15 Dose Calculation Algorithms

Transcription

1 Dose Calculation Algorithms Dose Calculation Algorithms Uwe Oelfke and Christian Scholz CONTENTS 15.1 Introduction Model-Based Algorithms Modeling of the Primary Photon Fluence Dose Calculation in Homogeneous Media TERMA Dose Kernels: Point-Spread Kernel and Pencil Beam Superposition and Convolution Algorithms Accounting for Tissue Inhomogeneities Pencil Beam and Pathlength Scaling Density Scaling of Point-Spread Kernels Collapsed Cone and Kernel Tilting A Simple Example: Doses at Cork Water Interfaces Dose Calculations and IMRT Optimization The Systematic Error The Convergence Error Conclusion 195 References Introduction The accurate and fast calculation of a 3D dose distribution within the patient is one of the most central procedures in modern radiation oncology. It creates the only reliable and verifiable link between the chosen treatment parameters, and the observed clinical outcome for a specified treatment technique, i.e., the prescribed dose level for the tumor, the number of therapeutic beams, their angles of incidence, and a set of intensity amplitudes obtained by a careful treatment plan optimization result in a distribution of absorbed dose which is the primary physical quantity available for an analysis of the achieved clinical U. Oelfke, PhD Deutsches Krebsforschungszentrum, Im Neuenheimer Feld 280, Heidelberg, Germany C. Scholz, PhD Siemens Medical Solutions, Oncology Care Systems, Hans- Bunte-Str. 19, Heidelberg, Germany effects of this specific treatment. The twofold application of dose calculation algorithms in radiation oncology practice, firstly for the plan optimization in the treatment planning process, and secondly for the retrospective analysis of the correlation between treatment parameters and clinical outcome, defines two mutually conflicting goals of the respective dose algorithms. Firstly, the dose calculation has to be fast such that the treatment planning process can be completed in clinically acceptable time frames, and secondly, the result of the dose calculation has to be sufficiently accurate so that the establishment of correlations between delivered dose and clinical effects remains reliable and meaningful. The conflict between high speed and high accuracy is one of the crucial challenges for the development of modern dose calculation algorithms. The accuracy of the dose calculation algorithms becomes a problem only for very heterogeneous tissues, where a very detailed modeling of the energy transport in the patient is required. The prediction of the dose around air cavities, e.g., often encountered for tumors adjacent to the paranasal sinuses or for solid tumors embedded in lung tissue, is intricate and time-consuming. Almost all new developments related to dose algorithms are specifically concentrating on these or equivalent areas of tissue heterogeneities, whereas for the majority of clinical cases with almost homogeneous tissues existing, simple calculation methods can be reliably applied. Naturally, dose algorithms for high-energy photon beams were first developed for the ultimate homogeneous patient a patient completely consisting of water (Schoknecht 1967). Measurements of a set of generic dose functions, e.g., tissue air ratios, tissue phantom ratios, output factors, and off-axis ratios, are measured in a water phantom for a set of regular treatment fields under reference conditions. The dose within a patient is then calculated by extrapolating these measurements to the specific chosen treatment fields and by the application of various correction algorithms, e.g., for the inclusion of missing tissues at the patient surface or the approximate consideration

2 188 U. Oelfke and C. Scholz of tissue heterogeneities. These phenomenological correction-based methods, which rely almost completely on a set of measurements, are very fast and have the further advantage of not needing to distinguish between the radiation field provided by the linear accelerator often labeled as the phase space of the respective linac and the subsequent energy transport by photons and electrons in the patient (see Fig. 15.1). Fig Typical patient setup at a clinical linear accelerator. The two main tasks of dose calculation in radiation therapy are shown here exemplarily by this clinical patient setup at a linear accelerator. Firstly, the properties of the particles emerging from the treatment head have to be modeled, and secondly, the energy transport through the patient has to be accounted for properly However, if one wants to account directly for the underlying physical processes responsible for the energy deposition within the patient, one has to introduce models which describe explicitly some