The prediction of the patient dose distribution is cen- tral to IMRT optimization and beam delivery processes.

Dose Calculations for IMRT

Jeffrey V. Siebers

6

Contents

6.1 Introduction . . . . 61 6.2 Dosimetric Characteristics of IMRT Beams . . . . 61 6.3 IMRT Dose Calculation Algorithms . . . . 62 6.4 Dose Calculation Within the IMRT Optimization Loop 64 6.4.1 IMRT Optimization Process . . . . 64 6.4.2 Dose Computation at Each Iteration . . . . 65 6.4.3 Precomputation of Sub-field or Bixel Dose

Distributions . . . . 65 6.5 Effects of Dose Accuracy on IMRT Plans . . . . 66 6.5.1 Plan Dose Distributions: Dose Prediction Error 66 6.5.2 Plan Optimality: Optimization Convergence

Error . . . . 67 6.6 Reduce IMRT Dose Calculation Time by Interlacing

Accurate and Fast Algorithms . . . 68 6.7 Outlook: Monte Carlo for IMRT Dose Calculations . . 69 References . . . . 69

6.1 Introduction

The prediction of the patient dose distribution is cen- tral to IMRT optimization and beam delivery processes.

The predicted dose for a candidate set of plan param- eters is used to evaluate the plan objective function during the iterative optimization, to adjust and deter- mine the beamlet intensities required to produce an optimal plan, and to judge the clinical acceptability of a plan. These tasks require accurate prediction of the IMRT dose distribution.

In competition with dose accuracy is IMRT dose calculation speed. IMRT plan optimization involves probing the solution space of beamlet intensities, num- ber and angle of beams, beam energies, and other free parameters for an optimal solution. The iterative optimization process can require from ten or fewer iterations to converge for simple bounded problems with a limited solution space to thousands of iterations for complex multiple-parameter optimization problems with complex objective functions. The time to predict

the dose for an individual candidate parameter set is often the rate-limiting component affecting the over- all optimization time. Hence, severe approximations are often imbedded in the calculation algorithm used within the optimization loop in order to allow the opti- mization to be completed in an acceptable time frame.

These approximations degrade the accuracy of the dose prediction.

This chapter focuses on the issues related to the cal- culation of dose for photon beam IMRT. The general dosimetric characteristics of IMRT beams and algo- rithms used for IMRT dose calculation algorithms are described, as are methods of incorporating dose cal- culations into the plan optimization loop. The balance and tradeoffs of dose calculation accuracy and dose cal- culation speed are considered throughout, with special attention paid to the consequences of inaccurate dose calculations on clinical dose distributions and the opti- mality of the final treatment plan. To aid the reader with respect to commercial systems, Table 1 summarizes the properties of several commercial IMRT systems with respect to the issues discussed in this chapter.

6.2 Dosimetric Characteristics of IMRT Beams

IMRT makes use of non-uniform radiation fluence dis- tributions incident upon a patient from numerous beam directions to create a desired three-dimensional dose distribution. Non-uniform fluence patterns are created by using a continuous or discrete sequence of small beam apertures formed using a multi-leaf collimator (MLC).

The consequences of this type of beam delivery method on the accuracy of the computed dose distribution are:

• Small beam apertures: most dose calculation algo-

rithms exhibit the greatest inaccuracies for small

treatment fields, particularly in regions of tissue het-

erogeneities [1–3]. For IMRT, the effects of small-field

dose calculation heterogeneity errors on the overall

dose distribution will depend on the intensity modu-

lation in the vicinity of the dose sampling. Errors can

be expected to be largest in regions that have large in-

Table 1.Comparison of commercial treatment planning system dose calculation algorithms and optimization strategies

System Optimization dose

algorithm

Deliverable optimization

Post optimization dose algorithm

Delivery method Segment weight re-optimization

Corvus 5.0 PB No MC (optional) sMLC and DMLC No

Eclipse 7.1.67 Fast 3D superposi- tion

No PB sMLC and DMLC No

CMS XiO 4.2 PB-based fast 3D superposition

No SC sMLC No (but can reopt

beam weight)

KonRad 2.1 PB Optional PB sMLC and DMLC No

Pinnacle 7.4 PB with SC corrections

Optional (direct leaf position optimization)

SC sMLC and DMLC Yes

Plato PB No PB sMLC and DMLC No

Oncentra 1.3 PB No PB or SC sMLC Yes

tensity gradients (spikes or dips) since the gradients behave like small fields superimposed on the main field. Conversely, in large regions of near uniform fluence, the heterogeneity correction induced error will be expected to be similar to that for 3DCRT al- gorithms. Dose calculation algorithm output factor errors are also typically greatest for small field sizes, which also contributes to IMRT dose errors.

