Abstract. Although the Breslow thickness provides the most important histologic information for prognosis in cutaneous melanoma, controversies and uncertainty.
Anatomic Pathology / TRANSFORMATION OF THE BRESLOW THICKNESS
Using a Continuous Transformation of the Breslow Thickness for Prognosis in Cutaneous Melanoma Robin T. Vollmer, MD, MS,1 and Hilliard F. Seigler, MD2 Key Words: Thickness; Mitotic rate; Age; Ulcer; Location; Cox model; Pathology
Abstract Although the Breslow thickness provides the most important histologic information for prognosis in cutaneous melanoma, controversies and uncertainty remain about how best to use thickness. It is unclear whether cut points should be used, or, if they are used, which are optimal. We studied new data collected from more than 1,000 patients followed up for a relatively long period. From Cox proportional hazards models of survival we learned that more cut points provide more prognostic information than using, for example, just 1 cut point at 1.7 mm. Nevertheless, a continuous transformation provides an effective alternative that captures the information that thickness provides, and it avoids the pitfalls of using multiple cut points. In a multivariate model, this transformation provided strong prognostic information, and the result produced a prognostic score for cutaneous melanoma. This score provides a practical way that Cox model results can be used, and we believe it consolidates the prognostic information provided by traditional histologic and clinical variables. When newer prognostic variables are introduced, we suggest that they be used with this continuous transformation of thickness rather than with cut points in thickness.
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Thirty years have passed since Alexander Breslow1 introduced his measurement of thickness in cutaneous melanoma. In that first article he wrote: “Tumor thickness was the most useful measurement” of several he had tried on 98 patients’ tumors, and now, 30 years later, the same can be said, because of all the histologic features we observe in cutaneous melanoma, thickness remains the most important.2-5 During these 30 years, controversies have arisen about the thickness; some have been resolved, while others persist. For example, initially it was uncertain which was more prognostically important, the Clark levels6 or the Breslow thickness. Now, multiple studies have demonstrated that thickness is more important and that levels in multivariable statistical models are most often not significant once thickness is accounted for.1,5,7-10 Furthermore, virtually all studies of prognosis in melanoma since 1989 have found thickness to be important.2-5,11-58 Nevertheless, the way thickness has been used has varied among investigators and over time. Most have used cut points in thickness, and these have ranged from as few as 1 at 1.7 mm11,12 to as many as 7.13 ❚Figure 1❚ shows a frequency distribution of the location of cut points used in thickness reported in 46 previous studies, and this histogram demonstrates that many have had difficulty deciding just how to partition thickness. In fact, the continuum in cut points in Figure 1 seems to reflect the continuum that thickness naturally expresses. The mean value of the cut points for the 46 studies in Figure 1 was 2.54 mm (range, 0.55-9.75 mm). The most commonly used were at 1.5 mm and 3.0 mm. Thus, one of the remaining questions about thickness has been whether natural cut points exist, and if they do which are ideal.13,14,59 In the present study, we used previously unreported data from more than 1,000 patients with cutaneous melanoma to Am J Clin Pathol 2001;115:205-212
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study the issue of cut points vs continuity of thickness as a prognostic factor for survival in melanoma, and in doing so, we found that a continuous transformation of thickness introduced by Stadelmann et al5 seems to provide a better prognostic tool than the cut points commonly in use.
Methods The patients included were 1,910 who were referred for consultation to the Duke University Melanoma Clinic, Durham, NC, during the late 1980s and early 1990s when the pathology of their cutaneous primary tumors was reviewed by 1 dermatopathologist (R.T.V.). All diagnostic biopsies had been performed before the patients arrived at Duke and were undertaken in a variety of clinical practices that make up the Duke Melanoma Clinic referral base. The median follow-up was 7.6 years with a 25th to 75th percentile range from 3.7 to 10.2 years. One pathologist (R.T.V.) measured the Breslow thickness on all these cases with a calibrated eyepiece micrometer and using a programmable Hewlett-Packard calculator (see the appendix). The frequency distribution of thickness for these tumors is shown in ❚Figure 2❚. In addition, the Clark levels and the presence of surface ulcer also were recorded. Because mitotic rate was recorded on only a subset of the tumors and because it has been omitted from recently published prognostic models, 16 it was not used in the following analyses. Furthermore, because the classification of melanomas as either “radial” or “vertical” became popular after many of these cases had been examined,11,12 this designation of phase was not in the database. Nevertheless, because 75% of our tumors had a Clark level of greater than 206
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❚Figure 2❚ Frequency distribution of Breslow thickness in 1,910 primary cutaneous melanomas.
