Analysis of Lymph Nodal Metastases in Malignant Melanoma Using the Poisson Probability Paradigm and Bayes Rule

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1 Anatomic Pathology / POISSON PARADIGM AND METASTATIC MELANOMA Analysis of Lymph Nodal Metastases in Malignant Melanoma Using the Poisson Probability Paradigm and Bayes Rule Robin T. Vollmer, MD Key Words: Probability; Melanoma; Metastasis; Lymph nodes; Poisson; Bayes theorem Abstract This article deals with and formalizes 2 notions common to the practice of pathology. The first is that the number of lymph nodes found positive for metastasis relates directly to the total number of lymph nodes examined. The second is that for any patient, there is a chance that the absence of lymph node metastases is a false-negative result. I introduce the Poisson probability density function to deal with the first notion and the Bayes probability rule to deal with the second. To illustrate the insight these 2 models provide, I apply them to data regarding lymph nodal metastases in malignant melanoma. In this preliminary study, the results of these 2 models correlate well with observed survival probabilities in patients with stage N0 melanoma and with observed rates of false-negative results in sentinel lymph node biopsy technology. With further development, the combination of these models should provide a way to estimate the probability of nodal metastasis when, in fact, none have been observed. Thus, these models might provide useful tools for evaluating patients with stage N0 malignant neoplasms. In malignant melanoma, clinical and pathologic stages are of greatest prognostic importance, and the status of regional lymph nodes is of special interest. 1,2 To evaluate the regional lymph nodes, sentinel node technology has become widely used, 3,4 and its success is based on the concept of an orderly progression of tumor cells from the primary site to key regional lymph nodes. Nevertheless, uncertainties remain, and falsenegative results occur In some cases, tumor cells are thought to bypass regional nodes via lymphatic-venous shunts. 12 Uncertainties also might be due to the multistep, multistage molecular processes that comprise metastasis, 13 because at each step and stage, tumor cells might fail to establish a viable focus of metastasis. For these reasons, it seems reasonable to view the observance of metastasis in any lymph node as a probabilistic event, especially if one considers the additional errors of surgical identification and removal of lymph nodes and of pathologic recognition of small foci of metastatic tumor in lymph nodes. In this article, I introduce and illustrate 2 models that deal with the probabilistic nature of identifying lymph nodal metastases in malignant melanoma: the Poisson probability density function and Bayes rule for conditional probabilities. 14 I have found that both lend insight into observed, or not observed, metastases in regional lymph nodes, so that both may provide useful tools for understanding nodal metastasis of malignant melanoma. Materials and Methods Study Cases The cases used in this study comprised 785 with cutaneous melanoma. All had lymph node biopsies or elective lymph node dissections; however, all were evaluated before Am J Clin Pathol 2005;123:

2 Vollmer / POISSON PARADIGM AND METASTATIC MELANOMA the use of sentinel node technology. I personally evaluated the histologic features of the primary lesion and the excised lymph nodes. Table 1 summarizes key characteristics of the patients, including sex and age. In what follows, I concentrate on 2 subsets of these patients. First is the subset with positive lymph nodes, which I use to develop the Poisson model and to estimate its parameter, and I will especially emphasize those who had at least 3 nodes excised. The second subset comprises those without positive nodes, and I will use this group to study the implications of the Poisson model and Bayes rule for understanding outcomes in melanomas in stages T2a and II. Poisson Probability Density Function The Poisson probability density function, symbolized here as f, deals with a counted, positive phenomenon like the number of positive lymph nodes (x) in a lymph node dissection and where the result for the i th lymph node number is statistically independent from the results for the (i 1) th and (i + 1) th nodes. 