Anatomic Pathology / NOMOGRAMS IN PROSTATE CANCER Information Content of Five Nomograms for Outcomes in Prostate Cancer Tarek A. Bismar, MD, 1 Peter Humphrey, MD, 2 and Robin T. Vollmer, MD 3 Key Words: Nomogram; Prostate cancer; Prostate-specific antigen; PSA; Gleason grade; Survival; Information; Entropy; Needle cores; Stage DOI: 1.139/XA419Q75F5D2TVJN Abstract In this study, we used 327 cases of localized prostate cancer to determine the information content provided by 5 popular nomograms for predicting outcomes in localized prostate cancer. All study patients underwent radical prostatectomy. For each case and each nomogram, we calculated the estimated probability of outcome, and, from this probability, we calculated the information content as 1 S, where S is the entropy. With this definition, information content is minimized at and maximized at 1. We found that the average information content ranged from.16 for the Partin tables to.44 for the recent Kattan nomogram for 1-year disease-free survival. Furthermore, the Kattan 1-year nomogram provided information content greater than.5 for 5% of study cases, so that among these 5 nomograms, we judged its performance the best. Nevertheless, because even this nomogram provided less than.5 information content for 5% of our cases, we believe that it can be improved and that additional measurements or markers observed on the biopsy tissues are likely to produce better nomograms. Many articles dealing with prostate cancer introduce the subject by quoting statistics on incidence and fatality. Herein, we do the same. The expected number of men to be diagnosed with prostate cancer in the United States in 27 is 218,89, and the expected number to die of this disease is 27,5. 1 To us, however, the most striking thing about these 2 numbers is not their magnitude, but their discrepancy. The difference between these 2 numbers underscores one of the many uncertainties regarding outcomes for prostate cancer. Many men get this disease, but most do not die of it. For most men with prostate cancer, their tumor is of localized stage, and most men treated by surgery or radiation therapy are cured. Yet, there remains uncertainty about whether such treatment is always necessary, and there is uncertainty about how to recognize men for whom neither surgery nor radiation treatment will be curative. To deal with such uncertainties and to provide useful information to men newly diagnosed with prostate cancer, some have derived algorithms or nomograms to predict the probability of important outcomes. 2 Most of these nomograms use pretreatment variables such as serum prostatespecific antigen (PSA) level and Gleason score, and many have validated their predictions with independent data. However, to our knowledge, no one has examined just how much uncertainty the nomograms resolve. Because uncertainty of outcome is directly related to the information theoretical concept of entropy, 3 we use here a measure of information derived from entropy. 4 Specifically, we evaluate the information provided by 5 popular nomograms for men with localized prostate cancer, and herein we report the results. Am J Clin Pathol 27;128:83-87 83 83 DOI: 1.139/XA419Q75F5D2TVJN 83
Bismar et al / NOMOGRAMS IN PROSTATE CANCER Materials and Methods Study Cases The study cases comprised 327, all of whom underwent radical prostatectomy. Of 327 men, 1 were diagnosed and treated at Washington University Medical Center, St Louis, MO; 132 at the Durham Veteran Affairs Medical Center, Durham, NC; and 95 at McGill University-Jewish General Hospital, Montreal, Canada. Other details about the patients and their tumors are given in Table 1. In this study, we used these 327 cases as prototypic cases to examine the amount of information to be expected from 5 nomograms that use preoperative variables to predict postoperative outcomes. Nomograms The nomograms we chose to examine comprised 5, which have proven popular to urologists and their patients and which provided sufficient details so that they could be executed on new data. For example, to apply the Partin nomogram we used the probability tables recently published. 5 These tables were based on the study of 5,79 men with localized prostate cancer, 5 and they rely on clinical stage, Gleason score, and the preoperative value of serum PSA to estimate the probability of organ-confined tumor status. Because the organ-confined state is significantly related to the long-term curative success of radical prostatectomy and because few patients nowadays undergo lymph node sampling, we restricted our analysis to the estimate of the probability of non organ confined tumor and did this for each study case. The second nomogram came from Shipley et al, 6 who studied 1,765 men treated with radiation therapy for localized prostate cancer. Shipley et al 6 derived a nomogram for predicting no evidence of tumor recurrence at 5 years based on follow-up values of