2, most of them were in the vertical phase. Clinical variables used included patient age, sex, and the body site of the primary tumor. Following previous results,16 we divided body site into 2 categories: extremities (excluding acral) and all other sites (including acral). Statistical Methods We used the Cox proportional hazards model60,61 to relate survival time to thickness. If we symbolize prognostic variables used as x1, x2, x3, … xk, then the Cox model provided estimates of the coefficients b1, b2, b3, … bk for a relative hazard score, hs, given as: ❚Equation 1❚
hs = b1 × x1 + b2 × x2 + b3 × x3 + … + bk × xk The way thickness appears in Equation 1 depends on whether cut points are used and, if used, on how many cut points are used. For example, a single cut point for thickness at 1.7 mm results in a single x1 variable for thickness that takes the value of 0 for thickness less than 1.7 mm and the value of 1 for thickness greater than or equal to 1.7 mm. In this way, the number of b coefficients and x variables for thickness equals the number of cut points, and the number of statistical degrees of freedom for thickness also equals the number of cut points. If thickness is used as a continuous variable, then just a single coefficient, x variable, and degree of freedom are required. In addition, we used the logistic regression model to relate binary outcomes to explanatory variables62,63 and the general linear model64 to relate continuous variables to covariates. All analyses were done with the S-PLUS software package (MathSoft, Seattle, WA), and reported P values were for 2-sided tests of the hypothesis. © American Society of Clinical Pathologists
Anatomic Pathology / ORIGINAL ARTICLE
Results Cut Points in the Breslow Thickness Because Buttner et al14 found that the more cut points than 1 used, the better the prognostic model, we examined this issue with our data, and the results are shown in ❚Table 1❚. Table 1 gives the results of 5 Cox model analyses for 1,910 of our patients with thickness and follow-up data. These included 633 who were followed until death (ie, uncensored). Among the 5, there is 1 each for a published set of cut points. The important results are under the headings “No. of Cut Points” and “Likelihood Ratio.” The number of cut points equals the numbers of coefficients in Equation 1, is 1 less than the number of intervals of thickness, and provides the number of degrees of freedom in the analysis. The likelihood ratio (LR) is the overall model likelihood ratio statistic. The larger the value of LR on a given set of data, the more information the model provides about survival and the better the model fits the data, and in general, increasing the degrees of freedom often increases the LR. The results demonstrate that the binary division of thickness into 2 categories—less than 1.7 mm and 1.7 mm or more— provided the lowest LR, that is, the least amount of prognostic information. The results also showed that as the number of cut points increased, so did the LR, with the maximum LR coming from the Keefe and Mackie13 model using 8 cut points. ❚Figure 3❚ demonstrates how cut points allow thickness to relate to prognosis. Figure 3 includes the results of the first and last models of Table 1, ie, those using 1 (Clark model) or 8 (Keefe and Mackie model) cut points. The stepped lines for each provide the coefficient values, ie, the contribution of thickness to the hazard score of Equation 1. Whereas the single cut point for thickness at 1.7 mm used in the Clark model produced 2 levels—1 at zero and 1 at 0.994—the 8 cut points in the Keefe and Mackie model allowed a more detailed relationship between thickness and hazard score to emerge. The Keefe and Mackie model demonstrated that there was a nonlinear, gradually changing relationship
❚Table 1❚ Cox Model Analysis of Survival Time in 1,910 Patients* Using Various Cut Points in the Breslow Thickness Reference
No. of Cut Points† Likelihood Ratio
Clark et al11; Elder and Murphy12 Sondergaard and Schou34 Schuchter et al3 Cascinelli et al36 Keefe and Mackie13 * †
1 2 3 5 8
147 149 177 186 199
The number of uncensored patients was 633. The number of cut points also equals the number of thickness variables, coefficients, and degrees of freedom in the Cox models.
