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放疗绝对剂量的数学算法模型

谢天赐 张彬 贺泊 李昊鹏 秦壮 钱金钱 石锲铭 LewisElfed 孙伟民

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放疗绝对剂量的数学算法模型

谢天赐, 张彬, 贺泊, 李昊鹏, 秦壮, 钱金钱, 石锲铭, LewisElfed, 孙伟民

Mathematical algorithm model of absolute dose in radiotherapy

Xie Tian-Ci, Zhang Bin, He Bo, Li Hao-Peng, Qin Zhuang, Qian Jin-Qian, Shi Qie-Ming, Lewis Elfed, Sun Wei-Min
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  • 本文提出一种通过物理模型计算放疗过程中每一个组织深度处绝对剂量的算法, 它可代替蒙特卡罗仿真的部分工作且耗费时间更少. 这个算法是基于对照射野内X射线产生电子的能量注量的积分运算, 并考虑了射线的能谱及二次散射线, 得到了后向散射对表面剂量的贡献比例, 同时得到前向散射、后向散射及原射线剂量贡献的关系. 比较了二次光子和二次电子的三维能谱, 得出该能谱是粒子注量关于粒子能量和粒子运动方向的函数. 为了得到每一深度处的光子注量, 计算了有连续能谱的X射线的期望质量衰减系数. 上述算法计算得到的绝对剂量与蒙特卡罗方式仿真的结果趋势一致, 两者的差异在于算法未考虑高于二次的散射线. 最后将算法应用到非均匀模体剂量计算, 能准确反映其中剂量分布特点且具有较小的误差.
    A algorithm of obtaining absolute dose at each tissue depth only by the mathematical calculation of formula is reported. The algorithm is based on integrating the energy flux of the electron generated by X-ray in the range of irradiation field, and the energy spectrum of ray and the secondary scattered ray are considered in this process. In this algorithm, the water phantom in the irradiation field is divided into several thin layers, and the energy flux of the electrons generated by interaction between the ray and thin layer reaching the calculation point is calculated. Finally, the absolute dose of the calculation point can be obtained by accumulating the energy flux contribution of all thin layers. For the X-ray with continuous energy spectrum, the expected mass attenuation coefficient is calculated for obtaining the photon flux at each depth in this process. The absolute dose calculated by this algorithm is verified by Monte Carlo simulation, and the difference between the algorithm and simulation is compensated for by a dose function about multiple scattering photons, and the function shows fast descent and then slow ascent. It is found that the ratio of the dose caused by backscatter to the surface dose, and the relationship among forward scatter, backward scatter and primary ray, and the relationship between the dose and the depth of the secondary scattered rays show a trend of first rising and then declining, and the depth of the peak value deviates from the position of the thin layer. Three-dimensional energy spectra of the secondary photon and the secondary electron are also compared with each other, and the spectrum is a function of particle flux about particle energy and particle direction. From the perspective of Compton effect, the physical meanings of different positions in the three-dimensional energy spectrum of the two particles are explained. It is found that the difference between algorithm percentage depth dose and simulation percentage depth dose is similar to the difference between small irradiation field percentage depth dose and big irradiation field percentage depth dose from simulation, and it is verified that the difference between algorithm and simulation comes from the increase of scattered rays. Finally, the algorithm is applied to the dose calculation of non-uniform phantom, which can accurately reflect the dose distribution characteristics and have less error.
      通信作者: 孙伟民, sunweimin@hrbeu.edu.cn
    • 基金项目: 黑龙江省自然科学基金(批准号: ZD2019H003)、国家自然科学基金与中国科学院合作设立的天文联合研究基金(批准号: U1631239, U1931206)和高等学校学科创新引智计划(批准号: B13015)资助的课题
      Corresponding author: Sun Wei-Min, sunweimin@hrbeu.edu.cn
    • Funds: Project supported by the Natural Science Foundation of Heilongjiang Province, China (Grant No. ZD2019H003), the Joint Research Fund in Astronomy of the National Natural Science Foundation of China and the Chinese Academy of Sciences, China (Grant Nos. U1631239, U1931206), and the Programme of Introducing Talents of Discipline to Universities (Grant No. B13015)
    [1]

    Yamamoto T, Mizowaki T, Miyabe Y, Takegawa H, Narita Y, Yano S, Nagata Y, Teshima T, Hiraoka M 2007 Phys. Med. Biol. 52 1991Google Scholar

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    Einstein A J, Henzlova M J, Rajagopalan S 2007 JAMA 298 317Google Scholar

    [3]

    Delaney G, Jacob S, Featherstone C, Barton M 2005 Cancers 104 1129Google Scholar

    [4]

    Das I J, Cheng C W, Watts R J, Ahnesjo A, Gibbons J, Li X A, Lowenstein J, Mitra R K, Simon W E, Zhu T C 2008 Med. Phys. 35 4186Google Scholar

    [5]

    van't Veld A A, van Luijk P, Praamstra F, van der Hulst P C 2001 Med. Phys. 28 738Google Scholar

