The Diagnostic Utility of the Lognormal Behavior of PET Standardized Uptake Values in Tumors

Data Features

Many investigations have taken place encompassing various cancer categories and involving the popular FDG tracer of PET. A small subset of these met our criteria for inclusion in Table 1. For this subset, additional distribution characteristics of results—especially information beyond the S̅U̅V̅ ± SD typically reported—have been calculated from published data.

A somewhat subjective ranking, in a decreasing importance order, for a distribution might be the mean, SD, and skew. The CVs are preferred to SUV SDs. Being normalized, the CVs' range of values among cancer categories was consequently more restricted than that of wide-ranging SUV SDs. Similarly, a more-or-less restricted range was found for the log₁₀SUVs' σ, as the latter is essentially the SUV CV ÷ 2.303 (Appendix). Nevertheless, statistical testing showed no commonality among σs of the different cancer types.

Tables 1 and 2 show that the logarithm operation on SUVs tended on average to remove skew. Moreover, the small positive and negative skews were equally likely for log₁₀SUV. On the other hand, SUV distributions always showed positive skews. For log₁₀SUVs, the largest ratio of skew to its SE found in a total of 40 investigations in Tables 1 and 2 was only 2.3. This skew removal is associated with the functional nature of taking a logarithm: compressing values, with larger ones causing the skew in SUVs being compressed more.

The global cerebral glucose metabolic rate is not in Table 1 because it is well reviewed elsewhere(11,12). This is closely related to the SUV. Thus, its statistical behavior might be compared with findings from Table 1. A review by Wang et al. (11)provided data for compiling Table 3: The data from multiple investigations of normal brain tissue were examined further in the manner of Table 1. Normal brain tissue shows a significantly smaller FDG σ̅ than all tumor types. The many factors influencing cerebral glucose uptake have been well studied and understood (11,12). Presumably aware of and accounting for these, investigators have been able to obtain a more-or-less intrinsic constancy in brain tissues' uptake in a well-defined population. Also, for both cancers and normal brain, the spread of the σs is about twice that expected from a hypothetic circumstance of all investigations assumed to have the same intrinsic variance of their σs. This corroborates Bartlett's test result, which indicated a lack of homogeneity among categories in Table 1.

Another question to examine is whether variations among S̅U̅V̅s in Table 1 (even factors of 2 or more) from investigations involving the same cancer category have statistical significance. This would show whether these have sufficient similarity in patient populations and PET protocols, including types of corrections (1,12–14) made. Within several categories (non-Hodgkin's lymphoma, breast, head and neck, and pancreatic cancers as well as normal liver), the Kruskal-Wallis P for the reported discordant results to be occurring by chance was always <0.0001. Thus, for a given category, 1 or more among its 3 or 4 investigations reported presumably had significantly different patient or protocol (or both) characteristics. Until there is better understanding, interinstitutional comparisons must evidently be made with caution.

Theoretic Reasons for Lognormal Distribution

Gaussian distributions fitting log₁₀SUVs is a significant observation. A variable with this type of distribution that is found so frequently in nature might be regarded as a more natural marker than a distribution that is non-Gaussian. However, because of limitations from available numbers of lesions or patients for a distribution, it has not been uniquely determined that the logarithm, rather than some other function, is best. Hence, the empiric evidence favoring logarithmic usage might be supplemented with some theoretic points.

There are underlying factors responsible for observed SUV variabilities in a given institution's investigation of a single cancer category, using the individual locally standardized procedures. A review by Carson (15) shows that these factors are attributed to separate influences: fundamental physiologic, test measurement, and data analysis. One approach is to express the observed SUV approximately as a product of factors, all of which contain their own internal sources of variability. This extends and quantifies a proposal by Bland and Altman (16) that rate constant products can lead to lognormal distributions: The first factor is the Gjedde-Patlak accumulation rate K. It and the second factor approximate the patient's intrinsic SUV in terms of basic physiologic quantities (17). Equation 1 contains the rate constants of the model of Sokoloff et al. (18). Here the product, PS, of capillary permeability P and its surface area per gram S is an explicit replacement of this model's rate constant k₁. V is an entire body distribution volume associated with a tracer plasma clearance rate k_p.

The PET protocol factor is a calibration factor, defined as the ratio of measured SUV to intrinsic SUV. It is the product of many (measured ÷ true) factors originating from many variability sources (1,12–14). These include counting statistics and corrections not fully made in interpreting the SUV measuring process. In particular, the absence of any partial-volume correction is a source of variability, with substantially different results possible from scanners of different resolutions, as shown by Grady (19).