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between thickness and the hazard score. For example, small increases in thickness at 2 mm produce greater changes in the hazard score than do the same changes at 7 or 8 mm. Because this richness of information was hidden in the model using 1 cut point, the LR for the single cut point model was smaller. On the other hand, the model using 8 cut points produced an unnaturally appearing stair-step effect with abrupt rises and falls along the continuum of thickness, and this suggested an overfitting of the model to the data. For example, using 8 cut points allows the model to reflect small numbers of patients in any one of the 9 thickness intervals. For this reason and to minimize the number of degrees of freedom used for thickness, we sought a continuous transformation of thickness that would nevertheless be capable of providing the same information as the model using 8 cut points. The shape of the stair-step result for the model using 8 cut points in Figure 3 reminded us of the transformation used by Stadelmann et al5 to predict 10-year mortality in cutaneous melanoma. Their transformation was based on results for a large number of patients, and it was validated with independent data. ❚Figure 4❚ shows this curve (adjusted for vertical scale) superimposed over the results for the model that uses 8 cut points. Specifically, the equation for the curve is: ❚Equation 2❚
hs = 1.9 × [1 – 0.966 × exp(–0.2016 × Thickness)] where the portion inside the brackets provided the estimated probability of 10-year mortality for Stadelmann et al,5 and where 1.9 is the scale factor. Clearly, the smooth curve seems to provide for the same kind of nonlinear relationship between thickness and hazard score, and it also avoids the abrupt points of discontinuities. Henceforth, we will symbolize this bracketed transformation of thickness as T(thick). A Cox model of survival for the data for patients used in the Table 1 analyses but with T(thick) instead of a cut point model produced an LR of 187—a result accomplished with just 1 coefficient and 1 degree of freedom. Because of this success, we next explored the use of T(thick) in a multivariate model. Multivariate Cox Model ❚Table 2❚ shows how T(thick) fits into a multivariate Cox model using presence of histologic ulcer, patient age, sex, and body site (nonacral extremity vs other). Because of missing data, this analysis was done on a subset of 1,851 of 1,910 patients, including 608 uncensored patients. The results demonstrate that T(thick) was the variable most closely associated with hazard of death, and T(thick) used just 1 degree of freedom. Furthermore, T(thick) provided a better model than the commonly used cut point at 1.7 mm, which also uses 1 degree of freedom. Whereas the LR for the Am J Clin Pathol 2001;115:205-212
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❚Figure 3❚ Plot of the contribution of thickness to Equation 1 (vertical axis) vs thickness (horizontal axis) for the first and last models in Table 1. The Clark model used a single cut point at 1.7 mm and produced a single step in the plot. The Keefe and Mackie model used 8 cut points and produced the more extensive stair-step pattern throughout the full range of thickness. The contribution to Equation 1 comes from the values of the coefficients estimated by a Cox model analysis of survival time for 1,910 patients.
❚Figure 4❚ Plot of the contribution of thickness to Equation 1 vs thickness for the model in Table 1 using the 8 cut points of Keefe and Mackie.13 The contribution comes from the values of the coefficients estimated by a Cox model analysis of survival time for 1,910 patients, and these form the contribution to Equation 1. The smooth curved line is an exponential transformation of thickness described by Stadelmann et al5 (see Equation 2).
