Evaluation of extrapolation chamber response for surface and buildup dose assessment in radiotherapy photon beams using Monte Carlo simulations

Abstract

Background

Surface dose assessment is essential in radiotherapy, but accurately measuring doses in the buildup region of megavoltage beams presents significant challenges. Extrapolation chambers are frequently regarded as the most suitable detectors for this purpose.

Purpose

To establish under what conditions extrapolation chambers can be used to measure surface doses and to use Monte Carlo calculations to prove that measured surface‐to‐maximum ionization ratios correspond to absorbed‐dose ratios. A secondary purpose is to understand why surface measurements with a Markus fixed parallel‐plate chamber are inaccurate.

Methods

EGSnrc applications were used for calculating the dose to the air cavity of two extrapolation chambers in a polystyrene phantom as a function of the electrode separation (also called gap), $s$, between 0.1 and 10 mm. Doses to the phantom, ${\mathrm{D}}_{\mathrm{pst}}$, at depths corresponding to the effective point of measurement (EPOM) of the chambers were also calculated. Calculations were performed using clinical photon beams (, 6, 10, and 25 MV) that were fully modeled using BEAMnrc. Calculated chambers' responses as a function of the gap were compared with experimental data from the literature. Variations of the replacement correction factor ($P_{\mathrm{repl}}$) and the wall perturbation factor ($P_{\mathrm{wall}}$) as functions of electrode separation and depth in the buildup region were also investigated.

Results

Calculated %dose to the air in the chamber's cavity (%${\mathrm{D}}_{\mathrm{ch}}$) and measured % ionization from the literature at the polystyrene's surface (i.e., at a depth z = EPOM) agreed to within 2%–3% across all gap sizes and beam qualities. Differences between measured % ionization and calculated %${\mathrm{D}}_{\mathrm{pst}}$ at the phantom surface, were less than 4% (relative to ${\mathrm{D}}_{\max }$). For small gaps $\le 1$ mm, the calculated chamber response (%${\mathrm{D}}_{\mathrm{ch}}$) closely matched the calculated dose to the phantom material (%${\mathrm{D}}_{\mathrm{pst}}$), differing by less than 0.9% (relative to ${\mathrm{D}}_{\max }$) with $\%D_{\mathrm{pst}}/\%D_{\mathrm{ch}}$ deviating from unity by 1%–6% depending on chamber's design and beam energy. At shallow depths (0.01 mm), $P_{\mathrm{repl}}$ exhibited the largest variation, increasing by 43% for gaps from 0.1 to 10 mm. In contrast, $P_{\mathrm{wall}}$ and stopping‐power ratios varied by less than 2.0% and 1.2%, respectively. All correction factors remained approximately constant for small electrode separations ($s\le 1$ mm). The Markus chamber exhibited maximum calculated and measured over‐responses of 16.2% and 18.2% for and 3.0% and 3.4% for 25 MV, all within the first half of the buildup region.

Conclusions

Surface dose assessment using extrapolation chambers can achieve accuracy within 1% when small electrode separations are used and depth is appropriately defined. The results confirm that the measured % ionization ratios for gaps $s\le 1$ mm are constant because perturbation effects are minimized. Extrapolation from larger gaps is unreliable due to increasing changes in $P_{\mathrm{repl}}$ and extrapolation is only needed to establish where % ionization ratios become constant for small gaps and are hence accurate. Surface measurements with fixed parallel‐plate chambers are inaccurate if the gap size is too large.

1. INTRODUCTION

1.1. Overview

Accurate dose measurement at the surface and within the buildup region of megavoltage photon beams is critical for ensuring proper dose delivery in radiotherapy. However, dosimetry in these regions remains challenging due to the lack of charged particle equilibrium and the stringent requirements for detectors, which must combine high spatial resolution, minimal perturbation of electron fluence, medium‐equivalent composition, and reproducible signal response. As a result, various approaches have been proposed in the literature over the years. ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ , ⁹ , ¹⁰ , ¹¹ The term surface dose does not correspond to the dose at z = 0 since that cannot be determined by direct measurements or even calculated by simulations. In addition there is a steep variation of the percentage dose (normalized to maximum dose) with depth close to the surface, as shown in Figure 1 where the calculated overall central‐axis dose to water increases by up to 55% within the first 0.1 mm (e.g., in ) and the photon component (i.e., no incident contaminant charged particles) dose increases by up to a factor of 4. Hence, it is important to accurately define the depth when using the term surface dose. Similarly, Figure 2 shows the response variation for an extrapolation chamber in polystyrene with 0.1 mm electrode separation $s$ (also called gap), where the percentage dose (relative to maximum response) to the chamber increases by roughly 48% (e.g., in ) for the overall component, and by a factor of 2.5 for the photon component within the same depth range.

FIGURE 1.

BEAMnrc Monte Carlo calculated central‐axis percentage dose‐versus‐depth (PDD) curves for the total beam (left) and the photon‐only component (right) for , 6, 10, and 25 MV x‐ray beams with a 10 x 10 ${\mathrm{cm}}^2$ field at SSD = 100 cm incident on a 30 x 30 x 30 ${\mathrm{cm}}^3$ water phantom. Photon‐only curves were obtained by removing contaminant charged particles from the linac head and air column above the phantom, leaving only the primary photon fluence. The curves are for superficial depths (0.5 $\mu \mathrm{m}$ $\le$ z $\le$ 0.1 mm).

FIGURE 2.

Same as Figure 1 except for EGSnrc calculated percentage dose to the air cavity of the chamber ($\%D_{\mathrm{ch}}$), normalized to its maximum value, as a function of depth for 0.01 mm $\le z\le$ 0.1 mm. Calculations were performed using an extrapolation chamber EPOM = 0.01 mm in a polystyrene phantom with a fixed electrode separation of 0.1 mm.

