Cluster model analysis of late rectal bleeding after IMRT of prostate cancer: A case–control study

Int. J. Radiation Oncology Biol. Phys., Vol. 64, No. 4, pp. 1255–1264, 2006 Copyright © 2006 Elsevier Inc. Printed in the USA. All rights reserved 0360-3016/06/$–see front matter

doi:10.1016/j.ijrobp.2005.10.029

PHYSICS CONTRIBUTION

CLUSTER MODEL ANALYSIS OF LATE RECTAL BLEEDING AFTER IMRT OF PROSTATE CANCER: A CASE– CONTROL STUDY SUSAN L. TUCKER, PH.D.,* MING ZHANG, PH.D.,* LEI DONG, PH.D.,† RADHE MOHAN, PH.D.,† DEBORAH KUBAN, M.D.,‡ AND HOWARD D. THAMES, PH.D.* Departments of *Biostatistics and Applied Mathematics, †Radiation Physics, and ‡Radiation Oncology, The University of Texas M.D. Anderson Cancer Center, Houston, TX Purpose: Cluster models are newly developed normal-tissue complication probability models in which the spatial aspects of radiation-induced injury are taken into account by considering the size of spatially contiguous aggregates of damaged tissue units. The purpose of this study was to test the validity of a two-dimensional cluster model of late rectal toxicity based on maximum cluster size of damage to rectal surface. Methods and Materials: A paired case– control study was performed in which each of 9 patients experiencing Grade 2 or higher late rectal toxicity after intensity-modulated radiation therapy of localized prostate cancer was paired with a patient having a similar rectal dose–surface histogram but free of rectal toxicity. Numeric simulations were performed to determine the distribution of maximum cluster size on each rectal surface for each of many different choices of possible model parameters. Results: Model parameters were found for which patients with rectal toxicity were consistently more likely to have a significantly larger mean maximum cluster size than their matched controls. These parameter values correspond to a 50% probability of tissue-unit damage at doses near 30 Gy. Conclusions: This study suggests that a cluster model based on maximum cluster size of damage to rectal surface successfully incorporates spatial information beyond that contained in the rectal dose–surface histogram and may therefore provide a useful new tool for predicting rectal normal-tissue complication probability after radiotherapy. © 2006 Elsevier Inc. Cluster models, Normal-tissue complication probability, Prostate cancer, Rectal bleeding.

INTRODUCTION

rectum exposed to doses of 75 Gy or more, was at least 40% (31), and of Storey et al., who found a higher incidence of Grade 2 or greater toxicity among patients with rectal V70 greater than 25% (32). In contrast, other studies have suggested that the volume of rectum receiving doses substantially lower than 60 Gy might be more strongly associated with rectal toxicity. For example, Skwarchuk et al. found that in patients with prescribed prostate doses of 70.2–75.6 Gy, the incidence of Grade 2 or greater rectal bleeding was significantly higher when the 50% isodose curve (corresponding to doses in the 35–38 Gy range) encompassed the entire rectal contour on the isocenter slice of the treatment planning CT scan (33). Similarly, a recent analysis from our group indicated that patients at highest risk of Grade 2 or greater late rectal bleeding were those for whom V32 for rectal wall was 80% or higher (34). Other studies have suggested that V50 may be the parameter most strongly associated with rectal bleeding (35, 36). The inconsistencies in the conclusions reached by various

Many studies have indicated that prostate carcinoma is a dose-responsive neoplasm and could therefore likely benefit from dose escalation (1–10). However, a major limitation to dose escalation for prostate cancer is the tolerance of rectum (11, 12). Information relating the risk of rectal toxicity to the rectal dose distribution is therefore vital for ongoing improvement in the design of three-dimensional conformal or intensity-modulated radiation therapy (IMRT) plans for radiotherapy to the prostate. Numerous studies have sought to determine the relationship between the radiation dose distribution to rectum and the associated rectal complication probability (13–30). Some of the published dose–volume analyses of rectal toxicity have suggested that late rectal toxicity is a consequence of high-dose (⬎60 Gy) exposure to portions of the rectal wall. Among these are the studies of Benk et al., who reported an increased rate of Grade 1 or higher late rectal bleeding among patients for whom V75, the percentage of Reprint requests to: Susan L. Tucker, Ph.D., Department of Biostatistics and Applied Mathematics, The University of Texas M.D. Anderson Cancer Center, 1515 Holcombe Blvd., Unit 237, Houston, TX 77030. Tel: (713) 792-2613; Fax: (713) 792-4262;

