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Stochastic modeling of carcinogenesis
Rafael Meza Department of Epidemiology
University of Michigan SPH-II 5533
[email protected] http://www.sph.umich.edu/faculty/rmeza.html
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
Armitage-Doll Model (1954)
Exponential waiting time
E0 E1 E2 En λ0 λ1 λ2 λn-1
Normal Stem Cell
Malignant Cell
Armitage-Doll Model (1954)
€
Let pk (a) be the probability that the cell is at stage kat age a
)()(
)()()(
)()(
11
11001
000
apdaadp
apapdaadp
apdaadp
nnn
−−=
−=
−=
λ
λλ
λ
Armitage-Doll Model (1954)
( )Nn apaSaP
N
)(1)(] ageby cancer No[
:cells stem esusceptibl are thereAssume
−==
Cancer hazard (age-specific cancer risk):
h(a) = −d ln S(a)[ ]
da≈Nλ0λ1λn−1a
n−1
(n−1)!
Armitage-Doll Model (1954) Age-Specific Incidence
)log()1()!1(
log))(log( 110 ann
Nah n −+⎟⎟⎠
⎞⎜⎜⎝
⎛
−≈ −λλλ …
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 …
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
TSCE Model
β(t)
Moolgavkar & Venzon (Math. Biosc, 1979); Moolgavkar & Knudson (JNCI, 1981)
Non-homogeneous Poisson Process
Birth-Death-Mutation Process
with initial condition Ψ(y,z,0)=1
,,
( , , ) ( ) j kj k
j ky z t P t y zΨ ≡∑
Let,
TSCE Model
€
∂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
As a continuous time Markov Process
Forward-Kolmogorov equation
TSCE Model
€
S(t) =q − p
qe− pt − pe−qt⎛
⎝ ⎜
⎞
⎠ ⎟
νXα
h(t) =νXα
pq e−qt − e−pt( )qe− pt − pe−qt
p,q =12−(α − β − µ)m (α − β − µ)2 + 4αµ[ ]
• 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
TSCE Model
β
Normal X Gatek+/- Gatek-/-
Premalig. Cancer
α
µ0 µ1 µ2
A simple 3-stage Model
Premalignant lesion Onset sojourn time s
€
τ
Φ1(u;a) Φ3(u;a) Φ2(u;a) Ψ(u;a)
3-stage Model
I1 I2 I3 X
Normal X Gatek+/- Gatek-/-
Premalig. Cancer
α
µ0 µ1 µ2
β
As a continuous time branching process
Probability Generating Functions
€
ψ(y1,y2,y3,u;a) = E[y1I1 (a )y2
I 2 (a )y3I 3 (a ) | I1(u) = 0,I2(u) = 0,I3(u) = 0]
Φ1(y1,y2,y3,u;a) = E[y1I1 (a )y2
I 2 (a )y3I 3 (a ) | I1(u) =1,I2(u) = 0,I3(u) = 0]
Φ2(y1,y2,y3,u;a) = E[y1I1 (a )y2
I 2 (a )y3I 3 (a ) | I1(u) = 0,I2(u) =1,I3(u) = 0]
Φ3(y1,y2,y3,u;a) = E[y1I1 (a )y2
I 2 (a )y3I 3 (a ) | I1(u) = 0,I2(u) = 0,I3(u) =1]
Backward Kolmogorov Eqns.
