Natalia Komarova

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Somatic evolution and cancer

• Natalia Komarova
• (University of California - Irvine)

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Plan

• Introduction The concept of somatic evolution
• Methodology Stochastic processes on
  selection-mutation networks
• Two particular problems
• Stem cells, initiation of cancer and optimal
  tissue architecture (with L.Wang and P.Cheng)
• Drug therapy and generation of resistance
  neutral evolution inside a tumor (with D.Wodarz)

3
Darwinian evolution (of species)

• Time-scale hundreds of millions of years
• Organisms reproduce and die in an environment
  with shared resources

4
Darwinian evolution (of species)

• Time-scale hundreds of millions of years
• Organisms reproduce and die in an environment
  with shared resources
• Inheritable germline mutations (variability)
• Selection
• (survival of the fittest)

5
Somatic evolution

• Cells reproduce and die inside an organ of one
  organism
• Time-scale tens of years

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Somatic evolution

• Cells reproduce and die inside an organ of one
  organism
• Time-scale tens of years
• Inheritable mutations in cells genomes
  (variability)
• Selection
• (survival of the fittest)

7
Cancer as somatic evolution

• Cells in a multicellular organism have evolved to
  co-operate and perform their respective functions
  for the good of the whole organism

8
Cancer as somatic evolution

• Cells in a multicellular organism have evolved to
  co-operate and perform their respective functions
  for the good of the whole organism
• A mutant cell that refuses to co-operate may
  have a selective advantage

9
Cancer as somatic evolution

• Cells in a multicellular organism have evolved to
  co-operate and perform their respective functions
  for the good of the whole organism
• A mutant cell that refuses to co-operate may
  have a selective advantage
• The offspring of such a cell may spread

10
Cancer as somatic evolution

• Cells in a multicellular organism have evolved to
  co-operate and perform their respective functions
  for the good of the whole organism
• A mutant cell that refuses to co-operate may
  have a selective advantage
• The offspring of such a cell may spread
• This is a beginning of cancer

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Progression to cancer
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Progression to cancer
Constant population
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Progression to cancer
Advantageous mutant
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Progression to cancer
Clonal expansion
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Progression to cancer
Saturation
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Progression to cancer
Advantageous mutant
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Progression to cancer
Wave of clonal expansion
18
Genetic pathways to colon cancer (Bert
Vogelstein)
Multi-stage carcinogenesis
19
Methodology modeling a colony of cells

• Cells can divide, mutate and die

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Methodology modeling a colony of cells

• Cells can divide, mutate and die
• Mutations happen according to a
  mutation-selection diagram, e.g.

u1
u4
u2
u3
(r3)
(r4)
(r2)
(1)
(r1)
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Mutation-selection network
u8
(r3)
u8
(r2)
(r6)
u8
u5
(1)
(r4)
(r1)
(r6)
u2
u2
u5
u8
(r1)
(r5)
(r7)
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Stochastic dynamics on a selection-mutation
network
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A birth-death process with mutations
Selection-mutation diagram
Number of is i
u
(1)
(r )
Number of is jN-i
Fitness 1
Fitness r gt1
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Evolutionary selection dynamics
Fitness 1
Fitness r gt1
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Evolutionary selection dynamics
Fitness 1
Fitness r gt1
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Evolutionary selection dynamics
Fitness 1
Fitness r gt1
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Evolutionary selection dynamics
Fitness 1
Fitness r gt1
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Evolutionary selection dynamics
Fitness 1
Fitness r gt1
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Evolutionary selection dynamics
Start from only one cell of the second
type. Suppress further mutations. What is the
chance that it will take over?
Fitness 1
Fitness r gt1
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Evolutionary selection dynamics
Start from only one cell of the second type. What
is the chance that it will take over?
If r1 then 1/N If rlt1 then lt
1/N If rgt1 then gt 1/N If r
then 1
Fitness 1
Fitness r gt1
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Evolutionary selection dynamics
Start from zero cell of the second type. What is
the expected time until the second type takes
over?
Fitness 1
Fitness r gt1
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Evolutionary selection dynamics
Start from zero cell of the second type. What is
the expected time until the second type takes
over?
In the case of rare mutations,
we can show that
Fitness 1
Fitness r gt1
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Two-hit process (Alfred Knudson 1971)
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A two-step process
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A two-step process
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A two step process
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A two-step process
Scenario 1 gets fixated first, and then
a mutant of is created
Number of cells
time
38
Stochastic tunneling
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Two-hit process
Scenario 2 A mutant of is created before
reaches fixation
Number of cells
time
40
The coarse-grained description
Long-lived states x0 all green x1 all
blue x2 at least one red
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Stochastic tunneling
Neutral intermediate mutant

