The Effect of Small Tumor Volumes on Studies of Intratumoral Heterogeneity of Tracer Uptake

DISCUSSION

For any tumor assayed via ¹⁸F-FDG PET, the tumor data are a distribution of gray-scale intensities. Data of this intensity distribution represent the biology of the tumor. We have demonstrated a feasible clinical scenario in which tumors following identical intensity distributions—that is, identical measured tumor biology—have heterogeneity measures that depend strongly on tumor volume. Therefore, differences in uptake heterogeneity observed between disparate tumor volumes may not indicate actual biologic differences between those tumors.

We chose to illustrate this point via the local entropy because that statistic has been proposed previously as a robust measure of uptake heterogeneity; however, we now argue that heterogeneity statistics generally are more sophisticated than the statistical moments familiar to most clinicians. Heterogeneity is a measure of the deviation from homogeneity. In image processing parlance, heterogeneity is the “texture” of the image; the differences from smoothness. In 1973, Haralick et al. described a comprehensive set of texture metrics for gray-scale images (13). Each of those fundamental metrics, one of which is the local entropy, is itself computed from gray-scale co-occurrence matrices. These matrices are simply the tallies of differences between pixel neighbors. That is, co-occurrence matrices track the probability that fixed-distance pixels are shaded differently. Over the entire image, these local variations accrue into the global texture statistics some propose to use as a measure of tumor heterogeneity. Precisely because they are accrued statistics, they measure only the information that actually is contained in the image data and, as such, must to some degree depend on sample size.

This dependence on sample size is not a failing of existing texture metrics. In quantifying texture, one is interested in the spatial patterns and intensity variations observed in image data. In ¹⁸F-FDG PET images, these variations ultimately are caused by some combination of the scanning process and tumor biology. If the total number of pixels is largely diminished, so too is the certainty that patterns and variations allowed by the underlying biology have had adequate opportunity to manifest. To put this another way, the set of intensities and intensity arrangements necessary to build a complete picture of the possible biology is itself incomplete for small volumes. Mathematically speaking, many texture metrics proposed to measure uptake heterogeneity are supposed to have completely different values for smaller volumes—values that may have no predictable relation to those computed for larger volumes. This is in stark contrast to the use of statistical moments such as the distribution variance as a first-order heterogeneity quantifier where, in the case of a common intensity distribution, moment values predictably regress to the mean values as sampling (of any size) is repeated across patients.

The minimum tumor volume we describe is a minimum with regard to the type of comparative heterogeneity analysis some researchers have proposed. That is, it is a minimum imposed by the desire for robust mathematical manipulation of intratumor statistics. Our point is that although very small tumor volumes may be sufficient for treatment planning or other clinical purposes, they do not necessarily contain enough intensity data to be further analyzed using the heterogeneity quantifiers earlier proposed.

We used a straightforward argument to estimate that the minimum tumor volume for adequate intensity sampling is about 700 voxels (45 cm³ for our image data). This argument is based ultimately on a tried-and-true criterion found in classic (14,15) and modern (16) textbooks alike regarding adequate sampling of unknown distributions that are to be compared via χ² goodness-of-fit testing. We chose this test because, in essence, that is what statistics derived from ¹⁸F-FDG PET images are—a comparison of intensity distributions. Without sufficient frequencies in every possible contingency, the χ² statistic does not regress to the χ² distribution, and table values regarding “significance” levels become moot. Although there are situations in which less strict sampling criteria are appropriate (14–16), for the present context, the reasoning against these lax criteria is clear from Figure 6. If, for example, a minimum intensity-bin population of only one were required, the corresponding volume (≈15 cm³; Fig. 5) would yield a local entropy value in the most steeply increasing portion of the h(v) curve. In other words, if the measured intensities have not sufficiently revealed the underlying intensity distribution, the heterogeneity metric is highly sensitive to tumor volume. Thus, inclusion of tumor volumes—or intratumor regions—so small as to essentially guarantee that under-sampling has occurred must bias the results of any comparative uptake heterogeneity study. Therefore, in the context of such studies, the default presumption should be that no statistical inference whatsoever may be made from small ¹⁸F-FDG PET volumes. The onus is on the researcher to demonstrate that a new heterogeneity result is not due to the effects of under-sampling.

