Effect of statistical fluctuation in Monte Carlo based photon beam dose calculation on gamma index evaluation

Let us consider two dose distributions to be compared, D_r and D_e. Mathematically, the γ-index at a comparison point (r_r, D_r(r_r))is defined as

$$\begin{equation} \gamma( {r_r }) = \mathop {\min }\limits_{r_e } \bigg\{ {\sqrt {\frac{{| {r_e - r_r }|^2 }}{{\Delta r^2 }} + \frac{{| {D_e (r_e ) - D_r (r_r )}|^2 }}{{\Delta D^2 }}} } \bigg\}. \end{equation} \tag{ 1 }$$

Here D_r(r_r) is the reference dose distribution at position r_r and D_e(r_e) is the evaluation dose distribution at position r_e. ΔD is dose difference criterion and Δr is the DTA criterion. If the γ-index is equal to or less than 1, the dose at the spatial point r_r is considered to pass the test.

The graphical interpretation of the γ-index has been established previously (Ju et al 2008) and illustrated in figure 1(a). Specifically, the γ-index is defined as the minimum Euclidian distance from the point O_r to the evaluation dose curve. In practice, the evaluation dose is usually defined at discrete spatial locations. To the first order approximation, Ju et al assumed a linear interpolation of the dose values between neighboring spatial points as shown in figure 1, and developed a simple and efficient γ-index calculation algorithm. This method is widely used nowadays (i.e. Gu et al 2011b). In this study, we keep this linear interpolation and our analysis in next section is built based on this implementation.

The γ-index value for a particular random realization of dose distributions is essentially a random variable. It is more meaningful to investigate how the γ-index test is affected on average, i.e. the average impact over all the random dose realizations. In this section, we outline a theoretical calculation in a simplified 1D model for the mean γ-index value when there exist MC statistical fluctuations in the dose distributions. Theoretical analysis based on this simplified model is intended to offer some theoretical insights rather than a strict theoretical proof. We then validate our theoretical conclusions for 3D cases using real clinical data.

2.2.1. MC statistical fluctuations in the reference dose

In figure 2, O_e1O_e2 is the line segment of the evaluation dose curve, O_r is the reference point, γ₀ is the original γ-index value and t parameterizes the deviation due to the statistical fluctuation. In this simplified model, the new γ-index value is always the minimum Euclidian distance from the point O_r to the line segment O_e1O_e2 of the evaluation dose curve, not to the other line segment of the evaluation dose curve. Here we first introduce the concept of signed-gamma index $\tilde \gamma$, such that its magnitude is the radius for a circle centered at the reference point on the reference dose and tangent to the evaluation dose curve, but its sign is positive if the center of the circle is below the evaluation dose curve (e.g. figure 2(a)) and negative if the center is above the curve (e.g. figure 2(b)). Since the number of particles in the MC simulation is usually very large to ensure a small uncertainty level of clinical relevance; and with the large number of particles, the dose to a voxel in an MC simulation is commonly considered following a Gaussian distribution (Sempau and Bielajew 2000). In this study, we assume that probability density function p(t) is symmetric around zero. Note that $\tilde \gamma = \gamma _0 - t\cos \theta$ where θ is the angle between γ₀ and vertical axis, it is straightforward that

$$\begin{equation} \gamma _0 = \mathop \int _{ - \infty }^{ + \infty } {\rm d}t\tilde \gamma (t)p(t), \end{equation} \tag{ 2 }$$

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For the average gamma index,

$$\begin{equation} \bar \gamma = \mathop \int _{ - \infty }^{ + \infty } {\rm d}t| {\tilde \gamma(t)}|p(t) = \mathop \int _{ - \infty }^{\frac{{\gamma _0 }}{{\cos \theta }}} {\rm d}t\tilde \gamma(t)p(t) + \mathop \int _{\frac{{\gamma _0 }}{{\cos \theta }}}^{ + \infty } {\rm d}t[ - \tilde \gamma(t)]p(t). \end{equation} \tag{ 3 }$$

Now, subtracting equations (2) and (3) leads to

$$\begin{equation} \bar \gamma - \gamma _0 = 2\mathop \int _{\frac{{\gamma _0 }}{{\cos \theta }}}^{ + \infty } {\rm d}t[ - \tilde \gamma(t)]p(t) \end{equation} \tag{ 4 }$$

