# An Empirical Model for Describing the Small Field Penumbra in Radiation Therapy

Authors: Shi-Qiang Tang; Yee-Min Jen; Jia-Ming Wu
Journal: BioMed Research International
 (2019)

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## Abstract

Purpose. We developed a mathematic empirical model for describing the small field penumbra in order to analyze the potential dose perturbation caused by overlapping field to avoid the dose calculation errors in linear accelerator-based radiosurgery. Materials and methods. A ball phantom was fabricated for measuring penumbra at 4 different gantry angles in the coplanar plane. A least square root estimation (LSRE) Model was created to fit the measured penumbra dose profile and to predict the penumbra dose profile at any gantry angles. The Sum of Squared Errors (SSE) was used for finding the parameters n and t for the best fitting of the LSRE model. Geometric and mathematical methods were used to derive the model parameters. Results. The results showed that the larger the gantry angle of the field, the more the expansion of the penumbra dose profile. The least square root estimation model for describing small field penumbra is as follows: PenumbraDš=T⋅1/2⋅1−š/n+š2+t where PenumbraDš denotes the dose profile D(š) at the penumbra region, T is the penumbra height (usually in scalar 100), n is the parameter for curvature, š = x − Wd/2 (x and š are the values in cm on x-axis), and t is the radiation transmission of the collimator. Geometric analysis establishes the correlation between the penetration depth of the exposure and its effect on the penumbra region in ball phantom. The penumbra caused by an exposure at any arbitrary angles can be geometrically derived by using a one-variable quadratic equation. Conclusion. The dose distribution in penumbra region of small field can be created by the LSRE model and the potential overdosage or underdosage owing to overlapping field perturbation can be estimated.


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## Body

## 1. Introduction

With the adoption of advanced technologies in modern radiotherapy such as stereotactic radiosurgery, stereotactic body radiation therapy, and intensity-modulated radiation therapy, there is an increased interest in the small-field dosimetry of photon beams. A beamlet used in linear accelerator-based radiosurgery is the smallest field formed by one single leaf of the MLC facing the opposite one and is only a portion of the target of dose delivery [1]. All these small fields are delivered and superimposed at different gantry angles during linear accelerator-based radiosurgery.The analysis of small field penumbra is important in linear accelerator-based radiosurgery. It includes geometric, transmission, and photon scattered components. Geometric penumbra originates from the radiation source when it is not a single point. Transmission penumbra occurs when the beam passes through the edge of the jaw or MLC before it reaches the full attenuation point of the jaw and the MLC. Scattered radiation from the former two components is added to form the total penumbra, namely, the physical penumbra. The physical penumbra width is defined as the lateral distance from the central axis between 20% and 80% of the central axis dose at a reference depth. The height of a penumbra is defined as the intersection point of the central axis with the dose profile and is usually normalized to be 100%.We design a mathematical model to fit the penumbra in small field used in linear accelerator-based radiosurgery. The penumbra perturbation was also investigated by our model when one field is overlapped by another segment created by MLC. The dose perturbation at the field edges could lead to a potential monitor unit calculation error; therefore, this penumbra perturbation has to be accounted for in order to obtain the correct dose modification especially in the situation of split field [2].


## 2. Materials and Methods

### 2.1. Experiment Design and Steps

The experiment was conducted in the following steps:(1)
Establishment of a calibration curve for future dose quantification of irradiated films(2)
Single exposures of the standard reference field and with double exposure at four different gantry angles(3)
Establishment of a mathematical model, the least square root estimation (LSRE) model, and using the LSRE model to fit the penumbra dose profiles obtained in the previous step(4)
Using the Sum of Square Errors (SSE) to find the values of the LSRE model parameters(5)
Geometric derivation of the LSRE penumbra parameters at any gantry angle by using a transformation equation(6)
Estimating the dose curve expansion owing to overlapping field perturbation in split field by using LSREThe details of each step are described in the following sections.


### 2.2. Establishment of a Calibration Curve

We used GAF Chromic EBT2 films (ISP Technology Inc., Wayne, NJ, USA; Lot #F05090901) for the experiment. The film has aZeff tissue equivalence of 6.98 with a fast polymerization in the image-forming process. Its absorption peaks are independent of radiation energy. Its higher sensitivity in the dose range of 2–800 cGy and an inhomogeneity smaller than 2% after radiation exposure make it suitable for dosimetry analysis. The film processing and dose profile measurements followed the international protocols [3]. A preexposure technique was used for the calibration curve derivation [4]. This was performed by giving each film a priming dose of 2 Gy to homogenize the film density using an Elekta 6 MV linear accelerator (Stockholm, Sweden) with a dose rate of 300 MU/min. We measured the dose homogeneity using a densitometer. The previously exposed films were embedded into a solid water phantom at a depth of 5 cm. Radiation was delivered with the same 6 MV linear accelerator with a 10 × 10 cm2 field size at an SAD of 100 cm. Graded doses of 5, 10, 15, 40, 60, 80, 100, 150, and 200 cGy were given to obtain the Hurter-Driffield calibration curve (H-D curve).All exposed films were then scanned with an Epson Expression 10000XL scanner in the 48-bit RGB mode (16 bits per color), and the data were saved as tagged image file format (TIFF) and analyzed by the VariSoft imaging procession software. The films were scanned in the landscape orientation to reduce the optical density (OD) variation to ≤2% [5]. The experiments and film analysis were performed at room temperature to reduce the temperature-dependent evolution of the films [6]. All OD measurements were performed 24 h after irradiation to avoid time effect on the films. GAF chromic film was handled with forceps to avoid scratching and fingerprints.The OD of the film was derived by the following equation:(1)OD=log10B0B,where B0 is the background density, namely, the scanner signal of the unexposed film, and B is the scanner signal of the exposed film.A red filter was placed on top of the GAF films before scanning to increase the slope of the H-D curve, thereby raising the resolution of the dose-OD curves [7].


