The MLP, described in section 3.3.2, is computed assuming the medium has both the scattering and the energy-loss properties of water. This approximation may induce some error, should the materials be much more or much less dense (such as bone or air for example). It has been shown [Wong et al.,2009] that in a mixed-slab geometry, the root mean square error made on the estimation of the lateral displacement is increased by 20% compared to an homogeneous phantom.
The potential of improving trajectory estimation by taking inhomogeneities into account was investigated. A segmentation in three kind of materials is considered: water, bone and air. This can easily be done using thresholds on a first reconstructed image.
This first image could be, for example, an analytical reconstruction used as the starting point for the iterative reconstruction using the MLP.
This section presents the investigation on a MLP formalism taking into account slabs of materials.
4.2.1 Scattering of particles in a non-uniform medium and implemen-tation in the MLP
In order to compute this “slab” version of the most likely path of a proton, the scattering matrices (Σ1 and Σ2, detailed in Equation3.21) need to be computed for a mixed-slab case.
Expressions of the scattering of charged particles in multi-material slabs geometries have been previously used for analytical modelling of beam propagation [Safai et al., 2008]. Consider a multi-slab, heterogeneous geometry consisting in N different slabs.
Let i denote the index corresponding to a slab of thickness ti and of radiation length X0,iwith its upstream face at the depthzi−1 and its downstream face at depthzi. Thus, ti = zi −zi−1. Let the depth at which the path is computed be considered as exit boundary of thejth slab, at positionzj. The elements of the variance-covariance matrix betweenzin and zj (Σ1 in Equation 3.28) can then be computed as:
σ2t1(zin, zj) = The star represents the referential in terms of energy of the current slab. A particle enters the slab i at position zi−1 with a given energy. The depth z∗i−1 represents the depth of ith material for which the particle would enter with the same energy. As a segmentation into three known materials is considered, this can be computed knowing the
4.2. IMPROVING SPATIAL RESOLUTION? MOST LIKELY PATH IN A NON-UNIFORM MEDIUM
Table 4.1: Coefficients of the fifth-degree polynomial fitting β2(u)p12(u) in c2/MeV2 divided by the appropriate powers of mm for air and bone. The coefficients of the polynomial computed for water are shown on Table 3.1.
Values for air Values for bone a0 7.516·10−6 7.450·10−6 a1 3.303·10−11 4.992·10−8 a2 1.487·10−16 −8.971·10−11 a3 5.686·10−22 1.313·10−11
a4 − −1.532·10−13
a5 − 9.164·10−16
relative stopping powers of the materials. As an example, an object with two slabs can be considered: the first slab is made of material 1 with a RSP = RSP1, the second is made of material 2 with a RSP = RSP2. The entrance point of the second slab, at depth z1 will correspond in the second material’s referential to the depthz1∗=z1·(RSP1/RSP2).
It can be noticed that, using this formulation, the sum of two slabs of the same material is not quite equal to the same depth in only one slab, because the common term to the three equations 1 + 0.038 lnXt0i
,i
2
differs. For our implementation, it was therefore chosen to replaceti by the total depth (zj−zin). Nevertheless, both versions were tested. As the difference only affects the logarithmic correction term, the effect was not significant.
As mentioned previously, a segmentation in three materials was considered: water (the current approximation, kept for soft tissues), air and bone. The polynomials approximating the energy loss of protons in air and bone were computed based on a Geant4 simulation, as was done for water (Section 3.3.2.2). As the energy loss of protons of tens of MeV in air is very small, a third order polynomial was used for the fit. The coefficients of the polynomial fits for air and bone are reported in Table 4.1.
5 10 cm
Figure 4.8: Sequences of materials in the different slabs geometries: (a)air gap in water, (b) bone insert in water, (c) air and bone in water for a nasal cavity (“nose”) and (d) multi-slab geometry used in [Wong et al.,2009].
4.2.2 Results
The multi-slab MLP was tested for different combinations of air, bone and water slabs, namely: an air gap in water (Figure 4.8(a)), a bone insert in water (Figure 4.8(b)), a
“nose” (Figure4.8(c)) and the multi-slab geometry ofWong et al.[2009] (Figure4.8(d)).
Geant4simulations of a 200 MeV, unidirectional, proton beam in 20 cm of material for each slab combination was performed. It was shown that cuts on the exit angles of the data allowed the removal of particles that underwent nuclear scattering [Schulte et al., 2005]. In order to reduce computation time, nuclear scattering was not included in the simulations.
For the first 2000 events of each geometry, the MLP was computed (i) using the standard all-water approximation, (ii) knowing the position and composition of each slab, using the multi-slab formulation. An example of proton path with the computed MLP and “slab” MLP are represented on Figure4.9 in the case of the air gap in water (Figure4.8(a)). It can be seen that the path of the proton estimated using the “slab”
version of the MLP is straight in the air slab. Nevertheless, the improvement in the path estimation seems minor.
Figure4.10shows the root mean square (RMS) of the displacement between the true proton path and the estimations using both MLP for the slab configurations detailed in Figure4.8. It can be seen that for some cases, such as the air insert in the water (Figure 4.10(a)) or the “nose” configuration (Figure4.10(c)), there is a slight improvement in the path approximation using the mixed slab MLP. In the other two cases, no improvement can be seen.
4.2. IMPROVING SPATIAL RESOLUTION? MOST LIKELY PATH IN A
Figure 4.9: A proton path in a medium consisting of water, air and water and the MLP computed considering only water and considering the slab media.
0
Figure 4.10: RMS error between the real and estimated proton paths using the MLP and slab-MLP for different configurations of slabs detailed in Figure4.8.
4.2.3 Discussion
Results show that in some slab geometries, a slight improvement of the path approxima-tion can be achieved by using the proposed multi-slab MLP. This is well illustrated by the cases with the bigger inserts, such as the air gap. This improvement seems however rather small (maximum 50µm on the RMS error for the tested geometries). The significance of this improvement needs to be estimated with respect to the size of the voxels of the reconstructed image. For the reconstruction study presented previously (in section4.1), cubic voxels of 1mm-side were chosen. In these conditions, such an improvement may not justify the additional computation time of the slab-MLP. However, different pCT image reconstruction studies have used smaller voxels (0.1 mm-side for Rit et al. [2013] for example), for which such an improvement may induce a difference in the reconstructed images. It can be noted that the 20% error increase in the “multi-slab” geometry [Wang et al., 2010] is not compensated by the slab implementation of the MLP. This may be an indication of the limitations of the most likely path approach: the most likely distribution is not very different in the presence of thin air or bone inserts, whereas the individual proton paths are.
The results shown here seem to indicate that, in the case of pCT imaging of a head, the gain in terms of path accuracy obtained with the slab MLP is limited. This assertion may need to be reassessed in other cases, should proton imaging be considered to produce images of a thorax for instance.
Finally, one disadvantage of the proposed multi-slab MLP is that it does not allow to take into account heterogeneities with a finite lateral dimension. One possibility to take into account the lateral dimensions of heterogeneities may be to perform this computation in an iterative fashion, computing depth at which a particle changes medium by already considering a curved trajectory.