Calculation of relative biological effectiveness for proton beams using biological weighting functions

Int. J. Radiation Oncology Biol. Phys., Vol. 37, No. 3, pp. 719-729, 1997 Copyright 0 1997 Elsevier Science Inc. Printed in the USA. All rights reserved 0360-3016/97 $17.00 + .OO

PII SO360-3016(96)00540-S

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Physics Contribution CALCULATION OF RELATIVE BIOLOGICAL EFFECTIVENESS FOR PROTON BEAMS USING BIOLOGICAL WEIGHTING FUNCTIONS HARALD PAGANETTI, PH.D.,* PAWEL OLKO, PH.D.,* HUBERT KOBUS,+ REGINA BECKER,* THOMAS SCHMITZ, PH.D.,* MICHAEL P. R. WALIGORSKI, PH.D.,*~~ DETLEF FILGES, PH.D.’ AND HANS-WILHELM MILLER-G~RTNER, M.D.* Institutes of *Medicine

IME and ‘Nuclear Physics IKP, Forschungszentrum Jiilich GmbH, Jiilich, Germany; *Institute of Nuclear Physics IFJ, Krakow, Poland; and §Center of Oncology, Krakow, Poland

Purpose: The microdosimetricweighting function approach is usedwidely for beam comparisonstudies.The suitability of this model to predict the relative biologicaleffectiveness(RBE) of therapeutic proton beamswas studied. The RBE, (i.e., linear approximation) dependenceon the type of biological end point, initial proton energy, energy spreadof the input proton beam,and depth of beampenetration wasinvestigated. Methods and Materials: Proton transport calculationsfor a proton energy range from 70 to 250 MeV were performed to obtain proton energy spectraat a given depth. The correspondingmicrodosimetricdistributionsof lineal energy were calculated.To thesedistributions the biologicalresponsefunction approach wasapplied to calculate RBE, the biological effectivenessbasedon a linear dose-response relationship. The early intestinal tolerance assessed by crypt regenerationin mice and the inactivation of V79 cellswere taken as biologicalend points. Results:The RBE, valuesapproachabout 1 in the plateau region and gradually increasewith the proton penetration depth. In the center of the Bragg peak, at the maximum dosedelivery, the valuesof RBE, range from 1.1 (250-MeV beam,early intestinal tolerancein mice) to 1.9 (70-MeV beam,ChinesehamsterV79 cellsin G,/S phase).Distal to the Bragg peak, where only a smallfraction of doseis delivered, the RBE, wasfound to beeven higher. For modulatedproton beamswe found an increasingRBE, with depth in the spread-outBragg peak (SOBP). Values up to 1.37at the distal end of the SOBP plateau (155MeV beam,SOBP between5.3 and 13.2 cm) were obtained. Conclusion:More experimental work on the determination of microdosimetrlcweighting functions is needed. The resultsof the presentedcalculationsindicate that for therapy planning it may be necessaryto accountfor a depth dependenceon proton RBE, especiallyfor lower energy. 0 1997Elsevier ScienceInc. Protons, Bragg peak, Microdosimetry, Relative biologicaleffectiveness.

Medical applications of charged particle beams have significantly increased worldwide during the last 20 years (32). The use of protons with energies between 70 and 250 MeV in cancer radiotherapy offers advantages over radiations conventionally employed. Target volumes in the body can be irradiated more precisely than is possible with photon, electron, pion, or neutron beams. Owing to the finite range and the sharpness of the dose gradients, the dose to well-defined target volumes can be delivered without excessive irradiation of nontarget tissue. In particle radiotherapy with proton beams, the question arises as to what extent the locally increased ionization density should be taken into account in therapy planning. In present practice, a single value of a thera-

peutic relative biological effectiveness (RBE) in the range of l-l.2 is adopted for the whole treatment plan.This RBE is further applied for recalculation of dose for a whole tumor volume, disregarding ionization density distributions in the tumor volume. For therapy planning, however, the variation of the biological dose in tissue should be known within an uncertainty not exceeding 5% (11). It has been argued that the microdosimetric weighting [or biological response (21)] function approach is able to predict the RBE of therapeutic proton beams (10, 16). Pihet, Menzel, Wambersie, and colleagues (20, 24, 35, 36) proposed a method to compare the beam quality of different neutron and proton therapeutic installations using measured microdosimetric distributions of lineal energy, yd(y). These distributions were next applied to calculate beam effectiveness using

