Individualized Predictions of Disease Progression Following Radiation Therapy for Prostate Cancer
http://www.100md.com
《临床肿瘤学》
the Departments of Biostatistics and Radiation Oncology, University of Michigan, Ann Arbor, MI
the Department of Biostatistics, Indiana University School of Medicine, Indianapolis, IN
ABSTRACT
PATIENTS AND METHODS: Data from 934 patients treated between 1987 and 2000 were used to develop a comprehensive statistical model to fit the clinical recurrence events and patterns of PSA data. A logistic model was used for the probability of cure, mixed models were used for serial PSA measurements, and a proportional hazards model was used for recurrences. Data available through February 2001 were fit to the model, and data collected between February 2001 and September 2003 were used for validation.
RESULTS: T-stage, baseline PSA, and radiotherapy dosage are all associated with probability of cure. The risk of clinical recurrence in those not cured is strongly affected by the slope of PSA values. We show how the model can be used for individual monitoring of disease progression. For each patient the model predicts, based on baseline characteristics and all post-treatment PSA values, the probability of future clinical recurrences and future PSA values. The model accurately predicts risk of recurrence and future PSA values in the validation data set.
CONCLUSION: This predictive information on future PSA values and the risk of clinical relapse for each individual patient, which can be updated with each additional PSA value, may prove useful to patients and physicians in determining post-treatment salvage strategies.
INTRODUCTION
In an effort to provide more rational selection of patients who are at high-risk for early disease recurrence following localized therapy, clinical models designed to predict or stratify patients based on clinical, pathologic, and biochemical criteria have been developed.1,2,3,4 These models have primarily focused on prediction of biochemical recurrence following therapy, based on information available at the time of initial therapy, and have not used post-treatment PSA values. Models are greatly needed that can best predict which patients are at high risk of early development of clinical recurrence, following a pattern of rising PSA for initiation of salvage therapy and enrollment onto clinical trials. Such models would be useful to accurately identify at an early time those men who are at high risk of early recurrence, and who may benefit from salvage treatment.5,6 Such models would also be useful to identify those unlikely to need therapy in the near future.
The correlation between changes in PSA and prostate cancer clinical recurrence (local recurrence or distant metastases) has been long recognized. A consensus on what constitutes a dangerous rise in PSA after radiation therapy and how to translate it into guidelines for clinical monitoring has not been fully reached. Defining biochemical recurrence based on a series of PSA measurements has been a problematic and controversial topic. Radiation reduces PSA, but not to an undetectable level, then at a later time, PSA levels may start to increase before any clinical evidence of disease is detectable. However, there is considerable variability in the amount by which PSA is reduced by radiation therapy, and the time interval before it starts to rise (if it does rise), and the rate at which it rises.7
To provide some uniformity to the way that rises in PSA levels following irradiation had been utilized, the American Society for Therapeutic Radiation Oncology (ASTRO) developed a consensus definition of biochemical recurrence.8 The definition was based on three consecutive rises in PSA levels. Although this definition is known to have a number of problems,9,10,11,12,13,14 it did provide a benchmark against which alternatives could be compared. The primary use of the ASTRO definition is for consistency in comparing the results among different studies. It is less useful for monitoring an individual patient and for deciding on the initiation of hormone therapy (HT) or other salvage therapy in an individual patient. Such decisions will usually be based, in the absence of clinical symptoms, on the value and the rate of rising PSA levels, together with other factors such as the patient’s age, and tumor stage and Gleason score of the tumor.
The purpose of this article is to demonstrate a means to provide quantitative information, by way of a statistical model, about the likely future course of disease for a patient. This information can be used to enhance the medical decision-making process. The quantitative information we provide will be derived from a large series of patients treated at the University of Michigan. We consider two aspects to the future progression of the disease, one is the predicted PSA pattern, and the other is the occurrence of clinical events, such as local recurrence or distant metastases, indicating definitive disease, typically requiring additional therapy.
Many previous studies on PSA profiles following radiation therapy have focused on identifying features of the pattern which are important or prognostic. Examples of this are rise above nadir, doubling time, or the ASTRO definition of recurrence. This can be regarded as leading to a two-stage approach, in which one first decides whether the data for a patient exhibits this summary feature, then one makes explicit or implicit predictions about the likely future course of disease. In this article, we bypass the intermediate step of consideration of specific features, instead we directly make predictions about the likely future course of disease based on all the available data, including all post-treatment PSA values. Thus, we demonstrate that, although consideration of summary intermediate features is convenient, it is not necessary to make predictions.
PATIENTS AND METHODS
Validation data set. The validation data set consists of all data collected on these 934 patients after February 2001 and was available in September 2003. There were 612 patients who were alive at the last contact time in the analysis data set, and were not known to have experienced a clinical recurrence or received HT before February 2001. We focussed on the 541 patients for whom new follow-up information was available. The baseline characteristics of these 541 patients were quite similar to the original 934 patients, except for a slightly lower percentage of patients with worse prognosis. Among these 541 patients, 472 were alive at the end of the new follow-up period, 63 patients died from other causes not related to prostate cancer, and six patients died from prostate cancer. The median additional follow-up time is 30 months. There were 329 patients with additional PSA values; these patients provided 999 PSA measurements within 3 years of the previous last follow-up date. Fifteen of the patients developed clinical recurrence in the new follow-up period, six of these 15 patients had HT before the recurrence, and an additional 14 patients have received HT without any clinical recurrence.