Since $\tilde \gamma (t) < 0$ when $t > \frac{{\gamma _0 }}{{\cos \theta }}$, we can conclude that

$$\begin{equation} \bar \gamma \ge \gamma _0 . \end{equation} \tag{ 5 }$$

This conclusion is valid for a general symmetric probability density function p(t). It has also been theorized that the statistical fluctuations of the dose distribution in an MC calculation, termed as 'noise' from here on, follow a Gaussian distribution (Sempau and Bielajew 2000). So in this case, the average gamma index is then specified to be

$$\begin{equation} \bar \gamma = \frac{1}{{\sqrt {2\pi } \sigma _t }}\mathop \int _{ - \infty }^{ + \infty } {\rm d}t|\tilde \gamma(t)|{\rm e}^{- t^2/}2\sigma _t ^2 , \end{equation} \tag{ 6 }$$

where σ_t is statistical uncertainty value on the reference point. Then the equation (6) can be rewritten as

$$\begin{equation} \fl \bar \gamma = \frac{1}{{\sqrt {2\pi } \sigma _t }}\mathop \int _{ - \infty }^{\frac{{\gamma _0 }}{{\cos \theta }}} {\rm d}t( {\gamma _0 - t\cos \theta } ){\rm e}^{- t^2/}2 \sigma _t ^2 + \frac{1}{{\sqrt {2\pi } \sigma _t }}\mathop \int _{\frac{{\gamma _0 }}{{\cos \theta }}}^{ + \infty } {\rm d}t( {t\cos \theta - \gamma _0 } )\,{\rm e}^{- t^2/}2 \sigma _t ^2 \end{equation} \tag{ 7 }$$

$$\begin{equation} {\hphantom{\fl \bar \gamma }} = \gamma _0\ {\rm erf}\bigg( {\frac{{\gamma _0 }}{{\sqrt 2 \sigma _t \cos \theta }}} \bigg) + \frac{2}{{\sqrt {2\pi } }}\sigma _t \cos \theta\ {\rm exp}\bigg( {\frac{{ - \gamma _0 ^2 }}{{2\cos ^2 \theta \sigma _t ^2 }}} \bigg). \end{equation} \tag{ 8 }$$

The derivative of equation (8) with respect to σ_t is

$$\begin{equation} \frac{{d\bar \gamma }}{{d\sigma _t }} = \sqrt {\frac{2}{\pi }} \cos \theta\ {\rm exp}\bigg( {\frac{{ - \gamma _0 ^2 }}{{2\cos ^2 \theta \sigma _t ^2 }}} \bigg). \end{equation} \tag{ 9 }$$

Since $\frac{{d\bar \gamma }}{{d\sigma _t }} > 0$, and when σ_t → 0, $\bar \gamma \to \gamma _0 ,$ we get the same conclusion as equation (5).

The increase of the average γ-index $\bar \gamma$ can be understood as following. When there is a finite deviation t, there are two scenarios resulting different impacts on the average γ-index. First, when t is small for the original γ-index γ₀, i.e. $t \le \frac{{\gamma _0 }}{{\cos \theta }}$, the average of the γ-index pair $| {\tilde \gamma (t)}|$ and $|\tilde \gamma ({ - t})|$ equals γ₀. Second, when t is large, i.e. $t > \frac{{\gamma _0 }}{{\cos \theta }}$, the average of the pair $| {\tilde \gamma (t)} |$ and $|\tilde \gamma ( { - t} )|$ is larger than γ₀ due to the flipped sign in one of them caused by the absolute value operation. It is the latter scenario that causes the increase of $\bar \gamma$. Hence when the original γ-index value is relatively large for the noise standard deviation, the increase of the average γ-index $\bar \gamma$ will be relatively small due to the small contributions from the second scenario.

2.2.2. MC statistical fluctuations in the evaluation dose

In figures 3(a) and (b), suppose without noises, O_e1O_e2 is the line segment of the evaluation dose curve. With the MC noises, the line O_e1O_e2 moves. As in the section 2.2.1, similarly we assume in this simplified model, the new γ-index value is only related to the line segment O_e1O_e2 of the evaluation dose curve, not to the other line segment of the evaluation dose curve. σ_t1, σ_t2 are the statistical uncertainties on the dose values at the pointsO_e1andO_e2, the average γ-index value $\bar \gamma$ can be calculated as,

$$\begin{equation} \bar \gamma = \mathop \int _{ - \infty }^{ + \infty } {\rm d}t_1 \mathop \int _{ - \infty }^{ + \infty } {\rm d}t_2 |\tilde \gamma( {t_1 ,t_2 } )|p( {t_1 ,t_2 }), \end{equation} \tag{ 10 }$$

