Degradation of proton depth dose distributions attributable to microstructures in lung-equivalent material

Purpose: The purpose of the work reported here was to investigate the influence of sub-millimeter size heterogeneities on the degradation of the distal edges of proton beams and to validate Monte Carlo (MC) methods’ ability to correctly predict such degradation. Methods: A custom-designed high-resolution plastic phantom approximating highly heterogeneous, lung-like structures was employed in measurements and in Monte Carlo simulations to evaluate the degradation of proton Bragg curves penetrating heterogeneous media. Results: Significant differences in distal falloff widths and in peak dose values were observed in the measured and the Monte Carlo simulated curves compared to pristine proton Bragg curves. Furthermore, differences between simulations of beams penetrating CT images of the phantom did not agree well with the corresponding experimental differences. The distal falloff widths in CT image-based geometries were underestimated by up to 0.2 cm in water (corresponding to 0.8–1.4 cm in lung tissue), and the peak dose values of pristine proton beams were overestimated by as much as ~35% compared to measured curves or depth-dose curves simulated on the basis of true geometry. The authors demonstrate that these discrepancies were caused by the limited spatial resolution of CT images that served as a basis for dose calculations and lead to underestimation of the impact of the fine structure of tissue heterogeneities. A convolution model was successfully applied to mitigate the underestimation. Conclusions: The results of this study justify further development of models to better represent heterogeneity effects in soft-tissue geometries, such as lung, and to correct systematic underestimation of the degradation of the distal edge of proton doses. C 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1118/1.4932625]


INTRODUCTION
Since Wilson proposed the use of protons for radiotherapy applications in 1946, 1 proton radiotherapy has gained more and more importance worldwide. 2The characteristic properties of the depth-dose curves of protons, a low entrance dose followed by the so-called Bragg peak, allow for the deposition of a large dose in the target while sparing healthy tissue.The dose decreases steeply to zero in the region distal to the Bragg peak.While no significant changes of these properties have been observed in many applications, a series of experiments has shown that inhomogeneities in the beam's path can cause a degradation of proton beams' Bragg peak, resulting in lowered peak-to-entrance dose ratios and significantly degraded distal falloffs.Degradation of proton dose distributions in lung can potentially lead to low dose regions in the target and to a large portion of healthy tissue receiving unwanted dose.The magnitude of the degradation is strongly correlated to material density variations along the protons' paths, chiefly those perpendicular to the beam direction (e.g., Refs. 3 and 4).These studies were mostly done using transmission experiments or Monte Carlo (MC) simulations of protons traversing heterogeneities, such as temporal lobes and the nasal cavity in the human head or stylized phantoms.Furthermore, these studies focused on bone/air or bone/soft-tissue interfaces on a millimeter scale, which can be resolved by contemporary CT scanners.Uncertainties in lung proton doses due to limitations in voxel size have been studied by Espana and Paganetti 5 in 2D Monte Carlo simulations and using swine lung.
In this work, we demonstrate that heterogeneities on a sub-millimeter scale, such as in human lung, may contribute significantly to the degradation of proton Bragg peaks.Specifically, we considered the effect of microscopic aspects of lung soft-tissue/air interfaces on proton beams.We conducted a series of measurements and MC simulations and evaluated the distal falloff widths (DFWs) from the 80% dose location to the 20% dose location as well as the reductions in peak height of Bragg curves and also in a spread-out Bragg Peak (SOBP).Our results indicate that distal edge degradation must be expected in proton treatment of lung tumors, given that the lung parenchyma is one of the most inhomogeneous tissues in a human body.Furthermore, we conclude that the use of CT images of heterogeneous structures in contemporary treatment planning systems may lead to noticeable underestimation of the degradation of the distal falloff of proton beams through highly heterogeneous, small-scale structures such as lung parenchyma.

MATERIALS AND METHODS
To demonstrate the existence of a proton dose degradation effect in lung, measured depth doses were acquired in water as well as downstream of a sample of plastinated human lung (see Fig. 1) donated by Von Hagens Plastination (Gubener Plastinate GmbH, Uferstr.26, 03172 Guben, Germany).The measurements were performed in one of the double scattering beam lines at the Proton Therapy Center at The University of Texas MD Anderson Cancer Center (Houston, TX).The lung sample was placed inside a protective container upstream of the water phantom where depth doses were determined by measuring at two different locations on the central beam axis.The sample was irradiated with unmodulated scattered proton beams with nominal proton beam energies of 140 and 200 MeV, corresponding to ranges of 9.8 and 18.9 cm at the 80% dose location in water, respectively.For both energies, the beam was collimated to a field size of 3×3 cm 2 at isocenter, and the range modulator wheel was stopped at its thinnest location to allow the delivery of quasi-monoenergetic beams.
