Dose perturbations due to contrast medium and air in mammosite treatment: an experimental and Monte Carlo study

In the management of early breast cancer, a partial breast irradiation technique called MammoSite (Proxima Therapeutic Inc., Alpharetta, GA) has been advocated in recent years. In MammoSite, a balloon implanted at the surgical cavity during tumor excision is filled with a radio-opaque solution, and radiation is delivered via a high dose rate brachytherapy source situated at the center of the balloon. Frequently air may be introduced during placement of the balloon and/or injection of the contrast solution into the balloon. The purpose of this work is to quantify as well as to understand dose perturbations due to the presence of a high-Z contrast medium and/or an air bubble with measurements and Monte Carlo calculations. In addition, the measured dose distribution is compared with that obtained from a commercial treatment planning system (Nucletron PLATO system). For a balloon diameter of 42 mm, the dose variation as a function of distance from the balloon surface is measured for various concentrations of a radio-opaque solution (in the range 5%-25% by volume) with a small volume parallel plate ion chamber and a micro-diode detector placed perpendicular to the balloon axis. Monte Carlo simulations are performed to provide a basic understanding of the interaction mechanism and the magnitude of dose perturbation at the interface near balloon surface. Our results show that the radio-opaque concentration produces dose perturbation up to 6%. The dose perturbation occurs mostly within the distances <1 mm from the balloon surface. The Plato system that does not include heterogeneity correction may be sufficient for dose planning at distances > or = 10 mm from the balloon surface for the iodine concentrations used in the MammoSite procedures. The dose enhancement effect near the balloon surface (<1 mm) due to the higher iodine concentration is not correctly predicted by the Plato system. The dose near the balloon surface may be increased by 0.5% per cm3 of air. Monte Carlo simulation suggests that the interface effect (enhanced dose near surface) is primarily due to Compton electrons of short range (<0.5 mm). For more accurate dosimetry in MammoSite delivery, the dose perturbation due to the presence of a radio-opaque contrast medium and air bubbles should be considered in a brachytherapy planning system.     

Dose perturbation effects in prostate seed implant brachytherapy with I-125

EGSnrc Monte Carlo simulation was used to investigate dose perturbation effects in prostate seed implant brachytherapy using 125I radioactive seeds used in implant brachytherapy. Dose perturbation effects resulting from the seed mutual attenuation in a prostate seed implant consisting of 27 seeds were investigated. The results showed that for 125I seeds implanted into the prostate at 1.00 cm, 0.75 cm and 0.50 cm apart (uniform spacing), the dose perturbation effects are up to 10%. The volume of the target occupied by the 10% dose difference between the full Monte Carlo simulation and the single seed superposition model decreases with increasing seed spacing. Despite the differences between the Monte Carlo simulation and the simple superposition, there was no significant change in the dose volume histogram for 1 cm and 0.75 cm seed spacing. However, there was a significant change in the dose volume histogram when the seed spacing was 0.5 cm. An analysis of the external volume index (EI), coverage index (CI) and homogeneity index (HI) also showed that there is no difference in these indexes for the 1.00 cm and 0.75 cm seed spacing between the simple superposition model and the full Monte Carlo simulation. Compared to the full Monte Carlo simulations, the simple superposition model overestimated EI, CI and HI by 7%, 5% and 4% respectively for the 0.50 cm seed spacing.     

A systematic evaluation of air cavity dose perturbation in megavoltage x-ray beams

The EGS4 Monte Carlo radiation transport code was used to systematically study the dose perturbation near planar and cylindrical air cavities in a water medium irradiated by megavoltage x-ray beams. The variables of the problem included x-ray energy, cavity shape and dimension, and depth of the cavity in water. The Monte Carlo code was initially validated against published measurements and its results were found to agree within 2% with the published measurements. The study results indicate that the dose perturbation is strongly dependent on x-ray energy, field size, depth, and size of cavity in water. For example, the Monte Carlo calculations show dose reductions of 42% and 18% at 0.05 and 2 mm, respectively, beyond the air-water interface distal to the radiation source for a 3 cm thick air slab irradiated by a single 5x5 cm2 15 MV beam. The dose reductions are smaller for a parallel-opposed pair of 5x5 cm2 15 MV x-ray beams, being 21% and 11% for the same depths. The combined set of Monte Carlo calculations showed that the dose reduction near an air cavity is greater for: (a) Smaller x-ray field size, (b) higher x-ray energy, (c) larger air-cavity size, and (d) smaller depth in water where the air cavity is situated. A potential clinical application of these results to the treatment of prostate cancer is discussed.     