14 Although this condition of independence might appear stringent, it derives from the notion that cells leaving the primary tumor behave independently of one another when they form sites for metastatic growth. Furthermore, in practice, the Poisson distribution often provides a good approximation for the distributions of many positive, counted phenomena. As used herein, the Poisson distribution also satisfies the plausible notion that the probability of finding a positive lymph node relates directly to the number of lymph nodes examined. In other words, the more lymph nodes examined, the more that should be found positive. Thus, the form of the Poisson distribution function I use is: Equation 1 f(x n, α) = [(α n) x exp( α n)]/x! Table 1 Characteristics of 785 Study Patients * Characteristic Result Sex F 309 (39.4) M 476 (6) Age (y) Mean 46 Range Location of primary tumor Nonacral extremity 263 (33.5) Other 522 (66.5) Tumor thickness (mm) Mean 2.6 Range Lymph node metastases 479 (6) No. of lymph nodes removed Mean 12.9 Range 1-63 No. of positive lymph nodes Mean 1 Range 0-20 * Data are given as number (percentage) unless otherwise indicated. In Equation 1, x is the number of positive lymph nodes, the product α n is the expected number of positive nodes, n is the total number of lymph nodes examined, and α gives the expected rate of positive nodes. In general, it is optimal to assume that α reflects individualistic factors, including the biologic potential of a particular patient s tumor for nodal metastasis, the palpability of lymph nodes, and the technology for identifying lymph nodes to be excised. In other words, the value of α should be thought specific for any given patient, and for any patient the maximum likelihood estimate of α is the sum of positive nodes divided by the sum of nodes examined. Nevertheless, where indicated, I also will use available data in the literature to derive a population-based estimate for α as the sum of positive nodes divided by the sum of nodes examined over all patients. In taking this second approach, I assume that the nodes across a population of patients are approximately independent of specific patient identity, because this broader assumption allows one to derive useful, approximate results from the literature. The approach here also assumes that patients with cutaneous melanoma fall into one of 2 biologic categories: (1) they do not have or imminently will develop nodal metastases or (2) they have or soon will manifest nodal metastases. In the first category α = 0, and in the second, α > 0. Thus, the probability of no metastasis is written as P(α = 0) and the probability of metastasis as P(α 0). When α = 0, f(x = 0 n, 0) in Equation 1 is taken to have the value of, because any variable raised to zero exponential has the value of and because 0! also has the value of. Furthermore, when α = 0, f(x > 0 n, 0) of Equation 1 always will equal 0. When α 0 (ie, α > 0), the Poisson probability of observing no positive lymph nodes simplifies to: Equation 2 f(0 n, α) = exp( α n) General Linear Model With Poisson Link Function To analyze how α relates to other key variables like the time of excising nodes and tumor thickness, I used the general linear model with Poisson link function. 15 In this analysis, the number of positive nodes, x, in each patient was the dependent variable, and y1, y2, etc, were the key explanatory variables. To adjust for n, the total number of nodes examined, I used the offset function in S-PLUS software (MathSoft, Seattle, WA). Thus, with this analysis, I obtained a linear model so that: Equation 3 α = exp(b0 + b1 y1 + b2 y2 + ) where b0 provides an intercept coefficient and b1, b2, etc, are regression coefficients estimated by the model software (S- PLUS). Where indicated, I also used linear regression. Bayes Rule The primary result I sought was an estimate of the probability of nodal metastases (Met) given that, in fact, no positive 708 Am J Clin Pathol 2005;123: Downloaded 708 from

