Optimization of ripple filter for pencil beam scanning

Transcription

1 Nuclear Science and Techniques 24 (2013) Optiization of ripple filter for pencil bea scanning YANG Zhaoxia ZHANG Manzhou LI Deing * Shanghai Institute of Applied Physics, Chinese Acadey of Sciences, Shanghai , China Abstract This paper presents a novel approach to seek the bar width for ripple filter used in pencil bea scanning proton therapy. A weight decay quadratic prograing ethod is eployed for the new optiization strategy. Copared to the coonly used iterative-least-square technique, the ripple filter derived by the proposed ethod not only has better depth dose unifority, i.e., the dose unifority is within 0.5%, but also has triangle-like vertical cross-sectional shape which is suitable for anufacture. Moreover, the new ethod has such good robust characteristics that it is also applicable to the real application with unavoidable easureent errors and noises. The siulation results of this study ay be helpful in iproving the design of the ripple filter. Key words Dose unifority, Proton therapy, Ripple filter, Weight decay quadratic prograing 1 Introduction One of the ain advantages of proton therapy is the depth dose profile, which has been naed Bragg peak. The few--wide pristine Bragg peak allows to irradiate a well localized region within the body, keeping low the dose release in the proxial (entrance plateau) and distal (tail) regions. If the tuor cells are bigger than the width of the Bragg peak, superposition of several Bragg peaks fro the different bea energies is introduced to ake a unifor radiation dose over the tuor. For pencil bea scanning, the target is irradiated layer by layer with the different bea energy. However, switching tie between energy layers is a aor contribution to the overall tie. Few energy layers ay result in large ripple produced on the flat-top depth-dose curve, which becoes serious for low energy. The ripple filter, which is of great interest in recent years [1 5], is used to address this proble. Fig. 1 shows the vertical cross-section of the coonly used ripple filter. After inserting the ripple filter perpendicular to the bea, the Bragg peak of the proton bea will becoe sooth [1,3]. Although Weber and Kraft [1] have presented a least square (LS) obective function to seek the width * Corresponding author. E-ail address: dli@sinap.ac.cn Received date: of the ripple bar, they have not entioned the ethod to solve it. In Ref.[2], iterative-least-square technique (ILST) is introduced to optiize the weights. In this paper, a new quadratic obective function is proposed for designing the width of the ripple filter bar. Besides, the ILST is also used to optiize the LS function for the coparison. Fig.1 Vertical cross-sectional view of ripple filter bar. 2 Optiization ethods The purpose of the treatent planning is to provide a desired biological dose which is calculated via ultiplying the physical dose by the relative biological effectiveness (RBE). As recoended by the ICRU [6], RBE is set to 1.1 in this paper. The principle of the analytical coputation for the ripple filter design used in Ref.[1] is adopted in this study. When the proton bea passes through the

2 ripple filter with the thickness t r, the dose distribution will be shifted to a certain water-equivalent value that corresponds to t r along the bea direction. Therefore, when the proton bea passes through the ripple filter perpendicularly to the bea direction, the pristine dose Bragg peak will be transfored by the ripple-bars into a superposition of displaced Bragg curves. Gaussian Bragg peaks were chosen to generate hoogeneous depth dose profiles with a low ripple for the soothing effect after superposition [1,3,5]. To for the dose profile close to Gaussian shape in the peak region, the weight of proton coponent has to be calculated. Since the weight of the proton coponent passing through the ripple-bar step is proportional to the width of the ripple-bar step, the weight deterines the shape of the ripple bar. Additionally, the difference of proton fluence caused by the nuclear interaction in the ripple filter and water is taken into account for the coputation. After proton bea passing through the ripple filter, the biological dose is odulated as follows: 2z dod ( z )/ RBE d ( z ) w f (1) ax i phy i in where d od (z i ) represents the odulated biological dose at a depth z i ; z is the step size of the dose points; λ is the period of the groove structure; in and ax are defined as the value of the iniu and axiu water-equivalent thickness of ripple bar divided by z, respectively; d phy (z i+ ) is the pristine physical dose at a depth z i+ ; w is the weight of the th segent of proton bea; f, the fluence difference introduced by the th step of ripple-bar, as described in Eq.(2) [7] : f t p exp( ) [1 t 1.19 R 1 rf p 1 ] (2) where t =t(z ) = 1.19t r (z ) is the water equivalent thickness of ripple filter in z direction; λ rf = c is the nuclear interaction length of plexiglass; p = c -1 ; R is the proton range in water. To siplify the analysis, (1) can be further written as the following atrix for: i1 ax D a w AW (3) od i in where D od is odulated biological dose; a i = 2Δz d phy (z i+ ) f /λ; is the diension of D od ; n= ax in +1, which is the diension of weight W; A is a n atrix; W is a n 1 vector and w 0. Since it is often the case that >n, then (3) is an over-deterined equation thus there ight be no unique solution. Besides, the coefficient atrix A is usually ill-conditioned so that the solution of (3) is derived by solving an optiization proble. In this paper, two optiization ethods with different obective function are used and copared for the ripple filter bar design. 2.1 Iterative least-squares technique ILST is a coonly used ethod for solving the least square obective function [8 10] with relatively fast converge rate and good perforance of averaging out the noisy data. In hadron therapy range odulation field, Schaffner et al. [2] utilized the ILST to produce the large, biologically unifor spread-out Bragg peak (SOBP) depth dose fro the sall SOBPs. The weight updating in two consecutive iterations (k th iteration and (k 1) th iteration) can be calculated by w w g d z 2 2 k, k, 1 i phy( i ) ax i1 i1 gd ( z ) 2 2 i phy i in pz ( ) d ( z ) f w i phy i, k1 (4) where g is set to 1 inside of the SOBP and 0 outside; P(z i ) is the prescribed biological dose at a depth z i. 2.2 Quadratic prograing ethod A large condition nuber of the atrix A, i.e., larger than 100, ay result in the different order of agnitudes of the derived weights value w. This will lead to the anufacture difficulties of the ripple filter because the noralized value of w is proportional to the width of the th ripple bar. For exaple, if w +1 is 100 ties saller than w and the width of the th segent is 10 μ, then the required width of the (+1) th segent is 0.1 μ which ight beyond the anufacture precision (1 μ). To prevent the weights fro growing too large, a new obective function is proposed in this study, it consists of a least square ite

