Optimal Design of Biomarker-Based Screening Strategies for Early Detection … · 2019-09-10 ·...

Optimal Design of Biomarker-Based ScreeningStrategies for Early Detection of Prostate Cancer

Brian Denton

Department of Industrial and Operations EngineeringUniversity of Michigan, Ann Arbor, MI

October 13, 2014

Michigan Brian Denton Prostate Cancer Screening 1

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


Biomarker-Based Cancer Screening

Physicians use biomarker tests toscreen healthy men for asymptomatic,early-stage cancer

Catching cancer at an early stage can increase a patient’s chance ofsurvival and decrease the cost of treatment

Prostate-specific antigen (PSA) test is the biomarker that is mostcommonly used for prostate cancer screening


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 beendiscovered.


Controversy

“Sophisticated new prostate cancer tests are coming to market that mightsupplement the unreliable PSA test, potentially saving tens of thousandsof men each year from unnecessary biopsies, operations andradiation treatments.” - The New York Times

Pollack A (2013) New Prostate Cancer Tests Could Reduce False Alarms. New York Times. 26 March 2013.

“The existing clinical biomarkers for PCa are not ideal, since they cannotspecifically differentiate between those patients who should betreated 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):12620-12649.


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,...)


Research Questions

Can optimization models help inform clinical screening decisions?

Can biomarkers be used for minimally invasive early detection ofprostate cancer?

What factors influence the effectiveness of prostate cancerscreening?


Typical Screening Process

PSA

Test

Result

PSA

Test?

PSA

Test?Epoch t Epoch t+1

Biopsy +

Biopsy -Y

N

Biopsy?

Y

N

Biopsy

Result

Y

N

Treatment

Men 40 years and older receive routine annual PSA tests

Physician (and patient) decides whether to refer for biopsy


Typical Screening Process

PSA

Test

Result

PSA

Test?

PSA

Test?Epoch t Epoch t+1

Biopsy +

Biopsy -Y

N

Biopsy?

Y

N

Biopsy

Result

Y

N

Treatment


Partially Observable Markov Chain

M

NC

D

C

OC

EP

LN

NC

C

OC

EP

LN

M

D

pt

(T|C)

NC

TC

pt

(C|NC)

pt

(D|NC)

pt

(D|C)

M

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


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)


POMDP Notation

t: decision epochs every 6 months from age 40 to age 95

st: health state at start of epoch t

ot: biomarker observation at start of epoch t

pt(st+1|st, at): transition probability between health states st to st+1

qt(ot|st), ∀st ∈ S, ot ∈ O: probability of observing PSA level ot inhealth state st in epoch t

rt(st, at): reward at epoch t


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 estimatedusing the entire history of observations:

Prt+1(st+1) =

qt+1(ot+1|st+1)∑

st∈Spt(st+1|st, at)Prt(st)∑

st+1∈Sqt+1(ot+1|st+1)

∑st∈S

pt(st+1|st, at)Prt(st)


Annual Rewards: Societal Perspective

rt(NC,DP) = β

rt(NC,DB) = β − cp

rt(NC,B) = β(1 − µ) − cb

rt(C,DP) = β

rt(C,DB) = β − cp

rt(C,B) = β(1 − µ − f ε) − cb − f · ct

f Biopsy detection rateµ Utility decrement of biopsyε Annual utility decrement after treatmentβ Societal willingness to pay per QALYcp Cost of a PSA testcb Cost of a biopsyct Cost of prostate cancer treatment


Optimality Equations

Decisions are based on risk of prostate cancer by selecting among thethree possibilities: defer PSA test, PSA test, or biopsy

vt(Prt(C)) = max {vt(Prt(C),DP), vt(Prt(C),DB), vt(Prt(C),B)}

Properties:

For any policy the sequence of Prt(C) is a Markov process

vt(Prt(C)) is piecewise linear convex


Optimality Equations

Select among three actions: defer testing, PSA test only, or biopsy

vt(Prt(C)) = max {vt(Prt(C),DP), vt(Prt(C),DB),Rt(Prt(C))} , ∀t

where

Rt(Prt(C)) = (1−Prt(C))R̄t(NC) + Prt(C)((1− f )R̄t(C) + f R̄t(T))−βµ− cb

vt(Prt(C),DB) = rt(Prt(C),DB) +∑

ot+1∈Ovt+1(Prt+1(C))pt(ot+1|Prt(C),DB)

vt(Prt(C),DP) = rt(Prt(C),DP) + vt+1(Prt+1(C))pt(Prt+1(C)|Prt(C),DP)


Model Analysis

Proposition

The incremental benefit of an additional PSA test in expected QALYs isnon-negative.

Theorem

The optimal biopsy referral policy is of control-limit type such that

a∗t (Prt(C)) =

{W, if Prt(C) ≤ Pr∗t (C)B, if Prt(C) > Pr∗t (C).

1Zhang, J., Denton, B.T., Balasubramanian, H., Inman, B., Shah, N., Optimization of Prostate Biopsy Decisions, MSOM, 14(4),

529-547, 2012


Model Analysis

Theorem

There exists a finite age, N, at which it is optimal to discontinue biopsyreferral if and only if the following condition is satisfied:

R̄N(T) − R̄N(C) ≤ µ/f .

Corollary

If a∗t (Prt(C)) = W, ∀Prt(C) ∈ [0, 1], ∀t ≥ N, PSA screening should bediscontinued.

1Zhang, J., Denton, B.T., Balasubramanian, H., Inman, B., Shah, N., Optimization of Prostate Biopsy Decisions, MSOM, 14(4),

529-547, 2012


Sampling Based Solution Method

Basic Idea: approximate the value function with a combination of inner andouter linearization subject to a fixed computing budget

(d) A minimal α-vector set

π0 1

v

(e) A lower bound

π0 1

v

(f) An upper bound

π0 1

v


Lower Bound

L∗t = arg minLt⊆Wt

∫x∈Π

(maxw∈Wt

w · x −max`∈Lt

` · x)

ft(x)dx

s.t.

|Lt| ≤ c

LB heuristic:

Step 1. Initialize L̂T as the true minimal α-vector set, LT . Sett = T − 1. Sample belief point sets, Nt, for t=1, ..., T.

Step 2. Using L̂t+1 find the dominating α-vector at all the belief pointsin Nt and estimate a probability of dominance.

Step 3. Select c vectors with highest probability. Let t = t − 1. If t = 1stop; otherwise return to Step 2.


Upper Bound

U∗t = arg minUt⊆Yt

∫x∈Π

mingu∈[0,1],∀u∈U

∑u∈U

vt(u)gu

∣∣∣∣∣∣∣∑u∈U guu = x,∑u∈U

gu = 1

− min

gy∈[0,1],∀y∈Y

{ ∑y∈Yt

vt(y)gy

∣∣∣∣∣∣ ∑y∈Y gyy = x,∑

y∈Ygy = 1

})ft(x)dx

s.t.

|Ut| ≤ k,

UB heuristic:

Step 1. Compute the value functions at all the points in NT . Lett = T − 1.

Step 2. Estimate the value function for all belief points in Nt usinginner linearization for the value functions of all the points inNt+1.

Step 3. Let t = t − 1. If t = 1 stop; otherwise return to start of Step 2.


Optimal PSA Screening Policies

40 50 60 70 80

0.0

0.2

0.4

0.6

0.8

1.0

Age

Pro

babi

lity

of h

avin

g P

Ca

B

DP

DB

40 50 60 70 80

0.0

0.2

0.4

0.6

0.8

1.0

Age

Pro

babi

lity

of h

avin

g P

Ca

B

DP

DB

From patient perspective

ε = 0.145

µ = 0.05

From societal perspective

β = 50, 000

ε = 0.145

µ = 0.05

1Zhang, J., Denton, B.T., Balasubramanian, H., Inman, B., Shah, N., Optimization of PSA Screening Policies,” Medical Decision

Making, 32(2), 337-349, 2012


Screening Benefit from the Patient Perspective

ε µ

Benefit over no Benefit over ScreeningPSA screening traditional guideline stopping(QALYs/person) (QALYs/person) age

0.050.01 0.289 0.292 > 840.05 0.271 0.312 > 840.1 0.253 0.340 > 84

0.1450.01 0.148 0.151 770.05 0.1311 0.165 760.1 0.116 0.204 75

0.240.01 0.044 0.047 610.05 0.032 0.073 600.1 0.025 0.113 60

Table: Comparison of optimal screening vs. no screening and the traditionalguideline for 40-year-old healthy men

1Base case: annual incremental benefit of 293,000 QALYs for U.S. populationMichigan Brian Denton Prostate Cancer Screening 23

Screening Benefit from the Societal Perspective

β costsCost of the Cost of the Screening

optimal policy traditional guideline stopping(U.S. $/person) (U.S. $/person) age

100,000−20% 904 764 73

base case 1104 956 73+20% 1274 1147 72

50,000−20% 768 764 71

base case 882 956 71+20% 989 1147 69

25,000−20% 583 764 68

base case 642 956 67+20% 658 1147 65

Table: Sensitivity analysis of the total costs of the optimal policy and thetraditional guideline for 40-year-old healthy men

1Base case: annual cost saving of 166 million dollars for the U.S. populationMichigan Brian Denton Prostate Cancer Screening 24

Sensitivity Analysis of Expected QALYs

37.688

37.678

37.658

37.617

37.662

37.518

37.556

37.378

35.877

37.714

37.721

37.753

37.739

37.784

37.704

37.897

38.195

39.957

35.800 36.800 37.800 38.800 39.800

d

t

w

t

ε

z

t

µ

γ

e

t

f

b

t

Figure: Expected QALYs of 40 year old male with Pr40(C) = 0Michigan Brian Denton Prostate Cancer Screening 25

Incorporating New Biomarker Tests

T2:ERG

Distinguishes betweenlow and high grade cancers

PCA3

FDA approved for usein combination with PSA


Expanded Prostate Cancer Treatment Options

Radical Prostatectomy

Surgical removal of the prostate

Appropriate when the cancer is contained in theprostate

Active Surveillance

Treatment option for patients with low-risk prostatecancer

Delays and possibly avoids curative treatment untilevidence of disease progression


Partially Observable Markov Chain

Depending on which core state the patient is in, we sample from the appropriateprobability distribution for their biomarker results


Multiple Biomarker-Based Screening

Sequential biomarker threshold-based policies:

Risk threshold based policies:

Pr(PCa) = logit−1(−β0 + β1 × PSA + β2 × PCA3 + β3 × T2:ERG)


Numerical Experiments

No Screening, PSA threshold, PSA+PCA3 threshold, PSA + T2ERGthreshold, 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 ScreeningLabel Ages (yr) Interval (yr) Source

S1 40-75 5 Ross et al. (2000)S2 50-75 2 Ross et al. (2000)S3 50-75 1 Ross et al. (2000),

Andriole et al. (2009)S4 40,45 - Ross et al. (2000)

50-75 2S5 40,45 - Ross et al. (2000)

50-75 1S6 55-69 1 Heijnsdijk et al. (2012)S7 55-74 1 Heijnsdijk et al. (2012)S8 55-69 4 Heijnsdijk et al. (2012)S9 55 - Heijnsdijk et al. (2012)

S10 60 - Heijnsdijk et al. (2012)S11 65 - Heijnsdijk et al. (2012)


Results: Public Health Perspective


Results: Patient Perspective


Conclusions

Fast approximations for POMDPs for PCa screening can achieve nearoptimal solutions

Combining multiple biomarkers can increase quality adjusted lifeyears, reduce the probability of metastasis, and reduce the probabilityof receiving a biopsy

Factors that most influence the effectiveness of prostate cancerscreening are: accuracy of high grade cancer detection, other causemortality, and cancer incidence


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 0844511.

Brian Denton, Department of Industrial and Operations Engineering, University of MichiganEmail: [email protected]: http://umich.edu/ btdenton


h

PSA Sampling Method

PSA growth model developed by Gulati et al. (2010):

log{yi(t)} = β0i + β1it + β2i(t − toi)I(t > toi) + ε,

where:

yi(t) is the PSA level for individual i at age t

t = 0 corresponds to age 35

toi is the age at onset of a preclinical tumor for individual i

Individual intercepts and slopes are given by βki ∼ N(µk, σ2k) for

k = 0, 1, 2

R. Gulati, L. Inoue, J. Katcher, W. Hazelton, R. Etzioni. Calibrating disease progression models using population data: a criticalprecursor to policy development in cancer control. Biostatistics, 11(4):707-719, 2010.


Model Validation

Statistic Model Estimate Literature Estimate Literature Source

Overall prostate cancer 0.028 (0.027, 0.029) 0.026 Howlader et al. (2012)death rate

Expected lifespan for 38.03 (37.92, 38.14) 37.7 Elizabeth (2010)40-year-old man (yr.)

Overall diagnosis rate 0.106 (0.104, 0.109) 0.159 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%


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}


Optimal Design of Biomarker-Based Screening Strategies for Early Detection … · 2019-09-10 ·...

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