5.3 Quantitative approach: a step towards stoichiometric composition?
5.3.2 Reconstruction using the scattering
Many approximations to multiple Coulomb scattering exist, amongst which the one used for the most likely path computation for example. Gottschalk [2009] reviewed these approximations of the scattering power used in proton transport calculations. The scattering powerT is a differential description of the Gaussian approximation to multiple scattering, with:
T ≡ dθ2
dx (5.10)
where θ2 represents the variance of the projected angular distribution (previously denoted σ2 in Equation 2.7). The advantage of such a differential formulation is that it takes a similar form to that of the stopping power S ≡ −dEdx, or more generally, to the formulation of the line integrals. As a consequence, the problem becomes particularly well-adapted to image reconstruction. However, an accurate formula for T must take into account the “history” of the protons, i.e. the competition between the Gaussian core and the single scattering tail of the distributions that will affect the rate of change of the Gaussian width. Thus, T should be non-local: it should not depend only on the local variables such as the mean proton energy, atomic properties and stopping power.
Different functions forT have been proposed, that recover more or less the Moli`ere scattering angle if integrated over x for any slab [Gottschalk, 2009]. In the paper, Gottschalk reviews three local formulas:
– TF R: Fermi-Rossi approximation , – TIC: based on ICRU report 35,
– TLD: based on the Linear displacement , and two non-local formulas
– TØS: Øvers˚as and Schneider , – TdH: differential Highland.
A new approximation is also proposed by Gottschalk[2009], named differential Moli`ere (TdM). Details of the approximations are given in AppendixB.
5.3.2.1 Reconstruction principle
The variance of the exit scattering angle of an initially parallel beam, after a slab of material of thickness xcan be written as:
θ2proj ≃ Z x
0 T x′dx′ (5.11)
All formulas listed previously except the Øvers˚as and Schneider approximation can be re-written as T =f(E)X1 withX the radiation length (X0), or the scattering length (XS), andf(E) a function of the energy of the protons. This makes it possible to isolate the X1 term. The energy-dependent term f(E) can be evaluated for each depth in the object by making use of the image of the RSP. As the term X0 is present many times
isolate it. Therefore, this approximation was not further considered.
By comparing the estimated variance of the angular scattering using a formula and the one obtained in the data, and knowing the energy loss properties of the materials, one can estimate the scattering or radiation length in each voxel. The reconstruction process goes as follows:
(i) Estimation of a trajectory. As the RMS of the scattering angle is considered, an average straight line path can be considered.
(ii) Computation of thef(E) term. For each voxel in the path, the average energy (or impulsion, or residual range) of the protons can be computed, knowing the initial energy and using the RSP image.
(iii) Reconstruction of the X1 term.
This process can be performed through an iterative reconstruction algorithm.
5.3.2.2 Model validation
In order to determine the most appropriate model, the five models with X1 term that can be isolated were compared to the GATE simulation data. It is possible to change the model used for multiple scattering in Geant4. GATE, however, only takes the default Urban simulation model [Urban,2006]. The Urban model is a condensed multiple Coulomb scattering model (Section 3.1.1.1). It is, however, much more complex than the approximations to multiple scattering listed previously, as it does not aim at only reproducing the Gaussian part of the MCS distribution, but also the tail. Gottschalk [2009] concludes that the differential Moli`ere formula is the best fit to real data. However, as this section aims at showing a proof of concept of the reconstruction principle, the model that is closest to the simulation data should be used.
Figure 5.17 shows the variance of the angular distribution after different depths in water, for the five models studied and the GATE simulation data.
For easier evaluation of the agreement between the models and the simulation, Figure 5.18 shows the normalized difference D between the calculated θproj2 and the variance obtained by Monte Carlo simulation for different depths in water, defined as:
D=
θsimulation2 −θmodel2
θ2simulation (5.12)
for the five models. θsimulation2 and the associated uncertainty was estimated using 100 simulations of 10000 protons for each depth.
The model which shows the best agreement with the simulation data is the differential Moli`ere. However, the discrepancies for thin objects (less that 5 cm depth) may induce some additional inconsistencies in the image reconstruction process. This behaviour was found to be the same for different materials tested, as illustrated on Figure 5.19 for water, adipose tissue, skull bone and muscle.