Measured depth-dose curves were acquired by scanning the beam's central axis with a Markus parallel plate ionization chamber.For comparison, depth-dose profiles of proton beams through the empty container were also acquired.Following these "proof of principle" measurements, a highly heterogeneous plastic phantom was designed and fabricated to approximate lung tissue and also provide a well-defined geometry for further investigations using the MC method.The phantom, designed to approximate soft-tissue/air interfaces (lung parenchyma), was a cube of size 5×5×5 cm 3 containing randomly distributed voxels consisting of either air or plastic (density ∼1.19 g cm −3 ).The randomization for the physical phantom was restricted by the requirement of connecting the plastic voxels to one another.
The voxels were cubic and equally sized (0.5 × 0.5 × 0.5 mm 3 ); the average density of the phantom was 0.27 g cm −3 (approximately equal to the density of inflated lung tissue), and the average water equivalent thickness of the phantom was 1.39 cm. Figure 2 shows an image of the phantom.The device was produced by a 3D printer, and the plastic material was a resin called Accura 25.
Depth-dose measurements with the same energies as in the plastinated lung experiments were performed in a water phantom with the plastic phantom located upstream at the water phantom surface (see Fig. 3).The depth-dose curves were recorded by using the Markus parallel plate ionization chamber for two unmodulated proton beams of nominal energies of 140 and 200 MeV, with and without the lung phantom present.
Finally, the distal falloff region of a SOBP with a range of 10 cm and a modulation width of 5 cm was recorded in the same way as already described, except that the range modulator wheel in the beam line was spinning.
MC simulations were performed by using  version 2.70. 6An exact model of the phantom, based on the design F. 2. One half of a lung like phantom with 0.5 × 0.5 × 0.5 mm 3 voxels of Accura 25 plastic.The average density of the voxelized structure was 0.27 g/cm 3 , and the overall size of the phantom is 5 × 5 × 5 cm 3 .blueprints of the physical device, was used for validation simulations.Depth doses were scored in water using circular energy deposition mesh tallies, distal to the phantom model, similar to the measurements.
To investigate changes to the depth-dose curves when the simulation of proton transport took place in a CT-image based phantom, we exchanged the exact phantom model with a variety of geometries derived from CT images of the lung phantom.The exact model served as a reference standard, i.e., a "perfect CT," and a set of CT image scans of the physical phantom, containing unavoidable volume-averaging effects, were produced and incorporated into the MC geometry.
The CT images were obtained with a CT scanner (General Electric LightSpeed 16RT C ), used for clinical treatment planning at MD Anderson Cancer Center.Two resolution settings were used, 0.98 × 0.98 × 1.25 and 0.19 × 0.19 × 1.25 mm 3 .Furthermore, another set of CT images was obtained using a small animal CT scanner with a resolution of 0.092 × 0.092 × 0.092 mm 3 .Because of the large memory requirements for this CT in a MC simulation, neighboring voxels in xand y-directions were coalesced (resulting in a resolution of 0.184 × 0.184 × 0.092 mm 3 ).All CT images were converted into MC geometries by applying a linear relationship between Hounsfield numbers and material densities.The Hounsfield number-to-material composition conversion was realized by assigning air to all voxels with −900 HU or less and plastic (Accura 25) to all those above this threshold.To assign the F. 3. Schematic description of the experimental and the Monte Carlo setup, showing the proton beam direction, the position of the lung-like phantom, and the water phantom containing a Markus type parallel plate ionization chamber.
material density, the HU of solid Accura 25 was determined at 20 different locations within the solid walls of the phantom.The readings were averaged, and a material density of 1.19 g cm −3 was assigned to this CT number.A density of 0.0012 g cm −3 (air) 7 was assigned to −1000 HU.A linear interpolation between these two points was then used for the conversion of CT numbers to material densities.Additional simulations were performed with the phantom rotated in the MC geometry to evaluate the influence of the asymmetry of the voxels on the depth-dose curves.Figure 4 shows a qualitative comparison of CT images from the clinical, the perfect, and the small animal data set.