As seen in Figure 6, h(v) is monotonic in v and therefore acts as a mere surrogate for tumor volume. This means that a decrease in volume must correspond to a decrease in heterogeneity. What is important here is the relative size of decrease. From Figure 6, it is seen that the derivative of h with respect to v is much less for the larger volumes than for the smaller ones. This indicates that h is much less sensitive to changes in v for large v. Thus, discovering a 20% change in heterogeneity between 2 large volumes actually could be significant since we suspect that h is not strongly affected by volume over the domain of only large volumes. Contrast this, for example, to the comparison of heterogeneities between an 80 cm³ tumor and a 20 cm³ tumor where a 20% change in h is seen to be caused by volume alone.

We feel that performing the analysis we presented in a higher dimension does not offer anything new while only serving to complicate subsequent discussion. The local entropy is extensible into 3 dimensions. However, because of the increased dimensionality, one must include much more detail as to which of the directional cooccurrence matrices are computed and how statistics derived from those matrices are implemented or combined. Furthermore, because these matrices already represent spatial averages over the entire image set (13), rearrangement of the image data is not likely to alter the qualitative dependence on volume. In specific cases of quantitative dependence, added dimensionality increases the number of spatial intensity configurations possible while the sampling (tumor volume) remains the same. Intuitively, however, as the number of unique scenarios a quantifier can describe increases—whether through data dimensionality or quantifier sophistication—the minimum sample size for meaningfully comparing those scenarios increases as well. We therefore predict the deleterious small-volume effect we describe to be even worse for 3-dimensional data and for higher-order heterogeneity metrics but caution that the specific dependence of a given metric on dimensionality quickly goes far beyond the intended scope of this work.

Another potential influence on the quantitative value of uptake heterogeneity metrics is the partial-volume effect. In the case of ¹⁸F-FDG PET, the partial-volume effect renders voxels at the tumor–microenvironment interface to be less bright than voxels filled with tumor (9). In fact, any intensities measuring less than the brightest biologically possible intensity could result solely from partial filling with background or necrotic regions. It is therefore feasible that the partial-volume effect creates at least some—if not all—of the heterogeneity that is visually evident in ¹⁸F-FDG PET images. For this reason, one might expect that partial-volume correction could influence our calculations. It is almost certain that such a correction will alter the overall distribution of measured ¹⁸F-FDG PET intensities. However, this alteration is unlikely to profoundly affect the bright end of the intensity histogram, because the brightest pixels likely are already the closest to being completely filled with tumor and therefore are the least affected by the partial-volume effect (or partial-volume correction). The nonsolid curves shown in Figure 4 represent the effect of plus or minus 1 SE around the probability of measuring the brightest intensities. The horizontal error bar in Figure 4 might therefore represent a reasonable bound on partial-volume effects (or any systematic measurement effects) because, by ensemble averaging many same-scale histograms, we have allowed the biology of similarly sized tumors numerous chances to manifest under the same scanning process.

The calculation of minimum volume we present is better thought of as a technique to be applied to distinct image datasets, rather than a justification for a rigid rule to be applied uniformly to all image data. What is crucial to our calculation is the number of samples of the underlying intensity distribution. Although this means that calculation of the probability at which at least L observations reside in the least populated intensity bin is independent of voxel size (Eq. 3), the input probability (ρ) is not. Because different PET scanners have voxels corresponding to different physical sizes, and because both partial-volume effects and uptake can depend nonlinearly on tumor size and biology, the distribution of measured intensities is itself likely unique to the particular combination of tumor type, scanner resolution, and scanner modality. In other words, the probability that any intensity resides in the least populated bin—as well as the bin size definition itself—is sensitive to the scanning process. In general, one can compute the ensemble intensity distribution for the image dataset; find the probability (ρ) that an intensity resides in the least populated intensity bin; use Equation 2 to construct a plot similar to Figure 3, from which the practical summation limit may be read; and compute the minimum number of voxels by setting P equal to 0.95 (or whatever confidence level is desired) in Equation 3 and numerically solving for v.