3 Anatomic Pathology / ORIGINAL ARTICLE nodes were observed among a total of n nodes examined. This is a conditional probability symbolized as P(Met x = 0, n). Using the Poisson model with parameter α, this conditional probability is equivalent to P(α 0 x = 0, n). Bayes rule 14 tells us that it can also be written as follows: Equation 4 P(x = 0 α 0, n) P(α 0) P(Met x = 0, n) = P(x = 0 α 0, n) P(α 0) + P(x = 0 α = 0, n) P(α = 0) Dividing both numerator and denominator of the right side of Equation 4 by P(x = 0 α 0, n) P(α 0) and noting that the Poisson probability for P(x = 0 α = 0, n) is simplifies Equation 4 to: Equation 5 1 P(Met x = 0, n) = P(α = 0)/P(α 0) 1 + P(x = 0 α 0, n) The Poisson model and Equation 2 allow substitution of exp( α n ) for P(x = 0 α 0, n). Furthermore, the term P(α = 0)/P(α 0) in Equation 5 is just the odds of no nodal metastases, which I estimated using a previously published Pnode function as follows 16 : Equation 6 P(α = 0)/P(α 0) = (1 Pnode)/Pnode and I adjusted Pnode to reflect specific values of thickness, presence of ulcer in the primary site, and tumor location (nonacral extremity vs other). Expected Value of Poisson Probability The Poisson probability of observing no positive lymph nodes when n are examined is given by Equation 2 and is used in Equation 5; however, to estimate this probability for published outcomes from studies of large numbers of patients with negative nodes (ie, stages I and II), I needed to use the expected value of the Poisson probability obtained for all possible values of n, because the publications of outcomes for stage I and II patients included neither exact numbers of lymph nodes examined nor even means. Nevertheless, the expected value for the Poisson probability can compensate for this lack of information by relying on the probability density function of the total number of examined nodes. The argument proceeds as follows. If the probability density function for n is symbolized as f(n), then the expected value, E, of the Poisson probability obtained for all possible values of n is defined as: Equation 7 E f(0 n, α) = exp( α n ) f(n) dn To complete this estimate, I used the subset of patients with negative nodes to derive a model for f(n) Appendix 1 and then used the result in Equation 7. Finally, I substituted this expected value for P(x = 0 α 0, n) in Equation 5. Results Figure 1 shows a histogram (bars) of the observed number of positive lymph nodes in the study subset of 476 patients with positive lymph nodes, and, for comparison, the points on the plot show a Poisson probability density function. Because the observed bars and the points are close, the Poisson model seems appropriate for this variable. Estimation of the Poisson Parameter α Table 2 shows the first of 2 general linear model analyses, the first performed on the same 476 cases illustrated in Figure 1. Table 2 demonstrates that 4 variables were related significantly to the number of positive lymph nodes. These were the clinical impression of a single abnormal lymph node, the time in months No. of Positive Lymph Nodes Figure 1 Frequency distribution of the observed number of positive lymph nodes (bars) in 476 patients with cutaneous melanoma and positive nodes. The points are the expected numbers from a Poisson distribution function. Table 2 General Linear Model Analysis of the Number of Positive Lymph Nodes * Variable Coefficient χ 2 P Clinical ~.00 Time Thickness Extremity * The analysis was done with the Poisson link function and with the number of examined nodes controlled for with the offset function in S-PLUS (for proprietary information, see text). Clinical indicates the clinical impression of a single positive node, so that just one was removed; time, the time in months between the biopsy of the primary tumor and removal of nodes; thickness, the Breslow measurement in millimeters; extremity, whether the primary tumor was located in an extremity site. Am J Clin Pathol 2005;123:

4 Vollmer / POISSON PARADIGM AND METASTATIC MELANOMA between the biopsy of the primary tumor and the lymph node excision, tumor thickness, and the presence of the primary tumor at an extremity site. The fact that all 4 variables had positive coefficients indicates that each was associated positively with an increase in the number of positive lymph nodes, and this was true even for an extremity primary site. The magnitude of the χ 2 statistics provides a measure of their relative importance and indicates that the clinical impression of a nodal metastasis was most important. Neither female sex nor presence of ulcer in the primary site provided further significant information (P >.05). Because the target of this study was to see how the Bayes rule and the Poisson model related to patients without