Stan Ulam once said: Branching Process Models of Cancer. I have sunk so low that my last paper contained numbers with decimal points.

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1 Branching Process Models of Cancer Rick Durrett... A report on joint work with Stephen Moseley (Cornell consulting in Boston) with Jasmine Foo, Kevin Leder (Minnesota), Franziska Michor (Dana Farber Cancer Institute), and John Mayberry (Cornell postdoc U. of the Pacific), and with Kaveh Danesh (Duke Undergraduate) Laura Havrilesky and Evan Myers (Obstetrics and Gynecology, Duke Medical Center). Stan Ulam once said: I have sunk so low that my last paper contained numbers with decimal points. Figure: Feynman, Ulam, von Neumann Rick Durrett (Duke) NESS 4/2 /3 Rick Durrett (Duke) NESS 4/2 2/3 Armitage and Doll (94) Noticed that log-log plots of cancer incidence data are linear for a large number of cancer types; for example, colorectal cancer incidence has a slopeof.8inmenand4.97inwomen. Multi-stage theory of carcinogenesis Armitage and Doll (94) use the observation that the slopes were.8 in men and 4.97 in women to argue that colon cancer is a six stage process. The math was very simple Suppose X i are independent and have an exponential distribution with rates u i.thesumx + + X k has a density function that is asymptotically t k u u k as t (k )! Rick Durrett (Duke) NESS 4/2 3/3 Rick Durrett (Duke) NESS 4/2 4/3 Incidence of Retinoblastoma Knudson s two hit hypothesis tumor-suppressor genes Progression to Colon Cancer Luebeck and Moolgavakar (22) PNAS fit a four stage model to incidence of colon cancer by age. Rick Durrett (Duke) NESS 4/2 /3 Rick Durrett (Duke) NESS 4/2 6/3

2 What are the stages? Multitype Markovian binary branching process In sporadic cases of colon cancer the first two stages are inactivation of the tumor suppressor gene APC adenomatous polyposis coli. KRAS is an oncogene (one mutation turns it on). Once it is turned on it recruits and activates proteins necessary for the propagation of growth factor The final stage is thought to involve the inactivation of TP3 the gene which makes p3. Mutant p3 can no longer bind DNA in an effective way, and as a consequence the p2 protein whose production it stimulates is not made available to act as the stop signal for cell division. Z i (t) is the number of type i cells Type i cell give birth at rate a i and die at rate b i. Yeswehavedeathsatrateb. Type i cells produce offspring of type i +atrateu i+. Rick Durrett (Duke) NESS 4/2 7/3 Rick Durrett (Duke) NESS 4/2 8/3 Type s are a branching process Type s: Durrett and Moseley (29) Birth at rate a, death at rate b, λ = a b. As t, e λt Z (t) W a.s. P(Z (t) =forsomet ) = b /a b W = d δ + λ exponential(λ /a ) a a exponential(r) has density re rt,mean/r. If we condition on nonextinction the limit is exponential(λ /a ). For simplicity suppose V is a constant EZ (t) = t V e λs u e λ(t s) ds Theorem. As t,e λt Z (t) W a.s. with EW = V u λ λ This overestimates Z (t) because the dominant contribution to the integral comes from s near,wheremutationsarerarebuthaveahugeeffect. Rick Durrett (Duke) NESS 4/2 9/3 Rick Durrett (Duke) NESS 4/2 / 3 A better approach Growth rate of type k s Suppose Z (t) =V e λt for t (, ) where V is exponential(λ /a ). Let Z (t) be the number of s when Z (t) =V e λt, t (, ). The expected number of mutations at times isv u /λ which is small for typical values u =, s =.2. Theorem. As t,e λt Z (t) V a.s. with E(exp( θv ) V )=exp( c h, u V θ α ) and α = λ /λ. (V V ) is one sided stable law with index α (, ). P(V > x) cx α so EV =. Let F k e λ kt Z k (t) V k a.s. be the σ-field generated by Zj (t), j k, t. E(e θv k F k )=exp( c h,k u k V k θ α k ) where α k = λ k /λ k and hence Ee θv k = ( +c θ,k μ k θ λ/λ ) k. c h,k = ( ) αk ak Γ(α k )Γ( α k ) a k λ k c θ,k = c θ,k c λ/λ k h,k, c θ, = a /λ and μ k = k j= uλ/λ j j. Rick Durrett (Duke) NESS 4/2 / 3 Rick Durrett (Duke) NESS 4/2 2 / 3