Statistical Model
We used a comprehensive statistical model to fit the clinical recurrence events and pattern of PSA data. The model used in this article was an adaptation and enhancement of the model previously described by Law et al17 and Yu et al.18 In this article, the model is used to fit the data in the analysis data set, and to make predictions of future clinical events and future PSA values for individual patients. These predictions were then compared with the real data from the validation data set.
The statistical model required three components, one to model the possibility of cure, one to model the serial PSA measurements, and one to model the hazard of a clinical event. A brief description is given below, with a fuller description in the Appendix. A logistic regression model was used for the probability of cure. We assumed that the log odds of the probability of cure is a linear combination of the baseline covariates, T-stage, Gleason, PSA, total dose, treatment duration, and age. Nonlinear hierarchical mixed models were used for serial PSA measurements. The longitudinal model for PSA for person i at time t is based on PSAi (t) = ai exp(–bit) + ci exp(dit).19,20 The first term represents the slow decrease in PSA after radiation, the second term represents the increase in PSA due to tumor regrowth. For cured patients, we assumed the value of di was close to zero. The coefficients (ai, bi, ci, di) can differ from one patient to the next, and they are also allowed to depend on baseline covariates. A time-dependent Cox proportional hazards model was used for the clinical recurrence events. During the follow-up period, the hazard of recurrence for noncured patients was assumed to depend on the current PSA, current slope of PSA, an indicator of whether salvage HT was given, and baseline covariates. The models described in this section are designed to include population average trends, to allow for between-patient heterogeneity in response, which can be explained by baseline patient and treatment variables, and to allow for extra unexplained heterogeneity in PSA and clinical response.
PSA data after HT are not used in fitting the model, but clinical recurrences after HT are used. The model is fit using Markov Chain Monte Carlo (MCMC) methods, as described in Yu et al.18 We present results of odds ratios from the cure model and relative hazards from the Cox model.
The model was used to predict future PSA values for patient i at any future time T. In addition, we gave the 2.5 to 97.5 percentile range of predicted values. For the prediction of future clinical events, the probability that patient i is event-free T months into the future can be calculated from the equations describing the model. Both the average probability and the range of probabilities are presented graphically. We only predicted future PSA values and clinical events for patients who were alive at last follow-up, had not had a clinical recurrence, or received HT. These predictions are compared with the data in the validation data set. We also presented graphically the estimated possible impact of HT on each patient’s time to clinical recurrence. This is calculated using the model, under the assumption that the patient starts HT at the time of last contact.
RESULTS
Figure 1 shows the pattern of PSA (on a log-transformed scale) and the predicted probability of being free of future clinical recurrence from the date of last contact, with and without the addition of HT at the last contact time, for three selected patients (Patients A, B, and C) with long follow-up times. The patients were selected to illustrate a range of PSA patterns and predictions. The magnitude of the potential impact of HT can be seen in Figures 1A, 1C, and 1E. Figures 1B, 1D, and 1F show the uncertainty of the prediction of clinical recurrence without HT. For Patients A and B there is a clear pattern of increasing PSA, suggesting eventual clinical recurrence. Patient A has a steeper rise than Patient B, which leads to a higher probability of recurrence within 4 years. For Patient C, the favorable pattern of PSA post-treatment suggests cure, which corresponds to the almost horizontal predicted clinical recurrence curve, however, there is a small probability of eventual recurrence (probability, 0.049) as seen in the uncertainty curves. Figure 2 shows the pattern of PSA and predicted probabilities of clinical recurrence for three patients with short or medium follow-up times. Patient D has an almost level pattern of PSA, but with the suggestion of an increase at the end of his follow-up. Thus, there is considerable uncertainty about the future for this patient (Fig 2B). Figure 2B shows horizontal curves reflecting cure (probability, 0.63) and decreasing curves reflecting eventual recurrence (probability, 0.37). Patient E has a clearly rising pattern of PSA, though the values are low; this leads to significant probability of recurrence within 4 years. Patient F has very short follow-up leading to considerable range of predicted probabilities of recurrence within 4 years. In general, patients with longer follow-up have less uncertainty in their prediction rates than do patients with shorter follow-up periods. It is also worth noting that despite a clearly increasing pattern of PSA for some of the patients in Figures 1 and 2, none have a predicted probability of recurrence within 4 years greater than 50%. We envision that graphs such as these could be useful to the clinician and the patient when deciding whether to initiate HT, or when PSA should next be measured.