aspects of the related energy transport. This is usually accomplished by the introduction of dose kernels, which describe in different levels the energy transport and deposition in water caused by a defined set of primary photon tissue interactions. For the application to inhomogeneous patient geometries these dose kernels are scaled in size according to the encountered local tissue densities. These model-based algorithms give a more realistic description of the absorbed dose in heterogeneous media simply because the patient anatomy, represented by the Hounsfield numbers of the patient CT, is sampled on a finer spatial grid. Besides the expected extended calculation times in comparison with the correction-based methods, the achievable higher spatial resolution of the absorbed energy in the patient requires a more accurate description of the radiation field provided by the linear accelerator, i.e., for model-based algorithms one generally has to invent an additional model for the radiation field emerging from the radiation source. Model-based algorithms in their various implementations constitute the standard algorithms provided by currently commercially available treatment planning systems. The simplest form, the so-called pencil-beam algorithm, is still the standard and fastest dose engine (Hogstrom et al. 1981; Schoknecht and Khatib 1982; Mackie et al. 1985; Mohan et al. 1986; Bortfeld et al. 1993). More sophisticated and accurate are the superposition algorithms (Mackie et al. 1985; Mohan et al. 1986; Ahnesjö et al. 1987; Mackie et al. 1988; Ahnesjö 1989; Scholz et al. 2003b), which are also becoming more widespread; however, as mentioned above, model-based algorithms still rely on approximations and only partly describe the physical processes involved in the microscopic absorption of the energy delivered by the radiation field. The most sophisticated approach to include almost all known physical features about the microscopic radiation tissue interactions is the Monte Carlo approach. Usually, a Monte Carlo dose calculation consists of two independent components: (a) the simulation of the already mentioned linac phase space based on the geometrical design of the treatment head for the considered machine and a few characteristic parameters for the electron beam before it impinges on the bremsstrahlungstarget of the linac; the respective radiation field serves as input for (b) the simulation of the energy absorption and transport within the patient; however, the implementation of these sophisticated algorithms and their verification is non-trivial and often relies on expert knowledge. A more detailed description of the involved principles is provided in section In the following section we focus on the description of the most prominent dose calculation models implemented in commercially available treatment planning systems, i.e., the concepts of pencil-beam and superposition algorithms is presented. Furthermore, the application of these models for different phantoms and clinical cases, especially for the optimization of IMRT cases with significant tissue inhomogeneities, is discussed Model-Based Algorithms In this section we briefly review the most prominent dose calculation concepts which include some explicit models for the relevant energy transport in the patient. The absorption of energy in the patient, i.e., the accumulation of dose, for these methods is sepa-

3 Dose Calculation Algorithms 189 rated into several steps. Before the energy absorption process is considered itself, one has to model the radiation output of the considered treatment machine. This is often accomplished by a simple model for the primary energy fluence of the photons emerging from the linear accelerator. The respective models are calibrated to measured dose data for simple treatment fields in water and are therefore not independent of the models employed for the actual energy absorption physics. The determined energy fluence of the primary photons serves as input for the subsequent calculation of the energy absorption and transport within the patient. Firstly, the absorption of the primary photons is considered and expressed by the total energy released per unit mass (TERMA; Ahnesjö et al. 1987). Then, the transport of this energy via secondary electrons and photons is accounted for by the introduction of specific dose kernels. The three mentioned components primary photon fluence, TERMA, and dose kernels are discussed below and are then employed in various forms for dose calculations in homogeneous and inhomogeneous media Modeling of the Primary Photon Fluence The radiation field delivered by a linear accelerator is a very complex mixture of primary photons, scattered photons, and