PSA. Their algorithm came from a recursive partitioning model, and it used just PSA and Gleason score to Table 1 Characteristics of Study Population * Characteristic Result Age, y 63.6 (42-76) No. of cases by clinical stage T1c 192 T2 135 Serum PSA level, ng/ml 7.6 (.1-147.9) Total No. of biopsy cores 6.6 (1-12) Biopsy Gleason score 6.3 (4-1) No. with pt3 stage 15 * Data are given as mean (range) unless otherwise indicated. All values were measured during the preoperative period except for pt3 stage, which was determined from the prostatectomy specimen. Values are given in conventional units; to convert to Système International units (µg/l), multiply by 1.. form 4 prognostic groups with expected disease-free probabilities respectively of.81,.69,.47, and.29. As before, we applied this algorithm to our study cases. The third and fourth nomograms came from D Amico et al, 7 who provided tables of probabilities for tumor recurrence (manifested by a rising PSA level) within 2 years after surgery or external beam radiation treatment for men with localized prostate cancer. 7 Like the Partin algorithm, the D Amico algorithms use clinical stage, Gleason score, and the preoperative value of serum PSA, and their algorithms were derived from the study of 892 men who underwent surgery and 762 men who were treated with radiotherapy. From the tables, we then estimated the probability of tumor recurrence within 2 years for each study case. The fifth nomogram was that of Stephenson et al 8 for the probability of tumor recurrence at several times during followup after radical prostatectomy, and their results were derived from the study of 1,978 men who underwent surgery. 8-1 Because Kattan first introduced this approach 11 and was senior author for its most recent results, we will subsequently refer to this algorithm as the Kattan nomogram. This nomogram also relies on clinical stage, serum PSA level, and Gleason grade; and in addition, it uses the number of cores with tumor and the total number of cores taken. Each of these variables contributes to points, which then are totaled to achieve a prognostic point score. Finally, this model uses the Cox proportional hazards model to relate probability of a tumor-free state at various times to the total number of points. In our application, we concentrated on the probability of being cancer-free at 5 years. Because clinical stage T2 in our study cases was not subdivided into T2a, T2b, and T2c, we used a weighted mean to derive the appropriate probability or points to apply to cases with stage T2. Briefly, for each substage of T2, we formed a fraction equivalent to the relative frequency of that substage in the data of Stephenson et al. 8 We then multiplied this fraction by the probabilities listed for T2 stages in the Partin and D Amico tables or the points from the Kattan algorithm, and, finally, we summed these products to obtain a weighted mean for study cases with clinical stage T2. Calculation of Information Content We used a recently described method to calculate information content from the entropy, S, for a binary outcome. 4 Briefly, if p symbolizes the probability of a binary outcome like the presence of organ-confined prostate cancer, then the entropy, S, is defined as follows 4 : Equation 1 S = p * log 2 (p) (1 p) * log 2 (1 p) Here, log 2 is the logarithm to the base 2, and we use it here because for binary outcomes, this base of 2 guarantees that S will always be between and 1. Furthermore, for any 84 Am J Clin Pathol 27;128:83-87 84 DOI: 1.139/XA419Q75F5D2TVJN
Anatomic Pathology / ORIGINAL ARTICLE number x, the log 2 can be easily calculated from the natural logarithm to the base e by dividing the natural logarithm by.693. As before, we define the information content as 1 S. With this definition, the information content is highest (ie, a value of 1) when the outcome is certain, and it is lowest (ie, a value of ) when the outcome is no more certain than a coin toss. For example, when a nomogram estimates that the probability of a binary outcome is or 1, it is providing maximum information, and in both of these circumstances, 1 S equals 1. By contrast, when a nomogram estimates that the probability of a binary outcome is.5, it is providing no more information than a coin toss, and the value of 1 S equals. Thus, with this definition, the information content of 1 S tells us just how far toward certainty of outcome the nomogram will take us. 12 1 8 6 4 2 Figure 1 Information content provided by the Partin 5 nomogram for postsurgical outcome of organ-confined tumor on the study patients. Results In our study population, the Partin nomogram estimated that the probability of organ-confined tumor averaged.58 (range,.9-.92), and this implies that the information content averaged.16 (range, -.6). In other words, for the average study case, the Partin algorithm provided limited information about organ-confined status. Figure 1 shows a