Disadvantageous intermediate mutant
Assume that and
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Stem cells, initiation of cancer and optimal
tissue architecture
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Colon tissue architecture
44
Colon tissue architecture
Crypts of a colon
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Colon tissue architecture
Crypts of a colon
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Cancer of epithelial tissues
Gut
Cells in a crypt of a colon
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Cancer of epithelial tissues
Cells in a crypt of a colon
Gut
Stem cells replenish the tissue asymmetric
divisions
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Cancer of epithelial tissues
Cells in a crypt of a colon
Gut
Proliferating cells divide symmetrically and
differentiate
Stem cells replenish the tissue asymmetric
divisions
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Cancer of epithelial tissues
Cells in a crypt of a colon
Gut
Differentiated cells get shed off into the lumen
Proliferating cells divide symmetrically and
differentiate
Stem cells replenish the tissue asymmetric
divisions
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Finite branching process
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What is known

• Normal cells undergo apoptosis at the top of the
  crypt, the tissue is renewed and cell number is
  constant

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What is known

• Normal cells undergo apoptosis at the top of the
  crypt, the tissue is renewed and cell number is
  constant
• One of the earliest events in colon cancer is
  inactivation of the APC gene

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What is known

• Normal cells undergo apoptosis at the top of the
  crypt, the tissue is renewed and cell number is
  constant
• One of the earliest events in colon cancer is
  inactivation of the APC gene
• APC-/- cells do not undergo apoptosis at the top
  of the crypt

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What is NOT known

• What is the cellular origin of cancer?
• Which cells harbor the first dangerous mutaton?
• Are the stem cells the ones in danger?
• Which compartment must be targeted by drugs?

?
?
?
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Colon cancer initiation

• Both copies of the APC gene must be mutated
  before a phenotypic change is observed (tumor
  suppressor gene)

X
X
X
APC-/-
APC/-
APC/
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Cellular origins of cancer
Gut
If a stem cell tem cell acquires a mutation,
the whole crypt is transformed
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Cellular origins of cancer
Gut
If a daughter cell acquires a mutation, it will
probably get washed out before a second mutation
can hit
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What is the cellular origin of cancer?
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Colon cancer initiation
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Colon cancer initiation
61
Colon cancer initiation
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Colon cancer initiation
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Colon cancer initiation
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Colon cancer initiation
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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Cellular origins of cancer

• The prevailing theory is that the mutations
  leading to cancer initiation occur is stem cells

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Cellular origins of cancer

• The prevailing theory is that the mutations
  leading to cancer initiation occur is stem cells
• Therefore, all prevention and treatment
  strategies must target the stem cells

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Cellular origins of cancer

• The prevailing theory is that the mutations
  leading to cancer initiation occur is stem cells
• Therefore, all prevention and treatment
  strategies must target the stem cells
• Differentiated cells (most cells!) do not count

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Mathematical approach

• Formulate a model which distinguishes between
  stem and differentiated cells
• Calculate the relative probability of various
  mutation patterns

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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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Stochastic tunneling in a heterogeneous population

• At least one mutation happens in a stem cell (cf.
  the two-step process)
• 2) Both mutations happen in a daughter cell no
  fixation of an intermediate mutant (cf tunneling)

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Stochastic tunneling in a heterogeneous population

Lower rate

• At least one mutation happens in a stem cell (cf.
  the two-step process)
• 2) Both mutations happen in a daughter cell no
  fixation of an intermediate mutant (cf tunneling)

83
Cellular origins of cancer

• If the tissue is organized into compartments with
  stem cells and daughter cells, the risk of
  mutations is lower than in homogeneous populations

84
Cellular origins of cancer

• If the tissue is organized into compartments with
  stem cells and daughter cells, the risk of
  mutations is lower than in a homogeneous
  population
• Cellular origin of cancer is not necessarily the
  stem cell. Under some circumstances, daughter
  cells are the ones at risk.

