3.3 Solid-state dosimeters and dose response compensation in external radiotherapy

This section shed light on describing the dosimetric response of solid-state dosimeters that are used for the dose measurement of external radiotherapy. Two approaches are presented for this purpose. The first approach, implementation of empirical method approach that considers the radiation beam, is separated into two components: primary and scattered beams. The spectral variation of radiotherapeutic beam is evaluated by their contribution in the dose to the medium that contains the region of interest. Solid-state dosimeters of high-density materials have an over-response issue that is commonly used in large and small fields. Hence, compensation factor should be calculated based on beam parameters such as energy, field size, depth, and other irradiation parameters. Dealing with overresponse issue is not an easy task; however, it generates a significant improvement in accuracy in dose measurements over non-compensated measurements.

The second approach is to implement a compensation method based on a modified cavity theory. In this method, dose response of solid-state dosimeter is described considering the local spectrum and monoenergetic response. The local spectrum could be obtained by convolution method of pencil beam kernels using a pre-evaluated database that considers different separated types of particles according to their history of interaction (primary photon and electron and secondary photons and electrons). On the other hand, monoenergetic response of solidstate dosimeter could be calculated using the Monte Carlo simulation using different codes like PENELOPE [47, 48]. The accuracy of compensation methods should be evaluated by comparing simulated data with the corresponding measurements. This approach could be applied in situations where there is no lateral electron equilibrium compared to the previous method of compensation. Since the compensation accuracy depends on the local reconstructed spectrum, it is possible to implement this process in more complex irradiation conditions such as small fields. However, this method requires specific information such as the field size, beam quality, and detector position. Yet using two dosimeters whose materials in the sensitive volume are different can be instead used without considering beam information and enough to evaluate over-response correction.

#### 3.3.1 The first approach: primer-released contribution separation

Cunningham [49] proposed a method for separation of primary beam component out of beam spectrum through dose calculation technique for irregular fields. He assumed that dosimeter placement does not introduce any local spectral disturbance in the volume of interest and the difference of dose response between solidstate dosimeter and water depends on the material difference and the ionization spectrum. The primary component of the spectrum photon beam is dependent on the design of the collimation system, for example, the primary collimator and flattening filter [50]. Accordingly, the difference between dosimetric response of

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field… DOI: http://dx.doi.org/10.5772/intechopen.89150

solid-state dosimeter and water is due to the primary component of the beam that remains almost invariant for the given beam quality and does not depend on the volume. On the other hand, the scattered component of the photon beam depends largely on the volume surrounding it; therefore, it depends on dosimeter depth, irradiated field size, etc. Hence, it is convenient to calculate the difference of the response caused by these two components separately and additively combine the two parts at the end of the calculation. To quantify the response rate provided by the primary and scattered components, a scattered factor (SF) could be introduced, defined as follows:

$$SF\_w = \frac{D\_w^{\kappa}}{D\_w^{pr}}\tag{4}$$

$$\text{CSF}\_{\text{SSD}} = \frac{D\_{\text{SSD}}^{\text{sc}}}{D\_{\text{SSD}}^{pr}} \tag{5}$$

where Dsc <sup>w</sup> and Dpr <sup>w</sup> correspond to the dose contributions in water by primary and scattered components of photon beam, respectively. A scattered factor of solid-state dosimeter may be defined in the same manner, as shown in Eq. (5). The scattered factor is dependent on the field size and depth position of the dosimeter.

The total dose of water ð Þ Dw or SSD (DSSD) is the sum of the two components, primary and scattered. This can be expressed in terms of scatter factor as follows:

$$D\_w = D\_w^{pr} (\mathbf{1} + \mathbf{S} F\_w) \tag{6}$$

$$D\_{\rm SSD} = D\_{\rm SSD}^{pr} (\mathbf{1} + \mathcal{S} F\_{\rm SSD}) \tag{7}$$

The response factor (RF) of SSD dose to water ratio is defined as:

$$RF\_w^{\text{SSD}} = \frac{D\_{\text{SSD}}}{D\_w} \tag{8}$$

Therefore, the dose response of SSD could be corrected and applied to all SSD measurement by the evaluation of RFSSD <sup>w</sup> before the measurement implementation. Eq. (8) can also be expressed in terms of scattered factor, combining Eqs. (6) and (7):

$$RF\_w^{\text{SSD}} = \frac{D\_{\text{SSD}}^{pr}(\mathbf{1} + \mathbf{S}F\_{\text{SSD}})}{D\_w^{pr}(\mathbf{1} + \mathbf{S}F\_w)}\tag{9}$$

In this way, expressing the response factor to fulfill the following objectives: The ratio between the primary dose of SSD to the dose of water (Dpr SSD=Dpr <sup>w</sup> ) can be considered relatively stable, because once the charged-particle equilibrium (CPE) is established, the local spectra of the primary electrons and photons remain invariant which are independent of irradiation condition variations. Therefore, the RFSSD w variation is due to the difference of the scattered component of both water and SSD. As a result, RFSSD <sup>w</sup> depends on the determination of the primary dose ratios and scattered factors (SFSSD, SFw). However, it could not be easy to evaluate the primary and scattered doses separately in experiments, especially for high-energetic photons, because it needs massive buildup of material, which is necessary to achieve CPE to introduce significant attenuation and wide contribution [51]. Nevertheless, the response factor could be evaluated in a small field where CPE is still achieved. So in small-field dosimetry, the evaluated RFSSD <sup>w</sup> is close to the Dpr SSD=Dpr <sup>w</sup> since the main

the technique of pulsed ultraviolet laser excitation. RPL dosimeters exhibit linear response, flat energy response in the energy range of keV and MeV, good reproducibility, good spatial resolution, and negligible fading of signal [45, 46].

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

• Alanine: Its macroscopic interaction coefficients and density are close to that of water. The exhibit volume averaging effect because of its large size, low sensitivity, and high doses of radiation is needed to be delivered to obtain

3.3 Solid-state dosimeters and dose response compensation in external

components: primary and scattered beams. The spectral variation of radio-

in accuracy in dose measurements over non-compensated measurements.

ified cavity theory. In this method, dose response of solid-state dosimeter is described considering the local spectrum and monoenergetic response. The local spectrum could be obtained by convolution method of pencil beam kernels using a

pre-evaluated database that considers different separated types of particles

mation and enough to evaluate over-response correction.