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where t_i, i = 1, 2, parameterizes the deviation of dose values at the two points O_e1 and O_e2 and p(t₁, t₂) is the probability density function. For some noise realizations, the signed-gamma index $\tilde \gamma$ may change its sign from positive to negative. For instance, in figure 3(a) when both t₁ and t₂ are large negative values. However, the probability of this situation is relatively small, given that σ_ti are small for γ₀. Hence, we first ignore the contributions of γ-indices from these situations. The consequence when this contribution cannot be ignored will be discussed later. Given this assumption, all $\tilde \gamma$ are positive regardless of the values of t_i.

As for the probability distribution, for the large number of particles simulated in an MC dose calculation, we assume that p(t₁, t₂) = p( − t₁, −t₂). When the noises at two points O_e1 and O_e2 are statistically independent, this assumption is apparently valid, as the noise distribution at each point is symmetric about zero. In reality, there exist correlations of noises between these two points. This correlation is caused by an electron track that passes through the two voxels. In one MC simulation, the number of electron tracks simultaneously passing the two voxels fluctuates about its average, and the probability of having more tracks is equal to that of having less tracks. Hence our assumption is still valid. Under this assumption, we can rewrite equation (10) as

$$\begin{equation} \bar \gamma = \frac{1}{2}\mathop \int _{ - \infty }^{ + \infty } {\rm d}t_1 \mathop \int _{ - \infty }^{ + \infty } {\rm d}t_2 [\tilde \gamma( {t_1 ,t_2 }) + \tilde \gamma( { - t_1 , - t_2 })]p( {t_1 ,t_2 }). \end{equation} \tag{ 11 }$$

From figure 3(c), we further separate the integral domain into four different quadrants,

$$\begin{equation} \bar \gamma = \frac{1}{2}\mathop \sum \limits_{i = 1}^4 \int\int_{I_i } {{\rm d}t_1 {\rm d}t_2 [\tilde \gamma( {t_1 ,t_2 }) + }\tilde \gamma ( { - t_1 , - t_2 })]p( {t_1 ,t_2 }). \end{equation} \tag{ 12 }$$

Since the integrals in the domain I₁ and I₂ equal to those in the domain I₃ and I₄, respectively, equation (12) reduces to

$$\begin{equation} \bar \gamma = \mathop \sum \limits_{i = 1}^2 \int\int_{I_i } {{\rm d}t_1\, {\rm d}t_2 [\tilde \gamma( {t_1 ,t_2 }) + }\tilde \gamma({ - t_1 , - t_2 })]p({t_1 ,t_2 }). \end{equation} \tag{ 13 }$$

Figure 3(a) illustrates the context when t₁ and t₂ are both positive (in domain I₁). Suppose O'_e1O'_e2 and $\widetilde{O_{e1} }\widetilde{O_{e2} }$ are the new evaluation dose lines, such that $O_{e1} \widetilde{O_{e1} }$ and O_e1O'_e1 are of the same length, similarly $O_{e2} \widetilde{O_{e2} }$ and O_e2O'_e2 are of the same length, and $\tilde \gamma ( {t_1 ,t_2 } )$ is represented as O_rD while $\tilde \gamma ( { - t_1 , - t_2 } )$ is O_rA. O_rB is the original γ-indexγ₀. If we set CF ⊥O'_e1O'_e2, we have

$$\begin{equation} O_r A + O_r D \le O_r A + O_r C + CF. \end{equation} \tag{ 14 }$$

Since O_rA = O_rE · cos (∠EO_rA) and CF = CE · cos (∠EO_rA), combining with equation (14), we can get

$$\begin{equation} O_r A + O_r D \le O_r C\cdot[ {1 + \cos (\angle EO_r A)}]. \end{equation} \tag{ 15 }$$

Moreover O_rB = O_rC · cos (∠EO_rB). From the geometric relationship, we also have ∠EO_rA = 180° − 2α − 2θ and ∠EO_rB = 90° − α − θ. Hence,

$$\begin{equation} O_r A + O_r D \le O_r C\cdot 2\cos ^2 (\angle EO_r B) \le 2\cdot O_r B. \end{equation} \tag{ 16 }$$