### 2.3. Single Exposures of the Standard Reference Field and with Double Exposure at Four Different Gantry Angles

The single exposure of a radiation field 4 × 2 cm2 on dimension 10 × 10 cm2 GAF film was conducted in a regular 30 × 30 cm2 polystyrene solid phantom at the depth of 5 cm with 100 cGy as the standard reference of penumbra profile. The other four films with dimension 10 × 10 cm2 were exposed each with a radiation field 4 × 2 cm2 on GAF film in a regular 30 × 30 cm2 polystyrene solid phantom at the depth of 5 cm with 100 cGy as the first exposure and then follows the 6 × 4 cm2 field at four different gantry angles in ball phantom separately. We chose a 4 × 2 cm2 field for two reasons: firstly, to simulate the commonly used calculation grid size in the treatment planning systems; secondly, to mark dose profile measurement direction for a smaller filed size used in linear accelerator-based radiosurgery. The penumbra dose profile produced by this single exposure was used as the original standard reference data.An acrylic ball phantom 22 cm in diameter was made for the second exposure. A wall laser was aimed at the ball equator plane to set the SAD at 100 cm isocentrically for the second exposure. The ball could be separated from its equator into two hemispheres. One of the two hemispheres has a 20 × 20 × 0.1 cm3 dent on the equator plane to hold the film (Figure 1). The four previously exposed 4 × 2 cm2 films were put separately into the film holder on the equator plane for the second exposure with a 6 × 4 cm2 field at 30°, 45°, 60°, and 90° gantry angles, respectively (Figure 1).Figure 1
The custom-made ball phantom for the second film exposure. Thex, y, and z axes of the film are defined on this figure. The second exposure with a 6 × 4 cm2 field was given to the film that was previously irradiated with a 4 × 2 cm2 field. The film was embedded in the equator plane with the longitudinal direction parallel to the y axis at different gantry angles for in-field penumbra perturbation study. Coplanar irradiation denotes that all the irradiations were delivered along the xz axis with the same isocenter.The penetration depth remains the same at the 2 cm side on the −y and +y axis of the 4 × 2 cm2 field at different gantry angles (Figure 1). On the other hand, the radiation penetration depth changes at the 4 cm side of the 4 × 2 cm2 field on the axis of −x and +x from the larger second field. Because the gantry rotated around the y axis on the xz plane, the penetration depth from the larger second field changes only at the 4 cm side of the 4 × 2 cm2 field on the axis of −x and +x (Figure 1). The penetration depth at the 2 cm side remains the same. This phenomenon led to different dose depositions on the penumbra region along the 4 cm side on the −x and +x axis compared to the 2 cm side on the −y and +y axis. We therefore analyzed only the 4 cm side.


### 2.4. Establishment of a Mathematical Model to Fit the Dose Profile of the 4 × 2 cm2 Field after Superimposition by a 6 × 4 cm2 Field at Different Gantry Angles in Coplanar Plane

A mathematical model, named the least square root estimation model (LSRE model), was developed for modeling the penumbra dose profile perturbation. The goal was to build a mathematical model which could predict the penumbra profiles at any gantry angles in coplanar irradiation. The LSRE model originated from the proportion functionyx=1/x. For details, please refer to Appendix A.The least square root estimation model is expressed as follows:(2)PenumbraDš=T⋅12⋅1−šn+š2+t,where PenumbraDš denotes the dose profile Dš at the penumbra region; T is the penumbra height representing the intersection of the beam central axis with the penumbra curve and is between 100 and 99.8%; n is the parameter which gives the curve the curvature, and the greater the n is, the flatter the penumbra would be; š = x − Wd/2, where š and x are numbers on the x axis in the unit of cm, Wd/2 is the half-field width or the width from the central axis to the 50% dose point of the penumbra region; and t is the radiation transmission of the collimator.The LSRE model was used to fit the penumbra of the single exposure standard reference curve and the 4 × 2 cm2 dose profile on the 4 cm side after 6 × 4 cm2 field double exposures at 4 different gantry angles in a coplanar irradiation. It fits the penumbra PenumbraDš dose profile from the central axis to the end of the curve.


### 2.5. Using the Sum of Square Errors (SSE) to Find the Optimaln and t of the LSRE Model

We need a tool to determine whether a best fit to the penumbra profile had been achieved or not by the LSRE. The Sum of Square Errors (SSE) provides an ideal measurement of the differences between the measured and calculated data by the LSRE model [8]. The lowest SSE indicates a minimum difference between the LSRE model prediction and the measured data.The SSE is defined as follows:(3)SSE=∑i=1nxi−xl¯2,where xi is the observation and xl¯ is the means of predicted values.The total SSE is(4)SSEtotal=SSE1+SSE2+SSE3+SSE4+⋯+SSEn.Then that leads to the lowest SSE would result in the best fit of the penumbra curvature; the t with the lowest SSE is derived separately and it affects the height of the penumbra tail.


### 2.6. Geometric Derivation of the Perturbation Effect on the Penumbra by a 6 × 4 cm2 Second Field at Arbitrary Angles in Coplanar Irradiation

