Modeling Prostate Cancer Detection Probability, with Applications Robert Serfling1 University of Texas at Dallas

Joint Statistical Meetings 2007

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www.utdallas.edu/∼serfling Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Background and Overview The Setting and Two Major Problems What Probability Modeling for Cancer Detection Provides Building and Using a Probability Model for Cancer Detection Building the Model – Some Geometry and Probability Using the Model – Some Plots Extended Model for Number of Distinct Biopsy Cores Hit Probability Distribution of Number of Distinct Cores Hit Application: a Posterior Distribution on Tumor Volumes Allocation of Biopsy Cores to Specified Zones of Prostate Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

The Setting and Two Major Problems What Probability Modeling for Cancer Detection Provides

Prostate Cancer – a Challenging Situation I

In the U.S., over 200,000 new cases diagnosed each year.

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Difficult decisions about treatment options and lifestyle changes – affecting both patients and family members.

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When detected and treated early, a cure rate of over 90%.

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Many cases missed, however, even by biopsies – false negative rates for first biopsies from 15% to 30%.

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Numerous studies investigating causes and treatments.

Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

The Setting and Two Major Problems What Probability Modeling for Cancer Detection Provides

The Urologist Wants Efficient Biopsy Sessions I

High detection probability, for example at least 90%.

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As few biopsy cores as possible, to limit risks of severe side effects.

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Assessment of total cancer present as well as tumor nodule sizes, to judge clinical significance before treatment selection.

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A good rule for choosing the number of biopsy cores.

Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

The Setting and Two Major Problems What Probability Modeling for Cancer Detection Provides

An Unexpected Finding in the Recent Prostate Cancer Prevention Trial (PCPT), 1994-2003 I

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The PCPT [Thompson, I. M. et al. (2003). The influence of finasteride on the development of prostate cancer. New Engl. Med. J. 349 211-220.] studied whether finasteride (generic drug) prevents or inhibits prostate cancer. Finasteride already successful treats benign prostate hyperplasia – it reduces prostate size. The finasteride arm of the PCPT exhibited a lower overall rate but, however, a higher rate of high-grade cancers. The possibility that finasteride induces high-grade prostate cancer has become a current major focus. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

The Setting and Two Major Problems What Probability Modeling for Cancer Detection Provides

Basic Problem: Finding a Needle in a Haystack I

Key idea: a tumor is easier to detect in a smaller prostate.

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How to quantify this usefully?

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A probability model has been developed [Serfling, R. et al. (2007). Quantifying the impact of prostate volumes, numbers of biopsy cores, and 5alpha-reductase inhibitor therapy on the probability of prostate cancer detection using mathematical modeling. J. Urology 177

], giving prostate cancer detection probability as a function of

2352-2356.

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prostate size number of biopsy cores assumed tumor nodule volumes. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

The Setting and Two Major Problems What Probability Modeling for Cancer Detection Provides

Major Applications of the Probability Model 1. Guides a urologist’s choice of number of biopsy cores, in a biopsy session for a patient with a given prostate volume and a given range of possible tumor volume. 2. Quantifies how prostate volume reduction by drugs increases cancer detection probability – I

Fully explains by volume reduction the higher rate of high-grade cancer for finasteride in the PCPT, yielding an adjusted rate lower than for placebo.

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Indicates a possible new role for finasteride and related drugs when prescribed over time – enhanced detection probability in case a biopsy is suggested and performed. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Building the Model – Some Geometry and Probability Using the Model – Some Plots

Finding a Single Tumor by a Single Biopsy Core I

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Detection of a spherical tumor by a cylindrical core occurs if the tumor center falls in the region extending the core in all directions by a distance equal to the tumor radius. For core of length ` and radius s, and tumor of volume v , the extended region has “effective core volume” Veff (v ) = π`s 2 + 2(3/4π)1/3 πs(` + s) v 1/3 + (3/4π)2/3 π(` + πs) v 2/3 + v .