Figure 3 shows predicted and observed future values of PSA for the six selected patients. The shaded areas represent 95% prediction intervals for each patient, these are derived from the model fit to the PSA data (as shown by the dots). We note that patients A, B, and C, who have lots of data, have fairly narrow prediction intervals, whereas patient F has less follow-up and thus a wider prediction interval. We envision a graph like this would also be useful in monitoring the progression of the patient. If a new PSA value is measured and it falls in the upper half of the shaded region then this is indicative of the patient doing worse than expected, if the PSA value falls in the lower half of the shaded region then this is indicative of the patient doing better than expected, and if it falls outside the shaded region then this is indicative of the patient doing either substantially worse than or substantially better than expected. After a new measurement is obtained, a new graph could be produced, thus allowing real-time monitoring of a patient’s progression.
The + symbols in the graphs (Fig 3) are PSA measurements obtained from the validation data set for these six patients; all the values fall within the 95% prediction intervals. We note that none of these six patients has had a clinical recurrence in the validation data set follow-up period. Table 3 shows the proportion of future PSA values among all available future data that were within the 95% prediction intervals; we see very good correspondence with the expected 95% level for all years.
Table 4 compares the predicted clinical recurrence with the observed data on clinical recurrences and salvage HT. For each of the 541 validation patients, the probability of clinical recurrence within 3 years after the last contact date is calculated, and divided into three groups as listed in the table. Within each of these groups the observed number of recurrences and salvage HT in the validation data set is shown, as well as the calculated Kaplan-Meier estimate of the 3-year recurrence or HT probability. The results show that a larger proportion of recurrences in the groups with the higher predicted probability, this provides support for the validity of the model. It is also clear that the model is predicting a slightly greater proportion of recurrences than were actually observed, however if HT is included, then the predictions are close to the observed data.
DISCUSSION
Patient C in Figure 1 illustrates this. He has been followed for 140 months after radiotherapy and his PSA level remains low. His risk of clinical recurrence—from the last PSA determination-—is near zero over the next 4 years, with or without HT (Fig 1E). The uncertainty of this estimate is also relatively low, as shown by the spread in the lines in Figure 1F. Compare this with patient F in Figure 2. His follow-up interval is shorter, only 25 months, and he has fewer post-treatment PSA values. Thus, while his risk of clinical recurrence is low, with or without HT (Fig 2E), the uncertainty of his estimate is large (Fig 2F). As time passes, the clinical recurrence estimates change with each PSA update. This gives the patient and caregiver either reassurance, if the estimate for recurrence is low, or raises concerns that might be addressed with salvage therapy, if the estimate for recurrence is high.
As with any model it contains a number of assumptions. We cannot hope to incorporate in the model all aspects of the known biology of prostate cancer progression, and in this sense our model is too simple. However, compared with most statistical models, this model would be regarded as complex, but this level of complexity was considered necessary to capture the variability in disease progression. An assessment of whether the assumptions are justified can be made empirically, by considering whether the model fits the current data, whether it can be validated using new data, and whether the parameter estimates are plausible. On all three counts the model appears to be adequate.
One of the clearest findings from the parameter estimates is the strength of the association of the slope of PSA with the hazard of clinical recurrence. This is consistent with the well-known observation that PSA doubling time is an important determinant of disease progression.21,22,4 Another substantive finding is that baseline variables, such as Gleason score and PSA, retain some information about disease recurrence even when the current value and slope of PSA are known.
The major contribution of this article is intended to be the illustration of the ability to predict future disease progression. We provide estimates and uncertainty ranges for these predictions. We believe that such quantitative information will enhance rational decision making concerning actions to take. In a clinical setting, one might base future clinical interventions on the likelihood of future clinical relapse. Possible actions that could be taken based on the quantitative predictions offered here, might include initiation of HT or return for next PSA measurement at a shorter or longer than usual interval.
Any statistical model that is to be used for prediction must adequately capture the variation that is observed in PSA data, and in the timing of clinical recurrences. This variation means that there will be uncertainty in any prediction; the model provides uncertainty ranges associated with those predictions. Two patients could have the same prediction: for example, the probability of clinical recurrence within 4 years is 0.3, but the uncertainty associated with the prediction could differ. One patient could have been followed for a short time and had wide uncertainty ranges, and the other patient could have been followed for a long time and had a narrow uncertainty range. This uncertainty range represents information that is additional to the estimated recurrence probability of 0.3, and might play a role in the clinical decision about initiating further therapy.
Another possible use for this model is as a method for defining biochemical recurrence. For example, one could define a patient as having biochemical recurrence if the estimated probability of clinical recurrence within 3 years is greater than 30%. We note that such a definition utilizes all the serial PSA and baseline data for the patient, not just the last four PSA values as in the current ASTRO definition.
We are not the first to develop statistical models to utilize a series of biomarker values for early detection of disease or to monitor disease progression. Other examples are using CA125 for early detection of ovarian cancer,23 using PSA for early detection of prostate cancer,24 and using PSA for detecting disease recurrence in prostate cancer.25 A key feature of all these publications is that the whole series of biomarker values is utilized and that they can handle unequally spaced observations, thus they make efficient use of all the information in the data. This is in contrast to simple methods that require the biomarker to be above a certain cutoff, or ad hoc methods, such as the ASTRO definition, that do not adapt well to heterogeneity in the time interval between measurements, and do not necessarily make efficient use of all the data.
