1 Estimation of three dimensional intrinsic dosimetric uncertainties 2 resulting from using deformable image registration for dose 3 mapping 4 Francisco J. Salguero 5 Department of Radiation Oncology, 6 Virginia Commonwealth University, 7 Richmond, Virginia, 23298 8 Nahla K. Saleh-Sayah 9 Department of Radiation Oncology, 10 Virginia Commonwealth University, 11 Richmond, Virginia, 23298 12 Chenyu Yan 13 Department of Radiation Oncology, 14 Virginia Commonwealth University, 15 Richmond, Virginia, 23298 16 Jeffrey V. Siebers 17 Department of Radiation Oncology, 18 Virginia Commonwealth University, 19 Richmond, Virginia, 23298 20 21 Abstract: 22 Purpose: This paper presents a general procedural framework to assess the point-by-point 23 precision in mapped dose associated with the intrinsic uncertainty of a deformable image 24 registration (DIR) for any arbitrary patient. 25 Methods: Dose uncertainty is obtained via a three step process. In the first step, for each voxel 26 in an imaging pair, a cluster of points is obtained by an iterative DIR procedure. In the second 27 step, the dispersion of the points, due to imprecision of the DIR method is used to compute the 28 spatial uncertainty. Two different ways to quantify the spatial uncertainty are presented in this 29 work. Method A consists of a one-dimensional analysis of the modules of the position vectors, 30 whereas method B performs a more detailed 3D analysis of the coordinates of the points. In the 31 third step, the resulting spatial uncertainty estimates are used in combination with the mapped 32 dose distribution to compute the point-by-point dose standard deviation. 33 demonstrated to estimate the dose uncertainty induced by mapping a 62.6 Gy dose delivered on 34 maximum exhale to maximum inhale of a 10-phase four dimensional lung CT 35 Results: For the demonstration lung image pair, the standard deviation of inconsistency vectors 36 is found to be up to 9.2 mm with a mean σ of 1.3 mm. This uncertainty results in a maximum 37 estimated dose uncertainty of 29.65 Gy if method A is used and 21.81 Gy for method B. The 38 calculated volume with dose uncertainty above 10.00 Gy is 602 cm3 for method A and 1422 cm3 39 for method B. 40 Conclusions: This procedure represents a useful tool to evaluate the precision of a mapped dose 41 distribution due to the intrinsic DIR uncertainty in a patient. The procedure is flexible, allowing 42 incorporation of alternative intrinsic error models. The process is 43 Introduction 44 Deformable image registration (DIR) algorithms allow consideration of physiologic changes in 45 patient anatomy during treatment, making possible techniques such as image guided adaptive 46 radiation therapy (IGART) and image guided radiation therapy (IGRT) [1, 2] among others. DIR 47 algorithms estimate the vectors between corresponding voxels in images that differ due to 48 morphological changes. There are numerous algorithms that use various approaches to calculate 49 the displacement vector field (DVF) that matches points in one image with points in another [3- 50 7]. Although several different basis functions have been utilized by DIR algorithms, no known 51 algorithm provides a DVF that maps tissue elements from between anatomic instances without 52 error [8-10]. The lack of a gold standard makes it difficult to assess the DVF accuracy for 53 arbitrary patient images; similarly, there is no universal or widely accepted method to evaluate 54 the uncertainty of an individual patient’s DVF. 55 Several approaches have been proposed to evaluate the accuracy of DIR algorithms. Some 56 authors have used real or simulated deformable phantoms where the displacements are known 57 [11-13]. Other tests include comparison between DIR mapped and physician-drawn contours 58 [14, 15], landmark tracking [15-17], energy conservation analysis [18] or self-consistency 59 evaluation [19]. Unfortunately, it is impractical to implement these tests in a clinical practice 60 routine and the information provided by these methods is not complete [19]. For example, a 61 deformable phantom does not provide any information about the actual errors made in a real 62 patient and it is impractical to identify a multitude of point-based landmarks for accuracy 63 evaluation. Even when limited automatic landmarks can be identified, they are limited in that 64 they provide no information distant to the landmarks. Furthermore, none of these methods are 65 intended to give the precision of the mapped doses, i.e. the reproducibility of the dose mapping 66 process. Accuracy is not the only measure that can be used to assess the confidence in a DVF. 67 Any physical measurement is also characterized by the precision: the degree to which repeated 68 measurements under the same conditions will show the same results. Precision does not measure 69 the true error of the measurement, but it gives a value of the spread of possible values around a 70 mean value. However, a precise algorithm is not necessarily accurate, so systematic errors must 71 be evaluated before relying in a precise DVF. 72 Although precision is not necessarily correlated with accuracy, errors reported in the literature 73 can be useful to estimate the order of magnitude of the expected precision of a DIR algorithm. 