gs4 : Flat slice selector for pi-pulse
gs4_times = [0, gs_ref.flat_time]
gs4_amp = [gs_ref.amplitude, gs_ref.amplitude]
gs4 = make_extended_trapezoid(channel=ch_ss,
times=gs4_times,
amplitudes=gs4_amp)
# gs5 : Bridged slice post-spoiler
gs5_times = [
0, gs_spr.rise_time, gs_spr.rise_time + gs_spr.flat_time,
gs_spr.rise_time + gs_spr.flat_time + gs_spr.fall_time
]
gs5_amp = [gs_ref.amplitude, gs_spr.amplitude, gs_spr.amplitude, 0]
gs5 = make_extended_trapezoid(channel=ch_ss,
times=gs5_times,
amplitudes=gs5_amp)
# gs7 : The gs3 for next pi-pulse
gs7_times = [
0, gs_spr.rise_time, gs_spr.rise_time + gs_spr.flat_time,
gs_spr.rise_time + gs_spr.flat_time + gs_spr.fall_time
]
gs7_amp = [0, gs_spr.amplitude, gs_spr.amplitude, gs_ref.amplitude]
gs7 = make_extended_trapezoid(channel=ch_ss,
times=gs7_times,
amplitudes=gs7_amp)
# gr3 : pre-readout gradient
gr3 = gr_preph
# gr5 : Readout post-spoiler
gr5_times = [
0, gr_spr.rise_time, gr_spr.rise_time + gr_spr.flat_time,
gr_spr.rise_time + gr_spr.flat_time + gr_spr.fall_time
]
gr5_amp = [0, gr_spr.amplitude, gr_spr.amplitude, gr_acq.amplitude]
gr5 = make_extended_trapezoid(channel=ch_ro,
times=gr5_times,
amplitudes=gr5_amp)
# gr6 : Flat readout gradient
gr6_times = [0, readout_time]
gr6_amp = [gr_acq.amplitude, gr_acq.amplitude]
gr6 = make_extended_trapezoid(channel=ch_ro,
times=gr6_times,
amplitudes=gr6_amp)
# gr7 : the gr3 for next repeat
gr7_times = [
0, gr_spr.rise_time, gr_spr.rise_time + gr_spr.flat_time,
gr_spr.rise_time + gr_spr.flat_time + gr_spr.fall_time
]
gr7_amp = [gr_acq.amplitude, gr_spr.amplitude, gr_spr.amplitude, 0]
gr7 = make_extended_trapezoid(channel=ch_ro,
times=gr7_times,
amplitudes=gr7_amp)
# Timing (delay) calculations
# delay_TR : delay at the end of each TSE pulse train (i.e. each TR)
t_ex = gs1.t[-1] + gs2.t[-1] + gs3.t[-1]
t_ref = gs4.t[-1] + gs5.t[-1] + gs7.t[-1] + readout_time
t_end = gs4.t[-1] + gs5.t[-1]
TE_train = t_ex + n_echo * t_ref + t_end
TR_fill = (tr - n_slices * TE_train) / n_slices
TR_fill = system.grad_raster_time * round(
TR_fill / system.grad_raster_time)
if TR_fill < 0:
TR_fill = 1e-3
print(
f'TR too short, adapted to include all slices to: {1000 * n_slices * (TE_train + TR_fill)} ms'
)
else:
print(f'TR fill: {1000 * TR_fill} ms')
delay_TR = make_delay(TR_fill)
# Add building blocks to sequence
for k_ex in range(n_ex + 1): # For each TR
for s in range(n_slices): # For each slice (multislice)
if slice_locations is None:
rf_ex.freq_offset = gs_ex.amplitude * thk * (
s - (n_slices - 1) / 2)
rf_ref.freq_offset = gs_ref.amplitude * thk * (
s - (n_slices - 1) / 2)
rf_ex.phase_offset = rf_ex_phase - 2 * np.pi * rf_ex.freq_offset * calc_rf_center(
rf_ex)[0]
rf_ref.phase_offset = rf_ref_phase - 2 * np.pi * rf_ref.freq_offset * calc_rf_center(
rf_ref)[0]
else:
rf_ex.freq_offset = gs_ex.amplitude * slice_locations[s]
rf_ref.freq_offset = gs_ref.amplitude * slice_locations[s]
rf_ex.phase_offset = rf_ex_phase - 2 * np.pi * rf_ex.freq_offset * calc_rf_center(
rf_ex)[0]
rf_ref.phase_offset = rf_ref_phase - 2 * np.pi * rf_ref.freq_offset * calc_rf_center(
rf_ref)[0]
seq.add_block(gs1)
