Armitage and Doll (1954) Probability models for cancer development and progression. Incidence of Retinoblastoma. Why a power law?

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1 Probability models for cancer development and progression Armitage and Doll (95) log-log plots of incidence versus age Rick Durrett *** John Mayberry (Cornell) Stephen Moseley (Cornell) Deena Schmidt (MBI) Jason Schweinsberg (UCSD) Figure: Slopes Stomach: 5.9 M, 5.7 F; Pancreas M 5.76, F 6.8 Rick Durrett (Cornell) Banff 9//9 /8 Rick Durrett (Cornell) Banff 9//9 /8 Why a power law? Incidence of Retinoblastoma Knudson s two hit hypothesis If mutations from stage i tostagei occur at rate u i, the probability density of reaching stage k at time t is t k u u u k (k )! so the slope is the number of stages. Slopes Stomach: 5.9 M, 5.7 F; Pancreas M 5.76, F 6.8 Rick Durrett (Cornell) Banff 9//9 /8 Rick Durrett (Cornell) Banff 9//9 /8 Progression to Colon Cancer The Problem Luebeck and Moolgavakar () PNAS fit a four stage model to incidence of colon cancer by age. Given a population of size N, how long does it take until τ k the first time we have an individual with a prespecified sequence of k mutations? Initially all individuals are type. Each individual is subject to replacement at rate. A copy is made of an individual chosen at random from the population. Type j mutatestotypej at rate u j. Rick Durrett (Cornell) Banff 9//9 5/8 Rick Durrett (Cornell) Banff 9//9 6/8

2 Theorem. If Nu andn u Idea of Proof P(τ > t/nu u ) e t, simulations of n =, u =, u = Since s mutate to s at rate u, τ will occur when there have been O(/u ) births of individuals of type. Probability Density Rescaled Time The number of s is roughly a (time change of ) symmetric random walk, so τ will occur when the number of s reaches O(/ u ). N >> / u guarantees that up to τ the number of s is o(n), so mutations occur at rate Nu, and s that have descendants occur at rate Nu u The waiting time from the mutation until the mutant appears is of order / u. For this to be much smaller than the overall waiting time /Nu u we need Nu <<. Rick Durrett (Cornell) Banff 9//9 7/8 Rick Durrett (Cornell) Banff 9//9 8/8 Waiting for k mutations Durrett, Schmidt, and Schweinsberg Total progeny of a critical binary branching process has P(ξ >k) Ck /,sothesumofm such random variables is O(M ). To get individual of type, we need of order /u births of type. / u mutations to type. /u u births of type. /u / u / mutations to type. /u u / u / births of type. /u / u / u /8 mutations to type. Probability type j has a type k descendant. r j,k = u / u/k j j+ u/ j+ k for j < k Theorem. Let k. Suppose that: (i) Nu. (ii) Forj =,...,k, u j+ /u j > b j for all N. (iii) Thereisana > sothatn a u k. (iv) Nr,k. Then for all t >, lim N P(τ k > t/nu r,k )=exp( t). Rick Durrett (Cornell) Banff 9//9 9/8 Rick Durrett (Cornell) Banff 9//9 / 8 Small time behavior Exponentially growing population, Most cancers occur in less than % of the population so we are looking at the lower tail of the distribution. Let g k (t) =Q (τ k t) where Q is the probability for the branching process started with one type. In the case u j μ g j (t) =μg j (t) ( μ)g j (t) μg j (t) One can inductively solve the differential equations and finds If t << μ / then g k (t) μ k t k /(k )! Schweinsberg (8) Electronic J. Probab., 78 Joint work with Stephen Moseley. Some chronic myeloid leukemia patients show resistance to imatinib at diagnosis, and many others develop resistance during the first year of treatment. Iwasa, Nowak, and Michor (6) and Haeno, Iwassa, Michor (7), both in Genetics. Model is a multi-type branching process in which type i cells have i mutations. Type i cells give birth at rate a i and die at rate b i. λ i = a i b i increases in i. Type i s mutate at rate u i+ becoming type i +. Rick Durrett (Cornell) Banff 9//9 / 8 Rick Durrett (Cornell) Banff 9//9 / 8