histogram of the information content for our entire study population, demonstrating in more detail the limited amount of information provided by the Partin algorithm for these typical prostatectomy cases. It also demonstrates that the most common information content was very nearly, ie, the information equivalent to a coin toss. The Shipley algorithm for PSA failure 5 years after radiotherapy yielded estimated probabilities averaging.77 in our study cases (range,.29-.81), and these results implied an average information content of.25 (range, -.3). A histogram of the information content provided by this nomogram is shown in Figure 2. The D Amico algorithm for PSA failure 2 years after surgery yielded estimated probabilities averaging.2 in our study cases (range,.4-.72), and these results implied an average information content of.32 (range, -.76). Thus, the D Amico algorithm provided more information for postsurgical outcome than did the Partin tables. A histogram of the information content provided by this nomogram is shown in Figure 3. The D Amico algorithm for PSA failure 2 years after external beam radiation therapy yielded estimated probabilities averaging.18 in our study cases (range,.3-.58), and these results implied an average information content of.37 (range, -.81), which was similar to the information provided by the D Amico postsurgical algorithm. A histogram of the information content provided by this nomogram is shown in Figure 4. 25 2 15 1 5 Figure 2 Information content provided by the Shipley 6 nomogram for postsurgical outcome of 5-year disease-free 8 6 4 2 Figure 3 Information content provided by the D Amico 7 nomogram for postsurgical outcome of 2-year disease-free Am J Clin Pathol 27;128:83-87 85 85 DOI: 1.139/XA419Q75F5D2TVJN 85
Bismar et al / NOMOGRAMS IN PROSTATE CANCER 12 8 1 8 6 4 2 6 4 2 Figure 4 Information content provided by the D Amico 7 nomogram for postradiation therapy outcome of 2-year disease-free Figure 5 Information content provided by the Kattan 11 nomogram for postsurgical outcome of 5-year disease-free In our study population, the Kattan nomogram estimated the probability for cancer-free recurrence at 5 years to average.83 (range,.23-.97), and this implies that the information content averaged.44 (range, -.8), a clear improvement over the preceding nomograms. Furthermore, for 5% of our study cases, this nomogram provided information content greater than.5, that is, greater than midway toward certainty. By contrast, for 29% of our study population, this nomogram provided information content less than.3. In other words, for these study cases, the information provided was just a fraction more than that provided by a coin toss. A histogram of the information content provided by this nomogram is shown in Figure 5. Discussion The results of this study, which are summarized in Table 2, suggest several conclusions. First, for an average patient with localized prostate cancer, none of these nomograms predicts a nearly certain outcome. Figures 1 through 5 demonstrate that for most of the study cases, the nomograms yielded information somewhere between that provided by a coin toss and midway to certainty. When compared with outcome information available without use of the prognostic variables in Table 2, these nomograms fare even worse. For example, the pooled 1-year disease-free survival obtained from several large studies of men after radical prostatectomy is 81%, 8,12-15 implying an information content of.3 for an average patient with localized tumor. Consequently, just 2 of the algorithms in Table 2 provide more information for an average patient than what is available without using the nomogram. From philosophical and religious points of view, this result should be no surprise, and perhaps even those favoring the collection of thousands of bits of molecular information will admit that predicting outcomes with near certainty is beyond human capability. The best we can hope for is to improve the average information content provided by the foregoing nomograms. Nevertheless, the results of this study are encouraging. For example, the information content provided by the Kattan nomogram was more than one third toward certainty for many of our study cases. The results also suggest that adding more variables increases information content. Because most of the information derived from pretreatment nomograms comes from things that Table 2 Summary of Variables Used and Results * Nomogram PSA Grade T Stage No. of Cores Mean 1 S Partin 5 5 4 Not used.16 Shipley 3 2 Not used Not used.25 D Amico 4 5 4 Not used.32 D Amico 4 5 4 Not used.37 Kattan C 4 5 C.44 * All listed variables were determined preoperatively. Entries for the variables give the number of defined categories of the variable used in the nomogram, and an entry of C means that the variable was used as a continuous, or nearly continuous, one. The T stage refers to the clinical assignment of stage. The entry of mean 1 S provides the average information content of the nomogram applied to our study patients. The first D Amico nomogram is for postsurgical outcome and the second for postradiation therapy outcome. 86 Am J Clin Pathol 27;128:83-87 86 DOI: 1.139/XA419Q75F5D2TVJN