So far, we have the best fits of parametersn and t only for the dose profiles at gantry angles of 30°, 45°, 60°, and 90°. Eventually, we aim to obtain ns and ts at any gantry angles. For this, we need to find the lengths of ed¯ at a specific gantry angle because the parameters n and t were dependent only on the penetration depth ed¯. ed¯ can be derived by using geometry (Figure 2(a)). ed¯ is the penetration length on the S′c¯ of the larger field at a gantry angle of θ. It is the distance where the radiation traverses inside the phantom before intersecting with the penumbra edge of the previous field. It is the only factor controlling the dose contribution from the larger superimposing field onto the smaller field along od¯. The larger ed¯ is, the more the attenuation of the passing radiation will be with less contribution to the penumbra. ed¯ is also the penetration depth on the S′c¯ of the 6 × 4 cm2 field. S′c¯ is the radiation line within the penumbra region from the source S′ in the larger field along the +x and −x at an arbitrary gantry angle of θ. For the detailed geometric derivation of ed¯, please refer to Figure 2(b) and Appendix B.Figure 2
(a) The calculation of the lengthed¯ at a gantry angle of θ on s′c¯ by geometric method after double exposures with a larger and a smaller field. The larger field was 6 × 4 cm2 and the smaller field was 4 × 2 cm2 field, respectively. S represents the linear accelerator radiation source position with a gantry angle of 0°, and S′ denotes the source position after a gantry rotation of θ degrees. o is the rotational isocenter; d is the field edge of the small field delivered at 0° at the 4 cm side of the 4 × 2 cm2 field; a indicates the field edge of the larger field at the 6 cm side of the 6 × 4 cm2 field when the gantry rotates to the angle of θ. od¯ is the half-field size of the smaller field on the 4 cm side of the 4 × 2 cm2 field while oa¯ is the half-field size of the larger field on the 6 cm side of the 6 × 4 cm2 field. fs/2 denotes the half-field size on the 4 cm side of the 4 × 2 cm2 field (od¯) and FS/2 denotes the half-field size on the 6 cm side of the 6 × 4 cm2 field (FS/2). (b) This figure shows the geometric derivation of ed¯. ed¯ is the key factor controlling the dose distribution from the larger field onto the penumbra region of the smaller field along od.¯ ed¯ = S′c¯ − cd¯ − S′e¯, where S′c¯=100/cosβ , and cd¯ = bd¯/cosβ. For details, please refer to Appendix B.
(a)
(b)


## 2.1. Experiment Design and Steps

The experiment was conducted in the following steps:(1)
Establishment of a calibration curve for future dose quantification of irradiated films(2)
Single exposures of the standard reference field and with double exposure at four different gantry angles(3)
Establishment of a mathematical model, the least square root estimation (LSRE) model, and using the LSRE model to fit the penumbra dose profiles obtained in the previous step(4)
Using the Sum of Square Errors (SSE) to find the values of the LSRE model parameters(5)
Geometric derivation of the LSRE penumbra parameters at any gantry angle by using a transformation equation(6)
Estimating the dose curve expansion owing to overlapping field perturbation in split field by using LSREThe details of each step are described in the following sections.


## 2.2. Establishment of a Calibration Curve

We used GAF Chromic EBT2 films (ISP Technology Inc., Wayne, NJ, USA; Lot #F05090901) for the experiment. The film has aZeff tissue equivalence of 6.98 with a fast polymerization in the image-forming process. Its absorption peaks are independent of radiation energy. Its higher sensitivity in the dose range of 2–800 cGy and an inhomogeneity smaller than 2% after radiation exposure make it suitable for dosimetry analysis. The film processing and dose profile measurements followed the international protocols [3]. A preexposure technique was used for the calibration curve derivation [4]. This was performed by giving each film a priming dose of 2 Gy to homogenize the film density using an Elekta 6 MV linear accelerator (Stockholm, Sweden) with a dose rate of 300 MU/min. We measured the dose homogeneity using a densitometer. The previously exposed films were embedded into a solid water phantom at a depth of 5 cm. Radiation was delivered with the same 6 MV linear accelerator with a 10 × 10 cm2 field size at an SAD of 100 cm. Graded doses of 5, 10, 15, 40, 60, 80, 100, 150, and 200 cGy were given to obtain the Hurter-Driffield calibration curve (H-D curve).All exposed films were then scanned with an Epson Expression 10000XL scanner in the 48-bit RGB mode (16 bits per color), and the data were saved as tagged image file format (TIFF) and analyzed by the VariSoft imaging procession software. The films were scanned in the landscape orientation to reduce the optical density (OD) variation to ≤2% [5]. The experiments and film analysis were performed at room temperature to reduce the temperature-dependent evolution of the films [6]. All OD measurements were performed 24 h after irradiation to avoid time effect on the films. GAF chromic film was handled with forceps to avoid scratching and fingerprints.The OD of the film was derived by the following equation:(1)OD=log10B0B,where B0 is the background density, namely, the scanner signal of the unexposed film, and B is the scanner signal of the exposed film.A red filter was placed on top of the GAF films before scanning to increase the slope of the H-D curve, thereby raising the resolution of the dose-OD curves [7].


## 2.3. Single Exposures of the Standard Reference Field and with Double Exposure at Four Different Gantry Angles

The single exposure of a radiation field 4 × 2 cm2 on dimension 10 × 10 cm2 GAF film was conducted in a regular 30 × 30 cm2 polystyrene solid phantom at the depth of 5 cm with 100 cGy as the standard reference of penumbra profile. The other four films with dimension 10 × 10 cm2 were exposed each with a radiation field 4 × 2 cm2 on GAF film in a regular 30 × 30 cm2 polystyrene solid phantom at the depth of 5 cm with 100 cGy as the first exposure and then follows the 6 × 4 cm2 field at four different gantry angles in ball phantom separately. We chose a 4 × 2 cm2 field for two reasons: firstly, to simulate the commonly used calculation grid size in the treatment planning systems; secondly, to mark dose profile measurement direction for a smaller filed size used in linear accelerator-based radiosurgery. The penumbra dose profile produced by this single exposure was used as the original standard reference data.An acrylic ball phantom 22 cm in diameter was made for the second exposure. A wall laser was aimed at the ball equator plane to set the SAD at 100 cm isocentrically for the second exposure. The ball could be separated from its equator into two hemispheres. One of the two hemispheres has a 20 × 20 × 0.1 cm3 dent on the equator plane to hold the film (Figure 1). The four previously exposed 4 × 2 cm2 films were put separately into the film holder on the equator plane for the second exposure with a 6 × 4 cm2 field at 30°, 45°, 60°, and 90° gantry angles, respectively (Figure 1).Figure 1
The custom-made ball phantom for the second film exposure. Thex, y, and z axes of the film are defined on this figure. The second exposure with a 6 × 4 cm2 field was given to the film that was previously irradiated with a 4 × 2 cm2 field. The film was embedded in the equator plane with the longitudinal direction parallel to the y axis at different gantry angles for in-field penumbra perturbation study. Coplanar irradiation denotes that all the irradiations were delivered along the xz axis with the same isocenter.The penetration depth remains the same at the 2 cm side on the −y and +y axis of the 4 × 2 cm2 field at different gantry angles (Figure 1). On the other hand, the radiation penetration depth changes at the 4 cm side of the 4 × 2 cm2 field on the axis of −x and +x from the larger second field. Because the gantry rotated around the y axis on the xz plane, the penetration depth from the larger second field changes only at the 4 cm side of the 4 × 2 cm2 field on the axis of −x and +x (Figure 1). The penetration depth at the 2 cm side remains the same. This phenomenon led to different dose depositions on the penumbra region along the 4 cm side on the −x and +x axis compared to the 2 cm side on the −y and +y axis. We therefore analyzed only the 4 cm side.


