**2.4 Category.4**

#### *2.4.1 Convolution-superposition algorithms*

The convolution-superposition algorithm is also a model-based algorithm and has two essential parts: 1) TERMA and 2) dose spread kernel. TERMA was first introduced by Ahnesjo et al. in 1987 [21] which is analogous to the Kerma, (the kinetic energy released in medium) and has the same unit as dose. The formula for the TERMA element (T) of the convolution method is given by the following equation

$$T(r') = \frac{\mu}{\rho} \left(\overrightarrow{r}', E\right). \Psi\left(\overrightarrow{r}'\right) \tag{7}$$

where μ/ρ is the mass attenuation coefficient and Ψ is the primary energy fluence. Then the convolution-superposition is the integration of the TERMA distribution times EDK over the entire volume. EDK is spatially variant and is deformed based on the local density environment to consider interface effects in regions of different densities. Also, to get a more accurate model of the scattering conditions, the kernels must be adjusted according to their direction and orientation at the site of interaction [22].

This method is widely used in TPS because computers are fast enough to do 3D dose calculations by using electron density data derived from CT images in a reasonable amount of time. According to AAPM report 85 (TG-65), the dose calculation accuracy of TPS algorithms should be within 2%. This goal serves as a useful benchmark to evaluate the capabilities of treatment planning algorithms to calculate the dose.

## *2.4.2 Anisotropic analytical algorithm*

Anisotropic Analytical Algorithm (AAA) (Varian Medical System, Inc) is a kernelbased convolution-superposition method. This algorithm was first designed by Ulmer

#### *Parameters Affecting Pre-Treatment Dosimetry Verification DOI: http://dx.doi.org/10.5772/intechopen.102517*

and Kaissl (2005) [23] in cylindrical coordinate and then improved by Tillikainen in 2008 [24]. The AAA dose calculation model has two main components, the configuration algorithm, and the actual dose calculation algorithm. Its configuration is based on the Monte Carlo simulations to determine basic physical parameters and match them with measured clinical beam data. The dose calculation algorithm utilizes separate models for primary photons, scattered extra-focal photons, and contamination electrons. The lateral distribution is adjusted according to the radiological distance to the calculation point for tissue heterogeneities corrections [16, 25]. For the most part, AAA is a pencil beam convolution-superposition algorithm where the pencil beam is compiled from Monte Carlo calculations and adjusted to fit measurements. In this case, two components need to be considered that contribute to final distributions; 1) longitudinal contribution of the pencil beam which is scaled according to Equivalent Length Path (EPL), and 2) contribution from the lateral extension of the pencil beam which is scaled with the density relative to water in directions normal to the pencil beam [26]. In this way, the changes in lateral transport of energy are modeled when the density varies in the irradiated object. Therefore, unlike the pencil beam algorithms, it can consider inhomogeneity correction on both longitudinal and lateral directions. However, many studies indicate the inability of AAA to accurately calculate doses at interfaces and for high atomic number materials such as bone and have shown that the deviation between AAA and measurements exceeds the goal of TG-65 [27–30].

The advantage of the AAA is its relatively short calculation time and its accuracy is better than the pencil beam convolution (PBC) model [30–32].

#### *2.4.3 Collapsed cone convolution*

In 1989 Ahnesjo [33] proposed collapsed cone convolution (CCC) method. The CCC algorithm uses the analytical kernel in polar coordinates represented by a set of cones. In this way, it is assumed that all energy is released into coaxial cones of equal solid angle and, from volume elements on the cone axis is approximated to be rectilinearly transported, attenuated, and deposited in volume elements on that axis [7]. The polyenergetic kernels can be described by

$$h(r, \theta) = \frac{A\_{\theta}e^{-a\_{\theta}r} + B\_{\theta}e^{-b\_{\theta}r}}{r^2} \tag{8}$$

where *Aθ*, *αθ*, *Bθ*, and *b<sup>θ</sup>* are fitting parameters depending on the scattering angle *θ* and *r* is radial distance. The first term mainly describes the primary dose and the second term is the scatter dose fraction.