Reprint requests to: Harald Paganetti, Ph.D.,Hahn-MeitnerInstitut Berlin GmbH, FI/ATT, Glienicker Str. 100, D-14103 Berlin, Germany.

Acknowledgements-This work was performed within the framework of the Polish-German Agreement on Cooperation in Science and Technology.

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the biological weighting function approach. As the RBE depends on the beam configuration and, therefore, on the energy distribution of the particles, differences can be described by microdosimetric quantities. In the present proton therapy installations, beams are shaped with range modulators (absorbers of dynamically varying thickness) to produce a uniform dose distribution within the target volume. However, the present methods of beam shaping using absorbers have some disadvantages. Scattering and energy loss straggling in the absorber reduce the sharpness of lateral and distal dose falloffs. Alternatively, a dynamic range modulation method may be envisaged in which monoenergetic particle beams of different energies are dynamically extracted over short time periods. Through this, the tumor volume will be irradiated with overlapping narrow monoenergetic pencil beams. In the present study we investigated the distribution of the biological effectiveness in semimonoenergetic and modulated proton beams for different incident proton energies and as a function of depth of beam penetration. The results of beam transport calculations in the form of proton energy fluence spectra at different depths were used to calculate microdosimetric yd( y) distributions. Next, distributions of RBE,, the biological effectiveness based on a linear dose-response relationship and on the microdosimetric single event spectra were calculated using the microdosimetric biological response function approach. The biological end points early intestinal tolerance assessed by crypt regeneration in mice and inactivation of V79 cells were considered. METHODS

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to a specific target-averaged ionization density. No general assumptions about the mechanisms of radiation action are made. When P(y) is normalized to y and to the response of a standard radiation, it is called a radiation weighting function, r(y) (20) or a specific quality function, q(y) (37). The determination of the weighting function rests on the assumption that the information of the single-event spectra d(y) is sufficient to assess the contribution of different radiation components to the overall biological effectiveness. d(y) can be interpreted as the distribution of dose deposited in a population of sensitive volumes with different lineal energy values. The demand for a linear dose-response relationship can be assumed for the zero-dose level [response function for the V79 cells (21)] and for higher dose levels [response function for the intestinal tolerance in mice for the LD50 level (1 S)]. In the latter case, one has to assume that the dose-response relationship is linear around a certain point of the dose-response curve. However, there are human cells where the radiation response is largely determined by the (Y term of the survival curve (i.e., fibroplasts, in particular Ataxia telangiectasia (9)]. In addition, in radiotherapy often doses of about 2 Gy/fraction are applied. Here, the p component is very small for most of the relevant end points (31). The second important assumption for the application of the microdosimetric model is that the information of the single-event spectra yd(y) has to be sufficient. This assumption is strictly only valid for low dose levels. The RBE for a linear dose-response relationship, RBE,i, can be defined as a ratio of slopes of dose-response curves for an investigated, i, and standard, x-ray radiation.

MATERIALS

Biological weighting function approach The biological weighting function approach is a biophysical model broadly applied to predict the biological effectiveness of ionizing radiation (21, 37). Within this approach, it is possible to characterize the biological effectiveness of a given radiation by microdosimetric parameters. The main assumption in the model is that a dose-biological effect relationship can be expressed as an integral of two separate functions of some stochastic variables describing ionization events: d(y), representing the distribution of energy deposited in the target; and P(y), representing the corresponding cellular response function. It is assumed that the dose-response relationship is linear around a given point on the dose-response curve. The biological effect per unit dose, (Y = E/D, can be presented in the following form:

(Eq. 1) P(y) describes, for a given biological end point, the cumulative probability that a subcellular target will respond

The approach can be used in the following way. First, results of radiobiological experiments available in the form of dose-response curves (e.g., survival curves) and obtained for N radiation modalities are fitted with linearquadratic equations to obtain slopes, LYE,of those curves and the corresponding RBEUj. For all radiation modalities used, the d(y) distributions have to be measured or calculated. Then, the following set of equations, which is usually underdetermined, is solved in the least-square sense to obtain r(y): RBE,i

= J r(y).d;(y).dy,+i

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This can be done by describing r(y) with a parametric function r(y) = r(y; al, a*, a,). This method of calculation of r(y) was applied by Morstin et al. (21). They used the following general expression for the response function r(y):

r(y) = a,.[1 - exp(-a,y - a2y2 - a3y3Wy, (Eq. 4)

Calculation of RBE for proton beams 0 H.

where a, gives the action cross section for the effect. The parameters a, and Ui were calcualted for a number of radiobiological end points (21). These response functions were originally not normalized to a reference radiation. For the purpose of this work, they were normalized with a response calculated for experimentally determined y *d(y) spectrum for 6oCo gamma rays (27). Alternative methods for solving Eq. 3 are nonparametric methods. Loncol et al. (18) applied a procedure which modified in an iterative process an initial guess function r(y) to best fit Eq. 3. They determined several weighting functions for different sets of RBE-d(y) data pairs combining high-energy y-rays, protons, and neutrons. Figure 1 shows the weighting function derived by Lonco1 et aZ.(18) normalized to the response of 6oCo y-rays. In addition, the figure contains response functions determined by Morstin et aZ.(21) analyzing results of experiments with V79 cells from Bird et uZ.(5) with different charged particles. Proton Monte-h-10 transport calculations If one neglects the differences in the proton+&ectron collision cross sections, the interaction of protons with tissue can be simulated by using cross sections for water vapor. Proton transport calculations with the Monte-Carlo code PTRAN (4) have been performed for water and for a set of energies between 70 and 250 MeV with small energy spreads (up to a momentum spread of Sp/p = 3 X lo-‘) as available at the Cooler Synchrotron (COSY) in Julich, Germany. The information obtained by PTRAN includes energy loss curves as well as energy spectra of protons at various depths. The program PIRAN accounts for energy-loss straggling in Coulomb collisions with atomic electrons, multiple-scattering deflections due to elastic scattering by atoms, and energy losses in nonelastic nuclear reactions. The inelastic nuclear interactions are treated as entirely absorptive effects. The energy lost by primary protons in Coulomb collisions is mostly transferred to secondary electrons, and to a lesser extent used for molecular excitation and dissociation. As the maximum reachable energy in those collisions depends on the proton energy, the energy transport by secondary electrons has a significant effect on the depth-dose curve only near the entrance face of the phantom. PTRAN disregards the energy transport by secondary electrons. Most of the energy lost by primary protons in nuclear reactions is transferred to secondary charged particles, neutrons, and -y-rays. Seltzer (28) estimated the number and energy spectra of secondary charged particles and neutrons from nonelastic reactions of protons with 160 for some proton energies (The fraction of y-rays was found to be unimportant). We used Seltzer’s results to determine a depth-dose curve from the calculated spectra of energy loss due to Coulomb and nuclear interactions (2). The contribution of nuclear interactions to dose almost vanishes at large depths, especially around the Bragg peak. On average, 90% of the energy is lost in Coulomb interactions.

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Fig. 1. Biological weighting functions r(y) for the biological end points early intestinal tolerance assessed by crypt regeneration in mice (LDSo survival level) (18) and inactivation of V79 cells (GJ S phase; late S phase) (low dose level) (21) for the reference radiation @‘Co.

