Quantification of the uncertainties of a biological model and their impact on variable RBE proton treatment plan optimization

Physica Medica 36 (2017) 91–102

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Original paper

Quantification of the uncertainties of a biological model and their impact on variable RBE proton treatment plan optimization A.F. Resch a,b,1, G. Landry a, F. Kamp b,c,d, G. Cabal e, C. Belka d, J.J. Wilkens b,c, K. Parodi a, G. Dedes a,⇑ a

Department of Medical Physics, Faculty of Physics, Ludwig-Maximilians-Universität München, Garching b. München, Germany Physik-Department, Technical University of Munich, Garching, Germany c Department of Radiation Oncology, Technical University of Munich, Klinikum rechts der Isar, Munich, Germany d Department of Radiation Oncology, LMU Munich, Munich, Germany e Clinica El Rosario, Medellin, Colombia b

a r t i c l e

i n f o

Article history: Received 14 July 2016 Received in Revised form 14 March 2017 Accepted 20 March 2017 Available online 30 March 2017 Keywords: Relative biological effectiveness Proton therapy Optimization

a b s t r a c t Purpose: In proton radiation therapy, a relative biological effectiveness (RBE) equal to 1.1 is currently assumed, although biological experiments show that it is not constant. The purpose of this study was to quantify the uncertainties of a published biological model and explore their impact on variable RBE treatment plan (TP) optimization. Methods: Two patient cases with a high and a low ða=bÞx tumor were investigated. Firstly, intensity modulated proton therapy TPs assuming constant RBE were derived, and subsequently the variable RBE weighted dose (RWD), including the uncertainty originating in the fit to the experimental data and the uncertainty of the ða=bÞx , were calculated. Secondly, TPs optimized for uniform biological effect assuming a variable RBE were created using the worst case tissue specific ða=bÞx . Results: For the nasopharyngeal cancer patient, the uncertainty of ða=bÞx corresponded to a CTV D98 confidence interval (CI) of (2, +4)% while for the fit parameter CI was ð2; þ1Þ%. For the standard fractionation prostate case the ða=bÞx CI was ð7; þ5Þ% and the fit parameter CI was ð3; þ3Þ%. For the hypofractionated case both CIs were ð1; þ1Þ%. In both patient cases, the RBE in most organs at risk (OARs) was significantly underestimated by the constant RBE approximation, whereas the situation was not as definite in the target volumes. Overdosage of OARs was reduced by using the biological effect optimization. Conclusion: For the two patient cases, the RWD uncertainty from the fit parameter in the biological model contributed non-negligibly to the total uncertainty, depending on the patient case and the organ. The presented optimization strategy is a basic method for robust biological effect optimization to reduce potential consequences caused by the ða=bÞx uncertainty. Ó 2017 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.

1. Introduction By the end of 2015 more than 130,000 cancer patients have been treated with high energy proton beams [1]. The main rationale for this technology is the proton depth dose profile, which is characterized by the Bragg peak with high dose deposition at the end of the proton range followed by a steep dose fall-off. This allows reducing the dose deposited in organs at risk (OARs). Fur-

⇑ Corresponding author. E-mail address: [email protected] (G. Dedes). Now affiliated with: Division Medical Radiation Physics, Department of Radiation Oncology, Christian Doppler Laboratory for Medical Radiation Research for Radiation Oncology, Medical University of Vienna/AKH Wien, Währinger Gürtel 18-20, 1090 Vienna, Austria. 1

thermore, the integral dose to the patient can approximately be reduced by a factor of 2 by using protons instead of photon intensity modulated radiation therapy [2]. The relative biological effectiveness (RBE) of protons is considered to be equal to 1.1 in clinical practice [3]. However, in vitro experiments show that the RBE is increasing towards the end of the beam range [4], where the linear energy transfer (LET) rises sharply [5–7]. As the RBE weighted dose (RWD) increases with dose averaged LET (LETd), the effective beam range increases by a few millimeters [8,9]. However, the RBE not only depends on LETd, but also on dose and tissue type [10,11]. For a certain cell type, the linear quadratic (LQ) model with the linear parameter a and the quadratic parameter b is commonly used to model in vitro cell survival experiments. The ratio of the two parameters, ða=bÞ, is clinically used to distinguish late and early reacting tissues (corresponding to a low and high ða=bÞ,

http://dx.doi.org/10.1016/j.ejmp.2017.03.013 1120-1797/Ó 2017 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.