The advantage of the CCC algorithm over standard convolution algorithms is that it can reduce the computation resources. The computation time for the CCC method in heterogenous media is proportional to MN3 where M is the number of cones and N is the number of voxels along one side of the calculation volume [16]. Different TPSs use the CCC algorithm such as Pinnacle (Philips Inc., Amsterdam, Netherlands), Oncentra MasterPlan (Nucletron, Inc., Columbia, MD, USA), CMS XiO (Elekta AB, Stockholm, Sweden), RayStation (RaySearch Laboratories AB, Stockholm, Sweden), etc.

#### *2.4.4 Monte Carlo*

Monte Carlo (MC) is a principle-based algorithm that almost includes all known physical features for photon interactions inside the patient body. Many MC codes

have been developed such as BEAMnrc, GEANT4, MNCP, PENELOPE, and XVMC. All of them have two main steps, first, modeling the linac head with all precise details of the target, component dimensions, geometry, locations, and material composition. The second step uses CT data to get morphological and chemical information in terms of mass density, electron density, and atomic composition, which are all required for accurate dose calculation in the tissue.

The MC has the capability of simulating all interactions, therefore it is expected to be accurate. However, its accuracy depends on correct and detailed geometry information of the linac head and the number of particle histories. This statistical uncertainty is proportional to the inverse square root of the generated event numbers [34, 35]. MC dose calculation is slow and time-consuming, so they are not yet applicable in clinics because the dose may recompute repeatedly during planning to get an optimized plan. A few vendors offer Monte Carlo methods in TPS as calculation options for the final dose calculation once the dose optimization is completed.

#### *2.4.5 Acuros XB*

Monte-Carlo (MC) dose calculation algorithm is widely considered as the golden dose calculation technique in radiation therapy; however, the calculation time of this method is still long especially where a greater number of particle histories should be used to reduce statistical noise and*/*or a high spatial resolution is required. An alternative method to MC is the linear Boltzmann transport equation (LBTE) method which solves LBTE refers to grid-based Boltzmann solver (GBBS). GBBS solves the LBTE through discretizing photon and electron fluences in space, energy, and angle to allow a deterministic solution of the transport of radiation through matter. Its calculation accuracy is comparable to MC, and both are convergent methods because the MC algorithm simulates an infinite number of particles, GBBS discretizes the LBTE variables into infinitely small grids, then the two methods should converge to the real solution. However, MC and GBBS have different sources of error, there is statistical noise due to simulating a finite number of particles in Monte Carlo, while most errors in GBBS methods are systematic and their main source is discretization of the solution variables in space, angle, and energy [36, 37]. An algorithm using this technique is based on Attila (Los Alamos National Laboratory, Los Alamos, NM, and Transpire Inc., Gig Harbor, WA). Attila employs linear discontinuous finite-element spatial differencing on a computational mesh consisting of arbitrary tetrahedral elements. The primary photon fluence is analytically transported through ray tracing, and the discrete ordinates method is used for angular differencing of the scattered fluence. Based on Attila, a dose calculation algorithm for external photon beams has been developed on the same methods and implemented in the Varian Eclipse external beam treatment planning system (Varian Medical Systems, Palo Alto, CA, USA) [38]. This new deterministic radiation transport algorithm is Acuros XB (AXB), and it has been well shown by several studies that the accuracy of dose calculation of AXB is more accurate than AAA and is very similar to MC dose calculations [36–38].

