An introduction to stochastic dynamics of cancer evolution · Cancer facts I Cancer: a family of...
Transcript of An introduction to stochastic dynamics of cancer evolution · Cancer facts I Cancer: a family of...
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An introduction tostochastic dynamics of cancer evolution
IMA Workshop on Careers for Women in Mathematics
Jasmine FooUniversity of Minnesota, USA
March 3, 2013
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Cancer factsI Cancer: a family of diseases in which abnormal cells divide
without control and are able to invade other tissues (>100 types).
I Many treatments effectively reduce tumor cell populations(surgery, chemotherapies, radiation, targeted therapies)
Figure: Prior to and 15 days after treatment with PLX4032, Photo NY Times 2010
I Tumors are evolving populations – drug resistance an obstacleto cure (e.g. Greaves and Maley, Nature Rev. Cancer 2012, Ding et al Nature 2012).
I Viewing cancer through the lens of evolution may lead to betterunderstanding of progression, treatment.
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Cancer is an Evolutionary Process
"Nothing in biology makes sense except in lightof evolution" - Theodosius Dobzhansky
Ingredients for natural selection:
I Variation in traits
I Differential fitness (reproductive rate)
I Heredity
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Overview of questions I: Fixed size models
Carcinogenesis: the process of cancer initiation from healthy tissue
I how does it happen? how long does it take?
I what are genetic characteristics of the initiating cell?
I what is the impact of tissue structure on initiation time, spatialpatterns? (why do cancers vary between sites in the body?)
I Can we prevent it?
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Multistep carcinogenesis
I Cells at risk of accumulating oncogenic mutations are organizedinto fixed size compartments of cells.
I Cancer arises when cells accumulate enough mutations toescape homeostatic mechanisms of the compartment.
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Classic Moran model of cancer initiation
Initiation driven by accumulation of specific mutations (e.g.tumor suppressor gene inactivation)
Moran process (Moran, 1961): compartment of N cells
I type 0 cells - both TSG active (fitness 1), type 1 - one TSGinactive (fitness r1), type 2 - both TSG inactive (fitness r2).
I At each event one cell chosen to replicate according to relativefitness
I Daughter cell replaces cell chosen at random to die.
I mutations: type 0 u1−→ type 1 u2−→ type 2
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Fixation and tunneling in Moran process
A mutation becomes fixed if present in all N cells (probability 1/N if r1 = 1).Tunneling occurs if type-2 cell arises before type-1 is fixed.
τi ≡ time that first cell accumulates i mutations.σi ≡ time to fixate first successful type-i .
sequential fixation: σ1 << τ2 − τ1
Analysis of τ2 in tunneling regime (neutral case): Komarova et al 2003, Iwasaet al 2004
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Clinical application
Retracing evolutionary steps in cancer (RESIC)Attolini et al PNAS 2010.
Used to determine temporal order of KRAS, TP53 mutations in coloncancer
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A spatial model of cancer initiation: interacting particlesystems
I Cells sit on rectangular lattice in Zd of total size N with periodic bcsI State of cell is its fitness, each cell divides at rate equal to fitness.I Daughter cell replaces one of the 2d neighbors at random.
I Type 0 (fitness 1)u1−→ Type 1 (1 + s)
u2−→ Type 2 (1 + s)2. (s > 0)I Klein et al (2008) compare model dynamics to mouse epidermis expts.
Joint work w/R. Durrett, K. Leder
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How fast do advantageous mutations spread throughtissue?
Only type-0 (fitness 1) and type-1 (fitness 1 + s) cells, selection dynamicsonly (mutations suppressed). –> biased voter process. Let ξt be the set oftype-1 occupied sites at time t .
Previous results (Bramson and Griffeath 81) show that ξt conditioned onsurviving grows linearly with asymptotic shape D, where D is convex andsymmetric.
Theorem Let e1 be the first unit vector and define the growth rate cd (s) suchthat the intersection of D with the x axis is [−cd (s)e1, cd (s)e1]. Then, ass → 0 we have
cd (s) =
O(s) d = 1O(
√s/ log(1/s)) d = 2
O(√
s) d = 3
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Graphical summary of σ2 regimes
Figure: log Γ,b = a− 2.
d = 2, s = .01
Let N = 10c , u1 = 10−a, u2 =10−b
Γ = (Nu1s)d+1(cdd u2s)−1
We are able to characterizethree regimes of behavior:
I Γ→∞I Γ→ g ∈ (0,∞)
I Γ→ 0
Durrett, F-, Leder. Spatial Moran models II, 2012 (preprint)
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A colorectal cancer applicationCells of colon subdivided into partiallyisolated subpopulations of proliferativeunits ‘crypts’
Consider initiation in sigmoid colon(approx 945000 crypts), cylindricalstructure (d=2)
Estimate of mutation rate in patients withpredisposing conditions u1 = 10−5
(Totarfumo et al 1987)
Inactivation of one APC allele induceschromosomal aberrations (Ceol et al,2007) u2 = 10−3. J = 2.17,K = 14.6(premalignant field)
Inactivation of APC -> mutations in p53or kRAS
Estimate of σ2:9.3 yrs
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Some ongoing work
I Understanding field cancerization: observations that multipleprimary tumors occur within the same area or cancer field
I Understanding heterogeneity - sampling guidelines, mutationprevalence (+ M. Ryser)
I Length-scale of heterogeneity: if the oncologist wants toknow the genetic portrait of a tissue specimen, how fine orcoarse should the sampling be, how many sections arerequired for a conclusive picture of the heterogeneouscancer?
I Excision margins: when removing a tumor, how much of theregion outside the visibly malignant clone is affected byprecursor lesions, i.e. how much tissue has to be removedto avoid a relapse?
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Overview of questions II: Exponential growth models
Better characterization of tumors for new drug design and prognosisprediction
I how to quantify the genetic composition of tumors?
I which mutations are important?
I what is the spatial scale of tumor heterogeneity of tumors?(sampling guidelines)
I can we predict the next steps of tumor evolution?
What do we do about it?
I How/when does drug resistance and metastasis emerge? Howdoes therapy impact this?
I Can we optimize treatment strategies via evolutionaryprinciples?
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Stochastic model of tumor growthModel growing tumor cell population as continuous time binarybranching process. Initially one cell of type-0 with net growth rate λ0.
Zi (t) ≡ type-i cells that have exactly i mutations at time t . Type-i cellsmutate at rate ui+1, creating type-(i + 1) cells.
Mutations confer a random additive change ν to the birth rate.
(i) ν has density g(·) on [0,b], g(b) > 0 and g is left-continuous at b.
Models genotypic or phenotypic variation. λk ≡ λ0 + kb, p ≡ b/λ0.
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Characterizing clonal heterogeneity in tumors
0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4100
102
104
106
Birth rate of clone
Size
of c
lone
Wave 1Wave 2Wave 3Wave 4Wave 5
Some results on growth and diversity in Z1 (can be generalized to Zk ):
I Weak limit of Z1 : t1+pe−(λ0+b)tZ1(t)⇒ V1, where V1 hasLaplace transform exp
(−u1c1(λ0,b)θλ0/(λ0+b)
)I Limit V1 has a point process representation w/ mean measureµ(z,∞) = A1(λ0,b)u1z−λ0/(λ0+b) .
I Structure reveals simple estimates for asymptotic populationdiversity. E.g. for Simpson’s Index, ESk = 1− αk , whereαk = λk−1/λk
Durrett, F-, Leder, Mayberry, Michor, Genetics 2011; Theoretical Population Biology 2010
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Cancer recurrence due to resistance mutations
I Despite initial response to therapy, drug resistance due to acquisition ofmutations a major obstacle in treatment
I Timing of recurrence exhibits large variations between patientsI Post-recurrence tumor composition also exhibits large variations
-> implications for prognostic prediction, drug evaluation, and treatmentdesign
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Two stochastic times
We characterize two important random times in cancer recurrence dynamics:Let Z0 and Z1 be the total population of sensitive and resistant cells.
I Turnaround time - when the total tumor size first begins to rebound (i.e.become supercritical) during treatment
τx = argmin{t ≥ 0 : Z0(t) + Z1(t)}.
-> Approximately observable in clinical setting using serial patientscans.
I θ-crossover time - first time that resistant cells make up fraction θ oftotal population
ξx,θ = inf{t ≥ 0 : Z1(t) ≥ θ
1− θZ0(t)}
-> Not easily observed in patients, but useful in designingcombination/sequential therapies.
F-, Leder. Dynamics of cancer recurrence, Ann. App. Probability, in press 2013.
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Application to Non-small cell lung cancer (NSCLC)In approximately 10-15 % of NSCLC cases, specific mutations in theEpidermal Growth Factor Receptor (EGFR) are associated with sensitivity totargeted drugs such as erlotinib and gefitinib (tyrosine kinase inhibitors).
A majority of patients experience large reductions in tumor burden inresponse to erlotinib/gefitinib therapy.
Drug resistance can be associated with a single point mutation (T790M)within EGFR. (Pao et al, 2006)
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Characterizing model parametersIsogenic sensitive (PC-9)/resistant pair of NSCLC lines developedwith and w/o T790M mutation (by W. Pao, J. Chmielecki)
Birth/death rates of sensitive/resistant cells vs. drug concentration,informs (inhomogeneous) growth kinetics of branching process.
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Optimized treatment schedule that delays resistance
I Current FDA approved schedule:continuous daily dose eliciting 3uM Cmaxconcentration in plasma.
I Validate model, evaluate a range ofpossible dosing strategies and search forstrategies that maximally delay resistance.
I We identify an alternate toleratedschedule that should delay resistance:
Oral intake eliciting 20uM pulse 1/wk (ormore potent inhibitor)+1 uM/day schedule.
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Validation in cell lines
Hypothesis: High dose pulse (BIBW-2992) 1 day/wk + Very Low doseErlotinib 6 days/wk will result in longer time to develop resistancethan the currently-used continuous dosing strategy.
Chmielecki, F-, et al, Science Translational Medicine, 2011
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That’s all - Thanks!
Collaborators:
I F. Michor (DFCI/Harvard)I K. Leder (Minnesota)I R. Durrett (Duke)I S. Mumenthaler (USC)I P. Mallick (Stanford)I J. Mayberry (Pacific)I W. Pao (Vanderbilt)I J. Chmielecki (Broad/Harvard)I M. Ryser (Duke)
Funding: National Cancer Institute (NCI), National ScienceFoundation (NSF)
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Model testing and validation
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Model testing and validation