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    van't Veld A A, van Luijk P, Praamstra F, van der Hulst P C 2000 Med. Phys. 27 923Google Scholar

    [7]

    Jette D 2000 Med. Phys. 27 1705Google Scholar

    [8]

    Jette D 1999 Med. Phys. 26 924Google Scholar

    [9]

    Tillikainen L, Helminen H, Torsti T, Siljamaki S, Alakuijala J, Pyyry J, Ulmer W 2008 Phys. Med. Biol. 53 3821Google Scholar

    [10]

    Jones A O, Das I J 2005 Med. Phys. 32 766Google Scholar

    [11]

    Aarup L R, Nahum A E, Zacharatou C, Juhler-Nottrup T, Knoos T, Nystrom H, Specht L, Wieslander E, Korreman S S 2009 Radiother. Oncol. 91 405Google Scholar

    [12]

    Krieger T, Sauer O A 2005 Phys. Med. Biol. 50 859Google Scholar

    [13]

    Beilla S, Younes T, Vieillevigne L, Bardies M, Franceries X, Simon L 2017 Phys. Medica 41 46Google Scholar

    [14]

    Jansen A A 1980 Icru Report 33: Radiation Quantities and Units (Washington)

    [15]

    胡逸民 1999 肿瘤放射物理学 (北京: 原子能出版社) 第38−44页

    Hu Y M 1999 Radiation Oncology Physics (Beijing: Atomic Energy Press) pp38−44(in Chinese)

    [16]

    Attix F H 1986 Gamma- and X-Ray Interactions in Matter (Vol.1) (Weinheim: Wiley-VCH Verlag GmbH & Co.KgaA) pp125−38

    [17]

    Price M J, Hogstrom K R, Antolak J A, White R A, Bloch C D, Boyd R A 2007 J. Appl. Clin. Med. Phys 8 61Google Scholar

    [18]

    Malataras G, Kappas C, Lovelock D M J 2001 Phys. Med. Biol. 46 2435Google Scholar

    [19]

    Cheng C W, Cho S H, Taylor M, Das I J 2007 Med. Phys. 34 3149Google Scholar

    [20]

    Carlone M, Tadic T, Keller H, Rezaee M, Jaffray D 2016 Med. Phys. 43 2927Google Scholar

    [21]

    Kinhikar R A 2008 Technol. Cancer Res. Treat. 7 381Google Scholar

    [22]

    Chow J C L, Grigorov G N, Barnett R B 2006 Med. Dosim. 31 249Google Scholar

    [23]

    Disher B, Hajdok G, Gaede S, Battista J J 2012 Phys. Med. Biol. 57 1543Google Scholar

    [24]

    Mesbahi A, Dadgar H, Ghareh-Aghaji N, Mohammadzadeh M 2014 J. Cancer Res. Ther. 10 896Google Scholar

  • 图 1  绝对剂量的算法模型示意图

    Fig. 1.  Schematic diagram of absolute dose algorithm model

    图 2  $ {D}_{\mathrm{s}\mathrm{f}}' $(a)和$ {D}_{\mathrm{s}\mathrm{b}}' $(b)算法模型示意图

    Fig. 2.  Schematic diagram of $ {D}_{\mathrm{s}\mathrm{f}}' $ (a) and $ {D}_{\mathrm{s}\mathrm{b}}' $ (b) algorithm model.

    图 3  蒙特卡罗PDD与真实测量数据的对比

    Fig. 3.  Comparison of PDD between Monte Carlo simulation and experimental data.

    图 4  6 MV射线能谱及其仿真函数

    Fig. 4.  6 MV X-ray spectrum and its fitting function.

    图 5  蒙特卡罗仿真的绝对剂量和计算绝对剂量$ D\left(z\right) $的比较

    Fig. 5.  Comparison of absolute dose between Monte Carlo simulation and calculation of $ D\left(z\right) $.

    图 6  比较由原射线产生的剂量$ {D}_{p}' $、前向散射产生的剂量$ {D}_{\mathrm{s}\mathrm{f}}' $、后向散射产生的剂量$ {D}_{\mathrm{s}\mathrm{b}}' $随深度的变化

    Fig. 6.  Comparison of dose caused by primary ray $ {D}_{\mathrm{p}}' $, forward scatter $ {D}_{\mathrm{s}\mathrm{f}}' $ and backscatter $ {D}_{\mathrm{s}\mathrm{b}}' $ with depth.

    图 7  向前和向后的二次射线产生的剂量随深度的变化

    Fig. 7.  Dose caused by scattered ray in forward and back directions with depth.

    图 8  二次电子的粒子注量随能量E和反冲角$ {\varphi }_{1} $的变化

    Fig. 8.  Particle flux of secondary electron with energy E and recoil angle $ {\varphi }_{1} $.

    图 9  二次光子的粒子注量随能量$ {hv}' $和散射角${\theta _{\rm{1}}}$的变化

    Fig. 9.  Particle flux of secondary photon with energy $ {hv}' $ and scattered angle $ {\theta }_{1} $.