3.3.1 The first approach: primer-released contribution separation

This section shed light on describing the dosimetric response of solid-state dosimeters that are used for the dose measurement of external radiotherapy. Two approaches are presented for this purpose. The first approach, implementation of empirical method approach that considers the radiation beam, is separated into two

therapeutic beam is evaluated by their contribution in the dose to the medium that contains the region of interest. Solid-state dosimeters of high-density materials have an over-response issue that is commonly used in large and small fields. Hence, compensation factor should be calculated based on beam parameters such as energy, field size, depth, and other irradiation parameters. Dealing with overresponse issue is not an easy task; however, it generates a significant improvement

The second approach is to implement a compensation method based on a mod-

according to their history of interaction (primary photon and electron and secondary photons and electrons). On the other hand, monoenergetic response of solidstate dosimeter could be calculated using the Monte Carlo simulation using different codes like PENELOPE [47, 48]. The accuracy of compensation methods should be evaluated by comparing simulated data with the corresponding measurements. This approach could be applied in situations where there is no lateral electron equilibrium compared to the previous method of compensation. Since the compensation accuracy depends on the local reconstructed spectrum, it is possible to implement this process in more complex irradiation conditions such as small fields. However, this method requires specific information such as the field size, beam quality, and detector position. Yet using two dosimeters whose materials in the sensitive volume are different can be instead used without considering beam infor-

Cunningham [49] proposed a method for separation of primary beam component out of beam spectrum through dose calculation technique for irregular fields. He assumed that dosimeter placement does not introduce any local spectral disturbance in the volume of interest and the difference of dose response between solidstate dosimeter and water depends on the material difference and the ionization spectrum. The primary component of the spectrum photon beam is dependent on the design of the collimation system, for example, the primary collimator and flattening filter [50]. Accordingly, the difference between dosimetric response of

reproducibility of less than 0.5%.

radiotherapy

primary contribution component is dominant. Rewriting Eq. (9) for the last condition of small-field irradiation as reference condition to the following equation:

$$RF\_w^{\rm SSD} = \frac{D\_{\rm SSD}^{r\notin}(\mathbf{1} + \chi\_{\rm SSD})}{D\_w^{r\notin}(\mathbf{1} + \chi\_w)}\tag{10}$$

spectral variation between SSD and ionization chamber is also affected by the depth of dosimeter. γSSD and γ<sup>w</sup> could be evaluated from representing normalized maxi-

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field…

The relationship between γSSD and γ<sup>w</sup> in small-field sizes may be linear due to the main contribution primary component. But the response factor linearity allows us to separate two sources that vary the local spectrum at the position of the detector: The size of the square field (A) and depth (z). Therefore, the response

where the factor k(z) is a function of depth and Aref corresponds to the size of the reference field (small-field size). Modeling of k(z) could be established from the

The previous characterization method of RFSSD can be applied for a series of square field sizes with orthogonal collimators. However, in the case of irregular or circular fields, it must be expressed in terms of an equivalent square field for the interpolation. The method of equivalent square field is a simple empirical method for calculating the dose of irregular field size [52]. There are several ways to calculate the equivalent square field according to the literature: Equivalent tables for rectangular fields [53], sum of the small rectangles [54], and Clarkson integration of [55, 56]. Sterling's formula [57] can be used to calculate the equivalent square fields

ESQ <sup>¼</sup> <sup>2</sup>WL

This method of empirical compensation consists primarily of establishing a response correction factors table by the experimental approach. It is based on the separation of primary and scattering contribution parts of photons and electrons in the beam. However, the primary contribution part to the SSD cannot be evaluated through the measurement in the air as the local spectral variation in the air with respect to that in the water causes a large SSD response difference. Therefore, an arbitrary square field should be selected as a reference field for a given energetic beam. In this reference field, the maximum tissue ratio of SSD is compared with that measured of water by ionization chamber. Although it is possible to apply this method of compensation in irregular fields, it is difficult to implement it in more complex fields such as IMRT or non-rectangular fields, because this method requires a lot of effort for measuring, adjustments, and approximations that could

where ESQ is the side length of the equivalent square field, W is the width of the

The cavity theory was originally developed to convert the absorbed dose in the ionization chamber to the absorbed dose in the medium of interest [58]. When the measurement is performed with a solid-state dosimeter, the material of the detector, in general, is different from that of the medium in which it is introduced. If we consider the detector as a cavity introduced into the uniform medium of interest, the absorbed dose in the detector Ddet is different with respect to the absorbed dose in the medium at this position in the absence of the detector, Dmed. The main

RFSSD <sup>¼</sup> <sup>1</sup> <sup>þ</sup> k zð Þ� <sup>A</sup> � Aref (13)

ð Þ <sup>W</sup> <sup>þ</sup> <sup>L</sup> (14)

mum dose ratio or percentage depth dose for both dosimeters.

factor can be modeled by:

of the two rectangular fields:

slopes of plotted RFSSDð Þ A against the depth.

DOI: http://dx.doi.org/10.5772/intechopen.89150

rectangle, and L is the length of the rectangle.

be uneasy in more complex fields.

3.3.2 Cavity theory approach

51

where Dref SSD=Dref <sup>w</sup> is the dose–response factor of SSD to the water in the reference condition (small field). γSSD andγ<sup>w</sup> are introduced instead of SFSSD and SFw in Eq. (9), respectively, and could be written as follows:

$$\chi\_{\text{SSD}} = \frac{D\_{\text{SSD}}}{D\_{\text{SSD}}^{\text{ref}}} - \mathbf{1} \tag{11}$$

$$\gamma\_w = \frac{D\_w}{D\_w^{r\circ f}} - \mathbf{1} \tag{12}$$

where DSSD and Dw correspond to the dose response of SSD and water in an arbitrary requirement. Both factors are used to describe primary component variation with respect to the scattering component in the reference condition. To calculate the response factor by interpolation or extrapolation, measure both the dose in water by ionization chamber and in SSD to establish a response factor table. So in experimental measurement setup as illustrated in Figure 5, choosing small-field area to be a reference condition that avoids scattering component for dose response of SSD. PMMA sheets could be arranged around the solid-state dosimeter to establish homogenous tissue equivalent material. Mentioning that, at least 10 cm thickness of PMMA should be placed below the detector to create a homogeneous volume for the backscattered radiation. However, the measurement of the dose at a reference point in water phantom is established by ionization chamber. In smallfield dosimetry beyond the buildup region, the relative difference of dose response between SSD and ionization chamber in water should be minimum, as low as possible as the result of the stability of primary component of radiation. Therefore, it is easy to explore the scattering factors in a large field. On the other hand, the

Figure 5. Geometrical configuration.

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field… DOI: http://dx.doi.org/10.5772/intechopen.89150

spectral variation between SSD and ionization chamber is also affected by the depth of dosimeter. γSSD and γ<sup>w</sup> could be evaluated from representing normalized maximum dose ratio or percentage depth dose for both dosimeters.

The relationship between γSSD and γ<sup>w</sup> in small-field sizes may be linear due to the main contribution primary component. But the response factor linearity allows us to separate two sources that vary the local spectrum at the position of the detector: The size of the square field (A) and depth (z). Therefore, the response factor can be modeled by:

$$RF\_{\rm SSD} = \mathbf{1} + k(\mathbf{z}) \times \left(\mathbf{A} - A\_{\rm ref}\right) \tag{13}$$

where the factor k(z) is a function of depth and Aref corresponds to the size of the reference field (small-field size). Modeling of k(z) could be established from the slopes of plotted RFSSDð Þ A against the depth.

The previous characterization method of RFSSD can be applied for a series of square field sizes with orthogonal collimators. However, in the case of irregular or circular fields, it must be expressed in terms of an equivalent square field for the interpolation. The method of equivalent square field is a simple empirical method for calculating the dose of irregular field size [52]. There are several ways to calculate the equivalent square field according to the literature: Equivalent tables for rectangular fields [53], sum of the small rectangles [54], and Clarkson integration of [55, 56]. Sterling's formula [57] can be used to calculate the equivalent square fields of the two rectangular fields:

$$\text{ESQ} = \frac{\text{2WL}}{(\text{W} + L)} \tag{14}$$

where ESQ is the side length of the equivalent square field, W is the width of the rectangle, and L is the length of the rectangle.