clinically suspicious nodes, I performed a second general linear model analysis on the subset of those with 3 or more nodes excised and without obvious clinical suspicion for metastasis (305 cases). Once again, the results demonstrated that time, tumor thickness, and extremity site were related significantly to the number of positive nodes, with P values, respectively, of , , and The coefficients of this second analysis allowed the Poisson parameter α to be estimated as follows: Equation 8 α = exp( Time Thickness Extremity) Here, time is given in months, thickness in millimeters, and extremity as 1 if the site was extremity (otherwise 0). Among these 305 cases, the mean estimate of α was 0.15 (range, ). Because the model explained just 10% of the variance in the data, clearly other variables unobserved in these data were likely to be important, and it is likely that such variables would include basic tumor biologic factors that relate to metastasis. Nevertheless, the result, as limited as it was, allowed further use of the Poisson model to study the situation when all excised nodes were found free of tumor. For example, using the model for α in Equation 8, the plots in Figure 2 illustrate how the probability of finding no positive lymph nodes should depend on the number of examined nodes. On the vertical axis is the Poisson probability calculated according to Equation 2 using Equation 8 to estimate α, and the 3 lines are for thicknesses of 1, 3, and 5 mm. In these plots, time was held constant at 0.5 months, and the tumor location was taken to be nonextremity. The graph shows that the probability of observing negative nodes is relatively high when fewer than 5 nodes are examined, even when metastases are assumed present (ie, when α 0). As the number of examined nodes increases, however, the probability of false-negative nodes decreases. The plots also demonstrate that the effect of tumor thickness on the Poisson probability is small. Application of Bayes Rule The derived probability of greatest interest here is the one using Bayes rule or Equation 5, because this formula estimates the probability of metastases, when in fact none are observed. Thus, Bayes rule deals with the issue of falsely negative lymph nodes when α takes on an expected value other than zero. Using Equations 5 and 8, Figure 3 shows plots of Bayes probability vs the number of examined nodes for respective thicknesses of 1, 3, and 5 mm. Time was held constant at 0.5 months, ulceration was considered absent, and the location was taken as other than extremity. The graph demonstrates that for Poisson Probability of Negative Nodes Figure 2 Plot of Poisson probability of observing negative nodes (vertical axis) vs the number of nodes examined (horizontal axis). The calculation of the probability comes from Equations 2 and 8 and from the results of the second general linear model (see text). The 3 lines are for tumor thicknesses of 1 mm (top), 3 mm (middle), and 5 mm (bottom). Time was held constant at 0.5 months, and the location was taken as other than extremity. Bayes Probability 5 mm 3 mm 1 mm Figure 3 Plot of Bayes probability of metastasis (ie, α > 0) given that all observed nodes are negative (vertical axis) vs the number of nodes examined (horizontal axis). The calculation of the probability comes from Equations 5 and 8. The 3 lines are for tumor thicknesses of 1 mm (bottom), 3 mm (middle), and 5 mm (top). Time was held constant at 0.5 months, ulceration was considered absent, and the location was taken as other than extremity. 710 Am J Clin Pathol 2005;123: Downloaded 710 from

5 Anatomic Pathology / ORIGINAL ARTICLE smaller numbers of total nodes, there is an appreciable probability of undiscovered metastases, whereas when many nodes are found negative, Bayes probability lessens. For example, Equations 5 and 8 indicate that a patient with a nonulcerated primary tumor of mm and with just 5 nodes sampled and found negative has a probability of 2 of undetected metastases. Greater thicknesses increased the estimated probability, and the presence of ulcer in a primary tumor mm thick (not shown on graph) raised the estimated probability to Bayes Probability and Survival for T Stages 2a to 4b Next, I used equations 5, 7, and 8 to calculate the expected Bayes probabilities for stages T2a, T2b, T3a, T3b, T4a, and T4b as defined by Balch et al. 2 Because the