E-mail: [email protected] Received July 5, 2005, and in revised form Oct 20, 2005. Accepted for publication Oct 25, 2005. 1255

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normal-tissue complication probability (NTCP) analyses of rectal toxicity may be explained, at least in part, by the strong correlations that tend to occur between the volumes treated to any two dose levels: V70 and V30, for example, or between any other pair, V(high dose) vs. V(lower dose). This is especially true for data from a single institution, where the similarity in treatment plans among patients contributes to such correlations. In dose–volume analyses, this phenomenon probably contributes to an element of chance as to whether V(high dose) or V(lower dose) is found to be more significantly associated with toxicity for a particular data set. However, the problem might also be due to the use of quantities derived from the dose–volume histogram (DVH) in analyses of complication risk. It has often been noted that the DVH suffers from the limitation that it does not retain spatial information about the dose distribution (37). In particular, the DVH does not indicate which portions of the organ receive which doses, or whether the high-dose region is a single contiguous volume or consists of two or more spatially separated subvolumes. The same criticism applies to the dose–wall histogram for rectal wall (38, 39) and the dose–surface histogram (DSH) for rectal surface (40 – 42). Some of the inconsistencies in the conclusions reached by the NTCP analyses of rectal toxicity cited above could be due to the failure of quantities such as V30 and V70 to incorporate relevant spatial information. Recently, there have been efforts to develop a new class of NTCP models called “cluster models,” in which the spatial distribution of dose to normal tissue is taken explicitly into account (43, 44). Such models are based on the hypothesis that the complication risk in normal tissues depends on the size of “clusters” of radiation damage. The organ is viewed as being made up of many small subunits of tissue, called “tissue units” or “functional subunits,” which may or may not correspond to physiologically defined structures such as nephrons in kidney or alveoli in lung. For example, they may correspond simply to voxels of tissue such as areas of size 0.1 cm2 in skin or volumes of 0.125 cc in lung. In cluster models, it is assumed that each tissue unit is either completely damaged or remains completely functional after radiotherapy, and the survival or destruction of each tissue unit is a random event whose probability depends on dose. In the simplest version of cluster models, the risk of organ toxicity is assumed to depend entirely on the maximum cluster size, i.e., the size of the largest damage cluster occurring as an aggregate of destroyed tissue units in spatial proximity to one another. In an organ such as the rectal surface, which can be modeled as a two-dimensional structure, the damaged tissue units can be viewed, conceptually, as small black squares placed randomly on a grid consisting of many such squares. The location of the black squares is random, with a higher density of squares in regions of the grid exposed to higher doses. By chance, many of the black squares may be spatially isolated, but, depending on the dose distribution, there may be regions where clusters of black squares occur, each

Volume 64, Number 4, 2006

cluster sharing one or more sides with other black squares. The maximum cluster size is a random variable depending on the dose distribution to the rectal surface but determined in part by chance. In the current study, we used a two-dimensional cluster model based on maximum cluster size to analyze late rectal toxicity data from a group of patients treated with IMRT for prostate cancer. Because there were too few events, as yet, to do a full NTCP analysis based on the cluster model, we performed a paired case– control study. The goal of the study was to validate the concept that a cluster model based on maximum cluster size can be used to distinguish between patients with and without rectal toxicity and therefore usefully incorporates spatial information not contained in the rectal DSH.