We consider a narrow pencil beam incident along the z axis onto a water medium. For each initial energy a set of 100,000 Monte-Carlo histories of primary protons was sampled and analyzed. The transport calculations were based on proton stopping powers in water as reported in the ICRU Report 49 (13). The whole Bragg curve has been divided into 45 scoring planes. We decreased the distance between consecutive planes with increasing depth to obtain more detailed information in the Bragg peak region. PTRAN is able to account only for monoenergetic beams. To obtain an incident proton energy spectrum, a number of suitably weighted monoenergetic beams is superposed. Simulation of modulated Bragg curves In the present work an initial spectrum of protons entering the water phantom and resulting in a given, measured, spread-out Bragg peak (SOBP), D&(z), was calculated by minimizing the following

expression for N data points:

where D&) are the calculated doses in a water phantom at depth z, for M proton beams entering the water phan-

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tom. Yk are the searched weighing factors for the individual dose distributions to form Dmod(z). This equation can be solved for a given set of dose distributions and a given SOBP plateau in the least square sense with the condition yk 2 0. For a set of dose distributions based on monoenergetic proton beams with a I-MeV energy difference between adjacent beams, an accuracy of 1% in terms of dose values over the SOBP plateau region, with respect to the average value over that region, can be reached easily. By simulating a measured SOBP using Eq. 5, the experimental method of modulation has to be taken into account. Range modulation and electronic modulation techniques result in different dose distributions at least in the entrance region of the Bragg curve. For an electronically modulated beam, the doses D!+(z) are built by energetically different monoenergetic proton beams. Simulating a range-modulated beam, one has to use incident monoenergetic proton beams, each with the same energy, but penetrating different depths in water (in a first approximation the range shifter can be assumed as consisting of water). In that case, the doses &(z) have to be calculated as Dk+l(z) = &(z - k).

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ary electrons of energies below 1 MeV. Such a secondary electron spectrum is comparable with the secondary electron spectrum produced in water by 6oCo y-rays and thus, the biological effectiveness of 200-MeV protons as compared to “Co y-rays is usually about 1. Twenty-MeV protons deposit 26% of the total dose to the l-pm spherical target by S-electrons. This dose fraction decreases from 0.071 for 5-MeV protons to 0.003 for l-MeV protons (22). Therefore, the &ray effects play only a marginal role in energy deposition of very slow protons.

RESULTS General results Transport calculations in water were performed for a number of monoenergetic proton beams with energies ranging from 70 to 250 MeV (with a step of 0.25 MeV), and the results were stored in a library containing the energy loss curves as well as the proton energy spectra at various depths. In Fig. 2 examples of calculated proton

(Eq. f-3

The depth of the absorbing material z has to be gradually increased by 6z to obtain the searched SOBP. A similar relation is valid for the corresponding initial proton energy spectra &&!Z,Z). Calculation of microdosimetric distributions The computer code PMIC was designed to obtain microdosimetric characteristics of the proton beam, such as the y. d(y) distributions and their moments from the proton energy fluence spectra resulting from the PTRAN proton transport calculations. PMIC reads a proton energy distribution at a given depth in a water phantom and calculates microdosimehic distributions of lineal energy for particles entering a sensitive site (crossers and stoppers) (12). The number of stoppers and crossers traversing the site is determined by the spectrum of protons at the surface of the cavity and the chord length distribution of the target. The lineal energy deposited in the target was calculated with the continuous slowing-down approximation (CSDA) and does not account for S-ray effects and straggling of energy loss. For CSDA calculations, identical stopping power and range tables were applied as available for PTRAN. Homogeneous spherical targets of a uniform sensitivity, 1 pm in diameter, were considered in our calculations. The method of the PMIC calculations is described in detail in Appendix E of the ICRU-Report 36 (12). The contribution of S-electrons to the target dose from protons passing outside the target (called passers or touchers) depends on proton energy and target diameter. For proton energies in the range of 200 MeV, the lineal energy spectra for a l-pm target are mostly produced by second-

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Fig. 2. Calculated proton energy spectra, normalized to one incident proton, for an incident monoenergetic beam with an energy of 150 MeV. To illustrate the respective positions, the Bragg curve is shown in addition. Except at large depths, the primary proton spectra are almost symmetric and have a shape close to a Gaussian distribution.