published tables for these stages omitted exact values of thickness, tumor location, and numbers of lymph nodes examined, I derived appropriate values to use in Equation 5 from the subset of 306 study cases with negative lymph node dissections. For example, among the stage T2a cases, the median thickness was 1.4 mm, so I used this value in Equations 5 and 8 to calculate the Bayes probability for stage T2a. Median values for thickness used for the other stages were determined in the same way and are listed in Table 3. To account for the number of nodes examined, I used the expected value of the Poisson probability in Equation 7 evaluated over the distribution of number of nodes examined in the node-negative population. This distribution seemed to follow a gamma probability density (see Appendix 1 for details). Finally, I weighted the Bayes probabilities for each stage to reflect the distribution of extremity vs other sites found in the study subset of 306 node-negative study cases. Table 3 shows the 5-year survivals expected for these stages as published by Balch et al, 2 and the last column provides the calculated Bayes probabilities of nodal metastases for the same groups derived from Equations 5, 7, and 8. The results also are plotted in Figure 4, which demonstrates a nearly linear, inverse relationship between 5-year survival probability and the calculated Bayes probability. Linear regression analysis showed that this relationship was significant (P =.001) and that it explained 94.9% of the variance. Impact of Poisson Probability Density Function on N Stages in Metastatic Melanoma Currently, stage N1 comprises those with 1 positive node, N2 those with 2 or 3 positive nodes, and N3 those with more than 3 positive nodes. 2 Thus, by definition, stage N3 can be found only when more than 3 nodes are examined. Because in general the number of lymph nodes found positive should depend on the total number examined, the observed N stage should depend on the number of sampled lymph nodes. The Poisson probability density function formalizes this idea with the following 3 expressions for the probabilities of observing stages N1, N2, and N3, given a particular value of α. Equation 9 P(N1 n, α) = f(1 n, α) Equation 10 P(N2 n, α) = f(2 n, α) + f(3 n, α) 5-Year Survival Bayes Probability Figure 4 Plot of reported 5-year survival probability (vertical axis) for patients with stage T2a and stages IIA-IIC malignant melanomas vs the calculated Bayes probability of nodal metastasis (horizontal axis) when no nodes are observed positive, ie, P(α > 0 x = 0) with α adjusted to match the thickness and presence of ulceration for the stage categories (see text). The straight line comes from a linear regression fit. Table 3 Bayes Probabilities and Reported Survival in Patients With Node-Negative Melanoma * Stage Thickness Ulcer 5-Year Survival Bayes Probability T2a 1.4 No T2b 1.4 Yes T3a 2.7 No T3b 2.7 Yes T4a 5.0 No T4b 5.0 Yes * The thicknesses listed were those used to calculate the Bayes probability, and these were medians observed in patients with node-negative disease according to the given stages. The 5-year survival probabilities come from Balch et al. 1,2 The Bayes probabilities are the probability of nodal metastasis given that no positive nodes were found, and they were calculated according to Equation 5 using an expected value of the Poisson probability in Equation 2 evaluated over a gamma distribution of total number of nodes examined (see Equation 7 and Appendix 1). Am J Clin Pathol 2005;123:

6 Vollmer / POISSON PARADIGM AND METASTATIC MELANOMA Equation 11 P(N3 n, α) = 1 f(0 n, α) f(1 n, α) f(2 n, α) f(3 n, α) The values for f in each of these expressions come from Equation 1 with the appropriate substitutions for x. To illustrate the relationship between the probability of observing stages N1, N2, and N3 and n, the number of examined nodes, Figure 5, Figure 6, and Figure 7 show plots, respectively, of the Poisson models for P(N1 n, α), P(N2 n, α), and P(N3 n, α) vs the number of examined nodes. In these plots, α was kept constant at the mean of 0.15 reported earlier. Figure 5 demonstrates that the expected probability of observing N1 stage rises then falls, because larger numbers of nodes examined makes it more likely that more than 1 node will be found positive. The same phenomenon happens in Figure 6, except at a higher value