Extrapolation chambers are parallel‐plate chambers with variable plate separations designed specifically for measuring surface doses. ¹² , ¹³ , ¹⁴ They are used in two different senses. One use is as part of an absolute dosimeter where, as will be shown below, the dose to the cavity is proportional to $Q/m$ where $Q$ is the charge released in the cavity and $m$ is the mass of the gas in the cavity. While $Q$ can be measured with high accuracy, $m$ is more difficult to determine. By using an extrapolation chamber and plotting $Q/m$ as a function of the cavity gap $s$ (where $m=\rho sA$ with $\rho$ the density of the cavity gas and $A$ the area of the collecting electrode), one can measure the slope of the line $\mathrm{d}Q/\mathrm{d}m$ with much greater accuracy, and, for a Spencer–Attix cavity, one has that $Q/m=\mathrm{d}Q/\mathrm{d}m$ (see below). The cavity is not a perfect Spencer–Attix cavity and hence corrections are needed. ¹⁵ , ¹⁶

The other application of extrapolation chambers is for measuring doses when charged particle equilibrium or even transient charged particle equilibrium do not exist, especially near the surface. Parallel‐plate chambers with fixed plate separation usually over‐respond due to electrons emitted from the side wall. ² , ⁴ , ¹⁷ Measurements are made with the extrapolation chamber as a function of gap thickness and extrapolated to zero gap. The assumption is that for zero gap thickness there are no side‐wall effects and the chamber is a Spencer–Attix cavity. However, as we describe below, the extrapolation to zero gap using large gaps does not yield the correct result.

The primary objective of this study was to determine the conditions under which extrapolation chambers can reliably measure surface doses and to use Monte Carlo calculations to investigate whether the ratio of measured ionization at the surface to that at the depth of maximum dose reproduces the corresponding absorbed‐dose ratio. A secondary objective was to clarify the physical reasons why surface measurements made with fixed parallel‐plate chambers, such as the Markus chamber, are inaccurate in the buildup region. The motivation for this work stems from the fact that, although extrapolation chambers are not routinely used in clinical practice, they are the reference‐class instruments for accurately assessing absorbed doses in the buildup region. ¹² , ¹⁸ Their performance at small electrode separations is therefore fundamental to accurate surface‐dose assessment. The present work provides new quantitative evidence of the accuracy of the measurements for sufficiently small gaps through a comprehensive Monte Carlo characterization of extrapolation‐chamber response in megavoltage photon beams, with particular emphasis on the surface and buildup regions where conventional detectors exhibit significant limitations. Although in principle there can be residual corrections factors required, they are shown to be small for the detectors and beams studied here.

1.2. Theory

Spencer–Attix cavity theory provides simple linear relationships between the dose to a point in the medium, $D_{\mathrm{med}}$, and the dose‐to‐air, ${\overline{D}}_{\mathrm{air}}^{\text{SA}}$, in a perfect Spencer–Attix air cavity positioned at that point, viz: ¹⁹ , ²⁰

$$D_{\mathrm{med}}={\overline{D}}_{\mathrm{air}}^{\text{SA}}{\left(\frac{{\overline{L}}_{\mathrm{\Delta }}}{\rho }\right)}_{\mathrm{air}}^{\mathrm{med}}=\frac{Q}{m}\left(\frac{W_{\mathrm{air}}}{e}\right){\left(\frac{{\overline{L}}_{\mathrm{\Delta }}}{\rho }\right)}_{\mathrm{air}}^{\mathrm{med}},$$ (1)

where ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{med}}$ is the Spencer–Attix medium to air mean restricted mass electronic stopping‐power ratio; $\mathrm{\Delta }$ is the cut‐off mean energy of electrons having projected range just large enough to cross the cavity; $Q$ is the charge collected in the cavity's air of mass, $m$ and $(W_{\mathrm{air}}/e)$ is the average energy deposited in dry air per coulomb of charge released as the electron slows to a stop. The value of ($W_{\mathrm{air}}/e$) is usually assumed to be constant, $33.97\pm 0.12$ J/C, ²¹ and independent of the electron energy for photon and electron beams used in radiotherapy, although evidence for this is indirect ²² , ²³ and there are indications that there is some ($\approx$ 0.3 %) variation in electron beams. ²⁴ , ²⁵ , ²⁶

Equation 1 requires several correction factors to account for a parallel‐plate ion chamber not being a perfect Spencer–Attix cavity leading to

$$D_{\mathrm{med}}={\overline{D}}_{\mathrm{air}}^{\text{SA}}{\left(\frac{{\overline{L}}_{\mathrm{\Delta }}}{\rho }\right)}_{\mathrm{air}}^{\mathrm{med}}={\overline{D}}_{\text{ch}}{P}_{\mathrm{repl}}{P}_{\mathrm{wall}}{\left(\frac{{\overline{L}}_{\mathrm{\Delta }}}{\rho }\right)}_{\mathrm{air}}^{\mathrm{med}},$$ (2)

where ${\overline{D}}_{\mathrm{ch}}$ is the dose to the air cavity of a real chamber; $P_{\mathrm{repl}}$ is the replacement correction factor, which accounts for perturbations in the electron fluence spectrum resulting from the presence of the cavity in the medium; $P_{\mathrm{wall}}$ is the wall perturbation factor, which compensates for differences between the chamber wall material and the surrounding medium. For extrapolation chambers with variable electrode separation $s$, the percentage dose to the medium, $\%D_{\mathrm{med}}$, at a given depth corresponding to the chamber's effective point of measurement (EPOM), can be obtained by normalizing each term in Equation 2 to its value at the depth of maximum dose, $D_{\max }$:

$$\begin{array}{cc}\hfill \%D_{\mathrm{med}}(z)& =\left[\frac{{\overline{D}}_{\text{ch}}(z,s)}{{\overline{D}}_{\text{ch}}(z_{\max },s)}\right]\left[\frac{P_{\mathrm{repl}}(z,s)}{P_{\mathrm{repl}}(z_{\max },s)}\right]\hfill \\ & \left[\frac{P_{\mathrm{wall}}(z,s)}{P_{\mathrm{wall}}(z_{\max },s)}\right]\left[\frac{{\left({\overline{L}}_{\mathrm{\Delta }}/\rho \right)}_{\mathrm{air}}^{\mathrm{med}}(z,s)}{{\left({\overline{L}}_{\mathrm{\Delta }}/\rho \right)}_{\mathrm{air}}^{\mathrm{med}}(z_{\max },s)}\right],\hfill \end{array}$$ (3)

where each term is evaluated as a function of depth $z$ in the medium and the gap $s$. The values of the restricted stopping‐power are evaluated for the different values of $\mathrm{\Delta }$ corresponding to the specific gap thickness $s$, since ${\left({\overline{L}}_{\mathrm{\Delta }}/\rho \right)}_{\mathrm{air}}^{\mathrm{med}}$ varies slightly as $s$ changes causing $\mathrm{\Delta }$ to change. ²⁷ Values of medium to air stopping‐power ratio may also change relatively quickly as a function of depth in the phantom, roughly 0.2% (4%) in the buildup region of a realistic 6 MV (25 MV) clinical beam.

