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Background: A previous study evaluated the intra-tumoral heterogeneity observed in the uptake of F-18 fluorodeoxyglucose (FDG) in pre-treatment positron emission tomography (PET) scans of cancers of the uterine cervix as an indicator of disease outcome. This was done via a novel statistic which ostensibly measured the spatial variations in intra-tumoral metabolic activity. In this work, we argue that statistic is intrinsically non-spatial, and that the apparent delineation between unsuccessfully- and successfully-treated patient groups via that statistic is spurious. Methods: We first offer a straightforward mathematical demonstration of our argument. Next, we recapitulate an assiduous re-analysis of the originally published data which was derived from FDG-PET imagery. Finally, we present the results of a principal component analysis of FDG-PET images similar to those previously analyzed. Results: We find that the previously published measure of intra-tumoral heterogeneity is intrinsically non-spatial, and actually is only a surrogate for tumor volume. We also find that an optimized linear combination of more canonical heterogeneity quantifiers does not predict disease outcome. Conclusions: Current measures of intra-tumoral metabolic activity are not predictive of disease outcome as has been claimed previously. The implications of this finding are: clinical categorization of patients based upon these statistics is invalid; more sophisticated, and perhaps innately-geometric, quantifications of metabolic activity are required for predicting disease outcome. Background smooth gradation of a single bright spot to a darker It is believed that cancerous tumors are intrinsically het- background is intuitively less heterogeneous than the erogeneous in many ways [1]. Experimentally quantified stark transitions seen by surrounding several clusters of properties that exhibit significant variation within the brightest pixels with only the darkest pixels. The tumors include: gene expression [2], cell proliferation goal of quantifying spatial heterogeneity is to objectively rate [3], degree of vascularization [4], and hypoxia [3,5]. calculate a single statistic that indicates one pattern is a When properties of tumors are assayed via an imaging certain percentage more or less heterogeneous than technique such as positron emission tomography (PET), another. the question of quantifying biologically-functional het- Although the applications of such a statistic to medical image processing and computational biology are broad, erogeneity becomes one of quantifying the spatial het- erogeneity observed in grayscale images. In this case, we focus our attention on the study of metabolic hetero- one describes the arrangement of the various pixel geneity observed within cancers of the uterine cervix. In intensities, with some arrangements subjectively appear- this case, cellular metabolism is assayed via the uptake of ing more heterogeneous than others. For example, the F-18 fluorodeoxyglucose (FDG), a glucose analog with a positron-emitting fluorine isotope [6]. Increased uptake * Correspondence: fbrooks@radonc.wustl.edu of FDG implies increased metabolism of glucose [7], Department of Radiation Oncology, Washington University School of which is then indicated by an increased pixel intensity Medicine, 4921 Parkview Place, Saint Louis MO 63110, USA in the grayscale PET image. Upon inspection of a Full list of author information is available at the end of the article © 2011 Brooks and Grigsby; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Brooks and Grigsby Radiation Oncology 2011, 6:69 Page 2 of 8 http://www.ro-journal.com/content/6/1/69 trans-axial, FDG-PET image of a typical cervical tumor study [9]. We briefly recapitulate the details of that (Figure 1), one can readily observe distinct regions of very prospective cohort study here. Patients underwent a bright pixel intensity near regions of lesser intensity, with pre-treatment, whole-body FDG-PET/CT scan. The each type of region being wholly contained within the pathologic diagnosis and histology were determined by bounds of the tumor. Since both the rate of proliferation pathologists at Washington University in St. Louis. All [8] and the rate of healthy tissue invasion [7] are related to patients were treated with concurrent chemotherapy the rate of cellular metabolism, the motivation to quantify and radiation. A post-therapy FDG-PET/CT scan per- the observed variation in regional metabolism is obvious. formed three months after completing the radiation One goal of such a study would be to investigate if this treatment was used to evaluate the response to treat- metabolic heterogeneity alone could serve as an predictor ment. For our re-analysis of the 73 total patients, the 14 of disease outcome. Indeed, the major conclusion of pre- with persistent disease were combined with the 9 exhi- cisely such a study is that intra-tumoral metabolic hetero- biting new metastases into a single group of those geneity observed in pre-treatment cervical tumors predicts having undergone unsuccessful