• MLC leakage radiation: the dose delivered to a given point in a patient consists of components due to fluence transmitted through the sequence of open MLC apertures and that due to MLC radiation leak- age, where leakage includes radiation transmitted through and scattered from the MLC and the MLC leaf tips. Figure 1 shows a profile through the fluence maps generated for a head and neck IMRT treat- ment plan delivered with the dynamic MLC (DMLC) technique, with the fluence from indirect sources (MLC leakage radiation) separated out from the total.

Within the narrow intensity spike at x = 1 cm, 50% of

Fig. 1. Sample profiles of the total and indirect (MLC leakage and scatter) fluence contributions for a typical head and neck IMRT treatment plan. Indirect sources consist of a substantial fraction of the total fluence in intensity valleys and in narrow intensity spikes

the fluence is from indirect sources; in the intensity valley at x = 2 cm, 100% of the fluence is from indi- rect sources; and in the relatively flat fluence portion from x = ^{4 to x} = 9 cm, the indirect component ac- counts for > 10

of the fluence [4]. The MLC leakage radiation sets the lower limit of the dose that can be delivered within the jaw boundaries. It has a harder energy spectrum than an open beam, with the per- cent depth-dose at 10 cm depth being 5% greater for an MLC blocked beam than for an open beam [5].

Most IMRT dose calculation algorithms approximate the effects of MLC leakage radiation on the total dose distribution.

Evidence of dose calculation errors in IMRT can be found in routine IMRT quality assurance in which com- puted and measured dose distributions in a phantom are compared. Such experimental checks of IMRT fields rou- tinely show discrepancies between the planned (desired) and actual dose, independent of the treatment-planning platform [6], particularly in regions of large dose gradi- ents. Oftentimes, IMRT QA measurements are, however, made in low dose gradient regions, precisely where the dose calculation error is expected to be smallest, possi- bly miss-leading the user regarding the accuracy of their dose calculation algorithm.

6.3 IMRT Dose Calculation Algorithms

The prediction of the absorbed dose delivered to a given

individual patient is a general problem encountered in

radiation therapy. For externally directed photon beams,

the problem can be stated as: for a given radiation flu-

ence incident upon the patient geometry, determine the

energy absorbed (the absorbed dose) within the patient

as a function of position. This simple formulation shows

that dose calculation problem consists of two distinct

components: fluence prediction and the determination

of the dose from that incident fluence. Each of these components plays an important role in an IMRT dose calculation.

Dose calculation algorithms are generally classi- fied as correction-based algorithms or model-based algorithms [7]. In correction-based algorithms, the dose computation effectively first computes dose to a homogeneous water phantom, then applies various correction strategies to account for source to patient surface distance (SSD) changes and tissue hetero- geneities within the patient. Model-based algorithms, on the other hand, generally perform the dose cal- culation directly in the patient geometry. The IMRT collaborative working group of AAPM and ASTRO has separated IMRT dose calculation algorithms into four categories [8], correction-based broad beam algorithms, correction-based pencil-beam algorithms, and model- based kernel-based and Monte Carlo algorithms.

• Broad beam algorithms are correction-based dose calculation algorithms [7] that utilize measured dose distributions to generate a parameterization of dose distributions in a homogeneous water phantom as function of field size, depth, off-axis distance, and surface to source distance. For a patient specific dose calculation, dose for the treatment conditions are first reconstituted for a homogeneous water phantom, and then patient anatomy specific corrections are applied to account for surface contours and tissue hetero- geneities. Broad-beam algorithms are designed for use with radiation beams that have nominally uni- form fluence distributions, such as for an open or simply blocked field, or for fields in which the flu- ence is a simple, smoothly varying function, such as that produced by a wedge. Broad beam algorithms rely on the fact that within a homogeneous patient or phantom, radiation equilibrium is approximated within the field boundaries. Heterogeneity correc- tions for broad beam algorithms, when used, are based upon density scaling equivalent path-length methods. Broad beam algorithms are not applicable to the variable intensity conditions of IMRT due to the lack of radiation equilibrium within IMRT fields, and are not recommended to be used, in general, for IMRT optimization. They are, however, occasion- ally used for aperture based IMRT optimization [9]

and can also be utilized as a secondary monitor unit checking programs for IMRT QA [10, 11].

