 Review
 Open Access
Monte Carlo simulations in radiotherapy dosimetry
 Pedro Andreo^{1}Email authorView ORCID ID profile
 Received: 30 April 2018
 Accepted: 14 June 2018
 Published: 27 June 2018
Abstract
Background
The use of the Monte Carlo (MC) method in radiotherapy dosimetry has increased almost exponentially in the last decades. Its widespread use in the field has converted this computer simulation technique in a common tool for reference and treatment planning dosimetry calculations.
Methods
This work reviews the different MC calculations made on dosimetric quantities, like stoppingpower ratios and perturbation correction factors required for reference ionization chamber dosimetry, as well as the fully realistic MC simulations currently available on clinical accelerators, detectors and patient treatment planning.
Conclusions
Issues are raised that include the necessity for consistency in the data throughout the entire dosimetry chain in reference dosimetry, and how BraggGray theory breaks down for small photon fields. Both aspects are less critical for MC treatment planning applications, but there are important constraints like tissue characterization and its patienttopatient variability, which together with the conversion between dosetowater and dosetotissue, are analysed in detail. Although these constraints are common to all methods and algorithms used in different types of treatment planning systems, they make uncertainties involved in MC treatment planning to still remain “uncertain”.
Keywords
 Monte Carlo
 Radiotherapy physics
 Radiotherapy dosimetry
 Monte Carlo treatment planning
Introduction
The use of the Monte Carlo (MC) method to solve problems in the field of radiotherapy dosimetry has increased almost exponentially since the 1970s [1–3]. The range of MC applications spans from the calculation of fundamental dosimetric quantities to simulations of radiotherapy treatment planning. Although computer power requirements restricted early applications to simple geometries, like infinite parallel slabs or cylinders, today’s availability of computing power allows the simulation of detailed 3D geometries like those used for clinical accelerator treatment heads, ionization chambers and other detectors, and patient treatments using CT data. In all cases the complete phasespace that characterizes the energy, position and direction of the particles reaching a detector or a given organ within a patient, including all particle generations, can be determined. Hence, absorbed dose and other dosimetric quantities like fluence, kerma, etc can be calculated directly or performing subsequent analytical calculations.
This paper describes important MC applications in the different areas of radiotherapy dosimetry. It is initiated with a background on basic dosimetric expressions and key quantities, followed by fluencebased calculations like stoppingpower ratios and mass energyabsorption ratios. Presented next is the influence of a detector placed within a homogeneous medium, causing deviations from BraggGray theory, which is followed by a description of calculations on perturbation correction factors. The sofar generally accepted approach to deal with perturbation effects has raised issues in relation with the dosimetry of small megavoltage photon beams; they have led to a widely accepted alternative that avoids computing perturbation factors, an issue also discussed. The general aspects and developments on MC in radiotherapy treatment planning are summarized, followed by a short discussion on the controversy between dosetowater and dosetotissue and the conversion between these two quantities. Finally, some conclusions are drawn.
Background
Basic dosimetry expressions
where D_{med}(P) corresponds to the dose at a point of interest in the homogeneous medium and \(\bar D_{\text {det}}\) is the mean absorbed dose within the detector. In a MC calculation it is relatively straightforward to calculate \(\bar D_{\text {det}}\) as the average energy deposited by all charged particles within a volume divided by its mass, but calculating the dose at a point, D_{med}(P), requires considering an infinitesimally small volume, an approach that relies on some kind of interpolation process.
where S_{el}(E)/ρ is the mass electronic stopping power (formerly called collision stopping power, see ICRU Report 85 [4]) at the chargedparticle energy E, μ_{en}(k)/ρ is the mass energyabsorption coefficient of the material at the photon energy k, and Φ_{ E } and Φ_{ k } are the chargedparticle and photon fluence differential in energy in the medium, respectively. Identical expressions can be formulated for the absorbed dose within a detector replacing “med” by “det”. The acronyms over the equal signs refer to the type of chargedparticle equilibrium, either that of all type of charged particles generated in the medium (CPE), or to “partial chargedparticle equilibrium” (PCPE) (often referred to as transient chargedparticle equilibrium, TCPE). These are important remarks, because if there is no CPE the quantity determined is not absorbed dose, but cema, C, or restricted cema, C_{ Δ }, and if there is not PCPE the quantity approximates kerma (note that strictly K involves \(\mu _{\text {tr}}(k) = \mu _{\text {en}}(k)/(1  \bar g)\), \(\bar g\) being the mean fraction of the kinetic energy of liberated charged particles lost in radiative processes, which depending on the photon energy might be nonnegligible).