## 2.4. Establishment of a Mathematical Model to Fit the Dose Profile of the 4 × 2 cm2 Field after Superimposition by a 6 × 4 cm2 Field at Different Gantry Angles in Coplanar Plane

A mathematical model, named the least square root estimation model (LSRE model), was developed for modeling the penumbra dose profile perturbation. The goal was to build a mathematical model which could predict the penumbra profiles at any gantry angles in coplanar irradiation. The LSRE model originated from the proportion functionyx=1/x. For details, please refer to Appendix A.The least square root estimation model is expressed as follows:(2)PenumbraDš=T⋅12⋅1−šn+š2+t,where PenumbraDš denotes the dose profile Dš at the penumbra region; T is the penumbra height representing the intersection of the beam central axis with the penumbra curve and is between 100 and 99.8%; n is the parameter which gives the curve the curvature, and the greater the n is, the flatter the penumbra would be; š = x − Wd/2, where š and x are numbers on the x axis in the unit of cm, Wd/2 is the half-field width or the width from the central axis to the 50% dose point of the penumbra region; and t is the radiation transmission of the collimator.The LSRE model was used to fit the penumbra of the single exposure standard reference curve and the 4 × 2 cm2 dose profile on the 4 cm side after 6 × 4 cm2 field double exposures at 4 different gantry angles in a coplanar irradiation. It fits the penumbra PenumbraDš dose profile from the central axis to the end of the curve.


## 2.5. Using the Sum of Square Errors (SSE) to Find the Optimaln and t of the LSRE Model

We need a tool to determine whether a best fit to the penumbra profile had been achieved or not by the LSRE. The Sum of Square Errors (SSE) provides an ideal measurement of the differences between the measured and calculated data by the LSRE model [8]. The lowest SSE indicates a minimum difference between the LSRE model prediction and the measured data.The SSE is defined as follows:(3)SSE=∑i=1nxi−xl¯2,where xi is the observation and xl¯ is the means of predicted values.The total SSE is(4)SSEtotal=SSE1+SSE2+SSE3+SSE4+⋯+SSEn.Then that leads to the lowest SSE would result in the best fit of the penumbra curvature; the t with the lowest SSE is derived separately and it affects the height of the penumbra tail.


## 2.6. Geometric Derivation of the Perturbation Effect on the Penumbra by a 6 × 4 cm2 Second Field at Arbitrary Angles in Coplanar Irradiation

So far, we have the best fits of parametersn and t only for the dose profiles at gantry angles of 30°, 45°, 60°, and 90°. Eventually, we aim to obtain ns and ts at any gantry angles. For this, we need to find the lengths of ed¯ at a specific gantry angle because the parameters n and t were dependent only on the penetration depth ed¯. ed¯ can be derived by using geometry (Figure 2(a)). ed¯ is the penetration length on the S′c¯ of the larger field at a gantry angle of θ. It is the distance where the radiation traverses inside the phantom before intersecting with the penumbra edge of the previous field. It is the only factor controlling the dose contribution from the larger superimposing field onto the smaller field along od¯. The larger ed¯ is, the more the attenuation of the passing radiation will be with less contribution to the penumbra. ed¯ is also the penetration depth on the S′c¯ of the 6 × 4 cm2 field. S′c¯ is the radiation line within the penumbra region from the source S′ in the larger field along the +x and −x at an arbitrary gantry angle of θ. For the detailed geometric derivation of ed¯, please refer to Figure 2(b) and Appendix B.Figure 2
(a) The calculation of the lengthed¯ at a gantry angle of θ on s′c¯ by geometric method after double exposures with a larger and a smaller field. The larger field was 6 × 4 cm2 and the smaller field was 4 × 2 cm2 field, respectively. S represents the linear accelerator radiation source position with a gantry angle of 0°, and S′ denotes the source position after a gantry rotation of θ degrees. o is the rotational isocenter; d is the field edge of the small field delivered at 0° at the 4 cm side of the 4 × 2 cm2 field; a indicates the field edge of the larger field at the 6 cm side of the 6 × 4 cm2 field when the gantry rotates to the angle of θ. od¯ is the half-field size of the smaller field on the 4 cm side of the 4 × 2 cm2 field while oa¯ is the half-field size of the larger field on the 6 cm side of the 6 × 4 cm2 field. fs/2 denotes the half-field size on the 4 cm side of the 4 × 2 cm2 field (od¯) and FS/2 denotes the half-field size on the 6 cm side of the 6 × 4 cm2 field (FS/2). (b) This figure shows the geometric derivation of ed¯. ed¯ is the key factor controlling the dose distribution from the larger field onto the penumbra region of the smaller field along od.¯ ed¯ = S′c¯ − cd¯ − S′e¯, where S′c¯=100/cosβ , and cd¯ = bd¯/cosβ. For details, please refer to Appendix B.
(a)
(b)


## 3. Results

### 3.1. Calibration Curve

When the film was scanned with a red color filter, the calibration curve was steeper and thus with a higher OD-dose resolution. Radiation dose was then derived by the OD method (please refer to Figure3).Figure 3
The calibration curve was steeper and thus more sensitive when the film was scanned with a red color filter.