Appendix
where PSAi(U1) is the value of PSA at time U1, Xi are baseline covariates, ti is the time of clinical recurrence or last follow-up, and {delta}i is the indicator for clinical recurrence. A logistic regression model was used for the probability of cure. Let Di be the latent cure indicator, with Di = 1 denoting cure and Di = 2 denoting not cure. Thus each patient is in one of two categories, either they are susceptible to recurrence or nonsusceptible to recurrence. The value of Di is unknown, except for patients who have had clinical recurrence. It is reasonable to assume that Di only depends on baseline and treatment variables, thus we assume
Non-linear hierarchical mixed models were used for serial PSA measurements. Let
where eij is measurement error, which we assume has a student’s t distribution with 5 degrees of freedom. The longitudinal model for PSA is
For cured patients we assume di is close to zero. The set (ai,bi,ci,di) are latent variable random effects, which have their own joint multivariate normal distribution for each value of Di, and are restricted to be positive. They are also allowed to depend on baseline covariates, through a regression model. The proportional hazards model for a clinical event for susceptible patients (Di = 2) is given by
In this model sli(t) is the square root of the slope of Ri(t), and HT(t) is a function which is non-zero on 0 to 60 months; it linearly decreases from one at t = 0 to zero at t = 60, and {tau}i is the initiation time of hormonal therapy for those who received it.
PSA data after hormonal therapy are not used in fitting the model, but clinical recurrences after hormone therapy are used. The model is fit using Markov Chain Monte Carlo (MCMC) methods, as described in Yu et al.18 The MCMC gives the posterior distribution of all the parameters and the latent variables.
The prediction of a future PSA value for patient i at time t is given by the formula
where
with (ai,bi,ci,di) produced by the MCMC method and e is a random draw from a T5 distribution. Consecutive iterates of the MCMC give different values of the parameters and hence of PSAi(t), thus giving a range of possible future PSA values. We use 2,000 draws from the MCMC and present the 2.5 to 97.5 percentile range.
For the prediction of future clinical events the probability that patient i is event free T months into the future can be calculated from the equations describing the model. Different iterates from the MCMC will give different probabilities; both the average probability and the range of probabilities (from 40 draws) are presented graphically.
Authors’ Disclosures of Potential Conflicts of Interest
NOTES
Supported by the National Institutes of Health grants P30 CA46592 and P50 CA069568.
Authors’ disclosures of potential conflicts of interest are found at the end of this article.
REFERENCES
1. Kattan MW, Wheeler TM, Scardino PT: Postoperative nomogram for disease recurrence after radical prostatectomy for prostate cancer. J Clin Oncol 17:1499-1507, 1999
2. Kattan MW, Pearn PA, Leibel S, et al: The definition of biochemical failure in patients treated with definitive radiotherapy. Int J Radiat Oncol Biol Phys 48:1469-1474, 2000
3. Kattan MW, Zelefsky MJ, Kupelian PA, et al: Pretreatment nomogram for predicting the outcome of three-dimensional conformal radiotherapy in prostate cancer. J Clin Oncol 18:3352-3359, 2000
4. D’Amico A, Whittington R, Malkowicz S, 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 17:168, 1999
5. Kuban DA, El-Mahdi AM, Schellhammer PF: PSA for outcome prediction and post-treatment evaluation following radiation for prostate cancer: Do we know how to use it Semin Radiat Oncol 8:72-80, 1998
6. D’Amico A, Moul JW, Carroll PR, et al: Surrogate end point for prostate cancer-specific mortality after radical prostatectomy or radiation therapy. J Natl Cancer Inst 95:1376-1383, 2003
7. Zagars GK, Pollack A: The fall and rise of prostate-specific antigen. Cancer 72:832-842, 1993
8. American Society for Therapeutic Radiology and Oncology Consensus Panel: Consensus Statement: Guidelines for PSA following radiation therapy. Int J Radiat Oncol Biol Phys 37:1035-1041, 1997
9. Horwitz E, Vincini F, Ziaja E, et al: Assessing the variability of outcome for patients treated with localized prostate irradiation using different definitions of biochemical control. Int J Radiat Oncol Biol Phys 36:565-571, 1996
10. Vicini FA, Kestin LL, Martinez AA: The importance of adequate follow-up in defining treatment success after external beam irradiation for prostate cancer. Int J Radiat Oncol Biol Phys 45:553-561, 1999
11. Hanlon AL, Hanks GE: Scrutiny of the ASTRO consensus definition of biochemical failure in irradiated prostate cancer patients demonstrates its usefulness and robustness. Int J Radiat Oncol Biol Phys 46:559-566, 2000
12. Taylor JM, Griffith K, Sandler H: Definitions of biochemical failure in prostate cancer following radiation therapy. Int J Radiat Oncol Biol Phys 50:1219-1229, 2001