electrons, of which the following physical properties have to be known in principle for an accurate dose calculation: the particle s energy spectrum, their spectrum of velocity directions, and their lateral distribution or fluence (number of particles per area) in a defined plane perpendicular to the central beam axis. The simplest approximation of this phase space of the linear accelerator (see Fig. 15.1) is provided by modeling an effective fluence of primary photons, based on a few phenomenological parameters, which are calibrated to simple measurements. A common assumption of the respective models is that the energy spectrum of the primary photons factorizes with the remaining phase space, i.e., the energy spectrum is independent of the lateral location of the photons with respect to the beam axis. There are two basic methods how these energy spectra are derived for clinical practice. Both of them require to different extent a calibration to measured dose data in water. Firstly, there is the approach to calculate the whole phase space of the emerging primary photon beam via Monte Carlo simulations (see Chap. 16; Mohan and Chui 1985; Rogers et al. 1995), i.e., the process of creating photons from a narrow electron beam impinging on the bremsstrahlung target of the linear accelerator and its subsequent attenuation and scattering in the treatment head is modeled from first physical principles. Provided that the geometrical location and physical properties of all relevant components in the treatment head are adequately known, these calculations depend only on a few parameters characterizing the initial electron beam of the linac, e.g., the average energy of the electrons, the variance of the energy spectrum, and the angular divergence of the electron beam. The complete phase space of the primary particles can then be used to derive an average energy spectrum of the respective photons and electrons. More details about these procedures are given in Chap. 16. A more practical-motivated method relies on a direct comparison of measured depth dose data and an energy weighted sum of pre-calculated depth dose curves (Scholz et al. 2003b; Ahnesjö and Aspradakis 1999). Each energy bin E i of the actual energy spectrum is assumed to contribute a depth dose D(E i,z) to the measured values D Spectrum (z) of a simple treatment field. The elementary, mono-energetic depth dose curves D(E i,z) are, for instance, also derived by Monte Carlo simulations. Mostly, a simple least-squares analysis is employed to derive weighting factors i such that D Spectrum (z)= i i D(E i,z), i.e., an energy spectrum is generated that reproduces the measured data (Scholz et al. 2003b; Altschuler et al. 1992). Admittedly, this phenomenological procedure only selects one possible energy spectrum of the photon beam which is compatible with the given experimental depth dose data. Moreover, it only accounts implicitly for any electron contamination of the photon beam. As an example, we show an effective energy spectrum for the primary photons delivered by a Siemens Primus in Fig In addition to the energy spectrum, the spatial distribution of the primary fluence also has to be modeled for the application in model-based algorithms. At least the following effects should be accounted for by the overall algorithm: (a) the spatial shape of the primary fluence distribution including so-called beam horns; (b) the broadening of the lateral penumbra through a spatially extended photon source; (c) the collimator scattering as described by the collimator scatter factor (CSF); and (d) the attenuation of the fluence inside the medium. As starting point for the modeling of the primary fluence one often uses the measured fluence in air for the largest aperture of interest (see Fig. 15.3). For the respective modification of this initial fluence, required by the aspects (b) (d),

4 190 U. Oelfke and C. Scholz all applications in radiotherapy. The energy fluence of the primary photons is to a first approximation determined by the linear photon absorption coefficient ( ) in water, i.e., at the interaction point of the primary photons r r one gets: r r Ψ( r) = Φ(r, ) E e - z 0 µ (E) Fig Energy spectra to generate multi-energetic kernels for 6- and 15-MV photon beams. Energy spectra of a Siemens Primus linear accelerator obtained by a phenomenological fit of depth dose curves (Scholz et al. 2003b) are shown. where r r denotes the coordinates perpendicular to the beam direction. The rate of the primary photon interactions in the medium determines the TERMA T( r r), i.e., the total energy per unit mass released by a radiation field interacting with a medium of density at a certain point r r: r µ r r Tr) ( = ( r) Ψ(r) ρ This locally