74 Published errors vary with the algorithm and evaluation method, with mean values generally in 75 the range of 2-4 mm and maximum errors larger than 15 mm[15, 16, 20]. Such large errors can 76 lead to unacceptable dose inaccuracies, limiting the applicability of 4D treatments. 77 A general DIR process has several sources of intrinsic errors. One specific intrinsic error source 78 is the lack of self-consistency in generating the DVF [19, 21]. In simple terms, this causes 79 composite transitive transformations, such as f◦f-1 and fAB◦fBC◦fCA to be not equal to the identity 80 transformation, with f-1 being the inverse of f, and fAB, fBC, and fCA the transformations from 81 image A to B, from B to C and from C to A, respectively. Self-consistency is a necessary but not 82 sufficient requirement for a precise DIR algorithm. 83 The clinical impact of DVF errors is not only related to the magnitude of such errors, but also to 84 their spatial location. Dosimetrically, a large displacement error may be of small importance if it 85 is located in a volume with homogeneous dose or with no dose at all. On the other hand, a small 86 DVF error may lead to a large dose error if it is located in a high dose gradient region. One 87 purpose of DIRs for a 4D treatment is to map the dose delivered on an image set to one or 88 several other sets and reconstruct the composite (total) delivered dose over all image sets [2, 22, 89 23]. Errors in dose mapping may or may not be consequential depending on whether they are 90 located within an important structure or not. 91 In this paper, a computational framework is presented to estimate the dose uncertainty in a 4D 92 treatment due to the intrinsic DVF uncertainty in a patient-per-patient basis. This procedure 93 obtains a set of mapped points for each voxel. The dispersion of these points is used to model 94 the uncertainty of the mapped point coordinates due to the inherent imprecision of DIR 95 algorithms by an iterative procedure. 96 imaging voxel from the standard deviation of the modules of inconsistency vectors (method A) 97 or from the covariance matrix of the coordinate of inconsistency vectors (method B). 98 This scheme provides an estimation of the precision for DIR mapped doses. A simplistic DVF 99 error model is used. It is not the aim of this work to propose a comprehensive DVF error model 100 but to present a method that, given any DVF error model, provides a dose uncertainty estimation. 101 In the form presented in this paper, random errors due to the intrinsic functioning of the DIR 102 algorithm are considered and therefore, the method is insensitive to systematic errors. 103 The intrinsic uncertainty algorithm takes advantage of the lack of self-consistency of the DIR to 104 obtain a set of similar images where each point in an image can be traced to the other ones. By 105 doing so, statistical dispersion due to intrinsic inconsistency can be calculated at each voxel. 106 Adaptation of this method for use with inverse consistent DIR algorithms is covered in the 107 discussion. Dose standard deviation is then calculated for each 108 Materials and Method 109 The procedure consists of three steps: an iterative registration method is used to obtain a cluster 110 of points associated with each voxel; the dispersion of the cluster is used to compute point-by- 111 point DIR uncertainties; and from the DIR uncertainties, dose error is evaluated. 112 Importantly, the process used in the first step to obtain the cluster of points is a simple first order 113 estimate of the point dispersion. Alternative processes to estimate DIR process point dispersion 114 can be simply substituted in this process. Ideally, the selected method should assess all sources 115 of intrinsic uncertainty of the whole registration procedure. The last two steps are general and 116 can be applied given any source or sub-source of intrinsic DIR errors. 117 Notation and terminology: 118 DIR algorithms map information (typically image intensity, but possibly contours) from a source 119 image to a target image. In this paper, the original source and target images are noted S0 and T0. 120 The uncertainty evaluation algorithm presented here generates a set of source and target images 121 in an iterative scheme. The generated images in step n are noted Sn and Tn and are referred as 122 mapped source and mapped target images. Every point in images Sn is the result of composite 123 mappings, S n f S n1T 0 124 the transformation from the resulting Tn to S0. The vector that goes from a point in S0 to the 125 corresponding point in Sn is noted the nth inconsistency vector or vn. 126 Two uncertainty analyses are used in this work. The first is the variance of the modules of 127 inconsistency vectors. 