Doses on the central axes of narrow 6-MV x-ray beams

The absorbed doses on the central axes of narrow beams (radii 0.07-2.5 cm) of 6-MV x rays have been studied by experiments and Monte Carlo simulations. The measurements were made in a geometry used for irradiation of intracranial lesions. For radii less than 1.0 cm the dose on the central axis is progressively reduced due to electron disequilibrium. This leads to measurement artifacts when the detector is too large, as was readily observed with ionization chambers. Radiographic and radiochromic films were used with densitometric evaluation to provide the resolution necessary to measure absorbed doses for the narrowest beams. The contribution by phantom-scattered photons is significant even at small field sizes, and scatter factors were determined from the experimental results. Photons scattered by the auxiliary collimator did not add appreciably to the dose on the central axis. The data were used to characterize the dose-to-kerma ratio as a function of beam radius. Differences between experimental results and those from Monte Carlo calculations were observed.     

Monte Carlo evaluation of a photon pencil kernel algorithm applied to fast neutron therapy treatment planning

When dedicated software is lacking, treatment planning for fast neutron therapy is sometimes performed using dose calculation algorithms designed for photon beam therapy. In this work Monte Carlo derived neutron pencil kernels in water were parametrized using the photon dose algorithm implemented in the Nucletron TMS (treatment management system) treatment planning system. A rectangular fast-neutron fluence spectrum with energies 0-40 MeV (resembling a polyethylene filtered p(41)+Be spectrum) was used. Central axis depth doses and lateral dose distributions were calculated and compared with the corresponding dose distributions from Monte Carlo calculations for homogeneous water and heterogeneous slab phantoms. All absorbed doses were normalized to the reference dose at 10 cm depth for a field of radius 5.6 cm in a 30 x 40 x 20 cm3 water test phantom. Agreement to within 7% was found in both the lateral and the depth dose distributions. The deviations could be explained as due to differences in size between the test phantom and that used in deriving the pencil kernel (radius 200 cm, thickness 50 cm). In the heterogeneous phantom, the TMS, with a directly applied neutron pencil kernel, and Monte Carlo calculated absorbed doses agree approximately for muscle but show large deviations for media such as adipose or bone. For the latter media, agreement was substantially improved by correcting the absorbed doses calculated in TMS with the neutron kerma factor ratio and the stopping power ratio between tissue and water. The multipurpose Monte Carlo code FLUKA was used both in calculating the pencil kernel and in direct calculations of absorbed dose in the phantom.     

On the discrepancies between Monte Carlo dose calculations and measurements for the 18 MV varian photon beam

Significant discrepancies between Monte Carlo dose calculations and measurements for the Varian 18 MV photon beam with a large field size (40 x 40 cm2) were reported by different investigators. In this work, we investigated these discrepancies based on a new geometry model ("New Model") of the Varian 21EX linac using the GEPTS Monte Carlo code. Some geometric parameters used in previous investigations (Old Model) were inaccurate, as suggested by Chibani in his AAPM presentation (2004) and later confirmed by the manufacturer. The entrance and exit radii of the primary collimator of the New Model are 2 mm larger than previously thought. In addition to the corrected dimensions of the primary collimator, the New Model includes approximate models for the lead shield and the mirror frame between the monitor chamber and the Y jaws. A detailed analysis of the phase space data shows the effects of these corrections on the beam characteristics. The individual contributions from the linac component to the photon and electron fluences are calculated. The main source of discrepancy between measurements and calculations based on the Old Model is the underestimated electron contamination. The photon and electron fluences at the isocenter are 5.3% and 36% larger in the New Model in comparison with the Old Model. The flattening filter and the lead shield (plus the mirror frame) contribute 48.7% and 13% of the total electron contamination at the isocenter, respectively. For both open and filtered (2 mm Pb) fields, the calculated (New Model) and measured dose distributions are within 1% for depths larger than 1 cm. To solve the residual problem of large differences at shallow depths (8% at 0.25 cm depth), the detailed geometry of an IC-10 ionization chamber was simulated and the dose in the air cavity was calculated for different positions on the central axis including at the surface, where half of the chamber is outside the phantom. The calculated and measured chamber responses are within 3% even at the zero depth.     