 (i)
The cavity is small compared to the chargedparticle ranges involved.
In this case f is identified with the stoppingpower ratio, and chargedparticle tracks are assumed to cross the cavity. Assuming also that the cavity does not perturb the primary chargedparticle fluence \(\Phi _{E}^{\text {prim}}\), i.e., that it conforms the fundamental BraggGray approximation Φ_{med}≈Φ_{det}, the socalled BraggGray stoppingpower ratio is defined as:$$ {}f_{\text{med,det}}(Q) \equiv s_{\text{med,det}}^{\text{BG}} =\frac {\int\limits_{0}^{E_{\max}} \left[ \Phi_{E}^{\text{prim}}\right]_{\text{med}} \left[ S_{\text{el}}(E)/\rho \right]_{\text{med}} \, \mathrm{d}E } {\int\limits_{0}^{E_{\max}} \left[ \Phi_{E}^{\text{prim}}\right]_{\text{med}} \left[ S_{\text{el}}(E)/\rho \right]_{\text{det}} \, \mathrm{d}E } $$(3)which is a quotient of fluenceweighted average mass stopping powers resulting in a ratio of absorbed doses (or of cemas if there is no CPE). Note that whereas the fluence is the same in the numerator and denominator, mass stopping powers correspond to each material, “med” and “det”, respectively. It is also emphasized that the primary chargedparticle fluence does not include knockon electrons (delta rays) or the secondary and higherorder charged particles created by the primary particles.
A refinement in the theory is made for SpencerAttix stoppingpower ratios, where the chargedparticle fluence, differential in energy, includes knockon electrons and any other generated charged particles having energies higher than a given threshold energy, Δ, related to the cavity size (its mean chord length), in addition to the primary particles. In this case the fundamental BraggGray approximation is \(\left (\Phi _{E}^{\text {tot}}\right)_{\text {med}} \approx \left (\Phi _{E}^{\text {tot}}\right)_{\text {det}}\). Additionally, a so called trackend term is added that accounts for the energy deposited by electrons with energies below Δ. The SpencerAttix stoppingpower ratio is defined as:$$ \begin{aligned} &f_{\text{med,det}}(Q) \equiv s_{\text{med,det}}^{\text{SA}} =\\ & \frac {\int\limits_{\Delta}^{E_{\max}} \left[\Phi_{E}^{\text{tot}}\right]_{\text{med}} \, \left[S_{\text{el}}(E,\Delta)/\rho\right]_{\text{med}}\, \mathrm{d}E \,+\, \left[\Phi_{E}^{\text{tot}}(\Delta)\right]_{\text{med}}\, \left[S_{\text{el}}(\Delta)/\rho\right]_{\text{med}} \, \Delta } {\int\limits_{\Delta}^{E_{\max}} \left[\Phi_{E}^{\text{tot}}\right]_{\text{med}} \, \left[S_{\text{el}}(E,\Delta)/\rho\right]_{\text{det}}\, \mathrm{d}E \,+\, \left[\Phi_{E}^{\text{tot}}(\Delta)\right]_{\text{med}}\, \left[S_{\text{el}}(\Delta)/\rho\right]_{\text{det}} \, \Delta } \end{aligned} $$(4)which is a quotient of fluenceweighted average restricted mass stopping powers resulting in a ratio of absorbed doses (or restricted cemas, C_{ Δ } [5] ^{2}, if there is no CPE), and the trackend terms include the fluence and the relevant (unrestricted) stopping power at the threshold energy Δ. Note that as the fluence includes now all kind of chargedparticles, is termed \(\Phi _{E}^{\text {tot}}\).
It should be emphasized that, according to the definition of linear energy transfer (LET) given in ICRU Report 85 [4], the restricted stopping power is not identical to the LET, L_{ Δ }, for very low values of the energy threshold or cutoff Δ, as is often formulated.