of nodes examined. P(N1) Figure 5 Plot of Poisson probability of observing stage N1 (1 node positive) vs the number of nodes examined. The probability was calculated from Equation 9, and α was held constant at Application of the Poisson Density Function and Bayes Rule to Sentinel Nodes At first glance, the concept of sentinel nodes may seem to violate the premise of the Poisson probability density function. For example, if i th lymph node is a sentinel node and the (i + 1) th node is a nonsentinel node, then the results for these two cannot be statistically independent from one another, because many have shown that sentinel nodes have a higher probability of demonstrating metastases than the nonsentinel nodes. Nevertheless, it seems reasonable to consider the individual nodal results within each of the 2 groups sentinel and nonsentinel as statistically independent from one another. Thus, in what follows, I assume this modification of statistical independence between lymph nodes. Because I had no personal data sufficient for the Poisson analysis of sentinel nodes, I relied on that published by others. For example, the maximum likelihood estimate of the Poisson parameter α for a population of patients can be estimated as the ratio of total number of positive sentinel nodes divided by the total number of removed sentinel nodes. In 25 previous studies with a total of 1,076 patients with positive sentinel nodes, 11,16-39 there were sufficient details reported to estimate 1 value of α for each study. The weighted mean for α across these studies was 0.59 (range, ). Thus, within the Poisson paradigm, the value of α for sentinel nodes was considerably higher than that for nodes in elective node dissections, which, in my study patients, averaged just Rounding the 0.59 to and inserting this value for α into Equation 2 demonstrates what the Poisson model predicts for the probability of observing no positive sentinel nodes when, in fact, α is not zero, and the results are given in Table 4. If just 1 sentinel node is examined, the Poisson probability of finding it negative is 0.55, which is just slightly better than even odds. However, if 3 nodes are examined, P(N2) P(N3) Figure 6 Plot of Poisson probability of observing stage N2 (2 or 3 nodes positive) vs the number of nodes examined. The probability was calculated from Equation 10, and α was held constant at Figure 7 Plot of Poisson probability of observing stage N3 (> 3 nodes positive) vs the number of nodes examined. The probability was calculated from Equation 11, and α was held constant at Am J Clin Pathol 2005;123: Downloaded 712 from

7 Anatomic Pathology / ORIGINAL ARTICLE the Poisson probability of finding all 3 negative should be just 0.17, and if 4 sentinel nodes are found negative, then the probability of finding all negative is just 9. Bayes rule also can be rewritten for sentinel lymph nodes, but to do this first requires an estimate of the a priori probability that the sentinel nodes will be positive. Whereas for any given patient the a priori probability should depend on variables such as tumor thickness, many studies have found that approximately 20% of patients will have positive sentinel nodes. For example, the median percentage of patients with positive sentinel lymph nodes in 49 previous studies involving more than 17,000 patients was 19.5% (references available on request). Rounding this to 20% and inserting it into Bayes rule of Equation 5 yields an equation for the overall estimate of the probability of sentinel node metastasis when, in fact, no metastases have been observed as follows: Equation 12 1 P(Met x = 0, n) = (1 )/ 1 + exp( n) The last column of Table 4 gives the overall Bayes probability estimates for occasions when 1, 2, 3, or 4 sentinel nodes are removed. Because the median number of lymph nodes removed per patient found in 44 previous studies of sentinel nodes was 2.1 (references available on request) and because the Bayes estimated probability of undiagnosed metastasis when 2 sentinel nodes are sampled in Table 4 is 7, the Poisson paradigm and Bayes rule together suggest that routine sentinel node sampling should be sufficient to find metastases in all but approximately 7% of patients. How does this estimate compare with observed results? Depending on the length of follow-up time, the false-negative rate for sentinel lymph nodes has been observed to range from 4% to 12%, 7 and with a median follow-up time of 42 months, 1 study reported a false-negative rate of 7%. 