### 3.2. The Dose Profile of the 4 × 2 cm2 Field with a Second Exposure by a 6 × 4 cm2 Field at 4 Different Gantry Angles in Coplanar Irradiation

Figure4 demonstrates the single exposure as the standard reference penumbra profile and the cumulative dose profiles after double exposures on 4 × 2 cm2 with a 6 × 4 cm2 field at 4 different gantry angles. The dose profiles on the + axis are slightly higher than the dose profile on the −x side. This is caused by smaller penetration depths ed¯s on the +x axis than e′d′¯s at the −x side in the ball phantom (Figure 2(a)). The Full-Width Half Maxima (FWHM, the width between 50% dose of the −x axis and the +x axis) was 19.8 mm after single irradiation at a gantry angle of 0 in the flat polystyrene solid phantom. The FWHM became 20.3 mm, 21.1 mm, 21.3 mm, and 21.6 mm after second exposures at gantry angles of 30°, 45°, 60°, and 90° in the ball phantom, respectively (Figure 4). The penumbra width between 80% to 20% of the central axis was 2.8 mm after a single exposure at 0° with a 4 × 2 cm2 field (Figure 4). After the second exposure, they were 4.7 mm, 6.2 mm, 6.7 mm, and 8.1 mm at gantry angles of 30°, 45°, 60°, and 90°, respectively (Figure 4). The renormalized dose profile shows, as expected, no change of the penumbra width after double exposures of any doses with a 6 × 4 cm2 field as long as the exposures were delivered at the same gantry angle as the first. The dose profile changes only after double exposures delivered at different gantry angles.Figure 4
The dose profile of the smaller 4 × 2 cm2 field irradiated at a gantry angle of 0° at the flat polystyrene solid phantom, followed by a second exposure in the ball phantom with a larger field of 6 × 4 cm2 at different gantry angles. The dose profile changes after second exposures at a gantry angle different from the first exposure. It also shows the full-width half maximum (the width at 50% dose of the −x axis and the +x axis) and the penumbra width from 80% to 20% of the central axis.


### 3.3. Dose Profile Fitting Using the LSRE Model

ParametersT, n, and t in the LSRE model determine the dose profile shape (equation (2)). The right-side dose profile after the first exposure at a gantry angle of 0° can be fitted accurately with T = 99.8, n = 90, and t = 0.2 (Figure 5(a)). n and t that result in the best fit, denoted as (n, t), at gantry angles of 30°, 45°, 60°, and 90°, are (310, 2), (410, 3), (490, 4), and (650, 5), respectively.Figure 5
(a) The right-side penumbra dose profiles vary with the gantry angles of the second exposure. The curvature of the right-side penumbra region can be fitted using the LSRE model by adjusting the (n) T and (t). (b) The left-side penumbra dose profiles vary with the gantry angles of the second exposure. The curvature of the left-side penumbra region can be fitted using the LSRE model by adjusting the (n) T and (t). The “90° measured curve” and “90° model estimated” curve denote the dose profile measured at the gantry angle of 90° in the ball phantom and the predicted curve at the same. They overlap each other completely.
(a)
(b)The measured curves of the left-side dose region can also be estimated by adjusting then, T, and t (Figure 5(b)). Parameters n and t of the left-side curve at 0 gantry angle after single exposure were (90, 0.2). After the second exposures, the parameters (n, t) were (320, 1.8), (430, 2.8), (510, 3.6), and (700, 4.8) at gantry angles of 30°, 45°, 60°, and 90°, respectively.


### 3.4. Sum of Square Errors (SSE): Finding the Bestn and t

Figures6(a) and 6(b) show n and t with the smallest SSE which gives the best LSRE model fitting of the measured data at the gantry angle of 45°. The best fit at other gantry angles can be done in the same manner.Figure 6
(a, b) These figures demonstrate the use of the SSE to find the bestn and t of the LSRE model at the gantry angle of 45°.
(a)
(b)