• Pencil-beam (PB) algorithms are correction-based algorithms [7] that utilize parameterized measured data from a limited number of irradiation conditions in addition to pencil beam energy deposition kernels in a water phantom derived from Monte Carlo calcu- lations [12] or measurements [13, 14] to reconstitute dose distributions in a homogeneous phantom. Pa- tient specific contours and heterogeneities are then accounted for as corrections to the homogeneous dose distribution [12, 15]. PB models account for

beam intensity modulations and field shapes, but utilize radiological path length scaling methods to account for heterogeneities and patient contours. PBs account for the radiation disequilibrium due to lateral transport of secondary radiation for modulated in- tensity distributions in homogeneous media, but not for internal heterogeneities and surface irregulari- ties. PB algorithms have significant speed advantages over kernel-based approaches (below) since they effectively pre-convolve the point kernels over the depth dimension. PB algorithms are the most com- monly used algorithms for IMRT optimization due to their fast dose calculation speed. However, their accuracy is dependent upon the heterogeneity of the patient geometry.

• Kernel-based algorithms are model-based algo- rithms [7] which can be used to compute directly the dose in a patient or phantom. Kernel-based approaches, typically called superposition or con- volution algorithms (SC), separate the effects of primary photons incident upon the patient from the effects of secondary radiations generated within the patient. In SC algorithms, the total energy re- leased per unit mass (TERMA) from primary photon interactions in the patient is computed and the ef- fect of secondary radiations generated in the patient are accounted for using pre-computed secondary en- ergy spread kernels which are superimposed over or convolved with the TERMA to yield the total dose dis- tribution [16–18]. SC algorithms account for tissue heterogeneities in the TERMA calculation, but typ- ically use radiological path length methods to scale the secondary energy spread kernels. They are much more accurate than PB in heterogeneous geometries and can accurately compute dose in regions of elec- tronic disequilibrium; hence they are applicable to IMRT dose calculations. SC dose calculation times are relatively long compared to PB approaches [19,20]

and, although they can be used within the IMRT op- timization loop [21], SC algorithms are usually only used after the end of optimization to compute the final deliverable dose distribution.

• Monte Carlo (MC) algorithms are model based al-

gorithms and can be used to compute directly the

dose to the patient. In the MC method, individual

photon and electron tracks through the accelerator

treatment head, multi-leaf collimator, and patient are

simulated. Since MC algorithms simulate a stochas-

tic process, the results have an inherent statistical

imprecision (noise) which generally decreases with

the square of the dose calculation time, but is inde-

pendent of the number of beams simulated, a distinct

advantage when many treatment angles are used for

patient treatment. MC algorithms are considered the

most accurate dose calculation algorithms since they

directly account for tissue heterogeneities and make

no assumptions regarding radiation equilibrium. For

IMRT, MC has the additional advantage that particles can be directly transported through the multi-leaf collimator segments or through a moving multi-leaf collimator, hence, radiation leakage and scatter ef- fects can be directly taken into account [22]. The major deterrent to implementation of MC for IMRT dose calculation, particularly during optimization, is the time required to complete the dose calcu- lation process. Additional discussion on MC dose algorithms is given at the end of this chapter.

In addition to the dose calculation algorithm type, a ma- jor factor influencing dose calculation accuracy is the user specific commissioning and tuning of the dose cal- culation model to match IMRT dose distributions for their particular accelerator. Both initial dosimetric qual- ity assurance under carefully controlled test conditions and routine patient specific quality assurance are useful to ensure dose calculation accuracy and determine the limits of a user’s specific implementation.

6.4 Dose Calculation Within the IMRT Optimization Loop

The choice of the dose calculation algorithm and how it is incorporated into the IMRT optimization loop af- fects the speed, accuracy, and optimality of the final dose distribution. The amount of realism used during the flu- ence optimization in terms of the whether the optimized fluences used by the dose calculation algorithm can be delivered by the accelerator hardware or not also affects the optimality of the plan.