This indicates that

$$\begin{equation} \fl {\int\int_{I_1 }} {{\rm d}t_1\, {\rm d}t_2 [\tilde \gamma ( {t_1 ,t_2 } ) + }\tilde \gamma ( { - t_1 , - t_2 } )]p( {t_1 ,t_2 } ) \le 2\gamma _0 \int\int_{I_1 } {p( {t_1 ,t_2 } )\,{\rm d}t_1\, {\rm d}t_2 } . \end{equation} \tag{ 17 }$$

Similarly, figure 3(b) illustrates a context where t₁ is negative and t₂ is positive (in domain I₂). $\tilde \gamma ( {t_1 ,t_2 } )$ is represented as O_rD while $\tilde \gamma ( { - t_1 , - t_2 } )$ is O_rA. O_rB is the original γ-index γ₀. With a similar derivation, we can generate the same conclusion as equation (17).

$$\begin{equation} \fl \int\int_{I_2 } {{\rm d}t_1\, {\rm d}t_2 [\tilde \gamma( {t_1 ,t_2 })} + \tilde \gamma( { - t_1 , - t_2 })]p( {t_1 ,t_2 }) \le 2\gamma _0 \int\int_{I_2 } {p( {t_1 ,t_2 })\,{\rm d}t_1\, {\rm d}t_2 }. \end{equation} \tag{ 18 }$$

Combine equations (13), (17) and (18), we then have

$$\begin{equation} \bar \gamma \le {\rm }2\gamma _0 \left[ {\mathop \sum \limits_{i = 1}^2 \int\int_{I_i } {p( {t_1 ,t_2 })\,{\rm d}t_1\, {\rm d}t_2 }{\rm }} \right]. \end{equation} \tag{ 19 }$$

Since $2\mathop \sum \limits_{i = 1}^2 \int\int_{I_i } {p( {t_1 ,t_2 } )\,{\rm d}t_1 \,{\rm d}t_2 = 1}$, we can conclude that

$$\begin{equation} \bar \gamma \le {\rm }\gamma _0 . \end{equation} \tag{ 20 }$$

This indicates that the presence of noise will lead to an underestimation of the γ-index value. It is important to note that when the probability for negative signed γ-index $\tilde \gamma ( {t_1 ,t_2 } )$ is not negligible, i.e. when the noise standard deviation is relatively large for the original γ-index value, equation (20) is no longer valid. The simplest case is when the reference dose distribution is the same as the evaluation dose and γ-index values are all zero. In this case, the MC noise in the evaluation dose distribution leads to an overestimation of γ-index values. However, as later shown in our numerical experiments, this situation is not clinically relevant because it only happens when the original γ-index values are very small and their variation due to the MC noise does not affect the γ-index passing rate.

We conducted numerical experiments on two realistic clinical cases for photon radiation therapy: a 7-beam IMRT prostate plan and a 2-arc VMAT HN plan. To study the effect of using an MC dose as the reference dose or the evaluation dose in γ-index tests, a non-MC dose distribution and a set of MC dose distributions at various σ levels (including zero σ level) are required for each clinical case. For the non-MC dose, we used the dose distribution in the patient plan extracted from a commercial treatment planning system (Eclipse, Varian Medical Inc., Palo Alto, CA) which was computed using the analytical anisotropic algorithm (AAA). The resulting dose from the Eclipse system is at a resolution of 2.5 × 2.5 × 2.5 mm³ and interpolated to the MC dose resolution of 1.953 × 1.953 × 2.5 mm³. The AAA algorithm is an analytical algorithm hence there is no noise in the dose. To get the set of MC dose distributions, we first extracted the MLC leaf sequences of all beam angles from the patient plan. Using a different number of particle histories in the simulation, the dose distributions of various σ level were calculated on the patient geometry using a GPU-based MC dose engine (gDPM) (Jia et al 2011). In this work we define the term σ level as the average σ value normalized to the maximum dose D_max within regions of dose values higher than 50% of D_max (VOI_50%). The dose distributions with σ level of 0.5%, 1%, 1.5%, and 2% were calculated, which are the most clinically relevant noise levels for MC dose calculations. An additional dose distribution of σ level of 0.2% was also computed and used to obtain the MC dose with zero σ level. As such we determined the de-noised dose value D(v) by solving such an optimization problem

$$\begin{equation} \fl D({v}) = {\rm argmin}_{D} E[D] = {\rm argmin}_{D} \int{{\rm d}{v}({D - \widehat D{\rm log}D})} + \frac{\beta }{2} \int {{\rm d}{v}|{\nabla D}|^2 } . \end{equation} \tag{ 21 }$$