It is of interest to recall that the restricted electronic stopping power S_{el}(E,Δ) provides the component of a chargedparticle kinetic energy lost in inelastic collisions with atomic electrons that is deposited “locally”. This corresponds to a volume whose dimensions are limited by the range of ejected secondary electrons having an energy Δ. The energy loss \(\mathcal W\) is then restricted to a maximum value Δ. Hence, while the unrestricted stopping power, S_{el}(E), considers the sum of all energy losses \(\mathcal W\) up to a maximum value \({\mathcal {W}}_{\text {max}}\) (E/2 for electrons and E for positrons), S_{el}(E,Δ) excludes energy losses in the interval \(\Delta < {\mathcal {W}} \le {\mathcal {W}}_{\text {max}}\). A secondary electron ejected from the atomic ishell has a kinetic energy \(\varepsilon _{i}={\mathcal {W}} U_{i}\), where U_{ i } is the electron binding energy of the shell. Note then that S_{el}(E) includes the sum of all the kinetic energies of secondary electrons, ε_{ke}, plus their binding energies U_{B}, i.e.,$$ S_{\text{el}}(E)=\varepsilon_{\text{ke}}+U_{\mathrm{B}}, $$(5)whereas S_{el}(E,Δ) includes the sum of secondaryelectron kinetic energies below Δ, ε_{ke≤Δ}, plus their binding energies, i.e.$$ S_{\text{el}}(E,\Delta)=\varepsilon_{\text{ke} \le \Delta}+U_{\mathrm{B}}, $$(6)The new definition of LET, L_{ Δ }(E), given by ICRU85 excludes from S_{el}(E) the kinetic energy of secondary electrons when is higher than Δ; it does not exclude their binding energy. Hence,$$ L_{\Delta} (E) = S_{\text{el}}(E)  \varepsilon_{\text{ke} > \Delta}, $$(7)where ε_{ke>Δ} is the sum of the kinetic energies of secondary electrons above Δ. As a consequence, L_{ Δ }(E)=S_{el}(E) for E≤Δ, and for low energies L_{ Δ }(E)>S_{el}(E,Δ). Reference [3] could be consulted for further details. Figure 1 shows ratios L_{ Δ }(E)/S_{el}(E) and S_{el}(E,Δ)/S_{el}(E) for different values of Δ, as a function of the incident electron kinetic energy, where significant discrepancies between the two ratios can be observed for values of Δ below 10 keV.  (ii)
The cavity is large compared to the electron ranges involved.
In this case f can be identified with the ratio of mass energyabsorption coefficients averaged over the photon spectrum, and chargedparticle tracks are assumed to be inside the cavity. Assuming again that the cavity does not perturb the photon fluence, i.e., (Φ_{ k })_{med}≈(Φ_{ k })_{det}, the mass energyabsorption coefficients ratio, [μ_{en}(k)/ρ]_{med,det}, is defined as:$$ \begin{aligned}f_{\text{med,det}}(Q) \equiv \left[\mu_{\text{en}}/\rho \right]_{\text{med,det}} =\frac {\int\limits_{0}^{k_{\max}} k \, \left[\Phi_{k} \right]_{\text{med}}\, \left[\mu_{\text{en}}(k)/\rho \right]_{\text{med}}\,\mathrm{d}k} {\int\limits_{0}^{k_{\max}} k \, \left[\Phi_{k} \right]_{\text{med}}\, \left[\mu_{\text{en}}(k)/\rho \right]_{\text{det }}\,\mathrm{d}k} \end{aligned} $$(8)where the different quantities have been defined above. Note that [μ_{en}(k)/ρ]_{med,det} is a quotient of energy fluenceweighted averaged mass energyabsorption coefficients.
Recall that the mass energyabsorption coefficient, μ_{en}/ρ, accounts for the local energy deposition by photongenerated charged particles, i.e., it excludes the radiative energy that escapes the local volume. The latter is, on the other hand, included in the mass energytransfer coefficient, μ_{tr}/ρ, which accounts for the photon energy transferred to kinetic energy of the generated charged particles (see, e.g., ref. [3]).
A third option exists for intermediate cases, based on Burlin’s cavity theory, which combines the approaches (i) and (ii) described above, but, from a MC calculation point of view, there is no difference in the way in which s_{med,det} and [μ_{en}/ρ]_{med,det} are evaluated.
Dosimetric key quantities
From the previous section one can infer that the key quantities needed for cavity theory are mass stopping powers, restricted S_{el}(E,Δ)/ρ and unrestricted S_{el}(E)/ρ, for different types of chargedparticles, and mass energyabsorption coefficients μ_{en}(k)/ρ, although Monte Carlo calculations involve also mass radiative stopping power S_{rad}(E)/ρ. Both S_{el}(E)/ρ and S_{el}(E,Δ)/ρ have an appreciable dependence with the fundamental quantity mean excitation energy, the so called Ivalue. Interested readers can find a detailed description of the formulation of the mean excitation energy in ICRU Report 90 [5]; the impact of the Ivalue in radiotherapy dosimetry has been discussed at length in refs. [3, 6, 7].