13. Horwitz E, Uzzo EM, Hanlon, et al: Modifying the ASTRO definition of biochemical failure to minimize the influence of backdating in patients with prostate cancer treated with 3D conformal radiation therapy alone. J Urol 169:2153-2159, 2003
14. Thames HD, Kuban DA, et al: Comparison of alternative biochemical failure definitions based on clinical outcome in 4839 prostate cancer patients treated by external beam radiotherapy between 1986 and 1995. Int J Radiat Oncol Biol Phys 57:929-943, 2003
15. Sandler HM, Dunn RL, McLaughlin PW, et al: Overall survival after PSA-detected recurrence following conformal radiation therapy. Int J Radiat Oncol Biol Phys 48:629-633, 2000
16. Symon Z, Griffith KA, McLaughlin PW, et al: Dose escalation for localized prostate cancer: Substantial benefit observed with 3D conformal therapy. Int J Radiat Oncol, Biol Phys 57:384-390, 2003
17. Law NJ, Taylor JMG, Sandler H: The joint modeling of a longitudinal disease progression marker and the failure time process in the presence of cure. Biostatistics 3:547-563, 2002
18. Yu M, Law NJ, Taylor JMG, et al: Joint longitudinal-survival-cure models and their application to prostate cancer. Statistica Sinica 14:835-862, 2004
19. Kaplan ID, Cox RS, Bagshaw MA: A model of prostatic carcinoma tumor kinetics based on prostate specific antigen levels after radiation therapy. Cancer 68:400-404, 1991
20. Cox RS, Kaplan ID, Bagshaw MA: Prostate-specific antigen kinetics after external beam irradiation for carcinoma of the prostate. Int J Radiat Oncol Biol Phys 28:23-31, 1994
21. Hanks GE, D’Amico A, Epstein BE, et al: Prostate-specific antigen doubling times in patients with prostate cancer: A potentially useful reflection of doubling time. Int J Radiat Oncol Biol Phys 27:125-127, 1993
22. Sartor CI, Strawderman MH, Lin XH, et al: Rate of PSA rise predicts metastatic versus local recurrence after definitive radiation therapy. Int J Radiat Oncol Biol Phys 38:941-947, 1997
23. Skates SJ, Pauler DK, Jacobs IJ: Screening based on the risk of cancer calculation from bayesian hierarchical change-point models of longitudinal markers. J Am Stat Assoc 96:429-439, 2001
24. Slate EH, Cronin KA: Changepoint modeling of longitudinal PSA as a biomarker for prostate cancer, in Gatsonis C, Hodges J, Kass R, et al (eds): Case Studies in Bayesian Statistics III. New York, NY, Springer-Verlag, 1997, pp 435-456
25. Pauler DK, Finkelstein DM: Predicting time to prostate cancer recurrence based on joint models for non-linear longitudinal biomarkers and event time outcomes. Stat Med 21:3897-3911, 2002(Jeremy M.G. Taylor, Mengg)
the Department of Biostatistics, Indiana University School of Medicine, Indianapolis, IN
ABSTRACT
PATIENTS AND METHODS: Data from 934 patients treated between 1987 and 2000 were used to develop a comprehensive statistical model to fit the clinical recurrence events and patterns of PSA data. A logistic model was used for the probability of cure, mixed models were used for serial PSA measurements, and a proportional hazards model was used for recurrences. Data available through February 2001 were fit to the model, and data collected between February 2001 and September 2003 were used for validation.
RESULTS: T-stage, baseline PSA, and radiotherapy dosage are all associated with probability of cure. The risk of clinical recurrence in those not cured is strongly affected by the slope of PSA values. We show how the model can be used for individual monitoring of disease progression. For each patient the model predicts, based on baseline characteristics and all post-treatment PSA values, the probability of future clinical recurrences and future PSA values. The model accurately predicts risk of recurrence and future PSA values in the validation data set.
CONCLUSION: This predictive information on future PSA values and the risk of clinical relapse for each individual patient, which can be updated with each additional PSA value, may prove useful to patients and physicians in determining post-treatment salvage strategies.
INTRODUCTION
In an effort to provide more rational selection of patients who are at high-risk for early disease recurrence following localized therapy, clinical models designed to predict or stratify patients based on clinical, pathologic, and biochemical criteria have been developed.1,2,3,4 These models have primarily focused on prediction of biochemical recurrence following therapy, based on information available at the time of initial therapy, and have not used post-treatment PSA values. Models are greatly needed that can best predict which patients are at high risk of early development of clinical recurrence, following a pattern of rising PSA for initiation of salvage therapy and enrollment onto clinical trials. Such models would be useful to accurately identify at an early time those men who are at high risk of early recurrence, and who may benefit from salvage treatment.5,6 Such models would also be useful to identify those unlikely to need therapy in the near future.
