Lognormal distributions of PSA levels: evaluation of PSA relevant parameters

Introduction:
Using the lognormal PSA concentration distribution to separate BPH from cancer

Probability Distributions and Receiver Operating Characteristic

• in the population k with cancer and
• in the population g without cancer,

f is the corresponding probability distribution. By definition the area under f (between -infinity and +infinity) is 1.

Source: Wikipedia

The Receiver Operating Characteristic (ROC) is a parametric curve with the total PSA concentration psa as parameter ("threshold"). Points on the ROC curve have the coordinates {TP[psa], FP[psa]}, where TP and FP are the areas under the normal distribution f[x, , ] to the right of the threshold x = ln[psa].

• TP[psa] is the corresponding area under f_k, the distribution for the population with cancer. ("true positive fraction").
• FP[psa] is the corresponding area under f_g, the distribution for the population without cancer ("false positive fraction").

Expressing the ROC in terms of normal distributions

• FP[x, , ] = 1 - Cdf[x, , ],
  where in the Cdf the distribution f_g is used,
• TP[x, , ] = 1 - Cdf[x, , ],
  where in the Cdf the distribution f_k is used.

Since {FP, TP}, and thus the ROC, is the primary information being generated from the medical data (i.e. from PSA measurements and biopsies), the lognormal distributions f_g and f_k are the result of a numerical fit. Figs. 1 and 3 show how this fit is done, and the result, distributions f_g and f_k, are shown in the shaded rectangle in the lower left of these figures.

In their Table 3 Punglia et al. specify the parameters of the resulting distributions ( = mean ln[PSA], = standard deviation). They further subdivided the populations k and g into 2 subpopulations depending on the findings of a digital rectal exam (DRE): DRE normal and DRE abnormal.

Figure 1. Connection of ROC curves (upper right corner) with lognormal-distributed PSA concentrations (shaded area in lower left corner). In the shaded area distributions for the subpopulation

• with normal (abnormal) DRE are shown in the upper (lower) half of the plot,
• without (with) cancer are represented by the solid (dashed) curves g (k).

Fig. 1 shows: The presence of BPH

• has little effect on the shapes of the solid and dashed curves and their relative positions, and thus on the ROC curve,
• shifts the solid-dashed curve ensemble towards higher PSA levels, thus changing the PSA threshold labels on the ROC curve.

In this sense, the effect of BPH on the PSA concentration is superimposed on the effect of cancer, rather than masking its effect. (Contrary to that, in Nuclear Magnetic Resonance Imaging (MRI) BPH and/or chronic prostatitis are known to mask cancer [Shukla-Dave A et al., "Chronic Prostatitis: MR Imaging and ¹H MR Spectroscopic Imaging Observations - Initial Observations", Radiology. 2004 Jun;231(3):717-24 (in cache)]

Using the lognormal PSA concentration distribution to separate mechanical prostate irritation from cancer

The level of sexual activity has been observed to correlate with tPSA concentrations. The corresponding tPSA elevation can be of about the same size as Punglia et al. showed for cancer. Reducing sexual activity will not readily reduce tPSA levels. tPSA relaxation can take weeks. This exceeds the urologists' recommendation to their patients to be sexually abstinent for several days before PSA measurement.

It is proposed here that the influence of sexual activity on the PSA test be treated similarly as the one of BPH, i.e. as a shift of the lognormal distributions g and k. In the final part of this draft two possibilities will be discussed:

1. If sexual activity acts similarly as BPH, its effect on the PSA test can be reduced by adjusting the test threshold.
2. If sexual activity leaves the location of the PSA concentration probability distribution for cancer unchanged the PSA test cannot discriminate between sexual activity and cancer.

Data

Figure 2: left diagram: Oscillations of the PSA concentration caused probably by variations of sexual activity (data). Data points are arbitrarily connected by straight lines. Vertical bars: heavy bar on curve near month 50 or 75: length = 2 x 0.2 ng/mL = 2 x standard deviation, as computed from coefficient of variation of PSA concentration measurements (less than 3 %) given by Labor 28, Angaben zur Messunsicherheit. 2 lines close to each other near month 90 mark the times at which Magnetic Resonance Imaging (MRI) and Fine Needle Aspiration Biopsy (FNAB) were performed: MRI and MRSI done at time marked by left line. MRI result: no prostate cancer. FNAB (with DNA-ploidy analysis) done at time marked by right line. Biopsy result: no prostate cancer. Gray line with slope 0.75 ng/(mL yr) represents the threshold PSA increase over time after Coakley FV, Chen I, Qayyum A, Westphalen AC, Carroll PR, Hricak H, Chen MH, Kurhanewicz J. Validity of prostate-specific antigen as a tumour marker in men with prostate cancer managed by watchful-waiting: correlation with findings at serial endorectal magnetic resonance imaging and spectroscopic imaging. BJU Int. 2007 Jan;99(1):41-5 (more). right diagram: Same as left diagram, except for higher temporal resolution and graphical representation of frequency of sexual activity or extended bicycle riding. Vertical (blue) lines mark days with sexual activity or extended bicycling (i.e. no bicycing/sexual activity on unmarked days). "no activity data" means: during this period sexual activity/bicycling was not recorded.

Disregarding the rise of the mean PSA concentration with age, the PSA concentrations in Fig. 2 will be approximated by a mean of 5 ng/mL about which it varies by roughly a factor of 2, i.e. from 2.5 ng/mL to 10 ng/mL. This will be used to construct the ROC curves stratified by sexual activity.

Alternatives

1. leaves the ensemble of both distributions g and k unchanged and shifts its position along the ln[PSA] axis towards higher PSA values, similarly as BPH does (Fig. 3a),
2. leaves both the position and shape of distribution k unchanged, but shifts distribution g towards higher ln[PSA] values while leaving its shape unaltered (Fig. 3b).

In Figs. 3 the shape of the lognormal PSA distributions are the ones published by Punglia et al. for normal DRE. Increased sexual activity will be taken into account by shifting corresponding distributions upwards by the mentioned factor of 2. (Note: ln2 = 0.7)

Alternative 1

• does not change the shapes and positions of the two lognormal distributions g and k relative to each other, thus leaving the "g-k ensemble" unaltered,
• does shift the "g-k ensemble" by a distance lnPSA = 0.7 to the right, which means a PSA level shift by a factor of 2.

Figure 3a Construction of ROC curve (upper right corner) from normal-distributed natural logarithms of PSA concentrations (f curves in shaded area, g means "without cancer", k means "with cancer"). Grey numbers on ROC curve are the PSA thresholds (units: ng/mL), numbers above the ROC curve give the thresholds for the average population, numbers below specify the thresholds for the subpopulation with increaed sexual activity (Notation).

Parameters used:

• The mean lnPSAs of the average population (with and without cancer) are chosen equal to the corresponding means given by Punglia et al. for the DRE normal population.
• The mean lnPSA of the "increased sex population" (with and without cancer) is chosen 0.7 higher than the corresponding mean given by Punglia et al. for the DRE normal population.

Then the

1. shape of the ROC curve -meaning the test performance- is independent from sexual activity (thus the ROC curves for average and increased sex coincide), and the
2. PSA threshold tags along the ROC curve (grey numbers) increase with sexual activity, in our example by a factor of 2.

• to the test threshold x = 3.3 ng/mL for the average population,
• to the test threshold 2 x = 6.6 ng/mL for the subpopulation with increased sex.

Result: For the subpopulation with increased sexual activity the PSA threshold needs to be raised by a factor of 2 as compared to the normal population, meaning in our example: the test gives the same result for a member of the subpopulation having a PSA level above 6.6 ng/ml as for a member of the normal population having a PSA level above 3.3 ng/ml.

Alternative 2

• leaves both the position and shape of distribution k unchanged, but
• shifts distribution g by a distance lnPSA = 0.7 towards higher ln[PSA] values while leaving its shape unaltered.