Nuclear Instruments and Methods in Physics Research B

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1 Nuclear Instruments and Methods in Physics Research B 69 () Contents lists available at ScienceDirect Nuclear Instruments and Methods in Physics Research B journal homepage: Depth-dose distribution of proton beams using inelastic-collision cross sections of liquid water C. Candela Juan, M. Crispin-Ortuzar, M. Aslaninejad Imperial College London, Blackett Lab, High Energy Physics Department, Prince Consort Road, London, United Kingdom article info abstract Article history: Received 3 July Received in revised form 8 September Available online 7 October Keywords: Bragg peak Proton therapy Complete physical processes contributing to the shape of the depth-dose distribution and the so-called Bragg peak of proton beams passing through liquid water are given. Height, width and depth of the Bragg peak through investigation of collisions of the projectile, nuclear processes and stochastic phenomena are discussed. Different analytical models for stopping cross sections which lead to Bragg peak are contrasted. Results are also compared with GEANT4 package simulation. Ó Elsevier B.V. All rights reserved.. Introduction One of the most important types of cancer treatments to control malignant cells is radiotherapy. A certain amount of ionising radiation is applied to the patient body, which deposits energy into the tissues. This may result in the disruption of DNA and the subsequent death of the cells. Radiotherapy research strives for higher precision and biological effectiveness of the applied dose. This minimises the damage to the healthy tissues. In the developed countries, every year about, oncological patients are treated with high-energy X-rays photons []. Usually, radiotherapists use electron linear accelerators (linacs) as sources of X-ray radiation. Despite this being so widely used, the absorbed dose actually reaches a maximum very rapidly after hitting the body and decreases slowly in an exponential manner. Thus, using X-rays is not an efficient way to reach the deep seated tumours. In addition, not only at the entrance of the body, but also after the tumour, healthy tissues are irremediably damaged, as the photons traverse our organs. To quantify the effects of ionising radiation it is important to know how its energy is distributed linearly. The linear energy transfer (LET) is a measure of the energy transferred to a material per unit length as an ionising particle travels through it. For photons, it shows a poor distribution, but protons and other charged particles show a very different depth-dose curve. They produce the so-called Bragg peak, a sharp increase in the deposited energy in the very last region of their trajectory, where they lose their whole energy and the deposited dose falls to zero. Hadron therapy uses collimated beams of hadrons to improve targeting and Corresponding author. addresses: m.aslaninejad@ic.ac.uk, morteza@theory.ipm.ac.ir, m.aslaninejad@imperial.ac.uk (M. Aslaninejad). achieve a larger effectiveness. When protons pass through a medium, they experience Coulomb interactions with the orbiting electrons and nuclei of the atoms in the medium. Electrons may be released from the atomic shells and the atoms become ionised. The freed electrons damage the DNA of the cells and can sometimes ionise other neighbouring atoms. As the electrons mass is small compared to that of protons, the protons lose in this process a very small amount of energy in a non-linear manner. In addition, protons may be involved in other inelastic processes, such as excitation of the medium or charge transfer. All these processes should be addressed in detail to gain a precise knowledge about the behaviour of proton beam in interaction with the patient body. The rate of change of energy as a function of traversed material is given by the Bethe-Bloch formula, which according to Groom and Klein [] is: " # de q dx ¼ Z Kz A b ln m ec b c T max b dðbcþ ðþ I where z shows the charge of the incident particle, Z and A are the atomic number and the atomic mass of the absorber, respectively. I is the mean excitation energy, dðbcþ is the density effect correction to ionisation energy loss and b and c are the usual relativistic factors. T max is the maximum kinetic energy which can be imparted to a free electron in a single collision. It is important to note that the deposited energy is inversely proportional to the square of the velocity of the particle b. Upon entering the medium, the protons have maximum energy and therefore, maximum velocity, but as they traverse the medium they interact with the orbiting electrons, lose energy and velocity and deposit more and more dose. Physically, the dependence on v arises from the time needed for a Coulomb interaction to take place. If the particle moves faster, there X/$ - see front matter Ó Elsevier B.V. All rights reserved. doi:.6/j.nimb...3