seq.add_block(gs2, rf_ex) # make sure gs2 has channel ch_ss
seq.add_block(gs3, gr3)
for k_echo in range(n_echo): # For each echo
if k_ex > 0:
phase_area = phase_areas[k_echo, k_ex - 1]
else:
# First TR is skipped so zero phase encoding is needed
phase_area = 0.0 # 0.0 and not 0 because -phase_area should successfully result in negative zero
gp_pre = make_trapezoid(channel=ch_pe,
system=system,
area=phase_area,
duration=t_sp,
rise_time=ramp_time)
# print('gp_pre info: ', gp_pre)
gp_rew = make_trapezoid(channel=ch_pe,
system=system,
area=-phase_area,
duration=t_sp,
rise_time=ramp_time)
seq.add_block(gs4, rf_ref)
seq.add_block(gs5, gr5, gp_pre)
# Skipping first TR
if k_ex > 0:
seq.add_block(gr6, adc)
else:
seq.add_block(gr6)
seq.add_block(gs7, gr7, gp_rew)
seq.add_block(gs4)
seq.add_block(gs5)
seq.add_block(delay_TR)
# Check timing to make sure sequence runs on scanner
seq.check_timing()
return seq, pe_order
def write_irse_interleaved_split_gradient(n=256,
fov=250e-3,
thk=5e-3,
fa=90,
te=12e-3,
tr=2000e-3,
ti=150e-3,
slice_locations=[0],
enc='xyz'):
"""
2D IRSE sequence with overlapping gradient ramps and interleaved slices
Inputs
------
n : integer
Matrix size (isotropic)
fov : float
Field-of-View in [meters]
thk : float
Slice thickness in [meters]
fa : float
Flip angle in [degrees]
te : float
Echo Time in [seconds]
tr : float
Repetition Time in [seconds]
ti : float
Inversion Time in [seconds]
slice_locations : array_like
Array of slice locations from isocenter in [meters]
enc : str
Spatial encoding directions; 1st - readout; 2nd - phase encoding; 3rd - slice select
Use str with any permutation of x, y, and z to obtain orthogonal slices
e.g. The default 'xyz' means axial(z) slice with readout in x and phase encoding in y
Returns
-------
seq : pypulseq.Sequence.sequence Sequence object
Output sequence object. Can be saved with seq.write('file_name.seq')
sl_order : numpy.ndarray
Randomly generated slice order. Useful for reconstruction.
"""
# =========
# SYSTEM LIMITS
# =========
# Set the hardware limits and initialize sequence object
dG = 250e-6 # Fixed ramp time for all gradients
system = Opts(max_grad=32,
grad_unit='mT/m',
max_slew=130,
slew_unit='T/m/s',
rf_ringdown_time=100e-6,
rf_dead_time=100e-6,
adc_dead_time=10e-6)
seq = Sequence(system)
# =========
# TIME CALCULATIONS
# =========
readout_time = 6.4e-3 + 2 * system.adc_dead_time
t_ex = 2.5e-3
t_exwd = t_ex + system.rf_ringdown_time + system.rf_dead_time
t_ref = 2e-3
t_refwd = t_ref + system.rf_ringdown_time + system.rf_dead_time
t_sp = 0.5 * (te - readout_time - t_refwd)
t_spex = 0.5 * (te - t_exwd - t_refwd)
fsp_r = 1
fsp_s = 0.5
# =========
# ENCODING DIRECTIONS
# ==========
ch_ro = enc[0]
ch_pe = enc[1]
ch_ss = enc[2]
# =========
# RF AND GRADIENT SHAPES - BASED ON RESOLUTION REQUIREMENTS : kmax and Npe
# =========
# RF Phases
rf_ex_phase = np.pi / 2
rf_ref_phase = 0
# Excitation phase (90 deg)
flip_ex = 90 * np.pi / 180
rf_ex, gz, _ = make_sinc_pulse(flip_angle=flip_ex,
system=system,
duration=t_ex,
slice_thickness=thk,
apodization=0.5,
time_bw_product=4,
phase_offset=rf_ex_phase,
return_gz=True)
gs_ex = make_trapezoid(channel=ch_ss,