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energy spectra as a function of depth of penetration are presented for an incident proton energy of 150 MeV. The results from PTRAN are normalized to one incident proton. Except at large depths, the primary proton spectra are almost symmetric and have a shape close to a Gaussian distribution. Proton energy spectra at a given depth, 4(&z), obtained from transport calculation of monoenergetic input pencil proton beams, were applied to compute the corresponding microdosimetric yd(y) distributions as a function of the depth in a water phantom (code PMIC, see above). With increasing depth of penetration, the dose was delivered at higher lineal energies caused by an increase of proton stopping power as protons were slowed down. The dose mean lineal energy, yD, increased from 0.6 keV/pm for a depth of 3.7 cm to 10.1 keV/pm in the Bragg peak (173MeV proton beam). The corresponding values of LET, as calculated by PTRAN were 0.5 and 8.0 keV/pm, respectively. In Fig. 3, a calculated yd(y) distribution in the center of the Bragg peak for a 173-MeV proton beam with an energy spread of Splp = lo-’ is compared with a measured yd(y) distribution. The input proton beam parameters were the same in the measurement and in the calculation The measurement was performed with a half-inch proportional counter filled with propane-based tissue equivalent gas simulating a site diameter of 1 pm. The agreement between the measured and calculated yd(y) distributions is fairly good when keeping in mind that the following neglections were made in the calculations: First, the CSDA was used to calculate the energy deposition in the l-pm sphere. This simplification results in a small shift of the calculated yd(y) spectra to lower y owing to a more narrow microdosimetric fly) spectrum [d(y) = y/ypf(y)]. This effect is illustrated clearly in Fig. 3. Second, no S-ray components were included. As discussed above, the S-ray effects play only a marginal role in the energy deposition of very slow protons. Third, no secondary particles from nuclear interactions were included. These particles may have high RBE values. As shown (2), the contributions from secondary charged particles other than protons increased the LETD significantly only at depths smaller than 0.9 times the CSDA range of the proton, which means at depths slightly smaller than the maximum of the Bragg peak. The influence of the secondary charged particles from nuclear reactions is therefore very small in the region where the LETD increased rapidly, so it can be expected to lead to only a small enhancement of biological effectiveness. However, the contribution of the secondary charged particles to the dose is included in our calculations. This contribution is small in the region of the Bragg peak. In the entrance region, nuclear interactions contribute about 56% to the dose for a 250-MeV beam. For a 70-MeV beam this value is only about 8%. In the Bragg peak this value is in the region of 1% for a 250-MeV beam and much lower for lower initial proton energies. Finally, one has to keep in mind that the proton energy distributions obtained by

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Fig. 3. Calculatedlinealenergyspectrain the centerof the Bragg peak for an incident beamwith an energy of 173 MeV and an energyspreadof 6plp = lo-’ (solidline). Shownfor comparison is a measuredlineal energy spectrumat the samedepth of penetration (dots with error bars).

PTRAN are distributions as a function of depth, thus integrating radially. In the experiment a half-inch counter was used. This effect should be very small. In Fig. 4, the calculated RBE, depth dependence is shown for the pencil beam energies of 70, 100, 150, 200, and 250 MeV and for the considered biological end points. With a decrease of the initial proton energy, the highest reached RBE, value increases, which is connected with a decrease of full width at half maximum (FWHM) of the Bragg peak. The sharp rise of the RBE, values at and beyond the Bragg peak is due to the presence of lowenergy protons. This sharpness decreases with initial proton energy. The rise in RBE, occurs mainly in the Bragg peak region (see Fig. 5) and is shifted toward the end of the Bragg peak as the incident proton energy increases. When y&y) distributions were calculated just for an average proton energy at a given depth, (Q&z)), and not for the whole spectrum, the calculated values of RBE, in the Bragg peak were found to be up to 10% lower. This is due to the asymmetric shapeof the proton energy distributions for greater depths (see Fig. 2).

