The effect of dose calculation uncertainty on the evaluation of radiotherapy plans

Transcription

1 The effect of dose calculation uncertainty on the evaluation of radiotherapy plans P. J. Keall a) and J. V. Siebers Department of Radiation Oncology, Medical College of Virginia Hospitals, Virginia Commonwealth University, Richmond, Virginia R. Jeraj Reactor Physics Division, Jožef Stefan Institute, University of Ljubljana, Slovenia R. Mohan Department of Radiation Oncology, Medical College of Virginia Hospitals, Virginia Commonwealth University, Richmond, Virginia Received 1 June 1999; accepted for publication 20 December 1999 Monte Carlo dose calculations will potentially reduce systematic errors that may be present in currently used dose calculation algorithms. However, Monte Carlo calculations inherently contain random errors, or statistical uncertainty, the level of which decreases inversely with the square root of computation time. Our purpose in this study was to determine the level of uncertainty at which a lung treatment plan is clinically acceptable. The evaluation methods to decide acceptability were visual examination of both isodose lines on CT scans and dose volume histograms DVHs, and reviewing calculated biological indices. To study the effect of systematic and/or random errors on treatment plan evaluation, a simulated error-free reference plan was used as a benchmark. The relationship between Monte Carlo statistical uncertainty and dose was found to be approximately proportional to dose. Random and systematic errors were applied to a calculated lung plan, creating dose distributions with statistical uncertainties of between 0% and 16% 1 s.d. at the maximum dose point and also distributions with systematic errors of 16% to 16% at the maximum dose point. Critical structure DVHs and biological indices are less sensitive to calculation uncertainty than those of the target. Systematic errors affect plan evaluation accuracy significantly more than random errors, suggesting that Monte Carlo dose calculation will improve outcomes in radiotherapy. A statistical uncertainty of 2% or less does not significantly affect isodose lines, DVHs, or biological indices American Association of Physicists in Medicine. S Key words: statistical uncertainty, accuracy, Monte Carlo, dose calculation I. INTRODUCTION The era of routine clinical Monte Carlo treatment planning for radiotherapy is imminent. Monte Carlo calculations are potentially more accurate than the best currently available commercial algorithms. However, with the advent of Monte Carlo treatment planning, some new challenges need to be addressed. One such challenge is how to deal with the statistical uncertainty inherent in all Monte Carlo calculations. With a Monte Carlo calculation, increased accuracy reduced systematic error is traded at the cost of statistical uncertainty random error. Due to the need to make assumptions about the distribution of the electrons above the target, imperfections in beam alignment, blemishes in the treatment head manufacture, etc., along with treatment setup errors and patient organ motion, Monte Carlo even ignoring limitations in the modeled physics is not perfectly accurate in computing dose. The variance of a Monte Carlo calculation result decreases inversely to the computer time used to solve the problem. 1 Hence the statistical uncertainty decreases with the square root of calculation time, which converges slowly. Obviously calculations with no uncertainty are most desirable, however, this would take infinite time to calculate. For a Monte Carlo dose calculation, we must accept a certain level of statistical fluctuations. Our aim in this research was first to determine the characteristics of Monte Carlo dose calculation uncertainty and, second, to use these characteristics to establish which statistical uncertainty level is acceptable for radiotherapy dose evaluation. II. METHOD AND MATERIALS A. Determining the characteristics of Monte Carlo uncertainty Monte Carlo calculations were performed to determine the properties of the statistical fluctuations as a function of the dose level in a Monte Carlo generated dose distribution. These calculations used EGS4 2 /BEAM 3 for the phase space creation and MCVRTP MCVRTP is a C Monte Carlo code provided by ADAC Laboratories ADAC Laboratories, Milpitas, CA that uses many of the published algorithms of EGS4 for the patient dose calculation. This configuration was used to obtain a dose distribution for a cm 2 open, 478 Med. Phys. 27 3, March Õ2000Õ27 3 Õ478Õ7Õ$ Am. Assoc. Phys. Med. 478