METHODS AND MATERIALS Patient population The data analyzed in this study are from patients with localized prostate cancer who were treated with IMRT at The University of Texas M.D. Anderson Cancer Center between January 2000 and November 2001. All patients included in this study received 42 fractions of 1.8 Gy per fraction (total dose 75.6 Gy) and received no adjuvant or neoadjuvant hormone therapy. The original treatment plans were designed by a commercial treatment planning system (Corvus; North American Scientific, Chatsworth, CA), and patients were treated with either the serial tomotherapy device (Peacock; North American Scientific, Chatsworth, CA) or the step-and-shoot multileaf collimator (Varian Medical Systems, Palo Alto, CA). All patients had ultrasonic prostate localization (BAT; North American Scientific, Chatsworth, CA) before each treatment. A subset of these patients was selected, as described below, to form a retrospective paired case– control study, with one matched control per case. This retrospective analysis was approved by the Institutional Review Board of The University of Texas M.D. Anderson Cancer Center. The cases consisted of all patients who developed Grade 2 or higher late rectal toxicity by the time of the present analysis (January 2005), excluding patients with diabetes mellitus or a history of rectal hemorrhoids. Patients with these factors are known to be at high risk for rectal bleeding after radiotherapy, and the rationale for excluding them was to minimize, as much as possible, the impact of nondosimetric confounding effects. Late rectal toxicity was defined as toxicity occurring at least 6 months after the end of treatment and was scored on a scale based on modifications of the Radiation Therapy Oncology Group and Late Effects Normal-Tissue Task Force scales, as described previously (32). There were 9 patients who qualified as cases for our study (all with rectal bleeding). As potential controls, we considered all remaining patients who had at least 3 years of follow-up after IMRT (n ⫽ 54). This time point was selected because the vast majority of patients who experience rectal toxicity after radiotherapy do so within 2 years, and nearly all do so within 3 years. Patients followed for 3 years without rectal complications can therefore reasonably be viewed as free of Grade 2 or higher rectal complications.

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Fig. 1. (Left panel) Contour points (connected by straight line segments) defining the location of the outer rectal surface, as entered by the attending physician on a cross-section of the rectum on the treatment planning computed tomography (CT) scan. The dot in the middle represents the average location (centroid) of the contoured points in this CT slice. (Right panel) Ray-contour points defined as the points of intersection between the rectal contour and rays emanating from the centroid at 5° intervals.

Dose–wall maps For each of the 63 patients mentioned above (9 cases and 54 potential controls), a “dose–wall map” was generated to represent the dose distribution to the two-dimensional rectal surface. The dose–wall map for the ith patient (i ⫽ 1, . . . , 63) is an array of dimension 72 ⫻ (2nzi ⫺ 2), where nzi is the number of slices in the z direction (superior-inferior) encompassing rectum in the treatment planning CT. The dose–wall map for each patient was generated as follows. Step 1: (Contouring). On each slice of the treatment planning CT (acquired at 3-mm intervals), the attending physician contoured the outer surface of the rectum by indicating the locations of discrete points around the perimeter of the rectum (Fig. 1, left panel). The coordinates of these contour points and the full threedimensional dose– distribution matrix were extracted from the commercial treatment planning system (Corvus; North American Scientific, Chatsworth, CA) and imported into our research computing system. Step 2: (Ray contour points). A computer program was written to calculate the centroid of the rectum (specifically, of the polygon defined by the contour points) on each CT slice and to compute the coordinates of the points of intersection between the rectal contour (the line segments connecting the contoured points) and each of 72 rays emanating from the centroid at evenly spaced 5° intervals (Fig. 1, right panel). The dose at each of the 72 resulting “ray contour points” was calculated based on interpolation from the doses in the three-dimensional dose– distribution matrix. Step 3: (Unwrapping). Conceptually, the resulting “cylindrical” array with 72 dose points per z level of rectum was then cut at the posterior-most position on each CT slice and “unwrapped” to form a flat rectangular 72 ⫻ nzi array representing the dose to the rectal surface. In reality, the “unwrapped” rectum does not lie flat, because the rectal surface is uneven, and each entry in the array represents the dose to one corner of an approximately trapezoidshaped area (voxel) of rectal surface.

Step 4: (z slice interpolation). Because, at this stage of the calculation, the voxels were found to be approximately twice as tall (3 mm) as they were wide (average width 1.3 mm), one additional row (z slice) was added by interpolation between each pair of adjacent rows in the 72 ⫻ nzi array, resulting in a 72 ⫻ (2nzi ⫺ 1) array with voxels that were more nearly square. Step 5: (Final dose–wall map). The voxels in the interpolated array correspond to the “tissue units” of the rectal surface in our application of the cluster model to the IMRT data. The dose to the center of each voxel was calculated by interpolation from the dose to each of its four corners, resulting in the final 72 ⫻ (2nzi ⫺ 2) dose–wall map for each patient.

Selection of control patients for the matched case– control study Because the purpose of this study was to test cluster models for their ability to extract spatial information from the dose distribution to rectal surface beyond that contained in the DSH, we sought to match each of the 9 cases with a control patient having the same DSH. Specifically, each case was matched with the control whose absolute cumulative DSH (calculated from the dose–wall map) had the smallest mean-squared difference from the DSH of the case, assessed at doses ranging from 0 to 100 Gy in increments of 0.1 Gy.