Extrapolation chambers are usually used to determine surface dose by extrapolating to zero gap thickness: such extrapolations are valid only within the range of gap sizes where perturbation corrections remain effectively constant. Under such ideal conditions, the normalized product of factors in Equation 3 would be approximately unity, and thus $\%D_{\mathrm{med}}(z_{\mathrm{EPOM}})/\%D_{\mathrm{ch}}(z_{\mathrm{EPOM}},s)\approx 1$.

2. METHODS

2.1. Radiation sources

A realistic model of the Elekta SL25 linear accelerator ²⁸ was simulated using the BEAMnrc package ²⁹ , ³⁰ for producing photon beams with nominal energies of 6, 10, and 25 MV. A BEAMnrc model of the Eldorado 6 treatment unit ³¹ was also employed. The realistic (i.e., including electron contamination) photon beams had a field size of 10 x 10 ${\mathrm{cm}}^2$ at the surface of the phantom at an SSD of 100 cm. The beams produced were used as an input to other EGSnrc applications. ³² Table 1 shows the radiation sources used in this study along with calculated beam quality specifiers.

TABLE 1.

Beam quality specifiers for the Elekta SL25 and Eldorado 6 BEAMnrc models. Values of $\%\text{dd}{(10)}_x$ are calculated with a 10 x 10 ${\mathrm{cm}}^2$ field on the surface of a cylindrical water phantom with 15 cm radius and 30 cm height using the DOSRZnrc application.

		Beam quality specifier
Beam	Nominal energy (MV)	$\%\text{dd}{(10)}_x$	${\text{TPR}}_{10}^{20}$
Eldorado 6	—	58.59(4)	0.571
Elekta SL25	6	68.02(6)	0.682
	10	73.90(5)	0.735
	25	84.30(5)	0.798

Note: The number in brackets represents the one standard deviation statistical uncertainty in the last digit. ${\mathrm{TPR}}_{10}^{20}$ values for a 10 x 10 ${\mathrm{cm}}^2$ field at the point of measurement, are calculated based on the calculated %dd(10) values with Kalach and Rogers formula. ³³

2.2. Extrapolation chamber calculations

The doses to the air cavity sensitive region in ionization chambers per incident fluence were calculated using the EGSnrc ³⁴ application egs_chamber by Wulff et al. ³⁵ and compared to experimental results from the literature. ¹ , ⁷ , ¹⁷ The extrapolation chambers used by Nilsson and Montelius ¹⁷ and by Gerbi and Khan, ¹ the latter employing a PTW 30‐360 chamber, for measuring surface doses in polystyrene phantoms as a function of electrode separation, $s$, in high‐energy photon beams were modeled using the egs++ geometry package. ³⁶ These two chambers are referred to as NM and PTW, respectively. The geometry of a fixed parallel‐plate PTW Markus chamber as described by Muir et al. ³⁷ was also modeled. Figure 3 illustrates the MC models of those chambers as simulated in this study.

FIGURE 3.

Schematic diagram of the Monte Carlo models of (a) NM and (b) PTW 30‐360 extrapolation chambers and (c) the PTW Markus chamber as simulated in this study. Dashed lines in the air cavity illustrate the cavity's sensitive region.

The EPOM was taken as the front face of the chamber's cavity at a depth taking into account the front wall thickness in mg/${\mathrm{cm}}^2$. This allows measurement of surface doses in polystyrene at z = 0.976 mg/${\mathrm{cm}}^2$ $\approx$ 0.01 mm (NM) and z = 2.61 mg/${\mathrm{cm}}^2$ $\approx$ 0.03 mm (PTW). The BEAMnrc models were used as a source input to the egs_chamber simulations. The egs_chamber application was used for running independent simulations for each of the different electrode separations of the chambers in a 30 x 30 x 30 ${\mathrm{cm}}^3$ polystyrene phantom. For the NM chamber, responses were calculated for five equally spaced gaps between 0.1 and 0.9 mm, and for ten equally spaced separations from 1.0 to 10.0 mm. The thickness of the A150 back material was adjusted between 2.5 and 3.45 cm to accommodate the varying gap sizes. For the PTW chamber, responses were calculated for five equally spaced separations between 0.1 mm and 0.5 mm, and five additional separations between 1.0 and 5.0 mm. The corresponding PMMA backwall thickness ranged from 2.6 to 3.06 cm to reflect the associated gap variations. Calculations were performed with the chambers at the surface of the phantom and at different depths to evaluate the influence of the electrode separation.

For gaps less than or equal to 2 mm, two approaches were initially evaluated for setting electron and photon transport thresholds in egs_chamber. In the first approach, fixed values of ECUT = 521 keV (corresponding to a 10 keV electron kinetic energy threshold) and PCUT = 10 keV were used. In the second, ECUT was varied between 512.5 and 520 keV (i.e., electron kinetic energies from 1.5 to 9 keV) so that the CSDA range of electrons in air approximately matched the cavity thicknesses ranging from 0.1 to 2 mm. The corresponding kinetic energies were determined using a fit to ICRU Report 37 data ³⁸ : $R=0.004238E^{1.754}$, where $R$ is the CSDA range in air (cm) and $E$ is the electron kinetic energy in keV. For gaps greater than 2 mm, fixed (10 keV) ECUT and PCUT values were applied. The resulting normalized dose to the air cavity of the chamber ($\%D_{\mathrm{ch}}$) obtained using gap‐specific electron transport cutoff parameters differed by less than 0.1% (relative to the maximum response) compared to those obtained using fixed cutoff values (e.g., ECUT = 521 keV). In all simulations, the global electron and photon production thresholds (AE and AP) were set equal to ECUT and PCUT, respectively.

