Dose to 'water-like' media or dose to tissue in MV photons radiotherapy treatment planning: still a matter of debate

As is well known, Monte Carlo-based treatment planning, being based on the computer simulation of radiation transport through a patient geometry using well established cross-section data, requires the detailed composition and mass density of the different sub-volumes (voxels) that compose the geometry.

The density of the different voxels is obtained from a CT examination, where an effective linear attenuation coefficient μ_eff for each voxel is determined. Note the emphasis on 'effective', as CT beams are far from narrow and scattering precludes the use of the conventional linear attenuation coefficient μ, obtained by measurement or calculation in narrow beam conditions. Thus the use of approximate expressions to derive a theoretical value of μ (see Schneider et al 1996) does not appear to be a proper choice. A voxel CT number, N_CT is derived as

$$\begin{eqnarray}&&{{N}_{\text{CT}}}=1000\frac{{{\mu}_{\text{eff},\text{t}}}-{{\mu}_{\text{eff},\text{w}}}}{{{\mu}_{\text{eff},\text{w}}}}\end{eqnarray} \tag{ 1 }$$

and as it is assumed that the Compton effect dominates photon interactions in human-body tissues at CT energies, N_CT is proportional to the voxel electron density $\rho _{\text{t}}^{\text{e}}$. A CT calibration is then performed using reference materials of known composition and electron and mass density so that CT numbers are related to density. Electron and mass density, $\rho _{\text{t}}^{\text{e}}$, and ρ_t, are related by the well-known expression

$$\begin{eqnarray}&&\rho _{\text{t}}^{\text{e}}={{\rho}_{\text{t}}}{{N}_{\text{A}}}\underset{i}{\sum} {{w}_{i}}\frac{{{Z}_{i}}}{{{A}_{i}}}\end{eqnarray} \tag{ 2 }$$

where N_A is the Avogadro constant, and w_i, Z_i and A_i are the fraction by weight, atomic number and atomic mass of the constituent atomic element i in the reference material. From a calibration curve, patient voxel CT numbers can then be related to a mass or electron density.

Some authors (see Seco and Evans (2006) and references therein) have pointed out that scaling distances and thicknesses by mass density in treatment planning leads to an overestimation of the dose, especially for bone and air, and it is the electron density that should be used for scaling. This is not the case with several treatment planning systems (TPS), and scaling by mass density is widely used. A figure from these authors illustrated the difference in the relation between CT number and density using the two density types.

In addition to the approximations mentioned it should also be noted that there might be different tissue types within a voxel, which still is related to a single density.

Assigning tissue types to the CT-densities obtained as above is known to be the weakest link of the calculation chain. A widely used method is to consider groups of different human-body tissues that are assigned to a given density (and CT number) range, using four or five groups. These vary between the different implementations and authors, for example the grouping by DeMarco et al (1998) included lung, fat, water, muscle and bone, whereas Ma et al (1999) used air, lung, tissue and bone. 'Standard' compositions (i.e. those given in ICRU or ICRP reports) are then used for each selected tissue type assigned to the voxel density.

An alternative to this method is to assign a given density (electron or mass) to a stopping power using a look-up table for 'standard' body-tissue compositions versus density, the latter being obtained as described above. The rationale for this approach is based on an early work by Kawrakow et al (1996) on MCTP for high-energy electron beams, which established a 'universal fit' linking cross sections and stopping powers for multiple materials, scaled relative to water, to a tissue density also relative to water. For the mass electronic stopping powers (formerly called mass collision stopping powers, see ICRU 2011) they gave the expression

$$\begin{eqnarray}\frac{{{S}_{\text{el}}}\left({{\rho}_{\text{t}}},E\right)/{{\rho}_{\text{t}}}}{S_{\text{el}}^{\text{w}}\left({{\rho}_{\text{w}}},E\right)/{{\rho}_{\text{w}}}}={{f}_{c}}\left({{\rho}_{\text{t}}}/{{\rho}_{\text{w}}}\right)=\left\{\begin{array}{*{35}{l}} {{\left({{\rho}_{\text{t}}}/{{\rho}_{\text{w}}}\right)}^{-0.17}} & {{\rho}_{\text{t}}}\geqslant 0.795\text{}\text{ g c}{{\text{m}}^{-3}} \\ 1.039 & \text{all other}\text{}{{\rho}_{\text{t}}} \\ \end{array}\right.\end{eqnarray} \tag{ 3 }$$

whereas for the mass radiative stopping powers the function was

$$\begin{eqnarray}\frac{{{S}_{\text{rad}}}\left({{\rho}_{\text{t}}},E\right)/{{\rho}_{\text{t}}}}{S_{\text{rad}}^{\text{w}}\left({{\rho}_{\text{w}}},E\right)/{{\rho}_{\text{w}}}}={{f}_{\text{rad}}}\left({{\rho}_{\text{t}}}/{{\rho}_{\text{w}}}\right)=\left\{\begin{array}{*{35}{l}} 1.13+0.56 \ln \left({{\rho}_{\text{t}}}/{{\rho}_{\text{w}}}-0.3\right) & {{\rho}_{\text{t}}}\geqslant 0.9\text{}\text{ g c}{{\text{m}}^{-3}} \\ 1.049-0.228~{{\rho}_{\text{t}}}/{{\rho}_{\text{w}}} & \text{all other}\text{}{{\rho}_{\text{t}}} \\ \end{array}\right.\end{eqnarray} \tag{ 4 }$$

where ρ_t is the mass density of a given material or tissue and ρ_w that of water. The latter is commonly taken to be one, which corresponds to 4 °C; at a room temperature of 20 °C ρ_w = 0.9982 g cm⁻³. It should be noticed that in the case of water, taking ρ_t ≡ ρ_w = 1, the expression for f_rad(ρ_t/ρ_w) does not yield one but 0.930; however, the $S_{\text{tot}}^{\text{w}}(E)/\rho$ value obtained is correct.

Equation (3) was plotted by these authors for electron energies of 5, 15 and 30 MeV, showing good agreement with data given by ICRU (1992) for different body tissues. It is emphasized that the plot was for electrons with a rather high energy, as their work dealt with MCTP of electron beams at these energies. To our knowledge, no attempt has been made to verify the validity of expressions (3) and (4) for the electron energies involved in photon radiotherapy, often at 6 MV, particularly for the low energies of the electron spectra they generate. These expressions are, however, used in some commercial TPSs without further validation.

A common aspect of the methods described above, and of any other existing so far, is that they neglect the patient-to-patient variation of tissue compositions, and assume that the compositions given by several ICRU or ICRP reports are 'standard'. This collides with the statement given by ICRU (1989): 'It is imperative that body-tissue compositions are not given the standing of physical constants and their reported variability is always taken into account'. The reported compositions have been obtained under different conditions for a rather limited number of human-body samples, and are expected to be approximate averages that may vary with gender and ethnics. In addition, even if a tissue composition were accurately known, information on a key physical parameter (its mean excitation energy, see below) would still be necessary and an approximate method is used for its estimation. This means that mass stopping power values for body tissues are average estimates having an uncertainty considerably larger than that of water, up to the order of 10–15% (see ICRU 1984b).