2 479 Keall et al.: The effect of dose calculation uncertainty 479 and cm 2 45 wedged 6 MV field at 100 cm SSD incident on a water phantom of size cm 3 divided into cm 3 voxels. Here particles were sampled from a BEAM-generated phase space file for a Varian Varian Oncology Systems, Palo Alto, CA C in 6 MV mode stored at the monitor chamber exit containing particles stored with cylindrical symmetry, and each particle incident on the patient was reused 17 times. The discrete electron AE and photon AP particle creation thresholds, and electron ECUT and photon PCUT transport thresholds were AE ECUT 0.7 MeV, AP PCUT 0.01 MeV. The variance note that variance in this work refers to the square of the standard error of the mean, ( x ) 2 was calculated by considering the energy deposition in each voxel of each individual particle history as a separate sample, which minimizes the dispersion of the variance. It was hypothesized that the variance for a Monte Carlo calculation is proportional to the dose see the results for justification of this hypothesis. This hypothesis assumes that the electron energy spectrum varies little throughout the phantom. To investigate the validity of this hypothesis statistical uncertainty was applied to a random error-free cm 2 6 MV dose distribution, calculated with convolution. To add statistical uncertainty to a dose distribution, the uncertainty in the maximum dose is specified, D max. For each voxel i, the simulated dose D i is calculated by sampling a Gaussian distribution centered on 0, (G( )), whose variance is proportional to dose, and adding the sampled value to the original dose, D i D i G D max D i /D max. To check the implementation of the variance/dose approximation, fields with added statistical uncertainty were compared to actual Monte Carlo calculated distributions. The uncertainties 1 in the maximum dose point were 0.6, 4.0, and 14.0%. B. Determining the uncertainty level acceptable for radiotherapy plan evaluation 1 A five-field radiotherapy plan to treat a lung tumor was used to assess the effect of Monte Carlo uncertainty on plan evaluation. The evaluation methods investigated were isodose curves, dose volume histograms DVHs, tumor control probability TCP, and normal tissue complication probability NTCP. The CT information, contours, beams, and beam weights used here were those used for the patient s treatment. A lung case was chosen, as plans with large density and/or atomic number variations are those for which Monte Carlo calculations will likely be most useful. Monte Carlo results without random errors are not possible the calculation would take infinite time, however, a reference plan with no statistical uncertainty was needed, as the benchmark with which to compare plans with added random or systematic errors. Hence, the reference plan was calculated using the best available random error-free algorithm in our department, the Pinnacle 3 ADAC Laboratories, Milpitas, CA collapsed cone convolution algorithm. Discrepancies between a convolution calculated plan and a Monte Carlo calculated plan are not important here, as relative dose differences as a function of added uncertainty are being investigated. Statistical uncertainty was added in the same manner as that described above for the cm 2 field incident on a water phantom. Statistical uncertainty levels simulated were 0%, 0.5%, 1%, 2%, 4%, 8%, and 16% of the maximum dose for each beam. The voxel size used for the calculation was mm 3. Note that for this particular case, the uncertainty at D max per beam approximately coincided with the uncertainty at D max for the treatment plan. To simulate systematic errors, each dose point was scaled by the magnitude of the systematic error. Such an error would occur if, for example, the machine output changed. Such a global scaling to simulate systematic errors between the true dose and that calculated with conventional dose algorithms is a crude approximation, however, is included here for comparison purposes. Systematic errors of magnitude 0%, 0.5%, 1%, 2%, 4%, 8%, and 16% were applied to the reference plan. For each plan at each statistical uncertainty level and systematic error level, target and critical structure DVHs were obtained. Biological indices were calculated from the DVHs. The TCP was calculated using TCP 0.5 exp 2 50 /ln 2 1 EUD/D 50, with a 50 value the percent increase in TCP per percent increase in dose at TCP 50% of 1.0 for a primary lung tumor. 