Monte Carlo simulations of the dosimetric impact of radiopaque fiducial markers for proton radiotherapy of the prostate

Many clinical studies have demonstrated that implanted radiopaque fiducial markers improve targeting accuracy in external-beam radiotherapy, but little is known about the dose perturbations these markers may cause in patients receiving proton radiotherapy. The objective of this study was to determine what types of implantable markers are visible in setup radiographs and, at the same time, perturb the therapeutic proton dose to the prostate by less than 10%. The radiographic visibility of the markers was assessed by visual inspection of lateral setup radiographs of a pelvic phantom using a kilovoltage x-ray imaging system. The fiducial-induced perturbations in the proton dose were estimated with Monte Carlo simulations. The influence of marker material, size, placement depth and orientation within the pelvis was examined. The radiographic tests confirmed that gold and stainless steel markers were clearly visible and that titanium markers were not. The Monte Carlo simulations revealed that titanium and stainless steel markers minimally perturbed the proton beam, but gold markers cast unacceptably large dose shadows. A 0.9 mm diameter, 3.1 mm long cylindrical stainless steel marker provides good radiographic visibility yet perturbs the proton dose distribution in the prostate by less than 8% when using a parallel opposed lateral beam arrangement.     

The value of radiochromic film dosimetry around air cavities: experimental results and Monte Carlo simulations

In this study we investigate radiochromic film dosimetry around air cavities with particular focus on the perturbation of the dose distribution by the film when the film is parallel to the beam axis. We considered a layered polystyrene phantom containing an air cavity as a model for the air-soft tissue geometry that may occur after surgical resection of a paranasal sinus tumour. A radiochromic film type MD-55 was positioned within the phantom so that it intersected the cavity. Two phantom set-ups were examined. In the first case, the air cavity is at the centre of the phantom, thus the film is lying along the central beam axis. In the second case, the cavity and film are located 2 cm offset from the phantom centre and the central beam axis. In order to examine the influence of the film on the dose distribution and to interpret the film-measured results, Monte Carlo simulations were performed. The film was modelled rigorously to incorporate the composition and structure of the film. Two field configurations, a 1 x 10 cm2 field and a 10 x 10 cm2 field, were examined. The dose behind the air cavity is reduced by 6 to 7% for both field configurations when a film that intersects the cavity contains the central beam axis. This is due to the attenuation exerted by the film when photons cross the cavity. Offsetting the beam to the cavity and the film by 2 cm removes the dose reduction behind the air cavity completely. Another result was that the rebuild-up behind the cavity for the 10 x 10 cm2 field, albeit less significant than for the 1 x 10 cm2 field, could only be measured by the film that was placed offset with respect to the central beam axis. Although radiochromic film is approximately soft-tissue equivalent and energy independent as compared to radiographic films, care should be taken in the case of inhomogeneous phantoms when the film intersects air cavities and contains the beam central axis. Errors in dose measurement can be expected distal to the air cavity due to attenuation in the film itself. This attenuation would not occur in the absence of the film. Both experiments and Monte Carlo computations support this conclusion.     

An evaluation of the AAPM-TG43 dosimetry protocol for I-125 brachytherapy seed

The EGSnrc Monte Carlo system has been used to calculate the dose distributions from 125I radioactive seeds (model 6711). The results showed that the agreement between EGSnrc and the American Association of Physicists in Medicine Task Group Report 43 (AAPM-TG43) dosimetry protocol is generally within +/-15% for radial distances less than 1.0 cm in both the transverse axis and longitudinal axis of the source. For radial distances between 1.0 and 2.5 cm the agreement between Monte Carlo simulations and the AAPM-TG43 dosimetry protocol is within +/-20%. In the longitudinal axis of the source the difference between Monte Carlo simulations and the AAPM-TG43 dosimetry is up to 40% for radial distances greater than 2.5 cm. The agreement between the EGSnrc/Monte Carlo simulation and the AAPM-TG43 dosimetry protocol improved significantly when recently published data of the anisotropic function were implemented (Weaver 1998 Med. Phys. 25 2271-8). The difference between Monte Carlo simulations and the AAPM-TG43 dosimetry protocol is not more than +/-10% in the transverse axis of the source up to a radius of 2.5 cm. The EGSnrc Monte Carlo simulation and the AAPM-TG43 with the Weaver anisotropic data were also used to investigate the differences in the dose distribution caused by small differences in the construction of individual seeds (Sloboda and Menon 2000 Med. Phys. 27 1789-99). The results show that a change in length of the silver rod containing the 125I radioactive material of 0.14 mm does not affect the dose distribution significantly in the transverse and longitudinal axes but a change of 0.13 mm in the thickness of the welded end of the encapsulation affected the dose significantly in the longitudinal axis of the source.     