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respectively) [10]. RBE models based on the linear quadratic formulation of cell survival proposed by Wilkens and Oelfke [12], Carabe-Fernandez [13], Wedenberg et al. [14] or McNamara et al. [15] describe the RBE as a function of dose and LETd. The latter two models additionally include the ða=bÞx of photons to account for RBE dependence on tissue type. However, ða=bÞx values exhibit large error bars, which cause considerable RBE variations when applied in biological models [16]. In addition to the uncertainty of the ða=bÞx , RBE modeling suffers from considerable uncertainties in experimental in vitro cell survival experiments [17]. Therefore, some authors suggest that the uncertainties in biological data are too high to apply RBE models in clinical practice [18], and suggest further investigations [17]. The clinical impact of the ða=bÞx uncertainty has been quantified in [16] in the framework of the CarabeFernandez model [13], but so far no study has reported on the uncertainty introduced by the fit to the experimental data. The purpose of this study was to investigate the uncertainty of the RBE weighted dose by considering both the uncertainty of the ða=bÞx and the intrinsic uncertainty of the biological model originating from the fit to experimental data. The Wedenberg et al. model was chosen as RBE model, since it was fitted to different cell types in contrast to the Carabe-Fernandez or Wilkens and Oelfke model, which were fitted solely on V79 cell data. Furthermore, it was statistically tested to represent the fitted data well [14]. To ensure optimal dose and LETd calculation, the treatment plans were created by a Monte Carlo simulation [19] based TPS using clinically applied fractionation schemes [11,20]. Additionally, we adapted the biological effect optimization method developed by Wilkens and Oelfke in order to yield a homogeneous RWD distribution when applying the Wedenberg et al. model [21,19,22]. Tissue specific ða=bÞx values were used and assigned in a way that aims to reduce the consequences of potential ða=bÞx uncertainties. 2. Materials and methods 2.1. Biological model The biological model used in this work was proposed by Wedenberg et al. [14]. Similar to other phenomenological models [12,13,23,15], the Wedenberg model is based on the linear quadratic model (LQ) in which the cell survival fraction (S) is described as a function of the absorbed dose (D)


 S ¼ exp aD  bD2 :

ð1Þ

The quadratic parameter b for protons (bp ) is assumed to be independent of LETd and the linear parameter (ap ) is considered to increase linearly with LETd [14]:

ap q :¼ 1 þ  L; bp :¼ bx ða=bÞx ax

ð2Þ

where ax ; bx and ða=bÞx are the parameters of the reference photon radiation, L the LETd and q a free parameter, which can be estimated by a fit to experimental cell survival data. Wedenberg et al. [14] fitted Eq. (2) to a set of 24 data points of 10 different cell lines from 6 cell survival studies, which resulted in q equal to 0.435 [CI 95% (0.366, 0.513)] ðGy lm ðkeVÞ Þ. Using the assumptions in (2) together with the definition of RBE ¼ Dx =Dp , RBE can be expressed as a function of ða=bÞx , LETd (L) and proton dose (D): 1

     a 1 a 1 RBE ; L; D ¼  þ 2D b x D b x sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2    1 a a  D þ D2 : þ qL þ 4 b x b x

ð3Þ

The functional form of Eq. (3) is plotted in Fig. 1. The RBE in the Wedenberg model increases monotonically with LETd and decreases with increasing ða=bÞx . There is only a small dose dependence for high ða=bÞx and a considerable increase in RBE with decreasing dose for low ða=bÞx . 2.2. Dose and LET calculation For this study, a Monte Carlo simulation-based treatment planning tool for clinical calculations in CT geometries was developed, following a similar concept as in [24]. A particle therapy extension of CERR ([25,22], Matlab/2014a) was used as TPS and Geant4 (version 10.01) [19] for dose and LETd calculation. Firstly, the lateral and axial pencil beam (PB) positions were determined in the TPS. The beam spots were placed on a 5 mm grid in lateral direction and with 3 mm spacing in axial direction. In order to get the energies of the PBs, the Hounsfield units of the CT image were converted into stopping power ratios (SPRs). From the SPRs along the central ray of the PB, the water equivalent path length (WEPL) and hence the PB energies using an energy-range look-up table generated with Geant4 were determined. Secondly, the geometrical positions and corresponding energies of the PBs were exported to Geant4 together with the patient CT, which was converted into density and elemental composition following Schneider’s approach [26] calibrated for the CT scanner (Toshiba Aquilion LB, Toshiba Medical Systems, the Netherlands) located at Klinikum Großhadern, and ionization potential according to ICRU report 49 [27]. Note that the conversion of CT numbers in the TPS and MC simulations need to yield the same SPR in order to prevent shifts between the range determination using the WEPL calibration in the TPS and intrinsic calculation in the Geant4 simulations. From extensive testing, linear interpolation of the SPR sampling points used for Geant4 input was found to yield the most consistent results and was hence used in the TPS. The dose and LETd were converted on the fly following the dose-to-water concept [28] and each PB was simulated in Geant4 with 105 primary protons (events). This corresponds to a total number of approximately 1:5  108 and 5  107 simulated events (Geant4), for the two studied patient cases, which are described in detail in Section 2.4.1 (patient settings). Both the CT and the dose scoring voxel sizes were 1:074  1:074  3 mm3. LETd scoring was successfully benchmarked against literature for different voxel sizes [29,30]. Finally, the three-dimensional dose and LETd distributions of each PB were passed back to the TPS and used for TP optimization (see Appendix A) [21,22]. 2.3. Biological effect optimization Wilkens and Oelfke presented a biological effect optimization method based on the LQ model [21] and a biological model which was fitted to Chinese hamster cell (V79) survival data [12]. This method was adapted in order to use the Wedenberg model instead of the original biological model and implemented into the TPS. Details about the implementation and the derivation can be found in the original paper [21] and the Appendix B. 2.4. Settings 2.4.1. Patient settings According to Eq. (3), the RBE depends on ða=bÞx . Therefore, two different tumor types were chosen for investigation, a nasopharyngeal cancer patient with a high ða=bÞx and a prostate cancer patient with a low ða=bÞx . As the debate on the optimal fractionation scheme for prostate cancer is not settled, a standard and a hypofractionation TP with 35 fractions of 2 Gy(RBE) and 5 fractions