Perturbation factors and polystyrene to air stopping‐power ratios were also calculated to fully evaluate the normalized responses of the chambers against percentage dose versus depth (pdd) curves in the buildup region. Values of ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}$ as a function of depth in the buildup region of a polystyrene phantom were calculated using the SPRRZnrc application. ³² Cutoff energy values for $\mathrm{\Delta }$ used in the calculation of restricted stopping powers were also selected based on gap, ranging from approximately 1.5 keV (for 0.1 mm gaps) to 22.5 keV (for 10 mm gaps), using the same CSDA‐based relation described above. Dose to polystyrene ($D_{\mathrm{pst}}$) versus depth was calculated using the egs_chamber application with cylindrical voxel elements of thickness 0.002 cm and radius equal to the active radius of each chamber cavity. These 0.002 cm thick disk‐shaped voxels were defined along the central axis of a 30 x 30 x 30 ${\mathrm{cm}}^3$ polystyrene phantom and the dose scored in each voxel was taken as the dose to the medium at the depth corresponding to the voxel center. Individual calculations of $P_{\mathrm{repl}}$ were obtained from equation 2 by considering a polystyrene‐walled chamber (and hence $P_{\mathrm{wall}}$ = 1) ³⁹ , ⁴⁰ , ⁴¹ so that:

$$P_{\mathrm{repl}}=D_{\mathrm{pst}}/\left({\overline{D}}_{\mathrm{air}}{\left(\frac{{\overline{L}}_{\mathrm{\Delta }}}{\rho }\right)}_{\mathrm{air}}^{\mathrm{pst}}\right),$$ (4)

where ${\overline{D}}_{\mathrm{air}}$ is the dose per incident fluence to the air cavity of the chamber when all chamber materials (except the air cavity) are made of polystyrene. Similarly, $P_{\mathrm{wall}}$ was calculated according to ³⁹ , ⁴⁰ , ⁴² :

$$P_{\mathrm{wall}}={\overline{D}}_{\mathrm{air}}/{\overline{D}}_{\text{ch}},$$ (5)

where ${\overline{D}}_{\mathrm{ch}}$ is the dose to the air cavity of the fully modeled chamber.

Photon cross‐section enhancement (XCSE), intermediate phase‐space scoring (IPSS), and correlated sampling (CS) are variance reduction techniques implemented within egs_chamber ³⁵ and were employed in this study to significantly improve dose calculation efficiency for ionization chambers. Russian Roulette was also applied to reduce computational load by terminating low‐weight particle histories. A summary of the key simulation parameters and variance reduction settings is provided in Table 2.

TABLE 2.

Default Monte Carlo simulation parameters used in this study, except where otherwise specified.

Parameter	Value	Description
AE	521 keV	Threshold for production of secondary electrons by charged particles
AP	10 keV	Threshold for production of secondary photons by charged particles
ECUT	521 keV	Charged particle cutoff energy
PCUT	10 keV	Photon cutoff energy
N	$2\times {10}^9$	Number of simulated histories
XCSE factor	256	Cross‐section enhancement factor
1/${\mathrm{N}}_r$	1/512	Survival probability for Russian roulette
Brems cross sections	NIST	Bremsstrahlung cross sections

In this study, when presenting or comparing values already expressed as percentages (e.g., %$D_{\mathrm{pst}}$), results are reported in the format %$D_{\mathrm{pst}}$ = (18 $\pm$ 1)%, where both the value and its associated “absolute” uncertainty are given in percentage units. Differences between such values are expressed directly in percentage points, relative to $D_{\max }$. For example, a 2% difference is reported between %$D_{\mathrm{pst}}$ = 18% (e.g., measured) and 16% (e.g., calculated). In contrast, statistical uncertainties associated with Monte Carlo–calculated values are explicitly described as fractional uncertainties, even when the reported quantity is a percentage. For instance, a result of %$D_{\mathrm{pst}}$ = (10.0 $\pm$ 0.1)% corresponds to a statistical (fractional) uncertainty of 1%.

3. RESULTS

3.1. Comparison to experimental data

3.1.1. NM chamber

Figure 4 shows the normalized percentage mass ionization as a function of the electrode separation, $s$, for the NM extrapolation chamber in and 6 MV x‐ray beams with the EPOM at z = 0.01 mm in a polystyrene phantom (i.e., with the chamber's front face at the surface of the phantom). Experimental results from Nilsson and Montelius ¹⁷ correspond to the mean ionization per unit mass (C ${\mathrm{kg}}^{-1}$) normalized to the value at the depth of maximum ionization for a fixed gap $s$ = 1 mm. Our calculated results for %$D_{\mathrm{ch}}(0.01,s)$ are the dose to the air cavity sensitive region of the chamber per incident fluence normalized by the same quantity at the depth of maximum dose for $s$ = 1 mm. Normalizing each chamber's response to its own maximum at the corresponding gap thickness, rather than to the fixed gap of 1 mm, resulted in differences of less than 0.1% relative to maximum response. Figure 4 also includes our Monte Carlo calculated values for the percentage dose to the polystyrene phantom, $\%D_{\mathrm{pst}}(0.01)$, for both beam qualities, normalized to the respective maximum dose, $D_{\max }$.

FIGURE 4.

Simulated $\%D_{\mathrm{ch}}(0.01,s)$ (closed symbols) and measured %I(0.01,s) (open symbols) for the NM ¹⁷ extrapolation chamber at z = 0.01 mm in a polystyrene phantom for (circles) and 6 MV (squares) beams as a function of electrode separation. The solid straight lines are linear fits to the experimental data for gaps $\ge$ 2 mm while the Xs on the y‐axis are the experimental “extrapolated” values for zero gap. Calculated values of percentage dose to polystyrene, $\%D_{\mathrm{pst}}$(0.01), at z = 0.01 mm are also shown (dashed lines).

Maximum differences relative to $D_{\max }$ between measured and calculated normalized chamber responses for a gap of 10 mm were approximately 1.6% in the beam and 2.1% in the 6 MV beam. For gap thicknesses $s\le 1$ mm, the maximum differences between the calculated chamber response, $\%D_{\mathrm{ch}}(0.01,s)$, and the corresponding percentage dose to polystyrene, $\%D_{\mathrm{pst}}(0.01)$, were approximately 0.5% for and 0.3% for 6 MV, both evaluated at $s=1$ mm. For the same gap range, maximum differences between the measured percentage ionization, $\%I_{\mathrm{ch}}(0.01,s)$, and calculated $\%D_{\mathrm{pst}}(0.01)$ were approximately 1.0% for and 1.3% for the 6 MV beam, both at $s=0.5$ mm.