system=system,
amplitude=gz.amplitude,
flat_time=t_exwd,
rise_time=dG)
# Refocusing (same gradient & RF is used for initial inversion)
flip_ref = fa * np.pi / 180
rf_ref, gz, _ = make_sinc_pulse(flip_angle=flip_ref,
system=system,
duration=t_ref,
slice_thickness=thk,
apodization=0.5,
time_bw_product=4,
phase_offset=rf_ref_phase,
use='refocusing',
return_gz=True)
gs_ref = make_trapezoid(channel=ch_ss,
system=system,
amplitude=gs_ex.amplitude,
flat_time=t_refwd,
rise_time=dG)
ags_ex = gs_ex.area / 2
gs_spr = make_trapezoid(channel=ch_ss,
system=system,
area=ags_ex * (1 + fsp_s),
duration=t_sp,
rise_time=dG)
gs_spex = make_trapezoid(channel=ch_ss,
system=system,
area=ags_ex * fsp_s,
duration=t_spex,
rise_time=dG)
delta_k = 1 / fov
k_width = n * delta_k
gr_acq = make_trapezoid(channel=ch_ro,
system=system,
flat_area=k_width,
flat_time=readout_time,
rise_time=dG)
adc = make_adc(num_samples=n,
duration=gr_acq.flat_time - 40e-6,
delay=20e-6)
gr_spr = make_trapezoid(channel=ch_ro,
system=system,
area=gr_acq.area * fsp_r,
duration=t_sp,
rise_time=dG)
gr_spex = make_trapezoid(channel=ch_ro,
system=system,
area=gr_acq.area * (1 + fsp_r),
duration=t_spex,
rise_time=dG)
agr_spr = gr_spr.area
agr_preph = gr_acq.area / 2 + agr_spr
gr_preph = make_trapezoid(channel=ch_ro,
system=system,
area=agr_preph,
duration=t_spex,
rise_time=dG)
phase_areas = (np.arange(n) - n / 2) * delta_k
# Split gradients and recombine into blocks
gs1_times = [0, gs_ex.rise_time]
gs1_amp = [0, gs_ex.amplitude]
gs1 = make_extended_trapezoid(channel=ch_ss,
times=gs1_times,
amplitudes=gs1_amp)
gs2_times = [0, gs_ex.flat_time]
gs2_amp = [gs_ex.amplitude, gs_ex.amplitude]
gs2 = make_extended_trapezoid(channel=ch_ss,
times=gs2_times,
amplitudes=gs2_amp)
gs3_times = [
0, gs_spex.rise_time, gs_spex.rise_time + gs_spex.flat_time,
gs_spex.rise_time + gs_spex.flat_time + gs_spex.fall_time
]
gs3_amp = [
gs_ex.amplitude, gs_spex.amplitude, gs_spex.amplitude, gs_ref.amplitude
]
gs3 = make_extended_trapezoid(channel=ch_ss,
times=gs3_times,
amplitudes=gs3_amp)
gs4_times = [0, gs_ref.flat_time]
gs4_amp = [gs_ref.amplitude, gs_ref.amplitude]
gs4 = make_extended_trapezoid(channel=ch_ss,
times=gs4_times,
amplitudes=gs4_amp)
gs5_times = [
0, gs_spr.rise_time, gs_spr.rise_time + gs_spr.flat_time,
gs_spr.rise_time + gs_spr.flat_time + gs_spr.fall_time
]
gs5_amp = [gs_ref.amplitude, gs_spr.amplitude, gs_spr.amplitude, 0]
gs5 = make_extended_trapezoid(channel=ch_ss,
times=gs5_times,
amplitudes=gs5_amp)
gs7_times = [
0, gs_spr.rise_time, gs_spr.rise_time + gs_spr.flat_time,
gs_spr.rise_time + gs_spr.flat_time + gs_spr.fall_time
]
gs7_amp = [0, gs_spr.amplitude, gs_spr.amplitude, gs_ref.amplitude]
gs7 = make_extended_trapezoid(channel=ch_ss,
times=gs7_times,
amplitudes=gs7_amp)
gr3 = gr_preph
gr5_times = [
0, gr_spr.rise_time, gr_spr.rise_time + gr_spr.flat_time,
gr_spr.rise_time + gr_spr.flat_time + gr_spr.fall_time
]
gr5_amp = [0, gr_spr.amplitude, gr_spr.amplitude, gr_acq.amplitude]
gr5 = make_extended_trapezoid(channel=ch_ro,
times=gr5_times,
amplitudes=gr5_amp)
gr6_times = [0, readout_time]
gr6_amp = [gr_acq.amplitude, gr_acq.amplitude]