The first term in the energy function E[D] is a data-fidelity term considering the Poisson noise, while the second term is a penalty term to ensure the smoothness of the de-noised dose D(v). Since the energy function is convex, the optimality condition is researched by solving

$$\begin{equation} 0 = \frac{{\delta E}}{{\delta D}} = \bigg( {1 - \frac{{\hat D}}{D}} \bigg) - \beta \nabla ^2 D. \end{equation} \tag{ 22 }$$

This model is solved using a gradient descent method. We would like to point out that the beam parameters in the gDPM code were not purposely tuned to match the Eclipse results. To analyze the situation when both the evaluation and reference doses are generated by the MC method, another set of MC dose distributions at various σ levels (including zero σ level) is desired. For this set of MC doses, we used the same MC dose engine gDPM, but shifted the isocenter position of all beams by 3 mm to the patient's left, posterior, and superior directions, respectively, to simulate the dosimetric effects due to a set up error in a clinical situation.

To study the impact of the MC noise on the γ-index value, we first needed to conduct a base comparison between the non-MC dose and the MC dose with zero σ level; the resulting γ-index values of a selected group of voxels are treated as the base values. Then the γ-index results for the same voxels were followed when the non-MC dose is compared with the MC doses of increasing σ levels. These voxels are selected as the ones with more clinical relevance. Since the passing rate within a region of interest (ROI) is the most common criteria used to compare two dose distributions, we focused on a range of γ-index values that contribute to the calculation of the gamma passing rate. We also noticed that the behavior of average γ-index variation due to the MC noise is similar for close γ-index values. Thus, based on the base comparison, we selected four groups of voxels in the reference dose distribution with γ-index values from value 0.6 to 0.8 (0.6, 0.8), from value 0.8 to 1 (0.8, 1.0), from value 1 to 1.2 (1.0, 1.2), and from value 1.2 to 1.4 (1.2, 1.4). For each group of voxels, we followed the variation of the average γ-index value with the increase of the σ level in the MC dose distribution. Furthermore, the γ-index passing rates were also reported for each comparison between the non-MC dose and the MC dose. In this study, in addition to VOI_50%, we also selected another ROI where dose values were higher than 10% of D_max (VOI_10%).

Since the MC noise is a random variable, for the comparison at each σ level, we repeated the γ-index test ten times for ten different random realizations of the dose distributions to obtain the mean of the γ-index test results. And during our experiments, all dose distribution comparisons were performed using the GPU-based fast γ-index algorithm (Gu et al 2011b).

To study the effect of using an MC dose as the reference dose in γ-index tests, we treated the set of MC doses as the reference doses and the non-MC dose as the evaluation dose. Figures 4(a) and (b) show the average γ-index values of voxels whose original γ-indices fall within the range of (0.6, 0.8), (0.8, 1.0), (1.0, 1.2) and (1.2, 1.4) as functions of the σ level in the reference dose ($\overline {\sigma _{{\rm ref}} }$) for both prostate and HN cases. For figures 4(a) and (b), we used the most common clinical γ-index test criterion: 3%-3 mm. We can see that the average γ-index value within each group slightly increases with $\overline {\sigma _{{\rm ref}} }$. Figures 4(c) and (d) show the passing rate within VOI_50% and VOI_10% as function of $\overline {\sigma _{{\rm ref}} }$ for two different γ-index test criteria: 3%-3 mm and 2%-2 mm. It is noted that, although the average γ-index value does not change much, the gamma passing rate decreases significantly with the increase of the σ levels in the reference dose, especially for VOI_50% for these two clinical cases.

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To better understand the effect of the MC noise on the gamma passing rate, we examined the voxels that contribute to the gamma passing rate calculation. We defined type-I voxels as those with γ-index values larger than one in the base comparison when the MC σ level is zero and γ-index values smaller than or equal to one when the MC σ level is 2%. Type-II voxels are those with the opposite situation, i.e., γ-index values increasing from below or equal to one to above one when the σ level increases from zero and 2%. Since the MC noise is a random variable, for a γ-index test with a random realization of the MC dose distribution, a particular voxel with the γ-index value around one can be either type-I or type-II. However, when running the γ-index test for many random realizations of the MC dose distributions, this voxel will have more chance to be type-II than type-I when using the MC dose as the reference dose. Table 1 summarizes the average percentages of type-I and type-II voxels within two ROIs for two comparison criteria after running the γ-index test for 10 different random realizations of the MC reference dose distributions. It is noted that the percentage of type-II voxels is higher than that of type-I. The net effect, type-II percentage minus the type-I percentage, is equal to the change of the gamma passing rate with the MC dose of 2% σ level, as shown in figures 4(c) and (d).