This method of empirical compensation consists primarily of establishing a response correction factors table by the experimental approach. It is based on the separation of primary and scattering contribution parts of photons and electrons in the beam. However, the primary contribution part to the SSD cannot be evaluated through the measurement in the air as the local spectral variation in the air with respect to that in the water causes a large SSD response difference. Therefore, an arbitrary square field should be selected as a reference field for a given energetic beam. In this reference field, the maximum tissue ratio of SSD is compared with that measured of water by ionization chamber. Although it is possible to apply this method of compensation in irregular fields, it is difficult to implement it in more complex fields such as IMRT or non-rectangular fields, because this method requires a lot of effort for measuring, adjustments, and approximations that could be uneasy in more complex fields.

#### 3.3.2 Cavity theory approach

The cavity theory was originally developed to convert the absorbed dose in the ionization chamber to the absorbed dose in the medium of interest [58]. When the measurement is performed with a solid-state dosimeter, the material of the detector, in general, is different from that of the medium in which it is introduced. If we consider the detector as a cavity introduced into the uniform medium of interest, the absorbed dose in the detector Ddet is different with respect to the absorbed dose in the medium at this position in the absence of the detector, Dmed. The main

primary contribution component is dominant. Rewriting Eq. (9) for the last condition of small-field irradiation as reference condition to the following equation:

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Dref

SSD 1 þ γ ð Þ SSD

(10)

<sup>w</sup> ð Þ 1 þ γ<sup>w</sup>

<sup>w</sup> is the dose–response factor of SSD to the water in the reference

� 1 (11)

� 1 (12)

RFSSD

Eq. (9), respectively, and could be written as follows:

where Dref

Figure 5.

50

Geometrical configuration.

SSD=Dref

<sup>w</sup> <sup>¼</sup> Dref

condition (small field). γSSD andγ<sup>w</sup> are introduced instead of SFSSD and SFw in

<sup>γ</sup>SSD <sup>¼</sup> DSSD Dref SSD

> <sup>γ</sup><sup>w</sup> <sup>¼</sup> Dw Dref w

where DSSD and Dw correspond to the dose response of SSD and water in an arbitrary requirement. Both factors are used to describe primary component variation with respect to the scattering component in the reference condition. To calculate the response factor by interpolation or extrapolation, measure both the dose in water by ionization chamber and in SSD to establish a response factor table. So in experimental measurement setup as illustrated in Figure 5, choosing small-field area to be a reference condition that avoids scattering component for dose response of SSD. PMMA sheets could be arranged around the solid-state dosimeter to establish homogenous tissue equivalent material. Mentioning that, at least 10 cm thickness of PMMA should be placed below the detector to create a homogeneous volume for the backscattered radiation. However, the measurement of the dose at a reference point in water phantom is established by ionization chamber. In smallfield dosimetry beyond the buildup region, the relative difference of dose response between SSD and ionization chamber in water should be minimum, as low as possible as the result of the stability of primary component of radiation. Therefore, it is easy to explore the scattering factors in a large field. On the other hand, the

objective of the cavity theory is to determine the response factor RFQ , given by Eq. (15):

$$RF\_Q = \left(\frac{D\_{det}}{D\_{med}}\right)\_Q \tag{15}$$

second condition ensures that the dosimetric contribution from the photon is negligible. This is a valid condition for high-energy photon beams in most situations

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field…

DOI: http://dx.doi.org/10.5772/intechopen.89150

<sup>0</sup> Φmedð Þ E ð Þ Scolð Þ E =ρ detdE

(16)

<sup>0</sup> Φmedð Þ E ð Þ Scolð Þ E =ρ meddE

where sð Þ colð Þ E =ρ det and sð Þ colð Þ E =ρ med are collision stopping power of the detector and the medium, respectively, and Emax is the maximum energy in the spectrum of

Secondary particles (delta rays) are considered prerequisite to assess the dose by the stopping power that reaches a balance in the cavity. Another way to express this requirement is that the electron is considered to lose energy in the continuous slowdown cavity. However, it could generate high-energy secondary electrons by hard collisions in the cavity. These secondary electrons will come out of the cavity, and thus the delta ray balance is no longer valid. To take into account the effects of delta rays in an approximate way, Spencer and Attix proposed an extension of the cavity of theory [62]. Spencer-Attix theory considered the separation of electron particles into two parts: The fast electrons with an energy greater than a threshold (Δ) and slow electrons with energy below the threshold. Slow electrons are considered to deposit the energy locally inside the cavity, while the fast electrons are considered completely capable of crossing the cavity. The dose contribution by fast electrons is estimated by the restricted stopping power, LΔ=ρð Þ E . The restricted stopping power is defined as the stopping power limited to lose energy below the

Ð Emax

A schematic illustration of small cavity behavior under high energetic photon irradiation.

Ð Emax

threshold energy (Δ). The total dose in the cavity can be written as:

<sup>Φ</sup>ð Þ <sup>E</sup> <sup>L</sup>Δð Þ <sup>E</sup> ρ

The first term on the right side of Eq. (17) corresponds to the dose deposited by fast electrons, and the second term takes into account the dose deposited by slow electrons, often termed as end track term (ET) suggested by Nahum [63] to esti-

� �dE <sup>þ</sup> ð Þ <sup>E</sup>:T: (17)

ðEmax Δ

D ¼

mate the contribution of slow electrons:

53

[61]. Under these conditions, Dmed is related to Ddet as:

Ddet Dmed ¼

ionization chamber fluence.

Figure 7.

where Q corresponds to a given beam quality. Figure 6 illustrates the schematic application of cavity theory to convert detector absorbed dose for a given beam quality to the dose in the medium of interest by RFQ [59].

There are two possible cavities where RFQ could be derived, the large cavity and small cavity. The terms "small and large" refer to the size of the cavity relative to the bearing surfaces of secondary particles, i.e., the electrons and positrons.

#### 3.3.3 Small cavity theory (SCT)

Small cavity theory is also referred to as Bragg-Gray cavity theory. First, William Bragg proposed it then Louis Harold Gray completed it [60]. Bragg-Gray proposed two conditions: (a) Cavity size should be small enough compared to the range of the charged particles inside the irradiated volume. So that, the fluence of charged particles and local fluence are not disturbed by the presence of the cavity in the middle (see Figure 7). (b) The absorbed dose in the cavity is completely deposited by charged particles which pass through the cavity.

The realization of the first condition ensures that the local influence is invariant, with or without the existence of this detector is to say Φdetð Þ¼ E Φmedð Þ E . The

#### Figure 6.

Application of the cavity theory: The detector's absorbed dose for a given beam quality converted to the dose in the medium of interest by RF<sup>Q</sup> [11].

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field… DOI: http://dx.doi.org/10.5772/intechopen.89150

Figure 7. A schematic illustration of small cavity behavior under high energetic photon irradiation.

second condition ensures that the dosimetric contribution from the photon is negligible. This is a valid condition for high-energy photon beams in most situations [61]. Under these conditions, Dmed is related to Ddet as:

$$\frac{D\_{det}}{D\_{med}} = \frac{\int\_0^{E\_{\text{max}}} \Phi\_{med}(E) (\mathcal{S}\_{col}(E)/\rho)\_{det} dE}{\int\_0^{E\_{\text{max}}} \Phi\_{med}(E) (\mathcal{S}\_{col}(E)/\rho)\_{med} dE} \tag{16}$$

where sð Þ colð Þ E =ρ det and sð Þ colð Þ E =ρ med are collision stopping power of the detector and the medium, respectively, and Emax is the maximum energy in the spectrum of ionization chamber fluence.

Secondary particles (delta rays) are considered prerequisite to assess the dose by the stopping power that reaches a balance in the cavity. Another way to express this requirement is that the electron is considered to lose energy in the continuous slowdown cavity. However, it could generate high-energy secondary electrons by hard collisions in the cavity. These secondary electrons will come out of the cavity, and thus the delta ray balance is no longer valid. To take into account the effects of delta rays in an approximate way, Spencer and Attix proposed an extension of the cavity of theory [62]. Spencer-Attix theory considered the separation of electron particles into two parts: The fast electrons with an energy greater than a threshold (Δ) and slow electrons with energy below the threshold. Slow electrons are considered to deposit the energy locally inside the cavity, while the fast electrons are considered completely capable of crossing the cavity. The dose contribution by fast electrons is estimated by the restricted stopping power, LΔ=ρð Þ E . The restricted stopping power is defined as the stopping power limited to lose energy below the threshold energy (Δ). The total dose in the cavity can be written as:

$$D = \int\_{\Delta}^{E\_{\text{max}}} \Phi(E) \left(\frac{L\_4(E)}{\rho}\right) dE + (E.T.) \tag{17}$$

The first term on the right side of Eq. (17) corresponds to the dose deposited by fast electrons, and the second term takes into account the dose deposited by slow electrons, often termed as end track term (ET) suggested by Nahum [63] to estimate the contribution of slow electrons:

objective of the cavity theory is to determine the response factor RFQ , given by

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

RFQ <sup>¼</sup> Ddet

quality to the dose in the medium of interest by RFQ [59].

by charged particles which pass through the cavity.

3.3.3 Small cavity theory (SCT)

Dmed 

where Q corresponds to a given beam quality. Figure 6 illustrates the schematic application of cavity theory to convert detector absorbed dose for a given beam

There are two possible cavities where RFQ could be derived, the large cavity and small cavity. The terms "small and large" refer to the size of the cavity relative to the bearing surfaces of secondary particles, i.e., the electrons and positrons.

Small cavity theory is also referred to as Bragg-Gray cavity theory. First, William Bragg proposed it then Louis Harold Gray completed it [60]. Bragg-Gray proposed two conditions: (a) Cavity size should be small enough compared to the range of the charged particles inside the irradiated volume. So that, the fluence of charged particles and local fluence are not disturbed by the presence of the cavity in the middle (see Figure 7). (b) The absorbed dose in the cavity is completely deposited

The realization of the first condition ensures that the local influence is invariant,

with or without the existence of this detector is to say Φdetð Þ¼ E Φmedð Þ E . The

Application of the cavity theory: The detector's absorbed dose for a given beam quality converted to the dose in

Q

(15)

Eq. (15):

Figure 6.

52

the medium of interest by RF<sup>Q</sup> [11].

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

$$E.T. = \Phi(\Delta) \frac{\mathbb{S}\_{col}(\Delta)}{\rho} \,\,\Delta \tag{18}$$

achieved in the border regions of the large cavity, due to the difference in the material around the border. However, the electronic balance is achieved in most of the cavity and the average absorbed dose in the detector can be determined using

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field…

μenð Þ E ρ � �

det

where ð Þ μenð Þ E =ρ det is the mass absorption coefficient of the detector and Ψdetð Þ E

To apply cavity theory, the first step is to obtain the spectrum at the position of interest. Fluence pencil kernels may be used to calculate the local spectrum in a homogeneous phantom. This model had been extensively used to calculate the dose in treatment planning system (TPS) [55, 67–69]. Its idea is to convolute the energy depositions for each pencil beam energy through wide beam spectrum. The most interesting approach to pencil beam fluence has been proposed by [47, 70] which calculated the local spectra via dividing it into high and low energies using small and large cavity theory approximations, respectively. However, it could be easy to calculate other physical quantities, such as the fluence spectrum in an irradiated water phantom as in Figure 9. Eklund and Ahnesjö [28, 70] use fluence pencil kernel database to evaluate the spectrum. In this database, the fluence pencil kernel has been defined as the spatial distribution of fluence, resulted from the irradiation of semi-infinite water slab with point of the monodirectional and monoenergetic beam in water phantom of infinite thickness. Monte Carlo simulation had a good feature of interaction for the evaluation of these spectra. As the energy deposited in the phantom is laterally symmetric, the parameters to describe the fluence pencil can be reduced to three parameters, as shown in schematic geometry for fluence

� Ψdetð Þ E dE (20)

D ¼

is energy fluence of photon energy E in the detector.

3.4.1 Spectrum convolution calculation

DOI: http://dx.doi.org/10.5772/intechopen.89150

3.4 Dose model for the SSD crystal in the photon beam

pencil kernel acquisition of monoenergetic beam (see Figure 10).

ðEmax 0

the following equation:

Figure 9.

55

A schematic geometry for fluence pencil kernel acquisition.

where Φ Δð Þ is the electron differentiated fluence energy valued at <sup>Δ</sup> and Scolð Þ <sup>Δ</sup> <sup>ρ</sup> is the nonrestricted stopping power evaluated at Δ. Instead of estimating the dose ratio in the detector and the medium by Bragg-Gray theory, it can be expressed as:

$$\frac{D\_{\rm det}}{D\_{\rm med}} = \frac{\int\_0^{E\_{\rm max}} \Phi\_{\rm med}(E)(L\_{\Delta}(E)/\rho)\_{\rm det} dE + \Phi(\Delta)(\mathcal{S}\_{\rm col}(\Delta)/\rho)\_{\rm det} \Delta}{\int\_0^{E\_{\rm max}} \Phi\_{\rm med}(E)(L\_{\Delta}(E)/\rho)\_{\rm med} dE + \Phi(\Delta)(\mathcal{S}\_{\rm col}(\Delta)/\rho)\_{\rm det} \Delta} \tag{19}$$

Δ is defined as the minimum energy needed for electron to pass through the cavity of interest. The value of Δ depends on the size and the material of the cavity. There are many studies based on determining Δ to apply the Spencer-Attix theory to some ionization chambers where Δ = 10 keV is used [64–66].

#### 3.3.4 Large cavity theory (LCT)

On the other hand, large cavity is opposite to small cavity, whenever the size of the detector is much larger than the range of the electron that passes through the cavity. In this case, the range of delta ray is small in large cavity compared to the size of the cavity (see Figure 8). Hence, electronic equilibrium is established in most of the cavity size [59]. If the radiation source is a photon, it interacts with the material in the cavity and hence produces secondary electrons. As these created electrons are unable to pass through the cavity, the electronic control is established in the cavity of the detector. It should be noted that the electronic balance is not

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field… DOI: http://dx.doi.org/10.5772/intechopen.89150

achieved in the border regions of the large cavity, due to the difference in the material around the border. However, the electronic balance is achieved in most of the cavity and the average absorbed dose in the detector can be determined using the following equation:

$$D = \int\_{0}^{E\_{\text{max}}} \left( \frac{\mu\_{en}(E)}{\rho} \right)\_{\text{det}} \cdot \Psi\_{\text{det}}(E) dE \tag{20}$$

where ð Þ μenð Þ E =ρ det is the mass absorption coefficient of the detector and Ψdetð Þ E is energy fluence of photon energy E in the detector.