gr6 = make_extended_trapezoid(channel=ch_ro,
times=gr6_times,
amplitudes=gr6_amp)
gr7_times = [
0, gr_spr.rise_time, gr_spr.rise_time + gr_spr.flat_time,
gr_spr.rise_time + gr_spr.flat_time + gr_spr.fall_time
]
gr7_amp = [gr_acq.amplitude, gr_spr.amplitude, gr_spr.amplitude, 0]
gr7 = make_extended_trapezoid(channel=ch_ro,
times=gr7_times,
amplitudes=gr7_amp)
t_ex = gs1.t[-1] + gs2.t[-1] + gs3.t[-1]
t_ref = gs4.t[-1] + gs5.t[-1] + gs7.t[-1] + readout_time
t_end = gs4.t[-1] + gs5.t[-1]
# Calculate maximum number of slices that can fit in one TR
# Without spoilers on each side
TE_prime = 0.5 * calc_duration(gs_ref) + ti + te + 0.5 * readout_time + np.max(
[calc_duration(gs7), calc_duration(gr7)]) + \
calc_duration(gs4) + calc_duration(gs5)
ns_per_TR = np.floor(tr / TE_prime)
print('Number of slices that can be accommodated = ' + str(ns_per_TR))
# Lengthen TR to accommodate slices if needed, and display message
n_slices = len(slice_locations)
if (ns_per_TR < n_slices):
print(
f'TR too short, adapted to include all slices to: {n_slices * TE_prime + 50e-6} s'
)
TR = round(n_slices * TE_prime + 50e-6, ndigits=5)
print('New TR = ' + str(TR))
ns_per_TR = np.floor(TR / TE_prime)
if (n_slices < ns_per_TR):
ns_per_TR = n_slices
# randperm so that adjacent slices do not get excited one after the other
sl_order = np.random.permutation(n_slices)
print('Number of slices acquired per TR = ' + str(ns_per_TR))
# Delays
TI_fill = ti - (0.5 * calc_duration(gs_ref) + calc_duration(gs1) +
0.5 * calc_duration(gs2))
delay_TI = make_delay(TI_fill)
TR_fill = tr - ns_per_TR * TE_prime
delay_TR = make_delay(TR_fill)
for k_ex in range(n):
phase_area = phase_areas[k_ex]
gp_pre = make_trapezoid(channel=ch_pe,
system=system,
area=phase_area,
duration=t_sp,
rise_time=dG)
gp_rew = make_trapezoid(channel=ch_pe,
system=system,
area=-phase_area,
duration=t_sp,
rise_time=dG)
s_in_TR = 0
for s in range(len(sl_order)):
# rf_ex.freq_offset = gs_ex.amplitude * slice_thickness * (sl_order[s] - (n_slices - 1) / 2)
# rf_ref.freq_offset = gs_ref.amplitude * slice_thickness * (sl_order[s] - (n_slices - 1) / 2)
rf_ex.freq_offset = gs_ex.amplitude * slice_locations[sl_order[s]]
rf_ref.freq_offset = gs_ref.amplitude * slice_locations[
sl_order[s]]
rf_ex.phase_offset = rf_ex_phase - 2 * np.pi * rf_ex.freq_offset * calc_rf_center(
rf_ex)[0]
rf_ref.phase_offset = rf_ref_phase - 2 * np.pi * rf_ref.freq_offset * calc_rf_center(
rf_ref)[0]
# Inversion using refocusing pulse
seq.add_block(gs_ref, rf_ref)
seq.add_block(delay_TI)
# SE portion
seq.add_block(gs1)
seq.add_block(gs2, rf_ex)
seq.add_block(gs3, gr3)
seq.add_block(gs4, rf_ref)
seq.add_block(gs5, gr5, gp_pre)
seq.add_block(gr6, adc)
seq.add_block(gs7, gr7, gp_rew)
seq.add_block(gs4)
seq.add_block(gs5)
s_in_TR += 1
if (s_in_TR == ns_per_TR):
seq.add_block(delay_TR)
s_in_TR = 0
# Check timing to make sure sequence runs on scanner
seq.check_timing()
return seq, sl_order
def calculate_kspace(self, trajectory_delay: int = 0) -> Tuple[np.array, np.array, np.array, np.array, np.array]:
"""
Calculates the k-space trajectory of the entire pulse sequence.