3 Transitions between waves τ k =min{t : Z k (t) > } z 2 (t) z (t) z (t) Population size /u β β P(τ k > t k /2 + x/λ ) +e x Simulated vs. Predicted CDFs of τ, τ 2, andτ 3. z k (t) = k L log+ Z k (t) λ k (t β k ) + L =log(/u) β k = /λ j j= Time Rick Durrett (Duke) NESS 4/2 3 / 3 Rick Durrett (Duke) NESS 4/2 4 / 3 Within Tumor Heterogeneity Intra-tumor diversity generated by model Problems in cancer treatment caused by intra-tumor diversity: Different subpopulations within a tumor may have varying types of response to any given treatment, making total tumor reduction and prevention of resistance difficult. Heterogeneity levels are associated with aggressiveness of disease (e.g., in Barrett s esophagus and prostate cancer). Studies have also shown mutational heterogeneity between primary tumors and metastases in breast cancer, explaining lack of response to EFGR antibody therapy in patients that appeared to have no mutation in KRAS. Size of clone 6 Wave Wave 2 Wave 3 4 Wave 4 Wave Birth rate of clone In this simulation, b i =., a =.2, a i a i U([,.]), u =.. Rick Durrett (Duke) NESS 4/2 / 3 Rick Durrett (Duke) NESS 4/2 6 / 3 Point process representation of V Flash back to stable laws Z (t) =V e λt, V nonrandom. Define a two dimensional point process X t with a point at (s, w) if there was a mutation to type at time s and the resulting type branching process Z (t) hase λ(t s) Z (t) w. A point at (s, w) contributese λs w to V =lim t e λt Z (t). V = (s,w) X t e λs w, the sum of points in a Poisson point process with mean measure μ(z, ) =A u V z α where α = λ /λ. True for (V k F k )withα = λ k /λ k. Let Y, Y 2,... be independent and identically distributed nonnegative random variables with P(Y i > x) cx α with <α<. Let S n = Y + + Y n.then S n /n /α V where V is the sum of points in a Poisson process with mean measure μ(z, ) =cx α Rick Durrett (Duke) NESS 4/2 7 / 3 Rick Durrett (Duke) NESS 4/2 8 / 3

4 Simpson s index t=7 We define Simpson s index to be the probability two randomly chosen individuals in wave k are descended from the same mutation. R 2 = i= where X > X 2 >... arepointsinthepoissonprocessandv k is the sum. In genetics this is the homozygosity. Theorem. ER 2 = α where α = λ k /λ k for wave k. Proof. Apply results of Fuchs, Joffe and Teugels (2) about convergence to stable laws. ER does not depend on V k. X 2 i V 2 k t= t= t= t= Simpsons Index Figure: Empirical distribution of Simpson s Index for wave at times t =7, 9,, 3,. Parameters:b i =., a =.2, a i a i U([,.]), mean is α =/. Rick Durrett (Duke) NESS 4/2 9 / 3 Rick Durrett (Duke) NESS 4/2 2 / 3 Poisson-Dirichlet Distribution(α,) Logan, Mallows, Rice and Shepp (973) The points Y i = X i /V k have this famous distribution introduced by Kingman (97) J. Roy. Stat. Soc. B. 37, -22 and which has been extensively studied, see 7 references in Pitman and Yor Ann. Prob. 2 (997), 8 9 and Pitman s 26 book Combinatorial Stochastic Processes. E f (Y i )= Γ(α)Γ( α) i= we find that R p = i X p i /V p k ER p = E i has f (u)u α ( u) α du Y p Γ(p α) i = Γ( α)γ(p) Consider the self-normalized sums n i= S n (p) = X i ( n j= X p S n (2) = R /2 j )/p n They proved convergence in distribution and identified the Fourier transform of the limit. which is α when p =2. Rick Durrett (Duke) NESS 4/2 2 / 3 Rick Durrett (Duke) NESS 4/2 22 / 3 Ovarian Cancer Ovarian cancer is the fifth leading cause of cancer death among women in the United States. 2,8 new cases and 3,8 deaths in 2. Rick Durrett (Duke) NESS 4/2 23 / 3 Rick Durrett (Duke) NESS 4/2 24 / 3

