This talk discusses the basic mathematical approaches and motivations underlying mechanistic models of carcinogenesis, specifically multi-stage models. After discussing simple ODE-based deterministic models, stochastic cancer models are introduced. On the simplest examples of the 1-stage (Poisson) process and a minimal 2-stage model, the basic features of such models are laid out. We then proceed to treat the widely used two-stage model with clonal expansion (TSCE), and its application to calculating risks due to external agents, such as radiation.

1. Looking inside mechanistic models of
carcinogenesis
Sascha Zöllner
Helmholtz Zentrum München (Germany)
Institute of Radiation Protection

2. Motivation
• Qualitative behavior?
• Why do we need
mechanistic model?

3. Outline
1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

4. 1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

5. Modeling cancer: Key ingredients
(epi)genetic transitions: X0
µ
→ X1
µ
→ ···
µ
→ Xk
dXk
dt
= µXk−1 =⇒ Xk(t) =
t
0
dt Xk−1µ
(...)
= X0(µt)k
/k!
[Armitage/Doll (1957)] polynomial growth
clonal expansion of pre-malignant cells: X1
γ
X1 +1
dX1
dt
= γX1 =⇒ X1(t) = X1(0)eγt
exponential growth

6. Modeling cancer: Key ingredients
(epi)genetic transitions: X0
µ
→ X1
µ
→ ···
µ
→ Xk
dXk
dt
= µXk−1 =⇒ Xk(t) =
t
0
dt Xk−1µ
(...)
= X0(µt)k
/k!
[Armitage/Doll (1957)] polynomial growth
clonal expansion of pre-malignant cells: X1
γ
X1 +1
dX1
dt
= γX1 =⇒ X1(t) = X1(0)eγt
exponential growth

7. k = 2 stages
˙X1 = Nµ0 +X1γ (γ ≡ α −β)
˙X2 = X1µ1
Link to medical data
• Hazard (↔Survival S)
h =
probability of new case in (t,t +∆t]
time step ∆t
=
−d lnS
dt
• Deterministic approximation:
h ≈
d
dt
X2 = µ1X1 =⇒ ˙h ≈ Nµ0µ1 +γh

8. Deterministic approximation
˙h ≈ Nµ0µ1 +γh =⇒ Solution: h(t) = Nµ0µ1
γ (eγt −1)
• early age: h(t) (Nµ0µ1)t∝ tk−1
• driven by mutations
• k = 1: h(t) = Nµ0 ∝ t0
always
• larger ages: h(t) ∼ Nµ0µ1
γ eγt
• governed by proliferation
• unbounded: h ∝ X1 → ∞ !? 0 10 20 30
t
0.2
0.4
0.6
0.8
1.0
h t

9. What’s the problem with h → ∞?
1 Accuracy
• typical data: slowing incidence at t t∗, w/ t∗ ∼ 60−90
• good approximation for t t∗
2 Conceptually...?
• Problem with cancer probability? Consider density ρ:
ρ(t) = h(t)× S(t)
=exp(− h)
→ 0 OK!
• ...but this implies near-total extinction of population:
S = O(e−eγt
) 1

10. What’s the problem with h → ∞?
1 Accuracy
• typical data: slowing incidence at t t∗, w/ t∗ ∼ 60−90
• good approximation for t t∗
2 Conceptually...?
• Problem with cancer probability? Consider density ρ:
ρ(t) = h(t)× S(t)
=exp(− h)
→ 0 OK!
• ...but this implies near-total extinction of population:
S = O(e−eγt
) 1

11. Can we fix it?
1 Go beyond deterministic approximation h = µ1X1
Stochastic model (Sec. 2)
2 Within deterministic model? Start from
h(t) = µ1(t)X1(t); ˙X1 = Nµ0 +γX1
Phenomenological modifications:
• Compensate X1 ∼ eγt? Age-dependent µ1(t) ∼ e−γt (as t → ∞)
• Keep X1 bounded? ˙X1
!
= 0
• Age-dependent γ(t), Nµ0(t)...?
• ad-hoc cell-cell interaction term: ˙X1 = Nµ0 +γX1−εX2
1
[Sachs, Rad. Research 164 (2005)]