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Figure 5 illustrates the depth dependence of the RBE, (end point early intestinal tolerance in mice) in the region of the Bragg peak for a 150-MeV proton beam with different energy spreads (momentum Splp = 0 and Splp = lo-‘). When increasing the energy spread, we obtained a decrease of the rise in RBE, and for the dose, besides a broadening of the Bragg peak, a shift of the Bragg peak maximum to lower depths. Whereas the Bragg peak maximum is shifted by 0.34 cm, the RBE, in the Bragg peak maximum decreases from 1.11 to 1.05 for a 150-MeV beam. Figure 6 shows the calculated RBE, values in the Bragg peak for the considered biological end points as a function of initial beam energy for beams without energy spread. The vanishing energy spread is a realistic situation looking at planned experiments at COSY. This accelerator is able to produce proton beams with momentum spreads lower than Splp = 10e3. For the maximum dose in the Bragg peak and the highest considered initial proton energy of 250 MeV, the calculated RBE, values varied from about 1.1 for the early intestinal tolerance assessed by crypt regen-

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Fig. 4. Dependence of the RBE, on the depth in a water phantom for monoenergetic proton beams of 70, 100, 150, 200, and 250 MeV. Both the maximum RBE, and the steepness of the RBE, increase are seen to decrease with increasing proton energy. The end points shown are the early intestinal tolerance assessed by crypt regeneration in mice (a), the inactivation of V79 cells in G1/S phase (b) and in late S phase (c). See Fig. 5 for a comparison of the RBE, and dose distribution.

eration in mice to 1.35 for the end point inactivation of V79 cells in G,/S phase, and to 1.25 for the end point inactivation of V79 cells in late-S phase. For the lowest considered initial proton energy of 70 MeV, the respective RBE, values were of about 1.3, 1.9, and 1.6. The RBE increase with decreasing initial proton energy is caused by lower straggling of energy loss for slow protons and consequently lower FWHM of the Bragg peak. The most probable proton energy in the Bragg peak maximum is about 20 MeV for a 250-MeV beam and about 6 MeV for a 70-MeV beam. The low-energy protons, with stopping powers above 10 keV/pm, contribute the most to the RBE increase because the biological weighting functions increase above that y limit and a part of the lineal energy distribution overlaps. In this article, the biological dose is defined as a product of the dose and the considered RBE. In Fig. 7, dose and biological dose (RBE, values were taken for early intestinal tolerance) produced in water by monoenergetic 150MeV protons are presented. The increase of the RBE, with depth results in a shift of the biological dose distribution to larger depths in comparison to the physical dose distribution. The shift of the biological dose is about 0.5 mm for the 70-MeV beam and increases with initial proton energy to about 0.9 mm for the 250-MeV beam. These

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SOBP decreases with increasing depth position of the SOBP. Figure 8 shows the calculated incident proton energy distributions on the water phantom which are necessary to build the SOBPs shown in Fig. 9. This figure gives the SOBP for the dose and for the biological dose for two end points considered. The biological dose has been obtained by adding the biological doses for the individual beams with the weightings used for building flat SOBPs for the physical dose. The effect of increasing biological dose with depth in the SOBP is greater for lowenergy proton beams. Figure 10 gives the calculated and the measured RBE values as a function of depth. From the front to the rear of the extended Bragg peak, we found an increasing RBE,. For the low-depth SOBP the RBE, increases up to 1.27 (V79 late S) and 1.37 (V79 Gr + S) (distal end of the SOBP plateau). For the medium-depth SOBP these values are 1.18 and 1.27, respectively. For the high-depth SOBP we found 1.16 and 1.24, respectively. This is due to a decrease in the relative proportion of high-LET contributions to the dose from slowing down protons. A comparison of the calculated and measured RBE values is difficult because Robertson et aZ. published only RBE values relative to the average RBE in the SOBP region. Second, their values are obtained at a survival level

Inactivation of V79 cells (Gl+S) Inactivation of W9 cells (late S) Early intestinal tolerance in mice