2 9 C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 69 () is less time for the electric fields of the projectile and the atoms in the medium to interact, and thus less energy is deposited. Although it is claimed that proton therapy is very efficient, this is only true if the position and the dose can be calculated very accurately. The greatest advantage of proton therapy, that is, the concentration of the deposited energy on a small region of the space, may be lethal if there is even a very slight shift in the position of the Bragg peak. Despite the fact that, one can obtain the right position of the Bragg peak via Monte Carlo based simulation packages like GEANT4, the full understanding of the whole processes contributing to the right height, width and position of the depth-dose distribution requires analytical and/or semi-empirical models. The present study investigates the physical processes involved in the passage of protons through matter, to study the accuracy in the determination of the position of the Bragg peak. We follow the same techniques used by Dingfelder et al. [3] to study the cross section for inelastic interactions of energetic protons with liquid water. These cross sections are used for determining the linear energy transfer of ions against the depth of their penetration according to the approach introduced by Obolensky et al. [4] and Surdutovich et al. [5]. While, in Refs. [4,5], the main study is on carbon ions, we focus on proton ions. Armed with inelastic cross section models and data through Ref. [3], we can in detail account for the different processes to which protons contribute, and study the contribution and effects of each process. We expect to establish a stable set of physical contributions which describes the depth-dose distribution of a certain projectile. Previous works in this direction have provided different models for the cross section terms, without obtaining the final curve for the linear energy transfer. Moreover, the literature does not combine the semi-empirical models for proton inelastic processes with effects like straggling and nuclear fragmentation, which we do here. In the current study, we analyse the effects of the addition of each contribution to the position, height and width of the Bragg peak, and compare with the Monte Carlo results obtained using GEANT4 [7] v.9.3, with the standard QGSP BIC physics package. To summarise, we try to provide a deep and thorough analysis of the depth-dose distribution of protons through liquid water through the basic physical phenomena. The paper is organised as follows. In Section we review and reproduce the inelastic processes that contribute to the total cross section for protons passing through liquid water. To this end, we follow the approach and use data mainly from Refs. [3,6] to obtain cross sections for processes like ionisation, excitation, stripping and charge transfer. Through calculation of the total stopping cross section and the linear energy transfer (LET) [4,5] for these phenomena we attempt to find the right position of the Bragg peak. On the other hand, particle fragmentation and stochastic processes like straggling, which characterise the height and the width of the peak are also discussed in this section. In Section 3, we compare the calculated stopping cross section using the foregoing Rudd model [6] with a more precise result obtained from Bethe-Bloch equation () for high energies. Then we discuss the validity of different approaches to find the right position of the peak. The results of our study are contrasted with GEANT4 simulations. Finally, we present a brief conclusion of our study.. The passage of protons through liquid water.. Cross section of inelastic processes We first review the passage of a single proton through liquid water in detail. When the energetic ion enters the medium, inelastic cross sections are typically small, but they grow to reach a maximum, namely the Bragg Peak, where the ion loses its energy at the highest rate. For our analysis, we follow mainly the approach Table Inelastic interactions of the projectile with the medium resulting in different contributions to the stopping cross section [3]. Contribution Excitation by the projectile Ionisation by the projectile Charge transfer Stripping of Hydrogen Ionisation by Hydrogen from Dingfelder et al. [3]. We initially consider impact ionisation, which is the dominant process contributing to the energy loss and resulting in the production of the secondary electrons, and then add several other terms; namely excitation, charge transfer, stripping and ionisation by hydrogen. Table gives an outline of the next steps. The secondary electrons are not produced in the excitation process. The charge transfer phenomenon consists of the transfer of