3.1.2. PTW 30‐360 chamber

Figure 5 presents the normalized response of the PTW extrapolation chamber (PTW 30‐360) as a function of electrode separation, $s$, in a 6 MV beam, with the EPOM at z = 0.03 mm in a polystyrene phantom. Experimental data from Gerbi and Khan ¹ and Reynolds and Higgins, ⁷ acquired using the same chamber model, represent the mean ionization per unit mass (C ${\mathrm{kg}}^{-1}$) normalized to its maximum value for each gap. Our calculated results for $\%D_{\mathrm{ch}}(0.03,s)$ are the dose per incident fluence to the air cavity sensitive region of the chamber also normalized by its value for each gap with the EPOM at depth of maximum dose, ${\mathrm{z}}_{\max }$. Relative statistical uncertainties on calculated chamber responses were 0.1% on average. The large error bars associated with the 1 mm gap measurement by Reynolds and Higgins was attributed to low signal‐to‐noise ratio and a significantly larger polarity correction. Monte Carlo calculated value for the percentage dose to polystyrene, $\%D_{\mathrm{pst}}(0.03)$, at z = 0.03 mm and normalized by $D_{\max }$ is also shown for comparison.

FIGURE 5.

Same as Figure 4 except for the PTW 30‐360 extrapolation chamber in 6 MV beam. Experimental data are from Gerbi and Khan (open squares) ¹ and Reynolds and Higgins (open triangles). ⁷ An offset of ‐0.1 mm is applied to the Reynolds and Higgins data for clarity.

Maximum differences relative to the maximum response between calculated and measured normalized chamber responses were approximately 1.2% at a 5 mm gap when compared to the measurements by Gerbi and Khan, and 2.1% at a 1 mm separation when compared to the data reported by Reynolds and Higgins. Within the small‐gap range ($s\le 1$ mm), the maximum differences between the measured percentage ionization, $\%I(0.03,s)$, and the corresponding percentage dose to polystyrene, $\%D_{\mathrm{pst}}(0.03)$, were 1.3% and 2.4% for Gerbi and Khan and Reynolds and Higgins, respectively, both evaluated at $s=1$ mm. In comparison, our calculated values of $\%D_{\mathrm{ch}}(0.03,s)$ for the same gap range differed from $\%D_{\mathrm{pst}}(0.03)$ by no more than 0.3% relative to $D_{\max }$, also at $s=1$ mm.

Figure 6 presents the differences between chamber responses obtained with a fixed parallel‐plate Markus chamber and the PTW extrapolation chamber as a function of normalized depth ($z/z_{\max }$) in a polystyrene phantom for and 25 MV beams. In the experimental work by Gerbi and Khan, ¹ the PTW chamber ionizations were extrapolated to zero plate separation, $\%I(z,0)$, to estimate the corrected percentage dose to polystyrene, $\%D_{\mathrm{pst}}(z)$. As such, the data shown in Figure 6 reflect the over‐response of the Markus chamber relative to the extrapolated PTW chamber values. In our Monte Carlo simulations, the PTW chamber normalized air‐cavity dose, $\%D_{\mathrm{ch}}(z,0.1)$, calculated using a plate separation of $s=0.1$ mm, were assumed to represent the true dose to the medium. These calculated values were then used as the reference to quantify the Markus chamber over‐response across the depth range for both beam qualities.

FIGURE 6.

Differences between responses (as a % of its maximum) of a PTW 30‐360 extrapolation chamber and a fixed parallel‐plate Markus chamber in the buildup region in polystyrene for (circles) and 25 MV (squares) beams with 10 x 10 ${\mathrm{cm}}^2$ fields at SSD = 100 cm as a function of depth $z$ normalized to $z_{\max }$. Experimental data (open symbols) are from Gerbi and Khan ¹ with '“absolute” uncertainties of approximately 1% (relative to maximum response). Monte Carlo calculated egs_chamber data (solid symbols) are also shown.

“Absolute” uncertainties in the experimental measurements were approximately 1% of $D_{\max }$, while statistical uncertainties in the Monte Carlo calculations averaged 0.4%. Maximum differences between calculated and measured over‐responses, relative to the maximum response, were 2.0% for the beam at $z/z_{\max }\approx 0.01$, and 2.6% for the 25 MV beam at $z/z_{\max }\approx 0.4$. For the beam, the maximum over‐response of the Markus chamber was 16.2% of $D_{\max }$ based on Monte Carlo calculations and 18.2% based on measurements, both at $z/z_{\max }\approx 0.01$. For the 25 MV beam, the corresponding maximum over‐responses were 3.0% (calculated) and 3.4% (measured), both observed at $z/z_{\max }\approx 0.001$.

3.1.3. Surface dose measurements and simulations

Table 3 summarizes the experimental results reported by Gerbi and Khan ¹ and Nilsson and Montelius ¹⁷ for the surface dose, $\%I(z_{\mathrm{EPOM}},0)$, in a polystyrene phantom. These values were obtained by extrapolating percentage ionization measurements for small gaps ($s\le 1$ mm) to zero gap thickness. The table also presents our Monte Carlo calculated chamber responses, $\%D_{\mathrm{ch}}(z_{\mathrm{EPOM}},0.1)$, normalized to their respective maximum values, along with the corresponding percentage dose to the medium, $\%D_{\mathrm{pst}}(z_{\mathrm{EPOM}})$, at depths $z=0.01$ mm and $z=0.03$ mm for the NM and PTW chambers, respectively.

TABLE 3.

Surface (z = EPOM) relative doses in polystyrene phantom for a 10 x 10 ${\mathrm{cm}}^2$ field in , 6 MV, 10 MV, and 25 MV photon beams. The second column shows the measured extrapolated values by Nilsson and Montelius ¹⁷ (NM, z = 0.01 mm) and Gerbi and Khan ¹ (PTW, z = 0.03 mm). egs_chamber calculated values using 0.1 mm plate separation are shown on third column. The calculated, $\%D_{\mathrm{pst}}(z_{\mathrm{EPOM}})$, values and are shown in the second last column at the corresponding depths (front window) of the chambers for comparison. The number in brackets represents the one standard deviation statistical uncertainty in the last digit. Last column shows the ratio between calculated values of $\%D_{\mathrm{pst}}(z_{\mathrm{EPOM}})$ and $\%D_{\mathrm{ch}}(z_{\mathrm{EPOM}},0.1)$.