Furthermore, a high number of particles (106protons) were used to test the agreement between the models and simulation, in order not to be affected by the statistical
5.3. QUANTITATIVE APPROACH: A STEP TOWARDS STOICHIOMETRIC COMPOSITION?
Figure 5.17: Scattering evaluation of the different models and the simulation for different thickness of water.
Figure 5.18: Normalized difference between models and the simulation for different thickness of water. The difference for the Fermi-Rossi approximation exceeds the range shown here for all thicknesses.
Figure 5.19: Comparison between the differential Moli`ere model and the simulation for different thickness of water, adipose tissue, skull bone and muscle.
uncertainty. In a realistic scan situation, between 100 and 1000 protons can be used per square millimetre of projection. Therefore the measures will be greatly affected by the statistical uncertainty: the statistical uncertainty on a measure ofθsimulation2 from 1000 protons is of the order of 4%. As a consequence, a rather high level of noise can be expected in the reconstructed images.
5.3.2.3 Reconstruction of the scattering length
Images of the inverse scattering length (1/XS) of the modified Zubal head phantom (Section 3.1.5.2) were reconstructed with an ART algorithm using the data binned into projections. 1000 protons/mm2 of projection and 256 projections were used to reconstruct images of 1×1×1 mm3 voxels. As the distribution of the angular scattering is fitted for each pixel in order to obtain the value ofθ2, the projections were naturally binned according to the positions of the particles upstream from the object.
Figure 5.20 shows a transverse view of (a)the FBP image of the RSP (b)expected inverse Xs and(c)reconstructed inverse Xs.
It can be seen that, as anticipated, the inverse scattering length image shows a high level of noise. Both the noise and spatial resolution seem degraded compared to the image produced using the scattering without any deconvolution of the energy contribution shown on Figure5.7(d)). Nevertheless, the two carcinoma can be distinguished from the brain, indicating a rather good contrast between the soft tissues.
The advantage of this reconstruction method rather than just using the scattering in a qualitative way (as was done in the previous section) is that it gives access to quantitative information on the materials. In order to evaluate the accuracy of this reconstruction, regions of interest (ROI) were drawn in the carcinoma, brain and in
5.3. QUANTITATIVE APPROACH: A STEP TOWARDS STOICHIOMETRIC COMPOSITION?
(a) (b) (c)
Figure 5.20: Transverse slice of (a) the FBP image of the RSP, (b) the phantom in inverse scattering length and (c) the reconstructed image in inverse scattering length.
The regions of interest studied are drawn on figure (a). Colour range of [0:1.5] for the RSP image, and of [0:4.5·10−3] for the inverse scattering length images (in mm−1).
0 0.001 0.002 0.003 0.004 0.005
right carcinoma left carcinoma brain bone 1/Xs (mm-1)
Expected valueReconstructed values
Figure 5.21: Expected and reconstructed values in the ROIs.
the skull (see Figure 5.20(a)). Figure5.21 shows the reconstructed values as well as the expected value for each ROI. The reconstructed values for the bone and brain regions are overestimated, whereas the two carcinoma regions seem to be accurately reconstructed.
This may be due either to the model used for the reconstruction or to the statistical uncertainty that introduces inconsistencies in the reconstruction algorithm.
5.3.2.4 Conclusion
These results show that quantitative image reconstruction of scattering length using a proton scanner is feasible. The use of the RSP image to estimate the energy of the protons make it possible to deconvolve the two contributions, and gives access to information directly related to the composition of the materials.
The statistical uncertainty on the measured θ2 generates a high level of noise in the reconstructed image. The spatial resolution of the inverse scattering length image
used for image reconstruction in this work.
The ROI study shows that the method is viable. The accuracy is currently limited to a few percent, which may be due to the differences between the models used for simulation and reconstruction introducing additional inconsistencies in the reconstruction process.
The use of a different reconstruction algorithm or the addition of a regularization scheme, such as the total variation (Section3.3.3.2), could probably help improve results.