An introduction tostochastic dynamics of cancer evolution

IMA Workshop on Careers for Women in Mathematics

Jasmine FooUniversity of Minnesota, USA

March 3, 2013

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.

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

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?

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.

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

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

Clinical application

Retracing evolutionary steps in cancer (RESIC)Attolini et al PNAS 2010.

Used to determine temporal order of KRAS, TP53 mutations in coloncancer

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

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

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)

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

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?

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?

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.

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

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

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.

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)

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.

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.

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

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)

Model testing and validation

Model testing and validation