The correlation between changes in PSA and prostate cancer clinical recurrence (local recurrence or distant metastases) has been long recognized. A consensus on what constitutes a dangerous rise in PSA after radiation therapy and how to translate it into guidelines for clinical monitoring has not been fully reached. Defining biochemical recurrence based on a series of PSA measurements has been a problematic and controversial topic. Radiation reduces PSA, but not to an undetectable level, then at a later time, PSA levels may start to increase before any clinical evidence of disease is detectable. However, there is considerable variability in the amount by which PSA is reduced by radiation therapy, and the time interval before it starts to rise (if it does rise), and the rate at which it rises.7
To provide some uniformity to the way that rises in PSA levels following irradiation had been utilized, the American Society for Therapeutic Radiation Oncology (ASTRO) developed a consensus definition of biochemical recurrence.8 The definition was based on three consecutive rises in PSA levels. Although this definition is known to have a number of problems,9,10,11,12,13,14 it did provide a benchmark against which alternatives could be compared. The primary use of the ASTRO definition is for consistency in comparing the results among different studies. It is less useful for monitoring an individual patient and for deciding on the initiation of hormone therapy (HT) or other salvage therapy in an individual patient. Such decisions will usually be based, in the absence of clinical symptoms, on the value and the rate of rising PSA levels, together with other factors such as the patient’s age, and tumor stage and Gleason score of the tumor.
The purpose of this article is to demonstrate a means to provide quantitative information, by way of a statistical model, about the likely future course of disease for a patient. This information can be used to enhance the medical decision-making process. The quantitative information we provide will be derived from a large series of patients treated at the University of Michigan. We consider two aspects to the future progression of the disease, one is the predicted PSA pattern, and the other is the occurrence of clinical events, such as local recurrence or distant metastases, indicating definitive disease, typically requiring additional therapy.
Many previous studies on PSA profiles following radiation therapy have focused on identifying features of the pattern which are important or prognostic. Examples of this are rise above nadir, doubling time, or the ASTRO definition of recurrence. This can be regarded as leading to a two-stage approach, in which one first decides whether the data for a patient exhibits this summary feature, then one makes explicit or implicit predictions about the likely future course of disease. In this article, we bypass the intermediate step of consideration of specific features, instead we directly make predictions about the likely future course of disease based on all the available data, including all post-treatment PSA values. Thus, we demonstrate that, although consideration of summary intermediate features is convenient, it is not necessary to make predictions.
PATIENTS AND METHODS
Validation data set. The validation data set consists of all data collected on these 934 patients after February 2001 and was available in September 2003. There were 612 patients who were alive at the last contact time in the analysis data set, and were not known to have experienced a clinical recurrence or received HT before February 2001. We focussed on the 541 patients for whom new follow-up information was available. The baseline characteristics of these 541 patients were quite similar to the original 934 patients, except for a slightly lower percentage of patients with worse prognosis. Among these 541 patients, 472 were alive at the end of the new follow-up period, 63 patients died from other causes not related to prostate cancer, and six patients died from prostate cancer. The median additional follow-up time is 30 months. There were 329 patients with additional PSA values; these patients provided 999 PSA measurements within 3 years of the previous last follow-up date. Fifteen of the patients developed clinical recurrence in the new follow-up period, six of these 15 patients had HT before the recurrence, and an additional 14 patients have received HT without any clinical recurrence.
Statistical Model
We used a comprehensive statistical model to fit the clinical recurrence events and pattern of PSA data. The model used in this article was an adaptation and enhancement of the model previously described by Law et al17 and Yu et al.18 In this article, the model is used to fit the data in the analysis data set, and to make predictions of future clinical events and future PSA values for individual patients. These predictions were then compared with the real data from the validation data set.
The statistical model required three components, one to model the possibility of cure, one to model the serial PSA measurements, and one to model the hazard of a clinical event. A brief description is given below, with a fuller description in the Appendix. A logistic regression model was used for the probability of cure. We assumed that the log odds of the probability of cure is a linear combination of the baseline covariates, T-stage, Gleason, PSA, total dose, treatment duration, and age. Nonlinear hierarchical mixed models were used for serial PSA measurements. The longitudinal model for PSA for person i at time t is based on PSAi (t) = ai exp(–bit) + ci exp(dit).19,20 The first term represents the slow decrease in PSA after radiation, the second term represents the increase in PSA due to tumor regrowth. For cured patients, we assumed the value of di was close to zero. The coefficients (ai, bi, ci, di) can differ from one patient to the next, and they are also allowed to depend on baseline covariates. A time-dependent Cox proportional hazards model was used for the clinical recurrence events. During the follow-up period, the hazard of recurrence for noncured patients was assumed to depend on the current PSA, current slope of PSA, an indicator of whether salvage HT was given, and baseline covariates. The models described in this section are designed to include population average trends, to allow for between-patient heterogeneity in response, which can be explained by baseline patient and treatment variables, and to allow for extra unexplained heterogeneity in PSA and clinical response.
PSA data after HT are not used in fitting the model, but clinical recurrences after HT are used. The model is fit using Markov Chain Monte Carlo (MCMC) methods, as described in Yu et al.18 We present results of odds ratios from the cure model and relative hazards from the Cox model.
