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Keywords:

  • prostate carcinoma;
  • prostate specific antigen (PSA);
  • model;
  • watchful waiting

Abstract

  1. Top of page
  2. Abstract
  3. METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

BACKGROUND

For the moment, there is uncertainty about the usefulness of early treatment of localized prostate carcinoma, uncertainty about whether some patients with early cancer can be managed expectantly, and uncertainty about how such patients might be recognized.

METHODS

The authors studied serial values of prostate specific antigen (PSA) in 94 Japanese men with diagnosed prostate carcinoma and who were managed by watchful waiting. Their median follow-up duration was 32 months (range, 1.6–118). The authors used a log-linear model to fit the values of PSA over time, and then they used the Cox survival model to relate the intercept (PSA amplitude) and slope (relative velocity) to observed local or systemic outcomes that were independent of PSA.

RESULTS

The authors found that the log-linear model fit the serial values of PSA during watchful waiting very well. Prostate specific antigen amplitude related significantly to T classification (P = 0.0006), but not to grade (P > 0.2), and the relative velocity related significantly to both T classification (P = 0.009) and to grade (P = 0.02). Although the T classification, histologic grade, and log(PSA) at diagnosis were associated significantly with time to outcome, the combination of amplitude and relative velocity provided more information. These 2 PSA parameters resulted in a higher model likelihood ratio, and their individual P values in the Cox model were 0.0005 and 0.005, respectively. With these two in the Cox model, T classification, grade, log(PSA), and PSA doubling time provided no further significant information.

CONCLUSIONS

A log-linear model seems to fit serial measurements of PSA during watchful waiting, and preliminary results suggest that both the amplitude and the relative velocity relate closely to clinical outcomes. Cancer 2002;94:1692–8. © 2002 American Cancer Society.

DOI 10.1002/cncr.10443

Prostate specific antigen (PSA) has been described as the ideal serum tumor marker,1 and its use as a screening tool has enabled the early diagnosis of prostate carcinoma. In many men whose disease was diagnosed by screening, PSA is the only indication of cancer (TNM classification T1c), and, in general, treatment of localized tumors by radical prostatectomy or radiation has produced long-term cures.2, 3 Nevertheless, controversy surrounds the use of radical treatment for localized prostate carcinoma discovered by PSA, because the treatments reduce quality of life4 and because some of these patients' tumors would not progress during their remaining years of life.5 In fact, early treatment for localized prostate carcinoma shares several similarities with expectant management, for example, with respect to disease specific survival and the way outcome depends on tumor grade and stage.6–9 Some researchers have gone so far as to suggest that cures of prostate carcinoma are caused by the biology of the cancer and/or lead time bias rather than by the treatment.10 Because of such issues, there are now in progress at least two prospective trials to study the impact of early diagnosis and treatment of prostate carcinoma on disease-related and overall survival.11, 12 Meanwhile, case studies of men during watchful waiting may provide useful ways to predict the likelihood of progression and thereby help select and time treatment for these men.

Factors likely to relate to progression include clinical stage, level of PSA, and the amount and grade of tumor in needle biopsies. In addition, several molecular markers measured on the tumor could help and are actively being studied.5 In this study, we concentrate on the information that serial measurements of PSA can provide. Previously, Schmid et al. studied serial measurements of PSA in men during at least 1 year of watchful waiting, and they found that PSA doubling time (DT) related significantly to tumor stage and grade.13 McLaren et al. found that PSA DT was the most powerful indicator of clinical progression in 113 men observed expectantly; whereas, T classification, histologic grade, and initial level of PSA were not important.14 By contrast, Bangma et al. found no relation between PSA DT and progression,15 and Gerber et al. concluded that it is unclear whether the serial PSA measurements for management of watchful waiting are useful.16 Because we have found previously that both the amplitude and the relative velocity of PSA were important for outcome after definitive radiation treatment of localized prostate carcinoma 17 or during treatment for hormone refractory prostate carcinoma,18, 19 we have studied the combination of these two dynamic PSA factors in a group of 94 Japanese men observed expectantly, and our results are discussed below.