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Fig. 1. RBE dependence of the Wedenberg model on ða=bÞx , LETd and dose. In the left and the middle figure, dose is kept constant at 2 Gy and LETd is kept constant at 3 keV/ lm in the right figure. The uncertainty bands correspond to the 95% CI and originate in the fit. The horizontal black line represents the RBE ¼ 1:1 approximation.

of 7.6 Gy(RBE) respectively [31] were investigated. The prescribed doses and treatment settings are listed in Table 1. TP optimization objectives were: brainstem (center: D2 < 54 Gy(RBE), surface: D2 <63 Gy(RBE)) [32,33], optic nerve (D2<56 Gy(RBE)) [34], rectum (D2<105%), bladder (D2<105%), where the index i of Di is % volume. The necessary biological parameters ax and bx including the 95% CI are listed in Table 2. In overlapping regions of OARs and the PTV, the ða=bÞx of the OARs were chosen for consistency with [16]. The structures were adopted from the intensity-modulated radiation therapy (IMRT) treatment scenario. 2.4.2. Investigating the uncertainties of the RWD In order to investigate the uncertainties of the RWD using the Wedenberg model, the TPs for the nasopharyngeal and the prostate cancer patient were optimized using the RWD resulting from a constant RBE equal to 1.1. Dose weighted by a constant RBE will further be referred to as RWD1.1. After the optimization procedure, the variable RWD according to the Wedenberg et al. model using Eq. (3) was calculated, denoted as RWDw in the following. The individual error bars were calculated separately using the lower and upper limits of the 95% CI interval of the fit parameter q and the ða=bÞx ratio, respectively. The upper limit of the total uncertainty was estimated by using the upper limit of q and the lower limit of ða=bÞx and vice versa for the lower limit. The ða=bÞx were assigned for each tissue individually according to Table 2. In this study, the RWD1.1 is said to be significantly different from the RWDw if it is not within the 95% CI originating in the uncertainty of either q, ða=bÞx or both depending on the origin of uncertainty we want to address. 2.4.3. Biological effect optimization The biological effect optimization following Wilkens’ approach was performed on the nasopharyngeal and prostate cancer patient. In order to take the uncertainty of the ða=bÞx ratio into account, a conservative strategy to assign ða=bÞx to each organ was chosen. All organs were separated into two types, targets and OARs. In a conservative strategy, the biological effect should be underestimated in a target and overestimated in an OAR by potential parameter errors. Therefore, the upper limit of the ða=bÞx CI as listed in Table 2 was assigned to targets and the lower limit of ða=bÞx was assigned to OARs (inverse dependence of RBE to increasing ða=bÞx in Eq. (3)). Due to the lack of literature values for ax and bx to the corresponding ða=bÞx ratio uncertainty, which were necessary for the biological effect optimization (Eq. (11), Appendix B), the ax and bx values had to be altered such that they matched the desired ða=bÞx ratio. This variation followed also a conservative approach. In the targets, the original bx value was lowered in order to increase the ða=bÞx ratio, whereas in the OARs, bx was increased in order to reduce the ða=bÞx ratio, while ax was kept constant in both cases. Since ax and bx of the bladder were not available in literature, the values of the rectum were used. TPs optimized using

the variable biological effect will be denoted as TPBEO, whereas TPs optimized using a constant RBE equal to 1.1 will be referred to as TP1.1. 3. Results 3.1. Uncertainties of the RBE weighted Dose (RWD) The TP1.1 in this section have been optimized using the constant RBE equal to 1.1, as described in Section 2.4.2 and according to the established clinical practice. The variable RWDw using the Wedenberg model was calculated after the aforementioned optimization procedure. 3.1.1. Nasopharyngeal cancer patient The RBE weighted dose volume histogram (DVH) for the nasopharyngeal cancer patient is shown in Fig. 2. The RWD1.1 shows the desired steep dose fall-off in the CTV. Additionally, the RWDw calculated by the Wedenberg formula is plotted together with the uncertainty band caused either by the uncertainty of the fit parameter q or the uncertainty of the ða=bÞx . The major contribution to the uncertainty band of the RWDw in the CTV, (left) optic nerve and the (left) eye was the uncertainty of ða=bÞx whereas the contributions of ða=bÞx and q in the brainstem were approximately equal. In the CTV, the fixed RBE value resulted in a consistently higher RWD1.1 than the nominal RWDw, calculated by the Wedenberg model including the uncertainty band originating in the uncertainty of q. However, the RWD1.1 was inside the uncertainty band caused by the uncertainty of ða=bÞx . In the case of the left optic nerve, the constant RBE was outside of the confidence interval of q and on the lower limit of the uncertainty band of ða=bÞx . In the left eye and the brainstem, the fixed RBE was significantly lower than the variable RBE regardless of the origins of uncertainty. In Fig. 2 (a), the corresponding DVH including the total uncertainty band, i.e., the combined uncertainties of q and ða=bÞx , is plotted. This figure demonstrates that the RWD1.1 was inside the total uncertainty band of the RWDw in the target, although the nominal RWDw value was consistently lower. Also in the left optic nerve, the RWD1.1 was within the total uncertainty band of the model, but further off the nominal value and in contrast to the target region, the RWDw was higher than the RWD1.1 due to a lower ða=bÞx . In the left eye and the brainstem, the RWD was significantly underestimated by the constant RBE approximation. Several characteristic values of the DVH plotted in Fig. 2 are listed numerically in Table 3. Note that the RBE was lower in the CTV than in the left eye although the LETd was mostly higher. This originates in the lower dose in the eye and the inverse dependency of RBE on dose (see Fig. 2). In the cumulative volume histogram, the LETd appears highest in the brainstem. As a consequence, the RWDw in the