128 inconsistency vector in step n is noted σn, while the standard deviation of the whole set of vectors 129 and the reconstructed initial standard deviation (the inferred standard deviation of the initial fT n S 0 , with f S n 1 0 T being the DIR transformation from Sn-1 to T0 and fT n S 0 For a given voxel, the standard deviation of the module of the 130 transformation) are noted σT and σ0, respectively. The second analysis is a generalization of the 131 first where the subscript notation remains but, instead of using the standard deviation σ of the 132 x2 modules of vectors, the covariance matrix xy xz 133 used, where σx2, σy2 and σz2 are the variances of the X, Y and Z coordinates and σxy, σxz and σyz are 134 the covariances. In all figures and the text, values of σ and |Σ|1/2 are given in mm. 135 Set of Points 136 The procedures to model the intrinsic DVF uncertainty in a first stage, determine σ0 and Σ0 in a 137 second stage, then determine dose uncertainty in a third stage are independent of one another. 138 The output of one stage is used as input for the next one, so the method to obtain the cluster of 139 points can be changed without modifying the other two stages, or the error model can be 140 replaced by an alternative model, either a more general model or one specific for a DIR 141 algorithm or anatomic region. The only link between the first and the second stage are the point 142 coordinates, and the only link between the second and the third phase are σ0 or Σ0. 143 In this paper DVF intrinsic errors are assumed to be Gaussian distributed. That is, the mapped 144 coordinates of each point are assumed to be Gaussian distributed around a mean value. This 145 assumption is not proven in this work and may be simplistic, although it is a reasonable a priori 146 assumption because the total DVF uncertainty is due to the sum of several sources of errors, so 147 their overall effect is expected to be nearly Gaussian. 148 To determine the precision of the DVF, the repeatability of the results must be assessed. 149 However, DIR algorithms are deterministic, so some ‘artificial’ perturbation must be introduced. xy xz y2 yz of the vector coordinates is yz z2 150 This perturbation must be large enough to cause the algorithm to provide a different solution, but 151 small enough for the differences to be due mainly to the intrinsic uncertainty of the algorithm. 152 This paper uses the inherent differences between the mapped image and the target image in a 153 DIR procedure. 154 Mapping a source image S0 into a target image T0 with a DIR algorithm implies determination, 155 for each point in S0, the associated position of the point in T0. Generally, this calculation is not 156 exact and thus, some uncertainty in the mapped position of the point is introduced by the DIR 157 algorithm. There are several sources of uncertainty, each of which contributes to the total 158 uncertainty. In the absence of systematic errors, the true value of the mapped position is located 159 in the neighborhood of the calculated point. As the total uncertainty arises from several causes, 160 the positional error of the mapped point can be assumed to follow a Gaussian probability density 161 function (PDF). Because of this uncertainty, the resulting mapped image T1 is not exactly the 162 same than T0. 163 The overall procedure utilized is shown schematically in Figure 1. After the initial mapping 164 from S0 to T0, and computation of T1, T1 is mapped back to S0. The coordinates of a point of the 165 new image S1 to the original point in S0 have a PDF that is the convolution of the forward and 166 the backward mapping uncertainties. Since the images are similar and the algorithm did not 167 change, we assume that the intrinsic uncertainty due to the DIR algorithm does not change. In 168 this case, the PDFs of the new point’s coordinates are the convolution of two identical Gaussian 169 distributions which is another Gaussian distribution with σ1,i2=2σ0,i2, where σ1,i and σ0,i are the 170 standard deviation of the coordinate xi. In the second step, image S1 is mapped to T0 (resulting 171 in T2). But the backward transformation is not from T2 to S1 but from T2 to S0. This assures that 172 the resulting images S1, S2, S3… do not diverge but tend to be similar to S0. In general, for any 173 n, the nth iteration consists of mapping Sn-1 to T0 and then Tn is mapped back to S0. The variance 174 of a coordinate of a point in Sn to the analogous point in S0 is 2nσ0,i2. 175 After N iterations, each point in S0 can be related to a cluster of N points. The total distribution 176 of this cluster is the sum of the PDFs of every point. The superposition of Gaussian distributions 177 is not in general a Gaussian, but if they have the same mean value, the variance of each 178 coordinate is given by: 179 T2,i xT2 ,i xT ,i 180 2 1 N N n2,i n 1 1 N N 2n n 1 2 0,i ( N 1) 0,2 i Where we have taken into account that xT2 ,i 1 N N n 1 xn2.i (1) 1 N N n 1 2 n ,i xn ,i 2 and chosen the 181 origin of the coordinate system so that xn ,i 0, n 1, N . 