Anatomic Pathology / ORIGINAL ARTICLE pathologists observe, the results suggest that we should seek new prognostic information from the samples of tumor or serum already available to us. Because the Kattan nomogram uses the number of cores with tumor and the total numbers of cores taken, the results suggest that we need to include this information in our reports. And we can hope that ongoing studies of molecular markers will evolve into immunohistochemical markers that could improve these nomograms. The relative success of the Kattan nomogram suggests that continuous variables like serum PSA values and nearly continuous variables like number of cores should be used in their natural state rather than with a few cut points. Breaking a continuous prognostic variable like PSA into 3 to 5 categories, as done in the Partin, Shipley, and D Amico nomograms, discards information. Furthermore, the success of the Kattan nomogram favors the use of the PSA level as a nonlinear variable because that is how his nomogram used the PSA value. We suggest that good nomograms should pass 3 successive tests. First, they should use variables that are significantly associated with important outcomes. Second, they should be validated by application to new data. In other words, their results should provide accurate predictions for new data. Finally, they should provide real information content. It is not sufficient that a nomogram accurately predicts that a probability of an outcome is, say,.5, because this prediction provides no useful information. When the predicted probability is.5, the information content is. Good nomograms should routinely provide information content greater than.3 and frequently more than.5. From the 1 Departments of Pathology and Oncology, McGill University-Jewish General Hospital, Montreal, Canada; 2 Department of Pathology, Washington University School of Medicine, St Louis, MO; and 3 Department of Pathology, Veterans Affairs and Duke University Medical Centers, Durham, NC. Address reprint requests to Dr Vollmer: Laboratory Medicine 113, VA Medical Center, 58 Fulton St., Durham, NC 2775. References 1. Jemal A, Siegel R, Ward E, et al. Cancer statistics, 27. CA Cancer J Clin. 27;57:43-66. 2. Ross PL, Scardino PT, Kattan MW. A catalog of prostate cancer nomograms. J Urol. 21;165:1562-1568. 3. Giasu S. Information Theory With Applications. New York, NY: McGraw-Hill; 1977. 4. Vollmer RT. Entropy and information content of laboratory test results. Am J Clin Pathol. 27;127:6-65. 5. Partin AW, Kattan MW, Subong EN, et al. Combination of prostate-specific antigen, clinical stage, and Gleason score to predict pathological stage of localized prostate cancer: a multiinstitutional update. JAMA. 1997;277:1445-1452 6. Shipley WU, Thames HD, Sandler HM, et al. Radiation therapy for clinically localized prostate cancer: a multiinstitutional pooled analysis. JAMA. 1999;281:1598-164. 7. D Amico AV, Whittington R, Malkowicz SB, et al. Pretreatment nomogram for prostate-specific antigen recurrence after radical prostatectomy or external-beam radiation therapy for clinically localized prostate cancer. J Clin Oncol. 1999;17:168-172. 8. Stephenson AJ, Scardino PT, Eastham JA, et al. Preoperative nomogram predicting the 1-year probability of prostate cancer recurrence after radical prostatectomy. J Natl Cancer Inst. 26;98:715-717. 9. Graefen M, Karakiewicz PI, Cagiannos I, et al. International validation of a preoperative nomogram for prostate cancer recurrence after radical prostatectomy. J Clin Oncol. 22;2:326-3212. 1. Greene KL, Meng MV, Elkin EP, et al. Validation of the Kattan preoperative nomogram for prostate cancer recurrence using a community based cohort: results from cancer of the prostate strategic urological research endeavor (CaPSURE). J Urol. 24;171:2255-2259. 11. Kattan MW, Eastham JA, Stapleton AMF, et al. A preoperative nomogram for disease recurrence following radical prostatectomy for prostate cancer. J Natl Cancer Inst. 1998;9:766-771. 12. Gerber GS, Thisted RA, Scardino PT, et al. Results of radical prostatectomy in men with clinically localized prostate cancer: multi-institutional pooled analysis. JAMA. 1996;276:615-619. 13. Krongrad A, Lai H, Lai S. Survival after radical prostatectomy. JAMA. 1997;278:44-46. 14. Dillioglogil O, Leibman BD, Kattan MW, et al. Hazard rates for progression after radical prostatectomy for clinically localized prostate cancer. Urology. 1997;5:93-99. 15. Pound CR, Partin AW, Epstein JI, et al. Prostate-specific antigen after anatomic radical retropubic prostatectomy. Urol Clin North Am. 1997;24:395-46. Am J Clin Pathol 27;128:83-87 87 87 DOI: 1.139/XA419Q75F5D2TVJN 87