Note

Go to the end to download the full example code.

Fitting models without motion correction#

This example illustrates how mdreg can be used to fit models without motion correction. The idea is illustrated for a 3D time series with variable flip angles (VFA), and a linear signal model fit.

Setup#

import numpy as np
import mdreg

# Example data included in mdreg
data = mdreg.fetch('VFA')

# Variables used in this examples
array = data['array']       # 4D signal data (x, y, z, FA)
FA = data['FA']             # The FA values in degrees

Perform model fit#

The linear VFA signal model is included in mdreg as the function mdreg.fit_spgr_vfa_lin, which requires the flip angle (FA) values in degrees as input:

fit, pars = mdreg.fit_spgr_vfa_lin(
    array,                      # Signal data to fit
    FA=FA,                      # Flip angle in degrees
    progress_bar=True,          # Set to True to show a progress bar
)

Fitting VFA model:   0%|                                                                                       | 0/269568 [00:00<?, ?it/s]
Fitting VFA model:   2%|█▊                                                                       | 6509/269568 [00:00<00:04, 57118.07it/s]
Fitting VFA model:   5%|███▌                                                                    | 13222/269568 [00:00<00:04, 59586.27it/s]
Fitting VFA model:   7%|█████▎                                                                  | 19878/269568 [00:00<00:04, 59780.59it/s]
Fitting VFA model:  10%|███████                                                                 | 26658/269568 [00:00<00:04, 59854.62it/s]
Fitting VFA model:  12%|████████▋                                                               | 32645/269568 [00:00<00:03, 59798.49it/s]
Fitting VFA model:  14%|██████████▎                                                             | 38626/269568 [00:00<00:03, 59455.26it/s]
Fitting VFA model:  17%|███████████▉                                                            | 44571/269568 [00:00<00:03, 59385.28it/s]
Fitting VFA model:  19%|█████████████▋                                                          | 51350/269568 [00:00<00:03, 59164.72it/s]
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Fitting VFA model:  23%|████████████████▉                                                       | 63282/269568 [00:01<00:03, 59059.52it/s]
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Fitting VFA model:  75%|█████████████████████████████████████████████████████▏                 | 201730/269568 [00:03<00:01, 61317.14it/s]
Fitting VFA model:  77%|██████████████████████████████████████████████████████▊                | 207926/269568 [00:03<00:01, 61472.92it/s]
Fitting VFA model:  79%|████████████████████████████████████████████████████████▍              | 214135/269568 [00:03<00:00, 61635.36it/s]
Fitting VFA model:  82%|██████████████████████████████████████████████████████████▎            | 221225/269568 [00:03<00:00, 61494.17it/s]
Fitting VFA model:  85%|███████████████████████████████████████████████████████████▉           | 227792/269568 [00:03<00:00, 61055.56it/s]
Fitting VFA model:  87%|█████████████████████████████████████████████████████████████▊         | 234591/269568 [00:03<00:00, 61394.70it/s]
Fitting VFA model:  90%|███████████████████████████████████████████████████████████████▌       | 241435/269568 [00:03<00:00, 61754.36it/s]
Fitting VFA model:  92%|█████████████████████████████████████████████████████████████████▍     | 248267/269568 [00:04<00:00, 61971.72it/s]
Fitting VFA model:  95%|███████████████████████████████████████████████████████████████████▏   | 254919/269568 [00:04<00:00, 61676.99it/s]
Fitting VFA model:  97%|████████████████████████████████████████████████████████████████████▉  | 261650/269568 [00:04<00:00, 61639.77it/s]
Fitting VFA model: 100%|██████████████████████████████████████████████████████████████████████▋| 268523/269568 [00:04<00:00, 62002.35it/s]
Fitting VFA model: 100%|███████████████████████████████████████████████████████████████████████| 269568/269568 [00:04<00:00, 60861.84it/s]
divide by zero encountered in divide

Plot the model parameters:

S0 = pars[...,0]
fig = mdreg.plot.par(S0, title='S0', vmin=0, vmax=5*np.amax(array))
S0 , Slice 1, Slice 2, Slice 3, Slice 4, Slice 5, Slice 6, Slice 7, Slice 8, Slice 9, Slice 10, Slice 11, Slice 12, Slice 13, Slice 14, Slice 15, Slice 16, Slice 17, Slice 18, Slice 19, Slice 20, Slice 21, Slice 22, Slice 23, Slice 24, Slice 25, Slice 26, Slice 27, Slice 28, Slice 29, Slice 30, Slice 31, Slice 32, Slice 33, Slice 34, Slice 35, Slice 36
T1 = -1/np.log(pars[...,1])
fig = mdreg.plot.par(T1, title='T1/TR', vmin=0, vmax=500)
T1/TR , Slice 1, Slice 2, Slice 3, Slice 4, Slice 5, Slice 6, Slice 7, Slice 8, Slice 9, Slice 10, Slice 11, Slice 12, Slice 13, Slice 14, Slice 15, Slice 16, Slice 17, Slice 18, Slice 19, Slice 20, Slice 21, Slice 22, Slice 23, Slice 24, Slice 25, Slice 26, Slice 27, Slice 28, Slice 29, Slice 30, Slice 31, Slice 32, Slice 33, Slice 34, Slice 35, Slice 36

Check the model fit:

anim = mdreg.plot.animation(
    fit,
    title='VFA model fit',
    vmin=0,
    vmax=np.percentile(array, 99),
)

Pixel-by-pixel fitting#

Alternatively, the function mdreg.fit_pixels can be used to fit any single-pixel model directly:

fit, pars = mdreg.fit_pixels(
    array,
    model=mdreg.spgr_vfa,
    xdata=FA,
    func_init=mdreg.spgr_vfa_init,
    p0=[1,0.5],
    bounds=([0,0], [np.inf,1]),
)

Check the model fit:

anim = mdreg.plot.animation(
    fit,
    title='VFA model fit',
    vmin=0,
    vmax=np.percentile(array, 99),
)

Total running time of the script: (64 minutes 33.466 seconds)

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