182 Accordingly, from the variance of the positions of an iteratively mapped point it is possible to 183 infer the expected variance in the first mapping, which enables estimation of the uncertainty 184 effects on clinically used DIR mappings. 185 DVF Uncertainty 186 Method A (Variance of Vector Modules) 187 The module of the inconsistency vector, vn, is given by v X 2 Y 2 Z 2 with X, Y and Z 188 being random variables with Gaussian PDFs. For the case of σ=1 and µ=0 (being σ the standard 189 deviation of v and µ the mean), the PDF of |vn| tends to the χ distribution with three degrees of 190 freedom. For the sake of simplicity, we can approximate the distribution of vn to a Gaussian 191 distribution with a good degree of accuracy. In this case, equation (1) can be applied to the 192 module of vn and we can approximate the uncertainty of the module of vn via σ02=σT2/(N+1). 193 This approximation is more accurate with larger values of the standard deviations of each 194 coordinate, σx, σy and σz. The standard deviation of this Gaussian distribution will be named σ0. 195 Method B (Coordinates Variance) 196 The analysis of the vector module variances has some drawbacks. The first problem is that it 197 lacks of directional information. For a point in S0, the variance of positions of the related cluster 198 of points in S1, S2... may have a predominant direction. Losing the directionality of variance 199 makes the estimation less accurate and ignores possible correlations between uncertainties in 200 coordinates. The second problem of the module analysis is related with the impact of DVF 201 uncertainty in the mapped dose precision. 202 Related to the previous drawbacks, dose uncertainty estimation can be improved if the 203 interaction between the directionality of the DVF uncertainty and the dose gradient is considered. 204 If dose gradient is perpendicular to the largest component of the DVF uncertainty, the effect of 205 the latter would be diminished. 206 To overcome these limitations, a generalization of the previous method is developed. The same 207 set of images S1, S2… are used as in method A. However, instead of looking at the modules of 208 the inconsistency vectors, the coordinates of the mapped points are analyzed. 209 position of the original point as origin of the reference system, the covariance matrix of the 210 coordinates of the associated cluster of points is calculated. 211 The covariance matrix of the coordinates retains information about the anisotropy of the cluster 212 of points. The probability that a point in S0 is mapped to a point in T0 can be evaluated by the Taking the 1 1 x T 0 x 2 213 3D Gaussian distribution f ( x ) 214 matrix, x is the coordinate column vector of the point, and x T is the transpose coordinate 215 vector. This 3D distribution can be viewed as the PDF that the mapped image of the point x0 in 216 S0 (the point where f ( x ) is calculated) is x . 217 Dose Uncertainty 218 After the variance for each point is computed, the impact of it on dose can be assessed. To first 219 order, dose uncertainty may be estimated by considering a sphere centered at a point with 220 diameter kσ0, being k a dimensionless parameter that depends on the desired degree of 221 confidence and σ0 the estimated standard variation of the DVF at each point. The variance of 222 dose values within this sphere can be used to estimate the dosimetric impact of the DIR 223 inaccuracies in this point. This procedure is the only suitable when obtaining σ0 by method A. 224 This simplistic process has several weaknesses. It is obvious that not every value within the 225 sphere has the same probability. The further the dose point is from the center, the less probable 226 it is. Furthermore, the dose uncertainty estimation depends on the arbitrary parameter k, the 227 diameter of the sphere to evaluate the dose uncertainty in. Changing k changes the calculated 228 variance. 229 uncertainty and no convergence is obtained for any value of k. This problem can be partially 230 overcome by weighting the dose points with a function that takes into account the radial distance, 231 however there is no universal appropriate function to use. 232 uncertainty may not be isotropic will result in misestimating the dose uncertainty within the 233 sphere. Furthermore, if the radius of the sphere is smaller than the size of the dose grid and (2 )3/ 2 0 1/ 2 e (Figure 2), where Σ0 is the covariance Increasing the value of k implies, in most cases, increasing the calculated dose Regardless, the fact that DIR 234 nearest neighbor interpolation is used to estimate dose values within the sphere, the estimated 235 uncertainty will be exactly 0 since only one dose value will be considered. 