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A.F. Resch et al. / Physica Medica 36 (2017) 91–102 Table 1 Treatment settings used in this study. Prescribed dose (Dpr ), number of fractions (n) and dose per fraction (D). Beam angles (/) are reported on the IEC-scale. qIn this study we investigated the first phase of the treatment. In the second phase a 20 Gy boost was applied to a smaller volume. Tumor

Dpr [Gy(RBE)]

n

D [Gy(RBE)]

/ [°]

Nasopharyngeal Prostate stand. Prostate hypo.

50q 70 38

25 35 5

2 2 7.6

0, 110, 260 90, 270 90, 270

Table 2 The tissue parameters used in this study [16,23,35,36]. Due to lack of literature values: *50% uncertainty assumed for the nasopharyngeal cancer and the eye, **ax and bx of the rectum used in the bladder. Tissue Prostate tumor Nasopharyngeal tumor Rectum Bladder Brainstem Optical nerve Eye

ax

[Gy1]

bx [Gy2]

ða=bÞx [Gy]

CI [Gy]

0.036 0.112 0.040 0.040** 0.053 0.051 0.040

0.024 0.011 0.010 0.010** 0.027 0.026 0.020

1.5 10 4.0 4.0 2.1 1.6 2.0

(1.2, (5.0, (2.5, (3.0, (1.5, (0.5, (1.0,

5.6) 15.0)* 5.0) 7.0) 3.9) 10.3) 3.0)*

Fig. 2. Graphs (a and b) show the cumulative DVH for the CTV, brainstem, left eye and left optic nerve of the TP1.1 for the nasopharyngeal tumor patient. In (a), the uncertainties caused by q and ða=bÞx are plotted as dark and lightly shaded area, respectively. In (b), the total uncertainty band is plotted. The RWD1.1 is shown as black dashed lines while the colored solid lines represent the nominal RWDw values. The corresponding dose averaged LET volume histogram is plotted in (c). Cumulative volumes may not sum up to 1 (a and c), as only voxels receiving at least 1% of the dose prescribed to the target (i.e., 0.02 Gy(RBE)/fraction) were considered.

overlapping region of the brainstem and the PTV was up to 20% higher than RWD1.1 (see Fig. 3). The optimization objective of decreasing RWD1.1 in this structure caused an increase in LETd

and consequently RWDw, as LETd is a free parameter in TP1.1 optimization. This LETd effect was additionally amplified by the lower ða=bÞx in the brainstem compared to the CTV. As expected, elevated

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Table 3 DVH parameters (DP) for several regions of interests (ROIs) for the nasopharyngeal and prostate cancer patient corresponding to Figs. 2 and 4. The CI caused by either the fit parameter q (CIq) or the ða=bÞx ratio (CIða=bÞx ) and the total RWDw range (CIt), respectively, are listed. The index i of Di indicates % volume. RWD1.1 [Gy(RBE)]

RWDw [Gy(RBE)]

CIq [Gy(RBE)]

CIða=bÞx [Gy(RBE)]

CIt [Gy(RBE)]

49.3 48.5 50.0 51.8 46.3 48.3 49.3 50.3 5.0 39.0 12.8 49.5

48.3 47.5 48.8 51.0 51.8 52.5 57.0 60.0 6.0 45.8 17.3 55.8

(47.5, 48.8) (46.8, 48.0) (48.3, 49.5) (50.3, 52.0) (50.0, 53.3) (50.8, 53.8) (55.0, 59.3) (57.5, 62.5) (5.8, 6.3) (44.0, 47.5) (16.3, 18.3) (53.8, 57.5)

(47.3, 50.3) (46.5, 49.3) (48.0, 50.8) (49.8, 53.5) (46.3, 54.5) (47.5, 55.0) (49.8, 60.8) (52.0, 64.3) (5.5, 7.0) (44.0, 48.0) (15.5, 18.3) (53.3, 56.8)

(46.7, 51.0) (46.0, 50.0) (47.5, 51.8) (49.2, 54.8) (45.5, 56.5) (46.7, 56.8) (48.7, 63.5) (50.7, 67.0) (5.2, 7.5) (42.7, 50.0) (14.7, 19.3) (51.5, 58.8)

Prostate cancer patient – standard fractionation TP  CTV 70.4 D D98 69.0 D50 70.4 D2 72.1 Rectum D2 70.7 Bladder D2 49.7

79.1 77.0 78.8 83.3 75.3 54.6

(76.7, (74.9, (76.3, (80.2, (73.2, (52.9,

81.6) 79.5) 81.2) 86.1) 77.0) 56.0)