	Measured $\%I(z_{\mathrm{EPOM}},0)$	Calculated %$D_{\mathrm{ch}}(z_{\mathrm{EPOM}},0.1)$	Calculated $\%D_{\mathrm{pst}}(z_{\mathrm{EPOM}})$	Ratio $\%D_{\mathrm{pst}}$ / $\%D_{\mathrm{ch}}$
Co
NM	16.1	17.27(6)	17.09(2)	0.99
PTW	21.2	19.2(1)	19.20(2)	1.00
6 MV
NM	11.6	12.69(2)	12.58(2)	0.99
PTW	15.2	14.2(1)	13.92(1)	0.98
10 MV
PTW	11.0	10.9(1)	10.6(2)	0.97
25 MV *
PTW	16.4	13.4(1)	12.56(2)	0.94

^*This experimental value from Gerbi and Khan is for a 24 MV Varian 2500 accelerator while the corresponding simulations are for a 25 MV Elekta SL25.

For the NM chamber, the differences between measured $\%I(z_{\mathrm{EPOM}},0)$ and calculated $\%D_{\mathrm{pst}}(z_{\mathrm{EPOM}})$ were approximately ‐1.0% of $D_{\max }$ for both and 6 MV beams. For the PTW chamber, the corresponding differences were 2.0%, 1.3%, 0.4%, and 3.8% for , 6 MV, 10 MV, and 25 MV beams, respectively. Reynolds and Higgins ⁷ reported a surface dose of (15.93 $\pm$ 2.1)% in polystyrene for a 6 MV beam, obtained by linearly extrapolating percent ionization values measured with the PTW chamber at gaps of 1, 3, and 5 mm (see Figure 5). This value differed by approximately 2.0% from our calculated $\%D_{\mathrm{pst}}(z_{\mathrm{EPOM}})$, and by 1.7% from our calculated $\%D_{\mathrm{ch}}(z_{\mathrm{EPOM}},0.1)$ at the same depth.

3.2. Perturbation factors for the NM chamber and stopping‐power ratios

Figure 7 shows the variation of the perturbation correction factors $P_{\mathrm{repl}}(0.01,s)$ and $P_{\mathrm{wall}}(0.01,s)$ for the NM extrapolation chamber, as well as the polystyrene‐to‐air stopping‐power ratios ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(0.01,s)$, as a function of electrode separation $s$ for both and 6 MV beams. All quantities were evaluated at a depth of 0.01 mm, corresponding to the chamber's EPOM in a polystyrene phantom. Additional plots showing the behavior of $P_{\mathrm{repl}}(z_{\max },s)$ and $P_{\mathrm{wall}}(z_{\max },s)$ at the depth of maximum dose ($z=z_{\max }$), along with the depth‐dependent variation of ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(z,s)$, are provided in the supplemental material (Figures S1–S3).

FIGURE 7.

Comparison of Monte Carlo calculated perturbation factors $P_{\mathrm{wall}}(0.01,s)$ (closed circles) and $P_{\mathrm{repl}}(0.01,s)$ (open squares) and polystyrene to air stopping‐power ratios ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{polyst}}(0.01,s)$ (closed triangles) as a function of the electrode separation, $s$, for the NM extrapolation chamber in polystyrene phantom with EPOM at z = 0.01 mm in (left panel) and 6 MV (right) photon beams.

A variation of approximately 43% in $P_{\mathrm{repl}}(0.01,s)$ was observed as the gap increased from 0.1 to 10 mm for both beam energies. Over the same range, $P_{\mathrm{wall}}(0.01,s)$ varied by 2.0% for the beam and 0.9% for the 6 MV beam. Within the narrower gap interval of 0.1–1 mm, $P_{\mathrm{repl}}(0.01,s)$ varied by 0.8% for and 0.4% for 6 MV, while the corresponding variation in $P_{\mathrm{wall}}(0.01,s)$ was 0.2% for both beam qualities.

As shown in the supplemental data, at $z=z_{\max }$, the variation in $P_{\mathrm{repl}}(z_{\max },s)$ for s values from 0.1 to 10 mm was 2.6% for and 1.7% for 6 MV (Figure S1). The corresponding changes in $P_{\mathrm{wall}}(z_{\max },s)$ were 0.6% and 0.4%, respectively (Figure S2). Over the limited range of 0.1–1 mm, $P_{\mathrm{repl}}(z_{\max },s)$ varied by 0.9% for and 0.3% for 6 MV, while $P_{\mathrm{wall}}(z_{\max },s)$ changed by 0.1% and 0.04%, respectively.

Values of ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(z,s)$ were calculated for cutoff energies $\mathrm{\Delta }$ ranging from 1.5 keV (corresponding to 0.1 mm gaps) to 22.5 keV (corresponding to 10 mm gaps). Over this full range of $\mathrm{\Delta }$, a consistent variation of approximately 1.2% was observed across all depths for both beam qualities. For cutoff energies between 1.5 and 6 keV, corresponding to air cavity thicknesses of 0.1–1 mm, the variation was approximately 0.7% for both and 6 MV beams. For a fixed $\mathrm{\Delta }$, depth‐dependent variations in ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(z)$ from $z=0.01$ mm to $z=z_{\max }$ were at most 0.05% for the beam and 0.3% for the 6 MV beam (Figure S3).

Table 4 summarizes values of $P_{\mathrm{repl}}(z,s)$, $P_{\mathrm{wall}}(z,s)$, and ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(z,s)$ at the surface of the polystyrene phantom (z = 0.01 mm) for a gap s = 0.1 mm, and at the depth of maximum dose (z = z ) for s = 1 mm, for both and 6 MV photon beams. Also included are the products of these factors and their corresponding ratios, calculated using Equation 3. For the beam, the product $P_{\mathrm{repl}}(z,s)$ $\times$ $P_{\mathrm{wall}}(z,s)$ $\times$ ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(z,s)$ was approximately 1.089 $\pm$ 0.006 at z = 0.01 mm with s = 0.1 mm, and 1.100 $\pm$ 0.001 at z = 4 mm with s = 1 mm, yielding a ratio of 0.990 $\pm$ 0.006. For the 6 MV beam, the corresponding products were 1.077 $\pm$ 0.003 at z = 0.01 mm and 1.087 $\pm$ 0.001 at z = 14 mm, resulting in a ratio of 0.991 $\pm$ 0.003.

TABLE 4.

Calculated perturbation correction factors $P_{\mathrm{repl}}(z,s)$ and $P_{\mathrm{wall}}(z,s)$ for the NM extrapolation chamber, and polystyrene‐to‐air restricted mass stopping‐power ratios ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(z,s)$ for and 6 MV beams, at the phantom surface ($z=0.01$ mm) and at the depth of maximum dose ($z_{\max }$), for selected electrode separations $s$.