METHODS

  1. Top of page
  2. Abstract
  3. METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

The patients of this study comprised 94 men seen at Kitasato University Hospital and with prostate carcinoma diagnosed between April 1991 and December 2000. All were observed prospectively by one physician (S.E.) with initial expectant management before any treatment was given. Their diagnoses was established from histologic examination of six sextant biopsies or transurethral resections, and the histology was reviewed and graded according to the Gleason system by one pathologist (S.K.).20 Patients were staged according to the 1997 revised TNM classification21 and observed with serial measurements of PSA conducted as previously reported.22 None received either endocrine therapy or chemotherapy during the time of our study, and all gave their written informed consent. Other details about our study patients are given in Table 1.

Table 1. Characteristics of Study Population
Characteristicn
  1. PSA: prostate specific antigen.

Age (yrs)
 Median75
 Range53–87
T classification
 149
 238
 37
Gleason score
 < 630
 629
 > 635
No. of PSA measurements
 Median10
 Range2–28
Mos of follow-up
 Median32
 Range1.6–118
No. with local outcome during study time10
No. with systemic outcome during study time6
Status at last follow-up
 Died of other causes10
 Treated for prostate carcinoma7
 Followed expectantly77
 Died of prostate carcinoma0

Because none of our study patients died of progressive prostate carcinoma, we chose two classes of clinical endpoints that could be obtained independently from the values of PSA.22 The first was local, and it consisted of a significant and progressive increase in size of the prostate, specifically, an increase of greater than or equal to 25% by digital rectal examination or of greater than 50% volume by ultrasound. The second outcome was systemic, specifically, evidence of new bone abnormalities by a positive bone scan or blastic lesions as seen on skeletal radiographs. Bone scans were repeated annually.

We used a log-linear regression model to analyze the dynamics of PSA measured serially during expectant management, because the time course of PSA often has followed an exponential model.13, 17, 23, 24 Specifically, we fit the equation:

  • equation image(1)

to the data of each patient. Here, y symbolizes PSA, and parameter a is the intercept and reflects the overall amplitude of PSA. Parameter b is the slope and also has been called the relative velocity of PSA, because it is the derivative with respect to time divided by the PSA.17–19 Specifically, in calculus b is given by:

  • equation image(2)

i.e., the PSA velocity relative to the value of PSA. For clarity, in the remainder of the article we symbolize these, respectively, as am and rv. Parameter rv also relates directly to DT by:

  • equation image(3)

We used the general linear model25 to analyze the relations between parameters am and rv and other factors, and we used the Cox proportional hazards model to analyze the relations between time to our clinical outcomes and a variety of factors, including parameters am and rv26 Here, time was measured from the point of the last PSA measurement used in curve fitting to observed outcome or, in the case of censored patients, to last time of follow-up. Unless otherwise specified, we used continuous variables continuously. For this study, follow-up time was terminated when patients developed an outcome, died, or were treated for prostate carcinoma, and if they did not have an observed outcome they were considered censored. All analyses were conducted with the S-PLUS software (MathSoft, Inc., Seattle, WA).

RESULTS

  1. Top of page
  2. Abstract
  3. METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

Figure 1 illustrates the serial PSA values of four typical patients in our study. The points are the observed values, and the lines indicate the fit obtained from the log-linear model of Equation 1. The two patients depicted in the upper two plots did not progress, whereas the two in the lower plots progressed. In general, we found that the log-linear model accurately reflected both the level and trend of PSA over time, and Figure 2 illustrates this further for all the patients and all the values of PSA. This plot compares the observed values on the horizontal axis with the fitted values from the model on the vertical axis, and the line indicates where perfect fit should occur. The proximity of the points to the observed line of perfect fit indicates that for most patients and for most values of PSA the model fits the data well. Furthermore, the residuals from the fit (expressed in terms of log[PSA]) were approximately normally distributed.

thumbnail image

Figure 1. Plots of PSA (ng/mL) versus time of follow-up for four patients. The points are the observed values of PSA, and the lines provide the fit from the log-linear model, i.e., exp(am + rv ∗ time). The top two plots are for two patients who did not experience progressive tumor during follow-up, and the bottom two plots are for patients who experienced local or systemic outcomes. PSA: prostate specific antigen.

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thumbnail image

Figure 2. Plot of observed (horizontal axis) values of PSA versus the values of PSA obtained from the fit of the log-linear model (vertical axis) of Equation 1 (i.e., exp[am + rv ∗ time]). The plot includes all values of PSA for all patients and all times of measurement, and the line indicates where perfect agreement between observed and fitted values should occur. PSA: prostate specific antigen.