treatment. response to therapy and risk of recurrence [9]. In this work, we re-analyze the identical FDG-PET- Segmentation of Additional FDG-PET Imagery derived data used in that previous study [9] and offer an The first task of analyzing imaged tumors is to delineate alternative interpretation. Specifically, we argue that the the tumors from the background (referred to as image novel measure employed in that work to quantify spatial segmentation). In the case of FDG-PET, the radiophar- heterogeneity of the grayscale PET images is intrinsically maceutical is also taken up and metabolized by noncan- independent of spatial arrangement, and indeed is a sur- cerous cells, although to a lesser extent [10,11]. The rogate for tumor volume. As such, it can offer no addi- typical result is an evidently stronger PET signal tional predictive capacity to that of tumor volume. (tumor) surrounded by a weaker signal (non-tumor), Thus, the delineation of patients into distinct groups of with the possibility of additional non-tumorous bright- post-treatment survival time via that heterogeneity mea- spots colocated with the bladder or rectum as undeliv- sure is invalid. Additionally, we examine a similar data ered radiopharmaceutical is cleared from the body [10]. set and demonstrate that fundamental, non-spatial mea- As may be seen in Figure 1, the interface between the sures of heterogeneity applied to the FDG-PET assay of healthy and tumorous regions may not be stark, but metabolic activity do not predict disease outcome. rather nebulous as tumor cells invade healthy tissue in a Finally, we discuss some implications of these results. diffuse fashion [12]. This is seen in the image as a smooth gradation from brighter pixels to dimmer ones. In order to objectively distinguish tumor from back- Methods Analysis of Previously Published Data ground, we employed the rule-of-thumb that, for a In this work, we first re-analyze the same data originally visually-selected, three-dimensional region of interest analyzed in a previous heterogeneity-quantification (ROI), any pixel brighter than 40% of the maximum Figure 1 Heterogeneity in an FDG-PET image. A typical FDG-PET image of a cancer of the uterine cervix. The artificial boundary delineates the region of activity above the 40% of maximum intensity threshold. The heterogeneity within the tumor is evidenced by the very bright regions (higher metabolic activity) juxtaposed with relatively dark regions (lower metabolic activity). The undelineated bright spot to the right is a lymph node and is thus not included in the main tumor volume. The vertical edge of this image represents a length of 10 cm. Brooks and Grigsby Radiation Oncology 2011, 6:69 Page 3 of 8 http://www.ro-journal.com/content/6/1/69 ROI pixel brightness is to be considered part of the tumor. This 40% rule is based upon the observation that tumors defined as regions of greater than 40% of the maximum standard uptake value (SUV) of FDG both: colocate with those independently identified via visual analysis of computed tomography scans; and yield volumes consistent with published surgical series [13]. The SUV is a PET intensity measure that first has been converted to proper radiation units, then corrected for both radioactive decay and patient body mass [11]. For each patient, the net result is that every grayscale image pixel is multiplied by a single, positive constant. Because we seek to quantify intra-tumoral variation and since there is some debate as to the usefulness and validity of standard uptake values [14,15], we apply the 40% rule directly to the grayscale intensities. A computer program to semi-automate the image seg- mentation process was written in Python v2.6.1 http:// www.python.org/. As is ubiquitous in the field, the raw FDG-PET images are first processed through a white- Figure 2 Volume versus Threshold Curves.Atypicalvolume balance-correcting, back-projection algorithm via the versus threshold curve (dots) from the data described in [9]. The proprietary software native to the PET machine. The tumor volume is defined to be those voxels with activity above 40% resulting DICOM image files are imported into our pro- of the maximum activity. The slope of the line (-0.37 cc/%) of best gram via the pydicom library v0.9.3 http://code.google. fit between 40% and 80% was then used as a measure of intra- tumoral heterogeneity. This is the slope which we now argue does com/p/pydicom/ and then converted to the 8-bit grays- not predict disease outcome as was claimed in [9]. For reference, cale images via the Python Imaging Library v1.17 http:// the best-fit exponential curve is also shown (dashed). www.pythonware.com/products/pil/. No additional image preprocessing was implemented. Our program enables the user to rapidly target a region of the whole- (inclusively) of the image maximum. The lower bound body, trans-axial PET image set. Next, the program was chosen to guarantee that the tumor could be distin- applies the 40% segmentation rule to all grayscale pixels guished from the background (see Methods) and the in the targeted region (e.g., the pelvic region). A