All MC simulations were performed with an extension for improved multiple Coulomb scattering (MCS).The MCS model was developed by Kuhn and Dodge 8 and first implemented in  by Stankowskiy et al. 9 Protons, deuterons, alpha particles, neutrons, and photons were produced and transported.The cutoff energy for protons, deuterons, and alpha particles was 5 MeV, corresponding to a range in water of 0.036 cm for protons, 0.022 cm for deuterons, and 0.004 cm for alpha particles.Because the depth dose curves were scored in a homogeneous water phantom with a resolution of 1 mm in beam direction, this choice was assumed to be reasonable.Below this threshold, the transport was aborted and the remaining kinetic energy was deposited locally.Other physics settings were kept at the default values. 6No variance reduction techniques were implemented.Proton beam energies are referred to as "nominal," i.e., they represent the protons' kinetic energy at nozzle entrance, upstream of any beam-modifying devices.The beam's energy at the phantom position is lower, as the protons have to travel through the double scattering system.All implemented tallies scored energy deposition per unit volume in water using a 0.5-cm diameter cylindrical mesh tally with 0.1-cm spacing along the beam's central axis to approximate the Markus-type parallel plate ionization chambers used in our experiments.Because of the density of water, i.e., 1 cm 3 corresponds to 1 g, the readings in units of energy per volume are equivalent to "dose to water."Hence, we also refer to dose for all  simulation results.The simulations were performed with 10 7 -10 9 source particles to ensure relative statistical uncertainties of less than 1% proximal to the 10% dose location at the distal edge.
The SOBP was simulated by weighted summation of 30 depth-dose curves, each computed with a 2 • increase in range modulator wheel rotation in the beam line setup.

RESULTS
The depth-dose curves of proton beams of 140 and 200 MeV nominal energy penetrating the plastinated human lung samples are presented in Fig. 5. Several important observations can be made.The beams through the empty container did not exhibit any measurable degradation or changes in the DFW, measured from the 80% to the 20% dose locations, compared to the undegraded, unmodulated beam without any object in its path (called "pristine" in the following).In the case of the 140 MeV beam, the DFW through the container wall was 0.22 cm and the DFWs through the wall and the lung sample were 0.31 and 0.35 cm at two different beam axis positions (Pos 1 and Pos 2 in Fig. 1), corresponding to increases of 40% and about 60%, respectively.
The maximum dose of the 140 MeV beam decreased by 22% and 26%, respectively, compared to the pristine peak value.The DFWs of the 200 MeV protons increased from 0.39 to 0.48 and 0.51 cm, respectively, at positions 1 and 2 (about 25% and 30%).The peak dose values were decreased by about 12%.
Results of validation measurements with the plastic phantom are presented in Fig. 6.Both measured and simulated depth-dose curves were normalized to the maximum dose of the corresponding profile.
Measurements of the pristine 140 MeV proton beam produced a range at the 80% distal falloff dose (r 80 ) of 9.74 cm in water and a DFW of 0.22 cm. simulations using the same geometry yielded a range of r 80 = 9.73 cm and a DFW of 0.22 cm.The ranges of the proton penetrating the randomized plastic phantom led to a measured range of r 80 = 8.40 cm and a simulated range of r 80 = 8.41 cm.The DFW was 0.48 cm measured and 0.51 cm in the simulation.
The range of the pristine 200 MeV proton was measured at r 80 = 18.96 cm and matched the simulation results exactly.Similarly, the DFW was measured and simulated to be 0.39 cm.After placing the randomized plastic phantom in the beam  path, the range was measured at r 80 = 17.57cm and the simulations yielded r 80 = 17.59 cm.The measured falloff width DFW was 0.61 cm, while the simulation results yielded a DFW of 0.63 cm.The maximum differences between the measured and simulated ranges in water were less than 0.02 cm in all cases considered, corresponding to maximum relative differences of 0.12% in range.The DFWs also differed by 0.02 cm, corresponding to a relative deviation of 6.25%.We concluded that these results represented acceptable validation of our MC methods.For readers' convenience, all DFWs and peak height values are shown in Table I.Considering the small differences, we will assume throughout the remainder of this paper that profiles from simulations of beams through the phantom (i.e., the "perfect" CT) and measurements are equivalent.