model with T(thick) was 262, the LR for the model with a cut point at 1.7 mm was just 237. The results also demonstrated that patient age, presence of ulceration, and body site were additional important variables, but sex was of borderline significance. In this analysis, we coded age as 0 if younger than 60 years and as age-60 for older than 60 years, because this use of age provided a better model than with a simple cut point at age 60 (LR of 262 vs 247 using 1 age cut point). Sex was less important in this analysis, probably because in our data, sex was associated significantly with thickness, presence of ulcer, age at diagnosis, and body site. Women had significantly lower thickness (P = 1.8 × 10–12 by general
linear model), a significantly lower incidence of ulceration (P = .0015 by logistic regression model), significantly younger age (P = .01 by general linear model), and significantly more favorable tumor locations (P = .00 by chi-square test). Although we did not categorize the tumors into radial and vertical phases (see the “Methods” section), we explored an approximation of the radial group by defining as radial the tumors of level 2 or with a thickness less than 0.4 mm, but we found that this approximation was not related to survival once thickness was in the model (P > .2). The coefficients and variables of Table 2 allowed us to write an equation for hs as: ❚Equation 3❚
hs = 2.28 × T(thick) + 0.472 × Ulcer + 0.299 × Site + 0.0622 × Age – 1.64 × Sex
❚Table 2❚ Multivariable Cox Proportional Hazards Model* Variable
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0.220 0.0927 0.0990 0.0105 0.0887
.00000 3.5 × 10–7 2.5 × 10–3 3.1 × 10–9 .065
Ulcer (0 or 1) Site§ Age|| Female sex *
The number of cases with sufficient data to be used in this analysis was 1,851, and the number uncensored was 608. SE of the estimate of the coefficient. ‡ T(thick) is a transformation of thickness (see Equation 2). § Site was 0 if nonacral extremity; otherwise, it was 1. || Age was coded as 0 for those younger than 60 years and as age-60 for those older than 60 years. †
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where T(thick) is defined by the bracketed part of Equation 2, ulcer is 1 for ulcerated (otherwise 0), site is 1 for axial or acral (otherwise 0), age is coded as 0 for younger than 60 years and as age-60 for older than 60 years, and sex is 1 if female. To illustrate the potential importance of the hazard score, ❚Figure 5❚ shows a plot of observed probability of 5year mortality due to melanoma in our data vs the hazard score as calculated with Equation 3. As the hazard score increased, so did the observed frequency of 5-year mortality and in a continuous fashion. Nevertheless, this model of © American Society of Clinical Pathologists
Anatomic Pathology / ORIGINAL ARTICLE
Observed Probability of 5-Year Mortality
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❚Figure 5❚ Plot of the observed probability of dying of melanoma in 5 years vs the hazard score given by Equation 3. The probability was estimated by the locally weighted scatterplot smoother (LOWESS) function of S-PLUS software package (MathSoft, Seattle, WA), which provides a moving average of the 0s and 1s along the horizontal axis. There were 1,419 patients used for this plot, that is, the plot used just those who were followed up for at least 5 years or who had died before 5 years of follow-up.
hazard score, which was based on only 5 variables, explained only 12% of the overall deviance in this plot, and for each value of the hazard score, there were some who survived 5 years and some who did not. For our data, the following equation provided a way to estimate this observed probability of 5-year mortality: ❚Equation 4❚
Estimated 5-Year Mortality =
1 1 + exp(2.59 – 1.42 × hs)
Discussion When Stadelmann et al5 introduced their transformation for the Breslow thickness, its purpose was to predict 10-year mortality. They had validated it with additional data, and our results validate it further and extend its use to outcomes other than 10-year mortality and in a multivariable setting. We have demonstrated that T(thick) clearly provides an advantage over using a single cut point in thickness, probably because it reflects the complete nonlinear relationship between thickness and outcome. T(thick) also avoids overfitting of the data that can happen with many cut points, and it does not produce unnatural discontinuities. Consequently, with T(thick), small changes such as from 1.6 to 1.7 mm © American Society of Clinical Pathologists
cannot translate into large steps in the hazard score for an adverse outcome. Finally, because we derived the definition of T(thick) from previously published studies, it was not subject to the “optimal cut point” method or other internally data-dependent definitions.59 Thus, we favor using this continuous transformation of thickness when forming prognostic models in cutaneous melanoma, and we believe that when one tests for the importance of new variables, T(thick) should be included as a control for thickness. The reasons that many have favored cut points for thickness rather than a continuous variable like T(thick) have seldom been made clear, but we guess that the preference for cut points may relate to a perceived need to match categories of thickness to treatments or other interventions, which by their nature are categorical. Nevertheless, to achieve this end, one need not