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The values of parameter am obtained from the fits of Equation 1 to the data ranged from −0.476 to 5.53 (median, 2.04), and in units of nanograms per milliliters these values correspond to a range of 0.6 to 252 (median, 7.7). The values of rv (in units of 1/months) ranged from −0.153 to 0.160 (median, 0.013). In general, the value of am was closely associated with the value of log(PSA) at the time of diagnosis (t statistic = 18; P ≅ 0 by linear regression), but rv was not related to PSA at diagnosis (P > 0.1 by linear regression). Parameters am and rv also appeared to be statistically independent from one another (P > 0.7 by linear regression). Parameter am related directly to T classification (P = 0.0005 by general linear model), but not to Gleason score (P > 0.2), and rv related significantly to both T classification (P = 0.047 by general linear model) and Gleason scores greater than or equal to 7 (P = 0.036). These relations are illustrated in Figure 3, which provides four box plots. The left two relate am to T classification (top left) and to Gleason score less than 7 versus greater than or equal to 7 (bottom left), and the right two relate rv to T classification (top right) and to Gleason score (bottom right). The top left box plot illustrates that there is an increase in am with increasing T classification, a result that is not surprising considering that am is closely tied to log(PSA) at diagnosis. Although the bottom left box plot suggests there is a mild elevation of am with Gleason scores greater than or equal to 7, after control for T classification this effect was not significant (P > 0.2). The two right box plots demonstrate that rv related only slightly to increasing T classification and to higher grade.

thumbnail image

Figure 3. Four box plots demonstrating the relations between either am or rv and T classification and grade. The left box plots are for am, and the right ones are for parameter rv. The top box plots are for T classification, and the bottom ones are for grade. Each box plot provides the 25th to 75th percentiles (dark rectangle), median value (white bar), and extreme values (top and bottom lines) for either am or rv according to values of T classification or binary category of grade. These relations also were examined using a general linear model (see “Results”).

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As Table 1 indicates, we observed only 16 patients who developed either a local or systemic outcome. Ten developed a significant increase in the size of their prostate, and 6 developed boney lesions. Because of this low number and to study the effect of parameters am and rv further, we pooled these two categories into one outcome for the next set of analyses, which appear in Tables 2 and 3. Table 2 provides the results of four Cox model analyses of time to first (either local or systemic) outcome, and each used a different set of covariates. For example, whereas the first Cox model used the three variables of log(PSA) at the time of diagnosis, T classification, and grade (i.e., Gleason score < 7 vs. ≥ 7), the second model added PSA DT to these three. The listed likelihood ratios provided us measures of how well each of these models related to the pooled outcome, because, in general, models that provide the most information about outcome are those with highest likelihood ratios. (Models with more variables generally also have higher likelihood ratio than ones with fewer variables.) Thus, the model using the three variables of T classification, grade, and PSA yielded a likelihood ratio of 9.9, and PSA DT failed to improve on this model, because the likelihood ratio remained at 9.9. In fact, we found that regardless of whether PSA DT was used as a continuous or as a binary variable, it did not significantly relate to our pooled outcome. The fourth model using am and rv provided the highest likelihood ratio.

Table 2. Cox Models of Time to Local or Systemic Outcomea
Variables includedModel likelihood ratio
  • PSA: prostate specific antigen.

  • a

    PSA was the value at the time of diagnosis. Parameters am and rv come from the fitted log-linear model, and rv is the relative velocity of PSA (see “Methods”). Doubling time was used as a binary variable, i.e., ≥ 24 mos vs. < 24 mos. Grade was used as Gleason score < 7 vs. ≥ 7.