7 Thus, the estimated Bayes probability and observed false-negative rate seem close. Discussion My purposes were to introduce the use of the Poisson probability density function and Bayes rule for analysis of lymph node metastases in malignant melanoma and to explore the insight these 2 models could provide. Although mathematical expressions like the Poisson probability density function and Bayes rule cannot be expected to perfectly model a complex biologic system like lymph node metastases in malignant melanoma, to the degree that they fit data like those in Figure 1 and to the degree that they predict observed outcomes like survival in node-negative patients and the falsenegative rate for sentinel lymph nodes, they provide insight. Not only do they formalize and concisely summarize reasonable Table 4 Application of Poisson and Bayes Probabilities to Sentinel Nodes * No. of Sentinel Nodes Poisson Probability Bayes Probability * The Poisson probability is the probability of observing no positive nodes, when in fact α is greater than 0. The value of α used was, which was a weighted mean of results from published studies of sentinel node results. The Bayes probabilities are the probability of nodal metastasis given that no positive nodes were found, and they were calculated according to Equation 12. and logical notions but they also suggest underlying principles that might not otherwise be obvious. The main notion that the Poisson model addresses for nodal metastases in melanoma is the idea that one finds more metastases when one examines more lymph nodes, and the results given herein suggest that this might be true for sentinel and nonsentinel nodes. Thus, the Poisson model suggests that one negative sentinel node should not be as persuasive for the absence of metastases as 3 or 4 negative sentinel nodes. If this result is validated by detailed application of the Poisson model to sentinel node data, then the prognostic information obtained from the sentinel node sampling will need to be adjusted for the total number of nodes sampled. The second notion related to the Poisson model is that its parameter, α, should reflect important clinical or biologic characteristics of the tumor such as the clinical impression of abnormality in a palpable lymph node, time lapse between the primary tumor and lymph node sampling, tumor thickness, and the technique for identifying and removing lymph nodes. I have found supportive evidence for all of these associations. It seems likely, then, that α should reflect other, as yet unstudied characteristics of the primary tumor or the way its nodal basins are examined. For example, it seems likely that α should be related directly to the number of step sections used for sentinel nodes. In this way, the Poisson model provides a tool for further study of metastases in malignant melanoma. Finally, the Poisson model also suggests that the current N staging system based on the number of positive lymph nodes could be flawed because it depends on the artifact of the number of examined nodes. Like the Poisson model, Bayes rule formalizes an idea that seems intuitive, namely that lymph nodes negative for tumor in a patient with melanoma can be falsely negative, especially when few nodes are excised. Furthermore, the Bayes rule provides a way to estimate the probability of a false-negative result. Applied to patients with stage N0 melanoma, Bayes rule suggests that observed survival might reflect in part undetected nodal metastases. Applied to sentinel nodes, Bayes rule suggests a small percentage of undetected metastases that seems close to the observed rate of false-negative results. Altogether, the results Am J Clin Pathol 2005;123:

8 Vollmer / POISSON PARADIGM AND METASTATIC MELANOMA of this preliminary study are sufficient to demonstrate important roles for the Poisson model and the Bayes rule in the understanding and analysis of lymph node metastases in malignant melanoma. Because the potential for using these 2 models probably is greatest in the node-negative patient population, future research should test whether Bayes probability in patients with negative nodes is predictive of tumor recurrence. If it is, then Bayes probability might provide useful supplemental information to that obtained from examination of sentinel lymph nodes. From Laboratory Medicine, Veterans Affairs Medical Center and Duke University Medical Center, Durham, NC. Address reprint requests to Dr Vollmer: Laboratory Medicine 113, VA Medical Center, 508 Fulton St, Durham, NC Appendix 1 The frequency distribution of the total number of lymph nodes examined, n, in patients with node-negative disease is given Figure 8. This frequency distribution seemed to follow a gamma probability density function, so that for f(n) in Equation 7, I used the following gamma probability density function. Equation A1 f(n) = λ 2 n exp( λ n) The value of λ used was 2/mean, which for the data was approximately 0.125, and this choice of f(n) is the line on the plot in Figure 8. Thus, the expected value for the Poisson probability of finding no positive nodes in Equation 7 becomes: Equation A2 E{f(0 n, α)} = λ 2 n exp( (λ + α) n) dn Performing the integration by parts in Equation A2 yields the final value for the expected value, now adjusted for all possible n, as: Equation A3 E{f(0 n, α)} = λ 2 /(λ + α) Total No. of Nodes Figure 8 Frequency distribution of the total number of lymph nodes examined in the subset of patients without nodal metastases. The smooth line is a gamma distribution function. References 1. Balch CM, Soong S-J, Gershenwald JE, et al. Prognostic factor analysis of 17,600 melanoma patients: validation of the American Joint Committee on Cancer Melanoma Stage System. J Clin Oncol. 2001;19: Balch CM, Buzaid AC, Soong S-J, et al. Final version of the American Joint Committee on Cancer Staging System for cutaneous melanoma. J Clin Oncol. 2001;19: McMasters KM, Reintgen DS, Ross MI, et al. Sentinel lymph node biopsy for melanoma: controversy despite widespread agreement. J Clin Oncol. 2001;19: Pawlik TM, Ross MI, Gershenwald JE. Lymphatic mapping in the molecular era. Ann Surg Oncol. 2004;11: Medalie NS, Ackerman AB. Sentinel lymph node biopsy has no benefit for patients with primary cutaneous melanoma metastatic to a lymph node: an assertion based on comprehensive, critical analysis, part I. Am J Dermatopathol. 2003;25: Jansen L, Nieweg OE, Petersen JL, et al. Reliability of sentinel lymph node biopsy for staging melanoma. Br J Surg. 2000;87: Muller MGS, Borgestein PJ, Pijpers R, et al. Reliability of sentinel node procedure in melanoma patients: analysis of failures after long-term follow-up. Ann Surg Oncol. 2000;7: Clary BM, Brady MS, Lewis JJ, et al. Sentinel lymph node biopsy in the management of patients with primary cutaneous melanoma: review of a large single-institutional experience with emphasis on recurrence. Ann Surg. 2001;233: Chao C, Wong SL, Ross MI, et al. Patterns of early recurrence after sentinel lymph node biopsy for melanoma. Am J Surg. 2002;184: Wagner JD, Ranieri J, Evdokimow DZ, et al. Patterns of initial recurrence and prognosis after sentinel lymph node biopsy and selective lymphadenectomy for melanoma. Plast Reconstr Surg. 2003;112: Estourgie SH, Nieweg OE, Olmos RAV, et al. Review and evaluation of sentinel node procedures in 250 melanoma patients with a median follow-up of 6 years. Ann Surg Oncol. 2003;10: Cady B. Lymph node metastases: indicators, but not governors of survival. Arch Surg. 1984;119: Nathanson D. Insights into the mechanisms of lymph node metastasis. Cancer. 2003;98: Freund JE. John E. Freund s Mathematical Statistics: With Applications. 7th ed. Englewood Cliffs, NJ: Prentice-Hall; McCullagh P, Nelder JA. Generalized Linear Models. 2nd ed. London, England: Chapman and Hall; Vollmer RT, Seigler HF. A model for the pretest probability of lymph node metastasis from cutaneous melanoma. Am J Clin Pathol. 2000;114: Morton DL, Wen D-R, Wong JH, et al. Technical details of intraoperative lymphatic mapping for early stage melanoma. Arch Surg. 1992;127: Reintgen D, Cruse CW, Wells K, et al. The orderly progression of melanoma nodal metastases. Ann Surg. 1994;220: van der Veen H, Hoekstra OS, Paul MA, et al. Gamma probe guided sentinel node biopsy to select patients with melanoma for lymphadenectomy. Br J Surg. 1994;81: Thompson JF, McCarthy WH, Bosch CMJ, et al. Sentinel lymph node status as an indicator of the presence of metastatic melanoma in regional lymph nodes. Melanoma Res. 1995;5: Am J Clin Pathol 2005;123: Downloaded 714 from

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