Figure 7 shows the Bragg curves of all simulations obtained by scoring the energy deposition in water, after the beams had penetrated the various implementations of CTs of the randomized plastic phantom upstream of the water container.The phantom geometries, based on clinical CT images, were each simulated twice using differently oriented CT images (one with the largest voxel dimension in z-direction and one rotated such that the largest voxel dimension would be along the y-direction).This was done to evaluate possible systematic differences in the depth-dose curves based on the different dimensions of the voxels in the z-and y-directions.No notable differences [see Figs. in a better agreement with the perfect CT; however, differences in DFWs and maximum peak heights, though significantly smaller, remained.The DFWs of the beams transported through the phantoms based on the clinical CT images yielded the following values: for 140 MeV protons, the values ranged from 0.29 to 0.33 cm in water, which corresponds to a relative increase of 32%-40% compared to the pristine DFW of 0.22 mm.The DFW of the perfect CT setup yielded 0.51 cm (232% of the pristine value).For the 200 MeV proton beams, we found that the clinical CT setups yielded DFWs between 0.45 and 0.46 cm, compared to a DFW of 0.39 cm for the pristine peak.This corresponds to an increase of about 15%-18%.The perfect CT setup produced a DFW of 0.63 cm (about 160% of the pristine value).The maximum dose values of the Bragg peaks were reduced in all cases of phantom penetration.In the case of the 140 MeV beam, the maximum dose was reduced by 10%-11% using the clinical CT based setups, whereas the maximum peak height was only 60% of the pristine peak dose value when the protons traversed the perfect CT phantom.The peak heights of the 200 MeV beams through geometries based on clinical CT images were lower by only 3.5%-5%, whereas the peak of the 200 MeV protons through the perfect CT geometry yielded 80% of the pristine peak value.
The phantom models based on small animal CT images yielded a DFW of 0.41 cm for the 140 MeV beam (an increase of 86%) and 0.52 cm for 200 MeV protons (33% larger than the DFW of the pristine peak), and the peak height reduction to about 76% of the pristine peak for 140 MeV and to 89% for 200 MeV, respectively.
Figure 8 shows the results for degradation of distal edge of the SOBP with a range of 10 cm in water and a modulation width of 5 cm.The distal falloff region is shown for several cases: a measured and a simulated SOBP in water without a lung phantom, a measured SOBP delivered through the phantom, and the corresponding simulated SOBP, F. 8. Distal dose falloff in a SOBP with a range of 10 cm in water and a modulation width of 5 cm.The solid line and the circles show the measured and simulated dose curves in water without the lung phantom in the beam line.The dotted curve shows the simulated, undegraded falloff with a 1.4 cm shift applied, corresponding to the water equivalent thickness of the phantom.A measured curve, delivered through the phantom, is shown by the dashed line, and triangles describe results of the convolution of Gaussian function with the undegraded SOBP, also with a shift of 1.4 cm applied.
T II.Distal falloff values from the 80% to the 20% dose locations and range at the 90% falloff location in a measured and a simulated SOBP.which was derived by convolving the Bragg curves from the simulations without a phantom with a Gaussian function (σ = 3.6 mm, see Sec. 4) and by shifting the curve by 1.4 cm (the water equivalent thickness of the lung-like phantom).Finally, a simulated SOBP with no convolution, which was shifted by 1.4 cm, is shown to emphasize the differences between the degraded and the undegraded falloff regions.All DFWs from the SOBP investigation are presented in Table II.

DISCUSSION
Measurements and simulations of Bragg curves of proton beams penetrating a sample of plastinated human lung and a custom-designed randomized plastic phantom led to noticeable changes in the distal falloff width of the proton dose scored in water downstream of the phantom.The energy deposited by degraded proton beams was clearly spread out over a larger volume than the doses of pristine Bragg peaks.
Comparing measured and simulated depth-dose profiles of pristine beams to those of beams through the plastic lunglike phantom, we found that the DFW increased by up to almost 0.3 cm in water and the peak dose decreased by up to 40% when beams penetrated the lung-like phantom.It is worth mentioning that these depth-dose curves were obtained in water.Comparison of the mass density of water to the average density of lung parenchyma 9 leads to estimates of dose falloff widths in lung tissue of about 4-7 times larger (i.e., on the order of up to 2 cm).The magnitude and energy-dependent behavior of the distal edge degradation was comparable to measurements performed with a sample of plastinated human lung tissue, indicating that the phantom design was suitable to approximate fine-structured, sub-millimeter-scale heterogeneities, such as those in lung parenchyma.