    图 10  比较随不同射野(射野大小3 cm × 3 cm和4 cm × 4 cm)变化的蒙特卡罗仿真数据$ {D}_{\mathrm{M}\mathrm{C}} $与计算的$ D{\left(z\right)}_{\%} $(射野大小3 cm × 3 cm)

    Fig. 10.  Comparison of Monte Carlo simulation PDD with different fields (field size 3 cm × 3 cm and 4 cm × 4 cm) and calculation PDD $ D{\left(z\right)}_{\%} $ (field size 3 cm × 3 cm).

    图 11  不同肺密度的水肺水模体的射野中心轴百分深度剂量

    Fig. 11.  Percentage depth dose of the central axis of the field of radiation in water-lung-water phantom with different lung densities.

    图 12  不同模体深度处的离轴比曲线 (a) 7 cm; (b) 11 cm

    Fig. 12.  Off-axis ratio curves at different phantom depths: (a) 7 cm; (b) 11 cm.

    表 1  不同肺模体密度ρ对应参数kρμ0

    Table 1.  Parameters kρ and μ0 of lung phantom densities ρ.

    ρ/g·cm–30.10.20.4ρT
    kρ–6–3.750.753
    μ06.86.14.74
    下载: 导出CSV
  • [1]

    Yamamoto T, Mizowaki T, Miyabe Y, Takegawa H, Narita Y, Yano S, Nagata Y, Teshima T, Hiraoka M 2007 Phys. Med. Biol. 52 1991Google Scholar

    [2]

    Einstein A J, Henzlova M J, Rajagopalan S 2007 JAMA 298 317Google Scholar

    [3]

    Delaney G, Jacob S, Featherstone C, Barton M 2005 Cancers 104 1129Google Scholar

    [4]

    Das I J, Cheng C W, Watts R J, Ahnesjo A, Gibbons J, Li X A, Lowenstein J, Mitra R K, Simon W E, Zhu T C 2008 Med. Phys. 35 4186Google Scholar

    [5]

    van't Veld A A, van Luijk P, Praamstra F, van der Hulst P C 2001 Med. Phys. 28 738Google Scholar

    [6]

    van't Veld A A, van Luijk P, Praamstra F, van der Hulst P C 2000 Med. Phys. 27 923Google Scholar

    [7]

    Jette D 2000 Med. Phys. 27 1705Google Scholar

    [8]

    Jette D 1999 Med. Phys. 26 924Google Scholar

    [9]

    Tillikainen L, Helminen H, Torsti T, Siljamaki S, Alakuijala J, Pyyry J, Ulmer W 2008 Phys. Med. Biol. 53 3821Google Scholar

    [10]

    Jones A O, Das I J 2005 Med. Phys. 32 766Google Scholar

    [11]

    Aarup L R, Nahum A E, Zacharatou C, Juhler-Nottrup T, Knoos T, Nystrom H, Specht L, Wieslander E, Korreman S S 2009 Radiother. Oncol. 91 405Google Scholar

    [12]

    Krieger T, Sauer O A 2005 Phys. Med. Biol. 50 859Google Scholar

    [13]

    Beilla S, Younes T, Vieillevigne L, Bardies M, Franceries X, Simon L 2017 Phys. Medica 41 46Google Scholar

    [14]

    Jansen A A 1980 Icru Report 33: Radiation Quantities and Units (Washington)

    [15]

    胡逸民 1999 肿瘤放射物理学 (北京: 原子能出版社) 第38−44页

    Hu Y M 1999 Radiation Oncology Physics (Beijing: Atomic Energy Press) pp38−44(in Chinese)

    [16]

    Attix F H 1986 Gamma- and X-Ray Interactions in Matter (Vol.1) (Weinheim: Wiley-VCH Verlag GmbH & Co.KgaA) pp125−38

    [17]

    Price M J, Hogstrom K R, Antolak J A, White R A, Bloch C D, Boyd R A 2007 J. Appl. Clin. Med. Phys 8 61Google Scholar

    [18]

    Malataras G, Kappas C, Lovelock D M J 2001 Phys. Med. Biol. 46 2435Google Scholar

    [19]

    Cheng C W, Cho S H, Taylor M, Das I J 2007 Med. Phys. 34 3149Google Scholar

    [20]

    Carlone M, Tadic T, Keller H, Rezaee M, Jaffray D 2016 Med. Phys. 43 2927Google Scholar

    [21]

    Kinhikar R A 2008 Technol. Cancer Res. Treat. 7 381Google Scholar

    [22]

    Chow J C L, Grigorov G N, Barnett R B 2006 Med. Dosim. 31 249Google Scholar

    [23]

    Disher B, Hajdok G, Gaede S, Battista J J 2012 Phys. Med. Biol. 57 1543Google Scholar

    [24]

    Mesbahi A, Dadgar H, Ghareh-Aghaji N, Mohammadzadeh M 2014 J. Cancer Res. Ther. 10 896Google Scholar

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出版历程
  • 收稿日期:  2020-06-26
  • 修回日期:  2020-08-30
  • 上网日期:  2020-12-12
  • 刊出日期:  2021-01-05

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