{"title":"关于蒙特卡洛剂量计算的不确定性及其与微剂量测定的关系的说明。","authors":"","doi":"10.1016/j.zemedi.2022.11.012","DOIUrl":null,"url":null,"abstract":"<div><h3>Purpose</h3><p>The Type A standard uncertainty in Monte Carlo (MC) dose calculations is usually determined using the “history by history” method. Its applicability is based on the assumption that the central limit theorem (CLT) can be applied such that the dispersion of repeated calculations can be modeled by a Normal distribution. The justification for this assumption, however, is not obvious. The concept of stochastic quantities used in the field of microdosimetry offers an alternative approach to assess uncertainty. This leads to a new and simple expression.</p></div><div><h3>Methods</h3><p>The value of the MC determined absorbed dose is considered a random variable which is comparable to the stochastic quantity specific energy, z. This quantity plays an important role in microdosimetry and in the definition of the quantity absorbed dose, D. One of the main features of z is that it is itself the product of two other random variables, specifically of the mean dose contribution in a ‘single event’ and of the mean number of such events. The term ‘single event’ signifies the sum of energies imparted by all correlated particles to the matter in a given volume. The similarity between the MC calculated absorbed dose and the specific energy is used to establish the ‘event by event’ method for the determination of the uncertainty. MC dose calculations were performed to test and compare both methods.</p></div><div><h3>Results</h3><p>It is shown that the dispersion of values obtained by MC dose calculations indeed depend on the product of the mean absorbed dose per event, and the number of events. Applying methods to obtain the variance of a product of two random variables, a simple formula for the assessment of uncertainties is obtained which is slightly different from the ‘history by history’ method. Interestingly, both formulas yield indistinguishable results. This finding is attributed to the large number of histories used in MC simulations. Due to the fact that the values of a MC calculated absorbed dose are the product of two approximately Normal distributions it can be demonstrated that the resulting product is also approximately normally distributed.</p></div><div><h3>Conclusions</h3><p>The event by event approach appears to be more suitable than the history by history approach because it takes into account the randomness of the number of events involved in MC dose calculations. Under the condition of large numbers of histories, however, both approaches lead to the same simple expression for the determination of uncertainty in MC dose calculations. It is suggested to replace the formula currently used by the new expression. Finally, it turned out that the concept and ideas that were developed in the field of microdosimetry already 50 years ago can be usefully applied also in MC calculations.</p></div>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0939388922001337/pdfft?md5=48e6762405035d2f14a0258e021b67b3&pid=1-s2.0-S0939388922001337-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Note on uncertainty in Monte Carlo dose calculations and its relation to microdosimetry\",\"authors\":\"\",\"doi\":\"10.1016/j.zemedi.2022.11.012\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><h3>Purpose</h3><p>The Type A standard uncertainty in Monte Carlo (MC) dose calculations is usually determined using the “history by history” method. Its applicability is based on the assumption that the central limit theorem (CLT) can be applied such that the dispersion of repeated calculations can be modeled by a Normal distribution. The justification for this assumption, however, is not obvious. The concept of stochastic quantities used in the field of microdosimetry offers an alternative approach to assess uncertainty. This leads to a new and simple expression.</p></div><div><h3>Methods</h3><p>The value of the MC determined absorbed dose is considered a random variable which is comparable to the stochastic quantity specific energy, z. This quantity plays an important role in microdosimetry and in the definition of the quantity absorbed dose, D. One of the main features of z is that it is itself the product of two other random variables, specifically of the mean dose contribution in a ‘single event’ and of the mean number of such events. The term ‘single event’ signifies the sum of energies imparted by all correlated particles to the matter in a given volume. The similarity between the MC calculated absorbed dose and the specific energy is used to establish the ‘event by event’ method for the determination of the uncertainty. MC dose calculations were performed to test and compare both methods.</p></div><div><h3>Results</h3><p>It is shown that the dispersion of values obtained by MC dose calculations indeed depend on the product of the mean absorbed dose per event, and the number of events. Applying methods to obtain the variance of a product of two random variables, a simple formula for the assessment of uncertainties is obtained which is slightly different from the ‘history by history’ method. Interestingly, both formulas yield indistinguishable results. This finding is attributed to the large number of histories used in MC simulations. Due to the fact that the values of a MC calculated absorbed dose are the product of two approximately Normal distributions it can be demonstrated that the resulting product is also approximately normally distributed.</p></div><div><h3>Conclusions</h3><p>The event by event approach appears to be more suitable than the history by history approach because it takes into account the randomness of the number of events involved in MC dose calculations. Under the condition of large numbers of histories, however, both approaches lead to the same simple expression for the determination of uncertainty in MC dose calculations. It is suggested to replace the formula currently used by the new expression. 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引用次数: 0
摘要
目的:蒙特卡罗(MC)剂量计算中的 A 类标准不确定性通常采用 "逐历史 "法确定。其适用性基于这样一个假设,即可以应用中心极限定理(CLT),从而使重复计算的分散性可以用正态分布来模拟。然而,这一假设的合理性并不明显。微观模拟领域使用的随机量概念为评估不确定性提供了另一种方法。这导致了一种新的简单表达方式:z 的主要特征之一是它本身是其他两个随机变量的乘积,特别是 "单一事件 "中的平均剂量贡献和此类事件的平均数量。单个事件 "是指在给定体积内,所有相关粒子对物质产生的能量总和。利用 MC 计算的吸收剂量与特定能量之间的相似性,可以建立 "逐个事件 "的方法来确定不确定性。为了测试和比较这两种方法,还进行了 MC 剂量计算:结果表明,通过 MC 剂量计算获得的数值的离散性确实取决于每个事件的平均吸收剂量与事件数量的乘积。应用获得两个随机变量乘积方差的方法,可以得到一个评估不确定性的简单公式,该公式与 "逐历史 "法略有不同。有趣的是,这两个公式得出的结果并无差别。这一发现归因于 MC 模拟中使用了大量的历史记录。由于 MC 计算的吸收剂量值是两个近似正态分布的乘积,因此可以证明所得到的乘积也是近似正态分布的:结论:逐个事件的方法似乎比逐个历史的方法更合适,因为它考虑到了 MC 剂量计算所涉及的事件数量的随机性。不过,在历史数据较多的情况下,这两种方法都能得出相同的简单表达式来确定 MC 剂量计算的不确定性。建议用新的表达式取代目前使用的公式。最后,事实证明,50 年前在微剂量测定领域提出的概念和想法也可以有效地应用于 MC 计算。
Note on uncertainty in Monte Carlo dose calculations and its relation to microdosimetry
Purpose
The Type A standard uncertainty in Monte Carlo (MC) dose calculations is usually determined using the “history by history” method. Its applicability is based on the assumption that the central limit theorem (CLT) can be applied such that the dispersion of repeated calculations can be modeled by a Normal distribution. The justification for this assumption, however, is not obvious. The concept of stochastic quantities used in the field of microdosimetry offers an alternative approach to assess uncertainty. This leads to a new and simple expression.
Methods
The value of the MC determined absorbed dose is considered a random variable which is comparable to the stochastic quantity specific energy, z. This quantity plays an important role in microdosimetry and in the definition of the quantity absorbed dose, D. One of the main features of z is that it is itself the product of two other random variables, specifically of the mean dose contribution in a ‘single event’ and of the mean number of such events. The term ‘single event’ signifies the sum of energies imparted by all correlated particles to the matter in a given volume. The similarity between the MC calculated absorbed dose and the specific energy is used to establish the ‘event by event’ method for the determination of the uncertainty. MC dose calculations were performed to test and compare both methods.
Results
It is shown that the dispersion of values obtained by MC dose calculations indeed depend on the product of the mean absorbed dose per event, and the number of events. Applying methods to obtain the variance of a product of two random variables, a simple formula for the assessment of uncertainties is obtained which is slightly different from the ‘history by history’ method. Interestingly, both formulas yield indistinguishable results. This finding is attributed to the large number of histories used in MC simulations. Due to the fact that the values of a MC calculated absorbed dose are the product of two approximately Normal distributions it can be demonstrated that the resulting product is also approximately normally distributed.
Conclusions
The event by event approach appears to be more suitable than the history by history approach because it takes into account the randomness of the number of events involved in MC dose calculations. Under the condition of large numbers of histories, however, both approaches lead to the same simple expression for the determination of uncertainty in MC dose calculations. It is suggested to replace the formula currently used by the new expression. Finally, it turned out that the concept and ideas that were developed in the field of microdosimetry already 50 years ago can be usefully applied also in MC calculations.