3 Math questions Growth of the s Compute the distribution of τ k bethetimeoftheoccurrenceofthefirst type k. k =, most relevant to development of immunity. Let Z k (t) be the number of type k cells at time t. Find the limiting behavior of e λkt Z k (t). t ) P(τ > t Z (s), s t, Ω )=exp ( u Z (s)ds (e λs Z (s) Ω ) V = exponential(λ /a )so ( ) P(τ > t Ω ) E exp u V e λt /λ ) λ = λ + a u e λt /λ (Zt, Zt,...Zt k ) is a decomposable Galton-Watson process, which Kesten and Stigum studied in discrete time. For any k M k t = e λ kt Z k (t) t Show that Mt is L bounded and conclude Theorem. e λt Z (t) W a.s. with u k e λ ks Z k (s) ds is a martingale EW = u /(λ λ ). Rick Durrett (Cornell) Banff 9//9 / 8 Rick Durrett (Cornell) Banff 9//9 / 8 W has a power law tail Proof of power law tail Let Zi (t) be the number of type-i s at time t in a system with Z (t) =eλt V for all t (, ). Theorem. e λt Z (t) V a.s. with Ee θv =/( + u c θ, θ λ/λ ) and hence P(V > x) c V, x λ/λ Figure: Simulated distribution Actually W does not have a power law tail but P(W V )=u a /λ is small. Rick Durrett (Cornell) Banff 9//9 5 / 8 Rick Durrett (Cornell) Banff 9//9 6 / 8 Results from Sjoblom et al (6): 5 tumors Results from Wood et al. (7) Science Figure: Last three columns: APC (), p5 (7), K-ras (6) Rick Durrett (Cornell) Banff 9//9 7 / 8 Rick Durrett (Cornell) Banff 9//9 8 / 8

4 Flawed Methodology? Exponentially growing population, Wood et al. (7): of the top 9 genes, selected based on the pathways in which they occur, were chosen for further sequencing. 5 of the genes (8%) were not mutated in any of the 96 tumors studied. False Discovery Rate of %?? Joint work with John Mayberry In some cancers, e.g., colon cancer, an early stage is the growth of a benign tumor, before progression to malignancy. Genetic Progression and the Waiting Time to Cancer Niko Beerenwinkel, Tibor Antal, David Dingli, Arne Traulsen, Kenneth W. Kinzler, Victor E. Velculescu, Bert Vogelstein, Martin A. Nowak PLoS Computational Biology (7) e5 Wright-Fisher model in exponentially growing population. Cells with k mutations have relative fitness ( + γ) k. Rick Durrett (Cornell) Banff 9//9 9 / 8 Rick Durrett (Cornell) Banff 9//9 / 8 Populations of fixed size Theorem. Suppose that X () = N and N = μ α for some α>. Let L =log(/μ). As μ, Y N k (t) L log+ (X k (Lt/γ)) y k (t) uniformly on compact subsets of (, ). The limit y k (t) is deterministic and piecewise linear. In applications γ is small, e.g.,., and the limit process is almost independent of γ. Figure: Simulation from Beerenwinkel et al. Rick Durrett (Cornell) Banff 9//9 / 8 Rick Durrett (Cornell) Banff 9//9 / 8 Inductive definition First regime α (, /). Limit when α =. Suppose we have defined the timit to time s n. Let m =max{j : y j (s n )=α}. Supposethat (i) y j (s n )=forj > k, y j (s n ) > form < j k, (ii) y j+ (s n ) y j (s n ) Let K = k if y k (s n ) <, and K = k +ify k (s n )=. γ m =(+γ) m. Then for t Δ n (y j (s n )+tγ j m /γ) + j < m y j (s n + t) = y j (s n )+tγ j m /γ m j K. j > K Rick Durrett (Cornell) Banff 9//9 / 8 Rick Durrett (Cornell) Banff 9//9 / 8

5 Second regime α (/, /6). Limit for α =.8 Third regime α (/6, 5/). Limit for α = Size of Y k Time (Units of L/γ) Rick Durrett (Cornell) Banff 9//9 5 / 8 Rick Durrett (Cornell) Banff 9//9 6 / 8 Exponentially growing population.5.5 y k (t) t Figure: Simulation from Beerenwinkel et al. Rick Durrett (Cornell) Banff 9//9 7 / 8 Rick Durrett (Cornell) Banff 9//9 8 / 8