Version 2 of the ESTAR computer code, from the US NIST (Berger 1993), for the calculation of stopping powers in elements, compounds and tissues, has been used. Compositions included in the code are adopted from ICRU (1984b), but it is possible to define other materials or make changes to the physical properties of those included in the package. For the present work, ESTAR has been extended for calculations at electron energies down to 1 keV, in the lower side of the limit of applicability of Bethe's theory for elements of low atomic number. Stopping powers for different tissues have been calculated, with average compositions taken from ICRP and ICRU reports, and changes in the density and mean excitation energy (I-value) have been made for some media to investigate the potential lack of constancy in stopping powers that would result from patient to patient. An arbitrary range of variation of 15% has been chosen for the I-values, based on the overall range for various body tissues given in ICRU reports 37, 44 and 46 ICRU (1984b, 1989, 1992). It should be emphasized that a 15% change in the I-value causes a mass stopping power variation much smaller than the ICRU-37 estimated uncertainty of 10–15%, as it enters logarithmicaly into the stopping-power formula (see equation (6) below).

It is important to recall that, in addition to the mass density ρ and average value of the ratio Z/A, their product being proportional to the electron density, see equation (2), the mean excitation energy of a material (its I-value) plays a key role on the electronic mass stopping-power values. Note that the ratio Z/A is almost constant (0.55 ± 1%) for most tissues except bone. The lack of accurate data on the I-value of tissues makes necessary an estimation based on the well-known Bragg-additivity rule

$$\begin{eqnarray}&&\text{ln}\text{}I=\frac{\underset{i=1}{\overset{n}{\sum}} {{w}_{i}}~{{\left(Z\text{/}A\right)}_{i}}\text{}\text{ ln}{{I}_{i}}}{\underset{i=1}{\overset{n}{\sum}} {{w}_{i}} {{\left(Z\text{/}A\right)}_{i}}}\end{eqnarray} \tag{ 5 }$$

where, as in equation (2), w_i, Z_i, A_i and I_i are the fraction by weight, atomic number, atomic mass and mean excitation energy of the constituent atomic element i in the tissue. It has been shown that the Bragg-additivity rule ignores aggregate effects in compound materials and is therefore rather approximate; substantial differences have been found in comparison with experimental determinations for many materials, see ICRU-37, thus justifying the large uncertainties quoted above for body tissues mass stopping powers.

Both ρ and I (or rather, the data used in the definition of the latter, see ICRU-37) influence the density effect that corrects the mass stopping power at the highest energies we will be dealing with. Recall that it is the logarithm of the I-value that enters into the electronic mass stopping power, i.e.

$$\begin{eqnarray}&&{{S}_{\text{el}}}/\rho \propto {{N}_{\text{A}}}\frac{Z}{A}\frac{1}{{{\beta}^{2}}}\left[f\left(\beta \right)- \ln I-\delta \left(\beta \right)\right]\end{eqnarray} \tag{ 6 }$$

where N_A Z/A is the number of electrons per gram of the medium, β is the electron velocity in units of the speed of light (β = v/c), f is a simple logarithmic function of β (or of the electron energy), and δ is the density-effect correction. It has been shown that the uncertainty of the I-value can modify substantially the dose distribution of protons and heavier charged particles in different tissues, adding uncertainty to the position of the Bragg peak and therefore to the heavy charged particle range (Andreo 2009, Besemer et al 2013). It remains to be seen if in the case of photon and electron beams its influence becomes as important for patient dose calculations, and the effects in megavoltage photons will be analyzed in this work.

The physical properties of materials and body tissues, with compositions taken from ICRU (1984b), and for which calculations have been made are given in table 1 of the Results section below. Departures from the average densities and I-values quoted by ICRU are indicated in the table.

Table 1. Ratios to the reference absorbed dose-to-water (D_w,ref) of doses to media calculated with the 'tissue' $\left(D_{\text{med}}^{\text{t}}\right)$, 'water-like' $\left(D_{\text{med}}^{\text{w}}\right)$, 'water-like-B' $\left(D_{\text{med}}^{\text{w-B}}\right)$ and 'tissue-fit' $\left(D_{\text{med}}^{\text{t}-\text{fit}}\right)$ approaches for 6 MV photons. The 'tissue' approach is based on the MC-calculated electron fluences and mass electronic stopping powers for each medium. The 'water-like' approach assumes all media to be identical to water but with different densities, electron fluences as in water and stopping powers for each medium. The 'water-like-B' approach is as the 'water-like', but with stopping powers from water scaled by the electron density of each medium. The 'tissue-fit' approach is based on the present MC-calculated electron fluences and stopping powers from the fit of Kawrakow et al (1996) given in equation (3).

Medium	Mass density g cm−3	Z/A	I-value eV	Ratio to Dw,ref of
				$D_{\text{med}}^{\text{t}}$	$D_{\text{med}}^{\text{w}}$	$D_{\text{med}}^{\text{w-B}}$	$D_{\text{med}}^{\text{t}-\text{fit}}$
water (Dw,ref)	1.00	0.5551	75	1.000	1.000	1.000	1.000
water_d2a	2.00	0.5551	75	0.989	0.989	1.000	0.889
water_d10b	10.0	0.5551	75	0.949	0.949	1.000	0.676
water_I86c	1.00	0.5551	86	0.987	0.987	1.000	1.000
ICRP-adipose	0.92	0.5585	63.2	1.006	1.023	1.006	0.998
ICRP-adipose_I55 c	0.92	0.5585	55	1.018	1.035	1.006	0.998
ICRP-bone (cortical)	1.85	0.5213	106.4	0.947	0.897	0.939	0.952
ICRU-bone (compact)	1.85	0.5301	91.9	0.955	0.926	0.955	0.929
ICRP-muscle	1.04	0.5494	75.3	0.989	0.989	0.990	0.993
ICRP-soft tissue	1.00	0.5512	72.3	0.996	0.996	0.993	1.000
ICRP-lung	1.05	0.5497	75.3	0.990	0.989	0.990	0.993
ICRP-lung inflated	0.26	0.5497	75.3	0.992	0.999	0.990	1.031

^aTwice the density of standard water. ^bTen times the density of standard water. ^cI-value 15% different from the standard medium.

A second piece of software used has been version 4.2.4 of the EGSnrc Monte Carlo system (Kawrakow et al 2011), particularly its user code FLURZnrc, for the calculation of electron spectra down to 1 keV generated at a depth of 10 cm in water by photons of 6 MV; at this position a cylindrical volume of 1 g cm⁻² height and 5 g cm⁻² radius filled with a given medium or body tissue is inserted, where the fluence is determined. Complementary calculations of the absorbed dose within the inserts using the code DOSRZnrc have also been done for comparison. The incident photon spectrum is one of the classic spectra calculated by Mohan et al (1985), which is included in the EGSnrc package. Note that EGSnrc is based on the ESTAR code to obtain the density-effect corrections used in its formulation of mass restricted electronic stopping powers, hence there is consistency between the two parts of our calculations.