### 3.5. Geometric Derivation of the Effects on the Penumbra by Second Exposures at Arbitrary Gantry Angles

We found that the relationship ofe−μϖ with the x is a straight line (Figures 7(a) and 7(b) and Table 1), where ϖ denotes the penetration depth ed¯,μ is the attenuation coefficient, and μ = 0.00494 cm−1 of 6 MV in water. The linear regression of e−μϖ versus x at different gantry angles can be described in the form of y=ux+v, where u is the slope of the regression line shown in Figures 7(a) and 7(b). Figure 7(a) demonstrates the 4 lines with their slopes at the gantry angles of 30°, 45°, 60°, and 90°. The slopes at any other gantry angles can be derived by interpolation from these 4 know angles.Figure 7
(a, b) The regression lines of e−μϖ, where ϖ = ed¯, versus the distance on the +x axis in (a) and the −x axis in (b), at different gantry angles. The slopes are 0.0021, 0.0019, 0.0015, and 0.0011 at gantry angles of 90°, 60°, 45°, and 30°, respectively.
(a)
(b)Table 1
The coplanar penetration depthed¯(ϖ) and e−μϖ on the right-side penumbra after double exposures.
Distance from central axis of pointd on the x axis (cm)
Gantry angles
0°
30°
45°
60°
90°
ed
¯
ϖ
e
−μϖ
ed
¯
ϖ
e
−μϖ
ed
¯
ϖ
e
−μϖ
ed
¯
ϖ
e
−μϖ
ed
¯
ϖ
e
−μϖ
0
11.00
0.95223#
11.00
0.95223
11.00
0.95223
11.00
0.95223
11.00
0.95223
0.1
11.00
0.95223
10.95
0.95233
10.93
0.95238
10.91
0.95241
10.90
0.95244
0.2
11.00
0.95223
10.90
0.95244
10.86
0.95253
10.83
0.95260
10.80
0.95265
0.3
11.00
0.95223
10.85
0.95255
10.79
0.95268
10.74
0.95278
10.70
0.95286
0.4
10.99
0.95223
10.80
0.95266
10.71
0.95283
10.65
0.95296
10.60
0.95308
0.5
10.99
0.95223
10.74
0.95277
10.64
0.95298
10.56
0.95315
10.50
0.95329
0.6
10.99
0.95223
10.69
0.95287
10.57
0.95313
10.48
0.95333
10.40
0.95350
0.7
10.98
0.95223
10.64
0.95298
10.50
0.95329
10.39
0.95352
10.30
0.95371
0.8
10.98
0.95223
10.58
0.95309
10.42
0.95344
10.30
0.95370
10.20
0.95393
0.9
10.97
0.95223
10.53
0.95321
10.35
0.95359
10.21
0.95389
10.10
0.95414
1
10.96
0.95223
10.47
0.95332
10.27
0.95375
10.12
0.95407
10.00
0.95435
1.1
10.96
0.95223
10.42
0.95343
10.20
0.95390
10.04
0.95426
9.90
0.95456
1.2
10.95
0.95223
10.36
0.95354
10.13
0.95406
9.95
0.95445
9.80
0.95477
1.3
10.94
0.95223
10.30
0.95365
10.05
0.95421
9.86
0.95463
9.70
0.95499
1.4
10.93
0.95223
10.25
0.95377
9.97
0.95437
9.77
0.95482
9.60
0.95520
1.5
10.92
0.95223
10.19
0.95388
9.90
0.95452
9.68
0.95501
9.50
0.95541
1.6
10.91
0.95223
10.13
0.95400
9.82
0.95468
9.59
0.95519
9.40
0.95562
1.7
10.90
0.95223
10.07
0.95411
9.75
0.95483
9.50
0.95538
9.30
0.95584
#Data are calculated from ed¯ (ed¯=ϖ) by using e−μϖ, μ = 0.00494 cm−1 at 6 MV in water.


### 3.6. Derivation of the Parametersn and t of the Penumbra Dose Profiles at Arbitrary Gantry Angles Using Mathematical Methods

ed
¯ = S′c¯ − cd¯ − S′e¯, where S′c¯=100/cosβ, cd¯ = bd¯/cosβ, anded
¯ is the penetration depth of the S′c¯ from the larger field in the ball phantom.S′e¯ can be derived from equations (5)–(7) (for details, please refer to Appendix B).(5)p2−2sp+s2+q2=R2,(6)p=2s±4s2−41+ξ2s2−R221+ξ2,(7)S′e¯=p2+q2=1+ξ2p2=s−s2−1+ξ2s2−R21+ξ2.Table1 shows the calculated ed¯s in Figure 2(b) in the ball phantom at different gantry angles in coplanar irradiation.The regression curves for slopes versusn and t were shown in Figures 8(a) and 8(b), respectively. Therefore, once we obtain the slope of a regression curve from Figures 7(a) and 7(b), n and t can be found in Figures 8(a) and 8(b). This enables us to derive the estimated curve with precision and thus the accurate effects on the penumbra of a second irradiation at any gantry angles without the labor conducting the actual experiment.Figure 8
(a-b) The curves of slope versusn and t were shown in (a, b), respectively. Once the relationships of gantry angle versus slope and slope versus n and t are known, the parameters n and t in the LSRE model can be found easily to fit the measured dose profile or to construct the estimated profile for any gantry angle at coplanar irradiation with good agreement.
(a)
(b)


## 3.1. Calibration Curve

When the film was scanned with a red color filter, the calibration curve was steeper and thus with a higher OD-dose resolution. Radiation dose was then derived by the OD method (please refer to Figure3).Figure 3
The calibration curve was steeper and thus more sensitive when the film was scanned with a red color filter.


## 3.2. The Dose Profile of the 4 × 2 cm2 Field with a Second Exposure by a 6 × 4 cm2 Field at 4 Different Gantry Angles in Coplanar Irradiation

Figure4 demonstrates the single exposure as the standard reference penumbra profile and the cumulative dose profiles after double exposures on 4 × 2 cm2 with a 6 × 4 cm2 field at 4 different gantry angles. The dose profiles on the + axis are slightly higher than the dose profile on the −x side. This is caused by smaller penetration depths ed¯s on the +x axis than e′d′¯s at the −x side in the ball phantom (Figure 2(a)). The Full-Width Half Maxima (FWHM, the width between 50% dose of the −x axis and the +x axis) was 19.8 mm after single irradiation at a gantry angle of 0 in the flat polystyrene solid phantom. The FWHM became 20.3 mm, 21.1 mm, 21.3 mm, and 21.6 mm after second exposures at gantry angles of 30°, 45°, 60°, and 90° in the ball phantom, respectively (Figure 4). The penumbra width between 80% to 20% of the central axis was 2.8 mm after a single exposure at 0° with a 4 × 2 cm2 field (Figure 4). After the second exposure, they were 4.7 mm, 6.2 mm, 6.7 mm, and 8.1 mm at gantry angles of 30°, 45°, 60°, and 90°, respectively (Figure 4). The renormalized dose profile shows, as expected, no change of the penumbra width after double exposures of any doses with a 6 × 4 cm2 field as long as the exposures were delivered at the same gantry angle as the first. The dose profile changes only after double exposures delivered at different gantry angles.Figure 4
The dose profile of the smaller 4 × 2 cm2 field irradiated at a gantry angle of 0° at the flat polystyrene solid phantom, followed by a second exposure in the ball phantom with a larger field of 6 × 4 cm2 at different gantry angles. The dose profile changes after second exposures at a gantry angle different from the first exposure. It also shows the full-width half maximum (the width at 50% dose of the −x axis and the +x axis) and the penumbra width from 80% to 20% of the central axis.


