Dark field for polychromatic sources
Outline
- (XNPIG 2017) calculating dark field values for lung samples
- dual phase contrast with CdTe detector
Ratio of the logarithms
- transmission \(A\)
- visibility reduction \(B\)
- log ratio \(R = \log(B) / \log(A)\)
Monochromatic case
dark field as a function of sphere diameter and photon energy
S. Lynch et al., Interpretation of dark-field contrast and particle-size selectivity in grating interferometers, 2011
\[
B \propto \mu_d = \frac{3\pi}{\lambda^2}f |\Delta n|^2 d
\begin{cases}
D' & \text{if } D' \leqslant 1\\[2ex]
\!\begin{align}
D' - \sqrt{D'^2 - 1}\\
(1 + D'^{-2}/2) \\
+ (D'^{-1} + D'^{-3} / 4) \\
\log\left(\frac{D' + \sqrt{D'^2 - 1}}{D' - \sqrt{D'^2 - 1}}\right)
\end{align} & \text{otherwise}
\end{cases}
\]
Monochromatic case
dark field as a function of sphere diameter and photon energy
S. Lynch et al., Interpretation of dark-field contrast and particle-size selectivity in grating interferometers, 2011
Polychromatic extension
\[ R(\energy) = \frac{\log B}{\log A} = \frac{\mu_d}{2k\beta} \]
\[ R \propto \frac{\sum_\energy w(\energy)|\Delta n(\energy)|^2 \energy u(\energy)}{\sum_\energy w(\energy) \energy \beta} \]
- \(w\) spectral weights
- \(u\) conditional statement
M. Abis et al., in preparation
Table-top grating interferometer with microspheres
20% volume fraction of SiO2 microspheres in glycerine
Investigating lung microstructures
- ground truth from synchrotron microtomography
- quantitative data from the table-top setup on a macroscopic scale
G. Lovrić et al., Dose optimization approach to fast X-ray microtomography of the lung alveoli, 2013
Acquisition of the ground truth
- critical point dried lungs
- tomographic scan at 21 keV
- 1.1 μm effective pixel size
- three mouse samples
G. Lovrić et al., Dose optimization approach to fast X-ray microtomography of the lung alveoli, 2013
Microtomography and postprocessing
- synchrotron microtomography
- segmentation
G. Lovrić et al., Automated computer-assisted quantitative analysis of intact murine lungs at the alveolar scale, in press 10.1371/journal.pone.0183979
Alveoli as spheres
- fit spheres in the lung microstructures
- plot diameter distribution
G. Lovrić et al., Automated computer-assisted quantitative analysis of intact murine lungs at the alveolar scale, in press 10.1371/journal.pone.0183979
The final model
sum over the sphere size distribution times the dark field response for each spere size
ground truth
dark-field response
Grating interferometry
measure \(R\)
Preliminary validation
| sample | expected | measured |
|---|---|---|
| \(1\) | \(22.2\) | \(21.3 \pm 1.8\) |
| \(2\) | \(13.7\) | \(16.5 \pm 1.6\) |
| \(3\) | \(14.4\) | \(17.7 \pm 1.4\) |
- sample 1 has smaller structures \(\rightarrow\) larger \(R\)
- sample 2/3 similar
- consistent values for the three samples
M. Abis et al., in preparation
Upcoming challenges
- the sphere fit program cannot accomodate full tomographic datasets
- possible workaround: sampling, splitting
- calculations take more than one day per sample (40 samples total)
Dual phase setup
Pushin et al. Far-field interference of a neutron white beam and the applications to noninvasive phase-contrast imaging- G0 with 10 μm slits every 100 μm (thanks Konstantins for the super fast manufacturing!)
- G1/2 phase gratings with 10-20 μm gold with 1.2 μm pitch (thanks Carolina)
- tried many distance combinations but no interference observed so far
Troubleshooting the current setup
- G0 might not be thick enough (running at 90 kVp should ensure at least 80% absorption)
- the phase gratings may not be consistent enough in their phase shift
- the alignment could be so far off that it's not possible to correct with the motors (these gratings can't be aligned through laser diffraction)
- trying different gratings with double the pitch at 2.4 μm also doubles the length but might be the only chance