flood- upper bound was chosen to exclude the relatively small fill algorithm is then applied to every pixel remaining in volumes represented by only the brightest pixels. The that region in order to determine the inter-pixel connec- remaining coordinates were fit to a line and the result- tivity (or lack thereof). The result of this algorithm is a ing slope was used as a measure of heterogeneity. set of distinctly-bounded, contiguous objects. The user Greater magnitude of slope was interpreted to indicate can then visually scan the objects and click to remove greater heterogeneity, although we now argue that this those few that are obviously (for sound anatomical rea- is not the case. sons) not tumors. The typical end result is a 10 - 20 Consider a perfectly homogeneous volume consisting of count stack of grayscale images representing trans-axial only a single grayscale value. An example curve for such a slices of a clearly-bounded tumor. scenario is shown as the solid curve in Figure 3. As the segmentation threshold is increased, no change is Results observed in the volume until the threshold becomes Theory greater than the single value. Here, a virtually discontinu- The original measure of heterogeneity presented in [9] ous drop to zero volume occurs. Next, consider a hetero- was derived from a volume versus threshold curve for geneous object, having the same volume as in the previous each tumor. In brief, a set of trans-axial image slices example, but with each of N > 1 grayscale values repre- comprise a virtual tumor object in three-dimensional sented in equal number. In this case, the same change in space. This object was segmented at increasingly high, volume is spread over a greater threshold change. We grayscale intensity thresholds and the volume recorded therefore observe that as more grayscale values are used, at each threshold. The result of this process is a curve heterogeneity increases and slope decreases. Because each likethetypical oneshown in Figure2. Thesecurves grayscale value is represented equally, the change in were then linearized by first restricting the domain of volume for a given change in percent threshold is constant the thresholding to be between 40 and 80 percent (Figure 3 (dashed)). Therefore, a perfectly linear volume Brooks and Grigsby Radiation Oncology 2011, 6:69 Page 4 of 8 http://www.ro-journal.com/content/6/1/69 -lT V e approximate a typical, observed volume (V) A0 versus percent threshold (T)curve forpatient A (see Figure 2). At zero percent threshold, V (0) = V ,the A A0 total volume of the initial target region. It is straightfor- ward to show that the slope of the line between a mini- mum, tumor-defining threshold T and twice that threshold(e.g, 40% and80%)is s =(V (T )/T )·(V A A m m A (T )/V - 1). We now wish to compare this slope m A0 (ostensibly a measure of heterogeneity) to that of a sec- -μT ond patient, B,where V (T)= V e .Fromthe 73 B B0 available V (T) curves, we observed that, save for extre- mely large tumor volumes (greater than 150 cm ), the total volume of tumor exhibiting pixel intensities greater than 80% of the maximum observed intensity is typically very small (≈3cm ). Thus, the end points of the lineari- zation are approximately equal for every patient. There- −λ2T −μ2T m m fore, , from which it is seen that V e ≈ V e A0 B0 2 2 V (T )/V ≈ V (T )/V . Proceeding as before, and m A0 m B0 A B employing this approximation, one may show that the change in slope is Δs ≡ |s - s |= |V (T )- V (T )|/ A B B m A m Figure 3 Schematic Heterogeneity Curves. The solid curve shows T ≡ ΔV (T )/T . In words, the previously published m m m the nearly discontinuous drop (large slope) that must occur for a measure of intra-tumoral heterogeneity is directly pro- perfectly homogeneous volume of single activity level. The dashed portional to the pre-treatment tumor volume. It is line shows the curve expected for a volume containing equal numbers of each activity level possible. This heterogeneous scenario important to note that this result depends only upon has a decreased slope. Thus, increasing slope implies increasing the measured 40% tumor volumes, and in no way homogeneity. This is counter to the interpretation given in [9]. depends upon the decay rate or closeness of fit of either exponential curve. versus threshold curve implies maximal heterogeneity over The linear proportionality derived above is seen in the multiple grayscale values. original FDG-PET data. As described in [9], we plotted Itis importanttopointoutthatinthe scheme the total volume (in cm ) of the target region with pixel described above, the numeric value of the slope is inde- intensities greater than a given percent threshold versus pendent of spatial arrangement. For example, the set of percent threshold. We then computed the least-squares grayscale values representing the tumor could be rear- linear regression for points between 40% and 80% ranged such that each value resides at a new 3D Carte- thresholds. The magnitude of the slope is plotted versus sian coordinate. In other words, it is possible to “draw” the tumor volume (i.e., that defined at 40% threshold) in