236 overcome by trilinear interpolation of doses on the 3D dose matrix. Due to the fact that method 237 A is merely a first order estimate and that trilinear interpolation is computationally expensive, 238 only nearest neighbor interpolation is used in this work with method A. 239 The problems related to the loss of directionality and sphere size inherent to method A can be 240 avoided by using the more detailed information provided by method B. The covariance matrix 241 Σ0 provides information not only about the magnitude but also about the directionality of the 242 uncertainty. In a DVF analysis, |Σ0|1/2 can be assimilated to σ0 but by doing so the advantages of 243 knowledge of DVF uncertainty directionality are lost. Furthermore, if the Gaussian PDF is 244 highly directional, |Σ0|1/2 can be very small even though some individual components are large. 245 The PDF calculated by method B can be used to calculate a weighted dose variance. Each dose 246 value is weighted by the value of the PDF at its position. In practice, for computational reasons, 247 only dose values whose statistical weight is above a threshold are considered for the calculation. 248 The dose uncertainty calculated in this way preserves the anisotropy of DIR errors, takes into 249 account the direction of the dose gradient and does not relay in an arbitrary parameter like k in 250 the previous procedure. The only selectable parameter is the Gaussian function threshold to 251 consider the dose point in the calculation. Importantly, the calculated dose variance converges as 252 the threshold value decreases, so if the chosen value is low enough, the calculated dose variance 253 will be accurate. 254 This approach has a further advantage when using nearest-neighbor interpolation in regions with 255 low spatial uncertainties. Unlike method A which yields zero dose uncertainty when kσ0 is less This can be 256 than the size of the dose grid, in method B the 3D Gaussian spatial uncertainty distribution 257 covers the whole patient volume. By setting a suitably low threshold for the minimum statistical 258 weight considered, multiple voxels are sampled in 3D, resulting in a sufficiently accurate 259 estimate of the dose uncertainty estimation without the need to resort to trilinear interpolation. 260 To validate that the threshold was set sufficiently low, a comparison of nearest-neighbor and tri- 261 linear dose estimates is presented. 262 In order to offer some insight to the differences between both methods and their accuracy, three 263 representative points are studied in detail. The points are chosen to be representative of voxels 264 where both methods show small, medium and large discrepancies. Specifically, the evolution of 265 the dose uncertainty in each point with the number of iterations used to determine σ0 and Σ0 are 266 analyzed and the Kolmogorov-Smirnov test is performed on the points coordinates to evaluate 267 normality. Results for these three points do not try to probe the accuracy of the DVF uncertainty 268 model used, but to show that it is a feasible model and may provide reasonable results. 269 Clinical Demonstration Case 270 A 4D CT of a lung case is used to demonstrate the procedures given above. The patient was 271 diagnosed as having a malignant neoplasm of right bronchus and lung. Treatment consisted of 272 an IMRT plan to deliver 50 Gy in 25 sessions, plus a boost of 12.6 Gy in 7 sessions with 6 and 273 18 MV beams. The main plan consisted of seven equispaced beams with 50 degrees steps. The 274 boost plan consisted of seven beams at angles of 15, 90, 130, 180, 230, 280 and 330 degrees. 275 For simplicity, only two breathing phase images (full inhale and full exhale) are used. The 276 general method is easily extendable to mapping multiple image phases with the total dose 277 uncertainty being the sum of the individual phase dose uncertainties added in quadrature. For 278 this demonstration, we compute the full 62.6 Gy delivery to the exhale phase, then map the dose 279 into the inhale phase using the point based thin plate spline (TPS) algorithm implemented in a 280 research version of Pinnacle planning system [7]. Volumes used as input by the TPS algorithm 281 were lungs, heart, cord, superior mediastinal lymphatic nodes, esophagus, PTV and GTV. The 282 number of vertices to create the volume meshes and the number of sample points for the TPS 283 algorithm are left at their default values. Fifteen iterative mappings are performed to estimate 284 the DIR induced point dispersion. The CT image voxel size is 0.97656×0.97656×2.99995 mm 285 and the dose calculation grid size is 2×2×2 mm. Both variance methods are applied in order to 286 illustrate the similarities and differences between them. Additional calculations with method B 287 are performed with a 1.5×1.5×1.5 mm dose resolution to show the insensitivity of the results to 288 sufficiently fine dose grid resolutions. 289 Results 290 The dose distribution is calculated on the exhale breathing phase and then mapped to the inhale 291 phase (Figure 3). Fifteen mapping iterations are performed, resulting in 15 mapped source 292 images and 15 evaluations of inconsistency vector vn in order to calculate uncertainty. In Figure 293 4 the spatial distribution of the inconsistency vectors is shown. Displacement uncertainties are 294 mainly clustered within a narrow band between the mediastinum and the left lung. Although not 295 shown here, this feature did not appear when a different set of DIR parameters were used. The 296 DIR set with the feature is used in this demonstration since it clearly shows the utility of the 297 method developed. For the case shown in Figure 4, the maximum σDVF is 9.2 mm and the mean 298 value within the patient’s body is 1.3 mm. The histogram of σDVF values is shown in Figure 5. 