## 3.3. Dose Profile Fitting Using the LSRE Model

ParametersT, n, and t in the LSRE model determine the dose profile shape (equation (2)). The right-side dose profile after the first exposure at a gantry angle of 0° can be fitted accurately with T = 99.8, n = 90, and t = 0.2 (Figure 5(a)). n and t that result in the best fit, denoted as (n, t), at gantry angles of 30°, 45°, 60°, and 90°, are (310, 2), (410, 3), (490, 4), and (650, 5), respectively.Figure 5
(a) The right-side penumbra dose profiles vary with the gantry angles of the second exposure. The curvature of the right-side penumbra region can be fitted using the LSRE model by adjusting the (n) T and (t). (b) The left-side penumbra dose profiles vary with the gantry angles of the second exposure. The curvature of the left-side penumbra region can be fitted using the LSRE model by adjusting the (n) T and (t). The “90° measured curve” and “90° model estimated” curve denote the dose profile measured at the gantry angle of 90° in the ball phantom and the predicted curve at the same. They overlap each other completely.
(a)
(b)The measured curves of the left-side dose region can also be estimated by adjusting then, T, and t (Figure 5(b)). Parameters n and t of the left-side curve at 0 gantry angle after single exposure were (90, 0.2). After the second exposures, the parameters (n, t) were (320, 1.8), (430, 2.8), (510, 3.6), and (700, 4.8) at gantry angles of 30°, 45°, 60°, and 90°, respectively.


## 3.4. Sum of Square Errors (SSE): Finding the Bestn and t

Figures6(a) and 6(b) show n and t with the smallest SSE which gives the best LSRE model fitting of the measured data at the gantry angle of 45°. The best fit at other gantry angles can be done in the same manner.Figure 6
(a, b) These figures demonstrate the use of the SSE to find the bestn and t of the LSRE model at the gantry angle of 45°.
(a)
(b)


## 3.5. Geometric Derivation of the Effects on the Penumbra by Second Exposures at Arbitrary Gantry Angles

We found that the relationship ofe−μϖ with the x is a straight line (Figures 7(a) and 7(b) and Table 1), where ϖ denotes the penetration depth ed¯,μ is the attenuation coefficient, and μ = 0.00494 cm−1 of 6 MV in water. The linear regression of e−μϖ versus x at different gantry angles can be described in the form of y=ux+v, where u is the slope of the regression line shown in Figures 7(a) and 7(b). Figure 7(a) demonstrates the 4 lines with their slopes at the gantry angles of 30°, 45°, 60°, and 90°. The slopes at any other gantry angles can be derived by interpolation from these 4 know angles.Figure 7
(a, b) The regression lines of e−μϖ, where ϖ = ed¯, versus the distance on the +x axis in (a) and the −x axis in (b), at different gantry angles. The slopes are 0.0021, 0.0019, 0.0015, and 0.0011 at gantry angles of 90°, 60°, 45°, and 30°, respectively.
(a)
(b)Table 1
The coplanar penetration depthed¯(ϖ) and e−μϖ on the right-side penumbra after double exposures.
Distance from central axis of pointd on the x axis (cm)
Gantry angles
0°
30°
45°
60°
90°
ed
¯
ϖ
e
−μϖ
ed
¯
ϖ
e
−μϖ
ed
¯
ϖ
e
−μϖ
ed
¯
ϖ
e
−μϖ
ed
¯
ϖ
e
−μϖ
0
11.00
0.95223#
11.00
0.95223
11.00
0.95223
11.00
0.95223
11.00
0.95223
0.1
11.00
0.95223
10.95
0.95233
10.93
0.95238
10.91
0.95241
10.90
0.95244
0.2
11.00
0.95223
10.90
0.95244
10.86
0.95253
10.83
0.95260
10.80
0.95265
0.3
11.00
0.95223
10.85
0.95255
10.79
0.95268
10.74
0.95278
10.70
0.95286
0.4
10.99
0.95223
10.80
0.95266
10.71
0.95283
10.65
0.95296
10.60
0.95308
0.5
10.99
0.95223
10.74
0.95277
10.64
0.95298
10.56
0.95315
10.50
0.95329
0.6
10.99
0.95223
10.69
0.95287
10.57
0.95313
10.48
0.95333
10.40
0.95350
0.7
10.98
0.95223
10.64
0.95298
10.50
0.95329
10.39
0.95352
10.30
0.95371
0.8
10.98
0.95223
10.58
0.95309
10.42
0.95344
10.30
0.95370
10.20
0.95393
0.9
10.97
0.95223
10.53
0.95321
10.35
0.95359
10.21
0.95389
10.10
0.95414
1
10.96
0.95223
10.47
0.95332
10.27
0.95375
10.12
0.95407
10.00
0.95435
1.1
10.96
0.95223
10.42
0.95343
10.20
0.95390
10.04
0.95426
9.90
0.95456
1.2
10.95
0.95223
10.36
0.95354
10.13
0.95406
9.95
0.95445
9.80
0.95477
1.3
10.94
0.95223
10.30
0.95365
10.05
0.95421
9.86
0.95463
9.70
0.95499
1.4
10.93
0.95223
10.25
0.95377
9.97
0.95437
9.77
0.95482
9.60
0.95520
1.5
10.92
0.95223
10.19
0.95388
9.90
0.95452
9.68
0.95501
9.50
0.95541
1.6
10.91
0.95223
10.13
0.95400
9.82
0.95468
9.59
0.95519
9.40
0.95562
1.7
10.90
0.95223
10.07
0.95411
9.75
0.95483
9.50
0.95538
9.30
0.95584
#Data are calculated from ed¯ (ed¯=ϖ) by using e−μϖ, μ = 0.00494 cm−1 at 6 MV in water.