electrons from the water molecules to the projectile in such a way that it ends up becoming neutral. When the projectile is a proton, which is the case being considered in this study, charge transfer involves the transfer of one single electron. The hydrogen atom formed can also ionise the medium or can become stripped, in which case it loses the previously absorbed electron.... Ionisation by protons The singly differentiated cross section, SDCS, is the basic quantity in the analysis of the ionisation process. It is the total ionisation cross section differentiated with respect to the energy, W, of the ejected electrons and is dependent on the kinetic energy of the projectile, T. We use a semi-empirical expression introduced by Rudd and co-workers [6], which was derived by adjusting the experimental data to Rutherford cross section, among other analytical models, drðw; TÞ dw ¼ X z i 4pa N i B i R B i F ðv i ÞþF ðv i Þx i ðþ ð þ x i Þ 3 ð þ expðaðx i x max i Þ=v i Þ where the sum is taken over the five electron shells of the water molecule, a ¼ :59 nm is the Bohr radius, R ¼ 3:6 ev is the Rydberg constant, N i is the shell occupancy, B i is the ionisation potential of the sub-shell i for water vapour, x i ¼ W=B i is the dimensionless normalised kinetic energy of the ejected electron. The dimensionless normalised velocity, v i, is given by [5] sffiffiffiffiffiffiffiffiffiffi mv v i ¼ ð3þ B i where m is the mass of the electron and V is the velocity of the projectile, which in the relativistic case is given by bc, sffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼! u Mc ¼ t ð4þ c Mc þ T M being the mass of the projectile. Eq. (3) uses the kinetic energy of an electron having the same velocity as the projectile, normalised by the corresponding binding energy. The expressions for the functions in Eq. () are F ðvþ ¼A lnð þ v Þ F ðvþ ¼C v D w max i B =v þ v þ A v þ B C v D C v D þ4 þ A v þ B ¼ 4v i v i R 4B i Reaction p þ H O! p þ H O p þ H O! p þ H O þ þ e p þ H O! H þ H O þ H þ H O! p þ e þ H O H þ H O! H þ H O þ þ e þ E v D þ4 ð5þ ð6þ ð7þ

3 C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 69 () Table Parameter sets for the SDCS of protons on water (from Refs. [3,6]). Parameter Outer shells Inner shell Liquid Vapour K-shell A B C D.8.3. E A.7.4. B C D.4.4. a Eqs. (5) (7) show original non relativistic expressions for Rudd model. V is calculated classically as T=M. The parameter set for the protons SDCS used in this work can be found in Table and were calculated by Dingfelder and co-workers [3]. Surdutovich et al. [5] introduce some modifications on F which account for the relativistic effects due to the high energies reached (up to b :6), ln F rel ðvþ ¼A þv b b B =v þ v þ C v D þ E v D þ4 ð8þ When we expand the Rudd s model to relativistic energies, we note that F and F tend to A lnðv Þ=v and A =v, respectively. Since F approaches zero faster, F absorbs the relativistic corrections. This results from the fact that as A A, for high velocity ðv Þ, F is always higher than F. In particular, the new F rel introduced by Surdutovich and co-workers [5] reproduces the same asymptotic behaviour as Bethe-Bloch formula. In this work, the Rudd model with relativistic corrections has been employed to account for the ionisation process. The parameter set for the SDCS of protons on water vapour is also given in 99 by Rudd et al. [6] and are also shown on Table. The division into inner and outer shells that Rudd suggests for choosing one or another set of parameters has been accepted. But, we followed the approach from Ref. [3], which consists of separating the K-shell from the rest, as seen on Table 3. In their investigation, Rudd and co-workers reviewed the different approaches for the total cross sections and suggested this semi-empirical approach. According to Ref. [3], the latter does not reproduce accurately the stopping cross section for liquid water due to the lack of a careful consideration of the water molecule s sub-shells. To overcome this, they suggest adding a partitioning factor G j which adjusts the contributions of the sub-shells to the results from the Born approximation. dr dw ¼ X dr j G j dw allj j Table 3 Ionisation energies (ev) for the different sub-shells in the water vapour (B j ) and liquid water (I j), the partitioning function G j and the number of electrons N j. The data are taken from Ref. [3]. Shell Outer shells Inner shell (K-shell) a b 3a b a I j B j G j N j ð9þ In this work, we reproduce the previous expressions for liquid water from Dingfelder et al. [3] directly, who obtained the parameter set for liquid water by modifying the water vapour set until the equations reproduced the recommended ICRU stopping cross section for the desired material [8]. It must be noted that Rudd and co-workers developed their model for vapour water, and took the binding energies for this phase as