299 The dose uncertainty, computed with k=1, shows that large dose uncertainty values exist where 300 large σDVF values matches up to high dose gradient regions, as was expected. In Figure 6, the 301 distribution of dose standard deviation is shown. Note that there are some regions (one of them 302 clearly visible in the axial slice, in the lower part of the right lung) where σ dose is exactly 0, and 303 there is a discontinuity between voxels where σdose>0 and voxels with σdose=0. Because of using 304 nearest neighbor interpolation, this discontinuity marks the point where σDVF value goes from 305 higher than twice the dose grid size to a lower value. If there is only one dose value within the 306 kσ sphere, the dose variance is exactly 0. As soon as the sphere is large enough to encompass 307 more points, the variance changes sharply from 0 to a non-zero value in neighbor voxels. The 308 maximum value of σdose is 29.66 Gy and the mean is 0.96 Gy. The volume where σdose is larger 309 than 310 V(σdose>0.10)=11084.2 cm3 (Figure 7). 311 The spatial obtained by method B (|Σ|1/2) differs significantly from that obtained by method A 312 (Figure 8). The uncertainty region between the mediastinum and the lung still exists but the 313 relative intensity of this region is much lower. The maximum |Σ|1/2 was 5.5 mm and the mean 314 value was 0.1 mm. The histogram of values of |Σ|1/2 is shown in Figure 9. Although |Σ|1/2 can be 315 viewed as a generalization of the standard deviation, its value is less meaningful since a small 316 value of |Σ| may hide a large variance in a dosimetrically important spatial direction. 317 Despite the differences between the distributions of σDVF and |Σ|1/2, the distribution of dose 318 uncertainties calculated by the method B (Figure 10) resembles roughly the distribution obtained 319 by the first procedure but magnitudes are much different. The largest uncertainties are located in 320 the same region, however the overall region is much smaller with method B and there is a 321 difference of an order of magnitude in the mean dose uncertainty. The maximum σdose found 322 with method B is 21.81 Gy and the mean is 0.07 Gy. 323 V(σdose>10.00)=1422.2 cm3, whereas V(σdose>1.00)=3035.7 cm3 and V(σdose>0.10)=5094.0 cm3 10.00 Gy is V(σdose>10.00)=602.3 cm3, whereas V(σdose>1.00)=5740.1 cm3 and The volume with σdose>10 Gy is 324 (Figure 11). 325 0.02% were not taken into account, that is voxels that are at 3.72σ from the Gaussian mean 326 value. Use of tri-linear interpolation in place of nearest-neighbor interpolation to evaluate dose 327 uncertainty was found to not significantly change the results. Using tri-linear interpolation, 328 maximum σdose was 21.81 Gy, mean σdose was 0.12 Gy. Estimation of V(σdose) agreed within 3% 329 for values of σdose lower than 3.50 Gy, however, difference increases for low values of σdose being 330 V(σdose>2.00Gy) 20% larger if trilinear interpolation was used. Use of trilinear interpolation 331 increased the computation time 4 fold. Using a 1.5x1.5x1.5 mm3 dose grid showed little 332 differences in uncertainty distributions. 333 The propagation of the estimated uncertainty for three points is shown in Figure 12 to illustrate 334 the differences between the methods. For each point the calculated dose standard deviation is 335 shown after 4 to 15 iterations. Method A tends to be more unstable and to provide larger 336 estimations of the dose standard deviation. This is likely due to the fact that the method A does 337 not take into account the directionality of spatial uncertainty, thus leading to an overestimation in 338 volumes where the largest component of the spatial uncertainty is perpendicular to the dose 339 gradient. Also, the assumption that the distribution of modules of inconsistency vectors can be 340 approximated to a Gaussian distribution is not always true and may lead to some errors. For 341 these three points, the Kolmogorov-Smirnov test was applied to compare the distribution of each 342 coordinate with the superposition of Gaussian distributions that was assumed in this work. All 343 the coordinates of the three points passed the test. In Table 1 the significance values for each 344 coordinate and point are shown. 345 For computational efficiency, voxels with cumulative statistical weight under 346 Discussion 347 The procedural framework presented in this work provides a dose uncertainty distribution that 348 may be useful to assess the quality of 4D dose mapping, hence also of a 4D treatment. The 349 system is flexible enough to be used with different error models. However, the results may be 350 highly dependent on the specific error analysis performed. 