($z$ mm, $s$ mm)	$P_{\mathrm{repl}}(z,s)$	$P_{\mathrm{wall}}(z,s)$	${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(z,s)$	$P_{\mathrm{repl}}\times$ $P_{\mathrm{wall}}\times$ ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}$
Co
(0.01, 0.1)	1.005(4)	0.967(4)	1.1205(1)	1.089(6)
(4, 1)	1.0001(4)	0.9888(3)	1.11240(1)	1.100(1)
ratio	1.005(4)	0.978(4)	1.0073(1)	0.990(6)
6 MV
(0.01, 0.1)	1.002(2)	0.971(2)	1.1070(2)	1.077(3)
(14, 1)	1.0001(7)	0.9906(5)	1.09673(2)	1.087(1)
ratio	1.002(2)	0.980(2)	1.0094(2)	0.991(3)

Note: The final column lists the product of the three factors and the corresponding ratio described in Equation 3. The number in brackets represents the one standard deviation statistical uncertainty in the last digit.

4. DISCUSSION

4.1. Agreement with experimental data

The calculated responses of the NM and PTW chambers at the surface as a function of electrode separation agreed well with experimental measurements. For both chamber models, differences between our Monte Carlo calculated normalized chamber doses, $\%D_{\mathrm{ch}}(z,s)$, and the experimentally reported normalized ionizations, $\%I(z,s)$, were within $\pm$2% relative to their values at $d_{\max }$, except for the PTW chamber in the 25 MV beam, for which the deviation reached $-3\%$. The comparison is intended primarily to evaluate the relative dependence on gap size rather than absolute agreement, since experimental measurements were performed on clinical beams that may differ from the modeled conditions, including electron contamination. Moreover, the shape of the response curves is governed predominantly by chamber geometry, which was explicitly and accurately modeled in the present simulations.

The observed increase in chamber response with increasing gap is explained by the increasing contribution of side‐wall scattered electrons, which become more likely to reach the sensitive volume as the gap increases. As shown in Figure 4, the calculated values for $\%D_{\mathrm{ch}}(z,s)$ versus $s$ reproduce the nonlinear response observed experimentally, with linear behavior only emerging for $s>2$ mm. It is worth noting that extrapolations based on these larger separations may introduce significant bias, especially in the absence of full electronic equilibrium. The critical point to recognize is that it is not a linear extrapolation of the normalized responses for larger gap sizes that is important. It is where this linearity breaks down for small gaps that tells us where the results now represent the relative surface dose.

The comparative analysis with the Markus chamber shown in Figure 6 confirmed the known overresponse in the buildup region, particularly at shallow depths. Calculated values were found to agree within uncertainties with the measured values. For the 25 MV beam differences were also within uncertainties except for $z/z_{\max }$ = 0.3 and 0.4 where differences were 1.6% and 2.6% respectively. However those two experimental values appear to exhibit disagreement with the rest of the curve. Both the calculated and experimental results demonstrate that the over‐response of fixed parallel‐plate chambers decreases with increasing photon beam energy. This trend is consistent with the broader angular spread of secondary electrons produced by lower‐energy beams such as , which contributes to the larger over‐response observed for that beam compared with 25 MV. Figure 6 also shows that the largest over‐response occurs within the first 50% of $z_{\max }$, where charged particle disequilibrium is strongest. This behavior helps explain why fixed parallel‐plate chambers provide inaccurate surface‐dose measurements in the buildup region.

4.2. Surface dose assessment

Surface dose estimates derived from our simulations were consistent with measured extrapolated values to within 1 to 3% of $D_{\max }$, depending on chamber type and beam quality, as summarized in Table 3. As already pointed out, it is important to keep in mind that the term surface dose as used by previous authors represents the dose at the inside of the chamber's front window (EPOM). In this sense, the experimental values shown in Table 3 are compared with the calculated values at the depths taking into account the respective window thickness of the chambers. The calculated values of $\%D_{\mathrm{pst}}(z_{\mathrm{EPOM}})$ were found to be in closer agreement with the simulated chamber doses, $\%D_{\mathrm{ch}}(z_{\mathrm{EPOM}},0.1)$, than with the measured extrapolated ionization values, $\%I(z_{\mathrm{EPOM}},0)$, for both NM and PTW chambers. Here, the term “measured extrapolated” refers to the ionization values obtained by extrapolating measurements within the small‐gap regime where the response becomes approximately constant. In agreement with the reported experimental findings, our results demonstrate that small gap responses are more reliable for assessing surface dose.

For small gaps ($s\le 1$ mm), the calculated chamber responses, $\%D_{\mathrm{ch}}(z_{EPOM},s)$, closely matched the corresponding dose to the medium, $\%D_{\mathrm{pst}}(z_{EPOM})$, with differences below 0.3% relative to $D_{\max }$, as shown in Figures 4 and 5 for and 6 MV beams, and below 0.9% for 10 and 25 MV beams as indicated in Table 3. Nevertheless, it is important to emphasize that the ratio $\%D_{\mathrm{pst}}(z)/\%D_{\mathrm{ch}}(z,s)$ in and 6 MV beams deviates from unity even in this small‐gap regime, with differences on the order of 1% to 2% and potentially larger depending on chamber‐specific geometry, construction materials, and photon beam energy. This highlights that, despite the absolute agreement being very good, the chamber response is not strictly perturbation‐free, and small corrections remain necessary to ensure accurate dose estimation, even for minimal electrode separations.

As shown in Table 3, our calculated values of $\%D_{\mathrm{ch}}(0.01,0.1)$ in both and 6 MV beams are in excellent agreement with the corresponding calculated surface doses to polystyrene, $\%D_{\mathrm{pst}}(0.01)$, of (17.09 $\pm$ 0.02)% and (12.58 $\pm$ 0.02)%, respectively. Likewise, for the PTW chamber, our calculated responses, $\%D_{\mathrm{ch}}(0.03,0.1)$, were within 0.3% difference, relative to $D_{\max }$, of the calculated $\%D_{\mathrm{pst}}(0.03)$ for , 6, and 10 MV beams. For the 25 MV beam, the difference remains below 0.9%, further confirming the close agreement between the percentage dose to the chamber at the surface of the phantom and the corresponding percentage dose to the medium.