#### 3.4 Dose model for the SSD crystal in the photon beam

#### 3.4.1 Spectrum convolution calculation

<sup>E</sup>:T: <sup>¼</sup> Φ Δð Þ Scolð Þ <sup>Δ</sup>

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where Φ Δð Þ is the electron differentiated fluence energy valued at <sup>Δ</sup> and Scolð Þ <sup>Δ</sup>

<sup>0</sup> Φmedð Þ E ð Þ LΔð Þ E =ρ detdE þ Φ Δð Þð Þ Scolð Þ Δ =ρ detΔ

<sup>0</sup> Φmedð Þ E ð Þ LΔð Þ E =ρ meddE þ Φ Δð Þð Þ Scolð Þ Δ =ρ detΔ

On the other hand, large cavity is opposite to small cavity, whenever the size of the detector is much larger than the range of the electron that passes through the cavity. In this case, the range of delta ray is small in large cavity compared to the size of the cavity (see Figure 8). Hence, electronic equilibrium is established in most of the cavity size [59]. If the radiation source is a photon, it interacts with the material in the cavity and hence produces secondary electrons. As these created electrons are unable to pass through the cavity, the electronic control is established in the cavity of the detector. It should be noted that the electronic balance is not

Δ is defined as the minimum energy needed for electron to pass through the cavity of interest. The value of Δ depends on the size and the material of the cavity. There are many studies based on determining Δ to apply the Spencer-Attix theory to

some ionization chambers where Δ = 10 keV is used [64–66].

Large cavity of high energetic photon beam deposition in a type of detector.

the nonrestricted stopping power evaluated at Δ. Instead of estimating the dose ratio in the detector and the medium by Bragg-Gray theory, it can be expressed as:

> Ddet Dmed ¼

3.3.4 Large cavity theory (LCT)

Figure 8.

54

Ð Emax

Ð Emax

ρ

Δ (18)

<sup>ρ</sup> is

(19)

To apply cavity theory, the first step is to obtain the spectrum at the position of interest. Fluence pencil kernels may be used to calculate the local spectrum in a homogeneous phantom. This model had been extensively used to calculate the dose in treatment planning system (TPS) [55, 67–69]. Its idea is to convolute the energy depositions for each pencil beam energy through wide beam spectrum. The most interesting approach to pencil beam fluence has been proposed by [47, 70] which calculated the local spectra via dividing it into high and low energies using small and large cavity theory approximations, respectively. However, it could be easy to calculate other physical quantities, such as the fluence spectrum in an irradiated water phantom as in Figure 9. Eklund and Ahnesjö [28, 70] use fluence pencil kernel database to evaluate the spectrum. In this database, the fluence pencil kernel has been defined as the spatial distribution of fluence, resulted from the irradiation of semi-infinite water slab with point of the monodirectional and monoenergetic beam in water phantom of infinite thickness. Monte Carlo simulation had a good feature of interaction for the evaluation of these spectra. As the energy deposited in the phantom is laterally symmetric, the parameters to describe the fluence pencil can be reduced to three parameters, as shown in schematic geometry for fluence pencil kernel acquisition of monoenergetic beam (see Figure 10).

Figure 9.

A schematic geometry for fluence pencil kernel acquisition.

that Ψ x<sup>0</sup>

Eq. (22):

Dcav <sup>¼</sup>

where Φ<sup>p</sup>

where Dcav

Dcav

57

ðEmax Δ

Φp

scattered photon fluence. <sup>L</sup>cav

cavð Þ� <sup>E</sup> <sup>L</sup>cav

<sup>Δ</sup> ð Þ E ρ

<sup>Δ</sup> ð Þ E

had been introduced, which is defined as follows:

K Eð Þ¼

Dcav <sup>p</sup> ð Þ E <sup>Ψ</sup>ð Þ� <sup>E</sup> <sup>μ</sup>cav

photons of energy E (or primary electrons) with a fluence of photons Ψð Þ E and its primary electrons. The denominator of this expression represents the collision kerma of the cavity, which is equivalent to the dose if CPE exists locally. Computing

<sup>p</sup> ð Þ E is only possible by a total particle transport calculation as by Monte Carlo

en ð Þ E ρ

<sup>p</sup> ð Þ E is the dose deposited in the cavity of the detector, by the primary

dE <sup>þ</sup> <sup>Φ</sup><sup>p</sup>

cavð Þ <sup>E</sup> is the primary electron fluence and <sup>Ψ</sup><sup>s</sup>

cavð Þ� Δ

mass absorption coefficient of material in the cavity. This model proved quite precisely the dose in water, verified by measurement [27]. Eklund and Ahnesjö [26] introduced some solution to calculate dose response through the two assumptions: (1) Ensuring all primary electrons satisfy the SCT condition if the detector size is rather small. Nevertheless, there are still low-energetic electrons in the spectrum that cannot pass through the cavity of the dosimeter. (2) The scattered photons are considered to satisfy the condition of LCT, indicating that CPE is assumed to be located in the cavity of the dosimeter. The validity of this assumption depends on the energy of the scattered photon. To solve this situation, Eklund and Ahnesjö [71] introduced two solutions for the condition of the hypothesis is closer to reality: Instead of calculating the primary low-energetic electron contribution by LCT, they calculate the contribution from primary photons in LCT that create low-energetic primary electrons. From this calculation of the fluence spectra, it is possible to find the low-energetic primary photons. Therefore, a partitioning of the primary electrons was performed, where the high-energetic primary electrons followed the SCT and the contribution of low-energy electron primary was calculated using their father or primary photons. Ideally, the scattered photons should be partitioned in the same way to treat low component of high energy differently. K(E) correction

Lcav <sup>Δ</sup> ð Þ Δ <sup>ρ</sup> � <sup>Δ</sup> <sup>þ</sup>

<sup>ρ</sup> is the restricted stopping power, and <sup>μ</sup>cav

ðEmax 0

Ψs

cavð Þ� <sup>E</sup> <sup>μ</sup>cav

cavð Þ E is the energy of the

en ð Þ E <sup>ρ</sup> is the

en ð Þ E ρ

dE

(22)

(23)

which is the case of IMRT.

DOI: http://dx.doi.org/10.5772/intechopen.89150

3.5 Response model of SSD

, y<sup>0</sup> ð Þ , E may vary depending on the dose deposition in the irradiation field,

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field…

In reality, the response behavior of any dosimeter deviates in two cavity theories, because the extreme conditions for both theories are never completely filled. To determine if the dosimeter cavity is large or small, the size of the cavity is compared with the range of the electrons. If the dosimeter is irradiated with a polyenergetic beam, the cavity of theories cannot be applied directly due to the varied response to particles of different energies. In 2004, Yin et al. [30] proposed a method to treat the primary and scattered components separately by different cavity theories, assuming that the primary particles satisfy conditions of SCT while the scattered particles satisfy the LCT. The total dose measured in the Si-diode dosimeter [30] considered the sum of primary and scattered contributions using

Figure 10. Particles categories as defined in [70].

Here r is the lateral distance of the axis of irradiation from the position of interest; z is the depth of the point of interest; E is the energy of monoenergetic ionization chamber beam. Also, these database separated particles into four categories, depending on their histories of interaction, as shown in Figure 5:


The first two types of particles (primary photons and electrons) are the main component of the beam, while the last two are the scattered component. This separation of particles is only possible with the feature of tracking the particle's interaction history in Monte Carlo simulation. To obtain the spectrum of charged particle fluence ΦEð Þ x, y, z at the point of interest (x, y, and z) at a given irradiation field size, a convolution integration on energy is applied, as follows:

$$\Phi\_E(\mathbf{x}, y, z) = \int\_0^{E\_{\text{max}}} \left[ \int \Psi(\mathbf{x}', y', E) \phi(\mathbf{x} - \mathbf{x}', y - y', z, E) d\mathbf{x}' d\mathbf{y}' dE \right] \tag{21}$$

where Ψ x<sup>0</sup> , y<sup>0</sup> ð Þ , E is the lateral distribution photon energy fluence of the beam and ϕ x � x<sup>0</sup> , y � y<sup>0</sup> ð Þ , z, E is the fluence pencil kernels to position (x, y, z). Note also Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field… DOI: http://dx.doi.org/10.5772/intechopen.89150

that Ψ x<sup>0</sup> , y<sup>0</sup> ð Þ , E may vary depending on the dose deposition in the irradiation field, which is the case of IMRT.