Local-damage function Use of the cluster model requires specification of a localdamage function describing the probability p(D) of tissue-unit damage as a function of dose D. The local-damage probability was modeled using a logistic function of dose with two unknown parameters, b1 and b2: p(D) ⫽ 1 ⁄ 兵1 ⫹ exp关b1 ⫺ b2共␣ ⁄ ␤ ⫹ x兲D兴其

(Eq.1)

In Eq. 1, the quantity (␣/␤ ⫹ x)D represents a transformation of

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Dose (Gy) Fig. 2. Cumulative dose–surface histogram curves for each of the nine pairs of matched cases (solid curves) and controls (dashed curves). dose using the linear-quadratic model (45) to correct for fractionation effects; x is the dose per fraction (here, x ⫽ D/42), and ␣/␤ is the ratio of parameters from the linear-quadratic model. Throughout this study, ␣/␤ was assumed to be equal to 5.4 Gy, based on the recent estimate of ␣/␤ for rectal wall derived by Brenner (46).

Estimation of cluster size distributions

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Fig. 4. Rectal dose–wall maps for three matched case– control pairs. Each dose–wall map is a 72 ⫻ (2nzi ⫺ 2) array of voxels representing the dose distribution to rectal surface, where nzi is the number of computed tomography slices encompassing rectum. The six dose–wall maps are reproduced to scale, to preserve the relative dimensions. Voxels are colored gray, green, black, yellow, or red to indicate areas receiving doses ⱖ10, 30, 40, 60, or 80 Gy, respectively.

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For each patient in the study, and for each choice of localdamage parameters (b1, b2) considered, the distribution of maximum cluster size was calculated from the patient’s dose–wall map by means of numeric simulation. Numeric simulations comprised 2,000 replicates each and were performed as described below.

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The size of the maximum cluster of radiation damage to the rectal surface is a random variable depending on the spatial dose distribution to rectum, the values of the local-damage parameters b1 and b2, and subject in part to chance because of the random nature of radiation-induced damage. That is, in a hypothetical population of patients with identical anatomies, radiation sensitivities, and dose distributions to rectum, variations in the maximum cluster size of damage to rectal surface would be expected after irradiation, based simply on stochastic effects.

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z-slice location (cm) Fig. 3. Rectal circumference as a function of z slice location (cm from the inferior-most computed tomography slice incorporating rectum) for each of the nine cases (solid curves) and their matched controls (dashed curves).

Each replicate in the numeric simulations consisted of randomly assigning each voxel in a dose–wall map to be either damaged (black) or undamaged (white), based on the local-damage probability p(D) (Eq. 1) for that voxel. The size of the largest occurring cluster of black voxels was determined and recorded before moving on to the next replicate. The calculation of cluster size requires a choice of connec-

Cluster model analysis of late rectal bleeding

tivity for defining when an aggregate of damaged tissue units has formed a cluster (44). The definition of connectivity is given in Appendix 1. For the present study, 2-connectivity was used to define clusters. If the rectal surface were a regular cylinder, each voxel in the dose–wall map would represent an equal area of the rectal surface. However, the rectal surface is irregular, with intrapatient and interpatient variations in rectal circumference. Therefore, the choice of a fixed number of dose points per circumference in the dose–wall map (n ⫽ 72) implies that individual voxels represent regions of the rectal surface having different areas. Variations in voxel area were taken into account in the cluster size calculations, as described below. First, we wished to ensure that equal-sized regions of rectal surface exposed to the same dose would develop clusters with equal probability, regardless of the number of voxels into which those regions were subdivided. To accomplish this, a reference voxel area (Aref) was selected, and the area of each voxel was expressed as a multiple, m, of the reference area: voxel area ⫽ m · Aref. In the numeric simulations, the local-damage probability p ⫽ p(D) (Eq. 1) was replaced in the numeric simulations by pm for a voxel of area m · Aref. Second, we wished to take into account the different voxel areas contributing to each cluster. Therefore, a voxel of area m · Aref, if blackened, was weighted by the multiple m in calculating cluster size. The rationale for both of these adjustments is explained more fully in Appendix 1.