Dose errors in the near field of an HDR brachytherapy stepping source

The dose rate at point P at 0.25 cm in water from the transverse bisector of a straight catheter with an active stepping source (Nucletron microSelectron HDR source) with a dwell length of 2 cm was calculated using Monte Carlo code MCNP 4.A. The source step sizes were 1 cm and 0.25 cm. The Monte Carlo (MC) results were used for comparison with the results calculated with the Nucletron brachytherapy planning system (BPS) formalism, first with BPS variants and then with its respective MC calculated radial dose function and anisotropy function. The dose differences at point P calculated using the BPS formalism and variants are +15.4% and +3.1% for the source step size of 1 cm and 0.25 cm respectively. This reduction in dose difference is caused by the increased importance of errors in the anisotropy function with the smaller step size, which counter the errors in the radial dose function. Using the MC calculated radial dose function and anisotropy function with the BPS formalism. 1% dose calculation accuracy can be achieved, even in the near field, with negligible extra demand on computation time.     

The use of radiographic film for linear accelerator stereotactic radiosurgical dosimetry.

The measurement of stereotactic radiosurgical dose distributions requires an integrating, high-resolution dosimeter capable of providing a spatial map of absorbed dose. Although radiographic film is an accessible dosimeter fulfilling these criteria, for larger radiotherapy photon fields the sensitivity of film emulsion exhibits significant dependencies on both depth in phantom and field size. We have examined the variation of film sensitivity over the ranges of depths and field sizes of interest in radiosurgery with a 6 MV photon beam. While for large (20cm x 20cm) fields the potential error in dose due to the variation of the film response with depth reaches 15%, the corresponding maximum error for a 2.5 cm diameter radiosurgical beam is 1.5%. This uncertainty was observed to be comparable in magnitude to that produced by variation in processing conditions (1.1%) and by varying the orientation of the film plane relative to the beam central axis (1.5%). The dependence of emulsion sensitivity on field size has been observed to be negligible for fields ranging in diameter from 1.0 cm to 4.0 cm. The source of the dependence of film sensitivity has been illustrated by using an EGS4 Monte Carlo simulation for large fields to illustrate significant increases in the photon spectrum below 400 keV with depth in phantom. In contrast, relative increase of this low-energy component is negligible for radiosurgical photon fields.     

Characterization of an extendable multi-leaf collimator for clinical electron beams

An extendable x-ray multi-leaf collimator (eMLC) is investigated for collimation of electron beams on a linear accelerator. The conventional method of collimation using an electron applicator is impractical for conformal, modulated and mixed beam therapy techniques. An eMLC would allow faster, more complex treatments with potential for reduction in dose to organs-at-risk and critical structures. The add-on eMLC was modelled using the EGSnrc Monte Carlo code and validated against dose measurements at 6-21 MeV with the eMLC mounted on a Siemens Oncor linear accelerator at 71.6 and 81.6 cm source-to-collimator distances. Measurements and simulations at 8.4-18.4 cm airgaps showed agreement of 2%/2 mm. The eMLC dose profiles and percentage depth dose curves were compared with standard electron applicator parameters. The primary differences were a wider penumbra and up to 4.2% reduction in the build-up dose at 0.5 cm depth, with dose normalized on the central axis. At 90 cm source-to-surface distance (SSD)--relevant to isocentric delivery--the applicator and eMLC penumbrae agreed to 0.3 cm. The eMLC leaves, which were 7 cm thick, contributed up to 6.3% scattered electron dose at the depth of maximum dose for a 10 × 10 cm2 field, with the thick leaves effectively eliminating bremsstrahlung leakage. A Monte Carlo calculated wedge shaped dose distribution generated with all six beam energies matched across the maximum available eMLC field width demonstrated a therapeutic (80% of maximum dose) depth range of 2.1-6.8 cm. Field matching was particularly challenging at lower beam energies (6-12 MeV) due to the wider penumbrae and angular distribution of electron scattering. An eMLC isocentric electron breast boost was planned and compared with the conventional applicator fixed SSD plan, showing similar target coverage and dose to critical structures. The mean dose to the target differed by less than 2%. The low bremsstrahlung dose from the 7 cm thick MLC leaves had the added advantage of reducing the mean dose to the whole heart. Isocentric delivery using an extendable eMLC means that treatment room re-entry and repositioning the patient for SSD set-up is unnecessary. Monte Carlo simulation can accurately calculate the fluence below the eMLC and subsequent patient dose distributions. The eMLC generates similar dose distributions to the standard electron applicator but provides a practical method for more complex electron beam delivery.     