351 The results obtained with the two error analyses presented in this paper show significant 352 differences. Although some structures are identifiable with both methods, the magnitudes of 353 displacement uncertainties are very different. For example, the band-shaped feature between the 354 mediastinum and the left lung is clearly the most prominent feature of the variance mapping 355 obtained by the first procedure (Figure 4), whereas it seems a much less prominent feature if we 356 look at the values of |Σ0|1/2. The reason for such differences is related with the directionality of 357 the DIR uncertainties. It is possible that inconsistency vectors of a point tend to be aligned along 358 a direction. If that happens the variance of modules of vn may be large, but the determinant of Σ0 359 will be small, since there is a coordinate system where all the elements of Σ0 but one are very 360 small. Analyzing the elements of Σ0 (Figure 13), it can be seen that the region between the 361 mediastinum and the lung shows a strong directionality near the z direction, which would explain 362 why the first procedure gives large deviations there but the second one does not. Note that 363 although |Σ0| may be very small, the information about the variances and covariances in each 364 direction is not lost and it is used when calculating the 3D Gaussian distribution of each point. 365 The effect of not considering the directionality of displacement uncertainty to calculate the dose 366 in method A are evident in comparing results with method B. Although the order of magnitude 367 of the maximum dose variances and the location of inaccurate regions are similar, the amount of 368 volume involved and the mean dose variance are very different. When dose variance is 369 estimated in method A from the unweighted variance of doses within a sphere the directionality 370 of dose gradient is not taken into account. However, in method B, since the anisotropy of the 371 DVF variance is taken into account, not only are the dose values accounted for, but the spatial 372 distribution of doses have an influence on the dose uncertainty estimation. 373 In general, method B provides a more accurate estimation of the dose uncertainty associated with 374 the DVF uncertainty because it takes into account the directionality of displacement uncertainties 375 and it weights dose values around each point. However, if one is interested only in the DVF 376 uncertainty, method A provides a simple index that can highlight potential inaccurate regions. 377 Method B does not provide such a simple insight of the spatial uncertainty, and a more complex 378 and less intuitive analysis of the covariance matrix for each point would be necessary. 379 With both methods, the procedure can estimate the impact on dose distribution of DIR intrinsic 380 inaccuracies. 381 contouring errors, CT image artifacts, or other errors. In general, systematic errors do not result 382 in a dispersion of points and thus, will not be detected by these algorithms. A low dose 383 uncertainty does not mean a correct dose, but that the DIR algorithm consistently maps the dose 384 and that the mapping shows little variability. The specific sources of errors to which these 385 procedures are not sensitive should be studied. 386 Intuitively, one might expect dose uncertainties to be largest in dose gradient or penumbra l 387 regions, however, this is not true in general. Relatively small spatial uncertainties in high 388 gradient dose regions can result in large uncertainties, however, the largest dose gradient region 389 is not necessarily the location with the largest dose uncertainty. Spatial uncertainties in high 390 gradient dose regions may be small, resulting in small dose uncertainties. Similarly, low dose However, they do not address the impact of systematic DIR errors due to 391 gradient regions with high spatial uncertainty can result in large dose uncertainties. Furthermore, 392 even large spatial uncertainties can result in low dose uncertainties in a large gradient region if 393 the spatial uncertainty is highly directional and normal to the gradient vector. The relationship 394 between the dose gradient and the sensitivity of the dose distribution to DVF errors is under 395 study [24]. In the case presented in this paper, the correlation coefficient between dose gradient 396 and dose uncertainty was 0.32, meaning a very weak correlation. 397 The presented algorithm takes advantage of the lack of self-consistency of most DIR algorithms. 398 Nonetheless it is still usable on DIR algorithms designed to be self-consistent [21, 25, 26]. Such 399 algorithms aim to minimize the lack of consistency but they do not eliminate it. Importantly, 400 computing the composite mappings S n f S n1T 0 401 allowing the evaluation of vn and application of the method. 