Investigation of local-damage parameter values The values of the local-damage parameters b1 and b2 (Eq. 1) govern the dose dependence of the damage probability of a tissue unit exposed to dose D. Appropriate values of the local-damage parameters for rectal surface are unknown. The goal of our study was to determine whether parameter values exist for which a simple two-dimensional cluster model based on maximum cluster size is consistent with the data from our case– control study.

RESULTS For each of the 9 cases in this study, it was possible to select a matched control with a very similar absolute rectal DSH, as shown in Fig. 2. None of the matched controls experienced any late rectal toxicity (including Grade 1) during 3 or more years of follow-up. Figure 3 illustrates the variability in rectal circumference across CT slices for the 18 patients in the study. The average rectal circumference per patient ranged from 8.2 cm to 11.8 cm (median 9.3 cm), with a coefficient of variation ranging from 9% to 34% (median 22%). The resulting distributions in voxel area were similar for all patients. The reference area, Aref, was chosen to be 0.02 cm2, corresponding to the mean voxel area overall. Figure 4 shows the dose–wall maps for 3 of the 9 pairs of patients in the matched case– control study. These pairs (Pairs 1, 7, and 9) correspond to the patients whose rectal

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Specifically, we sought to determine whether values b1 and b2 could be found for which the expected (mean) maximum cluster size for a patient experiencing Grade ⱖ2 late rectal bleeding (a case) was significantly larger than the mean maximum cluster size for the matched (control) patient without late rectal bleeding. Values of the local-damage parameters investigated in this study were all pairs b1, b2 in the range b1 ⫽ 1, 2, . . . , 40, and b2 ⫽ 0.01, 0.02, . . . , 0.2 Gy⫺2. For each choice of values for the local-damage parameters b1 and b2, the distributions in maximum cluster size generated for each case– control pair using numeric simulations were compared using a t-test, to test whether the mean maximum cluster size for the case was larger than the mean maximum cluster size for the control. The number of case– control pairs for which the comparison reached statistical significance (p ⬍ 0.05) was recorded.

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Fig. 5. Distributions of maximum cluster size obtained by numeric simulation for two of the matched case– control pairs using local-damage parameter values b1 ⫽ 3 and b2 ⫽ 0.03 Gy⫺2.

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panel of Fig. 5, and in this case the mean maximum cluster size for the case is actually significantly smaller than for the control patient. Table 1 lists, for each of various values of the localdamage parameters (b1, b2), the number of case– control pairs for which the mean maximum cluster size for the case was significantly larger than for its matched control. The maximum count observed for any of the parameter choices was 8 of the 9 case– control pairs, and there were 56 such instances, shown in boldface in Table 1. As an illustration, Table 2 lists the mean maximum cluster size obtained for cases and controls for one choice of parameter values b1 and b2. Although the differences were small, the mean maximum cluster size was nonetheless consistently larger for the case for all matched pairs except Pair 7.

DSHs are shown in the upper left, lower left, and lower right panels of Fig. 1, respectively. Each array in Fig. 4 is comprised of 72 ⫻ (2nzi ⫺ 2) voxels, where nzi is the number of slices encompassing rectum in the treatment planning CT, as described in “Methods and Materials.” The voxels in Fig. 4 have been colored to represent a few different dose levels. The distributions in maximum cluster size obtained by numeric simulation are illustrated in Fig. 5 for 4 patients (2 matched case– control pairs) using one choice of localdamage parameter values (Eq. 1). In the left-hand panel of Fig. 5, the distribution of maximum cluster size is displaced to significantly higher values for the case compared with the control (p ⬍ 0.0001, t-test). In fact, the probability is nearly 100% that a randomly selected maximum cluster size for the case (representing the maximum cluster of rectal damage that might occur by chance in this patient after radiotherapy) is larger than a randomly selected maximum cluster size for the matched control patient. In contrast, there is considerable overlap in the distributions shown in the right-hand

DISCUSSION In the present study, we investigated the use of a twodimensional cluster model to analyze rectal toxicity data

Table 1. Counts of the number of case– control pairs for which mean maximum cluster size is greater for the case than for the control for various values of the local damage parameters (Eq. 1) b1 and b2 b2 (Gy-2) b1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

.01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

7 7 5 4 3 3 3 4 4 4 3 3 5 5 5 5 5 3 3 3 4 2 2 2 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0