Build-up and surface dose measurements on phantoms using micro-MOSFET in 6 and 10 MV x-ray beams and comparisons with Monte Carlo calculations

This work is intended to investigate the application and accuracy of micro-MOSFET for superficial dose measurement under clinically used MV x-ray beams. Dose response of micro-MOSFET in the build-up region and on surface under MV x-ray beams were measured and compared to Monte Carlo calculations. First, percentage-depth-doses were measured with micro-MOSFET under 6 and 10 MV beams of normal incidence onto a flat solid water phantom. Micro-MOSFET data were compared with the measurements from a parallel plate ionization chamber and Monte Carlo dose calculation in the build-up region. Then, percentage-depth-doses were measured for oblique beams at 0 degrees-80 degrees onto the flat solid water phantom with micro-MOSFET placed at depths of 2 cm, 1 cm, and 2 mm below the surface. Measurements were compared to Monte Carlo calculations under these settings. Finally, measurements were performed with micro-MOSFET embedded in the first 1 mm layer of bolus placed on a flat phantom and a curved phantom of semi-cylindrical shape. Results were compared to superficial dose calculated from Monte Carlo for a 2 mm thin layer that extends from the surface to a depth of 2 mm. Results were (1) Comparison of measurements with MC calculation in the build-up region showed that micro-MOSFET has a water-equivalence thickness (WET) of 0.87 mm for 6 MV beam and 0.99 mm for 10 MV beam from the flat side, and a WET of 0.72 mm for 6 MV beam and 0.76 mm for 10 MV beam from the epoxy side. (2) For normal beam incidences, percentage depth dose agree within 3%-5% among micro-MOSFET measurements, parallel-plate ionization chamber measurements, and MC calculations. (3) For oblique incidence on the flat phantom with micro-MOSFET placed at depths of 2 cm, 1 cm, and 2 mm, measurements were consistent with MC calculations within a typical uncertainty of 3%-5%. (4) For oblique incidence on the flat phantom and a curved-surface phantom, measurements with micro-MOSFET placed at 1.0 mm agrees with the MC calculation within 6%, including uncertainties of micro-MOSFET measurements of 2%-3% (1 standard deviation), MOSFET angular dependence of 3.0%-3.5%, and 1%-2% systematical error due to phantom setup geometry asymmetry. Micro-MOSFET can be used for skin dose measurements in 6 and 10 MV beams with an estimated accuracy of +/- 6%.     

Normalized glandular dose (DgN) coefficients for arbitrary X-ray spectra in mammography: computer-fit values of Monte Carlo derived data

Normalized glandular dose (DgN) values have been reported by several investigators for specific spectra, however for unconventional or unanticipated x-ray spectra considered for use in mammography, practical methods are not available for DgN computation. In this study, the previously validated SIERRA Monte Carlo code was used to compute the normalized glandular dose coefficients for monoenergetic energies from 8 keV to 50 keV. The overall mammography geometry used was a 65 cm source to image distance, a 1.2 cm air gap between the breast and the detector, and breast thicknesses ranging from 2 to 9 cm. A 4 mm layer of skin was also modeled, and semicircular breast radii of 8.5 cm and 10.0 cm were studied. Breast compositions of 0% glandular, 50% glandular, and 100% glandular were evaluated. The Monte Carlo derived DgN results demonstrated coefficients of variation less than 0.3%. The monoenergetic DgN values, DgN(E), were computer fit using commercial software and the best fit equations are reported. All fits resulted in r2 values of 0.9999 or better. The computer fit equations, along with easy to use spectral modeling routines, are available electronically on the web.     

Tissue inhomogeneity correction for brachytherapy sources in a heterogeneous phantom with cylindrical symmetry.

