Stochastic modeling of carcinogenesis

Transcription

1 Stochastic modeling of carcinogenesis Rafael Meza Department of Epidemiology University of Michigan SPH-II

2 Outline Multistage carcinogenesis models Examples Potential projects

3 Multistage Carcinogenesis Models

4 Multistage Carcinogenesis Cancer is the consequence of the accumulation of genetic transformations in a single cell (or its descendants) Mueller (1951) & Nordling (1953) (before DNA structure discovery!) Armitage-Doll Model (1954); TSCE Model (1979) Mechanistic models (biologically based) Cancer epidemiology Laboratory experiments

5 Armitage P & Doll R, BJC 1954

6 Armitage P & Doll R, BJC 1954

7 Armitage-Doll Model (1954) λ 0 λ 1 λ 2 λ n-1 E 0 E 1 E 2 E n Normal Stem Cell Exponential waiting time Malignant Cell

8 Armitage-Doll Model (1954) Let p k (a) be the probability that the cell is at stage k at age a ) ( ) ( ) ( ) ( ) ( ) ( ) ( a p da a dp a p a p da a dp a p da a dp n n n = = = λ λ λ λ

9 Armitage-Doll Model (1954) Assume there are N susceptible stem cells: P [No cancer by age a ] = S ( a ) = ( 1 p a ) N n( ) Cancer hazard (age-specific cancer risk): h(a) = d ln S(a) [ ] da Nλ 0λ 1 λ n 1 a n 1 (n 1)!

10 Armitage-Doll Model (1954) Age-Specific Incidence Nλ 0λ1 λn 1 log( h( a)) log + ( n 1) log( a) ( n 1)!

11 Armitage P & Doll R, BJC 1954

12 Armitage P & Doll R, BJC 1954

13 Hazard or Incidence Function (Measure of Cancer Risk) The hazard is a theoretical representation of the observed incidence or mortality of cancer in the population (# of cases(a) / population(a)) Mathematically it measures the instantaneous probability of getting (dying from) cancer Carcinogenesis model à Derive hazard/survival à Estimate model parameters by fitting to cancer incidence/mortality dataà

14 TSCE Model (1979) Knudson s hypothesis (early 70 s): Two hits are needed for the retinoblastoma gene to cause a tumor, with this occurring at the somatic level in the sporadic form while one hit is inherited in the familial form Retinoblastoma gene identified in 1987 Two Stage Clonal Expansion Model Mathematical expression of Knudson s hypothesis Incorporates clonal expansion of pre-malignant cells Follows initiation-promotion-progression paradigm

15 TSCE Model Non-homogeneous Poisson Process β(t) Birth-Death-Mutation Process Moolgavkar & Venzon (Math. Biosc, 1979); Moolgavkar & Knudson (JNCI, 1981)

16 TSCE Model As a continuous time Markov Process Let, j Ψ( yzt,, ) P ( tyz ) jk, jk, k Ψ(y,z,t) dt = (y 1)ν(t)X(t)Ψ(y,z,t) + {[ µ(t)z + α(t)y (α(t) + β(t) + µ(t)) ]y + β(t) } Ψ(y,z,t) dy with initial condition Ψ(y,z,0)=1 Forward-Kolmogorov equation

17 TSCE Model S(t) = q p qe pt pe qt νx α h(t) = νx α pq( e qt e pt ) qe pt pe qt p,q = 1 [ 2 (α β µ) m (α β µ)2 + 4αµ ]

18 TSCE Model Analysis of population level data: Closed form expressions for the hazard and survival functions in case of constant and piecewise constant parameters. Heidenreich et al. (Risk Analysis, 1997) Numerical solution in case of general age-dependent parameters Analysis of experimental data: Number and size distribution of premalignant and malignant lesions

19 Luebeck-Moolgavkar (2002) Generalizations Tan (1986), Little (1996)

20 A simple 3-stage Model Normal X µ 0 µ 1 µ 2 Gatek+/- Gatek-/- Premalig. Cancer α β Premalignant lesion Onset τ sojourn time s