The reason for choosing 6 MV as a working incident spectrum is based on the predominant use of this energy worldwide, according to the most recent compilations of the International Atomic Energy Agency (IAEA) and the Imaging and Radiation Oncology Core (IROC) Houston QA Center (formerly the Radiological Physics Center, RPC). Their databases show that worldwide approximately 75% of the more than 9000 clinical accelerators registered have nominal accelerator potentials of 10 MV or less (Izewska 2014), a percentage that decreases to about 70% in the USA (Followill 2014). As will be discussed below, 50% of the absorbed dose at a depth of 10 cm in water by 6 MV and 15 MV photon beams is due to electrons with energies approximately below 0.6 MeV and 1.3 MeV, respectively. Considering that electrons in this low-energy range play a fundamental role in most of the calculations presented below, it is reasonable to assume that the general results of this work, explained in terms of the electron spectra at depth and their energy deposition, are applicable up to at least 15 MV beams.

We distinguish three scenarios for absorbed dose calculations, which will be compared to other approaches:

• (a)  
  Tissue approachFrom stopping powers and MC-calculated spectra, the absorbed dose is determined from the primary electron fluence, assuming that δ-ray equilibrium exists, according to

  $$\begin{eqnarray}&&{D_{{\rm{med}}}}\mathop = \limits^{\delta - {\rm{eq}}} {C_{{\rm{med}}}} = \mathop {\mathop \int \limits^{{E_{{\rm{max}}}}} }\limits_0 \Phi _{E,{\rm{med}}}^{{\rm{prim}}}{\left[ {{S_{{\rm{el}}}}\left( E \right)/\rho } \right]_{{\rm{med}}}}{\rm{d}}E\end{eqnarray} \tag{ 7 }$$

  where $\Phi _{E,\text{med}}^{\text{prim}}$ is the primary electron fluence, S_el(E)/ρ is the electronic stopping power and 'med' refers to water or tissue. Recall that equation (7) defines the quantity cema, C_med (see ICRU (2011)). The validity of the assumption on δ-ray equilibrium for the configuration described above will be verified with absorbed dose calculations using DOSRZnrc. Note that for photon beams, when transient charged-particle equilibrium (TCPE) exists, the absorbed dose to a medium calculated from photon spectra and mass energy-absorption coefficients coincides with that given by equation (7).Calculating dose (or cema) ratios is made according to

  $$\begin{eqnarray}&&\frac{{{D}_{\text{med}}}}{{{D}_{\text{w}}}}=\frac{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{med}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{med}}}\text{d}E}{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{w}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{w}}}\text{d}E}\end{eqnarray} \tag{ 8 }$$

  that should not be confused with a Bragg–Gray type of stopping-power ratio, $s_{\text{med},\text{w}}^{\text{BG}}$, in which $\Phi _{E}^{\text{prim}}$ is assumed to be the same both in the numerator and in the denominator. A variant to this approach will be based on the use of stopping-power data evaluated according to the fit by Kawrakow et al (1996) described above. Note that equation (8) is used to normalize the calculated dose-to-medium to a reference dose-to-water, which can be determined according to a dosimetry protocol (e.g. IAEA TRS-398, Andreo et al 2000). It is therefore assumed that the denominator of equation (8) is such a reference dose.
• (b)  
  Water-like approachIt is assumed that the different body tissues are 'water-like', i.e. they are identical to water but have different density, obtained from the relevant CT-derived mass density of the calculation volume. This assumes, according to Fano's theorem (see e.g. Attix 1986), that the electron spectrum $\Phi _{E}^{\text{prim}}$ is practically the same in water-like tissue and in water, i.e. that the fluence is approximately independent of the medium density. The expression corresponding to equation (8) thus coincides with a stopping-power ratio $s_{\text{med},\text{w}}^{\text{BG}}$

  $$\begin{eqnarray}&&\frac{{{D}_{\text{med}}}}{{{D}_{\text{w}}}}=\frac{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{w}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{med}}}\text{d}E}{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{w}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{w}}}\text{d}E}\end{eqnarray} \tag{ 9 }$$
• (c)  
  Water-like-B approachFor dose calculations in the case of 'water-like' tissues, some authors (see e.g. Ma and Li (2011) and references therein) have used an approach that differs substantially from (b), in which a tissue is replaced by 'water with tissue electron density', its $\rho _{\text{t}}^{\text{e}}$ being obtained from CT as in the previous cases. It involves solely water stopping powers, which are scaled by the electron density of the relevant tissue, and therefore [S_el(E)/ρ]_med values do not enter in the process. This means that, under this approach, ratios of stopping powers for body tissues to those for water are energy independent. Hence, for each medium or body tissue, dose ratios to water are equal to the constant

  $$\begin{eqnarray}&&\frac{{{D}_{\text{med}}}}{{{D}_{\text{w}}}}=\frac{\rho _{\text{med}}^{\text{e}}}{\rho _{\text{w}}^{\text{e}}}=\frac{{{\rho}_{\text{med}}}}{{{\rho}_{\text{w}}}}\frac{{{(Z/A)}_{\text{med}}}}{{{(Z/A)}_{\text{w}}}}\end{eqnarray} \tag{ 10 }$$

  The implications of this over-simplication of stopping-power physics will be discussed below and its results compared with other approaches. To emphasize the difference with the previous method, this will henceforth be termed the 'water-like-B' approach.

The differences between the 'tissue' and the two 'water-like' approaches are in principle expected to depend on the different stopping powers and spectra in the two media, water and tissue, as from the discussion above on the influence of different quantities on mass stopping-powers it is clear that not many tissues can strictly be considered to be 'water-like'.

Results for mass stopping powers of different materials and tissues, including variations in their physical properties, and a comparison with Kawrakow et al (1996) model and with the 'water-like-B' approach are presented below.

Figure 1 shows ratios of calculated mass total stopping powers for different tissues to those for water for mono-energetic electrons. The curve labelled 'ICRP soft tissue' corresponds to 'all tissues other than osseous tissue, teeth, hair and nails, as well as all the body fluids, muscle-like tissues and fatty tissues (e.g. adipose tissue)', ICRU (1992). Curve 'water_d2' corresponds to water with a mass density of 2 g cm⁻³, whereas 'water_I86' corresponds to water with ρ = 1 g cm⁻³ but having an I-value of 86 eV, i.e. approximately 15% higher than the currently recommended value (75 eV). Similarly, curves 'ICRP-adipose' and 'ICRP-adipose_I55' correspond to adipose tissue with the same density (ρ = 0.92 g cm⁻³) but 15% different I-values, the currently used 63.2 eV and a 15% lower value (55 eV). The two bone curves (lowest) have identical density, ρ = 1.85 g cm⁻³, but rather different composition (ICRP-cortical versus ICRU-compact) and therefore their I-values differ, being 106.4 eV and 91.9 eV, respectively. Recall that a 15% change in the I-value causes a stopping-power variation much smaller than the ICRU-37 estimated uncertainty of 10–15%, as it enters logarithmicaly into the stopping-power formula, see equation (6).

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It can be seen that for soft tissues, or for those tissues considered to be similar to water, like muscle, stopping powers are within a few per cent of those for water but differ substantially for tissues like adipose and bone, both in the zone of the extreme densities (except in the case of inflated lung with ρ = 0.26 g cm⁻³, which can be considered similar to water in its I-value, see table 1). This raises a well-known potential concern with these tissues, whereas values for low-density lung don't differ much except at the highest energies (where the onset of the density effect occurs). Additional remarks can, however, be pointed out:

• (a)  
  'Water_d2' results are practically identical to those for standard water except for a minor difference starting at the energy where the onset of the density effect correction occurs (a few hundred keV). This means that mass density does not change much the stopping power value, as should be expected from the use of mass stopping power values.
• (b)  
  'Water_I86', having the same density as ordinary water but an I-value 15% higher, differs by up to 5% at the lowest energies. 'Adipose_I55', also with the same density as the currently used adipose tissue but with a 15% lower I-value, follows the same pattern. This also occurs with ICRP-cortical and ICRU-compact bone, both having identical density but different I-values. It is then clear that the I-value of a tissue is a much more critical parameter than its mass or electron density. The latter can be derived from CT measurements, but the former depends considerably on the exact tissue composition, which, as already stated, may vary from patient to patient and it cannot be assumed that tissue composition is a physical constant. The restrictions indicated above on the validity of the Bragg-additivity rule to derive I-values should, in addition, be taken into consideration.

It is also of interest to verify the accuracy of Kawrakow et al (1996) 'universal fit' for electron energies as those included in the spectra generated by photons, that extend down to very low energies (see below), because their fit is used in several TPS for stopping power look-up tables. Figure 2 shows a comparison of the sum of equations (3) and (4), yielding tissue mass total stopping powers $S_{\text{tot}}^{\text{t}}(E)/\rho$ relative to water (thick lines), with the detailed calculations using ESTAR; the curves shown for the latter (thin lines) are ratios tissue/water from figure 1, and for clarity only values for some tissues are reproduced. As expected, at high electron energies the ratios are approximately correct in most cases but some observations can be made on the grounds of the sometimes large differences observed.

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The first remark to be made on the fit is that it yields a ratio of stopping powers practically independent of the electron energy, i.e. the sum of equations (3) and (4) predicts a total stopping power relative to water that is constant except at energies and materials where radiative effects are of importance (the solid lines bend at high energies). In addition, as the fit is based solely on mass densities, it fails

• (a)  
  When the relation between density and composition does not follow the 'standard' pattern used for the fit, e.g. the ratio for high-density water falls close to that of bone, whereas stopping powers for water should be practically independent of density. This poses a problem when media similar to water are considered, for example inflated lung, much closer to water from the point of view of composition and physical interactions (except scattering) than what the fit predicts, but having much lower density.
• (b)  
  When tissue compositions are different from the ICRP or ICRU average values (e.g. to take into account patient-to-patient variations), and their I-values are different. For example, the fit cannot distinguish between the two ICRU and ICRP bone compositions as their density is identical. The same occurs with adipose tissue or with water assuming a 15% different I-value, that results in substantial stopping power differences. Recall that, as emphasized earlier, a 15% change in the I-value causes a stopping power variation much smaller than the ICRU-37 uncertainty estimation of 10–15%.

Based on the above, users should be aware of the limitations of the Kawrakow et al (1996) 'universal fit' at the low electron energies encountered in photon radiotherapy, which in spite of having been developed in the early stages of MCTP is, as already mentioned, still included in some new TPSs (e.g. Elekta 2013) without further validation.

The second comparison of interest refers to the degree of agreement between water stopping powers scaled with tissues electron density, i.e. the 'water-like-B' approach that neglects the influence of I-values, density-effect corrections and energy dependences, as well as real stopping powers for different tissues. This is the approach used in some MCTP comparisons (see Ma and Li (2011) and references therein). Figure 3 illustrates the ratio of the two quantities as a function of electron energy. It can be observed that differences are substantial except for tissues really close to water in density and I-value. As expected, adipose tissue and bone show the largest discrepancies but it is also of interest to emphasize that

• (a)  
  The ratio of stopping powers for tissues having the same electron density but different I-value are quite different, in consistency with the remarks made above with respect to the influence of the I-value
• (b)  
  Even for true 'water-like' media as 'water_d2', having different density but the same I-value as ordinary water, the ratio starts deviating from one at the energy where the onset of the density-effect correction occurs. The deviation can be observed more clearly for the unrealistic case of 'water_d10', i.e. water with a mass density of 10 g cm⁻³.

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If the influence of the density-effect correction is ignored, and the effect of the I-value is disregarded, it is obvious to expect that any calculation of dose-to-tissue using the 'water-like-B' approach will yield results proportional to those of ordinary water and independent of the electron energy, as stopping power ratios to water will differ exclusively by the corresponding electron density ratios, see figure 4 that is plotted at the same scale as figure 1. One may, however, question the extent to which a numerical or MC calculation based on such over-simplification of the physics governed by the properties of different media can be considered to be adequate.

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Monte Carlo calculated spectra, differential in energy, for the primary and total electrons generated at a depth of 10 cm in water by photons of 6 MV are shown in figure 5. As already indicated, the determination of absorbed dose from this spectra, providing the reference dose-to-water with which other calculations are compared, is based on the use of the quantity cema, i.e. the primary electron spectrum in water and mass electronic stopping powers, see equation (7). In the 'tissue' approach the spectra are evaluated within the volume insert, filled with the corresponding tissue and situated at the depth of 10 cm in water. In the 'water-like' approach $\Phi _{E,\text{w}}^{\text{prim}}$ is assumed to be identical in water and in the various tissues (by virtue of Fano's theorem), and is used together with the corresponding [S_el(E)/ρ]_med to determine the absorbed dose to each medium or tissue.

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It is illustrative to compare the different primary electron spectra in various tissues and thus verify the extent to which the assumption made in the 'water-like' approach is appropriate. Figure 6 shows $\Phi _{E,\text{med}}^{\text{prim}}$ for various tissues, and $\Phi _{E,\text{w}}^{\text{prim}}$ is also included for comparison (filled circles), although differences with tissues that can practically be considered to be water-like, are almost indistinguishable. Note that the ordinate scale is linear, unlike in figure 5 (the most common way of representing electron fluence spectra), so that differences among the various spectra can be observed more clearly.

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A thick-like line can be seen in figure 6, which groups together tissues with rather similar physical characteristics, notably their I-value. These spectra are clearly independent of tissue density, in agreement with Fano's theorem, as it can be seen for the two lung tissue cases that are almost identical and practically coincide with the water spectrum. However, the two curves above this group, for ICRU and for ICRP bone, with identical density but having different composition and I-values, appear clearly separated, showing that the fluence is higher than in water and that it increases with the I-value (because the stopping power decreases). On the other extreme, adipose tissue (the lowest curve), with its standard I-value approximately 40% lower than that of water, appears clearly under the water-like group, demonstrating again the influence of this key quantity whose determination remains problematic even if we could obtain the composition of a tissue (due to the Bragg-additivity rule constraint).