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values were calculated at the FWHM position of the biological dose distribution and for the end point early intestinal tolerance. Consequently, using an average RBE results in an underestimation of the effectiveness of the beam at the end of its range. Comparison with biological experiments The results of a measurement on the proton RBE in SOBPs for the inactivation of V79 cells were published by Robertson et al. (25). For comparison we calculated three range-modulated Bragg curves using the method described above (Eqs. 5 and 6) replacing the experimentally used rotating stepped polycarbonate wheel by water of equivalent proton range. In the calculations, a thickness step of the degrading material of 1.5 mm was used. Following Robertson et aZ., the position of the three used modulated Bragg peaks are given as the positions of 90% dose, proximal and distal, and are 5.3; 13.2 cm, 13.0; 21.1 cm, and 23.1; 3 1.4 cm with proton beam energies of 155, 200, and 250 MeV, respectively. The calculated SOBPs are in accordance with the measured ones with a maximum difference of 1% in dose. As the pencil beam Bragg peak broadenswith increasing initial proton beam energy, the amount of individual beams necessary to build the

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plateau region to be 1.44 -+ 0.08 which is about 10% below our calculated values for RBE,. In addition, there are several experimental studies on the behavior of RBE as a function of depth of penetration using different biological end points (6, 8, 17, 19, 26, 30, 33, 34). They all found an increase of the proton RBE with depth especially for low-energy beams.

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Statistical and systematic errors Statistical and systematic errors occur in the MonteCarlo calculations to determine the dose distributions and proton energy spectra, and in the microdosimetric calculations to obtain the yd(y) spectra. In addition, there are uncertainties in the weighting functions. The statistical errors for the dosedistribution dD/dz determined by the Monte-Carlo code PTRAN are smaller than l%, independent of the initial proton energy, for depths smaller than 1.02 times the CSDA range of the proton beam, which is slightly behind the Bragg peak (3). For greater depths this error rises to 6%. These values are valid for 1,OOO,OOO proton histories. The systematic errors are estimated to be 2-3% (3). These systematic errors are due to the uncertainties of the stopping-power data, the uncertainties in the nuclear interaction cross sections, and the approximations in the PTRAN code (3). The uncer-

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of lo%, whereas the responsefunction which we used is based on low doses.Following the publication of Bird et al. (5) from which the V79 response functions are obtained, the calculated RBE values could be clearly lower when using a survival level of 10%. For 4-MeV protons Bird et al. found that the RBE is higher by about 30% and 60% at low doses (ratio of the LYterms of the survival curves) in comparison to the 10% survival level for V79 late-S cells and Gr/S cells, respectively. Robertson et al. used a heterogeneous cell population in their measurement. The cell responsefunctions we used from Morstin et al. (21) are determined for synchronized cell populations. An average sensitivity for a population of cells can be determined by different distributions of sensitivity among its components. In addition, the contribution of the certain cell cycle to the overall sensitivity is dose dependent (29). As we have not included the increased RBE of heavy secondary particles, the calculated values do not show an enhanced RBE at low depths as do the measured ones. For the inactivation of Chinese hamster cells irradiated with a proton beam of lOO-MeV, Wainson et al. (34) found the RBE in the Bragg peak in comparison to the

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Fig. 10. Calculated absolute RBE, values for the inactivation of V79 cells for modulated proton beams (lines, left scale; lowdose approximation; response function approach). The dots are experimentally determined relative RBE values at survival S/S0 equal to 0.1 (25) (right scale). See Fig. 9 for the SOBPs.

tainties in the proton energy distributions are comparable, except for depth far behind the Bragg peak where the number of events decrease drastically. Simplifications in the determination of the microdosimetric yd(y) spectra were discussed above. The most important error source is the usage of the CSDA within our calculations, because this results in a shift of the yd(y) distribution to lower y (see Fig. 3). As a result of the shape of the weighting functions, this affects a small underestimation of the RBE, values in the region where the gradient of r(y) is high. This effect is negligible for very small proton energies (< 10 MeV), where the CSDA is permitted (l), and for very high proton energies, where the weighting function is flat. Consequently, this underestimation of RBE, takes place mainly in the Bragg peak region. The credibility of the weighting functions depends on the experimental input data (microdosimetric and biological) from which they were obtained. In our case, the RBE calculations should have an accuracy of about 3% (18) (early intestinal tolerance in mice) and of about 10% (21) (inactivation of V79 cells) for the range of yd(y) values in our calculations.