Involved also in certain dosimetric expressions, like in the beam quality factor \(k_{Q,Q_{0}}\) for reference dosimetry (e.g., in the Code of Practice IAEA TRS398 [8]), is the mean energy to create an ion pair in air, the W_{air}value, which for highenergy electrons and photons has the value W_{air}=33.97 eV or W_{air}/e=33.97 J C ^{−1}, e being the elementary charge.
All the quantities above have recently been updated for water, air and graphite by ICRU Report 90 [5], superseding the values given in ICRU Reports 37 [9] and 49 [10], and now being adopted by standard laboratories. As a consequence, the basic data used in MC simulations, and for most of the available stoppingpower ratios s_{med,det} and ratios of mass energyabsorption coefficients [μ_{en}(k)/ρ]_{med,det}, should be updated to avoid breaking the consistency of the dosimetry chain. (Note that there is an ongoing IAEA project to update TRS398 on this regard).
Monte Carlo calculation of dosimetric quantities
Stoppingpower ratios for reference dosimetry
Once the dosimetric key quantities have been adopted, both in the MC code at hand and for solving the cavity integrals, the problem basically is restricted to the calculation of the relevant charged particle (primary or total) or photon fluence, differential in energy, by scoring tracklength spectra.
It is not an overstatement to claim that many of the MC advances in the field have been related to developments on electron transport algorithms, especially in the presence of interface boundaries, where tracklength segments may become so short that multiple scattering theories are no longer valid under the socalled condensed history technique developed by Berger more than 50 years ago [11]. Many of the current MC systems include such technique, rather than the interactionbyinteraction (single scattering) type of simulation often used for lowenergy transport simulation. An extreme case to deal with is that of an airfilled ionization chamber, whose detailed simulation has posed a considerable challenge for years, to the extreme that only two of the MC systems generally available, PENELOPE [12] and EGSnrc [13], can yield accurate results^{3}. Computer codes based on these systems can switch from multiple to singlescattering physical models whenever tracklength segments shorten at the proximity of an interface boundary. Except at low photon energies, where photoabsorption and atomic radiative and nonradiative transitions need to be accounted for properly, photons are in principle simulated in a rather straightforward manner, interactionbyinteraction. However, the large number of generated secondary and higherorder electrons brings us to the beginning of this paragraph: there is no reliable photon transport simulation without an accurate treatment of electron transport.
The calculation of dosimetric quantities and correction factors for radiotherapy measurements can be considered to have been initiated by the work of Berger and Seltzer [14], whose results served to benchmark other MC codes in the early days. As is wellknown, a basic dosimetric quantity for absorbed dose determination using ionization chambers is the watertoair stoppingpower ratio, s_{w,air}, which is determined from MCcalculated electron fluence using Eqs. (3) and (4). This type of calculations was pioneered by Berger et al. [15] for electron beams using slowingdown spectra at different depths; they were later improved by Nahum [16] for the evaluation of SpencerAttix s_{w,air} values including the trackend term. For photon beams, Nahum [16] calculated for the first time stoppingpower ratios, scoring electron fluence spectra at various depths and solving subsequently the relevant cavity integrals introducing the trackend term. Values of s_{w,air} correlated with clinical photon beam quality specifiers were computed by multiple authors (see, e.g., refs. [17–20]), as well as for electron beams [21], providing the data included in most dosimetry protocols like IAEA TRS398 [8] and AAPM TG51 [22]. It should be pointed that most of the currently available s_{w,air} data performs the calculations internally during the MC simulation, rather than computing Φ_{ E } first and evaluating subsequently the cavity integrals described above.
Influence of the detector: perturbation factors
As emphasized for Eqs. (3) and (4), the calculation of stoppingpower ratios is based on the fundamental BraggGray CPE approximation Φ_{med}≈Φ_{det} for the electron spectra. Inserting a detector in the medium results in a change in the electron spectrum within the detector radiation sensitive volume relative to that in the homogeneous medium, i.e., CPE strictly fails due to the influence of the detector size, shape and construction materials. The effect is known as a perturbation.
where p_{dis} accounts for the effect of replacing a volume of water by that of the detector, p_{wall} accounts for the presence of nonwaterequivalent materials in the detector body and walls, p_{fl} corrects for the intrinsic difference in fluence between water and the detector volumes, and p_{cel} and p_{stem} correct for the presence of a central electrode and stem, respectively, if they are relevant to the type of detector involved.