parameters. Hence, even though our approach is for liquid water, the binding energies used in the SDCS (Eqs. (), (3) and (7)) are those from water vapour. By integrating the SDCS over the secondary electron energy, W, the total cross section of impact ionisation by the ion with kinetic energy T is obtained: rðtþ ¼ Z Wmax drðw; TÞ dw dw ðþ The maximum kinetic energy is W max;i ¼ T I i ðþ Note that in this case, the binding energy corresponds to that of liquid water.... Excitation by protons For the case of excitation of the molecules of liquid water, we use a semi-empirical model which relates proton excitation cross section to electron excitation cross section, developed by Miller and Green [9]. The cross section for a single excitation of an electron in sub-shell k is given by the expression: r exc;k ðtþ ¼ r ðzaþ X ðt E k Þ m J Xþm þ T Xþm ðþ where r ¼ m, Z = is the number of electrons in the target material and E k is the excitation energy for the state k. a, X, J and m are parameters (shown in Table 4), but they have physical meaning: a(ev) and X (dimensionless) represent the high energy limit and J (ev) and m (dimensionless) the low energy limit. We have taken the values recommended by Dingfelder and his group, who in turn used the original suggestions in Ref. [9]...3. Charge transfer There exists the possibility that an electron from the water joins the projectile, the proton in this case. A neutral hydrogen atom is then formed either in a bound state or in a state in which the electron travels with the proton at the same velocity. This phenomenon is known as charge transfer. Again following Dingfelder s approach [3], who showed the contribution of the charge-transfer cross section by an analytical formula: r ðtþ ¼ YðXÞ where X ¼ logðtþ with T measured in ev and YðXÞ ¼½a X þ b c ðx x Þ d HðX x ÞŠHðx XÞ þða X þ b ÞHðX x Þ ð3þ ð4þ where HðxÞ is the Heaviside step function and the other parameters can be found in Table 5. These parameters were obtained by considering the experimental data on water vapour [ ] as a starting point and adjusting Table 4 Set of parameters for the excitation cross section for liquid water, taken from [3]. k Excited state E k (ev) a (ev) J (ev) X t A e B B e A Ryd A + B Ryd C + D Diffuse bands

4 9 C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 69 () Table 5 Parameter set for the charge transfer cross section, taken from [3]. a b c d a b x x them to fit the contribution of the charge transfer to the total cross section in the recommended values for liquid water. The subscripts, in the charge transfer cross section mean that the projectile goes from the state to the state, where corresponds to a proton and to a neutral hydrogen. total cross section (m ) 8 9 Ionisation (p) Excitation (p) Charge transfer (p) Ionisation (H) Stripping (H)..4. Stripping of neutral hydrogen The neutral hydrogen atom can undergo stripping or electronloss. As before, not many experimental data exist for liquid water. Hence we decided to use a semi-empirical formula given by Rudd et al. [3] for the cross section. In this expression the harmonic mean of a low energy and a high energy part is considered. Parameters used are those obtained by Dingfelder and his group to fit the recommended values for liquid water. r ðtþ ¼ þ : ð5þ r low r high The two parts of the stopping cross section may be calculated as D r low ðtþ ¼4pa C s R r high ðtþ ¼4pa R h A ln þ s i þ B s R ð6þ ð7þ where A, B, C and D are fitted parameters as given by Table 6. R = 3.6 ev is the Rydberg constant and s is the kinetic energy of an electron travelling at the same speed as the proton...5. Ionisation by neutral hydrogen The last effect to be taken into account is the ionisation by neutral hydrogen. Despite of the lack of direct experimental information on this process, Bolorizadeh and Rudd [4] concluded that the ratio of the SDCS for hydrogen impact to the SDCS for proton impact as a function of the energy of the secondary electron is independent on this parameter when the projectile energy is fixed. We can then use the SDCS for protons as the starting point and correct it by a factor gðtþ that depends only on the particle energy T (ev), that is: dr de hydrogen ¼ gðtþ dr de proton ð8þ where the correcting function is adjusted by Dingfelder and his group to fit the values for liquid water provided by ICRU 49 [8]. gðtþ ¼:8 þ exp logðtþ 4: þ :9 ð9þ :5.. From the total cross section to the LET The total cross sections for the different processes considered above for protons in liquid water are plotted in Fig.. From this figure, it can be observed that the charge transfer cross section is Table 6 Parameter set for the stripping total cross section, taken from [3]. A B C D higher for low energies of the projectile but it falls to zero very fast for energies around. MeV. It is then expected that only for low energies, the probability to