4 The dose yielding 50% TCP (D 50 ) of 58 Gy was calculated from the information given in Munzenrider et al. 4 that for a primary lung tumor the TCP 70% at 70 Gy with a 50 of 1.0. The equivalent uniform dose EUD was calculated from ln 1 D V i SF i 2 /2 i EUD 2. 3 ln SF 2 V is the volume composed of elements i, and SF 2, the surviving fraction of cells after irradiation of 2 Gy, was set to 0.5. Equation 2 can be derived from Niemierko. 5 It is possible that due to uncertainties in dose calculation and treatment delivery that the 50 value is greater than 1, so calculations for 50 3 were also performed. The dose for the reference plan was set to give a TCP of 50% for a calculation with no uncertainty. A TCP of 50% for this plan corresponds to a maximum dose of 62.6 Gy. The NTCP for pneumonitis in the lung was calculated using the Lyman model. 6 The volume dependence parameter n was 0.87, the slope of the complication probability versus the dose parameter m was 0.18, and the dose to the whole organ that would lead to a complication probability of 50% TD 50 was 24.5 Gy. These values were tabulated in Burman et al., 7 based on data taken from Emami et al. 8 2

3 480 Keall et al.: The effect of dose calculation uncertainty 480 FIG. 1. Variance versus dose level for the Monte Carlo calculated dose distribution for a a cm 2 open 6 MV beam and b a cm 2 45 wedged 6 MV beam. C. Worst case TCP calculations Only a lung plan has been studied thus far. For generality, we calculated TCP values for a variety of situations including those likely to be at the extremes of clinical reality. Due to the sigmoidal nature of the dose/response curve, TCP calculations are most strongly influenced by the lowest dose values in the volume. Hence, a step function DVH, where all the values are the lowest in the volume, represents the shape most susceptible to statistical uncertainty in dose. A parametric study was performed, where TCP for a step function DVH was calculated. Both D max from 0.9 to 1.4 D 50, and the statistical uncertainty at D max from 0% to 16%, were varied, for 50 values of 1, 3, and 5. III. RESULTS A. Determining the characteristics of Monte Carlo uncertainty Figures 1 a and 1 b show the variance versus absolute dose cgy/monitor unit for a cm 2 open, and a cm 2 45 wedged, 6 MV field at 100 cm SSD, respectively. This figure shows that almost all of the variance points fall along a line proportional to the dose. The variance is relatively lower in the dose buildup region see the points near the buildup label in Fig. 1 a than in other voxels with the same dose. This reduction is because there are more primary photon interactions at the surface than elsewhere in the beam especially of the low-energy components of the incident energy spectrum, and contaminant electrons in the beam interact at the surface. Variance is also relatively lower outside the beam points left of the vertical line near the outside field label in Fig. 1 a and in the penumbra region see the points near penumbra label in Fig. 1 a due to the lower average energy transfer per interaction for scattered photons as opposed to primary photons. Therefore, when using the variance/dose proportionality approximation to simulate Monte Carlo statistical uncertainty, the variance in the buildup region, in the penumbra and outside the field will be slightly overpredicted. For all Monte Carlo calculations performed in both homogeneous and heterogeneous phantoms, the correlation coefficient was greater than 0.9, indicating that the variance/dose proportionality assumption is a reliable description of the actual relation between dose and variance. Interestingly, for Monte Carlo calculations in heterogeneous geometries, lines of different slopes corresponding to voxels of different density were observed. The generality of the variance/dose proportionality approximation is shown in the comparison of Figs. 1 a and 1 b, which display the same constant of proportionality slope, even though the beam in Fig. 1 b has been modified by a wedge. FIG. 2. Monte Carlo left and simulated right dose distributions for statistical uncertainty levels of a 14%, b 4%, and c 0.6%. The outermost isodose line corresponds to 10 cgy for a 100 cgy prescription and the isodose levels increase by 10 cgy.

