What stars can teach us about cancer: Modeling metastasis through population genetics

What the study of the stars can teach us about cancer:
There’s no success like failure, and failure’s no success at all.
Jacob G. Scott
Key Factors in the Metas
from Populatio
Christopher McFarland1*, Jacob Scott2,3,*, David Bas
1Harvard-MIT Division of Health Science & Technology, 2H. Lee Moffitt
Oncology, *contributed equally to this workoorly understood process that
aths
not explained by deterministic
the genomic level and use
explore this phenomenon
Res
Cells deriv
Metastatic
Three regi
metastasis
which only
We further
Feature of Model Observed Phenomenon
Population size determined
by fitness of cells
Larger Tumors more likely to
metastasize
Cells can acquire passenger
mutations that are slightly
deleterious
Many micrometastases never
grown to macroscopic size
Cells with more mutations are
less likely to metastasize
Stromal environment reduces
efficacy of driver mutations
Certain stromal conditions
prohibit metastasis
Metastases with same
Key Factors in the Meta
from Populat
Christopher McFarland1*, Jacob Scott2,3,*, David B
1Harvard-MIT Division of Health Science & Technology, 2H. Lee Mo
Oncology, *contributed equally to this work
Background:
Metastasis is a highly lethal and poorly understood process that
accounts for the majority cancer deaths
Patterns of metastatic spread are not explained by deterministic
explain these patterns
We develop a stochastic model at the genomic level and use
population genetics techniques to explore this phenomenon
R
Cells
Meta
Three
metas
which
We fu
Feature of Model Observed Phenomenon
Population size determined
by fitness of cells
Larger Tumors more likely to
metastasize
Cells can acquire passenger
mutations that are slightly
deleterious
Many micrometastases never
grown to macroscopic size
Cells with more mutations are
less likely to metastasize
Stromal environment reduces
efficacy of driver mutations
Certain stromal conditions
prohibit metastasis
Radiation Oncology and
Integrative Mathematical
Oncology
further, the stem compartment can differentiate at a rate β. Each population als
growth (r) and death (d) rate, proportional to their population.
FIG. 3: To capture the behavior of a putative compartment system in which there is only e
from the stem compartment into the TAC compartment, death and growth in both, but
by a common carrying capacity.
We therefore write a system of ODEs as:
Stem compartment :
dN0
dt
=
growth

r0N0(1 −
N
K
) −
differentiation

β0N0 −
death

d0N0
Differentiated :
dN1
dt
=
growth

r1N1(1 −
N
K
) +
differentiation

β0N0 −
death

d1N1
Where:
Phys. Biol. 8 (2011) 015016 D Basanta et al
Table 1. The four phenotypes in the game are autonomous growth
(AG), invasive (INV), glycolytic (GLY) and invasive glycolytic
(INV-GLY). The base payoff in a given interaction is r and the cost
of moving to another location with respect to the base payoff is c.
The fitness cost of acidity is n and k is the fitness cost of having a
less efficient glycolytic metabolism. The benefits from having
access to the vasculature as a result of angiogenesis are reflected by
the parameter α.
AG INV GLY INV-GLY
AG 1
2
+ α
2
1 1
2
− n + α 1
2
− n + α
INV 1 − c 1 − c
2
1 − c
3
1 − c
3
GLY 1
2
− k + n + α 1 − k + α
2
1
2
− k + α
4
1 − k + α
2
INV-GLY 1
2
− k + n + α 1 − k + α
2
1 − c
3
− k + α
2
1 − k − c
6
+ α
2
Table 2. List of variables used by the model.
Value Affected phenotypes Meaning
c INV, INV-GLY Cost of motility
k GLY, INV-GLY Cost having a glycolytic metabolism
n AG, INV Cost of living in an acid
microenvironment
to leaky or otherwise defective vascularization [9]. This is
shown in the table by the fact that AG cells interacting with
other AG cells (assumed to produce only moderate amounts of
HIF-1α) receive a benefit of α
2
from the moderate angiogenic
vasculature. On the other hand, AG cells interacting with
GLY cells produce, in combination, an optimal amount of
HIF-1α and obtain in return the total benefit derived from
functioning vascularity (α). Finally, as IDH-1 mutant GLY
cells proliferate producing excessive amounts of HIF-1α, the
benefit of angiogenesis is a reduced α
4
, consistent with the
angiogenic vasculature being leaky and inefficient in this case.
Another notable difference with the previous model is that
the cost of motility is assumed to be smaller in the presence of
acid-producing glycolytic phenotypes. This is represented by
a cost of motility c
3
and represents the acid-mediated invasion
[21–23] of glioma cells throughout the brain, particularly along
the myelinated neuronal axons in the white matter of the brain
along which glioma cells are known to quickly invade [24, 25].
This reduced cost of motility also quantifies and models the
generally invasive characteristics of gliomas which are well
known for their diffuse invasion that has been quantified in
In one dimension, this becomes:
∂
∂x