find the projectile as a neutral hydrogen becomes important.... Probabilities of charge state and stopping cross section The combination of the contributions to the total stopping cross section has to be done in such a way that it takes into account the state of the projectile. In the case of a proton, we have just considered two possible charge states: neutral hydrogen (H, i = ) and proton (p,i = ). The total stopping cross section will be the sum over all charged states i of the particle [3], r st ¼ X U i ðr st;i þ r ij T ij Þ ðþ i;j;i j where the term T ij denotes the energy loss in the charge changing process, from state i to state j. Only one electron will be ripped out from the water molecule in the charge transfer process, and so we can suppose the most probable sub-shell to undergo the reaction is that with the smallest binding energy. This way we can establish a lower bound for the sum T þ T as the ionisation energy of water I plus the kinetic energy of the electron, T þ T ¼ I þ mv ðþ and the U i is the probability that the projectile is in state i. This is determined by the charge transfer and the stripping cross section: r U ¼ r þ r r U ¼ r þ r Eq. () may now be rewritten as r st ¼ U r st; þ U r st; þ r st;cc where r st;cc ¼ r r r þ r ði þ sþ r st;i ¼ Incident particle energy, T (ev) Fig.. Total cross sections of the considered inelastic processes of protons in liquid water: ionisation by protons and by neutral hydrogen, excitation by protons, charge transfer and stripping. The stopping cross sections are calculated as Z Emax E dr iðe; TÞ de de ðþ ð3þ ð4þ ð5þ ð6þ where E max is the maximum energy loss by the projectile in the corresponding inelastic process.

5 C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 69 () Stopping cross sections (ev m ) Fig. shows that from all the inelastic processes considered, for high particle energies, only proton ionisation is noticeable. Approximately below.3 MeV contributions from the other processes appear and for kev, ionisation by neutral hydrogen becomes the most important process, followed by the charge changing process. Note also that the energy loss by excitation is nearly negligible. The total stopping cross section is compared with the PSTAR values [5]. The reference data base for Stopping power and range tables for protons (PSTAR) is provided by the National Institute of Standard and Technology (NIST) of the US department of Commerce. These values are obtained, for high energies, through Bethe s stopping formula including shell corrections, Barkas and Bloch corrections [6,7] and density effect corrections. The mean excitation energy, I, used in PSTAR is taken from ICRU 37 [8], corresponding to I = 75 ev for liquid water. For low energies, experimental data in PSTAR are fitted to empirical formulas. By definition, the linear energy transfer is proportional to the stopping cross section, dt dx ¼ nr stðtþ ð7þ where n ¼ 3:343 8 molecules=m 3 is the number density of water molecules. The position of the projectile as a function of its kinetic energy is obtained by integrating the inverse of LET [4,5], xðtþ ¼ x 8 Z T T dt jdt =dxj Total Ionisation (p) Charge changing Ionisation (H) Excitation (p) PSTAR values Kinetic energy of the projectile, T (ev) Fig.. Total stopping cross section of protons through liquid water. ð8þ where T is the initial energy of the projectile. The parametric representation of LET versus depth is known as the depth-dose distribution. This process, used to calculate the energy transfer, is known as Continuous Slowing Down Approximation (CSDA). In this approximation, the projectile is assumed to lose energy in a continuous way. However, energy loss by a proton is a stochastic process, which produces variations in the range of the particles. In addition, the projectile might react with a water molecule and become fragmented. Then, for a real model we need to include two important contributions: straggling and nuclear fragmentation.... Straggling and fragmentation Straggling accounts for the stochastic fluctuations in the range of the particles which traverse the medium. The straggling model considered here claims that the ranges of the projectiles follow a Gaussian distribution. The so-called broad-beam depth-dose curve results from the convolution of the previously CSDA LET with this Gaussian range straggling distribution. The concept is known as depth scaling and was introduced by Carlsson et al. [9]. For each point x in the trajectory, it is assumed that every particle in the beam has an effective penetrated depth x. These effective depths are distributed as a Gaussian centred on the actual penetrated depth x, which coincides with the CSDA range,! pðx; x Þ¼pffiffiffiffiffiffi p rstrag ðxþ exp ðx xþ ð9þ r stragðxþ where r strag ðxþ is the variance, which may be obtained from a phenomenological formula given by Chu et al. [] and