The model was used to predict future PSA values for patient i at any future time T. In addition, we gave the 2.5 to 97.5 percentile range of predicted values. For the prediction of future clinical events, the probability that patient i is event-free T months into the future can be calculated from the equations describing the model. Both the average probability and the range of probabilities are presented graphically. We only predicted future PSA values and clinical events for patients who were alive at last follow-up, had not had a clinical recurrence, or received HT. These predictions are compared with the data in the validation data set. We also presented graphically the estimated possible impact of HT on each patient’s time to clinical recurrence. This is calculated using the model, under the assumption that the patient starts HT at the time of last contact.
RESULTS
Figure 1 shows the pattern of PSA (on a log-transformed scale) and the predicted probability of being free of future clinical recurrence from the date of last contact, with and without the addition of HT at the last contact time, for three selected patients (Patients A, B, and C) with long follow-up times. The patients were selected to illustrate a range of PSA patterns and predictions. The magnitude of the potential impact of HT can be seen in Figures 1A, 1C, and 1E. Figures 1B, 1D, and 1F show the uncertainty of the prediction of clinical recurrence without HT. For Patients A and B there is a clear pattern of increasing PSA, suggesting eventual clinical recurrence. Patient A has a steeper rise than Patient B, which leads to a higher probability of recurrence within 4 years. For Patient C, the favorable pattern of PSA post-treatment suggests cure, which corresponds to the almost horizontal predicted clinical recurrence curve, however, there is a small probability of eventual recurrence (probability, 0.049) as seen in the uncertainty curves. Figure 2 shows the pattern of PSA and predicted probabilities of clinical recurrence for three patients with short or medium follow-up times. Patient D has an almost level pattern of PSA, but with the suggestion of an increase at the end of his follow-up. Thus, there is considerable uncertainty about the future for this patient (Fig 2B). Figure 2B shows horizontal curves reflecting cure (probability, 0.63) and decreasing curves reflecting eventual recurrence (probability, 0.37). Patient E has a clearly rising pattern of PSA, though the values are low; this leads to significant probability of recurrence within 4 years. Patient F has very short follow-up leading to considerable range of predicted probabilities of recurrence within 4 years. In general, patients with longer follow-up have less uncertainty in their prediction rates than do patients with shorter follow-up periods. It is also worth noting that despite a clearly increasing pattern of PSA for some of the patients in Figures 1 and 2, none have a predicted probability of recurrence within 4 years greater than 50%. We envision that graphs such as these could be useful to the clinician and the patient when deciding whether to initiate HT, or when PSA should next be measured.
Figure 3 shows predicted and observed future values of PSA for the six selected patients. The shaded areas represent 95% prediction intervals for each patient, these are derived from the model fit to the PSA data (as shown by the dots). We note that patients A, B, and C, who have lots of data, have fairly narrow prediction intervals, whereas patient F has less follow-up and thus a wider prediction interval. We envision a graph like this would also be useful in monitoring the progression of the patient. If a new PSA value is measured and it falls in the upper half of the shaded region then this is indicative of the patient doing worse than expected, if the PSA value falls in the lower half of the shaded region then this is indicative of the patient doing better than expected, and if it falls outside the shaded region then this is indicative of the patient doing either substantially worse than or substantially better than expected. After a new measurement is obtained, a new graph could be produced, thus allowing real-time monitoring of a patient’s progression.
The + symbols in the graphs (Fig 3) are PSA measurements obtained from the validation data set for these six patients; all the values fall within the 95% prediction intervals. We note that none of these six patients has had a clinical recurrence in the validation data set follow-up period. Table 3 shows the proportion of future PSA values among all available future data that were within the 95% prediction intervals; we see very good correspondence with the expected 95% level for all years.
Table 4 compares the predicted clinical recurrence with the observed data on clinical recurrences and salvage HT. For each of the 541 validation patients, the probability of clinical recurrence within 3 years after the last contact date is calculated, and divided into three groups as listed in the table. Within each of these groups the observed number of recurrences and salvage HT in the validation data set is shown, as well as the calculated Kaplan-Meier estimate of the 3-year recurrence or HT probability. The results show that a larger proportion of recurrences in the groups with the higher predicted probability, this provides support for the validity of the model. It is also clear that the model is predicting a slightly greater proportion of recurrences than were actually observed, however if HT is included, then the predictions are close to the observed data.
DISCUSSION
Patient C in Figure 1 illustrates this. He has been followed for 140 months after radiotherapy and his PSA level remains low. His risk of clinical recurrence—from the last PSA determination-—is near zero over the next 4 years, with or without HT (Fig 1E). The uncertainty of this estimate is also relatively low, as shown by the spread in the lines in Figure 1F. Compare this with patient F in Figure 2. His follow-up interval is shorter, only 25 months, and he has fewer post-treatment PSA values. Thus, while his risk of clinical recurrence is low, with or without HT (Fig 2E), the uncertainty of his estimate is large (Fig 2F). As time passes, the clinical recurrence estimates change with each PSA update. This gives the patient and caregiver either reassurance, if the estimate for recurrence is low, or raises concerns that might be addressed with salvage therapy, if the estimate for recurrence is high.
