Optimal Design of Biomarker-Based Screening Strategies for Early Detection of Prostate Cancer

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1 Optimal Design of Biomarker-Based Screening Strategies for Early Detection of Prostate Cancer Brian Denton Department of Industrial and Operations Engineering University of Michigan, Ann Arbor, MI October 13, 2014 Michigan Brian Denton Prostate Cancer Screening 1

2 Prostate Cancer Prostate cancer is the most common cancer among men: 60-80% of men will eventually develop prostate cancer 1 in 7 men will be diagnosed during his lifetime 1 in 36 men will die of prostate cancer Michigan Brian Denton Prostate Cancer Screening 2

3 Biomarker-Based Cancer Screening Physicians use biomarker tests to screen healthy men for asymptomatic, early-stage cancer Catching cancer at an early stage can increase a patient s chance of survival and decrease the cost of treatment Prostate-specific antigen (PSA) test is the biomarker that is most commonly used for prostate cancer screening Michigan Brian Denton Prostate Cancer Screening 3

4 New Biomarker Tests T2:ERG Urine test In late stage clinical validation PCA3 Urine test FDA approved for repeat biopsy More than a dozen other prostate cancer biomarkers have been discovered. Michigan Brian Denton Prostate Cancer Screening 4

5 Controversy Sophisticated new prostate cancer tests are coming to market that might supplement the unreliable PSA test, potentially saving tens of thousands of men each year from unnecessary biopsies, operations and radiation treatments. - The New York Times Pollack A (2013) New Prostate Cancer Tests Could Reduce False Alarms. New York Times. 26 March The existing clinical biomarkers for PCa are not ideal, since they cannot specifically differentiate between those patients who should be treated immediately and those who should avoid over-treatment. Rigau et. al. (2013) The Present and Future of Prostate Cancer Urine Biomarkers. Int J Mol Sci., 14(6): Michigan Brian Denton Prostate Cancer Screening 6

6 Harms and Benefits Longer life expectancy with early detection and treatment Unnecessary biopsies and overtreatment High costs of biomarker tests, biopsy and treatment Michigan Brian Denton Prostate Cancer Screening 7

7 Harms and Benefits Longer life expectancy with early detection and treatment Unnecessary biopsies and overtreatment High costs of biomarker tests, biopsy and treatment Conflicting guidelines for PSA screening American Urological Association (AUA, 2013) American Cancer Society (ACS, 2010) National Comprehensive Cancer Network (NCCN, 2010) U.S. Preventive Services Task Force (USPSTF, 2008, revised 2011, revised 2012,...) Michigan Brian Denton Prostate Cancer Screening 7

8 Research Questions Can optimization models help inform clinical screening decisions? Can biomarkers be used for minimally invasive early detection of prostate cancer? What factors influence the effectiveness of prostate cancer screening? Michigan Brian Denton Prostate Cancer Screening 8

9 Typical Screening Process Biopsy + Treatment Y PSA Test Result Biopsy? N Y Biopsy Result Biopsy - Y PSA Test? N PSA Test? N Epoch t Epoch t+1 Men 40 years and older receive routine annual PSA tests Physician (and patient) decides whether to refer for biopsy Michigan Brian Denton Prostate Cancer Screening 9

10 Typical Screening Process Biopsy + Treatment Y PSA Test Result Biopsy? N Y Biopsy Result Biopsy - Y PSA Test? N PSA Test? N Epoch t Epoch t+1 Michigan Brian Denton Prostate Cancer Screening 9

11 Partially Observable Markov Chain NC OC NC OC NC p t (C NC) C EP C EP C p t (T C) T M LN M LN p t (D C) M D D p t (D NC) D Markov transitions between prostate cancer states: No cancer (NC) Unobservable Cancer present but not detected (C) Unobservable Cancer detected (T) Treated immediately after detected Death (D) Prostate cancer and other cause mortality Michigan Brian Denton Prostate Cancer Screening 10

12 Partially Observable Markov Decision Process (POMDP) Objective: maximize expected benefits minus costs: Societal willingness to pay, β dollars/qaly Reward = β QALY Cost of Screening and Treament Three decisions at each decision epoch: Defer biomarker testing (DP) Defer biopsy (DB) Biopsy (B) Michigan Brian Denton Prostate Cancer Screening 11

13 POMDP Notation t: decision epochs every 6 months from age 40 to age 95 s t : health state at start of epoch t o t : biomarker observation at start of epoch t p t (s t+1 s t, a t ): transition probability between health states s t to s t+1 q t (o t s t ), s t S, o t O: probability of observing PSA level o t in health state s t in epoch t r t (s t, a t ): reward at epoch t Michigan Brian Denton Prostate Cancer Screening 12

14 Sequential Updating of Cancer Risk Biomarker test results are observed over time: The probability a patient is in a given health state at epoch t is estimated using the entire history of observations: Pr t+1 (s t+1 ) = q t+1 (o t+1 s t+1 ) p t (s t+1 s t, a t )Pr t (s t ) s t+1 S s t S q t+1 (o t+1 s t+1 ) p t (s t+1 s t, a t )Pr t (s t ) s t S Michigan Brian Denton Prostate Cancer Screening 13

15 Annual Rewards: Societal Perspective r t (NC, DP) = β r t (NC, DB) = β c p r t (NC, B) = β(1 µ) c b r t (C, DP) = β r t (C, DB) = β c p r t (C, B) = β(1 µ f ɛ) c b f c t f Biopsy detection rate µ Utility decrement of biopsy ɛ Annual utility decrement after treatment β Societal willingness to pay per QALY c p Cost of a PSA test c b Cost of a biopsy Cost of prostate cancer treatment c t Michigan Brian Denton Prostate Cancer Screening 14

16 Optimality Equations Decisions are based on risk of prostate cancer by selecting among the three possibilities: defer PSA test, PSA test, or biopsy v t (Pr t (C)) = max {v t (Pr t (C), DP), v t (Pr t (C), DB), v t (Pr t (C), B)} Properties: For any policy the sequence of Pr t (C) is a Markov process v t (Pr t (C)) is piecewise linear convex Michigan Brian Denton Prostate Cancer Screening 15

17 Optimality Equations Select among three actions: defer testing, PSA test only, or biopsy where v t (Pr t (C)) = max {v t (Pr t (C), DP), v t (Pr t (C), DB), R t (Pr t (C))}, t R t (Pr t (C)) = (1 Pr t (C)) R t (NC) + Pr t (C)((1 f ) R t (C) + f R t (T)) βµ c b Michigan Brian Denton Prostate Cancer Screening 16

18 Optimality Equations Select among three actions: defer testing, PSA test only, or biopsy where v t (Pr t (C)) = max {v t (Pr t (C), DP), v t (Pr t (C), DB), R t (Pr t (C))}, t R t (Pr t (C)) = (1 Pr t (C)) R t (NC) + Pr t (C)((1 f ) R t (C) + f R t (T)) βµ c b v t (Pr t (C), DB) = r t (Pr t (C), DB) + o t+1 O v t+1 (Pr t+1 (C))p t (o t+1 Pr t (C), DB) Michigan Brian Denton Prostate Cancer Screening 16

19 Optimality Equations Select among three actions: defer testing, PSA test only, or biopsy where v t (Pr t (C)) = max {v t (Pr t (C), DP), v t (Pr t (C), DB), R t (Pr t (C))}, t R t (Pr t (C)) = (1 Pr t (C)) R t (NC) + Pr t (C)((1 f ) R t (C) + f R t (T)) βµ c b v t (Pr t (C), DB) = r t (Pr t (C), DB) + o t+1 O v t+1 (Pr t+1 (C))p t (o t+1 Pr t (C), DB) v t (Pr t (C), DP) = r t (Pr t (C), DP) + v t+1 (Pr t+1 (C))p t (Pr t+1 (C) Pr t (C), DP) Michigan Brian Denton Prostate Cancer Screening 16

20 Model Analysis Proposition The incremental benefit of an additional PSA test in expected QALYs is non-negative. Theorem The optimal biopsy referral policy is of control-limit type such that { a W, if Prt (C) Pr t (C) t (Pr t (C)) = B, if Pr t (C) > Pr t (C). 1 Zhang, J., Denton, B.T., Balasubramanian, H., Inman, B., Shah, N., Optimization of Prostate Biopsy Decisions, MSOM, 14(4), , 2012 Michigan Brian Denton Prostate Cancer Screening 17

21 Model Analysis Theorem There exists a finite age, N, at which it is optimal to discontinue biopsy referral if and only if the following condition is satisfied: R N (T) R N (C) µ/f. Corollary If a t (Pr t(c)) = W, Pr t (C) [0, 1], t N, PSA screening should be discontinued. 1 Zhang, J., Denton, B.T., Balasubramanian, H., Inman, B., Shah, N., Optimization of Prostate Biopsy Decisions, MSOM, 14(4), , 2012 Michigan Brian Denton Prostate Cancer Screening 18

22 Sampling Based Solution Method Basic Idea: approximate the value function with a combination of inner and outer linearization subject to a fixed computing budget (d) A minimal α-vector set (e) A lower bound (f) An upper bound v v v 0 1 π 0 1 π 0 1 π Michigan Brian Denton Prostate Cancer Screening 19

23 Lower Bound Lt = arg min s.t. L t c L t W t x Π ( ) max w x max l x f t (x)dx w W t l L t LB heuristic: Step 1. Initialize L T as the true minimal α-vector set, L T. Set t = T 1. Sample belief point sets, N t, for t=1,..., T. Step 2. Using L t+1 find the dominating α-vector at all the belief points in N t and estimate a probability of dominance. Step 3. Select c vectors with highest probability. Let t = t 1. If t = 1 stop; otherwise return to Step 2. Michigan Brian Denton Prostate Cancer Screening 20

24 Upper Bound Ut = arg min s.t. U t Y t U t k, UB heuristic: x Π min g y [0,1], y Y min v g u [0,1], u U t (u)g u g u u = x, g u = 1 u U u U u U { }) v t (y)g y g y Y t y y = x, g y = 1 f t (x)dx y Y y Y Step 1. Compute the value functions at all the points in N T. Let t = T 1. Step 2. Estimate the value function for all belief points in N t using inner linearization for the value functions of all the points in N t+1. Step 3. Let t = t 1. If t = 1 stop; otherwise return to start of Step 2. Michigan Brian Denton Prostate Cancer Screening 21

25 Optimal PSA Screening Policies Probability of having PCa B DB DP From patient perspective ɛ = µ = Age Probability of having PCa B DB DP From societal perspective β = 50, 000 ɛ = µ = Age 1 Zhang, J., Denton, B.T., Balasubramanian, H., Inman, B., Shah, N., Optimization of PSA Screening Policies, Medical Decision Making, 32(2), , 2012 Michigan Brian Denton Prostate Cancer Screening 22

26 Screening Benefit from the Patient Perspective ɛ µ Benefit over no Benefit over Screening PSA screening traditional guideline stopping (QALYs/person) (QALYs/person) age > > > Table: Comparison of optimal screening vs. no screening and the traditional guideline for 40-year-old healthy men 1 Base case: annual incremental benefit of 293,000 QALYs for U.S. population Michigan Brian Denton Prostate Cancer Screening 23

27 Screening Benefit from the Societal Perspective β 100,000 50,000 25,000 costs Cost of the Cost of the Screening optimal policy traditional guideline stopping (U.S. $/person) (U.S. $/person) age 20% base case % % base case % % base case % Table: Sensitivity analysis of the total costs of the optimal policy and the traditional guideline for 40-year-old healthy men 1 Base case: annual cost saving of 166 million dollars for the U.S. population Michigan Brian Denton Prostate Cancer Screening 24

28 Sensitivity Analysis of Expected QALYs d t w t ε z t µ γ e t f b t Figure: Expected QALYs of 40 year old male with Pr 40 (C) = 0 Michigan Brian Denton Prostate Cancer Screening 25

29 Incorporating New Biomarker Tests T2:ERG Distinguishes between low and high grade cancers PCA3 FDA approved for use in combination with PSA Michigan Brian Denton Prostate Cancer Screening 26

30 Expanded Prostate Cancer Treatment Options Radical Prostatectomy Surgical removal of the prostate Appropriate when the cancer is contained in the prostate Active Surveillance Treatment option for patients with low-risk prostate cancer Delays and possibly avoids curative treatment until evidence of disease progression Michigan Brian Denton Prostate Cancer Screening 27