6.4.1 IMRT Optimization Process

A flow diagram for a typical IMRT optimization pro- cess is shown in Fig. 2. Dose calculation algorithms appears twice in this flow diagram, once during the optimization process (Box 2) and once following the creation of MLC leaf sequences in computing the de- liverable dose distribution (Box 9). In many planning systems, these are different dose calculation algorithms.

The Box 2 dose calculation, which is repeated multi- ple times during optimization, is normally a fast dose calculation algorithm, such as a fast PB algorithm, to enable rapid completion of the optimization. Approxi- mations utilized within the Box 2 algorithm to achieve the dose calculation speed can result in dose inaccura- cies. To minimize the impact of these inaccuracies, the Box 9 dose calculation algorithm, which is executed only once per optimization, is often performed with a slower, more accurate algorithm (such as an SC algorithm) to determine the post-optimization deliverable dose dis- tribution. The differences in dose calculation accuracy between the Box 2 and Box 9 dose calculation algorithms

Fig. 2. In the typical IMRT optimization process, dose calcula- tion occurs both with the optimization loop (Box 2) and following conversion of the op- timized intensities to MLC leaf sequences (Box 9). These may be the same, or dif- ferent dose calculation algorithms

combined with the inability of the MLC to achieve the optimal fluence patterns results in differences between optimized (Box 5) and deliverable (Box 9) dose distri- butions for an IMRT plan, often with deterioration in the plan quality between Box 5 and Box 9. An example of this is shown in Fig. 3, which shows a T2N3M0 base- of-tongue cancer patient who was treated with dynamic MLC-IMRT from nine equiangle, coplanar beams. The original plan (computed with SC) showed good target coverage and dose uniformity, while the deliverable plan that considered the MLC delivery showed a substantial hot spot in the target volume and higher doses to crit- ical structures. Furthermore, the cord is spared by the 40 Gy line in the optimized plan, but is transected by the 45 Gy line for the deliverable plan. This type of devia- tion between optimized and deliverable results is typical in routine IMRT practice.

In clinical practice, techniques used to improve a deliverable treatment plan when the optimized and deliverable dose distributions disagree to such an ex- tent that the deliverable dose distribution is clinically unacceptable include:

• Adjusting the plan monitor units so as to produce an acceptable plan.

• Re-optimizing the IMRT beam segment weights for multi-segmental (sMLC) IMRT.

• Modifying the plan objectives and re-optimizing the

plan. This requires the planner to modify or over-

specify planning objectives for the optimized dose

a) Optimized b) Deliverable c) DV H comparison

Fig. 3. a–c Isodose profiles and DVH comparison for an optimized intensity distribution and the corresponding deliverable dose dis- tribution for DMLC delivery to a patient. Note the differences to

in the target coverage (red PTV) and normal tissue sparing (green cord)

distribution (D

, Fig. 2, Box 5) in order to achieve an acceptable deliverable (D

, Fig. 2, Box 9) dose distribution.

Optimization techniques that avoid the post- optimization plan degradation include:

• Using an accurate dose calculation throughout the optimization process, or for the final iterations of the plan optimization process [19].

• Including the MLC leaf sequencing within the optimization loop. This is termed deliverable- based optimization (DBO). DBO does not require post-optimization conversion to leaf sequencing, thereby avoiding plan the degradation from the post-optimization leaf conversion step. DBO can ac- complished by including the full leaf sequencing process within the optimization loop [22, 23] or by directly optimizing MLC leaf positions [24].

6.4.2 Dose Computation at Each Iteration

Most IMRT planning systems compute the entire 3D dose distribution during each optimization iteration (Fig. 2, Box 2). To account for the intensity modulation during the dose calculation, for each IMRT beam, the intensity variations are modeled as a 2D matrix of en- ergy fluence or energy fluence modifiers incident on the patient. This energy fluence is then used by the dose calculation algorithm to compute the 3D dose dis- tribution for each optimization iteration. A variety of algorithms can be used when the entire dose distribu- tion is computed at each iteration. A major advantage of this method is it is straight forward to implement into a treatment planning system since the intensity modu- lation can be considered as a transmission compensator matrix by the treatment planning system [21].