categorize thickness. Alternatively, one could categorize the composite hazard score into groups such as 0 to 1, 1 to 2, 2 to 3, and more than 3 and, thus, take advantage of additional variables of prognostic importance. Even better might be a categorization of the final estimated probability for an adverse outcome, such as 5- or 10-year mortality. In fact, patients could assist in the process because a priori they are more likely to understand categories of probability than categories of either Breslow thickness or hazard score. Many patients also will understand that a probability describes a naturally continuous process without magical cut points and that it reflects an uncertainty about outcome. Thus, we suggest using the probability of outcome as is or in convenient categories such as low, medium, and high. This approach makes unnecessary the use of cut points in thickness, and it postpones the categories until the multivariable algorithmic process is completed. A second reason some may have preferred using cut points for thickness may have been because of the nonlinear relationship between thickness and outcomes, but we have now demonstrated that the transformation of thickness developed by Stadelmann et al5 reflects this nonlinearity well and without resorting to cut points. Our most important results are about thickness, and we view the hazard score in Equation 3 as preliminary. Nevertheless, because we believe that using T(thick) can improve current prognostic models, we hope that others with data on melanoma will pursue this approach and, thus, form better hazard scores. Whereas tumor thickness, ulcer, tumor location, patient age, and sex are clearly important in melanoma, our results suggest that they explain only 12% of the observed variation in outcome, leaving substantial uncertainty for a disease that often follows a long course. The challenge we face is to discover new variables that are not simply correlates of thickness but that provide truly independent prognostic information. Our results suggest that unless one uses thickness optimally, one runs a risk of identifying Am J Clin Pathol 2001;115:205-212
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new variables that are not really so new. For the moment, the status of sentinel nodes seems important, and a useful serum marker for melanoma could help substantially. In this regard, preliminary results suggest that serum levels of melanomainhibitory activity may be useful,65 and reverse transcription–polymerase chain reaction technology for circulating tyrosinase messenger RNA or melanoma-inhibitory activity messenger RNA also could help.66,67 Thus, we encourage those studying these variables to use T(thick) in their analyses.
Appendix: Using the Hewlett-Packard Programmable Calculator 32SII for the Breslow Thickness A programmable calculator is very helpful for measuring the Breslow thickness, especially when examining many melanomas. For example, the Hewlett-Packard (Corvallis, OR) model 32SII can do this with ease. It has function keys labeled as A, B, C, and so on, which can be used to store the calibrating equation together with the appropriate calibration ratios. One proceeds as follows: First, using an eyepiece micrometer and glass slide with a scale in millimeters, determine the ratio of millimeter length/micrometer length for each objective one plans to use. (I [R.T.V.] have done this for 2×, 4×, and 10× objectives. Next, enter the calibration ratio for the 2× objective into the calculator simply by punching in the number. Then store this number in register “u” by pushing in sequence the keys “sto” and then “u.” Repeat this sequence for the calibration ratios for the 4× objective (stored in register “v”) and for the 10× objective (stored in register “w”). Once these numbers are stored in the memory, they need not be reentered unless they are replaced by others or unless the entire memory is cleared accidentally. Next, enter the program mode of the calculator with the “PRGM” key and enter the following 3 programs with the sequence of commands given in the following table: Microscope Objective Command 1 2 3 4 5 6
× 2×
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LBL A RCL u × RTN
LBL B RCL v × RTN
LBL C RCL w × RTN
Each function begins with the command “LBL” and ends with the command “RTN,” so that entering these 3 programs requires a total of 18 key strokes. The “×” in the fifth command stands for the product, or multiplying function. After entering the program, exit the program mode with the 210
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PRGM key. You are now ready to use the program for determining the Breslow thickness, and you should not have to reprogram the calculator. To use the program, first measure the tumor thickness on the scale of the micrometer, then enter this number into the calculator. Note the objective that was used. If you made the measurement with the 2× objective, then use the “XEQ” key followed by the “A” key, and the thickness will be displayed in millimeters. For the 4× and 10× objectives, use the “XEQ” key followed by the “B” or the “C” key, respectively. From 1Laboratory Medicine, VA Medical Center, and the Departments of 1Pathology and 2Surgery, Duke University Medical Center, Durham, NC. Address reprint requests to Dr Vollmer: Laboratory Medicine (113), VA Medical Center, Durham, NC 27705.
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