31 Partially Observable Markov Chain Depending on which core state the patient is in, we sample from the appropriate probability distribution for their biomarker results Michigan Brian Denton Prostate Cancer Screening 28

32 Multiple Biomarker-Based Screening Sequential biomarker threshold-based policies: Risk threshold based policies: Pr(PCa) = logit 1 ( β 0 + β 1 PSA + β 2 PCA3 + β 3 T2:ERG) Michigan Brian Denton Prostate Cancer Screening 29

33 Numerical Experiments No Screening, PSA threshold, PSA+PCA3 threshold, PSA + T2ERG threshold, PSA+PCA3+T2ERG Risk Score All combinations of biopsy thresholds for PSA, PCA3, T2ERG, increments of.05 for risk scores Screening frequency based on published schedules: Schedule Range of Screening Label Ages (yr) Interval (yr) Source S Ross et al. (2000) S Ross et al. (2000) S Ross et al. (2000), Andriole et al. (2009) S4 40,45 - Ross et al. (2000) S5 40,45 - Ross et al. (2000) S Heijnsdijk et al. (2012) S Heijnsdijk et al. (2012) S Heijnsdijk et al. (2012) S Heijnsdijk et al. (2012) S Heijnsdijk et al. (2012) S Heijnsdijk et al. (2012) Michigan Brian Denton Prostate Cancer Screening 30

34 Results: Public Health Perspective Michigan Brian Denton Prostate Cancer Screening 31

35 Results: Patient Perspective Michigan Brian Denton Prostate Cancer Screening 32

36 Conclusions Fast approximations for POMDPs for PCa screening can achieve near optimal solutions Combining multiple biomarkers can increase quality adjusted life years, reduce the probability of metastasis, and reduce the probability of receiving a biopsy Factors that most influence the effectiveness of prostate cancer screening are: accuracy of high grade cancer detection, other cause mortality, and cancer incidence Michigan Brian Denton Prostate Cancer Screening 33

37 Acknowledgments Christine Barnett, University of Michigan James Montie, MD, University of Michigan Todd Morgan, MD, University of Michigan Scott Tomlins, MD, PhD, University of Michigan Daniel Underwood, MSc, North Carolina State University John Wei, MD, University of Michigan Jingyu Zhang, PhD, Phillips Research This project is funded in part by the National Science Foundation through Grant Number: CMMI Brian Denton, Department of Industrial and Operations Engineering, University of Michigan Homepage: btdenton Michigan Brian Denton Prostate Cancer Screening 34

38 PSA Sampling Method PSA growth model developed by Gulati et al. (2010): where: log{y i (t)} = β 0i + β 1i t + β 2i (t t oi )I(t > t oi ) + ɛ, y i (t) is the PSA level for individual i at age t t = 0 corresponds to age 35 t oi is the age at onset of a preclinical tumor for individual i Individual intercepts and slopes are given by β ki N(µ k, σ 2 k ) for k = 0, 1, 2 R. Gulati, L. Inoue, J. Katcher, W. Hazelton, R. Etzioni. Calibrating disease progression models using population data: a critical precursor to policy development in cancer control. Biostatistics, 11(4): , Michigan Brian Denton Prostate Cancer Screening 35

39 Model Validation Statistic Model Estimate Literature Estimate Literature Source Overall prostate cancer (0.027, 0.029) Howlader et al. (2012) death rate Expected lifespan for (37.92, 38.14) 37.7 Elizabeth (2010) 40-year-old man (yr.) Overall diagnosis rate (0.104, 0.109) Howlader et al. (2012) Biopsy-detectable Age Prevalence Age Prevalence Haas et al. (2007) prostate cancer 50 8% 50 13% prevalence 60 19% 60 22% 70 33% 70 36% 80 49% 80 51% 89 62% 89 65% Michigan Brian Denton Prostate Cancer Screening 36

40 Experiments: Biopsy Thresholds PSA: {1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 6.5} PCA3: {19, 25, 35, 55, 75} T2:ERG: {7, 10, 30, 50, 100} MiPS: {0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50} Michigan Brian Denton Prostate Cancer Screening 37