Since the time required to complete an optimiza- tion is proportional to the product of the number of

iterations and the dose calculation time per iteration, to minimize optimization time, fast, approximate dose calculation algorithms are often used such as those based on ray-tracing the primary beam or those based upon fast PB algorithms [12, 25, 26]. Techniques such as using adaptive dose grids, [27] random sampling of dose points within structures of interest, [28, 29]

and pre-computation of radiological path-lengths and other quantities that do not change from one itera- tion to the next are often used to reduce the dose calculation time [19]. Each of these methods has the potential to reduce dose calculation accuracy; however, the goal of proper implementation of these techniques is to have the accuracy reductions become clinically in- significant. Post-optimization dose recalculation with and an accurate algorithm is used to reduce the clini- cal consequences of the approximations used during the optimization.

6.4.3 Precomputation of Sub-field or Bixel Dose Distributions

Instead of computing the entire dose distribution dur-

ing each iteration in the optimization, an alternative

method is to perform individual dose calculations for

sub-components of the radiation field prior to opti-

mization and then use weighted summations of these

sub-components to compute the dose at each optimiza-

tion iteration. The simplest of these approaches is to

pre-compute large sub-divisions of the radiation field

in what is called aperture-based optimization [9,30–32],

in which beams are sub-divided into multiple (possi-

bly overlapping) MLC apertures to be used for beam

delivery. The number of segments used limits the num-

ber of possible intensity levels and the complexity of

the delivery. The simplified IMRT planning technique,

in which a few segments are manually adjusted, is called

forward IMRT. Arbitrary 3D dose calculation algorithms

are suitable for both forward and inverse aperture-based approaches, subject to the field-size and heterogeneity limitations of the algorithm. The pre-selected MLC leaf positions can be used to incorporate head scatter or even MLC scatter radiation effects on the dose distribution, depending on the algorithm.

An alternative pre-calculation approach is to sub- divide an individual beam into a matrix of small (1 cm ×1 cm or smaller) equal-sized pencil-beam like elements, each with a corresponding intensity value [33, 34]. For each individual beam element, the contribution to the total dose distribution is computed and stored in what is termed a bixel. Since bixels are small, only PB, SC, or MC algorithms are suitable for computing them. The dose computation during plan optimization consists of summing the doses of the individual pre-computed bix- els weighted by their respective intensities, which are varied from one iteration to the next.

In the bixel approach, the time required to pre- compute the individual bixel dose distributions depends strongly on the algorithm used to create them. However, the bixel dose distributions are computed only once, and therefore the dose computation typically consumes only a small fraction of the total optimization time. Depend- ing on the beam size, substantial quantities of computer memory are necessary to store the bixel dose distribu- tions. For a typical IMRT case with 500 beamlets per beam, 9 treatment angles, and 200,000 2-byte integer dose volume elements in the dose scoring grid, 1.8 GB of memory is required to store the dose matrices. In early IMRT implementations, this memory requirement was a substantial barrier to implementation of bixel-based optimization. Although modern computers are now ca- pable of utilizing this much high speed RAM, alternative methods have been developed to reduce the memory re- quirements and reduce the overall computation time.

These include (1) storing and computing doses only for regions of interest (targets and critical structures), (2) computing dose only for a sub-set of points within regions of interest [28, 29], (3) specifying a cut-off ra- dius for pencil beams, beyond which the contribution is set equal to zero [35], and (4) sparsely sampling the bix- els (pencil beams) in the low dose region in such a way that the total energy deposited is conserved [36].

Unlike the aperture-based pre-computation ap- proach, the bixel-based approach inherently is a non- deliverable IMRT optimization approach. In the bixel-based approach, the MLC beam delivery and the effects of MLC leakage and scatter radiation are not incorporated into the optimization process. It may be possible to add in bixels representing the blocked beam (with a contributing fraction of F

where F

is the fraction of time that the MLC is closed for that bixel) for radiation passing through the closed MLC. However, even in this case, the effect of radiation passing through the rounded MLC leaf tips will not be incorporated.

6.5 Effects of Dose Accuracy on IMRT Plans

Approximations used in IMRT dose calculation op- timization algorithms and in post-optimization dose computations affect both the accuracy and the optimal- ity of IMRT treatment plans. The following sub-sections describe the accuracy component in terms of their effect on the dose distribution used for plan evaluation using what is called a dose prediction error and of the loss in optimality in terms what is called an optimization convergence error.

