Modeling Spatial Signatures of

Cooperation and Competition Within

TumorsRuchira S. Datta, PhD

Maley LabCenter for Evolution and Cancer

UCSF

Workshop on Game Theory and CancerJohns Hopkins University

August 13th, 2013

The Virtuous Cycle of Scientific Progress

Simulations DefinedPer John Maynard Smith, 1978:• Simulations:

• Predict effects of particular policies/interventions

• As much relevant detail as possible

• Most useful to analyze particular cases

• “The better a simulation is for its own purposes, by the inclusion of all relevant details, the more difficult it is to generalise its conclusions”

Models DefinedPer John Maynard Smith, 1978:• Models:

• For discovery of general ideas• “Whereas a good simulation should include as much

detail as possible, a good model should include as little as possible.”

• Answers questions such as “what patterns of interaction and of relative mobility are most likely to lead to stability?”

• “We adopt the method of the experimental scientist, which is to vary one factor at a time, and to do so in a system which is otherwise as simple as possible.”

• “Resort to a computer does not convert a model into a simulation”

• Levins (1968): “Given the essential heterogeneity within and among complex biological systems, our objective is not so much the discovery of universals as the accounting for differences.”

Modeling in Biology• Generate testable hypotheses• Find consequences of our

assumptionso That we might not have been aware we were

making

• Fitting a model to experimental data is just the startoOur model is now a possible explanationo Infinitely many models may exist that fit the same

data

The FriendsOrFoes Model

• Purposeo Clones competing, cooperating, coexisting – how would we

know?o Find signatures of cooperation and conflict in spatial patterns

• Start with model where relationships are known• Apply findings to images• Flexible or rigid lattice – does it make a difference?

• Todayo Model is under developmento Looking for:

• More cancer types equipped with data to which to apply modelo Maybe with modification

• Feedback/suggestions about model development

Evolution• Population with varying individuals• Descent with modification• Change in distribution of

genotypes/phenotypes over timeo Drift – Phenotypes change abundance

stochasticallyo Natural selection – Fitter phenotypes tend to

increase in abundance over generations• Differential survival (viability)• Differential reproduction (fertility)

Cancer Suppression Allowed Multicellular Organisms to

Evolve

About 600 Million Years Ago

www.palaeos.com D.W. Miller from American Scientist, March-April, 1997

The Multicellular Covenant

• Somatic cells curtail their reproduction

• Germ cells propagate the geneso Leo Buss, 1987. The Evolution of

Individualityo John Maynard Smith & Eörs

Szathmáry, 1995. Major Transitions in Evolution

• Cancer is the breaking of that covenant

Volvox: A model of the transition to multicellularity

Cancer as an Evolutionary Process

• Multicellular Organism: population of cells• Genetically identical, though phenotypically

(& epigenetically) distinct• Phenotypes cooperate for survival and

reproduction of whole organismo Social contract: Act for the fitness of the whole

• Cancer: Breaking the covenant• Clone: population of genetically identical

cells• Tumor: evolving population of clones

How Does Heterogeneity Arise?

• Normal cells are supposed to be genetically identical

• For cancer to arise, this condition had to fail

• It could fail once, and it does fail repeatedlyo Tumors have hundreds of mutations

• Multiple “driver” mutationso Mutations in p53, “the guardian of the genome”o Faulty DNA repair mechanism

• Thus: multiple distinct clones arising and coexisting

Organism As Ecological Community

• Selection is on phenotypes• In healthy organism, balance of cell phenotypes

serves organismal function• In tumor, cooperation is no longer a given

o Genetically normal cells (fibroblasts, macrophages): part of evolving community

o Tumor microenvironment is phenotypically heterogeneous

• See review by Basanta & Anderson, “Exploiting ecological principles to better understand cancer progression and treatment”, on arXiv 2013

Normal Cells Play a Role

Myeloproliferative Neoplasia Remodels the Endosteal Bone Marrow Niche into a Self-Reinforcing Leukemic Niche

Schepers et al, Cell Stem Cell 2013

How Does Heterogeneity Persist?

• Several possibilities:oNeutral coexistenceoCompetition taking place dynamicallyoCooperation

It Matters in the Clinic• Design therapy to target specific cell

populationso What will be the overall effect on the cancer process?o Knowing how the targeted and untargeted populations interact is

crucial

• Example: adjuvant therapy with bisphosphonateso “The development of skeletal metastases involves complex

interactions between the cancer cells and the bone microenvironment. The presence of tumor in bone is associated with activation of osteoclasts, resulting in excessive bone resorption. Bisphosphonates are potent inhibitors of osteoclastic bone resorption with proven efficacy in reducing tumor-associated skeletal complications.”

-- J.R. Gralow, Curr Onc Rep, 2001 Nov;3(6):506-15

Spatial Signatures of Cooperation & Competition

• Simulate various starting parameter setso Many replicates in eacho Sample the ensemble effectively

• Start from alternate hypotheses:o Neutral coexistenceo Competitiono Cooperation

• Identify patterns in the distributions resulting from these hypotheses

• Discover statistics that distinguish the hypotheseso Strong null hypothesis: neutral coexistence

• Apply them to images

Statistical Inference Using Agent Based

Models“Integrating Approximate Bayesian Computation with Complex Agent-Based Models for Cancer Research”, Andrea Sottoriva & Simon Tavaré, Proceedings of COMPSTAT 2010, 2010, 57-66• Run simulations in parallel using parameters sampled

from priors• Compute summary statistic X on observed data• For each simulation run:

o Compute summary statistic X’ on simulated datao Check that the summary measure |S(X)-S(X’)| is within toleranceo If so accept this as sample from posterior distribution

In particular: can we reject the null hypothesis of neutral coexistence?

Motivation:Barrett’s Esophagus

• Using precancerous condition to guide initial model choices

• Model is sufficiently general to apply to any epithelial sheet

Development of Cancer in

Barrett’s

Metaplasia Dysplasia CancerSquamous

Accumulation of genetic lesions

CDKN2A (p16), FHIT, TP53, ploidy abnormalities

Crypt

Imaging Heterogeneity

Leedham et al., Gut 2008

p53 mutation

Wild-type

c.473G>A het

c.473G>A hom

Shahab Khan

p16ki67DAPI

Trevor Graham

Chatelain and Flejou. Virchows Arch (2003)

metaplasia

LGD

HGD

cancer

For 62 patients counted:

% mutated crypts

mean patch size(Trevor Graham)

Our Model

Entities• Population consisting of• Individual Cells which belong to various clones• Clones

State Variableso Consider a cell c at time t

• Clonal identifier C(c): the clone to which this cell belongs

• Its coordinates x(c) and y(c)• Hexagonal or Voronoi grid• Topological tube

o Clone C• How do neighboring cells of a clone C’

impact the fitness of a cell of clone C?oE(C,C’): their effect by their mere

presenceoF(C,C’): their effect depends on their own

fitness

Process overview and scheduling

• Initialize: field of one clone, random single cell of another

• Initial fitness of each clone is specified• At each time step,

o For each cell c:• Reinitialize fitness of cell c to clonal fitness then loop through its

neighbors c’• If c’ is from clone C’, add E(C,C’) + F(C,C’) f(c’,t-1)• Probability of survival is proportional to f(c,t); check that the cell

survives. Constant of proportionality depends on clone C.o Pick a random ordering of the remaining cells

• For each cell c:o Do a binomial check on probability of reproduction proportional to

f(c,t). Constant of proportionality depends on clone C.o If so, check if there is space

Space to Reproduce?• Hexagonal grid:

o Is there an adjacent empty space?

• Voronoi grid:o Is the area of the polygon at least twice the

area threshold?• If not, there’s no space

o Go through the vertices of the polygon, drawing the perpendicular segments to the opposite side

o Pick the shortest of these to cut the polygono The new sites are the centroids of the

subdivided polygono Each daughter cell inherits half the fitness of

the mother cell

Outcomes• Ratio of perimeter to area of each clone• Average number of neighbors from a different

clone• Proportion of cells that are adjacent to a cell of

another clone• Whether or not a clone, initialized from a single

cell can invade the environment (reach 50%)o Compare with Ohtsuki, Nowak et al evolutionary graph theory results

• Time for a clone to reach majority o Compare with Ohtsuki, Nowak et al evolutionary graph theory results

• Rate of expansion of a clone over time

Spatial Statistics• Partial segregation index

o From ecology

• Lacunarityo Fractal image processing

• Clustering coefficiento Network theory

• Please send me more!

Demo

Additional Future Directions

• Allow crypts to mutate, leading to new subcloneso Keep track of clonal phylogeny

• Generalize to 3D geometries• Allow crypts to migrate?

Acknowledgments• Center for Evolution and

Cancer, UCSFo Carlo Maleyo Athena Aktipiso Aurora Nedelcuo Trevor Graham – Queen Mary’s University

Londono Aleah Caulino Amy Boddyo Viola Walther

Seeking faculty position for 2014!

Tumor HeterogeneityHow Does It Arise, and Why Does It Matter?

Why Does Heterogeneity Matter?

• Biodiversity in community yields resilience to changing environment

• Diversity in Barrett’s esophagus yields increased risk of EA (Merlo, Maley et al 2010)

• Diversity in lung cancer suggests poor prognosis for survival (Lui, Graham, Maley et al submitted)

• We expect diversity in a variety of cancers to yield resistance

Genetic diversity & prognosis

Homogeneous tumor

Genetically diverse tumor

Selective pressure(eg chemotherapy)

kills sensitive cells

Recurrence/resistance

Tumor eradication

Selective pressure

Somatic Evolution Drives

Progression

What We Know• Heritable (epi)genetic heterogeneity within a neoplasm

• Leads to variation in cell fitness (survival & reproduction)o Clonal expansions

Frequencywithin

theNeoplasm

Time

BE

p16+/-

p53-

p53-p16-/-

neutral neutral

neutral

HGD

CA

neutral

Not quite MCMC• Model is a Markov process

o Not necessarily reversible!

• Doing Monte Carlo simulation• MCMC:

o Simulate until mixing time: reach stationary distribution• Good to simulate for longer and longer times

o Do a number of starting points to make sure chain doesn’t get stuck

• Our modelo Simulate on biologically realistic time scale

• Not necessarily to stationarityo Sample distribution effectively

ODD Protocol• Standardized way of specifying individual-based

or agent-based models• Sections:

o Overview• Purpose• Entities, States Variables, and Scales• Process Overview and Scheduling

o Design Concepts• Emergence? Adaptation? Prediction? Sensing? Interaction?

Stochasticity? Collectives? Observation?• What outcomes will be measured?

o Details• Initialization• Input• Submodels

IID Random Variates• Common practice: parallelize simulation using

different seedso Not necessarily correct

• Pseudorandom number generation on deterministic computer is tricky

• Independence of parallel streams cannot be assumed unless explicitly guaranteed

• Use RngStream by Pierre L’Ecuyero An R package also exists