modelling longitudinal biomarkers of disease progression (natural history of prostate cancer)
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Modelling Longitudinal Biomarkers of Disease Progression (Natural History of Prostate Cancer). Donatello Telesca Stochastic Modeling Preliminary Exam. Preview. Prostate Cancer Background Natural History Models Modeling Different Views A case study (BLSA) - PowerPoint PPT PresentationTRANSCRIPT
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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
26
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