released energy of the radiation field is subsequently available for a further transport emerging from the interaction point r r, which is usually described by the concept of dose kernels Dose Kernels: Point-Spread Kernel and Pencil Beam Fig Primary fluence matrix. In this figure a 2D primary fluence matrix obtained by 1D measurements along the diagonals of square treatment fields in air is shown. It is normalized to 1.0 at the origin. The increase of fluence with the distance to the center is induced by the shape of the flattening filter inside the treatment head, which improves the dose profile at the isocenter plane. different technical procedures can be applied, such as those described, for instance, by Scholz et al. (2003b) Dose Calculation in Homogeneous Media TERMA We first consider the dose deposition of a mono-energetic, infinitely narrow photon beam in z-direction of energy E and initial fluence in a homogeneous medium, which is naturally chosen to be water for It is common to use two elementary dose kernels for model-based algorithms (Mohan et al. 1986; Bortfeld et al. 1993; Mackie et al. 1988; Ahnesjö 1989). The most elementary kernel k( r r, r r', E), the so-called point-spread kernel, indicates the distribution of absorbed energy in water at the coordinate r r which is created by interactions of primary photons of energy E at the coordinate r r' (see Fig. 15.4). The elemental mono-energetic dose deposition kernels can be taken from Monte Carlo simulations, e.g., from Mackie et al. (1988). The second class of dose kernels and most commonly used in current treatment planning systems is the pencil beam. A pencil-beam kernel is obtained through the integration of all point-spread kernels along an infinite ray of photons in the medium as indicated in Fig It is evident that the pencil-beam kernel uses the more condensed information about the dose in water along the central kernel axis, i.e., it provides a coarser sampling of the physical processes than the point-spread kernel and it is therefore harder to adapt the dose calculations based on pencil-beam kernels to regions with intricate tissue inhomogeneities. On the other hand, pencil-beam kernels bear the obvious advantage of reduced dose calculation times. The first pencil-beam-type dose calcula-

5 Dose Calculation Algorithms Superposition and Convolution Algorithms Fig Point-spread kernel. This figure indicates the dose distribution at the points r r deposited by primary photons that interact at the r r'. In clinical mega-voltage photon beams the macroscopic extent of this dose kernel is given mainly by the range of secondary electrons, typically several centimeters Fig Point kernels and pencil-beam kernels. In this figure on the left a point kernel or so-called point-spread function is pictured. It describes the energy spread around a single interaction point of primary unscattered photons. By the integration of this kernel inside the patient along an infinite thin ray a so-called pencil-beam kernel is derived, which is displayed on the right-hand side Finally, the two components of TERMA and dose kernels are combined to achieve the accurate calculation of absorbed dose. The most general approach of model-based dose calculation algorithms is the superposition method (Mackie et al. 1985; Ahnesjö 1989; Scholz et al. 2003b). It generates the dose delivered at a point r r by superimposing the dose contributions from all dose kernels k( r r, r r', E) of the defined energy spectrum originating from all primary interaction points r r' and weighs their contributions with the respective TERMA, i.e., r 3 r r r D(r) = de' d r' T( r', E') k(r, r',e'). The superposition approach certainly is a too sophisticated method to be applied for a dose calculation in homogenous media, but it can be used very well for dose calculations in regions of interest with tissue inhomogeneities. A reduction of the computational effort required for the superposition approach, is achieved, if one assumes that the shape of the dose kernel is translational invariant. Then, the kernel k( r r, r r', E) is only a function of the distance between the interaction r r' point and the coordinate r r where the dose is measured such that the superposition formula reduces to the well known convolution approach, i.e., r 3 r r r D(r) = de' d r' T r', E') k( r r',e'). ( The calculation times of the convolution algorithm are reduced dramatically compared with the more accurate superposition methods. The application of a pencil-beam kernel, usually employed with convolution algorithms, leads to a further substantial reduction of calculation times, e.g., with a simple single value decomposition of the kernel the dose calculation can be reduced to a few 2D convolutions (Bortfeld et al. 1993), so that a 3D dose calculation for a conventional treatment plan can be accomplished in seconds. tion was introduced by Mohan et al. (1986). Pencilbeam kernels can be either created again with Monte Carlo simulations or can be directly derived from a few standard measurements, e.g., output factors of regular fields and tissue phantom ratios. The technical details of this approach are given by Bortfeld et al. (1993) Accounting for Tissue Inhomogeneities In order to calculate the dose in regions with tissue inhomogeneities, the dose calculation has to account for the variations of electron densities derived from

