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Modelling Longitudinal Modelling Longitudinal Biomarkers of Disease Biomarkers of Disease

Progression Progression (Natural History of Prostate (Natural History of Prostate

Cancer)Cancer)

Donatello TelescaDonatello TelescaStochastic Modeling Preliminary ExamStochastic Modeling Preliminary Exam

2

PreviewPreview

Prostate Cancer Background Prostate Cancer Background Natural History Models Natural History Models Modeling Different Views Modeling Different Views A case study (BLSA)A case study (BLSA) Model Assessment and ConclusionsModel Assessment and Conclusions

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Prostate CancerProstate Cancer Most commonly diagnosed Most commonly diagnosed

form of cancer in USA.form of cancer in USA. Usually diagnosed in men Usually diagnosed in men

over 55 and slow growing.over 55 and slow growing. Second most common Second most common

cause of cancer death in cause of cancer death in American men (after lung American men (after lung cancercancer ) ) The prostate The prostate

gland plays a gland plays a role in the role in the male urinary male urinary and and reproductive reproductive systems.systems.

IncidencIncidencee

200/100,000200/100,000

MortalitMortalityy

50/100,00050/100,000

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Natural History of Prostate Natural History of Prostate CancerCancer

Histologic Grade

(Cell differentiation)

Clinical Stage

(Size and Extent of the tumor)

Local Metastasis

Gleason score 1

Gleason score 5

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Natural History ModelsNatural History Models Natural history models aim to chart the Natural history models aim to chart the

progression of a disease.progression of a disease. They provide critical information about They provide critical information about

the early stages of a disease.the early stages of a disease. They provide recommendations for They provide recommendations for

cancer screening and detection.cancer screening and detection. The challenge is related to the latency of The challenge is related to the latency of

the main events comprising disease the main events comprising disease progression.progression.

They usually rely on the availability of a They usually rely on the availability of a biomarker associated to the presence biomarker associated to the presence and progression of the disease.and progression of the disease.

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PSA and Prostate CancerPSA and Prostate Cancer PSA (Prostate Specific Antigen) is a protein produced by PSA (Prostate Specific Antigen) is a protein produced by

the prostate gland to keep the semen in a liquid state. the prostate gland to keep the semen in a liquid state.

0 20 40 60 80 100

050

100

150

PS

A

Leve

l

Puberty

AGE

Disease Onset

Normal

Cancer

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Different Views on Disease Different Views on Disease ProgressionProgression

Prostate adenocarcinomas have a Prostate adenocarcinomas have a different natures directly from onset. different natures directly from onset. Some are more aggressive (Low cell Some are more aggressive (Low cell differentiation), others are less differentiation), others are less aggressive (Good cell differentiation).aggressive (Good cell differentiation).

Prostate adenocarcinomas have a Prostate adenocarcinomas have a progressive nature. They start out as progressive nature. They start out as well differentiated tumor cells and they well differentiated tumor cells and they progress with time to more aggressive progress with time to more aggressive forms, with poorly differentiated tumor forms, with poorly differentiated tumor cells. cells.

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A model with no grade A model with no grade progressionprogression

AGE

40 50 60 70 80 90 100

-20

24

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Log

(PS

A+

α)

High Grade

Low Grade

Tc : Clinical Diagnosis

TM : Metastasis (Advanced)

T0 : Onset Time

TM : Metastasis (Local)

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PSA TrajectoriesPSA Trajectories

2

0 1

0 1 2 0

log( ) ,

(0, ),

,

( ) ,

ij ij ij

ij iid

i iij

i i i i

y const

Normal

b b t if normal patient

b b t b t t if cancer patient

2

22 2

2 23 3

2 3

( , ), 0,1

( , ),

( , ),

;

ik k k

i

b Normal k

Normal if low gradeb

Normal if high grade

with

Subject level

Population level

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Disease OnsetDisease Onset

0

0 0 0

200 0 00

0 0 0

( ) ,

( )2

( ) ( ) exp{ ( )}

i

i i

t

i i

i i i

h t t

H t u du t

f t h t H t

0 50 100 150

0.000

0.005

0.010

0.015 0 0.001

0 0.0002

0 0.0005

0it

0( )if tHazard

Cumulative Hazard

Density

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Time to Diagnosis and MetastasisTime to Diagnosis and Metastasis

0

0

00

1 2

( ) exp{ ( )} { }

0

( ) exp{ ( )} exp{ ( )}

iM M i iM iM i

iM i

iM i iM i iM iM i

i i

h t t I t t

for t t

H t t tfor t t

b b

0

0

00

1 2

( ) exp{ ( )} { }

0

( ) exp{ ( )} exp{ ( )}

ic c i ic iC i

ic i

ic i ic i ic ic i

i i

h t t I t t

for t t

H t t tfor t t

b b

Hazard

Cumulative Hazard

Hazard

Cumulative Hazard

Time to Metastasis

Time to Clinical Diagnosis

1 2 0i ib b

Monotonicity

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A causal diagram of grade A causal diagram of grade progressionprogression

t0 tG

tC tM

PSA

Onset Grade trans.

Diagnosis Metastasis

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A model with grade A model with grade progressionprogression

30 40 50 60 70 80 90 100

-20

24

6

AGE

Log

(PS

A +

α)

t0 : Onset

tg : Grade transition

tc : Diagnosis

tM : Metastasis

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PSA TrajectoriesPSA Trajectories

2

0 1

0 1 2 0 3

log( ) ,

(0, ),

,

( ) ( ) ,

ij ij ij

ij iid

i i

iji i i i i ig

y const

Normal

b b t if normal patient

b b t b t t b t t if cancer patient

2( , ), 0,1, 2,3ik k kb Normal k

Subject level

Population level

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Grade TransitionGrade Transition

0

0

2 2000

0 0 0

( ) ( ) , 0

0 0

( )( ) ( )( )

2

ig

ig g ig i ig

ig it

ig iig g ig i ig i ig i

h t t t t

t t

t tH t u t dut t t t t

igt

( )igf t

Hazard

0.05g

60 70 80 90 100

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.01g

0.005g

t0

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Time to MetastasisTime to Metastasis

0

0

00

1 2

0

1 2 3

( ) exp{ ( )} { }

0 0

exp{ ( )} exp{ ( )}( )

exp{ ( )} exp{ ( )}

iM M i iM im i

iM i

i iM i iiM M i iM ig

i i

i iM i iM iM ig

i i i

h t t I t t

for t t

t tH t for t t t

b b

t tfor t t

b b b

Hazard

Monotonicity: 1 2

1 2 3

0

0i i

i i i

b b

b b b

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LikelihoodLikelihood

0

0 0 00 0

0 0 0 00

( | , , , 2) ( , , 2 | , )

( , , , | , ) { }

( , ,| , , , ) ( | , , ) ( | , )ic ic

i

i i ic i ic i

i ic i iM i i iM ic iM i

t t

i ic i iM i iM i i i i iM it

L t y t x f y t x t

f y t t t t I t t t dt dt

f y t t t t f t t t f t t dt dt

yi: log(PSA + const) for individual i

θ : parameter vector x : stage(1=local, 2=metastasis)

0 0 00 0

0 0 0 00

( | , , , 1) ( , , 1| , )

( , , , | , ) { }

( , ,| , , , ) ( | , , ) ( | , )ic

ic

i i ic i ic i

i ic i iM i i ic iM iM i

t

i ic i iM i iM i i i i iM it

L t y t x f y t x t

f y t t t t I t t t dt dt

f y t t t t f t t t f t t dt dt

Local stage

Advanced Stage

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Bayesian EstimationBayesian Estimation

0 0 0

0 0

( | , , ) ( ) ( | , , , )

( | , , ) ( | , , , , ) ( , | , , )

( , | , , ) ( , | , , , ) ( | , , )

c c

c c i iM i iM c i iM

i iM c i iM c c

y t x L t y t x

y t x y t x t t t t y t x dt dt

t t y t x t t y t x y t x d

i) Given (t0(k-1), tM

(k-1)) , θ(k) ~ π(θ|y, tc, x, t0(k-1) ,tM

(k-1));

ii) Given θ(k) , (t0(k-1) , tM

(k-1)) ~ π(θ|y, tc, x, t0(k-1) ,tM

(k-1) );

iii) Iterate (i), (ii).

POSTERIOR

Chained data augmentation

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Dealing with constrained parameter Dealing with constrained parameter spaces (Example)spaces (Example)

{ } 4( | , ) ( , ), 0,...,3ikik i b n nf b y N k

-2 -1 0 1 2

-2-1

01

2

Growth rates full conditional:

With constraints:

-2 -1 0 1 2

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

i1 i2

i1 i2 i3

b + b > 0

b + b + b > 0

1. bi0 ~ bi0|yi,θ(-bi0)2. bi1 ~ bi1|yi,θ(-bi1) in (bi1>-bi2)3. bi2 ~ bi2|yi,θ(-bi2) in (bi2>-bi1)4. bi3 ~ bi3|yi,θ(-bi3) in (bi3>-

(bi1+bi2))

bi1

bi2

bi1

+b

i2

bi3

bi2=-bi1 bi3 = -(bi1+bi2)

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Case Study (BLSA)Case Study (BLSA)

GROUPGROUP Individuals Individuals MeasurementMeasurementss

Length of FULength of FU Age at last FUAge at last FU

NormalNormal 282282 6.24 (2.28)6.24 (2.28) 16.15 (8.6)16.15 (8.6) 60.05 (12.42)60.05 (12.42)

Local (Low Local (Low GS)GS)

3939 7.85 (3.47)7.85 (3.47) 18.09 (7.78)18.09 (7.78) 70.65 (7.82)70.65 (7.82)

Met. (Low Met. (Low GS)GS)

33 3.33 (1.52)3.33 (1.52) 8.33 (10.07)8.33 (10.07) 76.66 (12.10)76.66 (12.10)

Local (High Local (High GS)GS)

55 9.00 (1.87)9.00 (1.87) 18.12 (6.19)18.12 (6.19) 73.12 (4.98)73.12 (4.98)

Met. (High Met. (High GS)GS)

66 9.5 (1.87)9.5 (1.87) 18.68 (8.08)18.68 (8.08) 73.43 (7.78)73.43 (7.78)

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Model Fit ComparisonModel Fit ComparisonSubject with local disease and high grade

Log

(PS

A +

0,0

3)

Progressive grade No grade progression

AgeAge

Log

(PS

A +

0,0

3)

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Model Fit ComparisonModel Fit ComparisonSubject with advanced disease and high grade

Age Age

No grade progressionProgressive grade

Log

(PS

A +

0,0

3)

Log

(PS

A +

0,0

3)

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Model Fit ComparisonModel Fit Comparison

Subject with local disease and low grade

Log

(PS

A +

0,0

3)

Log

(PS

A +

0,0

3)

Progressive grade No grade progression

AgeAge

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Posterior Predictive AssessmentPosterior Predictive Assessment

Age

(High GS)

(Low GS)

Den

sity

Log

(PS

A +

0.0

3)

Posterior predictive distributions for transition times and median predictive PSA trajectories, assuming no grade progression .

4ng/ml

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Posterior Predictive AssessmentPosterior Predictive Assessment

4ng/ml

Posterior predictive distributions for transition times and median predictive PSA trajectory, assuming grade progression .

Age

Log

(PS

A +

0.0

3)

Den

sity

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Model AssessmentModel Assessment

1

2

( | )og 35.189

( | )

P D ML

P D M

→ Strong evidence against M2

● Bayes Factor

M1: No grade progressionM2: Grade progression

1

( ) 1

1

1ˆ( | ) ( | )K

ji

j

P D M P DK

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CPO AnalysisCPO AnalysisLog( CPO ) = Log( f(yi, tci, xi|y-i, tc,-i ,x-i) )

Subject

Log

(CP

O)

o● No Grade progression● Grade progression

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ConcludingConcluding We proposed a way to translate scientific We proposed a way to translate scientific

hypotheses about the progression of prostate hypotheses about the progression of prostate cancer into a statistical model for the disease cancer into a statistical model for the disease main biomarker (PSA). main biomarker (PSA).

The BLSA data provides evidence in favor of The BLSA data provides evidence in favor of the hypothesis of no grade progression as the hypothesis of no grade progression as opposed to that of grade progression. opposed to that of grade progression.

Limitations of this approach : Limitations of this approach : - Difficult validation of the hazard models for - Difficult validation of the hazard models for

the latent transition times.the latent transition times. - Prior sensitivity. - Prior sensitivity. Extensions may consider : Extensions may consider : - Misclassified diagnosis of the normal - Misclassified diagnosis of the normal

subjects. subjects. - Non-parametric formulation of the problem.- Non-parametric formulation of the problem.

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AcknowledgementsAcknowledgements

• Julian Besag Julian Besag • Lourdes Inoue Lourdes Inoue • Stat518(2005): Congley, Haoyuan, Stat518(2005): Congley, Haoyuan,

Liang, Nate, Yanming. Liang, Nate, Yanming.

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Adaptive Slice Sampling Adaptive Slice Sampling (R.M. Neal, 2000)(R.M. Neal, 2000)

-5 0 5 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

x0

y ~ U[0,f(x0)]

f(x0)

x1 ~ U(S)

S