As with any model it contains a number of assumptions. We cannot hope to incorporate in the model all aspects of the known biology of prostate cancer progression, and in this sense our model is too simple. However, compared with most statistical models, this model would be regarded as complex, but this level of complexity was considered necessary to capture the variability in disease progression. An assessment of whether the assumptions are justified can be made empirically, by considering whether the model fits the current data, whether it can be validated using new data, and whether the parameter estimates are plausible. On all three counts the model appears to be adequate.
One of the clearest findings from the parameter estimates is the strength of the association of the slope of PSA with the hazard of clinical recurrence. This is consistent with the well-known observation that PSA doubling time is an important determinant of disease progression.21,22,4 Another substantive finding is that baseline variables, such as Gleason score and PSA, retain some information about disease recurrence even when the current value and slope of PSA are known.
The major contribution of this article is intended to be the illustration of the ability to predict future disease progression. We provide estimates and uncertainty ranges for these predictions. We believe that such quantitative information will enhance rational decision making concerning actions to take. In a clinical setting, one might base future clinical interventions on the likelihood of future clinical relapse. Possible actions that could be taken based on the quantitative predictions offered here, might include initiation of HT or return for next PSA measurement at a shorter or longer than usual interval.
Any statistical model that is to be used for prediction must adequately capture the variation that is observed in PSA data, and in the timing of clinical recurrences. This variation means that there will be uncertainty in any prediction; the model provides uncertainty ranges associated with those predictions. Two patients could have the same prediction: for example, the probability of clinical recurrence within 4 years is 0.3, but the uncertainty associated with the prediction could differ. One patient could have been followed for a short time and had wide uncertainty ranges, and the other patient could have been followed for a long time and had a narrow uncertainty range. This uncertainty range represents information that is additional to the estimated recurrence probability of 0.3, and might play a role in the clinical decision about initiating further therapy.
Another possible use for this model is as a method for defining biochemical recurrence. For example, one could define a patient as having biochemical recurrence if the estimated probability of clinical recurrence within 3 years is greater than 30%. We note that such a definition utilizes all the serial PSA and baseline data for the patient, not just the last four PSA values as in the current ASTRO definition.
We are not the first to develop statistical models to utilize a series of biomarker values for early detection of disease or to monitor disease progression. Other examples are using CA125 for early detection of ovarian cancer,23 using PSA for early detection of prostate cancer,24 and using PSA for detecting disease recurrence in prostate cancer.25 A key feature of all these publications is that the whole series of biomarker values is utilized and that they can handle unequally spaced observations, thus they make efficient use of all the information in the data. This is in contrast to simple methods that require the biomarker to be above a certain cutoff, or ad hoc methods, such as the ASTRO definition, that do not adapt well to heterogeneity in the time interval between measurements, and do not necessarily make efficient use of all the data.
Appendix
where PSAi(U1) is the value of PSA at time U1, Xi are baseline covariates, ti is the time of clinical recurrence or last follow-up, and {delta}i is the indicator for clinical recurrence. A logistic regression model was used for the probability of cure. Let Di be the latent cure indicator, with Di = 1 denoting cure and Di = 2 denoting not cure. Thus each patient is in one of two categories, either they are susceptible to recurrence or nonsusceptible to recurrence. The value of Di is unknown, except for patients who have had clinical recurrence. It is reasonable to assume that Di only depends on baseline and treatment variables, thus we assume
Non-linear hierarchical mixed models were used for serial PSA measurements. Let
where eij is measurement error, which we assume has a student’s t distribution with 5 degrees of freedom. The longitudinal model for PSA is
For cured patients we assume di is close to zero. The set (ai,bi,ci,di) are latent variable random effects, which have their own joint multivariate normal distribution for each value of Di, and are restricted to be positive. They are also allowed to depend on baseline covariates, through a regression model. The proportional hazards model for a clinical event for susceptible patients (Di = 2) is given by
In this model sli(t) is the square root of the slope of Ri(t), and HT(t) is a function which is non-zero on 0 to 60 months; it linearly decreases from one at t = 0 to zero at t = 60, and {tau}i is the initiation time of hormonal therapy for those who received it.
PSA data after hormonal therapy are not used in fitting the model, but clinical recurrences after hormone therapy are used. The model is fit using Markov Chain Monte Carlo (MCMC) methods, as described in Yu et al.18 The MCMC gives the posterior distribution of all the parameters and the latent variables.
The prediction of a future PSA value for patient i at time t is given by the formula
where
with (ai,bi,ci,di) produced by the MCMC method and e is a random draw from a T5 distribution. Consecutive iterates of the MCMC give different values of the parameters and hence of PSAi(t), thus giving a range of possible future PSA values. We use 2,000 draws from the MCMC and present the 2.5 to 97.5 percentile range.
For the prediction of future clinical events the probability that patient i is event free T months into the future can be calculated from the equations describing the model. Different iterates from the MCMC will give different probabilities; both the average probability and the range of probabilities (from 40 draws) are presented graphically.
Authors’ Disclosures of Potential Conflicts of Interest
NOTES
Supported by the National Institutes of Health grants P30 CA46592 and P50 CA069568.
Authors’ disclosures of potential conflicts of interest are found at the end of this article.
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