€
∂Ψ u;a( )∂u
= µ0 a − u( )X a − u( )Ψ u;a( ) Φ1 u;a( )−1[ ]
∂Φ1 u;a( )∂u
= µ1 a − u( )Φ1 u;a( ) Φ2 u;a( )−1[ ]
∂Φ2 u;a( )∂u
= β a − u( )+α a − u( )Φ22(u;a)
− α a − u( )+β a − u( )+µ2 a − u( )(1−Φ3(u;a))[ ]Φ2(u;a)
∂Φ3 u;a( )∂u
= 0
€
∂S3,0 u;a( )∂u
= µ0 a − u( )X a − u( )S3,0 u;a( ) S3,1 u;a( )−1[ ]
∂S3,1 u;a( )∂u
= µ1 a − u( )S3,1 u;a( ) S3,2 u;a( )−1[ ]
∂S3,2 u;a( )∂u
= β a − u( )+α a − u( )S3,22 u;a( )
− α a − u( )+β a − u( )+µ2 a − u( )[ ]S3,2 u;a( )
1);0();0();0( 2,31,30,3 === aSaSaS
€
S3 a( )= S3,0 a;a( )
= exp µ00
t
∫ Xq − p
qe−p a−u( ) − pe−q a−u( )
⎛
⎝ ⎜
⎞
⎠ ⎟
µ1 /α−1
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ du
⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪
p,q= 12− α − β − µ2( )m α − β − µ2( )2 +4αµ2⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
h3 a( )= −d ln(S3(a))
da
= µ0X 1−q − p
qe− pa − pe−qa⎛
⎝ ⎜
⎞
⎠ ⎟
µ1 /α⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
3-stage Model Hazard
We can say more with some asymptotic analysis
€
µ0Xµ1p∞ a −Ts( )Premalignant lesion incidence
Ts
age-specific cancer incidence - explained
mean sojourn time
Age (a)
Can
cer I
ncid
ence
asymptotic value
€
exp α − β( )a{ }
Prem. lesion growth
age-specific cancer incidence - explained
Age (a)
Can
cer I
ncid
ence
€
µ0Xasymptotic value
€
µ0Xµ1p∞ a −Ts( )Premalignant lesion incidence
Ts €
exp α − β( )a{ }
Prem. lesion growth
age-specific cancer incidence - explained
power law
€
12
µ0Xµ1µ2a2
mean sojourn time
Age (a)
Can
cer I
ncid
ence
Meza R et al, PNAS 2008
Age Age
Slope=3.9
Slope=2.8
Ts=52.9 Ts=56.3
Age
-spe
cific
Inci
denc
e pe
r 100
K
Males Females
30 40 50 60 70 80 30 40 50 60 70 80
0
20
4
0
60
8
0
100
12
0 Pancreatic Cancer
(“adjusted” incidence)
Meza R et al, PNAS 2008
Age Age
Slope=20 Slope=16
Ts=56 Ts=57.5 Age
-spe
cific
Inci
denc
e pe
r 100
K Males Females
30 40 50 60 70 80 30 40 50 60 70 80
0
100
20
0 3
00 4
00
500
600
Colorectal cancer (“adjusted” incidence)
Meza R et al, PNAS 2008
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
Preclinical
Normal X Preini.ated Pre
malignant
αpm βpm
µ0 µ1 Preclinical
αpc βpc
µpm
IA1 IA2 IB II IIIA IIIB IV
Clinical Detec.on
λIA1 λIA2 λIB λII λIIIA λIIIB
δIA1 δIA2 δIB δII δIIIA δIIIB δIV
Michigan/FHCRC Lung Cancer screening model. By gender and histology (SC,AC,SQ,ONSCLC)
37
Infec.ous agents and cancer
• Two disease processes with very different scales – Popula.on vs individual
– Days vs years
– Persons vs cells/genes
38
S E I R βSI/N νE γI
b
d d
A. Population level (SEIR Model)
B. Individual level (Multistage Carcinogenesis Model)
Normal X Gatek+/- Gatek-/- Cancer
α β
µ0 µ1 µ2
INFECTIOUS AGENT
INFECTIOUS AGENT
increase cell division
reduce apoptosis
increase mutation rates
d d
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
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
Conclusions • Other applications:
– Radiation risk assessment
– Toxicology
– Developmental mutations and cancer risk
– Public health policy
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
• 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, 1978.
• 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. 66:1037-52, 1981
• Kopp-Schneider A. Carcinogenesis models for risk assessment. Stat. Methods Med. Res. 6: 317-340, 1997
• Luebeck EG & Moolgavkar SH. Multistage carcinogenesis and the incidence of colorectal cancer. PNAS 99:15095-15100, 2002
• Meza R, Luebeck EG & Moolgavkar SH. Gestational mutations and carcinogenesis. Mathematical Biosciences 197:188-210, 2005.
• Meza R, Jeon J, Moolgavkar SH & Luebeck EG. Age-specific incidence of cancer: Phases,
transitions, and biological implications. PNAS 105:16284-9, 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), 5419-5429