Accuracy of the Models

After performing the model fitting, the resulting parameters for the Bragg-Kleeman model can be compared to those obtained by other studies: see Table 2.7. The parameters from

2.2 Proton Range Calculations with Analytical Models 43

Figure 2.7:Range as a function of the initial energy, and the PSTAR energy-range data from both the training group and the control group.

Initial energy [MeV]

Figure 2.8: The accuracy of the proton range calculations using different models. The range error as shown is the relative and absolute difference between the estimated range and the PSTAR data. The results are presented with respect to the control group, and are visually limited downwards by the PSTAR dataset accuracy of 1 µm. From Pettersen et al. (2018).

44 2. Proton Range Calculations: Monte Carlo Simulations and Analytical Models

Number of data points in training group

3 4 5 6 7 8 910 20 30 40 50 100

75% percentile of errors [%]

0.001 0.01 0.1 1 10

100 Bragg-Kleeman

Sum of exponentials Linear interpolation Spline interpolation

Figure 2.9: The convergence of the models as a function of the number of data points used for model fitting. The error shown here is calculated as the 75^thpercentile of all the relative errors as shown in Fig. 2.8 for each of the models. The high accuracy of the spline interpolation is a result of its curvature. From Pettersen et al. (2018).

Depth in water [cm]

20 20.5 21 21.5 22 22.5 23 23.5 24

Energy loss [MeV/cm]

8 10 20 30 40 100

200 Bragg-Kleeman

Sum of exponentials Linear interpolation Spline interpolation PSTAR values

Figure 2.10: The energy loss curves for individual (190 MeV) protons obtained by dif-ferentiating the models obtained withNC=125. The range is kept constant by the choice of R0, in order to to facilitate a comparison between the curve shape. A curve showing the PSTAR energy loss data is also included. From Pettersen et al. (2018).

2.2 Proton Range Calculations with Analytical Models 45 α[MeV/cm] p Error [mm]

This work 0.00262 1.736 0.69

Bortfeld (1997) 0.00220 1.770 0.85

Boon (1998) 0.00256 1.740 1.50

Table 2.7: The parameters of the proton range calculation using the Bragg-Kleeman model, with the results found in this work and compared with other results. The (median) error between PSTAR ranges and the model-calculated ranges are included for each parameter set. From Pettersen et al. (2018).

the fit of the “sum of exponentials” model in are not as easy to compare, due to the many terms linearly added; the values are therefore not reproduced here.

The accuracy of the proton range determination in water using different models is shown in Fig. 2.8, with the 75^th percentile accuracy shown in Table 2.8. The Bragg-Kleeman model is the least accurate at a 75^th percentile value of 3%. The “sum of ex-ponentials” model and the linear interpolation model have a similar 75^thpercentile accu-racy at around 0.3%, while the spline interpolation model has a 75^thpercentile accuracy of 0.003%. The spline interpolation model yields the highest accuracy. A sub-percent range calculation accuracy is obtained for all models above 90 MeV, and for the spline interpolation model above 10 MeV.

Energy loss curves resulting from the different models is shown in Fig. 2.10. The method described in this work has also been applied on a sample of other materials. The result-ing deviations between the PSTAR ranges of different materials and the correspondresult-ing model-generated ranges are similar for the various materials, as shown in Table 2.8.

Range accuracy oscillation

An oscillatory behavior in the accuracy, with respect to the initial energy, is observed for three of the models, as seen in Fig. 2.8. The behavior has two different explanations, depending on the model in question. For the analytical models, the oscillation is due to the approximation of the energy-range relationship. Since the absolute error is shown, a sudden drop in the range error signifies that the model-calculated range curve “crosses”

the PSTAR-calculated range curve. In the observed energy range, this happens twice for the Bragg-Kleeman model (with two parameters), and three times for the “sum of exponentials” model (with five parameters). The quick oscillation of the linear interpo-lation model is seen because the linear approximation does not reproduce the curvature of the underlying data, and thus any interpolated values between two sampled points have higher errors than values close to the sampled points. The spline interpolation model reproduces the curvature, and no similar oscillation is observed.

46 2. Proton Range Calculations: Monte Carlo Simulations and Analytical Models Material Bragg-Kleeman Sum of exp. Linear interp. Spline interp.

Liquid Water 2.98% 0.30% 0.26% 0.003%

A-150 T. E. P. 3.02% 0.25% 0.25% 0.003%

Aluminum 2.55% 0.49% 0.26% 0.006%

Tungsten 1.25% 0.36% 0.22% 0.003%

Table 2.8: The deviation between the PSTAR control values and the model values for the models under study, applied on four different materials: Liquid Water, A-150 Tissue Equivalent Plastic (T. E. P.), aluminum and tungsten. The error shown is the 75^th per-centile of the absolute error over all energies in the range 1–250 MeV. The number of data points for model training is 25.

Model convergence

The numbers of training points needed for convergence of the different models are shown in Fig. 2.9. A larger number of measurements at different energies is required for an interpolation-based range calculation scheme compared to using the simple Bragg-Kleeman rule with two parameters, or the “sum of exponentials” with five parameters. Using 25 data points for model fitting, the accuracy is kept at an acceptable level for all models, and the 75^thpercentile of the errors in the range calculation is at 0.1% of the range for both interpolation schemes and the sum of exponentials.

Bragg Curve Reproduction

The Bragg curves obtained from the interpolations and from differentiation of the Bragg-Kleeman model are similar in shape. The curve obtained by using the “sum of expo-nentials” model exhibits differences close to the Bragg Peak, mimicking an exponential decay. While the area under the curve is the same (due to that their integrals, yielding the energy-range relationship, are similar to within a few permille), the curve deviates around the true energy loss-curve.