D
∂c
∂x
+ χc
∂a
∂x


 = Dc
∂2
c
∂x2
− cχ(a)
∂2
a
∂x2
− χ
∂c
∂x
∂a
∂x
(2.18)
Additionally, we must consider the creation and dispersal of this chemoattractant, a. To do this, we assume
Fickian diffusion as for the cells in the initial model as per 2.8, and a creation term that is linearly related
to the death of the cells by a coefficient, ω. Further, we introduce a consumption term, the rate at which
the chemoattractant is consumed by the cells, linearly related to the number of cells by a coefficient, µ.
Therefore, we can write down a full model for both the cellular concentration, as derived above, and for the
chemoattractant, a, thus:
rate of change of
glioma cell concentration

∂c
∂t
=
net dispersal
of glioma cells

∇ · (Dc∇c) +
net growth
of glioma cells

ρc(1 −
c
K
) −
chemotaxis
of glioma cells

∇ · (cχ(a)∇a) −
death
of glioma cells

λc , (2.19)
rate of change of
chemotactic factor

∂a
∂t
=
net dispersal of
chemotactic factor

∇ · (Da∇a) +
creation of
chemotactic factor

λcω −
consumption of
chemotactic factor

µca . (2.20)
And again in 1-dimension:
∂c
∂t
= D
∂2
c
∂x2
− cχ(a)
∂2
a
∂x2
− χ
∂c
∂x
∂a
∂x
+ c(1 −
c
Kc
) − λc, (2.21)
∂a
∂t
= Da
∂2
a
∂x2
+ λcωa − µca. (2.22)
While the death term has remained a constant, λ, times the population, this addition does little to effect the
overall dynamics. Only with very large parameter changes, likely large enough to be physically unrealistic,
?

Tortuous Path
•Hawken ’94
•US Naval Academy ’98, Physics
Major
•Navy Nuclear Reactor Engineer
‘98-’03
•High School Physics Teacher
Florida, ’03-’04
•Case Western Reserve MD
2004-2009
•Radiation Oncology Resident
Tampa, FL, ’09-Present
•Oxford D.Phil candidate -
mathematics

Net worth.
-300000
-225000
-150000
-75000
0
75000
1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013
$$
Hawken
Diploma
MSBS
Entrance to
medical school
MD

May 1783, John Goodricke - 100GBP

~10-7 meters - 1,000 Angstroms
~1021 meters (106 light years)

Important to understand your limitations:
Dogs are so cute when they
try to comprehend quantum
mechanics - I’m not.
As Mr. Dlugozs will tell
you, I am a terrible bench
biologist - yes, I tried.

Radiation
Oncology
The study of cancer -
a disease on the
human length scale

Ptolemy’s geocentric
solar system and
crystalline heavens
(c. AD 90 – c. AD 168)

measurement
meta-
phenomenological
laws
conserved laws
(14 December 1546 – 24 October 1601)
(December 27, 1571 – November 15, 1630)
(25 December 1642 – 20 March 1727)
mechanism
(14 March 1879 – 18 April 1955
(29 April 1854 – 17 July 1912

Hippocrates - the four
humors model of physiology

Mr. Dlugosz
The Cell Cycle - IPMAT
The cycle is broken in cancer!

suggest we all adopt ideologies such side of the majority, but to escape j.stebbing
Essay
Phase itrialist
There is a new breed of clinical trialist
in cancer research. You might not
have seen them yet—they will not
be knocking down your door in the
clinic. They do not know what HIPAA
stands for. They do not know what
to do in a code. They do not wear a
white coat, you will be lucky if they
wear a tie. They are not biologists—if
you ask them to change the media,
they will probably bring you some
music you have not heard. They are
the phase i trialists.
What is a phase i trial? It is a
preclinical trial, but one in which no
cells, mice, or rats will be harmed.
Before one begins killing cells in
a dish, there is the step to decide
how to treat those cells or mice in
a sensible, yet new way. It is in this
phase, before even stepping into
a laboratory, in which we are now
seeing an influx of other types of
scientists—physicists, engineers, and
mathematicians. Some of these folks
have run out of problems in their
field and have found fertile ground
for their tools and physical science
perspective in the dizzying biological
complexity of cancer. Others have
become frustrated by the esoteric
nature of their first specialty—it takes
a special mind to be happy studying
things in other galaxies, or things so
small that you need a super collider
spanning three countries to learn
anything new.
And then, some are just naturally
dreamers, or follow their hearts into
a specialty that has affected their
you turn
research
as app
to gene
energy
people w
be expla
we can b
or a com
how a tu
how a pe
phase i.
The bi
right no
and scie
with a
to bet
grant m
mathem
to think
they too
until the
from th
Cancer and Society
From experience, we believe that
doctors are far less accepting of such
ideas than patients or healthy non-
physicians. Oncologists are more
tolerant of the concept of ginger as a
treatment for chemotherapy-induced
nausea becausethese data come from
a large trial with sound statistical
analysis. However, its potential as an
anticancer drug directly conflicts with
the beliefs of most physicians, even
though no precise mechanism of
action has been confirmed for either
potential use. We certainly do not
suggest we all adopt ideologies such
as those advocated by the authors
of books such as How to Cure Almost
Any Cancer at Home for $5·15 a Day,
but perhaps many of us are guilty of
intolerance of alternative therapeutic
ideologies.
Albert Einstein is quoted as saying
that insanity was “doing the same
thing over and over again and
expecting a different result”, and
perhaps some cancer researchers
are guilty of this way of thinking.
Marcus Aurelius once said ”the
object of life is not to be on the
side of the majority, but to escape
finding o
insane”.
that if m
researche
philosoph
professio
worth of
ginger m
doctors m
idea of th
concentr
Jonathan
Imperial Co
j.stebbing@
Essay
Phase itrialist
There is a new breed of clinical trialist
in cancer research. You might not
have seen them yet—they will not
be knocking down your door in the
clinic. They do not know what HIPAA
stands for. They do not know what
to do in a code. They do not wear a
a dish, there is the step to decide
how to treat those cells or mice in
a sensible, yet new way. It is in this
phase, before even stepping into
a laboratory, in which we are now
seeing an influx of other types of
scientists—physicists, engineers, and
you turn
research?
as appl
to genet
energy t
people w
be explai

Molecular
Reductionism
Qolism
Cellular Organism
the current i
which invasi
cancer progr
There are
these hypoth
relevant to th
First, this is a
mathematica
cancer resear
type of insigh
perhaps mos
such quantit
used. Second
to experimen
conceived. In
experiments
conducted in
relevance of
quantitative
cult and the
it exposes a g
cal and expe
Figure 2 | Cancer is multiscale. Changes at the genetic level lead to modified intracellular signal-
lingwhichcauseschangesincellularbehaviourandgivesrisetocanceroustissue.Eventually,organs
and the entire organism are affected. We propose that a focus on the cell as the fundamental unit
PERSPECTIVES
~10-20 - 1020 meters

Each of these models explains only a small part of our
experience on this earth - and with caveats...
But can we just
add them all up
and recapitulate
life?

Mechanistic Modeling:
but at what scale?
Bioinformatics
498 I. J. Radiation Oncology d Biology d Physics Volume 75, Number 2, 2009

Build models!
What is Science?
What do Scientists do?
“All models are wrong, but some are useful”
George E P Box (Statistician)

Nutrients (c)
Signalling
Proteases (m)
Invasive front
Angiogenesis
Stem cell
Inflammatory response
Stromal cell
Immune cell
Normal cell
Matrix adhesion
Dm
2
m ni,j – m,
m
t
– m ,
t
Dc
2m – ni,jc – c.
c
t
Tumour cells (n)
Extracellular matrix (f)
plement
ch as
M)
s and
ge over
by a set of
es a
of cancer
n intuitive
ancer
tion.
behav-
nt?Each
riven by
quantified
escribes
ucially,
lattice
s context
fficients
We can
PERSPECTIVES

Cancer is not only a collection of mutated cells
A complex system of many interacting cellular and
microenvironmental elements that were once normal

•More complex models are better
•Something that looks similar is similar
•Biological facts should drive derivation
•Distill key components (Dialogue)
•Focus on mechanism
•Subsequent model refinement“All models are wrong, but some are useful”
George E P Box (Statistician)
“Models should be simple but no simpler”
Albert Einstein
Minimal modeling approach

We use a suite of mathematical and computational
models to bridge a range of spatial and temporal scales.
TIME/SPATIAL SCALE
CELLULAR DETAIL
Evolutionary
Game Theory
Reaction
Diffusion Models
Hybrid Cellular
Automata
Cellular Potts
Model
Immersed
Boundary Model
Hybrid Cellular
Automata
Non-spatial
continuum
Reaction
Diffusion
Network
Theory

Dead
P
QProliferating
Quiescent
Dead
Heterogeneous Population
Spatial Constraints
Nutrient/Growth Factor Constraints
Stromal Constraints/Interactions

Time
CellDensity
dN(t)
dt
= λN(t)
N(t), number of cells
at time t
λ, proliferation rate
http://math.dartmouth.edu/~klbooksite/3.02/302.html

N(t) = Aeλt
Doubling time
Here λ =0.04, hence
T=17.33 hrs
T =
ln2
λ

Time
CellDensity
dN(t)
dt
= λN(t) 1−
N(t)
K
⎛
⎝⎜
⎞
⎠⎟
N(t), number of cells
at time t
K, carrying capacity

Time
CellDensity
dN
dt
= −λN log
N
K
⎛
⎝⎜
⎞
⎠⎟
N, number of cells at
time t
K, carrying capacity

Cons
Over simplification (no cycle)
Proliferate with same rate at same time
Non-spatial
No mechanistic insight
Pros
Fit tumour growth data well
Compartmental models
Time
CellDensity

∂N(x,t)
∂t
= D
∂2
N(x,t)
∂x2
N, number of cells at
time t, position x
D, Diffusion coefficient
the rate of change of cell
number at position x and time t =
change in cell number
due to random dispersal

dN(x,t)
dt
= D
∂2
N(x,t)
∂x2
+ λN(x,t) 1−
N(x,t)
K
⎛
⎝⎜
⎞
⎠⎟
Sir Ronald Fisher
1890-1962
N, number of cells at time
t, position x
D, Diffusion coefficient
K, carrying cpacity
the rate of change of cell
number at position x and
time t
=
change in cell number
due to random dispersal + change in cell number
due to cell proliferation

du
dt
= Du
∂2
u
∂x2
− σv,
dv
dt
= Dv
∂2
v
∂x2
− u − v
Alan Turing
1912-1954
u, activator
v, inhibitor
D, Diffusion coefficients
, proliferation rateσ

Phys. Biol. 8 (2011) 015016 D Basanta et al
GF~0% GF~80% GF~100%
starting
t t t
control
raisereduce
AG
inv
gly
inv gly
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(INV-GLY). The base payoff in a given interaction is r and the cost
of moving to another location with respect to the base payoff is c.
The fitness cost of acidity is n and k is the fitness cost of having a
less efficient glycolytic metabolism. The benefits from having
access to the vasculature as a result of angiogenesis are reflected by
the parameter α.
AG INV GLY INV-GLY
AG 1
2
+ α
2
1 1
2
− n + α 1
2
− n + α
INV 1 − c 1 − c
2
1 − c
3
1 − c
3
GLY 1
2
− k + n + α 1 − k + α
2
1
2
− k + α
4
1 − k + α
2
INV-GLY 1
2
− k + n + α 1 − k + α
2
1 − c
3
− k + α
2
1 − k − c
6
+ α
2
Table 2. List of variables used by the model.
Value Affected phenotypes Meaning
c INV, INV-GLY Cost of motility
k GLY, INV-GLY Cost having a glycolytic metabolism
n AG, INV Cost of living in an acid
microenvironment
α AG, GLY, INV-GLY Benefit from angiogenesis
moving cells incur since they cannot proliferate whilst moving
[19, 20] or as the cost for degrading and detaching from the
extra cellular matrix. The parameter k represents the cost of
utilizing glycolysis as opposed to the more efficient oxidative
phosphorylation. The parameter n represents the penalty that
cells suffer for living in an acidic environment created by the
glycolytic cells. GLY cells will suffer this penalty less as they
are adapted to live in acidic environments. The parameter α
represents the benefit of the surrounding vasculature. One way
of envisioning variations in α is the increase in oxygen and
nutrients resulting from an optimized vascularization resulting
from the release of HIF-1α and downstream proteins. Table 2
lists all model variables. These variables are normalized and
assumed to be in the range [0:1].
The payoff table 1 assumes that non-motile phenotypes
(GLY and AG) will share existing resources with the cells they
interact with, whereas motile phenotypes can chose whether to
stay or move. In the case of INV cells, they will always move
and leave existing resources for the cell it is interacting with
unless the interaction happens with another INV cell, in which
Phys. Biol. 8 (2011) 015016 (9pp) doi:10.1088/1478-3975/8/1/015016
The role of IDH1 mutated tumour cells in
secondary glioblastomas: an evolutionary
game theoretical view
David Basanta1, Jacob G Scott1, Russ Rockne2, Kristin R Swanson2
and Alexander R A Anderson1
1
Integrated Mathematical Oncology, H Lee Moffitt Cancer Center and Research Institute, Tampa,
FL 33612, USA
2
Pathology and Applied Mathematics at the University of Washington, Seattle, WA 98104, USA
E-mail: david.basanta@kclalumni.net and jacob.scott@moffitt.org
Received 17 September 2010
Accepted for publication 10 January 2011
Published 7 February 2011
Online at stacks.iop.org/PhysBio/8/015016
Abstract
Recent advances in clinical medicine have elucidated two significantly different subtypes of
glioblastoma which carry very different prognoses, both defined by mutations in isocitrate
dehydrogenase-1 (IDH-1). The mechanistic consequences of this mutation have not yet been
fully clarified, with conflicting opinions existing in the literature; however, IDH-1 mutation
may be used as a surrogate marker to distinguish between primary and secondary glioblastoma
multiforme (sGBM) from malignant progression of a lower grade glioma. We develop a
mathematical model of IDH-1 mutated secondary glioblastoma using evolutionary game
theory to investigate the interactions between four different phenotypic populations within the
tumor: autonomous growth, invasive, glycolytic, and the hybrid invasive/glycolytic cells. Our
model recapitulates glioblastoma behavior well and is able to reproduce two recent
experimental findings, as well as make novel predictions concerning the rate of invasive growth
as a function of vascularity, and fluctuations in the proportions of phenotypic populations that
a glioblastoma will experience under different microenvironmental constraints.
1. Introduction
Our ability to tease apart pathologic differences in cancers
began with microscope and differential staining and has
progressed to the current age of molecular medicine.
The mantra of clinical medicine in the molecular age is
‘personalized medicine’—the hope that one day we will
be able to perfectly understand each person’s tumor at the
molecular and mechanistic level in order to prescribe the
perfect treatment. While we have made many advances
in subtyping many different cancers and even designed
molecularly targeted therapies, the results so far have been
disappointing. One cancer that has remained particularly
resistant to our therapies is glioblastoma multiforme (GBM),
which carries a prognosis of less than a year and certain
mortality.
It has been understood for several years that there
are different subtypes of glioblastoma characterized by
mutation pattern and cell of origin [1], but this knowledge
has not altered our treatment strategy, only our ability
to prognosticate outcome. That these subtypes all end
up looking the same under the microscope and end up
behaving very similarly as aggregates is an example of
convergent evolution—genotypically different cells with
similar phenotypic characteristics.
Most recently, two significantly different classes of
glioblastoma have been identified which carry very different
prognoses [2–4]. These two groups of glioblastoma are, for
the most part, differentiated by mutations found in a single
coding region of an enzyme involved in the Krebs cycle,
isocitrate dehydrogenase 1 (IDH1). This mutation is present
in the majority of secondary glioblastomas (sGBM) and low
grade gliomas (LGGs), many of which progress to become
1478-3975/11/015016+09$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK
IOP PUBLISHING
Phys. Biol. 8 (2011) 015016 (9pp) doi:10.1088
The role of IDH1 mutated tumour c
secondary glioblastomas: an evolut
game theoretical view
David Basanta1, Jacob G Scott1, Russ Rockne2, Kristin R Swanson2
and Alexander R A Anderson1
1
Integrated Mathematical Oncology, H Lee Moffitt Cancer Center and Research Institute, Tampa,
FL 33612, USA
2
Pathology and Applied Mathematics at the University of Washington, Seattle, WA 98104, USA
E-mail: david.basanta@kclalumni.net and jacob.scott@moffitt.org
Received 17 September 2010
Accepted for publication 10 January 2011

LETTER TO THE EDITOR
Production of 2-hydroxyglutarate by
isocitrate dehydrogenase 1–mutated
gliomas: an evolutionary alternative
to the Warburg shift?
Neuro-Oncology
Neuro-Oncology Advanc
Neuro-Oncology
NEURO-ONCOLOGY
Neuro-Oncology Advance Access published July 22, 2011
015016 D Basanta et al
Invasion
BeforeBevacizumabAfterBevacizumab
Loss of PET signal
~ Less Glycolysis
starting
t t t
reduce
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3. Plot of k = 0.1, n = 0.2, c = 0.1 and α = 0.3, 0.32, 0.35. The first panel shows two interesting dynamics: with increasing benefi
vasculature (increasing α), we see a more rapid progression as well as a higher overall proportion of cells with the GLY phenotype. Also
decreasing α promotes the INV phenotype (stars) which is recapitulated in recurrent glioblastoma after bevacizumab treatment. The two
panels below the control one show what happens after bevacizumab has been administered after 600 time steps without assuming wheth
the main effect would be a normalization of the angiogenic vasculature (which would increase α, shown in the second row) or the reduc
of the existing vasculature (which would have a negative effect on α, shown in the third row).
changes that we see in glioblastoma patients after failure of
bevacizumab (a monoclonal antibody to VEGF-α).
The recapitulation of known behaviors allows some
measure of confidence in our model and gives some credence
to predictions that the model can make. Now we can begin to
make observations about the mechanisms driving the behaviors
that were, otherwise, obscured by the biological complexity.
A recurring theme observed in the time-dependent behavior
of our model suggests an underlying mechanism driven by
interactions between different phenotypes. Specifically, the
emergence of the invasive phenotypes is always preceded
by a rise in the glycolytic fraction. This rise in the
glycolytic fraction is preceded by an overgrowth of AG cells.
cells grow into a viable proportion, the damage that they
to the local environment with their excessive acid product
begins to promote the benefit of cells that can move to a n
place (INV). We see this sequence reproduced in nearly
areas of the parameter space, and certainly in all the areas
are relevant to glioma. Further, these results agree nic
with earlier work done by this group suggesting that
glycolytic phenotype is necessary to bring about the emerge
of invasion [18].
In addition to this sequence, there was an interest
dynamic that emerged in some areas of the parameter spa
Figure 5 shows an example of two types of oscillatory behav
that our model can produce. Even though neither manag

Symmetric Division Rate
and live in a ‘continuous’ milieu described completely by
the experimenter
end result is a model that looks complicated, but is entirely
described by a minimum of parameters - allowing for
emergent phenomena
and the subtype driven by PDGF overexpression. For NF1-driven
cancers, we investigated bi-allelic loss of NF1 and a dominant
negative mutation of TP53 as the necessary driver mutations that
must be accumulated in a single cell to initiate tumorigenesis. For
PDGF-driven cancers, the necessary driving alterations are those
leading to PDGF overexpression and bi-allelic loss of INK4A/ARF.
We did not include the accumulation of passenger mutations in
this model since those alterations, by definition, do not influence
of cells carrying alterations in
similarly leads to increased pro
on the background of either T
INK4A/ARF2/2
NF12/2
cells
fitness RARF 6RNF1,mut and div
In contrast, cells mutated in NF
a fitness detriment, RNF1,wt0.
cell divisions beyond the norm
Figure 1. A mathematical model of the cell of origin of PDGF- and NF1-driven gliomas. Initially, th
cells (blue) and 2z+1
21 wild-type transit-amplifying non-self-renewing (TA) cells (purple). At each time ste
probability a, the SR cell divides symmetrically and one daughter cell replaces another randomly chosen SR
divides asymmetrically and one daughter cell remains a SR cell while the other daughter cell becomes comm
This new TA cell divides symmetrically z times to give rise to successively more differentiated cells (progres
becoming terminally differentiated. This restriction of the stochastic process ensures that the total number of
homeostatic conditions in the healthy brain. In the figure, the darkening purple gradations refer to successively
clarify a single time step of the stochastic process. We investigate the dynamics of only one cell cluster since the
given by the probability per cluster times the number of clusters; hence, a consideration of all clusters does n
cell of origin of brain cancer.
doi:10.1371/journal.pone.0024454.g001
Rounds of Transient
Ampliﬁcation
Vascular Density
0.01 0.05 0.1
!!!!! !! !!
!!
! !!!!
! !!!!! ! !!!!!! ! !!!!!! ! !!!!!!!

Loss of homeostasis
Symmetric division rate - 0.5
Rounds of Ampliﬁcation -11
Vascular density - .05

1 5 10 15 20
0
0.5
1
1.5
2
2.5
x 10
5
Divisions per progenitor cell
Cells
s/a 0.1
s/a 0.3
s/a 0.5
1 5 10 15 20
0
0.5
1
1.5
2
x 10
5
s/a 0.1
s/a 0.3
s/a 0.5
1 5 10 15 20
0
0.5
1
1.5
2
2.5
x 10
5
Cells
s/a 0.1
s/a 0.3
s/a 0.5
Vascular Density
Quantifying the unmeasurable
0.01 0.05 0.1
Background:
Metastasis is a highly lethal and poorly understood process that
accounts for the majority cancer deaths

90% of cancer death is from metastatic disease
Yet from the clinicians perspective, metastasis is a binary
event - the least understood process

Which patients will end up as M1 vs. M450??
We now have targetted local therapies (like SBRT) that
ablate these tumors...

Game
1
Game
2
Jacob Scott, Maciunas STEMM Keynote 2010

The cutoff frequency .[o is then defined as
1
./o= ~ (17)
Values afro for each segment are listed in Table 1.
5 Computational procedure
A digital computer program was written in FOR-
TRAN to operate on the branching configuration
multiple branching
parallel. Transmiss
node calculated fr
backward towards
pedance of the wh
final result is a com
ching configuration
and transmission p
every node. Hence
ejection waveform
flow waveforms m
53
52/~ 55
51~5136 3~
~7 7i ~4 the branching struc
0s 6o flow throughout th
---r-~ pheral resistance v57 35 47
~564~)]~0__.~33]1~ sistances and visco
The input data t
dimensions and
o~4' _ . ?2
42 i!/6; 2,1 ,e sS/.~ 9
8~8~70 15 49 61, ,, 3, ,, o2 :,, I
, }1 3200[ !/
.o, ,,oo,!,
,o8 ' ~'~, ,03 ~q
~o9 10cm ! ~L
' I 800
pH} c 0
113
,,8 y ' , , s
12it 125
humanarterialtree
Fig. 1 Schematic representation of the human arterial tree
with all lengths drawn to scale. Segment numbers
correspond to arteries listed in Table l
2
r 1-0
.
= -1.ot i
Fig. 2 Input impedanc
pedance in each
simultaneous re
ascending aorta
culatedfrom mo
Medical Biological Engineering Computing November

Simple
experiment
and
ODE
model

to

begin
a
conversa:on
Can
likely
measure
f(t)
in
a
mouse
model
by

injec:on
a
bolus
of
tumor
cells
into
a
tail

vein
and
measuring
CTCs
at
several
:me

points
tumor
Other
organ
•C
is
number
of
CTCs
•Alpha
and
beta
are
constants
•T(t)
is
a
func:on
describing
tumor
size
•z(t)
is
rate
of
tumor
cell
intravasa:on

“shedding
rate”
•f(t)
is
the
rate
of
ﬁltra:on
or
CTC
arrest

What would we need for Kirchoff’s rules?

!#$%'#()*'+,
-+'.#
!./*+
0,
-1'((*+'#(2+3%,',*
-3#*
β
η
γ
Unifying metastasis — integrating
intravasation, circulation and
end-organ colonization
Jacob Scott1,2
, Peter Kuhn3
and Alexander R. A. Anderson1
!#$%'$()(*+'+,$($+'-,./.01'/(23,'+#($-$(-3+(+,/+2($-+(4+#5%+4+,$(.6('1%'5/$1,0(
$54.5%('+//#(7898#:(1,(;$1+,$#(-3+(#;5%%+2(1,$+%+#$(1,($-+('1%'5/$.%(;-#+(.6(4+$#$#1#=(
9+'-,15+#($-$(2.(,.$(#./+/(%+/(.,((/..2(#4;/+(//.?(#5#$,$1/(1./.01'/(1,$+%%.0$1.,(
+.,2(#14;/('.5,$1,0(898#=
for
gy,
A.
In patients with advanced primary cancer, circulating
tumour cells (CTCs)1
can be found throughout the entire
vascular system2
. When and where these CTCs form
metastasis is not fully understood, and is currently the
subject of intensive biological study. Paget’s well-known
seed–soil hypothesis3
suggests that the ‘soil’ (the site of
a metastasis) is as important as the ‘seed’ (the metastatic
cells) in the determination of successful metastasis. The
mechanism by which seeds are disseminated to specific
soil has, to date, been a ‘known unknown’. We think that
it is during this poorly understood phase of metastasis
that we stand to answer important questions4
.
We hypothesize that the rich variety of possible meta-
static disease patterns not only stems from the physical
aspects of the circulation but also from CTC hetero-
geneity (FIG. 1). These seeds represent many different
populations that are derived from a diverse population
of competing phenotypes within the primary tumour5
.
Because such seeds need to pass through a system of
physical and biological filters in the form of specific
organs, the circulatory phase of metastasis could be
modelled as a complex deterministic filter. In theory,
until the evolution of a suitable seed, any number of
CTCs could flow through the circulation and arrest
at end organs without metastases forming. As tumour
heterogeneity is thought to expand as the tumour pro-
gresses, it follows that at some point a seed will come
into existence that is suited to a specific soil within that
patient’s body. If this seed is to propagate it must find
its soil, a process that we hypothesize is governed by
solvable physical rules that relate to the dynamics of
do not fit a model that is defined only by physical flow
and filtration.
To begin the process of physical interrogation, we
propose a model that represents the human circulatory
system as a directed and weighted network, with nodes
representing organs and edges representing arteries and
veins.The novelty is only fully realized when combined
withaheterogeneousCTCpopulation(drivenbyprimary
tumour heterogeneity) modulated by the complex organ
filter system (with physiologically relevant connections)
under dynamic flow. Four important biological processes
emerge from this representation. First, the shedding rate,
which is defined as the rate at which the tumour sheds
CTCs into the vasculature. Second, CTC heterogeneity,
which is defined as the distribution of CTC phenotypes
present in the circulation. Third, the filtration fraction,
which is defined as the proportion (and type) of CTCs
that arrest in a given organ. Fourth, the clearance rate,
which is defined as the rate at which cancer cells are
cleared from the blood and/or organ after arrest. Each of
these biological processes is probably disease- and even
patient-specific, and each is extremely poorly understood.
Using this representation to motivate the develop-
ment of a mathematical model we can define both the
concentration of CTCs and their phenotypic distribu-
tion at any given point in the network, as well as organ-
specific filtration values. To parameterize this model,
characterization and enumeration of CTCs taken from
a single patient at different time points and from differ-
ent points in this network will need to be undertaken.
A complete understanding of the model will also pro-
Unifying metastasis — int
intravasation, circulation
end-organ colonization
Jacob Scott1,2
, Peter Kuhn3
and Alexander R. A. Anders
!#$%'$()(*+'+,$($+'-,./.01'/(23,'+#($-$(-3+(+,/+2($
$54.5%('+//#(7898#:(1,(;$1+,$#(-3+(#;5%%+2(1,$+%+#$(1,($-+('1%
8@A
Nature Reviews Cancer | AOP, published online 24 May 2012; doi:10.1038/nrc3287