The slightly larger discrepancies of 3% and 3.8% observed for the 25 MV beam when comparing $\%I(0.03,0)$ with $\%D_{\mathrm{ch}}(0.03,0.1)$ and with $\%D_{\mathrm{pst}}(0.03)$, respectively, may be partially attributed to the use of different accelerator models in simulation and measurement (25 MV Elekta SL25 vs. 24 MV Varian 2500). Such differences affect the electron spectra and hence the dose buildup characteristics, which are particularly sensitive to beam quality near the surface. However, using the same BEAMnrc model the calculated $\%D_{\mathrm{ch}}(0.03,0.1)$ and $\%D_{\mathrm{pst}}(0.03)$ agreed within 0.9% of $D_{\max }$.

4.3. Influence of perturbation factors and stopping‐power ratios

The analysis of perturbation correction factors for the NM chamber presented in Figure 7 confirmed that $P_{\mathrm{repl}}(0.01,s)$ is the dominant factor influencing chamber response with changing electrode separation near the surface, with its variation far exceeding that of $P_{\mathrm{wall}}(0.01,s)$ and stopping‐power ratios. The largest correction (i.e., farthest from unity) due to fluence perturbation was obtained for the largest distance between the electrodes ($s=$ 10 mm) for both energy beams due to lack of charged particle equilibrium, consistent with the findings of Nilsson and Montelius, ¹⁷ as also illustrated in Figure 4. As the gap decreases, $P_{\mathrm{repl}}(0.01,s)$ approaches values closer to unity which indicates that for small gaps ($s\le 1$ mm), the fluence perturbation due to cavity size becomes minimal. Notably, $P_{\mathrm{repl}}(0.01,s)$ exhibited significantly smaller variations ($\le$ 0.8%) within the small‐gap range (0.1 mm $\le s\le$ 1 mm) compared to the much larger variation of approximately 43% observed over the full separation range (0.1 mm $\le s\le$ 10 mm) for both beam energies.

In contrast, $P_{\mathrm{wall}}(0.01,s)$ exhibited considerably smaller variations under all conditions. Within the range of small gaps relevant to surface dose measurements ($s\le 1$ mm), changes in $P_{\mathrm{wall}}(0.01,s)$ did not exceed 0.2% for either beam quality. Similarly, ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(0.01,s)$, varied by no more than 0.7% within the small‐gap regime and remained below 1.2% across the full range of gaps.

The over‐response of parallel‐plate chambers such as the Markus chamber is influenced not only by cavity thickness, but also by chamber radius, guard width, and side‐wall design. As shown by Gerbi and Khan, ¹ the small guard width of the Markus chamber largely explains its pronounced over‐response compared with other fixed parallel‐plate chambers. A detailed quantitative analysis of $P_{\mathrm{repl}}$ and $P_{\mathrm{wall}}$ for an extrapolation‐chamber geometry replicating the Markus design would provide further insight into the underlying mechanisms, but such chamber‐specific modeling is beyond the scope of the present work, which focuses on the extrapolation chambers of Nilsson and Montelius and Gerbi and Khan. Nevertheless, while geometrical details affect the magnitude of the response, there is no reason to expect qualitatively different behavior from the perturbation factors than that shown in Figure 7. In particular, $P_{\mathrm{repl}}$, which depends on both the guard width and the collecting volume diameter to thickness ratio, is expected to remain the dominant contributor to over‐response in parallel‐plate chambers.

The ability of extrapolation chambers to accurately estimate the relative surface dose, $\%D_{\mathrm{pst}}(z)$, without requiring additional corrections, can be evaluated using Equation 3 by examining the normalized product of perturbation correction factors. As shown in Table 4, the Monte Carlo–calculated values of $P_{\mathrm{repl}}(z,s)$, $P_{\mathrm{wall}}(z,s)$, and ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}(z,s)$ yield a product close to 0.99 for both and 6 MV beams. This result is consistent with the independently calculated ratio $\%D_{\mathrm{pst}}(z)/\%D_{\mathrm{ch}}(z,s)$, as discussed earlier and presented in Table 3. These findings demonstrate that, while extrapolation chambers operated at small electrode separations exhibit reduced perturbations, their response is not strictly perturbation‐free and requires Monte Carlo–based correction factors to accurately assess $\%D_{\mathrm{pst}}(z)$. The use of small gaps primarily serves to minimize the magnitude of these corrections, rather than to eliminate them, and the appropriate correction is given by the calculated ratio $\%D_{\mathrm{pst}}(z)/\%D_{\mathrm{ch}}(z,s)$ for the specific chamber geometry and beam quality.

5. CONCLUSIONS

Accurate assessment of surface dose in megavoltage photon beams requires precise specification of measurement depth due to the steep dose gradient near the phantom surface, where differences can exceed 10% over sub‐millimeter distances (e.g., 12.3% for and 10.7% for 6 MV between 0.01 and 0.03 mm). Monte Carlo simulations using the EGSnrc system successfully reproduced experimental chamber responses, including the known over‐response of fixed‐geometry parallel‐plate chambers such as the Markus.

The results confirmed that the replacement perturbation factor, $P_{\mathrm{repl}}$, is the primary contributor to the variation in chamber response with electrode separation near the surface. In contrast, the wall perturbation factor, $P_{\mathrm{wall}}$, and the restricted mass stopping‐power ratios, ${({\overline{L}}_{\mathrm{\Delta }}/\rho )}_{\mathrm{air}}^{\mathrm{pst}}$, showed minimal dependence on gap size. For small gaps ($s\le 1$ mm), the correction factors for the NM extrapolation chamber in and 6 MV beams exhibited minimal variation, and the chamber response approached Spencer–Attix cavity behavior, with residual differences on the order of 1%.

For the specific chambers investigated at and 6 MV beams in this work, the residual deviations correspond to an effective correction factor of approximately 0.99 $\pm$ 0.01 to obtain the dose to the medium at the surface. This value should not be regarded as universal, as the magnitude of the required correction is expected to depend on beam energy and chamber design, as illustrated by the larger deviations observed for the PTW chamber and at higher photon energies. The key observation is not the linear extrapolation of normalized responses obtained at larger gap sizes, but rather the identification of the gap region where this linearity breaks down, indicating conditions under which the chamber response most closely represents the true relative surface dose. However, rather than implying perturbation‐free operation at small gaps, the present results demonstrate that Monte Carlo calculations can be used to quantify chamber‐specific correction factors for any electrode separation employed experimentally. While smaller gaps minimize the magnitude of the correction, the gap dependence of chamber response is accurately reproduced, enabling reliable correction of measured surface doses across the full range of separations investigated.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

Supporting information

Supporting Information