6.5.1 Plan Dose Distributions: Dose Prediction Error The accuracy of a treatment planning systems’ dose calculation algorithm is a measure of agreement be- tween the dose distribution predicted by the treatment planning system and that that would be achieved in a pa- tient. The difference between the actual and predicted dose distributions can be called a dose prediction er- ror (DPE). DPE is a measurable quantity for phantoms in which 3D dose distributions can be measured. DPE, however, can not be determined under circumstances where the full 3D dose distribution cannot be meas- ured, such as within a patient. For practical purposes, in such cases, the DPE of a given algorithm can be esti- mated by comparing its dose calculation results with those of a dose calculation algorithm that is known to be superior (more accurate) based upon measurable DPEs.

There are several sources of DPEs for radiation ther- apy dose calculations. The major sources of inaccuracies and hence DPEs for IMRT dose calculations are:

1. Heterogeneities: due to improper or incomplete handling of patient heterogeneities by the dose- calculation algorithm. Heterogeneity errors can be estimated by comparing calculations with mea- surements in an anthropomorphic phantom, or by comparing with another algorithm that more accu- rately accounts for heterogeneities such as SC or MC.

In general, sites in which tissues are nearly homo- geneous (brain, prostate) little heterogeneity error would be expected independent of the dose calcula- tion algorithm. For heterogeneous geometries (lung, head and neck), on the other hand, the heterogeneity errors would expected to be larger for dose calcu- lation algorithms that use radiological path-length corrections to account for heterogeneities. The use of multiple beam angles may dilute the dose errors from individual beams such that the error in the to- tal dose distribution is acceptable, however, this has not been shown to be true for all cases.

2. Fluence: due to improper or incomplete prediction of the fluence incident upon the patient or phantom.

Dose calculation algorithms require accurate predic-

tion of the fluence to predict dose accurately. Fluence errors can be due to approximations or inaccura- cies in the conversion from MLC leaf-sequences to fluence or intensity-maps to be used by the dose cal- culation algorithm, or due to errors in the treatment machine delivery of the fluence. In phantom geome- tries, fluence errors can be measured by comparing calculated dose distributions with ones measured with films or portal images on a plane normal to the beam central axis. This is often completed as a part of routine IMRT QA. The impact of fluence er- rors on patient dose distributions can be estimated by transferring measured fluence values back into the treatment planning system to be used as input for dose re-calculation [37]. Alternatively, this can be done by re-computing doses using the leaf sequences reported by treatment machine (from their log files) with an algorithm that has been verified to have min- imal fluence errors. Generally, it is the purview of the IMRT quality assurance process (whether mea- surement or calculation based) to detect gross errors caused by fluence errors.

3. Patient geometry: due to improper or incomplete accounting for the patient set-up uncertainties, intra- fraction motion, or other patient anatomical changes in the dose calculation. Most dose calculation algo- rithms consider the patient to be a static geometry, with the single CT data set acquired at the begin- ning of treatment accepted to be representative of the geometry throughout the treatment. Margins are used to account for patient set-up uncertainties and intra-fraction motion. The adequacy of margins to ensure target coverage and normal tissue sparing for IMRT is the subject of on-going investigations. Re- computing doses for multiple patient set-ups [38], convolving dose distributions [39] or fluences [40]

with expected patient setup errors is also being ap- plied to estimate the impact of these errors. Also, multiple patient imaging studies are being applied to be able to better estimate the variability of patient setups and their impact of patient setup errors on IMRT dose distributions. It may be difficult or im- possible to evaluate the dose errors caused by patient geometry changes since patient setup-errors and day- to-day anatomic variations are impossible to predict, however, the user should be aware of such errors and avoid using IMRT in situations in which probable pa- tient geometry changes would result in unacceptable patient outcomes, such as in treatment of moving tar- gets such as a mobile lung tumor with only a static planning image.

4. Other: there can be other contributors to dose errors.

These can be specific to the calculation algorithm and to individual users’ commissioning of a treatment planning algorithm. It is generally the role of routine and patient specific quality assurance measurements to detect such errors.

Clinically, DPEs depend upon both the algorithms used for the dose calculation, the treatment site, and the beam configuration. Uncorrected DPEs will result in the dose delivered to a patient to differ from that which is predicted by the treatment planning algo- rithm. An example of a DPE is shown in Fig. 4, in which a lung IMRT treatment plan that was optimized using a fast PB algorithm that used a radiological path-length correction method to account for the tissue hetero- geneities is compared with the same plan recalculated with SC and MC dose calculation algorithms which inherently account for tissue heterogeneities. The PB algorithm which was used for optimization predicted the PTV D

to be 59.5 Gy, while the SC algorithm pre- dicts 55.5 Gy and the MC predicts 54.7 Gy. The cord DVHs for the three algorithms, however, are all very similar. Had the PB plan been used for the dose pre- scription, the target would have been underdosed by 4–5 Gy.