(72.8, (71.4, (72.8, (76.0, (73.9, (51.8,

79.8) 80.5) 79.5) 84.0) 77.0) 56.0)

(71.4, (69.7, (71.1, (73.9, (72.1, (50.4,

82.6) 81.6) 82.3) 87.2) 79.5) 57.8)

Prostate cancer patient – hypofractionation TP  CTV D D98 D50 D2 Rectum D2 Bladder D2

37.1 36.1 37.0 39.0 37.2 26.7

(36.7, (35.8, (36.6, (38.3, (36.9, (26.0,

37.5) 36.6) 37.4) 39.6) 37.6) 26.7)

(36.6, (35.8, (36.5, (38.3, (37.1, (26.0,

37.1) 36.2) 37.0) 39.1) 37.4) 26.7)

(36.3, (35.4, (36.7, (37.7, (36.8, (25.7,

37.6) 36.6) 37.4) 39.7) 38.0) 27.0)

ROI

DP

Nasopharyngeal cancer patient  CTV D D98 D50 D2  Optic nerve D

Eye Brainstem

D50 D10 D2 D50 D2 D50 D2

38.1 37.3 38.1 39.0 38.7 26.6

LETd values were observed towards the end of the beam range and the lateral penumbra (Fig. 3 (c)) and correspondingly increasing RWD values with discontinuities at tissue boundaries.

level. Along tissue boundaries the RWDw distribution showed discontinuities due to the discontinuity of ða=bÞx . 3.2. Biological effect optimization

3.1.2. Prostate cancer patient Fig. 4 shows the DVH of the standard fractionation (a and c) and hypofractionation TPs (b and d) for the prostate cancer patient. For both fractionation schemes and all ROIs, the magnitude of the uncertainty caused by either of the two sources of uncertainty was comparable. However, the RWDw uncertainty band reflects the asymmetry of the CI of ða=bÞx around the nominal value for prostate cancer (see Table 2). The uncertainty bands were distorted towards lower RWDw values. The uncertainty bands caused by q were almost symmetrical (but not completely, due to the square root in the RBE function in Eq. (3)). In Fig. 4 (c) and (d), the DVHs of the two TPs including the total uncertaintyband are plotted. The RWD1.1 was significantly underestimated in the standard fractionation TP. The hypofractionation TP exhibited a significant overestimation of RWD1.1 with respect to the total uncertainty of the Wedenberg et al. model. The bladder did not receive a crucial dose in both TPs. The LETd of the CTV in the standard fractionation TP ranged from 3 to 4 keV/lm. However, LETd were mostly lower for the hypofractionation TP and ranged from 2 to 5 keV/lm. In the rectum, LETd values were slightly higher than in the CTV ranging from 2.5 to 5 and 2 to 5 keV/lm for the standard fractionation and hypofractionation TP, respectively. Fig. 5 (e) and (f) show the LETd distribution of a representative CT slice. High LETd values could be found towards the end of the spread out Bragg peak and in the lateral penumbra, hence the rectum and bladder. The RWDw distribution in (c) and (d) follows the LETd distribution. The RBE increases towards the end of the beam range and caused an up to 25 % higher RWDw than the prescribed dose in the region between the CTV and the PTV, whereby the RWD1.1 was distributed homogeneously at the prescribed dose

Figs. 6 and 7 show the results of the TPs for the two patient cases using the biological effect optimization (TPBEO). The RBE weighted doses of the TPBEO (RWDw,BEO) and of the previous RBE¼ 1:1 optimized TP1.1 (RWDw) are shown.

3.2.1. Nasopharyngeal cancer patient Fig. 6 (a) shows the cumulative RWD volume histogram of TPBEO and TP1.1 for the nasopharyngeal cancer patient. The new optimization strategy (TPBEO) yielded a RWDw,BEO fall-off in the CTV at the prescribed dose level equal to 2 Gy(RBE), as it was obtained for RWD1.1 in the previous TP1.1. In the optic nerve, the two objective variables in the optimization, namely the RWDw,BEO in the TPBEO and RWD1.1 in the TP1.1, were also comparable. However, RWDw was noticeably higher than RWDw,BEO. Similarly to the optic nerve, the RWDw,BEO in the brainstem was noticeably lower than RWDw. Note that the RWDw,BEO was calculated using the worst case ða=bÞx , whereas the nominal ða=bÞx were used to calculate RWDw. Consequently, comparing only the nominal RWDw and RWDw,BEO would yield a bias in favor of RWDw and therefore the uncertainty band needs to be taken into account. The worst case ða=bÞx assignment was reflected in the RWDw,BEO uncertainty band, which suggests a potential underestimation of the RWDw,BEO in the CTV, whereas RWDw,BEO was potentially overestimated in the OARs. On the other hand, the uncertainty band around the nominal RWDw (see Fig. 2) was approximately symmetric. Fig. 6 (b) shows the corresponding RWDw,BEO distribution of a representative CT slice. RWDw,BEO was homogeneously distributed inside the CTV at the prescribed dose level. In contrast to RWDw, RWDw,BEO did not exhibit hot spots in the brainstem.