6 192 U. Oelfke and C. Scholz CT scans. The electron densities influence the dose calculation in two aspects: 1. The local TERMA depends on the path of the primary photons through the patient to their interaction point. 2. The energy distribution around the primary interaction point described by the dose kernels is also influenced by variations of the respective electron densities. The accurate calculation of the TERMA is accomplished by ray tracing the pathway of a photon to its interaction point. For this process the absorption rate of the photons along a considered trajectory is scaled by the ratio of the electron densities of the encountered media to the electron density of water. More important and more difficult to deal with is the influence of electron density variations on the dose kernels. This problem and some related practical aspects are briefly discussed below. the energy transport through the kernel from the interaction point r r' to the dose point r r occours along a straight line, i.e., within the kernel one also introduces an internal ray tracing. Along each internal ray the contribution of the kernel is scaled with the average electron density encountered along the line connecting r r' and r r. This leads to the so-called density-scaled dose deposition kernels. As an example we show in Fig a hypothetical dose kernel in water together with a density-scaled kernel including some tissue inhomogeneities. It can be clearly seen that the kernel extends further from the interaction point if the internal energy transport encounters a medium of an electron smaller than water on its way to the dose deposition point. The dimensions of the scaled dose kernel are shrinking in comparison with the original dose kernel in water if higher electron densities represent an additional obstacle for the energy transport within the dose kernel Pencil Beam and Pathlength Scaling The pencil beam is a dose kernel describing the 3D dose distribution of an infinitely narrow mono-energetic photon beam in water. The individual interaction points of the photons are all assumed to be on the central axis of the pencil beam. Here tissue inhomogeneities are accounted for by the same pathlength scaling with relative electron densities between tissue and water as applied for the calculation of the TERMA. The values of the pencil-beam kernel in water are used according to the radiological pathlength calculated along the central axis of the pencil beam. Electron variations perpendicular to the pencil-beam axis are not accounted for, i.e., this inhomogeneity correction assumes a slab geometry of tissue inhomogeneities, which are represented by the values of the electron densities along the pencil-beam axis. 2 > < 0 Fig Isodose curves of density-scaled kernels. Here the density-scaling of the Superposition kernel is displayed. The range of the energy spread behind tissue inhomogeneities is adapted to the radiological distance between the interaction point at r r' and the dose point r r. This leads to a deformation of the isodose lines behind inhomogeneities as shown in the right part of the figure Density Scaling of Point-Spread Kernels For the superposition algorithm the most accurate sampling of the primary interaction points within the patient is required. Consequently, this method also applies the most accurate technique of inhomogeneity corrections for the dose kernel. According to O Connor s theorem, Mohan et al (1986) devised a density scaling method which is applied directly to the point-spread kernel. Basically, it is assumed that Collapsed Cone and Kernel Tilting The problem of the superposition approach are its long computation times, which may still take hours for a complicated IMRT case even if performed with state-of-the-art hardware technology (Schulze 1995); therefore, various approximations have been suggested for accelerating the dose calculation with the superposition technique. One method introduced by Ahnesjö (1989) is the collapsed cone-beam tech-