Clinical consequences of many DPEs can be reduced by performing a final dose calculation with the most accurate dose calculation algorithm available in the planning system prior to plan evaluation and modify- ing the plan monitor units to meet the clinical goals. For the lung plan above, this would have involved increas- ing the monitor units to deliver the prescription dose.

This, however, would also have increased the cord dose.

6.5.2 Plan Optimality: Optimization Convergence Error

The optimality of an IMRT plan is a measure of how well the plan satisfies (minimizes or maximize) the objective function used during optimization. The dif- ference between the plan that best satisfies the plan objective function and the plan that is delivered to the patient can be called the optimization convergence er- ror (OCE). Determination of OCEs have some of the

Fig. 4. Example of a dose prediction error (DPE) on dose-volume histograms for a sample lung treatment plan which was opti- mized with a PB algorithm and recomputed with SC and MC dose algorithms. DVHs for the PTV and the cord are shown

same limitations as determination of DPEs since it may be impossible to determine the true optimal plan; how- ever, OCEs can be estimated by comparing plans with those in which a source of the OCE has been minimized or eliminated.

One source of OCE is DPEs during the iterative IMRT optimization. When a dose calculation algorithm with a DPE is used during optimization, the optimization can converge to a different intensity solution than the one that it would converge to if an algorithm without the DPE was used [41]. Consider the lung example previ- ously shown in Fig. 4. The PB algorithm used during optimization used a radiological path-length method to account for heterogeneities, a method known to overes- timate dose in lung tissue and at lung-tumor interfaces.

The optimizer, therefore, was misguided in its attempt to adjust intensities to correct for heterogeneity in- duced dose perturbations; hence, when the plan was re-computed with SC or MC, the PTV dose was un- derestimated. When the lung plan is optimized using an SC algorithm (Fig. 5), the optimizer is guided to ac- count more properly for the heterogeneity induced dose perturbations. Interestingly, the dose distribution and DVHs for the SC-optimized and PB-optimized plans are very similar, but with different intensity distributions for each beam. This indicates that the IMRT optimiza- tion process can account for heterogeneity induced dose perturbations.

Non-dose calculation related sources of OCE include post-optimization conversion of optimized intensity patterns to deliverable MLC leaf sequences (Fig. 2) and failures of the optimizer to find the global minimum, ei- ther because of becoming trapped in a local minima or failing to run to convergence.

Clinically, OCEs result in a sub-optimal plan being delivered to a patient. Provided that DPEs are eliminated by performing a final dose calculation with an accu-

Fig. 5. Optimization convergence error for the lung IMRT example of Fig. 4. When the PB optimized DVH (solid lines) is re-computed with SC (dotted lines), the DPE of the PB dose calculation is observed. When the plan is optimized using the SC algorithm (dot-dash lines), the OCE (the difference between the SC recalc and the SC optimized) is observed

rate dose calculation algorithm, the dose distribution used to evaluate the plan can be accurate and the sub- optimal plan may be found to be clinically acceptable.

The advantages to reducing OCEs are that better plans may be found, often with reduced dose to critical struc- tures [22]. Furthermore, using planning techniques such as deliverable optimization which minimizes OCE, the optimizer deals with realistic plan objectives. This can reduce the trial and error procedure of adjusting (over- specifying) the objective function and re-optimizing the plan often used in IMRT planning.

6.6 Reduce IMRT Dose Calculation Time by Interlacing Accurate and Fast Algorithms

Using accurate dose calculations for all iterations of an optimization process to minimize OCE is impractical due to the excessive dose calculation time required by accurate dose calculation algorithms. A solution to the dose calculation speed vs accuracy dilemma is to inter- lace the use of fast and accurate dose calculations during the plan optimization process. Fast but less accurate algorithms can be used for most of the optimization iterations and slower accurate methods for a smaller number of final iteration.

One of the simplest approaches is to use sequentially the different algorithms during the IMRT optimization.

For example, fast PB dose computations can be used in the initial stages of the optimization, with the results be- ing provided to a slower, more accurate SC algorithm.