depends in general on x. Nevertheless the shape of the depth-dose distribution allows us to assume that it is approximately constant, with rðxþ rðr Þ [], since most of the curve is approximately flat and will not be significantly affected by the convolution. The expression for r strag ðxþ is then r strag ¼ :x :95 A :5 ð3þ where A denotes the ion s mass number []. Straggling has two important consequences on the depth-dose distribution. Firstly, as fewer particles arrive to the same final position, the dose in the peak is considerably reduced. In addition, different particles have different ranges, and thus the peak is broadened. Nuclear fragmentation accounts for the possibility that when the projectile hits a target, it undergoes fragmentation. Two different effects need to be considered. The first one is the reduction of the beam fluence, lowering the height of the depth-dose distribution at the peak. On the other hand, fragments can be produced in the same direction of the original beam. If lighter elements are formed, as having a longer range, they produce a tail in the depth-dose distribution which goes clearly beyond the position of the Bragg peak maxima. For protons, fragments are not important and so a tail is not observed. Hence, in this work we only include fragmentation by considering the reduction in the number of particles of the beam, N. This model deals with a pencil beam with a single projectile, so the concept of fluence loses its meaning within this context (it would be a Dirac delta function). Instead, the fragmentation effect is considered through the decay of the total number of particles. If the projectile has probability p to react with a water nucleus, then: dnðxþ ¼ pnðxþ dx NðxÞ ¼N e px ¼ N e x k : ð3þ ð3þ where k is called mean free path length and N is the initial number of incident protons. From the prediction of Ref. [], we have k ¼ 435 mm, for protons in liquid water. The broad-beam depth-dose curve considering fragmentation and straggling is therefore calculated as dt dx ðxþ ¼ NðxÞ Z R pðx; x Þ dt dx dx ð33þ The results of adding fragmentation and straggling are illustrated on Fig. 3. Different depth-dose distributions for various initial energies have been compared on Fig. 4. It can be observed that not only the height of the peak, but also the entrance dose depends on the initial kinetic energy of the projectile. Protons with higher initial energies deposit less dose in the entrance since the cross section increases as the velocity of the projectile decreases. In addition, projectiles with a longer range suffer more straggling and nuclear fragmentation and hence their Bragg peaks are shorter and wider.

6 94 C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 69 () CSDA CSDA + straggling CSDA + straggling + fragmentation 6 5 GEANT4 This work (using PSTAR cross section) This work (using the simulated cross section) LET (AU) 3 LET (AU) 4 3 Linear Energy Transfer, LET (AU) Depth, x (mm) Fig. 3. Depth-dose distribution of protons in liquid water for T = MeV, considering the CSDA and adding straggling and nuclear fragmentation Depth, x (mm) There is one additional process which has not been considered, multiple scattering, which accounts for the divergence of the beam as it travels along the matter. It has been shown that multiple scattering produces changes on the shape of the Bragg peak [3], especially on the amplitude and the peak to entrance dose ratio, although not on the range. Usually, hadron therapy simulations merely apply normalisation procedures to fit the experimental data [4], as the effect is only due to a reduction of the fraction of primary particles. To compare with experimental data, the current model normalises the depth-dose distribution to the experimental entrance dose and so scattering is not included. 3. Results and discussion 8 MeV MeV 5 MeV MeV Fig. 4. Depth-dose distribution of protons in liquid water with initial energies of 8,, 5 and MeV. GEANT4 provides us with data for the depth-dose distribution of protons in liquid water. For an initial energy of MeV, which is within the usual range in proton therapy, the depth-dose distributions found through the different methods are shown on Fig. 5. The solid line shows the result from GEANT4. The position, height and width of the Bragg peak obtained with our approach, but using PSTAR values for the stopping cross section, is in excellent agreement with the result from GEANT4. However, using the values for the stopping cross sections obtained semi-empirically gives rise to a shift of the curve. The height of the peak is correct, as the straggling and fragmentation contributions remain the same, but the Depth, x (mm) Fig. 5. Bragg peaks obtained with different methods (GEANT4, our program and our program using PSTAR values) compared. The curves have been normalised to the entrance dose for a better