Parameters
----------
trajectory_delay : int, default=0
Compensation factor in millis to align ADC and gradients in the reconstruction.
Returns
-------
k_traj_adc : numpy.array
K-space trajectory sampled at `t_adc` timepoints.
k_traj : numpy.array
K-space trajectory of the entire pulse sequence.
t_excitation : numpy.array
Excitation timepoints.
t_refocusing : numpy.array
Refocusing timepoints.
t_adc : numpy.array
Sampling timepoints.
"""
# Initialise the counters and accumulator objects
count_excitation = 0
count_refocusing = 0
count_adc_samples = 0
# Loop through the blocks to prepare preallocations
for block_counter in range(len(self.dict_block_events)):
block = self.get_block(block_counter + 1)
if hasattr(block, 'rf'):
if not hasattr(block.rf, 'use') or block.rf.use != 'refocusing':
count_excitation += 1
else:
count_refocusing += 1
if hasattr(block, 'adc'):
count_adc_samples += int(block.adc.num_samples)
t_excitation = np.zeros(count_excitation)
t_refocusing = np.zeros(count_refocusing)
k_time = np.zeros(count_adc_samples)
current_duration = 0
count_excitation = 0
count_refocusing = 0
kc_outer = 0
traj_recon_delay = trajectory_delay
# Go through the blocks and collect RF and ADC timing data
for block_counter in range(len(self.dict_block_events)):
block = self.get_block(block_counter + 1)
if hasattr(block, 'rf'):
rf = block.rf
rf_center, _ = calc_rf_center(rf)
t = rf.delay + rf_center
if not hasattr(block.rf, 'use') or block.rf.use != 'refocusing':
t_excitation[count_excitation] = current_duration + t
count_excitation += 1
else:
t_refocusing[count_refocusing] = current_duration + t
count_refocusing += 1
if hasattr(block, 'adc'):
_k_time = np.arange(block.adc.num_samples) + 0.5
_k_time = _k_time * block.adc.dwell + block.adc.delay + current_duration + traj_recon_delay
k_time[kc_outer:kc_outer + block.adc.num_samples] = _k_time
kc_outer += block.adc.num_samples
current_duration += self.arr_block_durations[block_counter]
# Now calculate the actual k-space trajectory based on the gradient waveforms
gw = self.gradient_waveforms()
i_excitation = np.round(t_excitation / self.grad_raster_time)
i_refocusing = np.round(t_refocusing / self.grad_raster_time)
i_periods = np.sort([1, *(i_excitation + 1), *(i_refocusing + 1), gw.shape[1] + 1]).astype(np.int)
# i_periods -= 1 # Python is 0-indexed
ii_next_excitation = min(len(i_excitation), 1)
ii_next_refocusing = min(len(i_refocusing), 1)
k_traj = np.zeros_like(gw)
k = np.zeros((3, 1))
for i in range(len(i_periods) - 1):
i_period_end = i_periods[i + 1] - 1
k_period = np.concatenate((k, gw[:, i_periods[i] - 1:i_period_end] * self.grad_raster_time), axis=1)
k_period = np.cumsum(k_period, axis=1)
k_traj[:, i_periods[i] - 1:i_period_end] = k_period[:, 1:]
k = k_period[:, -1]
if ii_next_excitation > 0 and i_excitation[ii_next_excitation - 1] == i_period_end:
k[:] = 0
k_traj[:, i_period_end - 1] = np.nan
ii_next_excitation = min(len(i_excitation), ii_next_excitation + 1)
if ii_next_refocusing > 0 and i_refocusing[ii_next_refocusing - 1] == i_period_end:
k = -k
ii_next_refocusing = min(len(i_refocusing), ii_next_refocusing + 1)
k = k.reshape((-1, 1)) # To be compatible with np.concatenate
k_traj_adc = []
for _k_traj_row in k_traj:
result = np.interp(xp=np.array(range(1, k_traj.shape[1] + 1)) * self.grad_raster_time,
fp=_k_traj_row,
x=k_time)
k_traj_adc.append(result)
k_traj_adc = np.stack(k_traj_adc)
t_adc = k_time
return k_traj_adc, k_traj, t_excitation, t_refocusing, t_adc
def plot(self, label: str = str(), save: bool = False, time_range=(0, np.inf), time_disp: str = 's',
plot_type: str = 'Gradient') -> None:
"""
Plot `Sequence`.