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Gy(RBE) 2.4 2.0

1.5

1.0

0.5

0.0

(c) keV/μm 6.0 5.0 4.0 3.0 2.0 1.0 0.5

Fig. 3. Representative RWD distribution of the TP for the nasopharyngeal cancer patient applying a constant RBE during TP1.1 optimization. (a) RWD1.1 [Gy(RBE)], (b) RWDw [Gy(RBE)] and (c) LETd [keV/lm]. The CTV (purple), PTV (blue), eyes (red and green), brainstem (green) and the optic nerves (green and orange) are delineated. No values are displayed in the nasal cavities with CT values equal to air. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7 shows the DVHs of the TPBEO for the prostate cancer patient. For both fractionation regimens, the biological effect optimization yielded a RWDw,BEO fall-off at the prescribed dose levels (2 and 7.6 Gy(RBE)) in the CTV. The uncertainty band of the RWDw,BEO indicated potentially higher doses than prescribed in the CTV, and potentially lower doses in the OARs for both fractionation schemes (a and b). In contrast to the biological effect optimized TPBEO, RWDw was significantly over- and underestimated in the standard (a) and hypofractionation (b) RBE¼ 1:1 optimized TP1.1, respectively. Therefore, only the biological effect optimization yielded the prescribed RWD in the CTV when considering variable RBE. In the rectum, the nominal RWDw of the standard fractionation TP1.1 was noticeably higher than RWDw,BEO, even though the worst case ða=bÞx assignment disadvantages RWDw,BEO. The uncertainty band of the RWDw,BEO indicated potentially lower RWD, whereas the uncertainty band of the RWDw was approximately symmetric and did not favor lower RWD. In the hypofractionation regimen, the RWDw was lower than the RWDw,BEO in the rectum. However, RWDw was also lower than RWD1.1 and the two optimization objectives of TP1.1 (RWDw) and TPBEO (RWDw,BEO) were comparable.

Fig. 7 (c and d) show the RWDw,BEO distribution. RWDw,BEO was homogeneously distributed inside the CTV in the standard fractionation and hypofractionation TPs. 4. Discussion The generic beam model used in this study assumed a constant initial beam size for all energies, although this is not achievable in all centers. However, LETd dependence on spot size in a homogeneous field is not expected as the lateral LETd dependence of a single PB is small [37] and LETd enters the RBE in Eq. 3 in the square root. Therefore, it is assumed that a constant spot size is a valid approximation for the purpose of this study. The RBE in the Wedenberg formulation is a function of three variables, ða=bÞx , LETd and dose. The dependence on the first variable, ða=bÞx , can be observed when comparing the nasopharyngeal cancer patient with the standard fractionation TP of the prostate cancer patient, where the dose levels (2 Gy(RBE)) matched. Although the LETd in the former was slightly lower, the increase of RBE in the prostate cancer patient can be attributed to the lower ða=bÞx .

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Fig. 4. DVH of the standard fractionation (a and c) and hypofractionation (b and d) TP1.1 for the prostate cancer patient. In (a) and (b), the uncertainty bands caused by q and ða=bÞx are plotted as dark shaded and colored shaded area, respectively. The total uncertainty band is shown in (c) and (d). The RWD1.1 is shown by dashed black lines while the colored lines represent the nominal RWDw values. In (e), the dose averaged LET cumulative volume histogram of the standard fractionation (solid lines) and hypofractionation (dashed lines) TP1.1 is plotted. Volumes might not sum up to 1 as only doses higher than 1% of the prescribed dose were considered in all histograms.

The inverse dose dependence of RBE caused the RBE significantly greater than 1.1 in the standard fractionation TP1.1 and the RBE significantly lower than 1.1 in the hypofractionation TP1.1 for the prostate cancer patient. The LETd in the standard fractionation plan was slightly higher than in the hypofractionation plan, as the TPs were optimized independently. LETd is a free parameter in the RBE = 1.1 optimization and can therefore vary for similar dose distributions [38]. However, LETd enters the RBE formula in the square root and consequently small LETd variations cause even smaller RBE variations.

The RBE higher and lower than 1.1 for the standard fractionation and hypofractionation regimen differs from the results in [16], but agrees with the observations in [11]. The differences come from the different RBE dependence of the Carabe-Fernandez model on dose compared to the Wedenberg et al. model (see Fig. 3 in [39]). The study by Carabe et al. [16] reported the RWD uncertainties caused by ða=bÞx as an average over 5 patients, whereas the present study presented only 1 patient for each cancer type. For the standard fractionation prostate scheme, the D90 RWD CI in [16] was (+6, +55)% to the RWD1.1, whereas in this study the