3 and an extra weight decay ite [11]. Min 0.5 P-AW γ W 2 s.t w0, =1,2,,n (5) where ǁ ǁ represents Frobenius nor operation; P ( 1) is the prescribed biological dose; γ (γ 0) is a paraeter that deterines how strongly large weights are penalized. The larger γ is, the ore unifor weights are. Particularly, when γ = 0, Eq.(5) becoes in0.5 P-AW 2 s.t w0, =1,2,,n (6) The proble (6) is a special case of (5). Proble (6) is a least square optiization proble with nonnegative constrains. Proble (5) can be written in the standard quadratic for as follows: Min 0.5W T (A T A+γI)W-(P T A)W s.t w0, =1,2,,n (7) where I is a n n identity atrix. Proble (7) can be solved by a quadratic prograing ethod (QPM) [12]. Adding the weight decay ter in ripple bar design can iprove generalization [11]. 2.3 Perforance evaluation The optiized results with ILST [2] and QPM are copared in this paper. Moreover, the robustness (anti-noise ability) of two ethods which specified as the dose difference was also exained. r=(d n -d 0 )d 0 (8) where d n and d 0 represent the dose derived fro the noise data and original data, respectively; r is the relative dose variation, which represents the robustness of the filter. The saller r is, the stronger robustness is. 3 Results and discussion In this study, a typical bea delivery syste for proton therapy is siulated with Geant4. Bragg peaks obtained fro the siulation are used to evaluate the ethods. Since the Bragg peaks of proton beas becoes narrow with the decrease of energy, the relative low energy of 70 MeV is chosen in the siulation. 3.1 Siulated odel As shown in Fig. 2, the scheatic of the siulated odel consists of a kapton window located at the end of the bea transport line, a nozzle, and a water tank phanto. The coponents of the nozzle are scanning agnets, vacuu chaber, bea onitors, a ripple filter and etc. The scanning agnets are used to steer the bea. And the vacuu chaber is designed to reduce the bea scattering. The bea onitors, are ionization chabers with an equivalent water thickness of 1.1. The ripple filter is used to sooth the flat-top region of the SOBP. The distance between the isocenter and the ripple filter is 50 c. The oentu spread for protons is set as Δp/p=0.05%. Fig.2 Scheatic of the siulation odel. 3.2 SOBP dose unifority with ripple filter designed by QPM Fig.3 shows the dose unifority with and without ripple filters for proton beas of 70 MeV. As illustrated in Fig.3(a), for the sae scanning step (3.2 ), the dose unifority is as high as 24.7% without ripple filter and the dose unifority sharply falls down to less than 2.5% when a 3 ripple filter was added to the syste. In Fig.3(b), the dose unifority is ore than 20.7% without ripple filter, while with ripple filter the axiu dose scanning error decreases to less than 2.5% with the sae step (3 ). As described in Ref.[3], the widths of distal fall-off (80% 20% distal falloff) are both oderate (less than 3 ). 3.3 Coparison between ILST and QPM Fig.4 illustrates the siulation results with ILST and QPM when pencil bea scanning step is 2.14 (a rando value). As shown in Fig.4(a), the depth dose distribution with QPM (solid line) is ore unifor and closer to the ideal value (1.00 in the figure) than that with ILST (dashed line). Particularly, the axiu dose unifority with QPM is 0.5%, which is less than that with ILST 0.7%. In addition, the shape of ripple