The MC calculation of perturbation correction factors for ionization chambers and other types of detectors has received special consideration due to the electron simulation difficulties mentioned in the previous section. The first MC calculations on ionization chamber correction factors were made for ^{60}Co inair measurements, and the simulations by Bond et al. [23], Nath and Schulz [24] and McEwan and Smyth [25] deserve being mentioned. Their results showing a dependence with the chamber dimensions contradicted, however, BraggGray theory and prompted critical publications by others that revealed the importance of interface effects (mostly related to multiple scattering). These led to the development of an algorithm termed EGS4/PRESTA [26] that made simulations of chambers more accurate, obtaining uncertainties of the order of 1%, a very good figure in the late 1980s. The approach was improved later on by a PRESTA2 algorithm, which is included in the EGSnrc MC system). The PENELOPE system, on the other hand, uses a different approach and has never been affected by the kind of interfaces effects shown by the EGS4 system.
At the time when the interface effects were realized, Smyth [27] demonstrated that the conditions required by Fano’s theorem for CPE conditions in a medium could be simulated with a fictitious experiment. This considered a cavity filled with the same material as the surrounding medium, but in a gaslike form, i.e., having the same cross sections but a very large difference in mass density. Simulations of this experiment were made by Seuntjens et al. [28] using EGSnrc and by Sempau and Andreo [29] using PENELOPE. The agreement with Fano’s theorem was of the order of 0.1%, a level that no other MC system has been able to achieve so far. The stateoftheart for this type of calculations is that linacs phasespace data (see next section) are used as radiation sources to simulate the response of ionization chambers based on the detailed description of their geometry.
In recent years a large number of calculations similar to those described above have been made for the dosimetry of small megavoltage photon fields, where other types of detectors have been simulated. Correction factors have been calculated using MC for mini and micro ionization chambers, silicon diodes, natural and synthetic diamonds, etc. The resulting data have been included in the IAEA TRS483 Code of Practice for the dosimetry of small static megavoltage photon beams (c.f. ref. [31] and references therein).
Issues on calculated perturbation factors
The applicability of the BraggGray approximation commented above raises a rather special type of issue in the MC calculation of perturbation factors.
It should be stated first that an absorbed dose calculation made with MC does not require CPE. However, our current formulations for f_{med,det}(Q) (e.g., stoppingpower ratios) rely on CPEbased expressions. The condition for the current BraggGray approach, i.e., assuming that Φ_{med}≈Φ_{det} and that the different perturbation corrections p_{det,i} in Eq. (11) are independent, still requires small perturbation correction factors to be able to assume that they are independent of each other.
In recent years it has been realized that, in small megavoltage photon beams, the MC calculated correction factors for many specific detectors can be very large (up to ∼ 10% for small ionization chambers) and often CPE is lacking. This means that the BraggGray assumptions used so far break down.
where \(\bar D_{\text {chair}}\) and D_{w}(P) are the MCcalculated mean absorbed dose in the chamber cavity and the dose to a point in water (a very small volume), respectively. Note that for conventional field sizes (i.e., nonsmall beams) Eq. (12) corresponds to Eq. (9), and that no specific perturbation correction factors are explicitly included.
It can then be concluded that solving the fluencebased cavity integrals (always assumed to be under CPE), discussed in the Background section, is no longer needed for practical dosimetry. In addition, it can also be stated that the assumption of small and independent p_{det,i} should no longer be needed in dosimetry.
The procedure in Eq. (12), which can be referred to as a global f_{ch}(Q) that includes s_{w,air} and all possible perturbations, irrespective of their size or interrelation (e.g., not being completely independent), has become the currently accepted MC calculation approach. It differs from that used by other authors (e.g., refs. [33, 34]), where instead of the dose to a point, D_{w}(P), the dose to water was calculated in a volume identical to that of the chamber, D_{w}(vol); it should be recalled, however, that BraggGray theory yields the absorbed dose at one point in the medium.
Detailed fluence spectra and subsequent perturbationcorrection calculations will, however, continue to be useful for analysing the influence of different components in the design of detectors (or for pedagogic purposes). For this purpose, electron fluence inside detectors where the composition of certain components can be varied (see, e.g. ref. [35]), provides a very efficient MC tool.
Monte Carlo treatment planning (MCTP)
Since the 1990s, a number of fruitful MC developments have been made for the direct calculation of dose distributions within a patient using linacs phasespace data impinging on 3D CT images. There were some early developments (see, e.g., an early MC review by this author [1]), but realistic MCTP could not be implemented and become a reality as a clinical tool until today’s considerable computing power was available.