7 Dose Calculation Algorithms 193 nique which refers to a specific internal sampling of the dose kernels. Furthermore, it seems to be important that the axis of the point-spread kernels be aligned with the original photon rays employed for the determination of the TERMA within the patient (Ahnesjö 1989; Scholz et al. 2003b; Sharpe and Battista 1993). The neglect of the required kernel tilting saves considerable calculation time; however, it also can introduce significant dose errors (Sharpe and Battista 1993; Liu et al. 1997). Another effect which might have to be considered is the hardening of the photon-energy spectrum. Works of Liu et al. (1997) and Metcalfe et al. (1990) showed that these effects are usually minimal in routine clinical practice A Simple Example: Doses at Cork Water Interfaces In order to demonstrate the sensitivity of the various types of dose algorithms to inhomogeneous media, we show the results of calculations performed for a simple phantom geometry. A 4-cm slab of cork was placed at a depth of 6 cm into a water phantom. The irradiation geometry of a 6-MV photon field was fixed as a 3 3 cm 2 open field with a source-to-skin distance SSD=95 cm. This field size was chosen since the measurements could still be performed reliably and because a significant effect on the dose patterns was anticipated. The 3D dose distributions for this simple geometry were calculated with three different algorithms: a pencil beam, a superposition, and a Monte Carlo dose calculation algorithm (Scholz 2004). All applied algorithms were proven to give equivalent results for 3D dose distributions in water for various regular and irregularly shaped fields. Firstly, we show the result of the 2D lateral dose distributions on a central slice of the phantom in Fig It is clearly indicated in Fig. 15.7a that the pencil beam almost completely misses the effect of lateral scattering within the cork slab. The result of the superposition algorithm, displayed in Fig. 15.7b, accounts for most of the lateral scattering of the secondary electrons as indicated by the extended 10% isodose line in cork. The effect of energy transport through secondary electrons is even more pronounced if the results of the Monte Carlo calculation, shown in Fig. 15.7c, are considered. a b Fig. 15.7a d. Dose distribution of a 6-MV beam through water cork water slabs. Here the dose distributions through a slab phantom consisting of solid water, cork, and perspex are displayed. The perspex is not penetrated by the beam directly, so its influence on the dose distribution can be neglected. a c The 2D cuts refer to dose calculation based on a the pencil beam, b the superposition, and c the Monte Carlo technique. The figure d shows central-axis-depth dose curves of all three dose distributions compared with ionization chamber measurements c d

8 194 U. Oelfke and C. Scholz These results are consistent with the respective depth-dose curves shown in Fig 15.7d. The enhanced lateral scattering of secondary electrons in the lowdensity medium reduces the dose values on the central-beam axis, an effect which is almost completely missed by the pencil-beam calculation resulting in a severe overestimation of the respective dose of up to 12% of the maximum dose. This discrepancy is substantially reduced by the superposition algorithm. Only the Monte Carlo calculation was able to reproduce the experimental data with an accuracy of 2%, which was the estimated accuracy of the measurement. While these simple phantom experiments may reveal some generic features of the discussed dose calculation algorithms, it is not clear to what extent the various algorithms generate dose differences which are of clinical relevance. In the final section of this review we address this issue briefly by considering the influence of different dose calculation engines on IMRT dose optimization for clinical cases with abundant severe tissue inhomogeneities Dose Calculations and IMRT Optimization The role of advanced high-energy photon dose calculations for iterative IMRT treatment planning is still under investigation. Since conventional algorithms, such as pencil-beam methods with poor consideration of inhomogeneities, could produce substantial deviations in media different to water, the quality of intensity-modulated treatment plans generated through those dose calculation methods was recently reviewed by different groups (Jeraj et al. 2002; Siebers et al. 2001, 2002; Scholz et al. 2003a). Particularly two aspects are important referring to dose calculation in IMRT: firstly, one has to assess the accuracy of the respective algorithm as required for the determination and evaluation of the final dose pattern originating from a given set of fluence amplitudes. Deviations from a reference dose calculation can be