An important observation to be made in figure 6 is that, for the 6 MV photon beam investigated, the primary electron fluence distributions $\Phi _{E,\text{med}}^{\text{prim}}$ show a most probable value (the peak of the distribution) around 250 keV and the FWHM of the distribution extends from about 30 keV to 1 MeV. This means that the largest discrepancies shown in figures 1 and 2 for the lowest energies will not play a significant role, but those in the energy range mentioned will affect substantially the outcome of dose calculations. This aspect will be discussed further below in relation with the dose fraction due to low-energy electrons.

Once the electronic stopping powers and fluence spectra are determined, the calculation of the absorbed dose (or cema) for each approach proceeds using equations (7)–(9). As a verification of the numerical procedure used in the 'tissue' approach, which is based on a δ-ray equilibrium condition that strictly could be questioned, independent MC dose calculations were done with the DOSRZnrc user code for each media, using the same configuration as for the fluence calculations with FLURZnrc. Recall that MC calculations are independent of CPE, TCPE or δ-ray equilibrium conditions. It was found that the r.m.s. ratio of the quotients $D_{\text{med}}^{\text{t}}/{{D}_{\text{w}}}$ obtained with the numerical and MC calculations was 1.006, thus providing confidence on the numerical procedure used throughout this work.

Prior to comparisons of dose ratios, an initial analysis can illustrate some of the expected, and not so clearly expected, results that will be presented below. In preceeding sections it has been shown that for a given medium, when the I-value increases the mass stopping powers decrease and the primary electron fluence increases. Expecting that a balance between these opposite trends could be of importance, the product $\Phi _{E,\text{med}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{med}}}\text{d}E$ has been plotted in figure 7(a) as a function of energy for ICRP and ICRU bone, the tissues that so far have shown the largest discrepancies. The results show that even if there are still some small differences, the two curves are very close to each other, so that the resulting absorbed doses will differ much less than what a priori would have been expected solely from their stopping power differences. The ratio of these distributions is shown in the upper curve of figure 7(b), labelled 'fluences in bone', where it can be seen that significant differences appear only at both extremes of the energy range, where the values of the product represent very small contributions, see figure 7(a). This is not the case, however, if tissues are assumed to be 'water-like', see figure 7(b) curve labelled 'fluence in water', i.e. if the electron fluence is assumed to be identical in water and in all tissues irrespective of the different densities; note in addition that ICRP and ICRU bone tissues have identical density, but differ in their I-value.

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The cumulative integration of distributions like those illustrated in figure 7(a) yields what can be termed the Cumulative Dose Fraction (CDF), a distribution showing the fraction of absorbed dose due to electrons up to a given energy. The CDF for water is shown in figure 8 where it can be seen that 50% of the dose is due to the contribution of electrons with energies below approximately 0.55 MeV, thus confirming the importance of using correct mass stopping powers at low electron energies. Even for a 15 MV beam, using Mohan et al (1985) spectrum yields the CDF_50% approximately at 1.3 MeV; hence, low electron energies play also a key role in this energy range. It can then be expected that the general results on electron spectra at depth and dose ratios presented below are applicable up to about 15 MV.

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Results from the different dose determination approaches and approximations discussed so far are given in table 1 for 6 MV photons, which shows absorbed-dose ratios of different media to water; the latter will henceforth be referred to as the reference dose-to-water, D_w,ref. The different dose to media, columns (5)–(8), correspond to (a) the 'tissue' approach, based on MC-calculated electron fluences and mass electronic stopping powers for each medium using equation (8), (b) the 'water-like' approach, using equation (9), which assumes all media to be identical to water but having different densities, with electron fluences as in water and stopping powers for each medium, (c) the 'water-like-B' approach, identical to the 'water-like' approach, but for each medium it uses stopping powers from water scaled by the electron density of the medium, and (d) the 'tissue-fit' approach, identical to the 'tissue' approach with regard to the present MC-calculated electron fluences, but for each medium it uses stopping powers obtained from the scaling fit of Kawrakow et al (1996), equation (3). Note that, strictly, the 'tissue-fit' case should have been based on fluence calculations obtained from a MC simulation using look-up tables for the stopping powers from Kawrakow et al scaling fit; the last column in the table therefore corresponds to an approximate estimation.

It is not obvious which of the two 'water-like' cases should be considered more representative for the assumption that a tissue can be replaced by water, as arguments can be given in favour of one case or the other. Scaling electron stopping powers solely with the electron density yields incorrect stopping-power data, as the approach ignores tissues fundamental parameters, but obviously such over-simplified physics has made the procedure easier to handle in practice; this is probably the main reason why it has been widely used, e.g. to scale MC-calculated water kernels for convolution-superposition algorithms.

From the results in table 1, it can be observed in general that compared with the reference D_w,ref value

• (a)  
  Dose ratios with respect to mass density are generally close to one for media having not too different I-values. A clear example is water, for which comparing the value obtained for ρ_w = 1 g cm⁻³ with that obtained duplicating its density (water_d2, i.e. similar to bone) yields only 1% difference for $D_{\text{med}}^{\text{t}}$ and $D_{\text{med}}^{\text{w}}$, being identical for $D_{\text{med}}^{\text{w}-\text{B}}$ as expected, and considerable different for $D_{\text{med}}^{\text{t}-\text{fit}}$. One would need to increase the water density up to a value ρ_w = 10 g cm⁻³ to get differences of ∼5% for $D_{\text{med}}^{\text{t}}$ and $D_{\text{med}}^{\text{w}}$.
• (b)  
  For a given density but different I-values, like ICRP and ICRU bone tissues, the differences are around 5% for the 'tissue' approach where fluence and stopping powers changes cancel each other, but increase to 7–10% for the 'water-like', 5–6% for the 'water-like-B' and 5–7% for the 'tissue-fit' cases. This shows again a stronger dependence on the I-value than on the mass density.

For visualization purposes, figure 9 reproduces the percent difference between the dose-to-media under the different approaches and the reference dose-to-water, the data being extracted from table 1. It can be observed in the figure that

• (a)  
  There is a trend, common to all the approaches, to yield dose-to-media smaller than the reference dose-to-water (negative values) except for adipose tissue, where 2–3% higher values are obtained. The 'tissue-fit' approach for inflated lung is also 3% higher.
• (b)  
  The differences with the reference dose-to-water are found to be very similar in all the approaches except for bone (up to 10% underestimation for the 'water-like' approach) and of smaller importance for adipose tissue. In the 'water-like-B' case, all the water cases (having different density and/or I-value) agree with D_w,ref as expected, because only electron density plays a role; the closeness between this case and $D_{\text{med}}^{\text{t}}$ agrees with the results of Ma and Li (2011). Note that the 'tissue-fit' approach differs substantially as soon as the medium differs from the average ICRP or ICRU data. The bone underestimation obtained in all cases agree with the comparisons made by Dogan et al (2006), and by Ma and Li (2011) for the difference between $D_{\text{med}}^{\text{t}}$ - and $D_{\text{med}}^{\text{w}}$ -based MC calculated treatment plans.These results do not point at a clear advantage of one approach over another unless a substantial amount of bone is present, in which case the 'water-like' approach differs more than the other methods. Considering that in most clinical situations bone is not particularly critical to the dose received, the potential concern is strongly dependent on the individual clinical practice.
• (c)  
  None of the approaches show, however, the large dependence on I-values that has proven to be of great importance in all the independent comparisons of quantities made above, meaning that an effective cancelation of Φ and S/ρ, and their ratios, occurs. As above, the 'water-like' approach yields the largest difference for bone, whereas all the methods yield differences scattered within about 3–4% for adipose tissue.