At this point it is interesting to recall that the simulation of accelerator treatment heads was pioneered by the work of Petti et al. [36], Mohan et al. [37] and Udale [38], all using the EGS4 system [39]. Currently, the EGSnrcbased BEAM user code [40] is probably the most widely used piece of software for this purpose; it was developed within a major project called OMEGA, designed for treatment planning purposes [41, 42]. Other MC systems incorporating accurate geometry packages like MCNP6 [43], PENELOPE [12] and GEANT4 [44] have been used to simulate specific accelerator models, and user codes like the GEANT4based GAMOS [45], and the PENELOPEbased PENLINAC [46], PENEASYLINAC [47] and PRIMO [48] have been developed and are in current use.
“Contemporary” developments in MCTP include the already mentioned OMEGA project, the Macro MC (MMC) code designed for electron treatment planning [49], the VMC and XVMC codes [50, 51]), the PEREGRINE system [52] focused on photon calculations, and the PENELOPEbased DPM [53] and PRIMO [48]. The latter is rather unique in the sense of being a comprehensive system that includes in a single package the simulation of linacs and patient doseplanning calculations (plus a number of beam analysing graphical tools). Some of the MC codes or systems mentioned are implemented in commercial treatment planning systems (TPS), while DPM and PRIMO are freesoftware packages.
In favour of the use of MCTP it could be argued that while most analyticalbased algorithms for treatment planning are adequate for calculations in homogeneous media, they have been shown to be rather crude approximations whenever inhomogeneities are present. MC simulations are not constrained a priori by the composition and/or density of the medium, and their superior dose distributions over analyticalbased calculations have been thoroughly demonstrated (see the textbook [54] and references therein). Additionally, costfree MC packages and sufficient computer power are available today at most desks. MC has then become an ideal tool for the simulation of radiation transport using any media and geometry, and MCTP is claimed to yield results within the requirements for TPSs even with inhomogeneities (experimentally verified dose differences are smaller than 2%).
Unfortunately, for sake of speed, some of the commercial MCTPs are based on MC codes trimmed for lowZ media, limiting for instance the number of materials they can handle (i.e., grouping similar tissues). A related issue is that of the ALARA uncertainty in MCTP, which will be discussed next.
Issues on MCTP
 (i)
Should MCTP calculate absorbed dosetotissue (D_{tis}) or dosetowater (D_{w})?
 (ii)
Does MCTP inherently calculate D_{tis} accurately?
 (iii)
How accurate is the conversion between D_{tis} and D_{w}?
 (i)In favour of using dosetowater:
 (a)
D_{w} is the basis for current clinical experience and trials, meaning that compliance with experience, mainly developed with conventional TPSs, and with established criteria for therapeutic and normaltissue tolerance, is required.
 (b)
The calibration of radiotherapy beams is always made in terms of the reference absorbed dose to water, which is used for any TPS dose normalization.
 (a)
 (ii)In favour of using dosetotissue:
 (a)
D_{tis} is the quantity inherently computed exactly by MCTP.
 (b)
Differences between D_{tis} and D_{w} for “waterlike tissues” is small and likely to have minimal clinical impact.
 (c)
Converting between D_{tis} and D_{w} introduces additional uncertainty in the treatment planning process, but a relation between D_{tis} and D_{w} is still necessary because of the normalization to the beam calibration reference dose to water.
 (a)

With regard to the inherently exactcharacterof MCTP calculations one could argue that, in addition to the sometimes oversimplified physical models implemented in certain MCTP systems, all existing methods for tissue segmentation, where densities are obtained from CT data, and used on a lookup table to assign different tissue types, neglect patienttopatient variation of tissue compositions, and assume that these are patientindependent (they use ICRU or ICRP compositions). This approach, used by practically all TPS types, collides with the statement by ICRU Report 44 [57]: “It is imperative that bodytissue compositions are not given the standing of physical constants and their variability is always taken into account”. The rationale for this categorical statement is that tissue compositions given in ICRU or ICRP reports are average values obtained from a reduced set of humanbody samples, and is the main reason why stopping powers for tissues, tabulated for example in ICRU Reports 37 [9] (electrons and positrons) and 49 [10] (protons and alpha particles), are estimated to have an uncertainty of the order of 10–15%.
To understand this uncertainty estimate, one should recall that mass electronic stopping powers, S_{el}/ρ, unlike photon mass energyabsorption and energytransfer coefficients, depend on the material density entering into the densityeffect correction, δ, but, in addition, the full dependence of δ for a given medium is through \(\delta _{\text {med}} = \text {function} \left [(\rho \,Z/A)_{\text {med}},I_{\text {med}}^{2},E\right ]\), which shows that S_{el}/ρ depends considerably on the meanexcitation energy of the medium [7]. Obtaining this Ivalue requires the detailed atomic composition of the medium (electron distributions per shell) for a theoretical calculation, or an experimental determination using measurements with heavycharged particles, an approach unrealistic to accomplish for individual body tissues as is done, for instance, for some compounds. Furthermore, even if tissue compositions were known, for example through MRspectroscopy, the usual Braggadditivity rule is a crude approximation that ignores aggregate effects, justifying the large uncertainties estimated for bodytissues stopping powers.