called systematic errors. Secondly, the influence of the dose algorithm in the optimization process itself should be analyzed, since deviations in the dose calculation induces differences in bixel intensities in the fluence maps, which is usually referred to as convergence error. In order to demonstrate the nature of these two errors, we consider one of the most sensitive clinical cases with respect to dose calculations, the irradiation of small, solid lesions embedded in lung tissue The Systematic Error As mentioned previously, the systematic error indicates the difference in dose which arises from the application of different dose calculation algorithms for a given set of fluence matrices. A solid lung tumor of about 30-mm diameter located in the left lung is totally surrounded by low-density lung tissue. It is irradiated with five coplanar, intensity-modulated beams. As indicated by the dose distribution on a transversal CT slice of the patient in Fig. 15.8a, the original optimization based on the pencil-beam dose calculation seems to cover well the PTV by the prescribed dose of 63 Gy (100% isodose); however, the recalculation with the superposition algorithm shown in Fig. 15.8b reveals that only a small volume of the target is encompassed by the 60 Gy isodose (95% isodose) and that a severe underdosage of about 13 Gy in mean dose is observed for the PTV. The systematic error induced by the pencil-beam algorithm demonstrates that this dose calculation method severely overestimates the dose inside the tumor. For the considered case the respective underdosage of the tumor shown by the superposition algorithm is based on the fact that for high-energy photon fields the range of many secondary electrons is larger than the radius of the tumor volume shown above; thus, there is not enough material to absorb the total number of secondary electrons inside the target region. The remaining electrons simply are scattered into the low-density lung tissue. These findings are also well represented by the dose-volume histogram shown in Fig. 15.8d The Convergence Error The convergence error is defined by the dose difference which arises from the application of two different fluence matrices obtained by inverse planning with two different dose algorithms for which the same reference dose calculation is applied. For our example we compare the dose distributions in Fig. 15.8b, where the pencil-beam algorithm was used for the generation of the fluence matrices, and the dose distribution in Fig. 15.8c, where the superposition algorithm was employed for the optimization. Once the fluence matrices were established, the final dose distributions were obtained by a calculation with the superposition algorithm. As clearly indicated by the fluence matrices shown in Fig. 15.9, the optimiza-

9 Dose Calculation Algorithms 195 a b c d Fig. 15.8a d. Absolute dose distributions for a lung patient. a c A transversal slice of the 3D dose distribution in a lung tumor patient is shown exemplarily. a Pencil-beam (PB) optimization plus PB recalculation. b Pencil-beam optimization plus superposition recalculation. c Superposition optimization plus superposition recalculation. The 100% isodose line represents the prescribed dose of 63 Gy. The dose-volume histograms of the planning target volume (PTV), the left lung, and the spinal cord for the different plans are displayed in d. One clearly recognizes the reduction of the mean PTV dose by about 13 Gy produced by the original optimization with a pencil beam algorithm which was recalculated with superposition. Compared with this, the superposition-optimized plan reaches the prescribed mean dose of 63 Gy in the target volume at the expense of a slightly higher irradiation of the left lung tion with the superposition algorithm compensates the obvious underdosage of the PTV created by the fluence generated with the pencil-beam method. The fluence matrix on the right-hand side of Fig. 15.9, obtained with the superposition algorithm, enhances the bixel weights at the periphery of the projected tumor such that an adequate dose coverage of the PTV is guaranteed. This fact is also well reflected by the respective values of the dose-volume histogram shown in Fig. 15.8d. Fig Fluence matrices of the pencil beam (left) and superposition-optimized plan (right). In these images the bixel intensities of one beam of both plans for the lung tumor case shown in Fig are displayed. The enhanced fluence values in the peripheral regions of the plan optimized with the superposition algorithm are clearly recognized 15.7 Conclusion Dose calculation algorithms play a central role for the clinical practice of radiation therapy. They form