5 Can screening reduce mortality? Can screening reduce mortality? In the June 8, 2 issue of the JAMA, results were published of a study of 78,26 women aged 74. Screening used tests for the bio-marker CA2 (cancer antigen) and transvaginal ultrasound. n diagnosed death annual screening 39, 22 8 routine care 39, 76 In the June 8, 2 issue of the JAMA, results were published of a study of 78,26 women aged 74. Screening used tests for the bio-marker CA2 (cancer antigen) and transvaginal ultrasound. n diagnosed death annual screening 39, 22 8 routine care 39, 76 3,28 women in the screening group had false positive results, 8 had surgery and 63 had serious complications from surgery. Rick Durrett (Duke) NESS 4/2 2 / 3 Rick Durrett (Duke) NESS 4/2 2 / 3 Multitype model Brown and Palmers parameter estimation Lengyel (2) Am. J. Pathology 77, 3 64 An epidermal to mesenchymal transition in the tumor allows its cells to detach. Cells float in the peritoneal fluid as single cells or small groups. Some cells adhere to the omentum, followed by invasion of the extracellular matrix, angiogenesis etc. There are no significant genetic differences between the main tumor and metastases indicating that no additional mutations are needed beyond those that created the initial tumor. type = primary tumor, type = cells in peritoneal fluid, type 2 = metastasis. u i = rate at which cells change from type i totypei. Brown and Palmer (29) PLoS Medecine Vol. 6, Issue 7, e4 λ =(ln2)/4, λ 2 =(ln2)/2. permonth,λ <λ u i =? Figure: A. Growth in stages I and II, B. Stages III and IV, versus 62 parameter combinations Rick Durrett (Duke) NESS 4/2 26 / 3 Rick Durrett (Duke) NESS 4/2 27 / 3 Type Type Type s are a branching process. Z (t) W e λt b W = d δ + λ exponential(λ /a ) a a exponential(r) has density re rt,mean/r. Condition on not dying out and Z (t) V e λt, V = exponential(λ /a ). Time t since original mutation is not observable so we simplify. Assume Z (t) =e λt. Type s leave from the surface of the primary tumor at rate u times the surface area. t EZ (t) = u e 2λs/3 e λ(t s) ds u ( = e 2λt/3 e λt) (2λ /3) λ Let γ =2λ /3. To remove the unknown rate λ,wesetλ =. Dominant contribution comes from times near t. Manymutationsso result is deterministic. Theorem. Z (t)/ez (t) in probability as t. Rick Durrett (Duke) NESS 4/2 28 / 3 Rick Durrett (Duke) NESS 4/2 29 / 3

6 Type 2 Asymptotics What is the detection window? At time s mutations occur to type 2 at rate u 2 (u /γ )e γs so we let s 2 = ( ) γ log γ u u 2 bethetimeatwhichthemutationrateis. Theorem e λ2(t s2) Z 2 (t) V 2 where V 2 is the sum of points in a Poisson process with mean measure μ(x, ) =C 2 x α2 where α 2 = γ /λ 2, C 2 = ( ) α2 a2 Γ(α 2 ) a 2 λ 2 and Γ(r) = t r e t dt is the usual gamma function. A commonly quoted fact is that cm 3 = 9 cells. We define the window of opportunity for detection to be [T, T 2 ] where T =inf{t : Z (t) =6. 7 }, diameter. cm T 2 =inf{t : m 2 (t) > 9 }. one gram Recall λ =(log2)/4 =.733 and λ 2 =(log2)/2. = Setting e.733t =6. 7 gives T =.733 log(6. 7 ) = 3.8 months or 8.6 years. Rick Durrett (Duke) NESS 4/2 3 / 3 Rick Durrett (Duke) NESS 4/2 3 / 3 T = 3.8. To make a crude calculation note that the first mutation to type 2 occurs at time s 2 = ( ) γ log γ u u 2 From this point for the 2 s to grow to size 9.it will take time roughly λ 2 log( 9 )=74.76 months If we let u u 2 = 4 and note γ =. then s 2 = log() = 6.. so T 2 = = 38. months, and T 2 T = 32 months or 2.66 years. Need to do screening every one or two years. Size of the Primary at T =inf{t : Z (t) > 9 } In the previous calculations we ignored the fact that Z (t) V e λt. At this time T = λ ln( 9 /V ) Z (T )=exp(λ T )=( 9 V ) α α = λ /λ Brockwell and Brown (978) ZfW (aka PTRF), V α has density k= ( x) k Γ(k +)Γ( α αk) Rick Durrett (Duke) NESS 4/2 32 / 3 Rick Durrett (Duke) NESS 4/2 33 / 3 Probability of Progression vs. Size Summary Growth, progression, and metastasis of cancer can be modeled with multi-type branching processes, and these models can be used to evaluate screening strategies and treatment regimens. Results about stable laws can be used to obtain quantitative results about tumor heterogeneity and other quantities of interest, which in contrast to simulation, reveal the dependence on underlying parameters. Rick Durrett (Duke) NESS 4/2 34 / 3 Rick Durrett (Duke) NESS 4/2 3 / 3