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For prostate volume V , the detection probability is   Veff (v ) min ,1 . V Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Building the Model – Some Geometry and Probability Using the Model – Some Plots

Detection with n Cores and k Tumor Nodules I

For n biopsy cores with disjoint effective core regions relative, individually, to each of k tumors with volumes v1 , . . . , vk in a prostate of volume V , the detection probability P depends on n and V through R = n/V : k Y P = 1 − (1 − R Veff (vi ))+ . i=1

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A key consequence: the number of cores needed for specified P increases proportionally with V . For example, for given P, doubling V requires doubling n. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Building the Model – Some Geometry and Probability Using the Model – Some Plots

Detection Probability as function of R, for Selected Tumor Sizes and Numbers of Nodules I

It is instructive to examine plots of detection probability P as function of R = n/V , comparatively for selected cases of total tumor size and number of nodules: Plot A total tumor volume 3 cc, with 1, 2, and 4 nodules Plot B total tumor volume 3 cc, with 4 and 8 nodules Plot C total tumor volume 1 cc, with 1, 2, and 4 nodules.

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Supported by studies, we assume: for k tumor nodules the volumes decrease by halves from largest to smallest. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Building the Model – Some Geometry and Probability Using the Model – Some Plots

Detection Probability P as Function of Ratio R of Number of Cores to Prostate Gland Volume, for Total Tumor Volume 3 cc 1.0

P 0.5

0.0 0.0

0.1

0.2

R

3 cc tumor, 1 nodule 3 cc tumor, 2 nodules 3 cc tumor, 4 nodules

Plot A – note closeness of the 2 and 4 nodule curves. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Building the Model – Some Geometry and Probability Using the Model – Some Plots

Detection Probability P as Function of Ratio R of Number of Cores to Prostate Gland Volume, for Total Tumor Volume 3 cc with 4 and 8 Nodules 1.0 0.75 P 0.5 0.25 0.0 0.0

0.05

0.1

0.15

0.2

R

3 cc tumor, 4 nodules 3 cc tumor, 8 nodules

Plot B – note overlapping of the 4 and 8 nodule curves Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Building the Model – Some Geometry and Probability Using the Model – Some Plots

Detection Probability P as Function of Ratio R of Number of Cores to Prostate Gland Volume, for Total Tumor Volume 1 cc 1.0

P 0.5

0.0 0.0

0.2

0.4

R

1 cc tumor, 1 nodule 1 cc tumor, 2 nodules 1 cc tumor, 4 nodules

Plot C – like Plot A (3 cc) but over wider range of R Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Building the Model – Some Geometry and Probability Using the Model – Some Plots

What the Practicing Urologist Faces with a Patient I

The following present increasing difficulty of detection: (i) tumor volume T = 3 cc, with k = 4 nodules (ii) tumor volume T = 1 cc, with k = 4 nodules (iii) tumor volume T = 0.5 cc, with k = 1 nodule.

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A urologist facing these possibilities with a patient might seek the minimum number of biopsy cores achieving, for example, detection probability P = 0.90.

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The first step is to find the values of R = n/V that achieve P = 0.90, for (i), (ii), and (iii), respectively.

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These are found (next plot) to be 0.14, 0.30, and 0.40. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Building the Model – Some Geometry and Probability Using the Model – Some Plots

Detection Probability P as Function of Ratio R of Number of Cores to Prostate Gland Volume, for Selected Total Tumor Volumes 1.0

P 0.5

0.0 0.0

0.2

0.4

R

3 cc tumor, 4 nodules 1 cc tumor, 4 nodules 0.5 cc tumor, 1 nodule

Plot D – use it like finding a quantile from a cdf Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Building the Model – Some Geometry and Probability Using the Model – Some Plots

What the Urologist Can Do with the Value R I

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For a patient with a 20 cc prostate, the required number n of cores for 90% detection probability ranges from n = R × V = 0.14 × 20 = 3 through 0.30 × 20 = 6 to 0.40 × 20 = 8. At least 3 and at most 8 cores are needed. On the other hand, for a patient with a 50 cc prostate, the required n ranges from 0.14 × 50 = 7 through 0.30 × 50 = 15 to 0.40 × 50 = 20. At least 7 and possibly as many as 20 cores are needed. The model calibrates the obvious principle that finding a tumor of given size is harder for a larger prostate. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Probability Distribution of Number of Distinct Cores Hit Application: a Posterior Distribution on Tumor Volumes Allocation of Biopsy Cores to Specified Zones of Prostate

What Else Might the Practicing Urologist Want? I

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The practicing urologist acquires from a biopsy session not only whether cancer is detected but also the number D of distinct cores hit. Interpretion of D requires its probability distribution. For n cores in prostate volume V and k tumor nodules with volumes v1 , . . . , vk ,
 X k y −1 n d−1
P(H = y ), P(D = d) = d y =d n+yy −1 for d = 0, . . . , min{k, n}, with H the number of hits in all by the k nodules and P(H = y ) given in the next slide. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Probability Distribution of Number of Distinct Cores Hit Application: a Posterior Distribution on Tumor Volumes Allocation of Biopsy Cores to Specified Zones of Prostate

Lemma: Distribution of Total Number of Hits I

The total number H of hits of n cores by k nodules with volumes v1 , . . . , vk has probability distribution P(H = y ) = 8 k Y` > ´+ > > 1 − R Veff (vj ) , > >
> > > > :

Y

[1 − (1 − R Veff (vi ))+ ]

Cy ,k i∈{i1 ,...,iy }

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The sum

P

Cy ,k

Y

` ´+ 1 − R Veff (vj ) ,

1 ≤ y ≤ k.

j∈{i1 ,...,iy }k

is over combinations {i1 , . . . , iy } from

{1, . . . , k}, and {i1 , . . . , iy }k denotes the complement of {i1 , . . . , iy } in {1, . . . , k}. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Probability Distribution of Number of Distinct Cores Hit Application: a Posterior Distribution on Tumor Volumes Allocation of Biopsy Cores to Specified Zones of Prostate

A Bayesian Approach is Okay with a Valid Prior I

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As stated earlier, the urologist seeks ... An assessment of total cancer present as well as of tumor nodule sizes, to judge clinical significance prior to treatment selection. That is, estimate the number k of tumor nodules and their volumes v1 , . . . , vk . Bayesian Approach. Obtain a prior distribution on a patient’s (k, v1 , . . . , vk ) and use with the probability model for number of distinct cores hit, obtaining a posterior distribution on (k, v1 , . . . , vk ). Development of suitable priors for various sets of patient covariates is in progress via meta-analysis of past studies. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Probability Distribution of Number of Distinct Cores Hit Application: a Posterior Distribution on Tumor Volumes Allocation of Biopsy Cores to Specified Zones of Prostate

Extended Model for Partitioning of the Prostate

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The model has been developed in greater detail – to handle partitioning of the prostate into zones.

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A common setting is allocation of specified numbers n1 and n2 of biopsy cores to the peripheral and transition zones of the prostate, having volumes V1 and V2 .

Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Extension to Include Additional Available Data I

The practicing urologist conducting a biopsy session acquires not only the number of distinct cores hit but also the respective volumes of tumor found in these cores.

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Interpretation of this extended data set calls for the joint probability distribution of the number of cores hit and the respective volumes of tumor found, given n cores, prostate volume V , and k tumor nodules with volumes v1 , . . . , vk .

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This extension will yield a more refined application of Bayesian analysis to obtain a posterior on (k, v1 , . . . , vk ). Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

The Practical Role of Probability Modeling I

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Elementary probability modeling in collaboration with urologists can have high scientific impact. The statistician must learn the science and what urologists are wanting. Retrospective studies can yield data-based models for detection probability relative to prostate volume. Data-based and probability-based models complement and validate each other. The value of the probability-based model is generality beyond any particular data set. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

References I I

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This talk in full (pdf): www.utdallas.edu/∼serfling Serfling, R. et al. (2007). Quantifying the impact of prostate volumes, numbers of biopsy cores, and 5alpha-reductase inhibitor therapy on the probability of prostate cancer detection using mathematical modeling. J. Urology 177 2352-2356. I Editorial on this paper in same issue: Kibel, A. S. (2007). Optimizing prostate biopsy techniques. J. Urology 177 1976-1977. (Links to these are at www.utdallas.edu/∼serfling) Thompson, I. M. et al. (2003). The influence of finasteride on the development of prostate cancer. New Engl. Med. J. 349 211-220. Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati

Outline Background and Overview Building and Using a Probability Model for Cancer Detection Extended Model for Number of Distinct Biopsy Cores Hit Extended Model Including Tumor Volumes in Cores Hit Concluding Comments Acknowledgements

Acknowledgements

The speaker is very grateful to G. L. Thompson and Roger Rittmaster for extremely thoughtful and helpful remarks and for generous encouragement. Support by grants from GlaxoSmithKline is also greatly appreciated.

Robert Serfling

Modeling Prostate Cancer Detection Probability, with Applicati