4 7 7 7 7 5 4 6 6 4 3 3 3 3 2 4 2 2 3 3 3 4 3 4 3 2 3 2 2 1 0 0 0 0 0 0 0 0 0 0

4 4 6 7 7 7 7 7 6 5 3 4 5 6 6 4 3 3 3 3 4 2 2 2 3 3 3 3 2 3 3 3 3 3 1 1 0 0 0 0

4 5 4 6 7 7 8 7 7 7 6 6 5 3 3 4 5 5 5 6 6 4 3 3 3 3 4 3 1 2 3 3 3 3 2 2 3 3 3 2

4 5 5 4 5 6 7 8 8 7 7 7 7 6 6 6 5 3 3 4 4 5 5 5 6 6 6 3 3 3 3 4 4 4 3 1 3 3 3 3

4 4 5 5 4 5 6 7 6 8 8 7 7 7 7 7 6 6 6 5 4 3 3 3 4 5 5 5 5 5 6 6 6 3 3 3 3 4 4 4

6 4 4 5 5 4 4 6 6 6 6 8 8 7 7 7 7 7 7 6 6 6 6 4 3 3 3 3 4 4 5 5 5 5 5 5 6 6 6 3

4 4 4 4 5 4 4 4 5 6 7 6 6 8 8 7 7 7 7 7 7 6 6 6 6 6 4 4 3 3 3 3 4 4 5 5 5 5 5 5

4 5 4 4 5 5 4 4 4 5 6 6 6 6 6 8 8 8 7 7 7 7 7 7 7 6 6 6 6 6 4 3 3 3 3 3 4 4 5 5

4 4 4 4 4 5 5 3 4 4 5 5 6 7 6 6 7 8 8 8 7 7 7 7 7 7 8 7 6 6 6 6 6 4 4 3 3 3 3 3

4 5 5 4 4 4 5 5 3 3 4 4 5 6 6 6 6 6 7 8 8 8 7 7 7 7 7 7 7 8 7 7 6 6 6 6 5 4 4 3

4 5 3 4 4 4 4 5 5 3 3 4 4 5 6 6 7 6 6 6 7 8 8 8 7 7 7 7 7 7 7 8 8 7 7 6 6 6 6 5

5 4 4 5 4 4 4 5 5 5 3 3 4 4 5 5 6 6 6 6 6 6 7 8 8 8 7 7 7 7 7 7 7 7 8 7 7 7 6 6

5 4 4 5 4 4 4 4 5 5 5 3 3 4 4 5 5 6 6 7 6 6 6 7 7 8 8 8 7 7 7 7 7 7 7 7 8 8 7 7

5 4 5 4 4 4 4 4 4 5 5 5 3 3 4 4 4 5 6 6 6 6 6 6 6 7 7 8 8 8 7 7 7 7 7 7 7 7 7 8

5 4 5 4 5 4 4 4 4 4 5 5 5 3 3 4 4 4 5 5 6 6 6 6 6 6 6 7 7 8 8 8 7 7 7 7 7 7 7 7

5 4 4 4 5 4 4 4 4 4 5 5 5 5 3 3 4 4 4 5 5 6 6 6 6 6 6 6 6 7 7 8 8 8 8 7 7 7 7 7

5 5 4 4 4 5 4 4 4 4 4 5 5 5 5 3 3 4 4 4 5 5 6 6 6 5 6 6 6 6 6 7 7 8 8 8 8 7 7 7

5 5 4 5 4 5 4 4 4 4 4 4 5 5 5 5 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 6 7 7 7 8 8 8 8 7

5 5 4 5 4 3 5 4 4 4 4 4 4 5 5 5 5 3 3 4 4 4 4 5 5 6 6 6 5 6 6 6 6 6 7 7 7 8 8 8

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1 2 3 4 5 6 7 8 9

48.8 ⫾ 0.6 52.3 ⫾ 0.4 57.0 ⫾ 0.5 58.5 ⫾ 0.6 45.1 ⫾ 0.5 53.2 ⫾ 0.5 45.2 ⫾ 0.4 42.3 ⫾ 0.5 73.2 ⫾ 0.5

46.6 ⫾ 0.4 51.9 ⫾ 0.4 52.8 ⫾ 0.6 53.8 ⫾ 0.5 43.1 ⫾ 0.5 49.0 ⫾ 0.5 50.1 ⫾ 0.4 38.8 ⫾ 0.3 71.6 ⫾ 0.7

⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 1.000 ⬍0.0001 ⬍0.0001

from a small set of prostate cancer patients treated with IMRT. The goal of the study was to test the hypothesis that cluster models based on maximum cluster size have the ability to extract information from the spatial dose distribution beyond that contained in the DSH. Specifically, we sought to determine whether parameter values could be found for which the mean maximum cluster size for patients with Grade ⱖ2 late rectal bleeding was significantly greater than for the control patient matched by DSH. The results show that parameter values could be found for which the cluster model distinguishes between cases and controls for 8 of the 9 matched pairs (Table 1). The chance of successfully distinguishing cases from controls this often, if the cluster model were no better than a coin toss, would be p ⫽ 0.02. Therefore, the results of our study suggest that a cluster model based on maximum cluster size can, in fact, successfully distinguish between the spatial dose distributions to rectal surface in patients with and without rectal toxicity, but having very similar rectal DSH curves. A noticeable feature of Table 1 is that the parameter values for which the cluster model was most consistent with the data (shown in boldface in Table 1) lie close to a line in the (b1, b2) plane corresponding to an approximately constant value of the ratio b1/b2. The relevance of the quantity b1/b2 is that it determines the D50 value for the localdamage function (Eq. 1). Specifically, the dose D50 for which p(D) ⫽ 50% satisfies the relationship D50(␣/␤ ⫹ D50/42) ⫽ b1/b2. The 56 (b1, b2) pairs highlighted in Table 1 correspond to D50 values ranging from 27 to 43 Gy (median 32 Gy). Therefore, the analyses in the present study lend further support to the concept that intermediate doses (⬃30 Gy), not high doses (⬎60 Gy), may be most significantly associated with producing the type of damage to rectum that is manifested as Grade ⱖ2 rectal toxicity. However, the extent to which the model parameters depend on the choice of voxel size and on connectivity remains to be investigated. The observation that b1/b2 is related to the D50 of the local-damage function helps to explain some of the other

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results shown in Table 1. For example, the zeroes in Table 1 occur for very high values of b1/b2 corresponding to D50 ⬎140 Gy. For these large D50 values, no clusters of damage occurred for either cases or controls. Conversely, for very small values of b1/b2, corresponding to D50 ⬍10 Gy, almost all voxels were damaged, and the comparisons of mean maximum cluster size are dominated by small differences in voxel number between cases and controls, which go in favor of the cases about half the time and in favor of the controls about half of the time (4s and 5s in the first few rows of Table 1). Figure 6 shows the average entries of Table 1 plotted as a function of D50, further illustrating that the consistency between the cluster model and the case– control data is best for local-damage parameters corresponding to D50 values near 30 Gy. The 18 paired dose–wall maps generated as part of this study were examined visually to investigate what qualitative spatial information was being captured by the cluster model that was not included in the DSH. No obvious features were apparent, although it was observed that for 5 of the 9 matched pairs, the region receiving ⬎30 Gy contained a wider band wrapping entirely around the circumference of the rectum for the case than for the matched control, consistent with the findings of Skwarchuk et al. (33). Also, the one pair for which the control but not the case had a 30-Gy band around the circumference of rectum (Pair 7, cf. Fig. 4) was the pair always excluded when the cluster model was found to be inconsistent with just one matched case– control pair (i.e., the 8s in Table 1). However, for the remaining 3 pairs (including Pairs 1 and 9 shown in Fig. 4), there was no obvious difference in the circumferential shape of the 30-Gy region for case vs. control. It is clear from Table 2 that there is no fixed cluster size, in this study, at which rectal complications occur. There is likely to be a continuously increasing risk of late rectal

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Table 2. Mean maximum cluster size generated for the 9 case– control pairs using local-damage parameters b1 ⫽ 7 and b2 ⫽ 0.04 Gy⫺2

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Fig. 6. Average value from Table 1 (number of case– control pairs for which the case has greater mean maximum cluster size than the control), plotted as a function of D50. Data were grouped into 10 equal bins of 80 values each. Symbols indicate the average values of D50 and Table 1 entries in each subgroup; error bars represent ⫾1 standard deviation in each direction.