12. Can we fix it?
1 Go beyond deterministic approximation h = µ1X1
Stochastic model (Sec. 2)
2 Within deterministic model? Start from
h(t) = µ1(t)X1(t); ˙X1 = Nµ0 +γX1
Phenomenological modifications:
• Compensate X1 ∼ eγt? Age-dependent µ1(t) ∼ e−γt (as t → ∞)
• Keep X1 bounded? ˙X1
!
= 0
• Age-dependent γ(t), Nµ0(t)...?
• ad-hoc cell-cell interaction term: ˙X1 = Nµ0 +γX1−εX2
1
[Sachs, Rad. Research 164 (2005)]

13. 1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

14. 1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

15. Stochastic process
• So far: “Sharp” # of cells Xi (t)
• But cancer evolution is stochastic process:
Xi (t) are random, w/ probability
Prob{X1(t) = x1,...,Xk(t) = xk} =: Px1...xk
(t) ≡ Px (t)
Goal
• Find Px (t) with initial condition Px (0) = δx,0 (healthy cells only)
• More precisely, we want to model time evolution
P(t0) → P(t)

16. Markov process
• Markov’s condition: “Short-memory” time evolution,
i.e., {Px (t)} completely determines Px (t +∆t)
• leads to Chapman-Kolmogorov eq.
Px (t +∆t) = ∑
x
Px (t)px →x
• Transition probabilities
• normalized: ∑x px →x = 1
• completely define the time evolution (i.e., parametrize our model!)

17. Markov process
Continuous-time process
Px (t +∆t) = ∑
x (=x)
Px (t)px →x +Px (t) 1− ∑
x (=x)
px→x
= Px (t)+ ∑
x (=x)
(Px (t)px →x −Px (t)px→x )
Master equation
• Take ∆t → 0, assuming px →x(=x ) Ax ,x ∆t:
d
dt
Px (t) = ∑
x
Px (t)Ax ,x −Px (t)Ax,x
• Formal solution: ˙P(t) ≡ A P(t) =⇒ P(t) = eA tP(0)

18. 1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

19. 1-stage (Poisson) process
• States: x ≡ (x1) – #cells in stage 1
• Assume only 1 transition (from “healthy” → “malignant”)
px →x = Nµ0∆t if x = x −1
Px (t +∆t) = Px−1(t)Nµ0∆t +Px (t)(1−Nµ0∆t)
• Transitions between states:
(x1 = 0)
µ0
−→ (1)
µ0
−→ (2)
µ0
−→ ···
continuous transfer from (x1 = 0) toward (x1 → ∞)

20. 1-stage process: Master equation
˙Px (t) = Nµ0


 Px−1(t)
transfer from x−1→x
− Px (t)
transfer x→x+1


; Px (0) = δx,0
Solve:
˙Px=0 = −Nµ0P0 ⇒ P0(t) = e−Nµ0t
˙P1 = Nµ0(P0 −P1) ⇒ P1(t) = e−Nµ0t
Nµ0t
...
...

21. Solution: Poisson distribution
Px (t) = e−Nµ0t (Nµ0t)x
x!
• E (X1) ≡ ¯X1(t) = Nµ0t – probability “travels” w/ speed Nµ0
• Var(X1) ≡ ∆X2
1 = Nµ0t – spreads out
• Steady state as t → ∞? ˙Px (t) = Nµ0 [Px−1(t)−Px (t)]
?
= 0
0 2 4 6 8 10
x
0.2
0.4
0.6
0.8
1.0
P x;t