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Fig. 5. Representative RWD and LETd distributions of the standard fractionation (a,b,e) and hypofractionation TP1.1 (c,d,f) for the prostate cancer patient applying a constant RBE during TP1.1 optimization. In (a and c) RWD1.1 [Gy(RBE)], (b and d) RWDw [Gy(RBE)] and (e and f) LETd [keV/lm]. The CTV (green), PTV (red) and the rectum (pink) are delineated. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

corresponding CI was (+4, +16)% (only considering ða=bÞx ). As only averaged values were reported in [16], it cannot be determined whether the discrepancies originate in the different models or the averaging over different patients. Nevertheless, in accordance to [16], relative RBE uncertainties in this study decreased with increasing dose per fraction. Here, the relative RWDw CI of D98 was 2% in the hypofractionation TP1.1 for the prostate cancer patient, whereas it was ð10; þ6Þ% in the standard fractionation TP1.1. All these results are based on the assumption of the applicability of the Wedenberg model, which is based on the LQ model to

describe cell survival to irradiation, but complications in OARs are only partially correlated to cell death [36]. For OARs, approaches taking different biological endpoints into account, might be more suitable than the concept of ða=bÞx . Although the LQ model is well established, a recent study proposed a modified LQ model, which results in a slightly reduced RBE [40]. However, this RBE reduction becomes more pronounced with increasing LET and most LETd values in this study were rather low (mostly 2  4 keV/lm, see Figs. 2 and 4). Furthermore, the RBE model is fitted to in vitro cell survival experiments and cells might react differently to ionizing radiation in vivo. There are several models which

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Fig. 6. Biological effect optimized TP (TPBEO) for the nasopharyngeal cancer patient. In (a) the DVH is shown, where the nominal RWDw,BEO is represented with a colored straight line and the total uncertainty band as shaded area. For comparison, the TP1.1 optimized using the constant RBE equal to 1.1 is shown in black. The dashed black line represents RWD1.1 and the solid black line represents nominal RWDw. In (b) RWDw,BEO distribution for a representative CT slice.

Fig. 7. The biological effect optimized TPBEO for the prostate cancer patient. The standard fractionation and hypofractionation TPs are plotted in the left (a and c) and right (b and d) column, respectively. In (a and b) the DVH is shown including the total uncertainty bands. The TPBEO is drawn as colored line, whereas the RBE = 1.1 optimized TP is drawn in black for comparison. RWDw is shown with a solid line, RWD1.1 with a dashed line and the uncertainty band around RWDw as dark shaded area. Graphs (c) and (d) show the RWDw,BEO distribution in a representative CT slice.

try to describe the RBE and there is no proof that one model yields better agreement to the available data than the others. The disagreement between these models by a few percent [39,15] may be interpreted as additional systematic uncertainty, but is partially included in the uncertainty of the fit parameter. The Wedenberg et al. model was chosen for this study on the grounds that there is a statistical analysis which demonstrates that the model represents the fitted data well. However, it has been fitted to a consid-

erably smaller data set than the McNamara et al. model [15]. The Carabe-Fernandez and the Wilkens and Oelfke model have only been fitted to V79 cells whereas the model used in this study was fitted to different cell lines. Since all RBE models are fitted to experimental data, it is expected that all models exhibit uncertainties originating in the fit. The Carabe-Fernandez and McNamara et al. model have two free parameters unlike the Wedenberg et al. or Wilkens and Oelfke model, which have only one.

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Overlapping ROIs with different ða=bÞx values are problematic and it is not clear which value should be assigned. In this study and in [16], the ða=bÞx of the OAR was chosen for the intersections. Consequently, the RBE and RWD at the transitions are discontinuous and problematic in the biological effect optimization. This explains the reduced RWD in the PTV along the boundary to the OARs. The conservative way of assigning the ða=bÞx ratios as investigated in this study is a possibility to account for the ða=bÞx uncertainties during the optimization procedure. But the concept suffers from the lack of ax and bx values to the corresponding ða=bÞx ratio and although we used a reasonable approximation to derive these values, they are still artificial. The LQ parameters ax and bx can only be determined from in vitro cell survival experiments, whereas an effective, cohort-averaged ða=bÞx ratio can also be obtained from clinical studies [41]. Instead of optimizing a cost function of the biological effect, the RWD (i.e., RBE  Dose) can be used. The advantage of the RWD method is that the separate knowledge of ax and bx is not needed. The disadvantage is an increased computational effort (see Eq. 3). Therefore, the gradient of the RBE is typically neglected in the optimization, justified with its assumed small contribution to the total gradient [24]. However, optimizing the biological effect allows us to take the full gradient in the optimization procedure into account and therefore allows for a fast optimization. The RBE depends on the LETd and the dose, thus minimizing the LETd or the dose in an OAR separately does not necessarily result in a lower biological effect. Hence, the biological effect optimization is a possibility to directly account for that and additionally for the sensitivities of different tissue types. Additionally to the uncertainties in RWD, proton therapy suffers from setup and range uncertainties [42]. These uncertainties are typically accounted for in the derivation of the PTV margins [43]. In order to minimize the clinical consequences of these uncertainties, robust treatment planning techniques are currently investigated [44–47]. However, robust treatment planning could also improve by taking biological uncertainties into account. This study showed only 2 exemplary patient cases. For generalization of the results, further studies investigating more patients are required.

