Modeling the chemotherapy-induced selection of drug-resistant traits - - PowerPoint PPT Presentation

SLIDE 1

Modeling the chemotherapy-induced selection

• f drug-resistant traits during tumor growth

Doron Levy

Department of Mathematics Center for Scientific Computation and Mathematical Modeling University of Maryland, College Park

Thanks: NSF, NIH, UMD-NCI partership, LCB/NCI, Wolfgang Pauli Institute Guggenheim Foundation, Simons Foundation, Jayne Koskinas Ted Giovannis Foundation

Abarbanel Memorial Conference Tel Aviv University, December 2018

SLIDE 2

Joint work with

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Heyrim Cho Maryland

★

Matt Becker Johns Hopkins APL

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Cristian Tomasetti Johns Hopkins

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Michael Gottesman, MD NCI/NIH

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Orit Lavi NCI/NIH

★

Jim Greene Rutgers

SLIDE 3

Outline

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Drug Resistance in Cancer

• Mechanisms of MDR
• Mathematics & Drug Resistance

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Heterogeneity and Resistance

• Selection process – a continuous trait
• Heterogeneity in space (non-radially symmetric case)
• Drug combinations (including cell-cycle and non-cell-cycle specific)
• Competition between healthy cells and cancer cells

SLIDE 4

Mechanisms of MDR

Gillet & Gottesman, 2010

SLIDE 5

Mathematics and Drug Resistance

Lavi, Gottesman, Levy. The dynamics of drug resistance: A mathematical perspective. Drug Resistance Updates 15, 2012, 90-97.

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Q1: What is the optimal protocol for drug scheduling in terms of dose and timing? Goal: maximize the control of the tumor while minimizing toxicity (Norton & Simon / Goldie & Coldman,…)

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Q2: Continuous infusion vs. short pulses (Gardner, Panetta, Smieja,…)

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Q3: When several drugs are available, how many drugs should be used? Should they be used in combination or sequentially? (Komarova & Wodarz,…)

SLIDE 6

Tumor heterogeneity

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Experiment: sequence the genome of cells that are sampled from different locations within the tumor

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Result: the genomes of these cells are different

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Conclusion: tumors are heterogeneous

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What does it mean? Different things to different people

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For us: heterogeneity in response to drugs

SLIDE 7

The genome is not everything!

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There is more to the dynamics than the genome

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Genetics vs. epigenetics

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Example: AMIGOS experimental data (TNBCs, 3D, Peyton lab)

SLIDE 8

The ultimate goal

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Assumptions (science fiction):

• We can know the initial data (mapping the entire tumor with regards to

the level of resistance)

• We can fully control the treatment (which drug and how much of it

arrives at what area)

• We can fully (and dynamically) know what the cells are doing without

the treatment and in response to the treatment

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Question: What is the optimal treatment?

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A preliminary question: What is “optimal”?

SLIDE 9

Modeling Tumor Heterogeneity

(Greene, Lavi, Gottesman, DL, Cancer Research 2013, BMB 2014, Trends in Mol. Med. 2014)

Following Lorz, Lorenzi, Clairambault, Perthame, et al. 2013 (Calsina & Cuadrado,

Champagnat, Desvillettes, Diekmann, …)

After some scaling…. And assuming

ρ(t) = Z 1 n(x, t) dx. ∂n(x, t) ∂t = ⇣ f(ρ(t))[r(x)(1 − θ(x)) − h(D(t), x)] − g(ρ(t))d(x) ⌘ n(x, t) + f(ρ(t)) Z 1 θ(y)r(y)M(y, x)n(y, t) dy. ∂n(x, t) ∂t = (r(x)(1 − θ) − c(x) − G(ρ(τ))d(x)) n(x, t) + θ Z 1 r(y)M(y, x)n(y, t)dy D(t) ≡ 1, θ(x) ≡ θ

Resistance level is assumed to be a continuous parameter

SLIDE 10

Modeling Tumor Heterogeneity

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Under these assumptions + rescaling time

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Case I: Trait-based growth

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Case II: Density-dependent model

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Case III: The full selection-mutation model

∂n(x, t) ∂t = (r(x)(1 − θ) − c(x) − G(ρ(τ))d(x)) n(x, t) + θ Z 1 r(y)M(y, x)n(y, t)dy

τ = Z t f(ρ(s)) ds

∂n(x, t) ∂t = [r(x) − c(x) − d(x)]n(x, t). ∂n(x, t) ∂t = [r(x) − c(x) − G(ρ)d(x)]n(x, t).