85
Cellular origins of cancer

• If the tissue is organized into compartments with
  stem cells and daughter cells, the risk of
  mutations is lower than in a homogeneous
  populations
• Cellular origin of cancer is not necessarily the
  stem cell. Under some circumstances, daughter
  cells are the ones at risk.
• Stem cells are not the entire story!!!

86
Optimal tissue architecture

• How does tissue architecture help protect against
  cancer?
• What are parameters of the architecture that
  minimize the risk of cancer?
• How does protection against cancer change with
  the individuals age?

87
Optimal number of stem cells
m1
m2
Crypt size is n16
m4
m8
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Probability to develop dysplasia
One stem cell
Probability to develop dysplasia
Many stem cells
Time (individuals age)
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The optimal solution is time-dependent!
Optimum many stem cells
One stem cell
Probability to develop dysplasia
Many stem cells
Optimum one stem cell
Time (individuals age)
90
Optimization problem

• The optimum number of stem cells is high in young
  age, and low in old age
• Assume that tissue architecture cannot change
  with time must choose a time-independent
  solution
• Selection mostly acts upon reproductive ages, so
  the preferred evolutionary strategy is to keep
  the risk of cancer low while the organism is
  young

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Evolutionary compromise
Many stem cells
Probability to develop dysplasia
One stem cell
Time (individuals age)
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Evolutionary compromise
Many stem cells
While keeping the risk of cancer low at the
young age, the preferred evolutionary strategy
works against the
older age, actually
increasing
the
likelihood of cancer!
Probability to develop dysplasia
One stem cell
Time (individuals age)
93
Cancer vs aging

• Cancer and aging are two sides of the same coin..

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Drug therapy and generation of resistance
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Leukemia

• Most common blood cancer
• Four major types
• Acute Myeloid Leukemia (AML),
• Chronic Lymphocytic Leukemia (CLL),
• Chronic Myeloid Leukemia (CML),
• Acute Lymphocytic Leukemia (ALL)

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Leukemia

• Most common blood cancer
• Four major types
• Acute Myeloid Leukemia (AML),
• Chronic Lymphocytic Leukemia (CLL),
• Chronic Myeloid Leukemia (CML),
• Acute Lymphocytic Leukemia (ALL)

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CML

• Chronic phase (2-5 years)
• Accelerated phase (6-18 months)
• Blast crisis (survival 3-6 months)

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Targeted cancer drugs

• Traditional drugs very toxic agents that kill
  dividing cells

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Targeted cancer drugs

• Traditional drugs very toxic agents that kill
  dividing cells
• New drugs small molecule inhibitors
• Target the pathways which make cancerous cells
  cancerous (Gleevec)

100
Gleevec a new generation drug
Bcr-Abl
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Gleevec a new generation drug
Bcr-Abl
Bcr-Abl
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Small molecule inhibitors
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Targeted cancer drugs

• Very effective
• Not toxic

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Targeted cancer drugs

• Very effective
• Not toxic
• Resistance poses a
• problem

Gleevec
Bcr-Abl protein
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Targeted cancer drugs

• Very effective
• Not toxic
• Resistance poses a
• problem

Mutation
Gleevec
Bcr-Abl protein
106
Treatment without resistance
treatment
time
107
Development of resistance
treatment
108
How can one prevent resistance?

• In HIV treat with multiple drugs
• It takes one mutation to develop resistance of
  one drug. It takes n mutations to develop
  resistance to n drugs.
• Goal describe the generation of resistance
  before and after therapy.