4 481 Keall et al.: The effect of dose calculation uncertainty 481 FIG. 3. Simulated isodose distributions for uncertainty levels in each beam at D max of a 0% pure collapsed cone convolution calculation, b 1.0%, c 2.0%, d 4%, and e 16%. To check the implementation of the uncertainty modeling, dose calculations for a cm 2 6 MV field were calculated using Monte Carlo for three different uncertainty levels at D max, 14%, 4%, and 0.6% were performed. These same levels were simulated using the variance/dose proportionality approximation. The results of the Monte Carlo simulations and variance/dose proportionality approximation are shown in Fig. 2. This figure shows that for the cases studied, the uncertainty appears to have been correctly applied, is valid for a range of uncertainties, and therefore, along with the results from Fig. 1, can be used to determine the statistical uncertainty level acceptable for radiotherapy plan evaluation. B. Determining the statistical uncertainty level acceptable for radiotherapy plan evaluation Treatment plans are generally assessed in three ways: evaluating isodose lines plotted on a CT dataset, evaluating the DVHs for the target and critical organs, and/or evaluating predicted biological outcomes of the proposed plan. The effect of uncertainty on these three assessment criteria is discussed below. Figure 3 a shows a transverse slice of the plan calculated using collapsed cone convolution that we use as the reference plan. Figures 3 b 3 e show plans with simulated Monte Carlo uncertainty levels at D max of 1%, 2%, 4%, and

5 482 Keall et al.: The effect of dose calculation uncertainty 482 TABLE I. TCP calculations for various uncertainty levels calculated for both simulated Monte Carlo and systematic errors for a 50 1, and b The and in the systematic error columns indicate whether the error is negative or positive. Magnitude of error as a % of D max Statistical uncertainty Systematic error FIG. 4. Target dose volume histograms for various statistical uncertainty levels. a TCP 50% for error-free plan , , , , , , 65 b TCP 49% for error-free plan , , , , , , 87 16%, respectively. Obviously, increasing the uncertainty level increases the jaggedness of the isodose lines, and the ability to evaluate the plan is degraded. Although the acceptable uncertainty level for a visual plan is individual and subjective, up to a 2% level of uncertainty cf. Fig. 3 c at D max does not significantly affect the isodose lines. The target DVHs for uncertainty levels of 0%, 2%, 4%, 8%, and 16% are shown in Fig. 4. Adding statistical uncertainty to a DVH has the effect of smoothing the curve for a step function DVH adding statistical uncertainty is equivalent to a convolution with a Gaussian curve. From this graph it can be seen that a Monte Carlo uncertainty level of 2% or less does not significantly affect the DVH. Figure 5 shows lung DVHs for 0%, 8%, and 16% uncertainties. Comparing Figs. 4 and 5 shows that the effect of statistical uncertainty on a critical structure is much less than that of the target. This difference is because the effect of dose distribution uncertainty on a DVH is essentially the same as convolving the DVH with a Gaussian distribution. Hence, for a target DVH, which resembles a step function, the uncertainty has the greatest effect. However, for critical structure DVH curves, the slopes are generally shallow, and the convolution with a Gaussian has little effect. Unlike isodose plans, in which the evaluation of the acceptance level of the statistical uncertainty is subjective, the effect of uncertainty on DVHs can be quantified by calculating biological indices. The TCP and NTCP biological indices for the different uncertainty levels are displayed in Table I and Table II, respectively. In Table 1 a, TCP calculations are given for different error levels both random and systematic for 50% TCP at Dmax 0 and Table 1 b gives results of calculations with Table I shows the relative insensitivity of the TCP to random errors for these 50 values, with the calculation result not changing to 2 significant figures for a random error level of 2% or less. However, it is seen that any systematic errors have a direct effect on the predicted TCP. Note that if random errors are present, the TCP calculation will generally underpredict the error-free value. Table II gives the predicted lung NTCP for various uncertainty levels, both random and systematic. As can be inferred from the DVHs, the random error level has little effect on the NTCP calculations. Systematic errors show more difference in the calculated NTCP values than random errors. FIG. 5. Lung dose volume histograms for various statistical uncertainty levels. C. Worst case TCP calculations TCP as a function of D max /D 50, and the statistical uncertainty at D max for 50 5 is shown in Fig. 6. Plots of Dmax from 0 16, and from 0 6 are shown. The larger range shows the sharp drop in TCP when Dmax becomes greater than approximately 6%, and the narrower range shows that even

6 483 Keall et al.: The effect of dose calculation uncertainty 483 TABLE II. Lung NTCP calculations for various uncertainty levels calculated for both simulated Monte Carlo and systematic errors. The and in the systematic error columns indicate whether the error is negative or positive. Magnitude of error as a%of D max NTCP at TCP 49% for error-free plan Statistical uncertainty NTCP at TCP 80% for error-free plan NTCP at TCP 49% for error-plan plan Systematic error NTCP at TCP 80% for error-free plan , , , , , 5 13, , 8 7, 52 for a high 50 value the TCP surface remains relatively constant. Though not presented here, as expected, the slope of the TCP surface for 50 3 is less than that of 50 5, and the surface is flatter still for This indicates that for most clinical cases, a high statistical uncertainty will not significantly affect TCP calculations. IV. DISCUSSION Although only one treatment plan was investigated here, the results can be generalized, as isodose plans are essentially similar from an uncertainty perspective. Also, target DVHs generally have steep slopes, and critical structure DVHs generally have shallow slopes. The 50 value used for the TCP calculations is the maximum for many tumor sites 4 and hence will overpredict the effect of uncertainty on the TCP as compared to lower 50 values. The jittery nature of isodose lines for treatments plans containing statistical uncertainty is also dependent on the voxel size used for the calculation mm 3 was used here. Larger voxels exhibit less jaggedness for the same uncertainty level, whereas small voxels give more. Smaller voxels also need more particles and hence more CPU time to achieve the same uncertainty. Advanced isodose plotting algorithms that reduce the jaggedness of the isodose curves without loss of accuracy may reduce this problem, if such algorithms can be devised and implemented. The insensitivity of DVHs and TCP/NTCP calculations to statistical uncertainty mean that Monte Carlo calculations with high statistical uncertainty could be used as the dose engine for inverse planning, significantly reducing the overall computation time, while retaining the accuracy of the calculation. Several scientists 9 have also suggested that the statistical uncertainty can be removed from DVHs, which would allow the use of even higher uncertainties for inverse planning. FIG. 6. TCP as a function of statistical uncertainty and D max dose for 50 5.