The distal dose falloff of a measured depth-dose curve from an SOBP showed significant degradation, increasing the DFW from 0.28 to 0.68 cm (increase of 140%) in water, which may translate to several centimeter in lung.
By applying a convolution with a Gaussian function (and shifting by the water-equivalent thickness of the phantom), it was possible to approximate the measured depth-dose curve with a simulated, undegraded beam, which indicates that the degradation may be described with a relatively simple analytical model of range degradation.Range degradation arises from the fact that different protons take different paths through the lung-like phantom and therefore penetrate a different sequence of air and water voxels.This leads to statistical fluctuations in the range of protons, which are added to the wellknown range straggling effect.To calculate the magnitude of range degradation, we consider the following model: a proton penetrating the lung-like phantom traverses a sequence of N voxels.We assume that each voxel is randomly filled with either air of water.The radiological depth ζ as a function of the geometrical depth z is given by S(z) denotes the stopping power ratio at depth z.For the discretized structures in a phantom, this is simplified to where S i = S w = 1 for water and S i = S a = 0 for air.∆ is the voxel size (0.5 mm in our phantom), and z/∆ = N is the number of traversed elements.We assume that filling a single voxel is a Bernoulli process and that the filling of different voxels is statistically independent.The radiological depth is then binomially distributed and the variance of the radiological depth σ 2 (z) is given by The expectation value of the stopping power ratio ⟨S i ⟩ is given by where p is the probability of a voxel containing water.Note that p is determined by the average density of the lung-like phantom (i.e., p = 0.27 is our phantom).Analogously, we obtain S 2 i = p.The variance of the radiological depth is thus given by σ 2 (z) = p − p 2 z∆.
Entering the parameter values used in this study (p = 0.27, z = 5 cm, ∆ = 0.5 mm), we obtain σ = 2.2 mm.If the number of traversed voxels is large, the distribution of the radiological path length is approximately Gaussian.Thus, within the validity of this model, the depth-dose curve of a proton beam after traversing a lung-like phantom of thickness z can be obtained by convolving the depth-dose curve of an unperturbed proton beam with a Gaussian distribution of width σ 2 (z) = p − p 2 z∆.
We observe that the magnitude of range degradation increases with the thickness z of the traversed heterogeneous medium.In addition, for fixed thickness z, the range degradation effect increases with the voxel size ∆.It is intuitive that the range degradation effect will vanish if ∆ → 0, which corresponds to a homogeneous medium.Note, however, that the model is not applicable for very large ∆ (i.e., ∆ ≈ z), since no averaging over different paths would be realized.The increase of the range degradation with z and ∆ was confirmed in MC simulations (results not shown).It is further assumed that, for a fixed amount of traversed heterogeneous material, the range degradation is independent of the initial proton energy.While the theoretical value of σ = 2.2 mm differs moderately from the applied value of 3.6 mm, it must be taken into account that the model assumes each proton to encounter a truly random series of air and soft tissue voxels.In reality, this is not the case.Furthermore, the randomness of the phantom was modified so that it could be produced in a 3D-printer.Presumably, the agreement between model prediction and MC simulations could be improved by generating a new randomized phantom for the simulation of each individual proton.
Another contributor to the uncertainty associated with lung proton doses was encountered when the high-resolution plastic phantom was converted into CT images of various resolutions and implemented into the MC simulations.The results of MC beam transport simulations proved that the resolution of the CT has a significant impact on the dose distribution.Contemporary clinical CT scanners with resolutions on a millimeter scale did not resolve the fine structured phantoms, leading to Bragg curves that underestimated the degradation in the distal falloff region by as much as 0.2 cm in water (corresponding to underestimations of up to 1.4 cm in lung) and overestimated the peak dose values by as much as 35% compared to dose computations performed on a perfect CT model of the phantom.MC dose predictions in a phantom based on CT image scans of a small animal CT system provided better agreement with realistic predictions; however, differences of about 0.1 cm (up to 0.7 cm in lung tissue) in DFW and of 10% to about 20% in dose peak values remained when the predictions based on the CT images were compared to the predictions in a perfect CT geometry.Note that, because of prohibitive memory requirements in the MC model, it was not possible to use the highest available resolution of the image.However, the results clearly show the limitations in dose predictions based on CT images from clinically employed contemporary scanners.Rotating the phantom to assess possible differences in the dose curves due to asymmetric shape of the CT voxels did not yield any noticeable differences in peak dose or DFWs.These results are in excellent agreement with the findings of Ref. 5.