## 3.5 Response model of SSD

In reality, the response behavior of any dosimeter deviates in two cavity theories, because the extreme conditions for both theories are never completely filled. To determine if the dosimeter cavity is large or small, the size of the cavity is compared with the range of the electrons. If the dosimeter is irradiated with a polyenergetic beam, the cavity of theories cannot be applied directly due to the varied response to particles of different energies. In 2004, Yin et al. [30] proposed a method to treat the primary and scattered components separately by different cavity theories, assuming that the primary particles satisfy conditions of SCT while the scattered particles satisfy the LCT. The total dose measured in the Si-diode dosimeter [30] considered the sum of primary and scattered contributions using Eq. (22):

$$D^{\rm cav} = \int\_{\Delta}^{E\_{\rm max}} \Phi\_{\rm cav}^p(E) \cdot \frac{L\_{\Delta}^{\rm cav}(E)}{\rho} dE + \Phi\_{\rm cav}^p(\Delta) \cdot \frac{L\_{\Delta}^{\rm cav}(\Delta)}{\rho} \cdot \Delta + \int\_0^{E\_{\rm max}} \Psi\_{\rm cav}^\epsilon(E) \cdot \frac{\mu\_{\rm em}^{\rm cav}(E)}{\rho} dE \tag{22}$$

where Φ<sup>p</sup> cavð Þ <sup>E</sup> is the primary electron fluence and <sup>Ψ</sup><sup>s</sup> cavð Þ E is the energy of the scattered photon fluence. <sup>L</sup>cav <sup>Δ</sup> ð Þ E <sup>ρ</sup> is the restricted stopping power, and <sup>μ</sup>cav en ð Þ E <sup>ρ</sup> is the mass absorption coefficient of material in the cavity. This model proved quite precisely the dose in water, verified by measurement [27]. Eklund and Ahnesjö [26] introduced some solution to calculate dose response through the two assumptions: (1) Ensuring all primary electrons satisfy the SCT condition if the detector size is rather small. Nevertheless, there are still low-energetic electrons in the spectrum that cannot pass through the cavity of the dosimeter. (2) The scattered photons are considered to satisfy the condition of LCT, indicating that CPE is assumed to be located in the cavity of the dosimeter. The validity of this assumption depends on the energy of the scattered photon. To solve this situation, Eklund and Ahnesjö [71] introduced two solutions for the condition of the hypothesis is closer to reality: Instead of calculating the primary low-energetic electron contribution by LCT, they calculate the contribution from primary photons in LCT that create low-energetic primary electrons. From this calculation of the fluence spectra, it is possible to find the low-energetic primary photons. Therefore, a partitioning of the primary electrons was performed, where the high-energetic primary electrons followed the SCT and the contribution of low-energy electron primary was calculated using their father or primary photons. Ideally, the scattered photons should be partitioned in the same way to treat low component of high energy differently. K(E) correction had been introduced, which is defined as follows:

$$K(E) = \frac{D\_p^{av}(E)}{\Psi(E) \cdot \frac{\rho\_m^{av}(E)}{\rho}} \tag{23}$$

where Dcav <sup>p</sup> ð Þ E is the dose deposited in the cavity of the detector, by the primary photons of energy E (or primary electrons) with a fluence of photons Ψð Þ E and its primary electrons. The denominator of this expression represents the collision kerma of the cavity, which is equivalent to the dose if CPE exists locally. Computing Dcav <sup>p</sup> ð Þ E is only possible by a total particle transport calculation as by Monte Carlo

Here r is the lateral distance of the axis of irradiation from the position of interest; z is the depth of the point of interest; E is the energy of monoenergetic ionization chamber beam. Also, these database separated particles into four catego-

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

ries, depending on their histories of interaction, as shown in Figure 5:

pair production.

Particles categories as defined in [70].

Figure 10.

ΦEð Þ¼ x, y, z

where Ψ x<sup>0</sup>

and ϕ x � x<sup>0</sup>

56

ðEmax 0

ðð Ψ x<sup>0</sup>

• The incident photons, without any interaction are primary photons.

• All other photons are scattered that are created by primary photon or

• The electrons created by the scattered photons are scattered electrons.

, y<sup>0</sup> ð Þ , E ϕ x � x<sup>0</sup>

, y � y<sup>0</sup> ð Þ , z, E dx<sup>0</sup>

, y<sup>0</sup> ð Þ , E is the lateral distribution photon energy fluence of the beam

, y � y<sup>0</sup> ð Þ , z, E is the fluence pencil kernels to position (x, y, z). Note also

dy<sup>0</sup>

dE (21)

field size, a convolution integration on energy is applied, as follows:

The first two types of particles (primary photons and electrons) are the main component of the beam, while the last two are the scattered component. This separation of particles is only possible with the feature of tracking the particle's interaction history in Monte Carlo simulation. To obtain the spectrum of charged particle fluence ΦEð Þ x, y, z at the point of interest (x, y, and z) at a given irradiation

• The electrons created in interactions of primary photons are primary electrons.

secondary electrons such as Rayleigh and Compton effect, bremsstrahlung, and

simulation. With the introduction of partitioning primary electron and the approximation, factor CPE K Eð Þ in Eq. (21) gives:

$$\begin{split} D^{\alpha w} &= \int\_{\Delta}^{E\_{\max}} \Phi^{[E\_{\underline{A}}, E\_{\max}]}\_{\alpha w}(E) \cdot \left( \frac{L^{cav}\_{\Delta}(E)}{\rho} \right) dE + \Phi^{[E\_{\underline{A}}, E\_{\max}], p}\_{cav}(\Delta) \cdot \left( \frac{L^{cav}\_{\Delta}(\Delta)}{\rho} \right) \cdot \Delta \\ &\quad + \int\_{0}^{E\_{\max}} K(E) \Big( \Psi^{[0, E\_{\underline{A}}], p}\_{cav}(E) + \Psi^{s}\_{cav}(E) \Big) \cdot \left( \frac{\mu^{cav}\_{\underline{m}}(E)}{\rho} \right) dE \end{split} \tag{24}$$

where <sup>Φ</sup>½ � EA,Emax <sup>p</sup> cav ð Þ <sup>E</sup> is the fluence of primary electrons produced by the photon with a higher energy EA and <sup>Ψ</sup>½ � 0,EA ,<sup>p</sup> cav ð Þ <sup>E</sup> is the fluence of primary photons with smaller energy EA. Applying Eq. (24), one can calculate the water dose, SSD dose, and response factor of Eq. (19). In order to compare the calculated response factor with the measured one, it is necessary to normalize the response factor determined for a reference, which gives:

$$RF\_{norm}^{\text{calc}}(A, r) = \frac{RF^{\text{calc}}(A, r)}{RF^{\text{calc}}(A\_{ref}, r\_{ref})} \tag{25}$$

In general case of large radiation beam, the value of total perturbation factor is approximately 0.99. However, for small photon beams, these perturbation factors become extremely large and no longer remain independent. Hence, Monte Carlo calculation of perturbation factors must be preferred over the use of SCT. Along with this, the size of the detector with respect to the source size and incorrect alignment can result in large values of correction factors. The perturbation caused by the displacement error in calculation of absorbed dose using Monte Carlo for PTW 60012 diode is illustrated in Figure 12. Similar results have been reported by various authors, for the Monte Carlo-based studies for computation of different

Uncertainty observed in absorbed dose determination using PTW 60012 diode, due to uniformly distributed

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field…

DOI: http://dx.doi.org/10.5772/intechopen.89150

displacement error of 1 mm in all directions perpendicular to the beam axis.