4 filter with QPM is closer to triangle than that with ILST as shown in Fig.4(b). The reason is that the extra weight decay ite prevents the value of the individual weight which corresponds to the width of ripple filter bar fro growing too large. Therefore, the width of ripple filter bar tends to have the sae order agnitude. Therefore, its corresponding ripple filter bar is close to the triangle shape and so that is convenient for anufacture. ability of the QPM. The results of scanning step of 3.2 with the use of 3 ripple filter are siilar. Fig.4 Siulation results of two ethods with 2 ripple filter: (a) depth dose coparison; (b) ripple filter bar shapes. Fig.3 Depth dose distributions for 70 MeV proton beas: (a) coparison between with and without use of 3 ripple filter Scanning step of 3.2 ; (b) coparison between with and without use of 2.8 ripple filter, scanning step of 3. In the real application, the dose is easured in the experient instead of deriving by the siulation. Then the unavoidable easureent error of dose should be considered. The noise with Gaussian distribution was added to the siulation dose in this study. Fig.5 shows the relative dose variation and the ripple filter shape curve difference while the 5% noise is added in the siulation. The dose difference value with QPM and ILST is shown in Fig.5(a). The calculated standard deviation of the relative dose unifority with QPM (0.0039) is saller than that with ILST (0.0042). Fig.5(b) further copares the filter shape difference, the horizontal axes represents the shape difference with and without noise data. The saller values derived by QPM indicates its stronger robustness. In other words, the proposed QPM is capable of obtaining the siilar ripple filter even using the data with noise, which is preferred in the real application. The good robustness is due to good generalization Fig.5 Robustness coparison of two ethods with the use of 2 ripple filter: (a) depth dose difference; (b) shape difference. Fig.6 copares the individual proton bea Bragg curves and the corresponding lateral dose distribution

5 at different positions with and without ripple filter. The relative dose of the odulated curves are higher than that of the pristine Bragg curve at the depth of odulated Bragg peak as illustrated in Fig.6(a). This phenoenon is also reflected fro Fig.6(b), in which the lateral dose distribution of proton bea without ripple filters are both lower than that with ripple filters. Besides, the curves for A and B is close to each other in Fig.6(b), which indicates there is slight difference of lateral dose distribution at Bragg peak between two different ripple filters. This is due to the proton bea is scattered by different ripple filters ainly according to Moliere theory [13,14]. Fig.6 Coparison of the siulation results of dose distribution with and without 2 ripple filter at the energy of 70MeV. (a) individual proton Bragg curves. BQPM, BILST indicate the Bragg curves with ripple filter designed by QPM and ILST respectively, while PPB represents the pristine Bragg curve without ripple filter. Point A and B are the Bragg peak position of the BQPM and BILST, respectively. A and C are at the sae depth, and that for B and D. (b) lateral dose distribution corresponding to A, B, C, D. 4 Conclusion A new design approach for ripple filter bar was proposed in this paper. The new approach uses the new obective function with QPM. It has been successfully applied to calculate the weights, which were further converted to the ripple filter diension. Copared to the ILST, the proposed ethod not only achieves better perforance, i.e., better depth dose unifority and ripple filter s shape for anufacture, but also has better robustness characteristics. This ripple filter can be anufactured ore accurately and achieve a good flatness of the SOBP at the sae tie. References 1. Weber U, Kraft G. Phys Med Biol, 1999, 44: Schaffner B, Kanai T, Futai Y, et al. Med. Phys, 2000, 27: Takada Y, Kobayashi Y, Yasuoka K, et al. Nucl Instru Meth Phys Res A, 2004, 524: Fuitaka S, Takayanagi T, Fuioto R, et al. Phys Med Biol, 2009, 54: Hara1 Y, Takada1 Y, Hotta1 k, et al. Phys Med Biol, 2012, 57: DeLuca P M, Wabersie A, Whitore G. Journal of the ICRU, 2007, 7: Akagi T, Higashi A, Tsugai H, et al. Phys Med Biol, 2003, 48: N301 N Goitein M. Nucl. Instru. Methods, 1972, 101: Brooks R A, Di Chiro G. Phys Med Biol, 1976, 21: Loax A. Phys Med Biol, 1999, 44: Krogh A, Hertz J. Adv Neural Inforation Process Sys, 1992, 4: Colean T F, Li Y 1996, SIAM J Opti, 6: B. Gottschalk B, Koehler A M, Schneider R J, et al. Nucl Instru Meth Phys Res B, 1993, 74: Mustafa A A M, Jackson D F. Phys Med Biol, 1981, 26: