Free PSA and Prostate Cancer

1 Overview

I am interested in biomarkers for prostate cancer. Free PSA (aka fPSA, also expressed as %fPSA) appears to be the most useful and affordable extension beyond the standard PSA (aka total PSA, tPSA) test. 4KScore looks like the next most useful extension, but costs $760 and I have been unable to find details of the calculation. This paper provides a good comparison of models using the first two, three, and five of age, tPSA, fPSA, iPSA, and hK2. It would be interesting to compute AIC (etc.) for the various models to assess the benefit of adding the variables vs. the “cost.”

Predicting High-Grade Cancer at Ten-Core Prostate Biopsy Using Four Kallikrein Markers Measured in Blood in the ProtecT Study (Bryant et al. 2015)

This paper provides a good overview of various biomarkers. But even though they reference the (Oto et al. 2020) paper discussed below they don’t appear to discuss its best model. Which is odd.
Modern biomarkers in prostate cancer diagnosis

This page provides a good overview of prostate cancer screening even if I think they are overly negative on various tests. Remember, a test result is not a treatment decision. Do not blame a poor outcome from the latter on the former.
Prostate Cancer Screening (PDQ®)–Health Professional Version

2 Free PSA Lab Test

The following two tests are available from LEF. Note that the tests go on sale for 25% off every spring.

Prostate Specific Antigen (PSA) Free with Total Ratio Blood Test $61

Prostate Specific Antigen (PSA) Blood Test $31

Also note that the PSA test is available as part of a number of LEF panels. As far as I know, the Free PSA test is only available standalone.

3 PSA Trends

PSA trends are another possible easily available biomarker. Summarized in this section, but will not be discussed further in this document.

PSA velocity is the most common form of PSA trend used. Here is a sample paper.
Distribution of PSA Velocity by Total PSA Levels: Data From the Baltimore Longitudinal Study of Aging (Loeb et al. 2011)

Here are some high level PSA velocity guidelines
PSA velocity is an important concept. To calculate velocity, at least 3 consecutive measurements on specimens drawn over at least 18-24 months should be used. Guidelines from the NCCN suggest that PSA velocity be considered in the context of the PSA level. [63] The following PSA velocities are suspicious for cancer:
- PSA velocity of 0.35 ng/mL/y, when the PSA is ≤2.5 ng/mL
- PSA velocity of 0.75 ng/mL/y, when the PSA is 4–10 ng/mL

Here is a more sophisticated look at PSA trends.
Prostate-Specific Antigen Trends Predict the Probability of Prostate Cancer in a Very Large U.S. Veterans Affairs Cohort (Karnes et al. 2018)
Some details.

The overall PSA trend equation, PSA(t), is the sum of the no cancer baseline PSA and estimated PSA from cancer:
PSA(t) = PSAn + PSAc(t0) * EXP(PSAgr * (t−t0))
PSA velocity (PSAV) is often defined as the annual rate of change in PSA. For our trend equation, PSAV at any point in time is defined as the slope of the PSA trend:
PSAV(t) = PSAgr * PSAc(t)

Description of their variables.
For each man with two or more PSA tests prior to biopsy, we estimated consistent underlying PSA trends using the methods described above and determined values for five descriptive variables: last PSA before biopsy (PSA), growth rate in PSA from cancer above a baseline (PSAgr), PSA variability around the trend (PSAvar), number of PSA tests (Tests), and time span of the tests (Span).

Although interesting, on the whole I found their results unimpressive. The highest AUC for their PSA trends model was 0.67. It may be useful to consider their results as heuristics though (e.g. high variability and slow trend growth are negative cancer indicators).

Discussion of that research: University students bring big data to cancer screening in start-up launch

The last author of that paper used to have a website:
https://web.archive.org/web/20150509090134/http://www.prostatesmart.info/

4 Free PSA and Prostate Cancer Probability

Most of this document will be focused on the following paper. It is notable for fully presenting a model which can be evaluated based on standard lab test results.

A predictive model for prostate cancer incorporating PSA molecular forms and age (Oto et al. 2020)

Abstract
The diagnostic specificity of prostate specific antigen (PSA) is limited. We aimed to characterize eight anti-PSA monoclonal antibodies (mAbs) to assess the prostate cancer (PCa) diagnostic utility of different PSA molecular forms, total (t) and free (f) PSA and PSA complexed to α1-antichymotrypsin (complexed PSA). MAbs were obtained by immunization with PSA and characterized by competition studies, ELISAs and immunoblotting. With them, we developed sensitive and specific ELISAs for these PSA molecular forms and measured them in 301 PCa patients and 764 patients with benign prostate hyperplasia, and analyzed their effectiveness to discriminate both groups using ROC curves. The free-to-total (FPR) and the complexed-to-total PSA (CPR) ratios significantly increased the diagnostic yield of tPSA. Moreover, based on model selection, we constructed a multivariable logistic regression model to predictive PCa that includes tPSA, fPSA, and age as predictors, which reached an optimism-corrected area under the ROC curve (AUC) of 0.86. Our model outperforms the predictive ability of tPSA (AUC 0.71), used in clinical practice. In conclusion, The FPR and CPR showed better diagnostic yield than tPSA. In addition, the PCa predictive model including age, fPSA and complexed PSA, outperformed tPSA detection efficacy. Our model may avoid unnecessary biopsies, preventing harmful side effects and reducing health expenses.

Here is how they describe their model.

Our model substantially improves the predictive capacity of PCa compared to that of tPSA. It achieved an apparent AUC of 0.86 (95% CI: 0.83–0.89) and an optimism-corrected AUC of 0.86, compared to AUC of 0.71 (95% CI: 0.67–0.75) for tPSA (Fig. 1). The formula for predicting the probability (Pr) of PCa would be:
\[ {\rm{\Pr }}(PCa)=\frac{{e}^{-10.57+0.056\ast Age+3.116\ast \log (tPSA)-2.995\ast \log (fPSA)+0.268\ast \log (tPSA)\ast \log (fPSA)}}{1+{e}^{-10.57+0.056\ast Age+3.116\ast \log (tPSA)-2.995\ast \log (fPSA)+0.268\ast \log (tPSA)\ast \log (fPSA)}} \]

In Microsoft Excel format that equation is:
(EXP(-10.572+0.056Age+3.116LOG(tPSA,EXP(1))-2.995LOG(fPSA,EXP(1))+0.268LOG(tPSA,EXP(1))LOG(fPSA,EXP(1))))/(1+EXP(-10.572+0.056Age+3.116LOG(tPSA,EXP(1))-2.995LOG(fPSA,EXP(1))+0.268LOG(tPSA,EXP(1))LOG(fPSA,EXP(1))))

5 Exploring the Pr(PCa) Equation

That equation is not at all intuitive to me, so let’s create some plots in an attempt to convey appropriate intuition. Note that I have expressed %free PSA as a proportion 0-1 rather than a percentage 0-100%. That is a bit confusing.

5.1 Plots of Pr(PCa) vs. %free PSA and total PSA for a selection of ages

# Not sure if increasing Pr(PCa) for tPSA < 2 is realistic, so excluding

Pr_PCa <- function(Age, tPSA, pfPSA) (exp(-10.572+0.056*Age+3.116*log(tPSA)-2.995*log(pfPSA*tPSA)+0.268*log(tPSA)*log(pfPSA*tPSA))) /
  (1+exp(-10.572+0.056*Age+3.116*log(tPSA)-2.995*log(pfPSA*tPSA)+0.268*log(tPSA)*log(pfPSA*tPSA)))  

# Pr_PCa2 <- function(tPSA, pfPSA) Pr_PCa(Age, tPSA, pfPSA)
Pr_PCa2 <- function(r) Pr_PCa(Age, r[1], r[2])

# I don't see how to get axes labels in these plots.
# xlab and ylab did not work
# axis.title.x="Total PSA" did not work
  
Age <- 50
cf_func(Pr_PCa2, xlim = c(2, 10), ylim = c(0, 0.5), bar=TRUE, with_lines=TRUE, main="Pr(PCa) for Age=50 and %free PSA vs. total PSA")

Age <- 60
cf_func(Pr_PCa2, xlim = c(2, 10), ylim = c(0, 0.5), bar=TRUE, with_lines=TRUE, main="Pr(PCa) for Age=60 and %free PSA vs. total PSA")

Age <- 70
cf_func(Pr_PCa2, xlim = c(2, 10), ylim = c(0, 0.4), bar=TRUE, with_lines=TRUE, main="Pr(PCa) for Age=70 and %free PSA vs. total PSA")

Age <- 80
cf_func(Pr_PCa2, xlim = c(2, 10), ylim = c(0, 0.4), bar=TRUE, with_lines=TRUE, main="Pr(PCa) for Age=80 and %free PSA vs. total PSA")

5.2 Plots of Pr(PCa) vs. age and total PSA for a selection of %free PSA values

Be sure to notice the way the changing scales affect the color mappings. That can be confusing when comparing plots. In the previous series of plots all of the scales went from 0.0 to 1.0.

Pr_PCa2a <- function(r) Pr_PCa(r[1], r[2], pfPSA)

pfPSA <- 0.05
cf_func(Pr_PCa2a, xlim = c(50, 80), ylim = c(2, 10), bar=TRUE, with_lines=TRUE, main="Pr(PCa) for %free PSA=0.05 total PSA vs. Age")

pfPSA <- 0.10
cf_func(Pr_PCa2a, xlim = c(50, 80), ylim = c(2, 10), bar=TRUE, with_lines=TRUE, main="Pr(PCa) for %free PSA=0.10 total PSA vs. Age")

pfPSA <- 0.15
cf_func(Pr_PCa2a, xlim = c(50, 80), ylim = c(2, 10), bar=TRUE, with_lines=TRUE, main="Pr(PCa) for %free PSA=0.15 total PSA vs. Age")

pfPSA <- 0.20
cf_func(Pr_PCa2a, xlim = c(50, 80), ylim = c(2, 10), bar=TRUE, with_lines=TRUE, main="Pr(PCa) for %free PSA=0.20 total PSA vs. Age")

pfPSA <- 0.25
cf_func(Pr_PCa2a, xlim = c(50, 80), ylim = c(2, 10), bar=TRUE, with_lines=TRUE, main="Pr(PCa) for %free PSA=0.25 total PSA vs. Age")

6 Case Study

Here is a case study for an individual with a series of PSA tests, some with free PSA and some without. The idea is to illustrate how we might use these equations to assess the changing risk over time.

7 Prostate Cancer Genetics

This page provides a good overview. Genetics of Prostate Cancer (PDQ®)–Health Professional Version

https://impute.me/ has a polygenic risk score for prostate cancer.

Beyond that, genetics is beyond the scope of this document so not discussed further.

8 File History

File Initially created: Monday,July 19, 2021

File knitted: Mon Jul 19 20:38:16 2021

Bibliography

Bryant, Richard J., Daniel D. Sjoberg, Andrew J. Vickers, Mary C. Robinson, Rajeev Kumar, Luke Marsden, Michael Davis, et al. 2015. “Predicting High-Grade Cancer at Ten-Core Prostate Biopsy Using Four Kallikrein Markers Measured in Blood in the ProtecT Study.” JNCI: Journal of the National Cancer Institute 107 (7). https://doi.org/10.1093/jnci/djv095.

Karnes, R. Jeffrey, F. Roy MacKintosh, Christopher H. Morrell, Lori Rawson, Preston C. Sprenkle, Michael W. Kattan, Michele Colicchia, and Thomas B. Neville. 2018. “Prostate-Specific Antigen Trends Predict the Probability of Prostate Cancer in a Very Large U.s. Veterans Affairs Cohort.” Frontiers in Oncology 8 (August). https://doi.org/10.3389/fonc.2018.00296.

Loeb, Stacy, H. Ballentine Carter, Edward M. Schaeffer, Anna Kettermann, Luigi Ferrucci, and E. Jeffrey Metter. 2011. “Distribution of PSA Velocity by Total PSA Levels: Data from the Baltimore Longitudinal Study of Aging.” Urology 77 (1): 143–47. https://doi.org/10.1016/j.urology.2010.04.068.

Oto, Julia, ’Alvaro Fern’andez-Pardo, Montserrat Royo, David Herv’as, Laura Martos, C’esar D. Vera-Donoso, Manuel Martand Mary J. Heeb, Francisco España, Pilar Medina, and Silvia Navarro. 2020. “A Predictive Model for Prostate Cancer Incorporating PSA Molecular Forms and Age.” Scientific Reports 10 (1). https://doi.org/10.1038/s41598-020-58836-4.