Explaining the magnitude and shape of an observed SUV distribution would require a rigorous quantitative examination of individual statistical distributions of every variable component in Equation 1. However, the scope here is limited to commenting on the possible dominant causes of variability. The relative importance of protocol versus physiologic factors might be judged from the following:

• In studying lung cancer, Minn et al. (20) found excellent FDG PET reproducibility in the same patient, obtaining CVs for SUVs and glucose-corrected SUV-leans (i.e., calculated from lean body mass not total body weight) of only 0.1 and 0.06, respectively. Similarly, for a variety of malignancies, Weber et al. (21) found the same patient reproducibility with a 0.09 CV for SUVs.

• Many investigations of the cerebral glucose metabolic rate (11,12) typically show CVs as low as 0.19 (Table 3), which includes both patient and PET effects.

• Table 4 from the work of Avril et al. (22), showing possible SUV definitions, illustrates variations from this source. These are responsible for, at most, a CV associated with methods of about the 0.20–0.22 shown in the last row of Table 4.

The average CV of 0.55 in Table 1 is much larger than the CVs of the foregoing. This suggests that individual physiologic factors are more important than PET factors.

With data showing the left side of Equation 1 to be lognormal, the product of the first 2 (dominating the variability) explanatory factors on the right must now be lognormal. If variations in individual subfactors in these are lognormal, then a mathematic consequence is that the entire combination is lognormal. It is tempting to theorize perhaps a variability dominance from the vascular surface area per gram S: in a specific tissue (as a factor in k₁) and in all body tissues (as a factor in k_p). This is because it is well established (23,24) that morphologic aspects of vasculature have lognormal distributions. Also corroborating the ideas are in vitro data from an ovarian carcinoma cell line (25): FDG uptake was proportional to cell density, whose distribution is lognormal (6) and relates to S.

The range noted for the CVs or σs of FDG among malignancies can also be discussed within the context of Equation 1. From Bartlett's test result, ranging was wider than would be expected if all cancers had the same intrinsic CV. However, the test showed that this range, 0.11–0.32 for σ in Table 1, was not excessively large for 25 investigations. Some insight might come from isoleucine uptake (26) in normal rat brain as a well-defined tissue type: CVs remaining essentially constant in spite of PS values being made, by concentration changes, to span a factor of 200. Perhaps in line with this, diverse cancer categories, with no doubt widely varying PS values, might be expected to have at least not a drastically wide range of CVs.

Potential Applications in Diagnoses

Considering our results and the above discussion, specific applications in the diagnostic process arise.

Receiver-Operating-Characteristic Curves.

Receiver-operating-characteristic (ROC) curves now can be constructed more accurately over wider ranges with fewer patient data. After transforming SUVs to log₁₀SUVs, a 2-parameter fit to a Gaussian distribution is possible. This provides a basis for extrapolations into distribution tails beyond data. If need be, it is even possible with limited data to use historical information on σ. This would permit approximating a distribution when only its measured mean is available. Once the best distribution curves for available numbers of patients for both malignant and benign or normal categories are gotten, the construction of an ROC is straightforward (Appendix).

Figure 3 illustrates generic ROCs on double probability paper (Appendix). Noteworthy is its primary dependency on only a ratio of S̅U̅V̅s. Secondarily, there is also a σ dependence that is not shown because this ROC uses its generic 0.23 value for both malignant and benign or normal distributions.

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FIGURE 3.

Generic ROCs for use in approximately describing any cancer category. Parameter shown identifying line is log₁₀(S̅U̅V̅_mal/S̅U̅V̅_ben), where subscripts indicate malignant and benign, respectively. Better straight-line descriptions are possible, provided there is sufficient number of patients. These would improve on 0.23 used for SDs of both malignant and normal log₁₀SUVs in calculations here.

A 2-tailed diagnostic test using 2 SUV cutoffs to discriminate among 3 conditions is a unique example. In a lung cancer investigation (27), adequate benign (with lower SUVs) and malignant (with moderately higher SUVs) cases existed for construction of a reliable ROC. However, 2 cases of blastomycosis were encountered with very high SUVs (10.8 and 16). The lognormal distribution of the latter (based on a geometric mean of 13 and a representative 0.23 σ for logarithms) was then approximated. When used with that of the malignant cases, it then becomes feasible to construct a preliminary additional ROC—that is, an ROC for discriminating malignant cases from blastomycosis—using an additional high SUV cutoff.