Because the PB is very fast, it can be used for any initial optimization tasks such as selection of the number of beams and of gantry, couch, and collimator angles. Fol- lowing determination of the optimal beam configuration and convergence of the PB optimization, resultant inten- sity distributions can be used as input to an optimization which uses SC dose algorithms. Because the initial PB calculation provides a good initial guess to the SC-based optimization, the number of SC iterations is reduced, thus, reducing the optimization time compared to an optimization using only the SC algorithm. It has been demonstrated that this technique yields plans that are equivalent to using the accurate algorithm throughout the optimization [19].

More efficient processes utilize fast algorithms throughout the optimization process. These are termed hybrid dose computation approaches since they entail combining or mixing of dose calculation algorithms.

One such approach uses an SC dose algorithm to calcu-

late an approximate scatterless dose kernel based upon

the ray-traced dose for each element of the intensity ma-

trix, then uses this kernel, implemented as a fast table

look-up, for subsequent dose calculations to compute

dose during the optimization. The kernel is periodi- cally updated by re-computing with the SC calculation to reduce the inaccuracies of using the scatterless ker- nel [20]. The scatterless kernel dose calculation is more than 100 times faster than the SC dose computation, thus the speed is dominated by the number of SC dose computations required.

Other hybrid methods use a periodically updated beam-by-beam dose correction matrix to correct dose values computed with a fast algorithm. The dose cor- rection matrix has the same dimensions as the patient dose matrix and is used to describe the deviation be- tween the fast dose algorithm results and the accurate dose algorithm results at a fixed point during opti- mization. Both multiplicative and additive correction matrices have been used. A flow diagram for the correc- tive additive dose-correction method is given in Fig. 6 using PB as the fast algorithm and SC as the slower, more accurate algorithm, although other algorithms could be used in this loop. At the beginning of the op- timization, all elements in the dose correction matrix are set equal to zero (Box 1). The optimization process (Fig. 6 Box 2), which consists of an optimization pro- cess such as one from Fig. 2, proceeds using the fast PB algorithm with the corrected dose D

=

+ C.

Following optimization convergence, the dose is re- computed with the more accurate algorithm (Box 4,

) and results are compared with D

. If the re- sults differ by more than the convergence criteria, then the correction matrix is updated using C =

^–D

and the optimization continues. Otherwise, the cor- rection matrix has converged and the optimization is completed.

Fig. 6. Flow diagram for a correction-based dose scheme for use in IMRT optimization

It is important to note that for the correction based methods, immediately after the correction matrix is set to C =

–D

, the initial dose in the optimization loop (Box 2) is computed with D

=

+ C =

, that is, the optimization occurs with the SC as the basis dose. It can be shown that, using the correction methods, the fast (PB) algorithm only effectively operates on the change in the fluence (or intensity) ∆

converges (as ∆ ^ln → 0), D

→ D

. The equivalence of plans developed using the correction methods and using an accurate algorithm throughout optimization has been empirically demonstrated [42].

6.7 Outlook: Monte Carlo for IMRT Dose Calculations

The desire for highly accurate optimized IMRT dose distributions will likely lead to widespread clinical im- plementation of MC algorithms into IMRT systems in the future. To date, the major use of MC for IMRT has been to verify plans developed with non-MC algo- rithms [43–45]. Although general agreement between the treatment planning systems dose calculation al- gorithm and MC is observed in these studies, dose differences of 10–20% for some patients have been observed.

For IMRT plan optimization, even though MC has been used for plan optimization in a test study [46], it is currently considered too slow for routine IMRT optimization. Within the next few years, several key components are coming into place which will enable future MC-based IMRT optimization including:

• Fast, new generation of fast MC dose computation algorithms such as XVMC [47,48], VMC++ [49], and DPM [50], which reduce the MC by factors of 4–20.

• Denoising techniques [51, 52], in which the statisti- cal noise inherent in MC dose computations can be reduces, reducing the MC dose calculation time to by a factor of 4–10.

• Hybrid dose calculation strategies, which can reduce the number of MC dose computations required to be three or less per optimization.

Combinations of these techniques should allow accu- rate MC dose calculation during optimization, which was previously considered too time consuming. MC optimized plans should minimize dose prediction and optimization convergence errors, resulting in improve- ment in the quality of deliverable IMRT plans.

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