Parameters
----------
label : str, defualt=str()
save : bool, default=False
Boolean flag indicating if plots should be saved. The two figures will be saved as JPG with numerical
suffixes to the filename 'seq_plot'.
time_range : iterable, default=(0, np.inf)
Time range (x-axis limits) for plotting the sequence. Default is 0 to infinity (entire sequence).
time_disp : str, default='s'
Time display type, must be one of `s`, `ms` or `us`.
plot_type : str, default='Gradient'
Gradients display type, must be one of either 'Gradient' or 'Kspace'.
"""
mpl.rcParams['lines.linewidth'] = 0.75 # Set default Matplotlib linewidth
valid_plot_types = ['Gradient', 'Kspace']
valid_time_units = ['s', 'ms', 'us']
valid_labels = get_supported_labels()
if plot_type not in valid_plot_types:
raise ValueError('Unsupported plot type')
if not all([isinstance(x, (int, float)) for x in time_range]) or len(time_range) != 2:
raise ValueError('Invalid time range')
if time_disp not in valid_time_units:
raise ValueError('Unsupported time unit')
fig1, fig2 = plt.figure(1), plt.figure(2)
sp11 = fig1.add_subplot(311)
sp12, sp13 = fig1.add_subplot(312, sharex=sp11), fig1.add_subplot(313, sharex=sp11)
fig2_subplots = [fig2.add_subplot(311, sharex=sp11), fig2.add_subplot(312, sharex=sp11),
fig2.add_subplot(313, sharex=sp11)]
t_factor_list = [1, 1e3, 1e6]
t_factor = t_factor_list[valid_time_units.index(time_disp)]
t0 = 0
label_defined = False
label_idx_to_plot = []
label_legend_to_plot = []
label_store = dict()
for i in range(len(valid_labels)):
label_store[valid_labels[i]] = 0
if label.upper() == valid_labels[i]:
label_idx_to_plot.append(i)
label_legend_to_plot.append(valid_labels[i])
if len(label_idx_to_plot) != 0:
p = parula.main(len(label_idx_to_plot) + 1)
label_colors_to_plot = p(np.arange(len(label_idx_to_plot)))
for block_counter in range(len(self.dict_block_events)):
block = self.get_block(block_counter + 1)
is_valid = time_range[0] <= t0 <= time_range[1]
if is_valid:
if hasattr(block, 'label'):
for i in range(len(block.label)):
if block.label[i].type == 'labelinc':
label_store[block.label[i].label] += block.label[i].value
else:
label_store[block.label[i].label] = block.label[i].value
label_defined = True
if hasattr(block, 'adc'):
adc = block.adc
# From Pulseq: According to the information from Klaus Scheffler and indirectly from Siemens this
# is the present convention - the samples are shifted by 0.5 dwell
t = adc.delay + (np.arange(int(adc.num_samples)) + 0.5) * adc.dwell
sp11.plot(t_factor * (t0 + t), np.zeros(len(t)), 'rx')
sp13.plot(t_factor * (t0 + t),
np.angle(np.exp(1j * adc.phase_offset) * np.exp(1j * 2 * np.pi * t * adc.freq_offset)),
'b.')
if label_defined and len(label_idx_to_plot) != 0:
cycler = mpl.cycler(color=label_colors_to_plot)
sp11.set_prop_cycle(cycler)
label_store_arr = list(label_store.values())
lbl_vals = np.take(label_store_arr, label_idx_to_plot)
t = t0 + adc.delay + (adc.num_samples - 1) / 2 * adc.dwell
p = sp11.plot(t_factor * t, lbl_vals, '.')