Test Figure of Merit.

The diagnostic capabilities of 2 tests having lognormal test values x, such as SUVs, may be compared using a simple figure of merit (FOM). Suggested is (Appendix, geometric mean usage) Eq. 2 where Eq. 3 may be used to characterize test effectiveness. The larger the FOM, the larger the ROC area, which, using the probability function (Appendix), is Φ(FOM). Figure 4 shows this function. When lacking better information, the σ_t may be approximated with an average over all tissue types, (0.23² + 0.23²)^1/2 = 0.33, based on Tables 1 and 2.

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FIGURE 4.

ROC area for comparing tumor diagnostic protocols. Parameters identifying these lines are total SDs—that is, square root of sum of squares of σs belonging to benign and malignant log₁₀SUV distributions. For central line, its (0.23² + 0.23²)^1/2 = 0.33 is typical σ_t of cancer categories of investigations here.

An example of this FOM is using Equations 2 and A7 for the calculated set of results appearing in Table 4. The intention is to find the most appropriate SUV definition for breast cancer diagnosis. Table 4 shows that the simply obtained analytic results agree well with areas from special ROC curve-fitting software. Moreover, formulas can help to understand why a particular method is superior to another.

Lognormal Distribution and Associated Parameters

A value x (such as SUV) is lognormal (28) if the distribution of n values of its logarithms can be adequately described by the Gaussian distribution probability density Eq. A1The lognormal distribution of xs is then Eq. A2Here σ is the SD of the log₁₀(x) values and x̅ is their geometric mean having relationships Eq. A3

The CV of the xs can be related to σ when variations are not too large. Eq. A4 Eq. A5For data of the type in Table 1, the error of this approximation is <10% in about two thirds of its applications.

For any distribution of values y (whether SUVs or log₁₀SUVs), skew is defined as Eq. A6It is 0 for perfectly symmetric distributions but otherwise is a quantitative measure of asymmetry. Its SE when there are n samples is (6n/[(n − 1)(n − 2)])^1/2.

Cumulative Distribution Function and Associated Graphs

The integral of Equation A1 or Equation A2 up to the value x is the cumulative distribution function nΦ(x). The probability integral Φ appears commonly in mathematic tables and software. nΦ(x) is the number of values occurring below a cutoff x. For a malignant population, 1 − Φ(x) is the sensitivity; for a benign or normal population, Φ(x) is the specificity.

Plots of Φ against its argument can become straight lines if the Φ axis is distorted into a probability paper's axis. This ordinate (Figure 2), invisibly marked off in uniform units of numbers of SDs, has the corresponding values of 1 − Φ visibly marked at desired locations. If the abscissa's independent variable on this probability paper is Gaussian, data points will then define a straight line. The slope (in SD units per abscissa unit) is the reciprocal of the independent variable's SD. For lognormal data, a logarithmic paper's axis can be convenient, as in Figure 2.

ROCs

An ROC can be constructed from 2 cumulative distribution function plots, a malignant distribution plot and a benign or normal distribution plot. If overlapping Gaussian distributions are involved, their Φs are straight lines on probability paper. This linear behavior then carries over to the ROC. Figure 3 shows 2 probability axes used. Thus, straight-line graphic ROC fitting is an alternative to more complex fitting requiring special software.

A commonly used measure of diagnostic capabilities, especially in comparing tests, is the area under an ROC. It has been shown (29,30) that for overlapping Gaussian distributions Eq. A7where σ_t is the square root of the sum of squares of the 2 SDs. The means' difference for lognormal SUVs is log₁₀S̅U̅V̅_mal − log₁₀S̅U̅V̅_norm = log₁₀(S̅U̅V̅_mal/S̅U̅V̅_norm), with geometric means used. Thus, a large means ratio and a small σ_t result in a large ROC area and a highly discriminating test.

Law of Proportionate Effects

A set of y_is can have its distribution of values influenced by how changes among these individual values occur. In many processes, small changes are proportional to existing values (28). Eq. A8 Eq. A9For example, in the angiogenesis process y would be S, the total capillary vessel area per gram, which increases by some random fractional amount r_i in each time interval. When Equation A9 is summed on both sides, the result on the left, ln(final y ÷ initial y) and, hence, ln(final y), is Gaussian. The reason stems from the central limit theorem for generating Gaussians from random numbers: When many of the latter are summed, the total is Gaussian. Thus, in the example here, lnS is Gaussian because many generations of random growth factors occurred.