402 This procedure also has the advantage of estimating the DVF and dose uncertainty in three 403 separable phases. Thus, more general methods to estimate the DIR induced point dispersion can 404 readily be used within this framework with little modification. Similarly, cause specific point 405 dispersion models can be used to investigate dose uncertainty induced by sub-components of an 406 DIR uncertainty. 407 Conclusion 408 A procedural framework to assess point-by-point dose precision in dose mapping is presented. 409 The procedure is conceived in a modular way so that different error models and analysis can be 410 used within it. In this work, two different analyses are used. 411 Although the second analysis is more complete and more reliable, the first method, based on the 412 variance of the modules of inconsistency vectors, may be useful for a more simple analysis when fT n S 0 , results in an image S n S 0 , thus 413 dose uncertainty accuracy is not a concern. 414 generalization of the first method takes into account more complex details of the impact of the 415 DVF uncertainty in the dose precision. 416 Although no clinical estimation can bethe clinical impact of dose uncertainties resulting from 417 intrinsic DIR uncertainties should notbe inferred from the single case presented in this work t, 418 the methods can be a useful tool to assess the precision of dose mapping in 4D treatments or the 419 potential impact of DIR uncertainties on dose conformation within planning volumes such as 420 PTVs and ITVs, and the clinical impact of such imprecision on a per-patient basis. 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Computer Methods and Programs in Biomedicine, 2009. 95(2): p. 105-115. 484 485 486 487 488 489 490 491 492 493 Figures: Figure 1 494 495 496 Figure 2 497 498 499 500 501 502 503 504 505 506 507 508 Figure 3 509 510 511 512 513 514 515 516 517 518 519 520 521 Figure 4 522 523 524 525 526 Figure 5 527 528 529 530 531 532 533 534 535 536 537 538 539 540 Figure 6 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 Figure 7 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 Figure 8 574 575 576 577 578 579 Figure 9 580 581 582 583 584 585 586 587 588 589 Figure 10 590 591 592 593 594 595 596 597 Figure 11 598 599 600 601 Figure 12 602 603 604 605 606 607 608 609 Figure 13 610 611 612 613 614 615 Figure captions: 616 Figure 1: Flowchart of the procedure to obtain a set of variant images as a first step to estimate the DVF 617 uncertainty. In every loop, the image resulting from the previous loop is warped toward the original 618 target image T0. The image obtained is warped back to the original source image S0 and the resulting 619 image will be used as input for the next loop. 620 Figure 2: Cluster of points generated by iteratively mapping a point in the source image. The original 621 point is depicted in red. Numbers show the iteration at which each point was obtained. The yellow 622 ellipsoid represents the 50% isosurface of the initial 3D Gaussian distribution calculated from the cluster. 623 Figure 3: Axial and coronal slices of the treatment case dose distribution mapped into the inhale phase. 624 Figure 4: Distribution of standard deviations of the modules of inconsistency vectors by using method A. 625 σ is given in mm. 626 Figure 5: Histogram of DVF standard deviations for the test case. Frequency is given in arbitrary units. 627 Figure 6: Axial and coronal slices showing the spatial distribution of the dose standard deviation 628 obtained by the method A. σ is given in Gy. 629 Figure 7: Dose standard deviation histogram obtained by the method A. Standard deviation is given in 630 Gy. 631 Figure 8: Square root of the determinant of the covariance matrices of coordinates by using method B. 632 |Σ|1/2 is given in mm. 633 Figure 9: Histogram of square root of the determinant of the coordinate covariance matrices for the test 634 case. Frequency is given in arbitrary units. 635 Figure 10: Axial and coronal slices showing the spatial distribution of the dose standard deviation 636 obtained by the method B. σ is given in Gy. 637 Figure 11: Dose standard deviation histogram obtained by the method B. Standard deviation is given in 638 Gy. 639 Figure 12: Dose standard deviation in three representative points as a function of the number of 640 iterations. Dashed lines represent calculations made with method A and solid lines with method B. 641 Figure 13: Mapping of the values of the independent elements of the covariance matrices obtained in the 642 test case. Note that each image has a different color scale. Predominance of one component in some 643 regions indicates that variance have a strong directionality. 644 645 646 647 648 Tables: 649 Table 1 Point 1 Point 2 Point 3 650 651 652 653 X Coordinate 20.59% 86.18% 64.68% Y Coordinate 75.59% 41.30% 43.81% Z Coordinate 26.14% 45.08% 37.73% 654 655 656 657 658 Table captions: Table 1: P-values resulting from applying the Kolmogorov-Smirnov test to the spread of the values of each coordinate for the three sample points. The null hypotesis was that the distribution of the coordinates comes from the superposition of 15 Gaussian distributions.