7 484 Keall et al.: The effect of dose calculation uncertainty 484 V. CONCLUSIONS The characteristics of Monte Carlo dose uncertainty as a function of the dose level have been studied. The results indicate that the variance can be approximated as being proportional to the dose to simulate statistical uncertainty in Monte Carlo dose calculations. The application of random errors to dose distributions to simulate Monte Carlo uncertainties is significantly less time consuming than actual Monte Carlo simulations. The variance/dose proportionality approximation to simulate Monte Carlo generated statistical uncertainty was applied to a five-field treatment plan for a lung tumor. The effect of uncertainty on three radiotherapy plan evaluation criteria has been studied. Statistical uncertainty is most evident in isodose distributions, while DVHs are less sensitive. TCP and NTCP calculations are even more resistant to uncertainty than DVHs. Though the acceptable level of uncertainty varies depending on the plan evaluation criteria, it is the opinion of the authors that a statistical uncertainty of 2% or less at D max does not significantly affect the results of any of the plan evaluation criteria. The sensitivity of TCP/NTCP calculations to systematic errors highlights the importance of using accurate dose calculation algorithms, both to simulate the transport of the radiation in patients, and also the description of the radiation source incident on the patient. The measurements used in the commissioning of the dose calculation algorithm must also be as true as possible. ACKNOWLEDGMENTS The authors wish to thank Dr. James Satterthwaite and Dr. Cyrus Amir for useful statistical discussions. The MCV authors acknowledge the financial support of NIH Grant No. CA a Corresponding author: Department of Radiation Oncology, MCV Hospitals, Virginia Commonwealth University, P.O. Box , Richmond, Virginia Phone ; fax: ; electronic mail: pjkeall@hsc.vcu.edu 1 A. F. Bielajew and D. W. O. Rogers, Variance-reduction techniques, in Monte Carlo Transport of Electrons and Photons, edited by T. M. Jenkins, W. R. Nelson, and A. Rindi Plenum, New York, 1988, pp W. R. Nelson, H. Hirayama, and D. W. O. Rogers, The EGS4 Code System, Stanford Linear Accelerator Center, Report No. SLAC-265, D. W. O. Rogers et al., BEAM: A Monte Carlo code to simulate radiotherapy units, Med. Phys. 22, J. E. Munzenrider et al., Numerical scoring of treatment plans, Int. J. Radiat. Oncol., Biol., Phys. 21, A. Niemierko, Treatment evaluation, in Teletherapy: Present and Future, edited by T. R. Mackie and J. R. Palta Advanced Medical Publishing, Madison, WI, J. T. Lyman, Complication probability as assessed from dose volume histograms, Radiat. Res. 104, S13 S C. Burman et al., Fitting of normal tissue tolerance data to an analytic function, Int. J. Radiat. Oncol., Biol., Phys. 21, B. Emami et al., Tolerance of normal tissue to therapeutic radiation, Int. J. Radiat. Oncol., Biol., Phys. 21, T. R. Mackie personal communication 1999 ; also in presentations by Dr. C.-M. Ma and Dr. J. Sempau at the Electron and Photon Transport Theory Workshop, Indianapolis, August 1999.