Modelling Longitudinal Biomarkers of Disease Progression (Natural History of Prostate Cancer)

1. Modelling Longitudinal Biomarkers of Disease Progression (Natural History of Prostate Cancer) Donatello Telesca Stochastic Modeling Preliminary Exam

2. Preview • Prostate Cancer Background • Natural History Models • Modeling Different Views • A case study (BLSA) • Model Assessment and Conclusions

3. Prostate Cancer • Most commonly diagnosed form of cancer in USA. • Usually diagnosed in men over 55 and slow growing. • Second most common cause of cancer death in American men (after lung cancer ) • The prostate gland plays a role in the male urinary and reproductive systems.

4. Natural History of Prostate Cancer Clinical Stage (Size and Extent of the tumor) Histologic Grade (Cell differentiation) Local Metastasis Gleason score 1 Gleason score 5

5. Natural History Models • Natural history models aim to chart the progression of a disease. • They provide critical information about the early stages of a disease. • They provide recommendations for cancer screening and detection. • The challenge is related to the latency of the main events comprising disease progression. • They usually rely on the availability of a biomarker associated to the presence and progression of the disease.

6. PSA and Prostate Cancer • PSA (Prostate Specific Antigen) is a protein produced by the prostate gland to keep the semen in a liquid state. Disease Onset Cancer PSA Level Normal Puberty AGE

7. Different Views on Disease Progression • Prostate adenocarcinomas have a different natures directly from onset. Some are more aggressive (Low cell differentiation), others are less aggressive (Good cell differentiation). • Prostate adenocarcinomas have a progressive nature. They start out as well differentiated tumor cells and they progress with time to more aggressive forms, with poorly differentiated tumor cells.

8. A model with no grade progression TM : Metastasis (Advanced) TM : Metastasis (Local) High Grade Low Grade Log(PSA+α) Tc : Clinical Diagnosis T0 : Onset Time AGE

9. PSA Trajectories Subject level Population level

10. Disease Onset Hazard Cumulative Hazard Density

11. Time to Diagnosis and Metastasis Time to Metastasis Hazard Cumulative Hazard Monotonicity Time to Clinical Diagnosis Hazard Cumulative Hazard

12. A causal diagram of grade progression Onset Grade trans. t0 tG PSA tC tM Metastasis Diagnosis

13. A model with grade progression tg : Grade transition Log(PSA + α) tM : Metastasis   t0 : Onset tc : Diagnosis AGE

14. PSA Trajectories Subject level Population level

15. Grade Transition Hazard t0

16. Time to Metastasis Hazard Monotonicity:

17. Likelihood yi: log(PSA + const) for individual i θ : parameter vector x : stage(1=local, 2=metastasis) Local stage Advanced Stage

18. Bayesian Estimation POSTERIOR Chained data augmentation • i) Given (t0(k-1), tM(k-1)) , θ(k) ~ π(θ|y, tc, x, t0(k-1) ,tM(k-1)); • Given θ(k) , (t0(k-1) , tM(k-1))~ π(θ|y, tc, x, t0(k-1) ,tM(k-1) ); • Iterate (i), (ii).

19. Dealing with constrained parameter spaces (Example) Growth rates full conditional: With constraints: • bi0 ~ bi0|yi,θ(-bi0) • bi1 ~ bi1|yi,θ(-bi1) in (bi1>-bi2) • bi2 ~ bi2|yi,θ(-bi2) in (bi2>-bi1) • bi3 ~ bi3|yi,θ(-bi3) in (bi3>-(bi1+bi2)) bi2=-bi1 bi3 = -(bi1+bi2) bi2 bi1+bi2 bi1 bi3

20. Case Study (BLSA)

21. Model Fit Comparison Subject with local disease and high grade Progressive grade No grade progression Log(PSA + 0,03) Log(PSA + 0,03) Age Age

22. Model Fit Comparison Subject with advanced disease and high grade Progressive grade No grade progression Log(PSA + 0,03) Log(PSA + 0,03) Age Age

23. Model Fit Comparison Subject with local disease and low grade Progressive grade No grade progression Log(PSA + 0,03) Log(PSA + 0,03) Age Age

24. Posterior Predictive Assessment Posterior predictive distributions for transition times and median predictive PSA trajectories, assuming no grade progression . (High GS) Log(PSA + 0.03) (Low GS) Density 4ng/ml Age

25. Posterior Predictive Assessment Posterior predictive distributions for transition times and median predictive PSA trajectory, assuming grade progression . Log(PSA + 0.03) Density 4ng/ml Age

26. Model Assessment M1: No grade progression M2: Grade progression ● Bayes Factor → Strong evidence against M2

27. CPO Analysis ● No Grade progression ● Grade progression o Log( CPO ) = Log( f(yi, tci, xi|y-i, tc,-i ,x-i) ) Log(CPO) Subject

28. Concluding • We proposed a way to translate scientific hypotheses about the progression of prostate cancer into a statistical model for the disease main biomarker (PSA). • The BLSA data provides evidence in favor of the hypothesis of no grade progression as opposed to that of grade progression. • Limitations of this approach : - Difficult validation of the hazard models for the latent transition times. - Prior sensitivity. • Extensions may consider : - Misclassified diagnosis of the normal subjects. - Non-parametric formulation of the problem.

29. Acknowledgements • Julian Besag • Lourdes Inoue • Stat518(2005): Congley, Haoyuan, Liang, Nate, Yanming.

30. Adaptive Slice Sampling (R.M. Neal, 2000) f(x0) S y ~ U[0,f(x0)] x1 ~ U(S) x0