While only one single phantom was used in this study, real lung parenchyma will present variations in density as well as complexity of the heterogeneities.In addition to the discussion of how degradation depends on heterogeneity parameters using an idealized model, we refer to the study performed by Sawakuchi 4 for the interested reader.
Furthermore, the study did not take into account any organ motion.
Finally, the topic of scanned beam vs scattered beam was not discussed so far.While this study was performed on a double scattering beam line, proton therapy with scanned beams has become the modality of choice.However, measurements with scanned beams were not available for this research project.We therefore approximated scanned beam conditions as well as possible by running the beam through a range modulator wheel with the angular position fixed at the thinnest point.The resulting Bragg peaks are comparable to contemporary scanned beam proton depth doses, and the results from the SOBP simulations (comprising a weighted sum of Bragg peaks) indicate that the underlying mechanisms may be applicable to both modalities.
In conclusion, the uncertainties associated with proton lung doses can be large in the distal falloff region, where significant degradation must be expected.This study proves that, in a typical thoracic treatment beam, several centimeter along the beam path may be affected, potentially leading to undesired and unanticipated dose to tissue distal to the target and to under dosage to parts of the target volume.Current treatment planning practice aims to mitigate such large uncertainties by application of large safety margins distal to the target.The dimensions of these margins are based on the geometric proton beam's range in the patient and are calculated by estimating 3.5% of the proton range in the patient and finally adding another 3 mm.In practice this can be 3-4 cm, depending on the specific patient geometry and beam arrangements.With a more accurate estimate of the dose distributions, especially in the distal edge, it may be possible to optimize and possibly reduce these safety margins to spare more healthy tissue.
While the findings of this study are limited to a few beam energies and a single SOBP delivered to a limited number of geometries, the results are evidence that a more detailed investigation of distal edge degradation and mitigation strategies is imperative.Future work should focus on development of mitigation methods to properly account for range degradation in treatment planning.

F. 1 .
Plastinated human lung sample in a Lucite container.The markings (Pos 1 and Pos 2) point out two locations used for positioning the sample on the proton beam axis.

F. 4 .
CT image of the plastic phantom obtained with a clinical CT scanner (left) compared to a perfect CT image (center) and an image obtained with the small animal CT scanner (right hand side).Note the images are from different locations; however, they demonstrate the differences in the fine structure which eventually led to variations in the DFWs.Medical Physics, Vol.42, No. 11, November 2015 F. 5. Dose, D, as a function of depth, d, measured downstream of a plastinated human lung sample and compared to the depth doses through an empty container and the pristine Bragg curves.Pos 1 and Pos 2 depict depth dose curves measured with the lung phantom located at different positions with respect to the central beam axis (see Fig. 1).(a) shows the 140 MeV data and (b) shows the depth doses of the 200 MeV beams.

F. 6 .
Measured and Monte Carlo simulated depth dose curves for the 140 MeV beam (a) and for the 200 MeV beams (b) with and without the lung-like plastic phantom in the beam.The simulated data are shown as solid and dashed lines, while measured curves are shown as scatter plots.Medical Physics, Vol.42, No. 11, November 2015 T I. Distal falloff values from the 80% to the 20% dose locations in measured and simulated Bragg curves and peak dose values relative to pristine values.
Figure7shows the Bragg curves of all simulations obtained by scoring the energy deposition in water, after the beams had penetrated the various implementations of CTs of the randomized plastic phantom upstream of the water container.The phantom geometries, based on clinical CT images, were each simulated twice using differently oriented CT images (one with the largest voxel dimension in z-direction and one rotated such that the largest voxel dimension would be along the y-direction).This was done to evaluate possible systematic differences in the depth-dose curves based on the different dimensions of the voxels in the z-and y-directions.No notable differences [see Figs.7(a) and 7(b)] were encountered when the simulations were performed with a rotated phantom.The Bragg curves of proton beam simulations through the phantom constructed from the small animal CT resulted