Calculations reported in ref. [55] for certain tissues, where their Ivalues were changed by ± 15 % with respect to the nominal values given in ICRU37, showed substantial discrepancies in the respective stopping powers; they are illustrated in Fig. 7a. The differences, shown for a change in I_{adipose} from 63.2 to 55 eV, in bone for I_{bone} values of 91.9 and 106.4 eV (as given by ICRU and ICRP, respectively, both for the same ρ_{bone}=1.85 g cm ^{−3}), and in water for I_{w} from 75 to 86 eV plus a fictitious case for water with ρ_{w}=2.0 g cm ^{−3}, are evident and reach up to several percent. The differences are clearly higher at low electron energies but, as is shown in Fig. 7b, for a 6 MV photon beam 50% of the dose in a water cylindrical volume of 5 cm diameter and 1 cm height at 10 cm depth is due to electrons below 0.75 MeV approximately.The answer to the inherently exact character of MCTP calculations is therefore negative. Generic Ivalues represent a major limitation on any MTCP (and even on full MC systems), as individual bodytissue stopping powers are required. One can only state that MCTP has an intrinsic uncertainty (type B) which is ALARA (As Low As Reasonably Achievable), or rather, ALAHBA (As Low As the Human Body Allows). Hence, this issue questions the claimed lowuncertainty of MCTP, even if the method is still superior to that of analytical algorithms.

The remaining important issue is the conversion between D_{tis} and D_{w}, applicable to MCTP or to any other type of TPS, to relate a calculated D_{tis} to the reference dose to water obtained at beam calibration. The ratio between the two absorbed doses can be written as:$$ \frac{D_{\text{tis}}}{D_{\mathrm{w}}} \approx \frac {\int\limits_{0}^{E_{\max}} \Phi_{E,\text{tis}}^{\text{prim}}\; \left[S_{\text{el}}(E)/\rho\right]_{\text{tis}}\;\mathrm{d}E} {\int\limits_{0}^{E_{\max}} \Phi_{E,\mathrm{w}}^{\text{prim}}\; \left[S_{\text{el}}(E)/\rho\right]_{\mathrm{w}}\;\mathrm{d}E} $$(13)
which is strictly a ratio of cemas, and only under CPE conditions the approximate sign can be replaced by equal (note that cema is used instead of restricted cema for simplicity in the formulation). This equation should not be confused with a BraggGray stoppingpower ratio, see Eq. (3), as it now includes the electron fluence in tissue and in water in the numerator and denominator, respectively, as fluence is different in water and in “not so waterlike tissues” like bone or adipose matter. Hence, there is an implicit statement on \(s_{\text {tis,w}}^{\text {BG}} \ne D_{\text {tis}}/D_{\mathrm {w}}\).