To study the effect of MC noise in the evaluation dose in γ-index tests, we treated the non-MC dose as the reference dose and the set of MC doses as the evaluation doses. Figures 5(a) and (b) show the average γ-index values of voxels whose original γ-indices fall within each range as functions of the σ level in the evaluation dose ($\overline {\sigma _{{\rm eva}} }$) for prostate and HN cases. The average γ-index value within each group decreases dramatically with $\overline {\sigma _{{\rm eva}} }$. Figures 5(c) and (d) show, for two different γ-index test criteria, 3%-3 mm and 2%-2 mm, the passing rate within VOI_50% and VOI_10% as function of $\overline {\sigma _{{\rm eva}} }$. We observed that, the gamma passing rates saturated for 3%-3 mm criterion, while the gamma passing rates for 2%-2 mm criterion increases with the increase of $\overline {\sigma _{{\rm eva}} }$, especially for VOI_50%. Table 2 summarizes the average percentage of type-I or type-II voxels within the ROIs. For the non-MC reference dose versus the MC evaluation dose of 2% σ level, the percentage of type-I voxels is higher than that of type-II which means that, statistically, there are more voxels where the γ-index value sinks below one than voxels where the γ-index value rises above one. The net difference between type-I percentage and type–II percentage, is the same as the change of the gamma passing rate with the MC dose of 2% σ level, shown in figures 5(c) and (d).

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In section 2.2.2, we have noticed that, when the statistical standard deviation in the MC evaluation dose distribution is relatively large for the original γ-index value, the γ-index value will be overestimated on average due to the noise. Figures 6(a) and (b) show the average γ-index value of voxels with small original γ-index values, ranging from 0.1 to 0.24, as functions of $\overline {\sigma _{{\rm eva}} }$ for the prostate and HN cases. From figure 6 we can see that when the original γ-index value is large enough, i.e. larger than 0.2, the average γ-index decreases with $\overline {\sigma _{{\rm eva}} }$ (shown in blue and dashed lines); however, when the original γ-index value is very small, i.e. smaller than 0.14, the average γ-index increases with $\overline {\sigma _{{\rm eva}} }$ (shown in green and dotted lines). For the in-between original γ-index value, the average γ-index first decreases and then increases with $\overline {\sigma _{{\rm eva}} }$ (shown in red and dashed/dotted lines). We would like to point out that voxels with this range of original γ-index values do not contribute to the passing rate change.

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To analyze the situation when both the evaluation and reference doses are generated by the MC method, we considered the set of MC doses as the reference doses and the other set of MC doses with the shifted isocenter as the evaluation doses. Figure 7 shows the color maps of the gamma passing rate in high dose region VOI_50% under 3%-3 mm criterion. The x-axis is the σ level in the evaluation dose, while the y-axis is the σ level in the reference dose. The values at origin of the two maps are the base value with zero σ level in both reference and evaluation doses. Along the x-axis, the results correspond to the cases for the non-MC reference dose versus MC evaluation doses, same as in figures 5(c) and (d). Along the y-axis, the results correspond to the cases for MC reference doses versus the non-MC evaluation dose, same as in figures 4(c) and (d). The black lines in figure 7 illustrate iso-value lines on which the passing rate is the same as the base value for MC doses of zero σ level. The iso-value line splits the map into two regions: the upper-left region, where the MC noise level is relatively high in the reference dose leading to the underestimation of the gamma passing rate, and the lower-right region, where the MC noise level is relatively high in the evaluation dose leading to the overestimation of the gamma passing rate. The shape of iso-value line and the way that it splits the map are case-dependent. When both doses are the MC doses, we redefine type-I voxels as those with γ-index values larger than one when the σ level is zero in both the evaluation and the reference doses and less than or equal to one when the σ level is 2% in both doses; type-II voxels are those with the opposite situation, i.e., γ-index values increasing from below or equal to one to above one when both σ levels increase from zero and 2%. From table 3, the average percentage of type-I is higher than the percentage of type-II voxels and the net contribution matches with the increased gamma passing rate in the right upper corner with 2% σ level in both reference and evaluation doses.

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