comparison. peak appears around 5 mm further away. This distance increases to around cm if Rudd model without relativistic corrections is used. A first look at the stopping cross section distribution on Fig. shows some differences between the PSTAR and the calculated values for energies below.5 MeV, which could be considered in principle to be the cause of the shift. However, a physical analysis of the whole picture of protons going through the matter shows the real problem. As we have already mentioned, highly energetic protons have a lower probability to interact with the matter and hence the energy per unit length they deposit is also lower. When their energy decreases (approximately below MeV) the linear energy transfer increases considerably. They lose their whole remaining energy in a very short distance, usually only a few millimetres. Therefore, the final range of the projectile is mostly determined by the distance it travels before reaching the MeV threshold. From Fig. 5, we observe a difference about 5 mm in the position of the peaks. This shift is too large to result from an error in the stopping cross section for energies below MeV. Neither Rudd equations for the ionisation cross section nor the corrections introduced by Refs. [3] or [4] give the precision required in the high energy limit (see Eq. (8)) of the stopping cross section to obtain the right position of the Bragg peak. As discussed on Section, Bethe-Bloch equation is believed to give the right values in the high energy limit. As this equation includes ionisation and excitation by the projectile, and these are the only contributions for energies above MeV (see Fig. ), it is worth using Bethe-Bloch equation for that range, keeping the semiempirical models introduced in Section for energies below MeV. Fig. 6 shows the stopping cross section using this remedy, which appears to be very similar to Fig., as expected. However, a closer look for high energies shown on Fig. 7, presenting the total stopping cross section from the combined semi-empirical and Bethe-Bloch model using a linear scale, proves otherwise. We observe that the Rudd equation with relativistic corrections, which seemed precise on Fig., turns out to deviate from the PSTAR results by about 5%. The discrepancy is even higher (around 7%) if no relativistic correction is considered. What we now show on Fig. 7 is a more precise version of the stopping cross sections plots. The observed error in the position of the Bragg peak is now understood through comparison with Bethe-Bloch formula in this limit. Note that only the first two terms in Eq. () are needed in this approach to obtain the desired precision in the final results. Fig. 8 shows the new Bragg peak, obtained by using the semiempirical model for energies below MeV and the Bethe-Bloch correction above this energy. It is compared with the Bragg peak

7 C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 69 () Stopping cross sections (ev m ) x 8 PSTAR values Total (p) Ionisation (p) Charge transfer Ionisation (H) Excitation (p) Kinetic energy of the projectile, T (ev) Fig. 6. Stopping cross section of protons in liquid water considering the semiempirical model for T < MeV and Bethe-Bloch formula for T > MeV. Range, R (mm) T. Bortfeld This work Initial kinetic energy of the projectile, T (MeV) Fig. 9. Range of protons in liquid water as a function of its initial kinetic energy. The range given using the semi-empirical model and the Bethe-Bloch formula is compared with the range given in Ref. [6], which fits experimental results. Stopping cross sections (ev m ).5 x 9 PSTAR values Semi empirical model Semi empirical (T< MeV) & Bethe Bloch formula (T> MeV) Kinetic energy of the projectile, T (ev) x 7 Fig. 7. Stopping cross section of protons in liquid water using two different models (Bethe-Bloch formula and the semi-empirical formulae) compared with the PSTAR results. LET (AU) GEANT4 Semi empirical Semi empirical + Bethe Bloch approach Depth, x (mm) Fig. 8. Depth-dose distribution of protons in liquid water for T ¼ MeV, considering the semi-empirical model, the Bethe-Bloch equation for T > MeV and GEANT4. The curves have been normalised to the entrance dose for a better comparison. obtained by the semi-empirical formulae in the whole energy range. These plots are also compared with the results of the simulation from GEANT4. Note that Bethe-Bloch formula uses the mean excitation energy of the medium as a parameter (see Eq. ()). In order to have consistency in the following comparisons, the value I = 78 ev is used in our simulation, which is the one employed by GEANT4 version 9.3 and recommended at the revised tables of ICRU 73 [5]. Through applying this new approach for the total stopping cross section, the right position of the depth-dose distribution of protons in the liquid water in the energy range for proton therapy (from 7 to 5 MeV) is achieved. Bortfeld [6] shows that the relation between the range, R, and the initial kinetic energy of the projectile, T, is given by: R ¼ at p ð34þ where a ¼ : mm MeV p and p is around.77 for the 7 5 energy range. Eq. (34) is constructed so as to fit the experimental results. Fig. 9 shows the range of protons in liquid water for different initial kinetic energies using Eq. (34) and the current model. We observe a great agreement for the range of protons in liquid water. We have therefore obtained an explanation for the passage of protons through the matter which reproduces accurately the results from Monte Carlo simulations. In order to further address the physics behind the Bragg peak features, we now analyse the individual contribution of the different inelastic processes to the depth-dose curve. For a typical proton beam with initial kinetic energy of MeV, it is observed that if we neglect stripping, charge transfer and also hydrogen ionisation terms, the change produced in the position of the peak is lower than mm; a negligible distance. In addition, the linear energy transfer deposited in the peak is reduced around.5%. If only the excitation term is removed and the other contributions are all kept, we observe a shift in the position of the peak of around.5 mm, whereas, the change in the amount of energy deposited per unit length is negligible. Finally, removing just proton ionisation makes the Bragg peak vanish. This shows that, from all the physical processes taking place in this situation, only proton ionisation and excitation (the latter in a smaller scale) contribute to the properties of the Bragg peak. Thus, the other terms, affecting only low energies, produce all together solely a very small variation in the height and width of the peak. 4. Conclusion In this study, we explain qualitatively and quantitatively, the processes which affect the passage of a proton beam through liquid water. Different effects contributing to the total stopping cross section including ionisation and excitation of the medium, charge transfer, stripping and ionisation by the neutral projectile, have been taken into account. We tried to find a consistent semi-analytical

8 96 C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 69 () approach to obtain a good numerical approximation. The role of the most relevant processes which contribute to the total cross section for protons is explained. From the stopping cross section, we obtained several depth-dose distributions and compared them with the results from GEANT4 Monte Carlo simulations, which have been shown to be in agreement with experimental data. We obtain a realistic height for the peak via phenomena such as straggling and fragmentation. We show that the small error in the values of stopping cross section, which has been overlooked in other studies and phenomena leads to a considerable shift in the position of the Bragg peak. This means that Rudd model with relativistic corrections for high energy protons is not precise enough to obtain the right position of the Bragg peak in the therapeutic energy range. For high energies, corrections from the Bethe-Bloch equation gives rise to more precise results for the stopping cross section. The PSTAR data for the stopping cross section and the correction from Bethe-Bloch equation result in the right position of the Bragg peak. We note that the position of the peak is mainly determined by the stopping cross section for high energies. Low energies also contribute to the position of the peak, but in a very small scale. Changes in the stopping cross section for small energies only produce variation in the height of the peak of some ev/mm. For hadron therapy precise knowledge about the different processes contribution to the height, width and the position of the peak is of vital importance. The model presented here for the stopping cross section (Bethe- Bloch corrections for high energy terms like ionisation and excitation, and low energy semi-empirical contributions for charge transfer, stripping and ionisation by Hydrogen) gives the right position for the Bragg peak. The right values for the height and the width of the peak are also achieved from fragmentation and straggling phenomena. The present work tries to continue the path towards the full semi-empirical derivation of the characteristics of the Bragg peak without having to use Monte Carlo methods such as GEANT4, whose simulations take a considerably long time. This would allow medical physicists to perform individual simulations for each patient, taking into account the particular inhomogeneities of each case, which are the main sources of divergence with the average results. References [] U. Amaldi, G. Kraft, Radiotherapy with beams of carbon ions, Rep. Prog. Phys. 68 (5) [] D.E. Groom, S.R. 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