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It is emphasized that these results and comparisons are based on the a priori knowledge of tissue compositions and their estimated I-values, and thus on the use of 'correct' stopping powers for each procedure. This is of course not the case in a real clinical situation, where generic tissue types are selected from CT-derived density and no detailed information is available for the mean excitation energies of the various tissues.

An important aspect that remains to be discussed is the widely used retrospective conversion from a Monte Carlo-calculated $D_{\text{med}}^{\text{t}}$ to $D_{\text{med}}^{\text{w}}$, justified on the grounds that most current clinical experience is based on the assumption of water-like tissues (the 'water-like-B' approach). This conversion is usually implemented using a method proposed by Siebers et al (2000) and the well-known property that, due to the approximate constancy with depth of their electron spectra, stopping-power ratios for photon beams are practically independent of depth except in the build-up region (Andreo and Brahme 1986); thus a constant stopping-power ratio can be used at all depths.

Notwithstanding the clarity of the conversion from a dosimetric point of view (converting dose from one medium to another using Bragg–Gray cavity theory), the findings above show that as electron fluence in media not equivalent to water are rather different from the electron fluence in water, see figure 6, a factor to correct for this difference should strictly be needed in addition to the correction for stopping-power differences proposed by Siebers et al (2000). This additional correction would parallel the fluence perturbation correction factor used in ionization chamber reference dosimetry and will be discussed below.

Two routes will be used to determine the stopping-power ratios necessary for the conversion

$$\begin{eqnarray}&&D_{\text{med}}^{\text{w}-\text{conv}}=D_{\text{med}}^{\text{t}} s_{\text{w},\text{med}}^{\text{BG}}\end{eqnarray} \tag{ 11 }$$

where $D_{\text{med}}^{\text{w--conv}}$ is the converted dose to a water-like medium and $s_{\text{w},\text{med}}^{\text{BG}}$ is the Bragg–Gray water-to-medium mass electronic stopping-power ratio, i.e. evaluated for the primary electron fluence. Note that now we deal with 'water/medium' ratios whereas in previous sections the ratios of interest were 'medium/water'. The first route is based on a situation where the electron spectral distribution in the volume of interest is not known, which corresponds to a realistic clinical situation where a MCTP has calculated the absorbed dose to tissue in each voxel and the user requires the equivalent dose to water-like voxels. The second route makes use of the MC-calculated primary electron fluence distributions discussed in the preceding sections. The converted dose-to-water for the different approaches will be compared with the reference dose-to-water D_w,ref (as in table 1 and figure 9) and with absorbed doses calculated using two commercial treatment planning systems.

In the first route a commonly used approximation is implemented, following (Attix 1986). A rough estimate of the equivalent monoenergetic photon energy, hν_eq, of a MV spectrum is usually obtained as MV/3, which corresponds to the energy of a monoenergetic photon having the same $s_{\text{w},\text{med}}^{\text{BG}}$ as the megavoltage beam (at the same depth in water). Assuming that all the secondary electrons are produced in Compton interactions, the mean initial secondary electron energy is obtained as

$$\begin{eqnarray}&&{{\bar{E}}_{0}}=h{{\nu}_{\text{eq}}}\times \frac{\sigma _{\text{tr}}^{\text{K}-\text{N}}\left(h{{\nu}_{\text{eq}}}\right)}{{{\sigma}^{\text{K}-\text{N}}}\left(h{{\nu}_{\text{eq}}}\right)}\end{eqnarray} \tag{ 12 }$$

where $\sigma _{\text{tr}}^{\text{K}-\text{N}}$ and σ^K − N are the Klein–Nishina cross-sections for energy hν_eq (see data e.g. in appendix D.1 of Attix 1986). The mean energy in the equilibrium electron slowing-down spectrum ${{\bar{E}}_{z}}$ is approximated by $0.5{{\bar{E}}_{0}}$. For the photon beam of 6 MV the result is ${{\bar{E}}_{z}}=0.53$ MeV. The value of

$$\begin{eqnarray}&&s_{\text{w},\text{med}}^{\text{BG}}\equiv s_{\text{w},\text{med}}^{\text{H}}=\frac{{{\left[{{S}_{\text{el}}}\left({{{\bar{E}}}_{z}}\right)/\rho \right]}_{\text{w}}}}{{{\left[{{S}_{\text{el}}}\left({{{\bar{E}}}_{z}}\right)/\rho \right]}_{\text{med}}}}\end{eqnarray} \tag{ 13 }$$

is then obtained from the ESTAR data for water and med, respectively. Note that this is the so-called Harder approximation to Bragg–Gray stopping-power ratios, see ICRU-35 (ICRU 1984a), therefore the superscript 'H' has been used.

In the second route the Bragg–Gray stopping-power ratios are evaluated using the standard expression (see ICRU 1984a)

$$\begin{eqnarray}&&s_{\text{w},\text{med}}^{\text{BG}}=\frac{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{w}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{w}}}\text{d}E}{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{w}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{med}}}\text{d}E}\end{eqnarray} \tag{ 14 }$$

with the primary electron fluence distribution in water shown in figure 5. It is emphasized that the same $\Phi _{E,\text{w}}^{\text{prim}}$ is used in the numerator and in the denominator, and therefore the MC calculated $\Phi _{E,\text{med}}^{\text{prim}}$ for the various media do not participate in the conversion procedure. Note that equation (14) corresponds to the inverse of the $D_{\text{med}}^{\text{w}}$ column of table 1, see also equation (9).

The results for both calculations are given in table 2, where columns (2) and (3) show that the two routes yield practically identical stopping-power ratios except in the case of water_d10 and inflated lung, both having the extremest densities, where differences are still within 1% . This shows the suitability of the crude Compton plus Harder approximations for the purpose of a retrospective conversion, making MC calculations unnecessary for estimating the required stopping-power ratios for the $D_{\text{med}}^{\text{t}}$ to $D_{\text{med}}^{\text{w}}$ conversion, contrary to a statement made by Siebers et al (2000). It is worth noting that our ICRP cortical bone and inflated lung tissues coincide with tissues used by Siebers et al, and for these media the agreement in stopping-power ratios is excellent despite the fact that they used $\Phi _{E,\text{med}}^{\text{prim}}$ instead of $\Phi _{E,\text{w}}^{\text{prim}}$ to calculate the $s_{\text{w},\text{med}}^{\text{BG}}$ of equation (14); these authors also used a different 6 MV incident spectrum and MC code (MCNP in their case, Briesmeister 1997).

Table 2. Stopping-power ratios using the two routes described in the text and conversion of $D_{\text{med}}^{\text{t}}$ and $D_{\text{med}}^{\text{t}-\text{fit}}$ (from table 1) to $D_{\text{med}}^{\text{w--conv}}$ using $s_{\text{w},\text{med}}^{\text{BG}}$, relative to the reference absorbed dose to water D_w,ref. The last three columns correspond to $D_{\text{tissue}}^{\text{w}}$ and $D_{\text{tissue}}^{\text{t}}$ calculated with commercial TPS.