DISCUSSION We calculated the proton RBE, for the biological end points early intestinal tolerance assessed by crypt regen-

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eration in mice and inactivation of V79 cells for different cell-cycle stages, using biological response functions from the literature. As the microdosimetric response function model is applied to the intercomparison of neutron beams, it becomes important as a tool for an intercomparison for proton therapy installations as well. We conclude from our calculations that RBE values depend on the biological end point, initial proton energy, depth of beam penetration, and initial energy spread of the input proton beam. The RBE, increases with depth in a water phantom. This rise decreases with increasing beam energy. Furthermore, the depth dependence of the RBE, results in a shift of the biological dose distribution to greater depths in comparison to the physical dose distribution. Use of an average RBE therefore underestimates the effectiveness of the beam at the end of its range. One should be aware of the biological extension of the peak in those applications where exact positioning of the end of the beam is required. This effect increases with decreasing initial proton energies. Even for modulated proton beams we found an increase of the RBE, values with depth in the SOBP. For an SOBP built by a 155MeV beam at a depth between 5.3 and 13.2 cm, the RBE, increases up to 1.27 (V79 late S) and 1.37 (V79 G1 + S) (distal end of the SOBP plateau). For a SOBP built by a 250-MeV beam at a depth between 23.1 and 3 1.4 cm, these values are 1.16 and 1.24, respectively. The gradient along the SOBP implies that despite a homogeneous physical dose, the distal end of a irradiated volume receives a higher biological dose than the proximal end. Our major interest was to clarify whether the response function approach is able to predict the RBE of proton beams. If it can, measured microdosimetric lineal energy spectra could thus be used for beam comparison studies. The model is strictly applicable only where only linear terms contribute to the survival curve and where the microdosimetric single-event spectra can be used. However, there are certain end points and dose rates where the (Y term is sufficient to describe the radiation response (9,3 1). The accuracy of our calculated data is highly dependent on the accuracy of the biological weighting function used. Error sources for these functions are the biological and the microdosimetric measurements. More accurate RBE values are required to determine the exact peak region of the weighting function. The calculated RBE is very sensitive to that region because of its overlap with the y. d(y) spectra in the region of the Bragg peak. Furthermore, it is interesting to unfold biological weighting functions at different survival levels and for other end points, especially for those which can be verified in radiobiological experiments. When the response function is properly determined, the RBE for any new proton beam could be derived for the considered biological systems, provided the microdosimetric spectra had been measured. Up to now, there has been a lack of sufficient and adequate radiobiological data suited for the analysis to

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obtain a weighting function by unfolding procedures. For most of the radiobiological experiments, no adequate characterization of the radiation field is available. Biological and dosimetric experiments at COSY in Jtilich, Germany, are planned to contribute to the solution of this problem. In addition, it has to be proven that a single response function is applicable to the whole range of initial proton energies presented here. As long as a single RBE value is still used in therapy planning for proton beams, more detailed investigations on the depth dependence of the RBE have to be done, especially to answer the question of whether this depth dependence should be taken into account in therapy planning.

Volume 37, Number 3, 1997

To continue our studies on the proton RBE, we started to do calculations with the track structure model of Katz and Sharma (7, 14, 15) and to compare our results with the predictions from this biophysical model (23). This model starts from completely different assumptions, e.g., from a zero initial slope of the survival curves. Calculations with a newly designed Monte-Carlo code with a detailed treatment of nuclear interactions are planned, to look for the influence of secondary particles on the RBE. Although the ratio of energy loss due to nuclear interactions is small, these interactions may produce heavy charged particles with a high RBE, especially at high initial proton energies and in the entrance region of the Bragg curve.

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