21 Ψ(u;a) 3-stage Model As a continuous time branching process Φ 1 (u;a) Φ 2 (u;a) Φ 3 (u;a) Normal X µ 0 µ 1 µ 2 Gatek+/- Gatek-/- Premalig. Cancer α β X I 1 I 2 I 3

22 Probability Generating Functions ψ(y 1, y 2, y 3,u;a) = E[y 1 I 1 (a ) y 2 I 2 (a) y 3 I 3 (a ) I 1 (u) = 0,I 2 (u) = 0,I 3 (u) = 0] Φ 1 (y 1,y 2, y 3,u;a) = E[y 1 I 1 (a ) y 2 I 2 (a ) y 3 I 3 (a ) I 1 (u) =1,I 2 (u) = 0,I 3 (u) = 0] Φ 2 (y 1, y 2,y 3,u;a) = E[y 1 I 1 (a ) y 2 I 2 (a) y 3 I 3 (a ) I 1 (u) = 0,I 2 (u) =1,I 3 (u) = 0] Φ 3 (y 1, y 2,y 3,u;a) = E[y 1 I 1 (a ) y 2 I 2 (a) y 3 I 3 (a ) I 1 (u) = 0,I 2 (u) = 0,I 3 (u) =1]

23 Backward Kolmogorov Eqns. Ψ( u;a) = µ u 0 ( a u)x( a u)ψ( u;a) Φ 1 ( u;a) 1 Φ 1 ( u;a) = µ u 1 ( a u)φ 1 ( u;a) Φ 2 ( u;a) 1 ( ) [ ] [ ] Φ 2 u;a = β( a u)+α( a u)φ 2 2(u;a) u [ α( a u)+β( a u)+µ 2 ( a u)(1 Φ 3 (u;a))]φ 2 (u;a) ( ) Φ 3 u;a u = 0

24 S 3,0 ( u;a) u = µ 0 ( a u)x( a u)s 3,0 ( u;a) S 3,1 ( u;a) 1 [ ] S 3,1 ( u;a) u = µ 1 ( a u)s 3,1 u;a ( ) S 3,2 ( u;a) 1 [ ] S 3,2 ( u;a) 2 = β( a u)+α( a u)s u 3,2 ( u;a) [ α( a u)+β( a u)+µ 2 ( a u) ]S 3,2 u;a ( ) S3,0( 0; a) = S3,1(0; a) = S3, 2(0; a) = 1

25 S 3 ( a)= S 3,0 a;a ( ) t = exp µ 0 X 0 qe p a u q p ( ) pe q ( a u ) µ 1 /α 1 du p,q = 1 2 ( α β µ 2)m α β µ 2 ( ) 2 +4αµ 2

26 3-stage Model Hazard h 3 ( a)= dln(s 3(a)) da µ q p 1 /α = µ 0 X 1 qe pa pe qa We can say more with some asymptotic analysis

27 age-specific cancer incidence Cancer Incidence Age (a)

28 age-specific cancer incidence - explained asymptotic value µ 0 X Cancer Incidence Age (a)

29 age-specific cancer incidence - explained Cancer Incidence µ 0 Xµ 1 p ( a T s ) Premalignant lesion incidence Ts mean sojourn time Age (a)

30 age-specific cancer incidence - explained asymptotic value Cancer Incidence Prem. lesion growth { } exp ( α β)a Age (a)

31 age-specific cancer incidence - explained Cancer Incidence 1 2 µ 0Xµ 1 µ 2 a 2 power law Age (a)

32 age-specific cancer incidence - explained asymptotic value µ 0 X Cancer Incidence 1 2 µ 0Xµ 1 µ 2 a 2 Prem. lesion growth exp ( α β)a µ 0 Xµ 1 p ( a T s ) { } Premalignant lesion incidence power law Ts mean sojourn time Age (a) Meza R et al, PNAS 2008