In brachytherapy it is customary to perform dose calculations for an implant assuming that the tumor and surrounding tissues constitute a uniform, homogeneous medium equivalent to water. In this work, the validity of the above assumption is studied quantitatively for points along the transverse axis of 103Pd, 125I, and 241Am brachytherapy sources, using measured and Monte Carlo calculated dose rates in homogeneous and heterogeneous media with cylindrical symmetry. The irradiation geometry chosen was a single source implanted in a Solid Water phantom which had a 1- or 2-cm-thick cylindrical Solid Water shell replaced by a polystyrene shell. The Monte Carlo simulations were performed using the integrated tiger series CYLTRAN Code. Experimental data were obtained for the same geometry to test the validity of the Monte Carlo calculations for a heterogeneous phantom. Measured dose rates just beyond a 2-cm-thick polystyrene heterogeneity were observed to be greater than those in a homogeneous Solid Water phantom by about 130%, 55%, and 10% for 103Pd, 125I, and 241Am, respectively. Thus the effect of a relatively small polystyrene heterogeneity in Solid Water can be substantial for lower energy photons. This perturbation of dose was found to increase steeply with decreasing energy and increasing size (thickness) of inhomogeneity. A simple dose calculation formalism has been developed to predict dose rate in a heterogeneous phantom with cylindrical symmetry, which uses as input the radial dose functions of the uniform media comprising the heterogeneous phantom. Dose rate predictions using this formalism are in reasonable agreement with the experimental data and the Monte Carlo calculated values.(ABSTRACT TRUNCATED AT 250 WORDS)     

Influence of phantom material and phantom size on radiographic film response in therapy photon beams

The energy dependence of radiographic film can introduce dosimetric errors when evaluating photon beams. The variation of the film response, which is attributed to the changing photon spectrum with depth and field size, has been the subject of numerous publications in recent years. However, these data show large unexplained differences in the magnitude of this variation among independent studies. To try to resolve this inconsistency, this study assesses the dependence of radiographic film response on phantom material and phantom size using film measurements and Monte Carlo calculations. The relative dose measured with film exposed to 6 MV x rays in various phantoms (polystyrene, acrylic, Solid Water, and water; the lateral phantom dimensions vary from 25 to 50 cm square; backscatter thickness varies from 10 to 30 cm) is compared with ion chamber measurements in water. Ranges of field size (5 x 5 to 40 x 40 cm2) and depth (dmax to 20 cm) are studied. For similar phantom and beam configurations, Monte Carlo techniques generate photon fluence spectra from which the relative film response is known from an earlier study. Results from film response measurements agree with those derived from Monte Carlo calculations within 3%. For small fields (< or = 10 x 10 cm2) and shallow depths (< or = 10 cm) the film response variation is small, less than 4%, for all phantoms. However, for larger field sizes and depths, the phantom material and phantom size have a greater influence on the magnitude of the film response. The variation of film response, over the ranges of field sizes and depths studied, is 50% in polystyrene compared with 30% in water. Film responses in Solid Water and water phantoms are similar; acrylic is between water and polystyrene. In polystyrene the variation of film response for a 50 cm square phantom is nearly twice that observed in a 25 cm square phantom. This study shows that differences in the configuration of the phantoms used for film dosimetry can explain much of the inconsistency for film response reported in the literature.     

An empirical method for the determination of wall perturbation factors for parallel-plate chambers in high-energy electron beams

The calibration of ion chambers in high-energy electron beams in terms of absorbed dose to water at the National Physical Laboratory requires knowledge of the ratio of perturbation factors in graphite and water phantoms. During a review of data required for the NPL calibration procedure an empirical model was developed to calculate the perturbation due to the rear wall, pwall, of a well-guarded ion chamber in a high-energy electron beam. The overall uncertainty in this method is estimated to be 0.4%, which is the lowest value reported to date. The model reproduces measured data at the 0.1% level or better and indicates that the NACP ion chamber has a nonzero perturbation factor in electron beams due to backscatter from the rear wall. The effect is small (<0.5%) at high energies (R50>4 cm, E0>10 MeV) but becomes large at low energies-up to 1.4% at E0=4 MeV (R50=1.2 cm). The model indicates that there is a nonzero correction for the NACP chamber in both a graphite and water phantom and that material adjacent to the air cavity has a significant effect on the measured ionization. These values are consistent with previous measurements and recent Monte Carlo calculations. The model could be used in the design of ion chambers and in the estimation of corrections for non-homogeneous systems, especially in the absence of accurate Monte Carlo simulations.