10 196 U. Oelfke and C. Scholz the basis for any treatment plan optimization, a feature which becomes increasingly important with the development of complex treatment techniques such as IMRT. The role of highly accurate and therefore mostly time-consuming dose algorithms, such as superposition algorithms or Monte Carlo simulations, in clinical radiation therapy is still under investigation. Their increased accuracy offers substantial advantages for clinical cases which involve intricate tissue inhomogeneities. Even if in many radiotherapy centers the treatment plans are still based on the pencil-beam method, its general applicability to inhomogeneous clinical cases has to be questioned. On the other hand, inside quite homogeneous regions, as in the central head region or the abdomen, the pencil beam generates dose distributions with excellent precision and provides the best trade-off between accuracy and calculation times. In the case of severe tissue inhomogeneities the superposition method produces dose distributions which fairly cover the target region, even if minor differences are observed in comparison with Monte Carlo calculations. These offer the best prediction of the deposited dose inside arbitrary tissue types. Some Monte Carlo-based programs already offer computation times comparable to those of superposition algorithms, and therefore their applicability in clinical practice will probably further increase for a small and special class of clinical cases. References Ahnesjö A (1989) Collapsed cone convolution of radiant energy for photon dose calculation in heterogeneous media. Med Phys 16: Ahnesjö A, Aspradakis MM (1999) Dose calculations for external photon beams in radiotherapy. Phys Med Biol 44: R99 R155 Ahnesjö A, Andreo P, Brahme A (1987) Calculation and application of point spread functions for treatment planning with high energy photon beams. Acta Oncol 26:49 56 Altschuler MD, Bloch P, Buhle EL, Ayyalasomayajula S (1992) 3D dose calculation for electron and photon beams. Phys Med Biol 37: Bortfeld T, Schlegel W, Rhein B (1993) Decomposition of pencil beam kernels for fast dose calculations in threedimensional treatment planning. Med Phys 20: Hogstrom KR, Mills MD, Almond PR (1981) Electron beam dose calculations. Phys Med Biol 26: Jeraj R, Keall PJ, Siebers JV (2002) The effect of dose calculation accuracy on inverse treatment planning. Phys Med Biol 47: Liu HH, Mackie TR, McCullough EC (1997) Correcting kernel tilting and hardening in convolution/superposition dose calculations for clinical divergent and polychromatic photon beams. Med Phys 24: Mackie TR, Scrimger JW, Battista JJ (1985) A convolution method of calculating dose for 15 MV X-ray. Med Phys 12: Mackie TR, Bielajew AF, Rogers DW, Battista JJ (1988) Generation of photon energy deposition kernels using the EGS Monte Carlo code. Phys Med Biol 33:1 20 Metcalfe PE, Hoban PW, Murray DC, Round WH (1990) Beam hardening of 10-MV radiotherapy X-rays: analysis using a convolution/superposition method. Phys Med Biol 35: Mohan R, Chui C (1985) Energy and angular distributions of photons from medical linear accelerators. Med Phys 12: Mohan R, Chui C, Lidofsky L (1986) Differential pencil beam dose computation model for photons. Med Phys 13:64 73 Rogers DWO, Faddegon BA, Ding GX, Ma C-M, We J, Mackie TR (1995) BEAM: a Monte Carlo code to simulate radiotherapy treatment units. Med Phys 22: Schoknecht G (1967) Description of radiation fields by separation of primary and scatter radiation. I. The tissue air ratio in 60 Co fields. Strahlentherapie 132: Schoknecht G, Khatib M (1982) Model calculations of the energy distribution of scattered radiation in a patient (author s transl). Röntgenblätter 35: Scholz C (2004) Development and evaluation of advanced dose calculations for modern radiation therapy. PhD thesis, University of Heidelberg Scholz C, Nill S, Oelfke U (2003a) Comparison of IMRT optimization based on a pencil beam and a superposition algorithm. Med Phys 30: Scholz C, Schulze C, Oelfke U, Bortfeld T (2003b) Development and clinical application of a fast superposition algorithm in radiation therapy. Radiother Oncol 69:79 90 Schulze C (1995) Entwicklung schneller Algorihmen zur Dosisberechnung für die Bestrahlung inhomogener Medien mit hochenergetischen Photonen. PhD thesis, DKFZ, Heidelberg Sharpe MB, Battista JJ (1993) Dose calculations using convolution and superposition principles: the orientation of dose spread kernels in divergent X-ray beams. Med Phys 20: Siebers JV, Tong S, Lauterbach M, Wu Q, Mohan R (2001) Acceleration of dose calculations for intensity-modulated radiotherapy. Med Phys 28: Siebers JV, Lauterbach M, Tong S, Wu Q, Mohan R (2002) Reducing dose calculation time for accurate iterative IMRT planning. Med Phys 29: