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Modeling Spatial Signatures
of Cooperation and
Competition Within Tumors
Ruchira S. Datta, PhD
Maley Lab
Center for Evolution and Cancer
UCSF
Workshop on Game Theory and Cancer
Johns Hopkins University
August 13th, 2013
The Virtuous Cycle of
Scientific Progress
Simulations Defined
Per 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 Defined
Per 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 assumptions
o That we might not have been aware we were making
• Fitting a model to experimental data is
just the start
o Our model is now a possible explanation
o Infinitely many models may exist that fit the same data
The FriendsOrFoes Model
• Purpose
o 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?
• Today
o Model is under development
o Looking for:
• More cancer types equipped with data to which to apply model
o Maybe with modification
• Feedback/suggestions about model development
Evolution
• Population with varying individuals
• Descent with modification
• Change in distribution of
genotypes/phenotypes over time
o Drift – Phenotypes change abundance stochastically
o 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 genes
o Leo Buss, 1987. The Evolution of Individuality
o 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 organism
o 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 repeatedly
o Tumors have hundreds of mutations
• Multiple “driver” mutations
o 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 SelfReinforcing Leukemic Niche
Schepers et al, Cell Stem Cell 2013
How Does Heterogeneity
Persist?
• Several possibilities:
o Neutral coexistence
o Competition taking place dynamically
o Cooperation
It Matters in the Clinic
• Design therapy to target specific cell populations
o 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 bisphosphonates
o “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 tumorassociated skeletal complications.”
-- J.R. Gralow, Curr Onc Rep, 2001 Nov;3(6):506-15
Spatial Signatures of Cooperation
& Competition
• Simulate various starting parameter sets
o Many replicates in each
o Sample the ensemble effectively
• Start from alternate hypotheses:
o Neutral coexistence
o Competition
o Cooperation
• Identify patterns in the distributions resulting from these
hypotheses
• Discover statistics that distinguish the hypotheses
o 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 data
o Check that the summary measure |S(X)-S(X’)| is within tolerance
o 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
Squamous
Metaplasia
Dysplasia
Cancer
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
p16
ki67
DAPI
Trevor Graham
metaplasia
LGD
HGD
cancer
Chatelain and Flejou. Virchows Arch (2003)
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 Variables
o 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?
o E(C,C’): their effect by their mere presence
o F(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 polygon
o The new sites are the centroids of the subdivided polygon
o 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
• Lacunarity
o Fractal image processing
• Clustering coefficient
o Network theory
• Please send me more!
Demo
Additional Future Directions
• Allow crypts to mutate, leading to new subclones
o Keep track of clonal phylogeny
• Generalize to 3D geometries
• Allow crypts to migrate?
Seeking faculty position for 2014!
Acknowledgments
• Center for Evolution and Cancer, UCSF
o
o
o
o
o
o
o
Carlo Maley
Athena Aktipis
Aurora Nedelcu
Trevor Graham – Queen Mary’s University London
Aleah Caulin
Amy Boddy
Viola Walther
Tumor Heterogeneity
How 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
Selective pressure
(eg chemotherapy)
kills sensitive cells
Tumor
eradication
Recurrence/
resistance
Genetically diverse tumor
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
neutral
neutral
Frequency
within
the
Neoplasm
neutral
neutral
p16+/-
p53-
p16-/-
CA
p53-
BE
HGD
Time
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 model
o Simulate on biologically realistic time scale
• Not necessarily to stationarity
o 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 seeds
o 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’Ecuyer
o An R package also exists