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 n1T 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 n1T 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. ThisSuch
421
evaluation is currently being performed by our group.are ripe topics for future study.
422
Acknowledgments
423
This work was supported by funding from National Institutes of Health (NIH-P01-CA116602) and
424
in part by a research contract with Philips Medical Systems.
425
Bibliography
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
1.
2.
3.
4.
5.
6.
7.
8.
9.
The second alternative, a more accurate
Xing, L., et al., Overview of image-guided radiation therapy. Medical Dosimetry, 2006. 31(2): p.
91-112.
Mackie, T.R., et al., Image guidance for precise conformal radiotherapy. International Journal of
Radiation Oncology Biology Physics, 2003. 56(1): p. 89-105.
Christensen, G.E., R.D. Rabbitt, and M.I. Miller, Deformable templates using large deformation
kinematics. Ieee Transactions on Image Processing, 1996. 5(10): p. 1435-1447.
Johnson, H.J. and G.E. Christensen, Consistent landmark and intensity-based image registration.
Ieee Transactions on Medical Imaging, 2002. 21(5): p. 450-461.
Lu, W.G., et al., Fast free-form deformable registration via calculus of variations. Physics in
Medicine and Biology, 2004. 49(14): p. 3067-3087.
Yan, C., et al., A pseudoinverse deformation vector field generator and its applications. Medical
Physics, 2010. 37(3): p. 1117-1128.
Kaus, M.R., et al., Assessment of a model-based deformable image registration approach for
radiation therapy planning. International Journal of Radiation Oncology Biology Physics, 2007.
68(2): p. 572-580.
Wang, H., et al., Validation of an accelerated 'demons' algorithm for deformable image
registration in radiation therapy. Physics in Medicine and Biology, 2005. 50(12): p. 2887-2905.
Foskey, M., et al., Large deformation three-dimensional image registration in image-guided
radiation therapy. Physics in Medicine and Biology, 2005. 50(24): p. 5869-5892.
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
727.
26.
Webb, S., Does elastic tissue intrafraction motion with density changes forbid motioncompensated radiotherapy? Radiotherapy and Oncology, 2006. 81: p. S98-S98.
Serban, M., et al., A deformable phantom for 4D radiotherapy verification: Design and image
registration evaluation. Medical Physics, 2008. 35(3): p. 1094-1102.
Kerdok, A.E., et al., Truth cube: Establishing physical standards for soft tissue simulation. Medical
Image Analysis, 2003. 7(3): p. 283-291.
Kashani, R., et al., Technical note: A physical phantom for assessment of accuracy of deformable
alignment algorithms. Medical Physics, 2007. 34(7): p. 2785-2788.
Pevsner, A., et al., Evaluation of an automated deformable image matching method for
quantifying lung motion in respiration-correlated CT images. Medical Physics, 2006. 33(2): p.
369-376.
Brock, K.K., D.R.A. C, and, Results of a Multi-Institution Deformable Registration Accuracy Study
(Midras). International Journal of Radiation Oncology Biology Physics, 2010. 76(2): p. 583-596.
Castillo, R., et al., A framework for evaluation of deformable image registration spatial accuracy
using large landmark point sets. Physics in Medicine and Biology, 2009. 54(7): p. 1849-1870.
Gu, X.J., et al., Implementation and evaluation of various demons deformable image registration
algorithms on a GPU. Physics in Medicine and Biology, 2010. 55(1): p. 207-219.
Zhong, H.L., T. Peters, and J.V. Siebers, FEM-based evaluation of deformable image registration
for radiation therapy. Physics in Medicine and Biology, 2007. 52(16): p. 4721-4738.
Bender, E.T. and W.A. Tome, The utilization of consistency metrics for error analysis in
deformable image registration. Physics in Medicine and Biology, 2009. 54(18): p. 5561-5577.
Kashani, R., et al., Objective assessment of deformable image registration in radiotherapy: A
multi-institution study. Medical Physics, 2008. 35(12): p. 5944-5953.
Christensen, G.E. and H.J. Johnson, Consistent image registration. Ieee Transactions on Medical
Imaging, 2001. 20(7): p. 568-582.
Keall, P.J., et al., Monte Carlo as a four-dimensional radiotherapy treatment-planning tool to
account for respiratory motion. Physics in Medicine and Biology, 2004. 49(16): p. 3639-3648.
Heath, E. and J. Seuntjens, A direct voxel tracking method for four-dimensional Monte Carlo dose
calculations in deforming anatomy. Medical Physics, 2006. 33(2): p. 434-445.
Saleh-Sayah, N.K., et al., SU-GG-T-03: A Distance to Dose Difference Tool for Estimating the
Required Spatial Accuracy of a Displacement Vector Field. Medical Physics, 2010. 37: p. 3183.
Shi, Y.G., et al., Inverse-Consistent Surface Mapping with Laplace-Beltrami Eigen-Features.
Information Processing in Medical Imaging, Proceedings, 2009. 5636: p. 467-478
Tao, G., et al., Symmetric inverse consistent nonlinear registration driven by mutual information.
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.