Matrices are important - pay attention in AMH!!

“Whereas a good simulation should
include as much as possible, a good
model should include as little as possible.”

Lung



outflow
∂CL
∂t
=
inflow
IL −
filtering
ηLCL
buildup
∂OL
∂t
=
arresting
ηLCL −
clearing
γLOL
Liver



outflow
∂CLi
∂t
=
inflow *A
αLiCLi +
inflow *B
αGCG −
filtering
ηLiCLi +
shedding
β
buildup
∂OLi
∂t
=
arresting
ηLiCLi −
clearing
γLiOLi
Make everything as
simple as possible,
but no simpler

Brain
Liver
Gut
Bone
Venous Arterial
Portal System
CTCflow
Lung
Breast
Primary Seeding
Primary
Tumor
(a) Primary Seeding cartoon
Brain
Liver
Gut
Bone
Venous Arterial
Portal System
CTCflow
Lung
Breast
Primary
Tumor
Secondary
Tumor
Secondary
Tumor
Secondary Seeding
(b) Secondary Seeding cartoon
A
Time (cell cycles)
Logtumourmass B
Removal rate λ
Returnprobabilityp
growthacceleration(logscale)
primary
seeding
secondary
seeding
Figure 2: Simulating the dynamics of primary seeding.(A) shows the total tumour b
for three different conditions where the removal rate was fixed at λ = 10−5 and return proba
was taken to be p = 10−2,10−3 and 10−4 respectively. (B) illustrates the model dynamics
the parameters λ and p are varied systematically, and shows that accelerated tumour growth
, 20130011, published 20 February 2013102013J. R. Soc. Interface
Jacob G. Scott, David Basanta, Alexander R. A. Anderson and Philip Gerlee
growth
secondary metastatic deposits as drivers of primary tumour
A mathematical model of tumour self-seeding reveals
Open Access

science society
B
iology has long been the stepchild
of the natural sciences. Compared
with mathematical proofs, physical
formulae and the molecules of chemistry,
biology, like life itself, has often seemed
unquantifiable, unpredictable and messy.
Yet, scientists have striven gallantly to pin
biology down through the application of the
of inspiration for mathematicians. “In my
40 plus years of research, I have found that
problems in biomathematics almost always
uncover unexplored and undeveloped areas
of mathematics,” he said. “These are areas
that mathematicians have not even thought
about exploring. New mathematics.”
involves events a
gene expression o
take place in nano
nisms or body-wi
minutes, hours o
between people
tions that last mon
understand these s
different layers an
Jost points out, ha
mathematical biolo
T
he applicatio
logy itself i
back at least
on the inheritanc
nineteenth centur
the theory of Men
foundation of mod
ally reproducing o
Biology is the new physics
The increasing use of mathematics in biology is both inspiring
research in mathematics and creating new career options for
mathematical biologists
Philip Hunter
EMBO reports VOL 11 | NO 5 | 2010350
“For many years the inspiration for
innovation in applied mathematics has
come from physics, but in my opinion,
in this century it will come from the bio-
logical sciences, broadly defined,” Mackey
explained, adding that this switch has been
taking place slowly over several decades.
While physics has stagnated, waiting for
new theoretical insights to make progress
against fundamental problems such as
quantum gravity, Mackey argued, theoreti-
cal biology has emerged as a new source
complexes;
able to accou
tle difference
infection am
ous approac
growing use
biological an
capable of a
the systems u
of a protein,
essence, inf
geometry wi
changing, co
Other tech
information
tions betwee
a huge ran
The human
continue to the extent that it
might even become the main
driving force behind innovation
and development in mathematics

What stars can teach us about cancer: Modeling metastasis through population genetics

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to What stars can teach us about cancer: Modeling metastasis through population genetics

Similar to What stars can teach us about cancer: Modeling metastasis through population genetics (20)

Recently uploaded

Recently uploaded (20)

What stars can teach us about cancer: Modeling metastasis through population genetics