5. Conclusion In this study the uncertainties of the RWD estimation in the framework of the Wedenberg model have been investigated for exemplary cases of nasopharyngeal and prostate cancer tumors. For the nasopharyngeal cancer patient, the uncertainty of ða=bÞx dominated. Nevertheless, the uncertainty originating in q was not negligible. For both fractionation schemes in the prostate cancer patient, the uncertainty of either of the two sources of uncertainty was of comparable magnitude. Therefore, neglecting the uncertainty originating in the fit to experimental data would lead to a considerable underestimation of the total uncertainty. Biological modeling can hence benefit by a reduction of both q and ða=bÞx uncertainties. In the CTV, the RWD1.1 was higher than the RWDw in the nasopharyngeal cancer patient, but within the total uncertainty band. Although dose and LETd were comparable in the standard fractionation TP of the prostate cancer patient (low ða=bÞx ), the RWD was significantly underestimated by the RBE = 1.1 approximation. However, in the hypofractionation plan for the prostate cancer patient, RWD was significantly overestimated by the RBE = 1.1 approximation. In contrast to the CTVs, the RWD1.1 significantly underestimated the RWDw in all OARs apart from the optic nerve with respect to the total uncertainties.

In the conservative optimization strategy presented in this paper, the worst case tissue specific ða=bÞx was used in the biological effect optimization. The RWDw to OARs could be reduced while yielding a homogeneous RWDw distribution in the target. Since it is unlikely that the accuracy of the ða=bÞx ratio increases in a short term, a possibility to go beyond RBE¼ 1:1 could be to consider the uncertainties in a robust optimization strategy. Conflict of interest The authors claim no conflict of interest. Acknowledgment This work was supported by the Federal Ministry of Education and Research of Germany (BMBF), Grant No. 01IB13001 (SPARTA), and by the German Research Foundation (DFG) Cluster of Excellence Munich-Centre for Advanced Photonics (MAP). This work was additionally supported by the Bavaria California Technology Center (BaCaTeC). We further acknowledge support by Dr. Kerstin Hofmann (Department of Radiation Oncology, Technical University of Munich) in the usage of CERR and Dr. Barbara Knäusl (Division Medical Radiation Physics, Department of Radiation Oncology, Medical University of Vienna/AKH Wien) for helpful discussions. Appendix A. Dose and LET scoring In the Geant4 simulations, the reference physics list QGSP_BIC_HP was used and the dose in each voxel was calculated according to

Dw ¼

qm Sw D qw Sm m

ð4Þ

where Dm is the dose to material, qm and qw the mass density of the material in the voxel and the density of water (G4_Water) respectively. The unrestricted stopping powers of water (Sw ) and of the material (Sm ) were computed at the proton energy by the Geant4 function ComputeTotalDEDX. The LETd in voxel i (Li ) was the average over all steps Sn for all particles N

XN XSn 2 sn lsn Li ¼ XNn¼1XSs¼1 ; n n¼1

s¼1

ð5Þ

sn

where sn is the energy deposition (GetTotalEnergyDeposit) and lsn the length of step s of event n (GetStepLength). Since the dose-towater concept was used for dose calculation, the energy deposition  was scaled with (SSmw ), which is equivalent to Eq. (4). While the energy deposition of any particle type was scored in dose calculation, only primary and secondary protons were taken into account for LETd calculation [48]. In Geant4, the incident lateral PB shape was modeled as a two-dimensional Gaussian with a FWHM equal to 9.4 mm and the initial energy spread was assumed to be normally distributed with a standard deviation equal to 0.5% of the nominal energy, which is a realistic value in modern proton therapy facilities [49,50]. In CERR, the weights of the PBs were determined by the minimization of a Chi-squared cost function

v2 ¼

X


Di ðwÞ  DT

2

ð6Þ

i2T

with the Matlab/2014a routine fmincon as described in [22]. DT denotes the desired dose in structure T and Di ðwÞ the dose in a voxel i for a current set of M spot weights w ¼ ðw1 ; w2 ; . . . ; wM Þ [21].

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Appendix B. Biological effect optimization In the presence of a set of M beam spots (PBs), the dose in a voxel i (Di ) can be expressed as

Di ðwÞ ¼

M X wj Dij ;

ð7Þ

j

where the Dij matrix contains the dose deposited for unit fluence by beam j in voxel i and w denotes the relative fluence weights of all M beam spots. Accordingly to the Dij matrix the Lij matrix is defined for LETd. Hence the total dose average LET for a set of M PBs in the voxel i can be calculated by

Li ðwÞ ¼

M 1X wj Dij Lij ; Di j

Li ðwÞ ¼ 0;

for Di > 0

ð8Þ

for Di ¼ 0:

ð9Þ

Instead of optimizing for a uniformly distributed RBE weighted dose (RWD), a desired biological effect is optimized. In the LQ model, the biological effect is the negative logarithm of the survival fraction (Eq. (1))  ¼  lnðSÞ ¼ aD þ bD2 . Therefore the desired biological effect in a tissue T (T ) can directly be determined from clinical experience with photon irradiation (here, using the values from Table 1 and 2). A quadratic deviation cost function can be formulated as

F  ðwÞ ¼

X

i ðwÞ  T

2

;

ð10Þ

i2T

using Eq. (2) to calculate the ap and bp parameter. Hence,

F  ðwÞ ¼

X


aTx Di ðwÞ þ q bTx Li ðwÞDi ðwÞ þ bTx D2i ðwÞ  T

2

ð11Þ

i2T

and the first partial derivative

 @F  ðwÞ X
T ¼ 2 ax Di ðwÞ þ q bTx Li ðwÞDi ðwÞ þ bTx D2i ðwÞ  T @wk i2T    aTx Dik þ q bTx Lik Dik þ 2bTx Di ðwÞDik ;

ð12Þ

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