if len(label_legend_to_plot) != 0:
sp11.legend(p, label_legend_to_plot, loc='upper left')
label_legend_to_plot = []
if hasattr(block, 'rf'):
rf = block.rf
tc, ic = calc_rf_center(rf)
t = rf.t + rf.delay
tc = tc + rf.delay
sp12.plot(t_factor * (t0 + t), np.abs(rf.signal))
sp13.plot(t_factor * (t0 + t), np.angle(rf.signal * np.exp(1j * rf.phase_offset)
* np.exp(1j * 2 * math.pi * rf.t * rf.freq_offset)),
t_factor * (t0 + tc), np.angle(rf.signal[ic] * np.exp(1j * rf.phase_offset)
* np.exp(1j * 2 * math.pi * rf.t[ic] * rf.freq_offset)),
'xb')
grad_channels = ['gx', 'gy', 'gz']
for x in range(len(grad_channels)):
if hasattr(block, grad_channels[x]):
grad = getattr(block, grad_channels[x])
if grad.type == 'grad':
# In place unpacking of grad.t with the starred expression
t = grad.delay + [0, *(grad.t + (grad.t[1] - grad.t[0]) / 2),
grad.t[-1] + grad.t[1] - grad.t[0]]
waveform = 1e-3 * np.array((grad.first, *grad.waveform, grad.last))
else:
t = np.cumsum([0, grad.delay, grad.rise_time, grad.flat_time, grad.fall_time])
waveform = 1e-3 * grad.amplitude * np.array([0, 0, 1, 1, 0])
fig2_subplots[x].plot(t_factor * (t0 + t), waveform)
t0 += self.arr_block_durations[block_counter]
grad_plot_labels = ['x', 'y', 'z']
sp11.set_ylabel('ADC')
sp12.set_ylabel('RF mag (Hz)')
sp13.set_ylabel('RF/ADC phase (rad)')
sp13.set_xlabel('t(s)')
for x in range(3):
_label = grad_plot_labels[x]
fig2_subplots[x].set_ylabel(f'G{_label} (kHz/m)')
fig2_subplots[-1].set_xlabel('t(s)')
# Setting display limits
disp_range = t_factor * np.array([time_range[0], min(t0, time_range[1])])
[x.set_xlim(disp_range) for x in [sp11, sp12, sp13, *fig2_subplots]]
fig1.tight_layout()
fig2.tight_layout()
if save:
fig1.savefig('seq_plot1.jpg')
fig2.savefig('seq_plot2.jpg')
plt.show()
# Pre-phasing gradients
pre_time = 1e-3
gx_pre = make_trapezoid(channel='x', system=system, area=-gx.area / 2, duration=pre_time)
gz_reph = make_trapezoid(channel='z', system=system, area=-gz.area / 2, duration=pre_time)
gy_pre = make_trapezoid(channel='y', system=system, area=-(Ny / 2 - 0.5 - (Ny - Nyeff)) * delta_k, duration=pre_time)
# Phase blip in shortest possible time
gy = make_trapezoid(channel='y', system=system, area=delta_k)
dur = math.ceil(calc_duration(gy) / seq.grad_raster_time) * seq.grad_raster_time
# Calculate some times constant throughout the process
# The time(gy) refers to the number of blips, thus we substract 0.5 since the number of lines is always even.
# The time(gx) refers to the time needed to read each line of the k-space. Thus, if Ny is even, it would take half of the lines plus another half.
duration_center = (calc_duration(gx)*(Ny/2 + 0.5 - (Ny - Nyeff)) + dur * (Ny/2 - 0.5 - (Ny - Nyeff)) + calc_duration(gx_pre, gy_pre))
rf_center_with_delay = rf.delay + calc_rf_center(rf)[0]
rf180_center_with_delay = rf180.delay + calc_rf_center(rf180)[0]
# Group variables
seq_sys_Dict = {"seq" : seq,
"system" : system}
grads_times_Dict = {"rf180" : rf180,
"rf180_center_with_delay" : rf180_center_with_delay,
"rf_center_with_delay" : rf_center_with_delay,
"gz_spoil_1": gz_spoil_1,
"gz_spoil_2" : gz_spoil_2,
"gz" : gz,
"duration_center" : duration_center,
"pre_time": pre_time}