109
Mutation network for developing resistance
against n3 drugs
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During a short time-interval, Dt, a cell of type
Ai can

• Reproduce faithfully with probability
• Li(1-Suj) Dt

111
During a short time-interval, Dt, a cell of type
Ai can

• Reproduce faithfully with probability
• Li(1-Suj) Dt
• Produce one cell identical to itself, and a
  mutant cell of type Aj with probability Liuj Dt

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During a short time-interval, Dt, a cell of type
Ai can

• Reproduce faithfully with probability
• Li(1-Suj) Dt
• Produce one cell identical to itself, and a
  mutant cell of type Aj with probability Liuj Dt
• Die with probability Di Dt

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The method
Assume just one drug. xij(t) is the probability
to have i susceptible and j resistantcells at
time t.
F(x,yt)Sxij(t)xjyi is the probability
generating function.
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The method
xij(t) is the probability to have i susceptible
and j resistant cells at time t.
F(x,yt)Sxij(t)xjyi is the probability
generating function.
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For multiple drugs
xi0, i1, , im(t) is the probability to have is
cells of type As at time t.
F(x0,x1,,xmt) S xi0, i1, , im(t) x0im
xmi0 is the probability generating function.
F(0,1,,1t) is the probability that at time t
there are no cells of type Am
F(0,0,,0t) is the probability that at time t
the colony is extinct
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The method
The probability that at time t the colony is
extinct is F(0,0,,0t)
xnM(t), where M is the initial of cells and
xn is the solution of
The probability of treatment failure is
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The questions

1. Does resistance mostly arise before or after the
   start of treatment?
2. How does generation of resistance depend on the
   properties of cancer growth (high turnover DL vs
   low turnover DltltL)
3. How does the number of drugs influence the
   success of treatment?

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1. How important is pre-existence of mutants?
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Single drug therapy
120
Single drug therapy
Generation during treatment
Pre-existance
121
Single drug therapy
Unrealistic!
Generation during treatment
Pre-existance
122
Single drug therapy
Pre-existance gtgt Generation during
treatment
123
Multiple drug therapies
Fully susceptible
Partially susceptible
Fully resistant
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Development of resistance
Fully susceptible
Partially susceptible
Fully resistant
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1. How important is pre-existence of resistant
mutants?

• For both single- and multiple-drug therapies,
• resistant mutants are likely to be produced
  before start of treatment, and not in the
  course of treatment

126
2. How does generation of resistance depend on
the turnover rate of cancer?

• Low turnover (growth rategtgtdeath rate)
• Fewer cell divisions needed to reach a certain
  size
• High turnover (growth ratedeath rate)
• Many cell divisions needed to reach a certain
  size

127
Single drug therapy
Low turnover cancer, DltltL
128
Single drug therapy
High turnover cancer, DL
More mutant colonies are produced, but
the probability of colony survival is
proportionally smaller
129
2. How does generation of resistance depend on
the turnover rate of cancer?

• Single drug therapies the production of mutants
  is independent of the turnover

130
2. How does generation of resistance depend on
the turnover rate of cancer?

• Single drug therapies the production of mutants
  is independent of the turnover
• Multiple drug therapies the production of
  mutants is much larger for cancers with a high
  turnover

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3. The size of failure

• Suppose we start treatment at size N
• Calculate the probability of treatment failure
• Find the size at which the probability of failure
  is d0.01

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3. The size of failure

• Suppose we start treatment at size N
• Calculate the probability of treatment failure
• Find the size at which the probability of failure
  is d0.01
• The size of failure increases with of drugs and
  decreases with mutation rate

133
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
134
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
135
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
136
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
137
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
138
CML leukemia

• Gleevec
• u10-8-10-9
• D/L between 0.1 and 0.5 (low turnover)
• Size of advanced cancers is 1013 cells

139
Log size of treatment failure
u10-8
u10-6
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Application for CML

• The model suggests that 3 drugs are needed to
  push the size of failure (1 failure) up to 1013
  cells

141
Conclusions

• Main concept cancer is a highly structured
  evolutionary process
• Main tool stochastic processes on
  selection-mutation networks
• We addressed questions of cellular origins of
  cancer and generation of drug resistance
• There are many more questions in cancer research

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Multiple drug treatments

• For fast turnover cancers, adding more drugs will
  not prevent generation of resistance

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Size of failure for different turnover rates