### **3. Beam Modeling of commercial treatment planning systems**

In radiotherapy, the ability of TPS to do accurate dose calculation is important. This capability depends on the algorithm of TPS as discussed before and beam

#### *Parameters Affecting Pre-Treatment Dosimetry Verification DOI: http://dx.doi.org/10.5772/intechopen.102517*

modeling. For beam modeling, several dosimetric parameters (e.g., PDDs, profiles, output factors) and non-dosimetric parameters such as MLC design, flattening filter, wedges etc. must be defined precisely. Then the dose calculation algorithm applies the beam model to the patient body or phantom to calculate the dose. The challenge of the beam model is becoming more and more crucial due to advanced treatment techniques such as IMRT and VMAT. In these treatment techniques, each beam consists of multiple segments or control points that are shaped with MLC. Using multiple control points provides this opportunity to deliver conformal dose to the target, however, delivering dose through small segments arises a challenge to accurately calculate the dose due to the complexity of MLC modeling in TPS. Many studies indicate the importance of accurate MLC modeling in TPS for IMRT. In 1998, LoSasso et al. [39] showed an MLC error gap of 1 mm may result 10% error in dose calculation in the sliding window IMRT technique. Cadman et al. [40] reported 12% discrepancy between calculation and measurement due to MLC leaf gap error in step-and-shoot IMRT. Because different commercial TPS have their own features for beam modeling, many guidelines have been published regarding TPS commissioning for IMRT [41, 42]. For example, TG-119 [43] based on the IMRT QA results of five institutions for a set of test cases provides a reference baseline for the accuracy of IMRT commissioning.

In Eclipse, leaf transmission factors and dosimetric leaf gaps (DLGs) are required to model the MLC. The DLG is a beam configuration parameter used to model the effects of rounded MLC leaf ends. Many research papers indicate the effects of DLG on the accuracy of dose calculation in Eclipse TPS [44–47].

In RayStation, modeling of MLC is different from other commercial TPS. The MLC model requires four parameters: leaf-tip offset, leaf-tip width, average transmission factor, and tongue and groove. The leaf-tip width is used for the MLC leaf-end transmission modeling instead of using dosimetric leaf gap (DLG) or rounded leaf-tip radius, and the MLC leaf radiation transmission is modeled using average transmission factor instead of intra-leaf and inter-leaf transmission [48, 49]. According to Chen et al. tongue-and-groove has a minimal effect on IMRT dose calculation, but transmission plays a significant role in this commercial TPS [49].

### **4. Measurement methods for pre-treatment verification**

The process of patient-specific QA usually involves applying an optimized plan using the same beam parameters as those of the patient plan and delivered in the phantom. This process can be done in a number of different ways but according to TG-218 [50], there are three common methods for performing pre-treatment QA. 1) True Composite (TC), 2) Perpendicular field-by-field (PFF), and 3) Perpendicular composite (PC).

#### **4.1 True composite**

In this method, phantom or measurement device is placed on the treatment couch and treatment plan is delivered using actual parameters such as MUs, couch, gantry, collimator angles, MLCs, and jaws positions. The phantom or measurement device has been used to integrate dose from all beams of a plan which result in a single dose image for comparison, therefore, this method is a comparison of planned dose vs. measured dose.

### **4.2 Perpendicular field-by-field**

The gantry is fixed at zero degree and the collimator is fixed at the nominal angle in the PFF technique. Therefore, beams are always perpendicular to the phantom surface and are comparing the dose of each beam with each measured beam dose.

#### **4.3 Perpendicular composite**

This method is similar to the PFF method, but this is not a comparison of field-byfield. This is the integration dose of all perpendicular field which result in one dose image for analysis.