Medium	Compton Harder	Electron spectra	Ratio $D_{\text{med}}^{\text{w--conv}}/{{D}_{\text{w},\text{ref}}}$		Ratio Dtissue/Dw,ref from TPS
	$s_{\text{w},\text{med}}^{\text{H}}$	$s_{\text{w},\text{med}}^{\text{BG}}$	from $D_{\text{med}}^{\text{t}}$	from $D_{\text{med}}^{\text{t}-\text{fit}}$	$D_{\text{tissue}}^{\text{w}}$ (E)a	$D_{\text{tissue}}^{\text{w--conv}}$ (M)b	$D_{\text{tissue}}^{\text{t}}$ (M)b
water (Dw,ref)	1.000	1.000	1.000	1.000
water_d2	1.011	1.011	1.000	0.898
water_d10	1.062	1.054	1.000	0.712
water_I86	1.015	1.014	1.000	1.014
ICRP-adipose	0.978	0.978	0.984	0.975	0.996	0.982	0.996
ICRP-adipose_I55	0.968	0.966	0.984	0.964
ICRP-bone (cortical)	1.115	1.115	1.056	1.061
ICRU-bone (compact)	1.081	1.080	1.032	1.004	1.030	1.068	0.961
ICRP-muscle	1.011	1.012	1.000	1.005
ICRP-soft tissue	1.004	1.004	1.000	1.004
ICRP-lung	1.011	1.011	1.001	1.004
ICRP-lung inflated	1.010	1.000	0.992	1.032

^aVarian Eclipse AAA; yields dose to water-equivalent tissue. ^bElekta Monaco 5; yields dose to tissue and optionally converts it to dose to water-equivalent tissue using Siebers et al approach.

Columns (4) and (5) of table 2 show ratios $D_{\text{med}}^{\text{w--conv}}$ to D_w,ref, where the former are obtained converting MC calculated dose-to-tissue to dose-to-water using equation (11) with the $s_{\text{w},\text{med}}^{\text{BG}}$ values given in the table. The dose-to-tissue corresponds to the 'tissue' and 'tissue-fit' approaches in table 1, $D_{\text{med}}^{\text{t}}$ and $D_{\text{med}}^{\text{t}-\text{fit}}$. These absorbed-dose ratios are shown as per cent differences relative to D_w,ref in figure 10 along with the corresponding values for the two 'water-like' cases discussed above. As in figure 9, all the different approaches yield very good agreement with the reference dose-to-water for tissues similar to water, except for a 3% discrepancy in the case of inflated lung using the $D_{\text{tissue}}^{\text{t}-\text{fit}}$ method (right-most triangle). Hence, we will focus on the differences for the two adipose and bone tissues.

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Recall that current tissue segmentation cannot distinguish tissues of the same type having different physical properties other than density, as it would occur with real patients. When these properties are (theoretically) fully taken into account, the values for the two $D_{\text{tissue}}^{\text{t}}$ pairs, adipose and bone tissues, are rather similar (filled circles). Converting them to dose-to-water using equation (11) yields differences (the open circles) practically symmetrical with the zero per cent level, i.e. for bone (long arrows) the dose changes from −5% to +5% (ICRP cortical bone) or from −4% to +3% (ICRU compact bone). Converted dose-to-water from $D_{\text{tissue}}^{\text{t}-\text{fit}}$ (triangles) is not better, and in addition leads to contradictory results, as for compact bone the agreement with D_w,ref is remarkable, whereas for cortical bone the difference is about the same as for converted dose-to-water from $D_{\text{tissue}}^{\text{t}}$ (open circles).

It can be noticed in figure 10 that, in general, converted dose-to-water values are not even close to the reference dose-to-water except for the tissues considered to be water-like. In all cases, the difference between converted and non-converted doses, relative to D_w,ref, is quite similar in magnitude but have opposite signs, pointing at an anomaly in the conversion process. The two 'water-like' cases (filled and open squares) also deserve being commented. The results for the approach based on the assumption of identical electron fluence (by virtue of Fano's theorem), combined with correct body-tissues stopping powers, prove that the influence of the I-values is too large to assume that any type of body tissue can be considered to be similar to water; its utility rests in providing accurate stopping-power ratios but it cannot be used for patient dose calculations. Note that the differences between $D_{\text{tissue}}^{\text{w}}$ (filled squares) and D_w,ref are identical in size to the length of the arrows shown in the figure for the $D_{\text{med}}^{\text{t}}$ to $D_{\text{med}}^{\text{w--conv}}$ conversion. The simplicity of the 'water-like-B' approach (open squares) does not improve results either; on the contrary, compared with real dose-to-tissue MC calculations (filled circles), differences are so small that may lead to the erroneous conclusion that dose-to-water and dose-to-tissue are practically identical if the level of the approximations involved is not kept in mind.

To estimate the expected differences following a conversion when the user wishes to report dose-to-water (and possibly establish a clinical criteria accounting for the difference), it is of importance to compare the converted doses to water above with converted values calculated using commercial treatment planning systems. This has been done for the Varian Eclipse AAA (Anisotropic Analytical Algorithm) version 11.0.31 and the Elekta Monaco version 5 systems, for the two tissues that in general show the largest discrepancies, adipose and bone, using the same geometry configuration as in previous calculations. The Eclipse AAA calculation engine is based on a convolution-superposition algorithm and yields directly dose to water-equivalent tissue. The Monaco is MCTP-based (XVMC, Fippel 1999, see also Elekta 2013) and yields dose to tissue; the calculated plan can subsequently be converted to dose to water-equivalent tissue using the Siebers et al (2000) approach of equation (11). Both systems use analytical models to describe the radiation source, but their infuence on the dose calculations presented in this work is difficult to estimate. It is to be noted that the Varian Acuros system, based on numerical radiation transport methods to calculate doses to tissues, relies on tissue segmentation procedures and assignment of cross section and stopping power data for tissues subject to constrains identical to those in the Elekta Monaco system (see Failla et al (2011) and Varian (2011)). Although results with Acuros have not been available for the present work, it goes without saying that all the issues discussed in relation with I-values, densities, stopping powers, electron fluence spectra etc also appy to this system, which also relies on Siebers et al (2000) approach to optionally convert dose to tissue into dose to water-equivalent tissue.