Equation (13) points at that the widely used conversion of Siebers et al. [58], where the “converted dosetowater” is calculated as \(D_{\mathrm {w}}^{\text {conv}} = D_{\text {tis}}^{\text {MC}}\; s_{\text {w,tis}}\), does not seem to be correct. To illustrate this statement, one can be write$$ \begin{aligned} s_{\text{w,tis}} \,=\, \frac{\bar s_{\mathrm{w}}}{\bar s_{\text{tis}}} \!= &\frac{{\int\limits_{0}^{{E_{\max }}} {\Phi_{E,{\mathrm{w}}}^{{\text{prim}}}{{\left[ {{S_{{\text{el}}}}(E)/\rho} \right]}_{\mathrm{w}}}{\mathrm{d}}E} \;{\Big /}\;\int\limits_{0}^{{E_{\max }}} {\Phi_{E,\mathbf{w}}^{{\text{prim}}}{\mathrm{d}}E} }}{{\int\limits_{0}^{{E_{\max }}} {\Phi_{E{\mathrm{,tis}}}^{{\text{prim}}}{{\left[ {{S_{{\text{el}}}}(E)/\rho} \right]}_{{\text{tis}}}}{\mathrm{d}}E} \;{\Big /}\;\int\limits_{0}^{{E_{\max }}} {\Phi_{E,{\text{tis}}}^{{\text{prim}}}{\mathrm{d}}E} }}\\ \!= &\frac{{\int\limits_{0}^{{E_{\max }}} {\Phi_{E,{\mathrm{w}}}^{{\text{prim}}}{{\left[ {{S_{{\text{el}}}}(E)/\rho} \right]}_{\mathrm{w}}}{\mathrm{d}}E} }}{{\int\limits_{0}^{{E_{\max }}} {\Phi_{E,{\text{tis}}}^{{\text{prim}}}{{\left[ {{S_{{\text{el}}}}(E)/\rho} \right]}_{{\text{tis}}}}{\mathrm{d}}E} }}\;\frac{{\int\limits_{0}^{{E_{\max }}} {\Phi_{E,{\text{tis}}}^{{\text{prim}}}{\mathrm{d}}E} }}{{\int\limits_{0}^{{E_{\max }}} {\Phi_{E,{\mathrm{w}}}^{{\text{prim}}}{\mathrm{d}}E} }} \equiv \frac{{{D_{\mathrm{w}}}}}{{{D_{{\text{tis}}}}}}\;\frac{{\Phi_{{\text{tis}}}^{{\text{prim}}}}}{{\Phi_{\mathrm{w}}^{{\text{prim}}}}} \end{aligned} $$(14)which shows that a fluence correction factor is required for converting between D_{tis} and D_{w}, leading to$$ D_{\mathrm{w}}^{\text{conv}} = D_{\text{tis}}^{\text{MC}}\; s_{\text{w,tis}}\;\frac{\Phi_{\mathrm{w}}^{\text{prim}}}{\Phi_{\text{tis}}^{\text{prim}}} $$(15)This is a conclusion that parallels the wellknown expression for reference dosimetry given in Eq. (10), where the corresponding perturbation factor can now be identified with a ratio of fluences in both media. The fluence correction factor, written as$$ k_{\Phi} = \frac{\Phi_{\mathrm{w}}}{\Phi_{\text{tis}}} $$(16)is shown in Fig. 8 for a 6 MV photon beam onto various media and tissues, including those mentioned above where their Ivalue was modified. It also includes values for “fictitious water” having densities of 2 and 10 g cm ^{−3}, demonstrating that, like the mass stopping powers discussed above, fluence does not depend substantially on the density of materials provided they have identical composition.The magnitude of the correction factors, particularly for bone, and to a lesser extent for adipose tissues, points at the need for using a fluence correction factor, k_{ Φ }, if differences of up to approximately 5% are clinically relevant. Hence, the argument on differences between D_{tis} and D_{w} “being small and not having much clinical impact” is a clinical decision based on the degree of accuracy required at a given radiotherapy facility.Similar calculations to those described for highenergy photons have been done in ref. [59] for dose conversions in brachytherapy dosimetry with ^{125}I, ^{131}Cs and ^{103}Pd sources, where the short electron ranges involved (corresponding to the case of a large detector in the Background section), make the correction factor to be in terms of an energyfluence ratio, i.e.,$$ \begin{aligned} D_{\mathrm{w}}^{\text{conv}} =& D_{\text{tis}}^{\text{MC}}\; \left (\bar \mu_{\text{en}}/\rho \right)_{\text{w,tis}}\;\frac{\Psi_{\mathrm{w}}}{\Psi_{\text{tis}}}\\ k_{\Psi} =& \frac{\Psi_{\mathrm{w}}}{\Psi_{\text{tis}}} \end{aligned} $$(17)
Conclusions
The use of the Monte Carlo method for calculations in radiotherapy dosimetry has become the most efficient and consistent tool for simulations in most of the fields related to the speciality, from basic dosimetric quantities, like stoppingpower ratios and perturbation correction factors for reference ionization chamber dosimetry, to fully realistic simulations of clinical accelerators, detectors and patient treatment planning. Its accurate use requires consistency in the data throughout the entire dosimetry chain, and the recent updates of key dosimetric data by ICRU Report 90 are necessary in reference dosimetry. Although data consistency is probably less critical for treatment planning, their implementation also in this field is advised. There are, however, a number of other issues raised throughout this work to conclude with the recommendation that no MC calculation should be considered free of errors. This is particularly important with regard to applications in MC treatment planning, where the uncertainties involved still remain “uncertain”, a general problem that is also applicable to other methods and algorithms used in different types of treatment planning systems.
Recall that, strictly speaking, the quantity absorbed dose cannot be measured; it is always determined from measurements of related quantities, like charge, current, heat, chemical changes, etc., using appropriate correction and conversion factors.
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