4. Energy spectrum and beam quality for small photon beams

The collimating devices utilized to project small photon beams result in blockage

of photon source and scattered component of photon beam generated from the interaction of primary photon beam with other components of linear accelerator head, as a consequence of which the low energy photons are removed from the central axis of the beam. However, there may be probable increase in the amount of secondary component of beam for off-axis fields. The material composition of flattening filter is a deciding factor about whether the radiation beam will be softened or hardened. Along with this, there is a decrement in phantom scatter in small beams in comparison to the large field sizes. However, the decrement in phantom scatter is more noticeable than head scatter. Both effects are responsible for making the photon spectrum hard along the central axis of the beam. As a result the mass energy coefficient ratio and stopping power ratio of water and material of the detector are changed. Also in the small beams and the absence of LCPE, the

perturbation factors [37, 72–77].

4.1 Energy spectrum

59

Figure 12.

The reference value of field size is square field of 10 cm � 10 cm, and the reference position from the axis is at a depth of 10 cm in the phantom.

Crop et al. [2] had conducted one of the most detailed studies on the response of air-filled detectors in small photon beams. Author's considered the effect of different perturbation effects: (a) perturbation caused by differences in the composition of detector with respect to water (pwall), (b) perturbation caused by replacement of water by detector (pa,w), (c) effect caused by the existence of central electrode of the air-filled detectors, and (d) volume averaging effect for two detectors with different volume. The results of the study are illustrated in Figure 11; it was a Monte Carlo-based study for 6 MV photon beam considering photon beams down to 0.8 cm � 0.8 cm. The maximum variation was reported for Pvol and Pa,w.

#### Figure 11.

Results reported by Crop et al. for different perturbation effects. Maximum deviation was reported for the volume averaging effect and perturbation caused by replacement of water by detector media.

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field… DOI: http://dx.doi.org/10.5772/intechopen.89150

#### Figure 12.

simulation. With the introduction of partitioning primary electron and the approx-

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

cavð Þ E

where <sup>Φ</sup>½ � EA,Emax <sup>p</sup> cav ð Þ <sup>E</sup> is the fluence of primary electrons produced by the photon

with a higher energy EA and <sup>Ψ</sup>½ � 0,EA ,<sup>p</sup> cav ð Þ <sup>E</sup> is the fluence of primary photons with smaller energy EA. Applying Eq. (24), one can calculate the water dose, SSD dose, and response factor of Eq. (19). In order to compare the calculated response factor with the measured one, it is necessary to normalize the response factor determined

The reference value of field size is square field of 10 cm � 10 cm, and the

Results reported by Crop et al. for different perturbation effects. Maximum deviation was reported for the

volume averaging effect and perturbation caused by replacement of water by detector media.

Crop et al. [2] had conducted one of the most detailed studies on the response of air-filled detectors in small photon beams. Author's considered the effect of different perturbation effects: (a) perturbation caused by differences in the composition of detector with respect to water (pwall), (b) perturbation caused by replacement of water by detector (pa,w), (c) effect caused by the existence of central electrode of the air-filled detectors, and (d) volume averaging effect for two detectors with different volume. The results of the study are illustrated in Figure 11; it was a Monte Carlo-based study for 6 MV photon beam considering photon beams down to 0.8 cm � 0.8 cm. The maximum variation was reported for Pvol and Pa,w.

dE <sup>þ</sup> <sup>Φ</sup>½ � EA,Emax ,<sup>p</sup> cav ð Þ� <sup>Δ</sup>

� <sup>μ</sup>cav en ð Þ E ρ � �

RFcalcð Þ <sup>A</sup>,<sup>r</sup> RFcalc Aref ,rref

Lcav <sup>Δ</sup> ð Þ Δ ρ � �

� � (25)

� Δ

dE (24)

<sup>Δ</sup> ð Þ E ρ � �

� �

imation, factor CPE K Eð Þ in Eq. (21) gives:

<sup>Φ</sup>½ � EA,Emax <sup>p</sup> cav ð Þ� <sup>E</sup> Lcav

K Eð Þ <sup>Ψ</sup>½ � 0,EA ,<sup>p</sup> cav ð Þþ <sup>E</sup> <sup>Ψ</sup><sup>s</sup>

RFcalc

normð Þ¼ A,r

reference position from the axis is at a depth of 10 cm in the phantom.

<sup>D</sup>cav <sup>¼</sup>

Figure 11.

58

ðEmax Δ

þ ðEmax 0

for a reference, which gives:

Uncertainty observed in absorbed dose determination using PTW 60012 diode, due to uniformly distributed displacement error of 1 mm in all directions perpendicular to the beam axis.

In general case of large radiation beam, the value of total perturbation factor is approximately 0.99. However, for small photon beams, these perturbation factors become extremely large and no longer remain independent. Hence, Monte Carlo calculation of perturbation factors must be preferred over the use of SCT. Along with this, the size of the detector with respect to the source size and incorrect alignment can result in large values of correction factors. The perturbation caused by the displacement error in calculation of absorbed dose using Monte Carlo for PTW 60012 diode is illustrated in Figure 12. Similar results have been reported by various authors, for the Monte Carlo-based studies for computation of different perturbation factors [37, 72–77].

### 4. Energy spectrum and beam quality for small photon beams

#### 4.1 Energy spectrum

The collimating devices utilized to project small photon beams result in blockage of photon source and scattered component of photon beam generated from the interaction of primary photon beam with other components of linear accelerator head, as a consequence of which the low energy photons are removed from the central axis of the beam. However, there may be probable increase in the amount of secondary component of beam for off-axis fields. The material composition of flattening filter is a deciding factor about whether the radiation beam will be softened or hardened. Along with this, there is a decrement in phantom scatter in small beams in comparison to the large field sizes. However, the decrement in phantom scatter is more noticeable than head scatter. Both effects are responsible for making the photon spectrum hard along the central axis of the beam. As a result the mass energy coefficient ratio and stopping power ratio of water and material of the detector are changed. Also in the small beams and the absence of LCPE, the

low-energy electrons reaching the axis of the beam will be reduced. Hence, the mean electron energy is increased, as a result of which stopping power ratio is also affected.

calibrate the detector in beam quality similar to that of user, and hence the values for all combination of machine/detector are not available. In that case the question arises, whether it is fine to use TPR20,10(10,10)x or %dd(10,10)x for dose measurements in fields smaller than 10 cm 10 cm on the same radiation emitter. As discussed above the stopping power ratio variation with change in field size is small. Hence, the beam quality indices measured for large photon beams can be utilized for small beams. The variation of stopping power ratios with beam size and other perturbation factors can be merged together into an output correction factor.

Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field…

In the conditions where the conventional beam size of 10 cm 10 cm, a number

a. A concept of machine-specific reference field (msr), fmsr, has been proposed

b. Machine-specific beam quality index has been proposed for TomoTherapy [82]. According to this methodology beam quality index is measured using similar methods as used for determination of the %dd(10, 10)x in the conditions achievable in case of TomoTherapy. In this proposed

methodology, authors calculated correction factors for beam quality with the help of Monte Carlo techniques and compared them with measurements as a function of conventional index to establish a relation between the machinespecific beam quality index and the conventional beam quality index [83].

c. In the third method, it is proposed to measure TPR20,10(S), the ratio of dose deposited at 20 and 10 g/cm<sup>2</sup> depths for an S cm S cm square field size at source-to-detector distance of 100 cm. The measurement of TPR20,10(S) is performed on the machine where 10 cm 10 cm field size is not achievable

for series of square field sizes S, and comparison is made with the measurements performed using radiation emitter where 10 cm 10 cm beam size is achievable; the measurement data is extrapolated [78, 79, 84]. Using this extrapolated data relationship for the beam quality index of 10 cm 10 cm beam size, TPR20,10(10) and TPR20,10(S) are derived [85].

It was observed by Sauer et al. that the third methodology to measure beam quality index in nonstandard field sizes is effective for circular or rectangular fields using the concept of equivalent square field method [79, 85] and for flattening filter-free beams by surety of correction factor for deficiency in the lateral scatter because of conical beam profiles. It must be noted that the relation between the stopping power ratio and beam quality index in case of FFF radiation fields and WFF beams is not similar [86–89]. The relations to calculate TPR20,10(10) and %dd (10,10)x can be derived for small beams (S lying between 4 and 12 cm) [90]. Figure 13 illustrated the variation of TPR20,10(S) for beam size of S cm S cm for field sizes ranging from 4 cm 4 cm to 12 cm 12 cm with energy of photon beam ranging from 4 and 10 MV [79, 90] (squares representing the measurement data

The determination of TPR20,10(10) from the measurements obtained for TPR20,10(S), where S is the equivalent square fmsr, by using the measurement data in the analytic expression given by Palmans (Eq. (26)) [90]. Figure 14 shows the experimental set-up to be used for measurement of TPR20,10(S), with source-to-

.

detector distance of 100 cm and at a depth of 20 g/cm<sup>2</sup> and 10 g/cm<sup>2</sup>

[79] and curved representing Monte Carlo results [90]).

4.3 Measurement of TPR20,10(10)

61

of methods have been proposed to determine the beam quality:

by Alfonso et al. [81].

DOI: http://dx.doi.org/10.5772/intechopen.89150

However, various Monte Carlo-based studies have revealed that the charged particle spectrum generated inside water is not much affected by the change in photon fluence. Hence, the stopping power ratio of water to air does not vary more than 0.5% at 10 cm of depth for 6 MV photon beam for field sizes ranging from 10 cm 10 cm to 0.3 cm 0.3 cm or circular fields of 0.3 cm diameter [78, 79], for depth ranging up to 30 cm maximum variation of 1% has been reported [70]. However, the response of diode detectors is affected by this hardening of the photon beam due to the noticeable change in the mass-energy absorption coefficient ratio of water and silicon. For field sizes ranging from 10 cm 10 cm to 0.5 cm 0.5 cm, the variation of 3–4% has been reported in the response of the unshielded diodes as a result of reduction in phantom scatter [71, 80].

## 4.2 Beam quality

For reference dosimetry in photon beams of high energy and large field sizes of beam quality Q using the air-filled detectors calibrated with respect to beam quality Q0, the radiation quality correction factor is required. There are two methods defined to account for beam quality [1, 3, 7]. First is the tissue phantom ratio at the depth of 20 and 10 g/cm<sup>2</sup> using water as a medium for 10 cm 10 cm beam size and source-to-detector distance (SDD) of 100 cm, TPR20,10(10,10)x [1]. The second method is based on percentage depth dose at a depth of 10 cm to 10 cm 10 cm beam size and source to surface distance of 100 cm, %dd(10,10)x. These beam quality indices are utilized to calculate .

For some calibration laboratories, it is possible to provide calibration of air-filled detectors using clinical linear accelerator photon beam from calibration laboratories. This methodology for calibration of measurement equipment is much more realistic as there are small variations on the absorbed dose to water calibration factor for radiation equipment of the same kind, as the quality of beam varies moderately between the modern equipment of the same type. Therefore, it is possible to use the same radiation beam quality correction factor for similar model of air-filled detectors and radiation emitting equipment of the same kind. Hence, the dosimetric measurements on such machines can be performed without correction for beam quality. This methodology has been applied at some level for Gamma Knife® (Elekta AB, Stockholm), Cyberknife®, and TomoTherapy® (Accuray Inc., Sunnyvale, CA) radiation generators. Also, the components of the clinical linear accelerators such as secondary jaws and multi-leaf collimators are employed for having better machine uniformity and accurate small-field size definition [25–27]. It is important to remember that by the above method of calibration of equipment, there is no requirement for beam quality correction factors, and even then beam quality indices are crucial from commissioning and quality check procedure perspectives. Since the nominal photon beam energies used for intensity-modulated radiation therapy (IMRT), volumetric modulated arc therapy (VMAT), and stereotactic methods are below 10 MV, and the variation of kQ,Q0 to quality of the beam is small [1, 2]. A large number of add-ons are utilized in IMRT and stereotactic radiotherapy treatment methods, which makes it impossible to prepare tables for beam quality correction factors for each and every combination of radiation emitters, add-ons, and detector types. Hence, kQ,Q0 is not available in all machine/ detector combinations. As a result the beam quality index or beam quality correction factor is required to relate the beam quality used for the detector calibration and the beam quality of the user machine. Since, it is sometimes not possible to

## Prospective Monte Carlo Simulation for Choosing High Efficient Detectors for Small-Field… DOI: http://dx.doi.org/10.5772/intechopen.89150

calibrate the detector in beam quality similar to that of user, and hence the values for all combination of machine/detector are not available. In that case the question arises, whether it is fine to use TPR20,10(10,10)x or %dd(10,10)x for dose measurements in fields smaller than 10 cm 10 cm on the same radiation emitter. As discussed above the stopping power ratio variation with change in field size is small. Hence, the beam quality indices measured for large photon beams can be utilized for small beams. The variation of stopping power ratios with beam size and other perturbation factors can be merged together into an output correction factor.

In the conditions where the conventional beam size of 10 cm 10 cm, a number of methods have been proposed to determine the beam quality:


It was observed by Sauer et al. that the third methodology to measure beam quality index in nonstandard field sizes is effective for circular or rectangular fields using the concept of equivalent square field method [79, 85] and for flattening filter-free beams by surety of correction factor for deficiency in the lateral scatter because of conical beam profiles. It must be noted that the relation between the stopping power ratio and beam quality index in case of FFF radiation fields and WFF beams is not similar [86–89]. The relations to calculate TPR20,10(10) and %dd (10,10)x can be derived for small beams (S lying between 4 and 12 cm) [90]. Figure 13 illustrated the variation of TPR20,10(S) for beam size of S cm S cm for field sizes ranging from 4 cm 4 cm to 12 cm 12 cm with energy of photon beam ranging from 4 and 10 MV [79, 90] (squares representing the measurement data [79] and curved representing Monte Carlo results [90]).