22. A toy model
Imagine the #cells, x, were continuous: Px (t) =: P(x;t)
˙P(x;t) = −Nµ0 [P(x;t)−P(x −1;t)] → −Nµ0
∂
∂x
P(x;t)
Solution: Any “traveling wave” with P(x;t) = f (x −Nµ0t)
• Proof: ∂tf (x −Nµ0t) = f (x −Nµ0t)
=∂x f
×[−Nµ0]
• Gives “central” dynamics, but no diffusion

23. Back to “deterministic” model
So far, solved whole problem, Px (t)
What is specific dynamics of mean cell #, ¯X(t)?
d
dt
¯X(t) = ∑
x
x ˙Px (t)
= ∑
x
x Nµ0 (Px−1 −Px )
= Nµ0 X +1−X = Nµ0
∴ Heuristic model in Sec. 1 ⇐⇒ Exact dynamics of ¯X(t)

24. Link to risk model
What is the hazard / survival probability for this model?
S(t) = Prob(Tcancer > t) =?
Simplest model: Interpret person as healthy:⇔ X1(t) = 0
S(t) = Prob{X1(t) = 0} = P0(t)
• Survival: S(t) = e−Nµ0t
• Hazard: h(t) = − d
dt lnS(t) = +Nµ0
• Same as deterministic model!
• Age-independence not realistic for cancer data/biology – let’s move on!

25. Link to risk model
What is the hazard / survival probability for this model?
S(t) = Prob(Tcancer > t) =?
Simplest model: Interpret person as healthy:⇔ X1(t) = 0
S(t) = Prob{X1(t) = 0} = P0(t)
• Survival: S(t) = e−Nµ0t
• Hazard: h(t) = − d
dt lnS(t) = +Nµ0
• Same as deterministic model!
• Age-independence not realistic for cancer data/biology – let’s move on!

26. 1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

27. Two mutation steps
• States: x ≡ (x1,x2) – #cells in stage 1 (pre-) and 2 (malignant)
• 2 possible transitions:
px →x =
Nµ0∆t if x = (x1 −1,x2)
x1µ1∆t if x = (x1,x2 −1)
• Now 2-D transition chain:
(x1 = 0,x2 = 0)
µ0
−→ (1,0)
µ0
−→ (2,0)
µ0
−→ ···
↓µ1 ↓µ1
(1,1)
µ0
−→ (2,1)
µ0
−→ ···
↓µ1 ↓µ1

28. 2-stage process: Master equation
˙Px1x2 (t) = Nµ0 (Px1−1,x2 (t)−Px1,x2 (t))
+x1µ1 (Px1,x2−1(t)−Px1,x2 (t))
If we are only interested in the hazard, h = − ˙S/S, with
S(t) = Prob{X2(t) = 0} = ∑
x1
Px1,0
then we only need the x2 = 0 entries:
˙Px1,0(t) = Nµ0 [Px1−1,0(t)−Px1,0(t)]−x1µ1Px1,0(t)
Same as 1-stage process, but w/ loss term for high #1-cells (1→2)

29. Hazard
˙S = ∑
x1
˙Px1,0 = −µ1 ∑
x1
x1Px1,0
h = −
˙S
S
= +µ1 ∑
x1
x1
Px1,0
∑x Px,0
= µ1E (X1|X2 = 0)
• Looks like deterministic approximation,
but w/ mean ¯X1|0 conditional on X2 = 0!
• Relevant probability distribution for X1:
Px1|0 ≡ Px1,0/∑x Px,0 (normalized)

30. Conditional X1 distribution
Px1|0 obeys the Master-like equation
˙Px1|0(t) = Nµ0 Px1−1|0(t)−Px1|0(t) − x1 − ¯X1|0(t) µ1Px1|0(t)
• 1st term: push toward x1 → ∞ (µ0)
• 2nd term: redistribute to x1 < ¯X1|0 (µ1)
• Approaches steady state, ˙Px1|0 = 0: Balance between x1-input (from
healthy cells) and output (to malign cells)
• Explicit solution (const. parameters): Poisson
Px|0(t) = e− ¯X1|0(t)
¯X1|0(t)x
x!
, with ¯X1|0(t) =
Nµ0
µ1
1−e−µ1t

31. Back to hazard
Since h = µ1
¯X1|0, all we need is effective equation for ¯X1|0:
d
dt
¯X1|0 = ∑
x1
x1 Nµ0 Px1−1|0 −Px1|0 − x1 − ¯X1|0 µ1Px1|0
= Nµ0 X1 +1−X1 |0
− X2
1|0 − ¯X2
1|0 µ1
d
dt
¯X1|0 = Nµ0 −∆X2
1|0µ1
• “Deterministic” term (from 1st step) – describes mean X1
• “Stochastic” fluctuation term (2nd step)
• leads to steady state!
• similar effect as phenomenological term inhibiting cell growth (Sec. 1)

32. Hazard: Constant parameters
• For const. parameters: Px1|0 Poissonian ⇒ ∆X2
1|0 = ¯X1|0
d
dt
¯X1|0 = Nµ0 − ¯X1|0µ1
• strong “damping” if many pre-malign cells (likely already malign –
discount, since no longer cause new cancer)
• Rewrite in terms of h = ¯X1|0 × µ1:
˙h = µ1(Nµ0 −h) h(t) = Nµ0 1−e−µ1t

33. Summary so far
Deterministic model
• Exact equations for mean #cells ˙X1 = µ0X0 +γ1X1
• Approximation for hazard: h ≈ µ1X1
• works well for earlier ages: polynomial / exp. growth
• growth unbounded!
Stochastic model
• Exact hazard: h = µ1E (X1|X2 = 0) ≡ µ1
¯X1|0
• ¯X1|0 obeys similar equation as ¯X1:
• same deterministic term (more 1-cells due to µ0)
• extra fluctuation term (fewer 1-cells due to µ1)
• Equilibrium at older age: Hazard saturates

34. 1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

35. 1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

36. 2-stage model w/ clonal expansion
• Deterministic model: ˙h = Nµ0µ1 +(α −β)h
• Include stochastic term −∆X2
1|0 (constant parameters):
˙h = Nµ0µ1 +(α −β − µ1
=:γ
)h−
α
Nµ0
h2
• 1st-order term −µ1h (“Poisson contribution ∆X2
= ¯X”)
• 2nd-order term ∝ −αh2
(“high α X1 ↑ increased loss to 2-cells”)
[Moolgavkar (1979-81)]

37. Phases
h
h'
deterministic
Nµ0µ1/q
Nµ0µ1
exact
0 20 40 60 80
t
0.2
0.4
0.6
0.8
1.0
h t
What can we learn from ˙h = Nµ0µ1 +γh − α
Nµ0
h2?
• initially: ˙h Nµ0µ1 +γh =⇒ h(t) Nµ0µ1
γ [eγt −1]
• max. growth: ¨h(t∗) = 0 =⇒ h(t) h(t∗)+ ˙h(t∗)(t −t∗)
• steady state: ˙h → 0 =⇒ h(t) → Nµ0µ1/q
w/ q(q +γ) ≡ αµ1; γt∗ ≡ ln γ+q
q

38. Effective parameters
˙h = Nµ0µ1 +γh −
α
Nµ0
h2
• Scaling invariance only 3 parameters “identifiable” from h:
• Nµ0µ1; γ (deterministic – early age)
• αµ1 (stochastic)
• One interpretation:
• time scale: t = γt
• hazard scale: h = hγ/Nµ0µ1
• functional shape: ε ≡ αµ1/γ2 ˙h = 1+h −εh 2

39. 1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

40. Multi-stage models
...
X0=N
healthy
cells
Xk-1
initiated
cells
µ0
μk-1
αk-1
division
βk-1
inactivation/
differentiation
Xk
malignant
cells
X1
initiated
cells
µ1µ1µ1µ1 µk-2
α1
division
β1
inactivation/
differentiation
k-stage model w/ clonal expansion (k = 2 ⇐⇒ TSCE)
• Parameters
• transitions (µ0,...,k−1): k parameters
• proliferation (α1...k−1;β1...k−1): 2(k −1) – too many!
• Qualitative behavior?
[Little, Int. J. Rad. Biol. 78 (2002)]

41. Example: 3-stage “pre-initiation” model
Analytic solution
h(t) = Nµ0

1−
qe(γ+q)t +(γ +q)e−qt
γ +2q
−µ1/α


• 4 (out of 5) identifiable parameters: Nµ0, γ, q, plus µ1/α...
• Phases: polynomial ∝t2, exponential, linear, saturation to Nµ0
[Luebeck, PNAS 99 (2002); Meza, PNAS 105 (2008)]

42. What can we really learn?
Can we identify the # of stages from data?
• Cancer biology: Many different stages (pathways) involved (k 1)
— why use 2-stage model?
• Modeling: Consider 2 models
• simple 2-stage: h2(t) = Nµ0(1−e−µ1t
)
• 1-stage: h1(t) = Nν(t) w/ ν(t) ≡ µ0(1−e−µ1t
)
Multistage dynamics not unambiguously observable
• Plausible assumptions (rates constant unless motivated, #stages)
• Only slow enough (“rate-limiting”) steps leave imprint
• Need large enough data set to detect qualitative difference

43. What can we really learn?
Can we identify the # of stages from data?
• Cancer biology: Many different stages (pathways) involved (k 1)
— why use 2-stage model?
• Modeling: Consider 2 models
• simple 2-stage: h2(t) = Nµ0(1−e−µ1t
)
• 1-stage: h1(t) = Nν(t) w/ ν(t) ≡ µ0(1−e−µ1t
)
Multistage dynamics not unambiguously observable
• Plausible assumptions (rates constant unless motivated, #stages)
• Only slow enough (“rate-limiting”) steps leave imprint
• Need large enough data set to detect qualitative difference

44. 1 Basic ingredients of multi-stage models
2 Stochastic models
Stochastic processes
Simplest case: 1-stage process
2-stage process (w/o clonal growth)
3 2-stage model (and beyond)
Qualitative features: Hazard
Beyond the 2-stage model
Time-dependent parameters

45. What about radiation...?
• Radiation-induced (epi-)genetic effects
• non-repair: apoptosis β ↑
• misrepair: additional mutations µi ↑, (α −β) ↑
• ...
• Toy model to understand effects:
µi (t) = µ
(0)
i [1+f (t)], etc.
f (t) =
const. t ∈ [t1,t2]
0 else
How could we solve that?

46. What about radiation...?
• Radiation-induced (epi-)genetic effects
• non-repair: apoptosis β ↑
• misrepair: additional mutations µi ↑, (α −β) ↑
• ...
• Toy model to understand effects:
µi (t) = µ
(0)
i [1+f (t)], etc.
f (t) =
const. t ∈ [t1,t2]
0 else
How could we solve that?

47. µ1(t): Deterministic model
h(t) = µ1(t)X1(t); ˙X1 = Nµ0 +γX1
µ1 = µ
(0)
1 (1+f ) =⇒ h = h(0)
(1+f )
• Jump at t1,2: ∆h = ∆µ1X1 — instant effect!
• Excess relative risk = f

48. µ1(t): Stochastic corrections
d
dt X = Nµ0 +γX − µ1∆X2
• µ1 increases at t1 ⇒ Damping term enhanced
• µ1 decreases at t2 ⇒ Damping term reduced
• Hazard “pulled back” toward baseline
[Heidenreich, Risk Anal. 17 (1997]

49. Now for µ0(t):

50. µ0(t)
d
dt X = Nµ0 +γX − µ1∆X2
• Kinks (no jumps): ∆˙h = µ1 ×N∆µ0 – extra stage delays effect
• Slow return to baseline:
Kink at t2 neglible compared to accumulated clonal growth

51. Excercise: α(t)