The results of the calculations with Eclipse AAA and Monaco 5, made by Södergren (2014), are included in table 2 and correspond to two different 6 MV linacs, from Varian and Elekta, respectively. Note that none of the TPSs can distinguish between two identical tissues having differences other than their density; thus, for the two types of adipose and bone tissues used so far, the two TPSs yield the same dose for each pair of tissues. The comparison is shown in figure 11 from where the following observations can be made:

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• (a)  
  For the two tissue types, Eclipse AAA yields equivalent dose-to-water closer to the reference dose-to-water D_w,ref than the Monaco 5 converted dose-to-water. The Eclipse AAA difference for bone does not exceed 3% . This is an important remark for an institution where current clinical dose constraints are mostly based on the Eclipse AAA system or similar.
• (b)  
  The Monaco 5 MCTP results for the two tissue types agree within about 2% with the MC results $D_{\text{tissue}}^{\text{t}}$ of this work despite being based on the $D_{\text{tissue}}^{\text{t}-\text{fit}}$ approach, in consistency with the results shown in figure 9. For bone, the difference with D_w,ref is approximately −4%.
• (c)  
  It would be logical to expect that Monaco 5 converted doses to water are close to the reference dose-to-water. However, Monaco's conversion using Siebers et al method (corresponding to the transition from filled squares to open squares) results even in worst agreement with D_w,ref (and of opposite sign, up to about 6%) than its dose-to-tissue. Note that this was also the case for converted doses from $D_{\text{tissue}}^{\text{t}}$, see figure 10. Hence, for bone, the difference between Monaco 5 converted dose-to-water and Eclipse AAA is around 3%, Monaco being 6% away from the reference dose-to-water.

When the overall constraints on CT density conversion to tissue and tissue segmentation described throughout this work are taken into consideration, it becomes apparent that the process of converting $D_{\text{med}}^{\text{t}}$ to $D_{\text{med}}^{\text{w--conv}}$ based solely on a ratio of stopping powers as in equation (11) does not go beyond being a rather crude estimation that, especially for adipose and bone tissues, mostly adds uncertainty to the overall patient treatment planning process. As noted in the introduction this uncertainty increase was commented e.g. by AAPM TG-105 and Ma and Li (2011), but figure 11 illustrates the expected range of change based on the widely used conversion method of Siebers et al (2000).

In addition to jeopardizing the advantage of having used a MCTP system, the retrospective conversion yields in some case a converted dose-to-water that differs substantially from the reference dose-to-water, as shown in figures 10 and 11. Considering that the use of MCTP calculated dose-to-tissue does not show large differences with the reference dose-to-water except for bone, the retrospective conversion appears to be unnecessary except when dose-to-bone is considered to be of high clinical relevance. For cases where dose-to-bone might be of importance, a pragmatic solution would be to take the average between dose-to-tissue and converted dose-to-water, which in the worst case will yield a dose value differing from current practice by not more than 3% for the Monaco 5 MCTP.

It has been shown so far that, as a result of the $D_{\text{med}}^{\text{t}}$ to $D_{\text{med}}^{\text{w--conv}}$ conversion using stopping-power ratios $s_{\text{w},\text{med}}^{\text{BG}}$, the difference between the reference dose-to-water D_w,ref and converted doses to water becomes particularly large in the case of adipose and bone tissues, while the expectation is that they should be practically identical. As has been mentioned above, this apparent anomaly points at the need for an additional correction factor, to be used in equation (11), which will take into account the electron fluence differences shown in figure 6 between water and the various tissues. An expression for such correction can simply be derived as follows.

The stopping-power ratio water-to-medium can be expressed as a ratio of the respective stopping powers, each averaged over the electron spectrum in the relevant medium, i.e.

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{s}_{\text{w},\text{med}}} & =\frac{{{{\bar{s}}}_{\text{w}}}}{{{{\bar{s}}}_{\text{med}}}}=\frac{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{w}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{w}}}\text{d}E/\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{w}}^{\text{prim}}\text{d}E}{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{med}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{med}}}\text{d}E/\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{med}}^{\text{prim}}\text{d}E} \\ {} & =\frac{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{w}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{w}}}\text{d}E}{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{med}}^{\text{prim}}{{\left[{{S}_{\text{el}}}(E)/\rho \right]}_{\text{med}}}\text{d}E}\frac{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{med}}^{\text{prim}}\text{d}E}{\underset{0}{\overset{{{E}_{\text{max}}}}{\int}} \Phi _{E,\text{w}}^{\text{prim}}\text{d}E} \\ \end{array}\end{eqnarray} \tag{ 15 }$$

The first ratio can be identified with the inverse of the ratio of doses (cemas) in equation (8), and the second is the ratio of total fluences in each medium obtained from the integration of fluence differential in energy, i.e.

$$\begin{eqnarray}&&{{s}_{\text{w},\text{med}}}\equiv \frac{{{D}_{\text{w}}}}{{{D}_{\text{med}}}}\frac{\Phi _{\text{med}}^{\text{prim}}}{\Phi _{\text{w}}^{\text{prim}}}\end{eqnarray} \tag{ 16 }$$

Therefore equation (11) should be modified to

$$\begin{eqnarray}&&D_{\text{med}}^{\text{w}-\text{conv}}=D_{\text{med}}^{\text{t}} s_{\text{w},\text{med}}^{\text{BG}} {{k}_{\Phi}}\end{eqnarray} \tag{ 17 }$$

where k_Φ is a fluence correction factor defined by

$$\begin{eqnarray}&&{{k}_{\Phi}}=\frac{\Phi _{\text{w}}^{\text{prim}}}{\Phi _{\text{med}}^{\text{prim}}}\end{eqnarray} \tag{ 18 }$$

Values of k_Φ calculated from the above MC distributions of primary electron fluence differential in energy are given in table 3 for the media and tissues used throughout this work, and for visualization purposes they are illustrated in figure 12. Their effect is shown in figure 13, where it can be seen that all converted doses to water from MC calculated dose-to-tissue now agree with the reference dose-to-water within 1% or better even for bone tissues, hence demonstrating the need for a fluence correction factor. For users switching from a TPS like Eclipse AAA to a MCTP-based system, but still wishing to be able to refer to dose-to-water, the changes in dose criteria for the various tissues would therefore become minimal (a maximum difference of approximately −2% for bone).

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Table 3. Values of the fluence correction factor k_Φ and the product $s_{\text{w},\text{med}}^{\text{BG}}{{k}_{\Phi}}$ for media and tissues.

Medium	kΦ	$s_{\text{w},\text{med}}^{\text{BG}}{{k}_{\Phi}}$
water (Dw,ref)	1.0000	1.0000
water_d2	1.0000	1.0108
water_d10	1.0000	1.0537
water_I86	1.0000	1.0135
ICRP-adipose	1.0131	0.9906
ICRP-adipose_I55	1.0131	0.9791
ICRP-bone (cortical)	0.9563	1.0665
ICRU-bone (compact)	0.9768	1.0555
ICRP-muscle	1.0000	1.0116
ICRP-soft tissue	1.0000	1.0039
ICRP-lung	0.9989	1.0101
ICRP-lung inflated	1.0045	1.0050

The discussion above seems to point at a situation where the issue of relating dose-to-tissue and dose-to-water can be completely avoided by the choice of a treatment plan based on dose-to-tissue. The final step in the treatment planning process requires, however, that the dose to a selected point, calculated by the TPS, be normalized to the reference dosimetry of the treatment unit, calibrated in terms of absorbed dose to water irrespective of the formalism or dosimetry protocol used for this purpose. This means that a conversion procedure cannot be avoided.

When the selected point belongs to a volume (voxel) filled with water-like tissue the conversion is straightforward using the procedure and data currently available (e.g. Siebers et al 2000). Nevertheless, as demonstrated in the preceding section, this conversion fails for a number of tissues and in those cases the modified expression (17), taking into account fluence differences in the various media, together with the data of table 3, should be used.