Log(PSA), T classification, grade9.9
Log(PSA), T classification, grade and doubling time9.9
rv, log(PSA)14.2
rv, am16.2
Table 3. Cox Model of Time to Local or Systemic Outcomea
VariableCoefficientStandard errorP value
  • a

    The analysis refers to a model with am and rv already included, that is, the individual P values for the other listed variables refer to their values when a and rv are already present in the model. The standard error is for the coefficient. Grade was defined here as Gleason score < 7 vs. ≥ 7.

am0.9510.270.0005
rv15.65.50.0048
T classification0.3
Grade0.72

Table 3 provides the model coefficients, standard errors of coefficients for a Cox model using am and rv, and individual P values for two variables when used in combination with am and rv, i.e., when added to the fourth model of Table 2. The results indicate that whereas both am and rv were significantly associated with the pooled outcome, none of the remaining parameters provided significant additional information. The coefficients for am and rv also suggested a practical way to combine the information from parameters am and rv. If we form a hazard score (HS) defined as:

  • equation image(4)

then the design and results of the Cox model suggest that HS should relate closely to outcome. In our study population, the calculated HS described a continuum ranging from −0.8 to 6.2 (median, 2.1). To illustrate its impact, we divided our patients into three categories according to their values of HS, namely, HS less than 1, HS = 1 to 3, and HS greater than 3. Figure 4 shows three Kaplan–Meier plots of the probability of being free of either outcome versus the entire time of follow-up, and it demonstrates that those with HS less than 1 had a very low incidence of either local or systemic outcome in comparison to the groups with HS between 1 and 3 or HS greater than 3.

thumbnail image

Figure 4. Kaplan–Meier plot of the probability of being free of either local or systemic outcome versus time of follow-up in months for our study population. The three curves correspond to those with hazard scores (HS) of < 1, 1–3, and > 3, with HS defined by Equation 4.

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DISCUSSION

  1. Top of page
  2. Abstract
  3. METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

Despite controversies about the use of PSA to screen for early prostate carcinoma or the treatment of these tumors with radical surgery or radiation, there is consensus that prostate carcinoma comprises a spectrum in biology and outcomes. Some prostate carcinomas are unlikely to progress or cause significant problems during the remaining lives of some men; some are likely to progress so rapidly that local treatment will be ineffective; and others sit between these extremes. Furthermore, despite criticisms of how particular thresholds of PSA have been used as decision points, few would contest that PSA relates closely to the presence and amount of prostate carcinoma. Thus, it would seem obvious that we are mandated to somehow maximize the information that PSA can provide about where tumors fit into the biologic spectrum. In this regard, serial measurements of PSA over time appear to provide more information than just a single measurement. Furthermore, it seems obvious that there might be at least two roughly independent pieces of information that could come from serial measurements of PSA. Specifically, when one measures a single value of PSA all one has are units of concentration, but when one measures values of PSA over time, there are two sets of units available—concentration and time. Thus, a priori one could expect there might be two pieces of critical information produced. The log-linear model now demonstrates that serial measurements of PSA provide two types of information, namely, the amplitude and the relative velocity. Whereas previous investigators studying serial levels of PSA have neglected the amplitude of PSA in favor of PSA velocity or PSA DT, we have found that the amplitude adds information. The amplitude has units of log(ng/mL), and the relative velocity has units of 1/time, and in our analyses both related significantly to clinical outcome. Together they formed a composite HS whose spectrum matched our expectation for a spectrum in biology of prostate carcinoma.

Although our study population was small and its outcomes limited, we believe our results sufficient to demonstrate that during watchful waiting the log-linear model can be an effective tool for capturing the information provided by serial values of PSA. The model fits the data well, and it demonstrated that both the amplitude and relative velocity of PSA are important to outcomes. Whereas PSA DT also may relate to outcome, this measure of the change in PSA with time is not sufficient, because it does not include the amplitude of PSA. In other words, two patients with a DT of 50 months but with different starting levels of 5 and 10 ng/mL, respectively, are likely to have different outcomes. Doubling time and relative velocity are directly related to one another by Equation 3, and the relation between PSA velocity and relative velocity is given by Equation 2. Among these three, we have consistently found the relative velocity to be the most helpful. One advantage that relative velocity of PSA has over DT is its ability to describe both increasing and decreasing values of PSA. Thus, whereas DT must be truncated when PSA values decrease, the relative velocity simply moves continuously from positive to negative values. Because the HS offers a way to combine the information from amplitude and relative velocity of PSA, it may be helpful and we encourage those collecting data for randomized trials regarding watchful waiting11, 12 to consider composite scores of the amplitude and relative velocity of PSA.

REFERENCES

  1. Top of page
  2. Abstract
  3. METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES