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@ARTICLE{Hartmann:186489,
      author       = {G. Hartmann$^*$ and H. G. Menzel},
      title        = {{N}ote on uncertainty in {M}onte {C}arlo dose calculations
                      and its relation to microdosimetry.},
      journal      = {Zeitschrift für medizinische Physik},
      volume       = {34},
      number       = {3},
      issn         = {0939-3889},
      address      = {Amsterdam},
      publisher    = {Elsevier, Urban $\&$ Fischer},
      reportid     = {DKFZ-2022-03210},
      pages        = {468-476},
      year         = {2024},
      note         = {2024 Aug;34(3):468-476 / #EA:E040# / Short Communication},
      abstract     = {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.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.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 on 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.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.},
      keywords     = {Absorbed dose (Other) / Microdosimetry (Other) / Monte
                      Carlo calculation (Other) / Specific energy (Other) /
                      Uncertainty (Other)},
      cin          = {E040},
      ddc          = {610},
      cid          = {I:(DE-He78)E040-20160331},
      pnm          = {315 - Bildgebung und Radioonkologie (POF4-315)},
      pid          = {G:(DE-HGF)POF4-315},
      typ          = {PUB:(DE-HGF)16},
      pubmed       = {pmid:36577627},
      doi          = {10.1016/j.zemedi.2022.11.012},
      url          = {https://inrepo02.dkfz.de/record/186489},
}