various artificial objects by purposefully placing selected Figure 4. As predicted, it is clearly seen that the slope magnitude is linearly proportional to tumor volume. grayscale values at desired coordinates. However, the Therefore, the previously published delineation between number of each distinct grayscale value remains con- unsuccessfully- and successfully-treated patient groups stant, regardless of where in the object those values may is based exclusively upon tumor volume, not upon any reside. Since the volume of the tumor object ultimately additional measure of heterogeneity. Larger volumes was calculated by counting pixels above a given thresh- intuitively imply long-duration, aggressive tumor pro- old, that volume does not change even when the tumor object is destroyed via rearrangement. Thus, any mea- gress. Thus, the simplest explanation of a statistically- sure of heterogeneity given by the slope is only of the significant, predictive result (in [9]) is that the relatively diversity of intensity values, not in spatial arrangement small number of patients with new or persistent cancer of those values. tended to have larger pre-treatment tumor volumes. In other words, the apparent statistical significance is no Critique of Previously Published Results more than the expected artifact arising from the inap- In a stack of trans-axial, FDG-PET images, a region of propriate use of the standardized permutation test interest fully containing the tumor is first selected by a (p-test) upon groups with greatly differing numbers of trained clinician. This is the region of interest that is members. successively thresholded and the volume of the region An important consequence of the finding that Δs ∝ remaining after thresholding is computed. Let V (T)= ΔV is that the slopes computed for similar volumes A Brooks and Grigsby Radiation Oncology 2011, 6:69 Page 5 of 8 http://www.ro-journal.com/content/6/1/69 Figure 4 A Volume Surrogate. A previously published measure of Figure 5 No Predictive Value. Histograms of the volume- intra-tumoral heterogeneity is plotted versus tumor volume for detrended slopes for patients who underwent successful (light patients who underwent successful (circles) or unsuccessful shading) or unsuccessful (dark shading) therapy. The overlapping (triangles) therapy. Observe that the heterogeneity measure is histograms indicate that the ostensible measure of distinguishing directly proportional to volume and there is a lack of clustering of intra-tumoral heterogeneities actually has the same mean value for patients into distinct groups with differing disease outcome. As every patient, differing only by random noise, and thus does not seen in the inset, the trend persists over three orders of magnitude. predict disease outcome. The inset axes have the same units as in the primary plot. segmented as described in the Methods section. We should themselves be similar, differing only by random computed the volume-detrended slopes as before. noise. To see this, we first detrended the slopes by Again, we found no distinguishing capacity whatsoever dividing each by the 40% tumor volume. This is identi- between the successfully treated patients, where the cal to having first plotted the percent volume versus per- mean slope is 2.20, and the unsuccessfully treated cent threshold and computing the slope of the best-fit patients where the mean slope is 2.23. line. The dimensionless, volume-detrended slopes were pooled and then a histogram bin width of 0.1 was com- Extended Heterogeneity Analysis puted via a commonly-used, optimal bin-width formula Previous arguments imply that the volume versus [16]. The slopes were separated into distinct groups threshold slope is sensitive to the distribution of grays- based upon apriori knowledge of patient outcome. A cale intensities of the trans-axial image stack. We there- histogram of volume-detrended slopes was created for fore chose to investigate the relation between these each group and is shown in Figure 5. There, it is clearly distributions and disease outcome via the fundamental seen that the group which underwent successful treat- quantifiers of distributions: the standard deviation, skew- ment (light shading) almost completely overlaps that ness and kurtosis. Each of these quantifiers describes a which underwent unsuccessful treatment (dark shading). unique quality of non-spatial heterogeneity. The stan- Each group differs from a single mean of 2.3 by the dard deviation indicates the number of unique grayscale same standard deviation, 0.13. This important observa- values comprising the image stack; that is, the number tion, that the volume-detrended slopes are essentially of different levels of metabolic activity observed. The identical for every patient, implies that the previously kurtosis indicates the relative strength of those meta- published measure of intra-tumoral heterogeneity is not bolic levels since a distribution with only a single, sharp in any way predictive of disease outcome. peak (higher kurtosis) indicates a favored metabolic In an effort to verify this result, we studied the