33 Pancreatic Cancer ( adjusted incidence) Age-specific Incidence per 100K Males Slope=3.9 Ts=52.9 Females Slope=2.8 Ts= Age Age Meza R et al, PNAS 2008

34 Colorectal cancer ( adjusted incidence) Age-specific Incidence per 100K Males Slope=20 Ts=56 Females Slope=16 Ts= Age Age Meza R et al, PNAS 2008

35 CURRENT PROJECTS

36 Lung cancer screening Does LC screening among heavy smokers reduce LC mortality Yes, with low dose CT screening One big expensive -trial has shown Extrapolate results of the trial Model relationship of smoking and LC Effects of screening Impact of radiation dose

37 Michigan/FHCRC Lung Cancer screening model. By gender and histology (SC,AC,SQ,ONSCLC) Normal X µ 0 µ 1 Preini.ated Pre malignant µ pm Preclinical α pm β pm α pc β pc Preclinical λ IA1 λ IA2 λ IB λ II λ IIIA λ IIIB IA1 IA2 IB II IIIA IIIB IV δ IA1 δ IA2 δ IB δ II δ IIIA δ IIIB δ IV Clinical Detec.on 37

38 Infec.ous agents and cancer Two disease processes with very different scales Popula.on vs individual Days vs years Persons vs cells/genes 38

39 A. Population level (SEIR Model) b βsi/n νe γi S E I R d d d d B. Individual level (Multistage Carcinogenesis Model) INFECTIOUS AGENT increase mutation rates Normal X µ 0 µ 1 µ 2 Gatek+/- Gatek-/- Cancer increase cell division α β INFECTIOUS AGENT reduce apoptosis

40 Cancer evolution Use new genetic data to infer the natural history and the dynamics of carcinogenesis Constrained by what s known at the population level

41

42 Conclusions Multistage carcinogenesis models - powerful framework for cancer risk analysis Complement to traditional statistical and epidemiological approaches mechanistic models Allows direct interpretation of results in terms of potential biological mechanisms Nice applied math area : stochastic modeling, dynamical systems, PDEs, ODEs, numerical analysis, statistics

43 Conclusions Other applications: Radiation risk assessment Toxicology Developmental mutations and cancer risk Public health policy

44 Conclusions Second cancers after radio- and chemo-therapy Cancers with infectious disease etiology Genomic, epigenomic and proteomic data Link between biological complexity and simplicity observed in public level data multi-scale modeling integrative cancer biology

45 Armitage P & Doll R. The age distribution of cancer and multistage theory of carcinogenesis. British J. Cancer 8:1-12, 1954 Whittemore A & Keller JB. Quantitative theories of carcinogenesis. SIAM Review 20, Moolgavkar SH & Venzon DJ. Two-event models for carcinogenesis: incidence curves for childhood and adult tumors. Mathematical Biosciences 47:55-77, 1979 Moolgavkar SH & Knudson A. Mutation and cancer: a model for human carcinogenesis. J Natl Cancer Inst. 66: , 1981 Kopp-Schneider A. Carcinogenesis models for risk assessment. Stat. Methods Med. Res. 6: , 1997 Luebeck EG & Moolgavkar SH. Multistage carcinogenesis and the incidence of colorectal cancer. PNAS 99: , 2002 Meza R, Luebeck EG & Moolgavkar SH. Gestational mutations and carcinogenesis. Mathematical Biosciences 197: , Meza R, Jeon J, Moolgavkar SH & Luebeck EG. Age-specific incidence of cancer: Phases, transitions, and biological implications. PNAS 105: , 2008 Meza R, Jeon J, Renehan AG, Luebeck EG (Jul 2010) Colorectal Cancer Incidence Trends in the United States and United Kingdom: Evidence of Right- to Left-Sided Biological Gradients with Implications for Screening., Cancer research, 70 (13),