## 3.6. Derivation of the Parametersn and t of the Penumbra Dose Profiles at Arbitrary Gantry Angles Using Mathematical Methods

ed
¯ = S′c¯ − cd¯ − S′e¯, where S′c¯=100/cosβ, cd¯ = bd¯/cosβ, anded
¯ is the penetration depth of the S′c¯ from the larger field in the ball phantom.S′e¯ can be derived from equations (5)–(7) (for details, please refer to Appendix B).(5)p2−2sp+s2+q2=R2,(6)p=2s±4s2−41+ξ2s2−R221+ξ2,(7)S′e¯=p2+q2=1+ξ2p2=s−s2−1+ξ2s2−R21+ξ2.Table1 shows the calculated ed¯s in Figure 2(b) in the ball phantom at different gantry angles in coplanar irradiation.The regression curves for slopes versusn and t were shown in Figures 8(a) and 8(b), respectively. Therefore, once we obtain the slope of a regression curve from Figures 7(a) and 7(b), n and t can be found in Figures 8(a) and 8(b). This enables us to derive the estimated curve with precision and thus the accurate effects on the penumbra of a second irradiation at any gantry angles without the labor conducting the actual experiment.Figure 8
(a-b) The curves of slope versusn and t were shown in (a, b), respectively. Once the relationships of gantry angle versus slope and slope versus n and t are known, the parameters n and t in the LSRE model can be found easily to fit the measured dose profile or to construct the estimated profile for any gantry angle at coplanar irradiation with good agreement.
(a)
(b)


## 4. Discussion

We developed the LSRE model for studying the penumbra perturbation incurred by field overlapping in IMRT. This model has several advantages for penumbra profile fitting. Several functions, such as cosine [9] and exponential [10] functions, have been used in commercially available planning systems to model the penumbra dose profiles of megavoltage X-ray beams. However, both functions have the disadvantage of being noncontinuous. For example, the exponential function describing the distribution of penumbra dose profile can be expressed as the edge-spread function (ESF) or the line-spread function (LSF) [11]. According to their mathematical models, the penumbra is fitted by(8)ESFx=Aeαx+Beβxxat the negative direction,ESFx=1−Aeαx−Be−βxxat the positive direction,LSFx=Aαe−αx+Bβe−βx,where x is the distance from the beam edge, namely, center of penumbra, and A + B = 0.5.A, B, α, andβ are constants. The weak point of this type of penumbra function is the discrete nature in describing the penumbra curve. On the contrary, the LSRE is based on an inverse square root function [12, 13] and is a continuous equation that can fit the entire dose profile along the x axis (Figure 9).Figure 9
The dose profile in the penumbra region on thex axis can be modified by adjusting the T s, t, and n in the LSRE model to fit the measured dose profile.Using our model,n and t can be derived easily for the measured and predicted penumbra at any gantry angles once given the gantry angle and then the slope and eventually n and t. For example, if we replace x with the slope from Figure 7(a) into the equation y = 283467x − 3.1361, then we get the y value as n (Figure 8(a)); if we replace x with the slope of a known gantry angle from Figure 7(b) into the equation y = 63.098x2 + 8.9481x + 0.2101, then we get the y value as t (Figure 8(b)). The parameters n and t in the LSRE model can then be used to predict the dose profile with precision for any gantry angles at coplanar irradiation. This method enables radiation physicists and treatment planning system programmers to obtain penumbra dose perturbation data by using a mathematical process.Penumbra region dose data are important in radiotherapy planning systems for monitor unit calculation especially in linear accelerator-based radiosurgery. The precise position of the MLC and the penumbra shape are two important factors for treatment monitor unit calculation. The MLC-related dose penumbra in the treatment planning system must be calibrated to avoid underdosage or overdosage.For example, the penumbra expansion perturbed by overlapping fields in 4 × 2 cm2 split field calculated in Figure 10 leads to an intersection change with the standard undisturbed 50% field edge dose profile of 70.5%, 67.1%, 62.5%, and 58.2% at 90°, 60°, 45°, and 30° gantry angles, respectively. According to this result, if we ignore tentatively the penumbra position offset correction, the split field monitor unit calculated by the escalated output could result in an overdosage up to 10.5% in linear accelerator-based radiosurgery technique [14].Figure 10
The intersection change of the dose profile perturbed by overlapping fields with beam central axis in 4 × 2 cm2 split field was calculated by LSRE in this study.Many lung cancer patients who undergo radiation therapy are treated with higher-energy photons such as 10 MV or higher to obtain a deeper penetration and better dose uniformity. However, lower energy such as 6 MV photon beams should be preferred over higher energies photons because of the significant loss of lateral dose equilibrium for high-energy beams in the low-density medium. Any gains in radial dose uniformity across steep density gradients for higher-energy beams must be weighed carefully against the lateral beam degradation due to penumbra widening. The LSRE model can be applied to predict the penumbra perturbation of both high- and low-energy photon beams in lesions surrounded by low-density organ such as lung.


## 5. Conclusion

Our study shows how to perform an accurate calculation of the penumbra perturbation caused by an overlapping field at different gantry angles by using a mathematical model in linear accelerator-based radiosurgery. The LSRE model is the first continuous equation which describes the dose profile across the entire penumbra. The clinical significance of this study is that the treatment monitor units can be overestimated or underestimated during linear accelerator-based radiosurgery if the penumbra dose profile is not correctly calculated in the treatment planning system. The LSRE model offers a mathematical approach to correct the penumbra and makes it unnecessary to do tedious physics experiments.



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*Source: 7584743-2019-12-07.xml*