FDG- activity level. The skewness indicates the pervasiveness PET imagery of 47 recently-examined patients that did of activity levels. For example, an overall brighter distri- not appear in the previously published study. The bution (negatively skewed) implies that the majority of images were again obtained as described in [9] but tumor volume exhibits relatively higher metabolic Brooks and Grigsby Radiation Oncology 2011, 6:69 Page 6 of 8 http://www.ro-journal.com/content/6/1/69 activity whereas a skewness of zero indicates equal covariance matrix to obtain the single variable repre- volumes of activities above and below the mean activity. senting the maximal use of information within the initial Since each of the fundamental quantifiers describing variables. We found that a new variable, ψ = 0.999 stan- the distribution of FDG-PET intensities represents an dard deviation - 0.010 skewness - 0.033 · kurtosis, best independent, biological aspect of the tumor, it seems described the variation in phase space. Since the disease reasonable to us that they are members of a basis set of outcomes are known, we computed the value of ψ for heterogeneity-describing statistics. In other words, we each patient and performed a standardized permutation test of significance (p-test). The mean values of ψ for suggest that any feasible non-spatial indicator of hetero- patients undergoing successful or unsuccessful treat- geneity would have to in some way depend upon the standard deviation, skewness and kurtosis. We com- ment are 30.4 (p = 0.36) and 28.8 (p = 0.24), respec- puted these quantifiers for the 8-bit grayscale intensity tively. The two-sided p-values given here indicate that distributions for each of the 47 recently-examined our default assumption that the mean of one group patients. We then constructed a three-dimensional equals the mean of the other cannot be rejected. In phase space where each patient is represented by a other words, these relatively large p-values are consis- point having a standard deviation, skewness and kurtosis tent with our earlier observation (seen in Figure 6) that coordinate. Each point in that space is then given a there is no substantial difference between the values of unique symbol corresponding to patient outcome after ψ for each treatment group. Thus, our conclusion is chemoradiotherapy with curative intent. In Figure 6, it that the optimal linear combination of the non-spatial is seen that the patients free of cancer after therapy (cir- metabolic quantifiers does not predict disease outcome cles) are well-mixed with those for whom therapy was any better than random chance. unsuccessful (triangles), and no obvious clustering of From the corresponding eigenvalues, we compute that the patient groups is apparent. To explore whether any ≈98% of the total variation in phase space is represented predictive information can be obtained from the non- by the standard deviation alone. This high percentage spatial metabolic activity quantifiers, we performed a indicates that more sophisticated, non-spatial measures principal component analysis. The standard deviation, of heterogeneity–which we assert ultimately are based skewness, and kurtosis for each of 47 patients comprise upon the fundamental quantifiers–are unlikely to the rows of the 3 × 47 matrix of observations. As is improve upon the standard measure of uncertainty. In described in many textbooks [17], we then compute the other words, the standard deviation alone is a reason- unit-magnitude eigenvectors of the mean-detrended able non-spatial measure of the variation in metabolic activity. Thus, we suggest that the textbook usage of the standard deviation as the uncertainty in the mean value is adequate when computing statistics, such as the total glycolytic volume, which are spatially averaged over the entire tumor volume. A potential concern lies in our definition of patient groups, where the unsuccessfully treated group is the union of those patients having post-treatment persistent cancer with those having post-treatment new metas- tases. In an effort to avoid any bias due to pre-existing metastases, we performed both the re-analysis of exist- ing data as well as our entire principal component ana- lysis again. We first eliminated those with new metastases from the unsuccessfully treated group. We then computed the volume-detrended slopes described earlier and again found that mean value for the success- fully treated group (2.28) is nearly identical to that (2.32) of the unsuccessfully treated group. Thus, bias due to inclusion of patients with new metastases does not explain the lack of predictive capacity of the pre- Figure 6 Quantifier phasespace. A phase space of intuitive, non- viously published measure of heterogeneity. We now spatial quantifiers of heterogeneity is shown. Each point has a explore the potential effect of this bias in our principal standard deviation, skewness and kurtosis coordinate. As is evident in the plot, and confirmed via principal component analysis, there is component analysis. Proceeding as before, we compute no delineation between patients who underwent successful (circles) anew ψ variable for the truncated matrix of observa- or unsuccessful (triangles) radiotherapy. tions, excluding patients with new metastases. The Brooks and Grigsby Radiation Oncology 2011, 6:69 Page 7 of 8 http://www.ro-journal.com/content/6/1/69 mean values of ψ for patients undergoing successful or common to image texture analysis [24] or the physics of unsuccessful treatment are then 30.4 (p = 0.51) and 31.7 particle systems [25]. (p = 0.38), respectively. We again see no substantive dif- ference between the mean values for each group and Conclusions thus conclude that patients with new metastases did not We have shown that neither the currently accepted bias our previous result that non-spatial metabolic quan- measure, nor other reasonable non-spatial measures, of tifiers do not predict disease outcome. intra-tumoral metabolic heterogeneity within cervical cancer are predictive of disease outcome. This is directly Discussion counter to a previously published claim. We have given It is important that we immediately point out that we a brief mathematical explanation of why that claim is are not claiming that intra-tumoral metabolic heteroge- erroneous and have supported our argument with the neity does not exist. Indeed, we presume that metabolic results of both a re-analysis of the originally published activity can vary significantly throughout a tumor. In a data and a fundamental statistical analysis of a similar younger, pre-vascularized tumor, such variations are data set. Our findings have immediate impact upon clin- likely due to a non-constant, diffusion-limited nutrient ical research and treatment. The use of currently- density [18]. In a mature tumor, these variations could accepted, non-spatial quantifiers of intra-tumoral meta- be due to necrosis [18] or even steric constraints bolic heterogeneity as a means to categorize patients imposed by the spatially-randomized, densely-packed into groups predicted to be successfully or unsuccess- nature of newly-formed vascularization networks [19]. fully treated is invalid. Thus, more sophisticated, and Inordertomeasureagenuineheterogeneityinastack perhaps innately-geometric, quantifications of metabolic of images, one must be able to distinguish a single activity are required for predicting disease outcome. volume element (voxel) from another. The minimum detectable inter-voxel difference is determined by the Acknowledgements noise intrinsic to the FDG-PET assay. The noise in a We would like to thank Scott Brame and Bruce Davis for illuminating typical 3D FDG-PET image reconstructed via filtered discussions and the latter for carefully reviewing the manuscript. This work was supported by the National Institutes of Health under Grant 1R01- back-projection has been estimated to be 1.5 kBq/mL CA136931-01A2. [20]. This is only 3% of the ≈50 kBq/mL mean activity of all tumor voxels defined above 40% intensity thresh- Author details Department of Radiation Oncology, Washington University School of old in our extended heterogeneity study. This implies Medicine, 4921 Parkview Place, Saint Louis MO 63110, USA. Division of that the FDG-PET assay can distinguish relatively small Nuclear Medicine, Mallinckrodt Institute of Radiology, Medical Center, Saint changes in the metabolism of tumor cells averaged over Louis MO, USA. Department of Obstetrics and Gynecology, Washington University Medical Center, Saint Louis MO, USA. Alvin J. Siteman Cancer a typical PET image voxel. We therefore conclude that Center, Washington University Medical Center, Saint Louis MO, USA. the non-predictive nature of bulk heterogeneity statistics is not due to either a genuine lack of variation in meta- Authors’ contributions FJB conceived and drafted the manuscript as well as performed all bolic activity or the poor resolution of this variation. mathematical analyses. PWG designed the protocol for the interpretation the Instead, our results imply that that quantification of FDG-PET images, acquired the volumetric data presented, and provided tumor composition via FDG-PET remains a challenging, crucial medical and anatomical insight into the analyzed data and imagery. Both FJB and PWG read and approved the final manuscript. open problem to be solved. We maintain that a shift of focus from tumor composition to shape and location Competing interests offers immediate potential for improved clinical therapy. Frank J. Brooks has no conflicts of interests. Perry W. Grigsby has no conflicts of interests. Consider that the uncertainty in the anatomical place- ment of brachytherapy radiation sources via a standard Received: 7 December 2010 Accepted: 9 June 2011 gynecological implant is at least several millimeters. Published: 9 June 2011 This is the same order of spatial uncertainty in FDG- References PET-assayed tumors where the side length of a cubical 1. Heppner GH: Tumor heterogeneity. Cancer Res 1984, 44(6):2259-65. voxel is typically ≈4 mm. Also, as the computation of 2. 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Radiation Oncology – Springer Journals
Published: Jun 9, 2011
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