### **5. Gamma index**

For the purpose of dose comparisons between calculated and measured dose gamma index have been used. Low et al. [51] developed a gamma index (γ) for the quantitative evaluation of dose distributions. This index checks dose difference and distance-to-agreement (DTA) simultaneously in a space that also includes dose, and provides quantitative value which indicates disagreement in the regions that fail the acceptance criteria. A γ comparison is performed between two dose maps: one distribution is the 'reference dose distribution' and the other is the 'evaluated dose distribution'. The *reference dose distribution* is referred to as true distribution so it is usually measured data using devices such as ion chamber, film, diode array detector etc., and the *evaluated dose distribution* is analyzed for its agreement with the reference and can be the predicted TPS dose distribution. To avoid any confusion, low replaced reference and evaluated terms by measured and calculated respectively. The gamma index calculation is based on Eq. (9):

$$\Gamma(r\_{\mathcal{R}}, r\_{\mathcal{E}}) = \sqrt{\frac{\Delta r^2(r\_{\mathcal{R}}, r\_{\mathcal{E}})}{\delta r^2} + \frac{\Delta D^2(r\_{\mathcal{R}}, r\_{\mathcal{E}})}{\delta D^2}}\tag{9}$$

where *rR* and *rE* are reference points and evaluated point respectively, *δr* is distance difference criterion and *δD* is the dose difference criterion. *ΔD* is dose difference which is calculated using Eq. (10):

$$
\Delta D(r\_R, r\_E) = D\_E(r\_E) - D\_R(r\_R) \tag{10}
$$

*DE* and *DR* are the doses at a point in evaluated dose distribution and reference dose distribution respectively.

The γ is the minimum value calculated overall evaluated points:

$$\gamma(r\_{\mathbb{R}}) = \min \left\{ \Gamma(r\_{\mathbb{R}}, r\_{\mathbb{E}}) \right\} \forall \{ r\_{\mathbb{E}} \} \tag{11}$$

Regions where γ is less than or equal to 1 corresponds to locations where the calculation meets the acceptance criteria. According to TG-218, criteria for tolerance limit is 2 mm/3% with 95% passing rate [50].

There are two types of gamma calculation which depends on how the percent dose difference (%Diff) is normalized: 1) local normalization method which %Diff is

normalized to the doses at each evaluated point, 2) global normalization method which %Diff is normalized usually to the maximum dose within the reference dose distribution. Each method has its own advantages and disadvantages. For example, local gamma will exaggerate %Diff and highlighted failures in low dose regions because in low dose regions the percent dose difference between calculated and measured may exhibit a very large value which results in more failings points. However, in the global method, the dose discrepancies in the low-dose regions could be underestimated which results in a higher passing rate than the local method [52, 53].

#### **5.1 Effect of planned grid size on gamma passing rate**

Low et al. [51] presented a powerful tool for dose distribution comparisons in a continuous environment; however, clinical comparisons are usually made between two dose distributions which are sampled at different spatial resolutions. The importance of spatial resolution was first analyzed by Depuydt *et al* in 2001 [54]. They indicated that the pixel size of the compared image needs to be small with respect to acceptance criteria and showed that large grid spacing in the discrete dose distribution, especially in high dose gradient regions causes overestimation of gamma values. Several investigators introduced different solutions to resolve this issue [54–56]. For example, Low and Dempsey [57] showed that by decreasing grid size to 1 1 mm2 , the error in γ reduced to less than 0.2 even in high dose gradient areas. Furthermore, Schreiner et al. [58] reported changing the resolution of the evaluated distribution (from 2.5 mm to 0.24 mm) increase the pass rate from 80.9% to 91.3%. These results are attributed to the behavior of gamma search. When the pixel size of the evaluated distribution is large compared to the reference distribution, many reference pixels would be far away from the nearest evaluated pixel which results in more failing points. Thus, the γ value for many reference pixels reflects significant spatial misalignment purely as an artifact of the coarse evaluated resolution. When the resolution of the evaluated distribution is increased to match that of the reference distribution, this spatial artifact is eliminated because each reference point has a directly corresponding pixel in the evaluated distribution. Increasing the evaluated resolution also provides each reference point with a greater range of dose values for comparison. Based on TG-218 [50], there is a rule of thumb that the resolution of the evaluated (calculated) should be no greater than 1/3 of the DTA and the straightforward solution for reducing artifact in gamma calculation is interpolation when planned grid size is greater than 1 mm (for DTA =3 mm).