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bleeding with increasing mean maximum cluster size. Such a relationship cannot be determined from a case– control study; an analysis with the goal of model fitting will be performed in the future using an unselected population after accrual of more patients treated with IMRT to prostate. Finally, it should be noted that all calculations in the present study were based on the planning CT, although it is known that organ movements may confound the contour information obtained from it. This is a possible interpretation of the fact that maximum cluster size was larger in the control than in the matched case for Pair 7 using “optimal” parameter values (Table 2). Future work will include techniques to incorporate organ motion in calculation of the dose–wall map. Also, it is important to note that the dose distributions to rectum in the present study all had the form of roughly concentric regions of increasing dose (Fig. 4). That is, the dose–wall maps did not include large disjoint

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dose areas of the type most likely to lead to divergent predictions from cluster models as compared with DVH- or DSH-based modeling. Data sets in which disjoint dose regions do occur will be of particular interest in further testing the utility of the simple two-dimensional cluster model based on mean maximum cluster size presented here. CONCLUSIONS Our case– control study of rectal toxicity in prostate cancer patients treated with IMRT suggests that a cluster model based on maximum cluster size successfully incorporates spatial information beyond that contained in the rectal DSH. We therefore expect cluster models to provide a useful new tool for predicting normal-tissue complication probability after radiotherapy as a function of the spatial dose distribution.

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APPENDIX 1 Connectivity Determining whether two adjacent voxels belong to the same cluster of damaged tissue units requires a choice of connectivity for defining clusters. In the two-dimensional setting considered here (rectal surface), 1-connectivity corresponds to the requirement that each black voxel in a cluster share an edge with another black voxel in the cluster. Two-connectivity requires that: (1) Each black voxel in a cluster sits at the corner of a 2 ⫻ 2 block of black voxels (a 2-square), and (2) If a black voxel in the cluster shares an edge with another black voxel, the second voxel is in the same cluster only if there is a second path between the two voxels, in addition to the immediate adjacency. For example, if the second voxel lies along the eastern edge of the first voxel, then either the two voxels to the north and northeast of the first voxel or the two voxels to the south and southeast of the first voxel must also be black. This amounts

to saying that the second black voxel is part of a 2-square that overlaps, by at least one voxel, the 2-square containing the first voxel. Higher connectivities can be defined in an analogous way and correspond to “thicker” clusters. Because 1-connectivity tends to lead to long, thin “snakelike” clusters, we used 2-connectivity for calculating cluster size in the present study. The appropriate choice of connectivity remains an open question, and further studies are under way to determine how the choice of connectivity affects the behavior of cluster models, and to what extent the optimal choice of connectivity depends on the number of voxels into which the area of the organ is subdivided [here, 72 ⫻ (2nzi ⫺ 2)]. Adjustment for voxel size in the cluster size calculation The voxels of the rectal surface represented by the entries of the dose–map array have varying areas owing to varia-

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Fig. 7. Example of two areas of rectal surface, each divided into 16 voxels. On the left, each square voxel has area equal to the reference area, Aref. On the right, each voxel is twice as wide as the voxels on the left and therefore has area 2Aref. An equal area (4Aref) is damaged in each case (black squares), but is comprised of four voxels on the left and only two voxels on the right.

tions in rectal circumference along the inferior-superior axis, both within and between patients (Fig. 3). These variations in voxel area require that adjustments be made per voxel in the calculation of maximum cluster size. To illustrate the necessary adjustments, consider the two areas of rectal surface shown in Fig. 7. The surface on the left is divided into 16 square voxels, each having area equal to the reference area, Aref. The surface on the right also consists of 16 voxels, each twice as wide as the voxels on

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the left and therefore having area 2Aref. After exposure to dose D, assume that the four black squares on the left and the two black rectangles on the right were damaged. Let p denote the local-damage probability, after dose D, for each tissue unit on the left. The probability that the block of black squares was damaged was therefore p4. Because the black regions in the two panels of Fig. 5 have the same area, we wish to assume that they had the same damage probability, although the black area on the right happened to be subdivided into fewer voxels. Therefore, each of the two blackened voxels on the right is assumed to have had damage probability p2, corresponding to damage probability p4 for the pair of rectangles. In general, a voxel with area m · Aref exposed to dose D is assumed to have damage probability pm, where p is the damage probability of a reference voxel (area Aref) at dose D. Furthermore, in calculating cluster sizes, we wish to count each black voxel on the right as contributing twice as much area as a black voxel on the left. In general, a voxel with area m · Aref will be counted as contributing m times as much area to a cluster as a reference voxel.