Lorz - cancer cells Lorz - a specific form

• f the eq. for healthy cells

SLIDE 11

Case I: Trait-Based Growth

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Only two different growth phenomena: extinction or unbounded growth

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Yet - a selection model: a selection of the traits with the maximum net growth rate (the x’s that maximize r(x)-c(x)-d(x))

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x r(x) c(x) d(x) 0.2 0.4 0.6 0.8 1 10 20 30 40 50 60

x

n(x,t)/ρ(t)

t=0 t=1.3442 t=25

∂n(x, t) ∂t = [r(x) − c(x) − d(x)]n(x, t).

ρ(t) → ∞, n(x, t) ρ(t) → X

i

aiδ(x − xi)

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SLIDE 12

Case II: Density Dependent Model

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Assumption:

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Theorem: The density remains bounded. The limit distribution is a collection of Dirac masses

∂n(x, t) ∂t = [r(x) − c(x) − G(ρ)d(x)]n(x, t). lim

ρ→∞ G(ρ) = ∞

0.2 0.4 0.6 0.8 1 −5 5 10 15 20 25 30 35 40 45

x n(x,t)

t=0 t=0.15557 t=0.53721 t=2 0.2 0.4 0.6 0.8 1 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3

x

r(x)−c(x) G(ρ)d(x) at t=0 G(ρ)d(x) at t=0.15557 G(ρ)d(x) at t=0.53721 G(ρ)d(x) at t=2

ρ(t) ≤ ρM, n(x, t) → X

i

aiδ(x − xi)

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SLIDE 13

Case III: Selection/Mutation Model

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Assumption:

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Theorem: The density remains bounded.

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The Dirac masses are replaced by distributions with nonzero variance

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Hence: a continuum of traits that are stable for all time.

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This is interpreted as tumor heterogeneity

lim

ρ→∞ G(ρ) = ∞

∂n(x, t) ∂t = (r(x)(1 − θ) − c(x) − G(ρ(τ))d(x)) n(x, t) + θ Z 1 r(y)M(y, x)n(y, t)dy

0.2 0.4 0.6 0.8 1 −5 5 10 15 20 25 30 35 40

x n(x,10)

ε=0.01 θ=0.1, ρf=2.9313 θ=0.4, ρf=2.917 θ=0.7, ρf=2.9089 θ=1, ρf=2.9025

Increasing the mutation ratio:

✦

increases heterogeneity

✦

smaller tumor size

SLIDE 14

Adding space: The Lorz et al model

(radially symmetric, BMB 2015)

∂tn(t,r, θ) = [R(t,r, θ) − D(ρ(t,r)) − µ1(θ)c1(t,r)] n(t,r, θ) − αs∆s(t,r) +

• γs +

1 p(θ)n(t,r, θ)dθ

• s(t,r) = 0,

− αc1∆c1(t,r) +

• γc1 +

1 µ1(θ)n(t,r, θ)dθ

• c1(t,r) = 0,

− αc2∆c2(t,r) +

• γc2 + µ2

1 n(t,r, θ)dθ

• c2(t,r) = 0.

R(t,r, θ) = R(t,r, θ; s, c2) = p(θ) 1 + µ2c2(t,r)s(t,r).

ρ(t,r) =

• n(t,r, θ)dθ,

∂rs(t,r = 0) = 0, s(t,r = 1) = S1, ∂rci(t,r = 0) = 0, ci(t,r = 1) = Ci(t), i = 1, 2.

Cytotoxic drug Cytostatic drug Nutrients Cancer cells Boundary conditions

SLIDE 15

An improved radially-symmetric model

(Cho-DL, BMB 2017)

Cytotoxic drug Cytostatic drug Nutrients Cancer cells

∂tn(t,r, θ) = [((1 − w)R(t,r, θ) − D(ρ(t,r)) − C(t,r, θ))] n(t,r, θ) + αn(ρ(t,r))∆n(t,r, θ) + w

• R(t,r, ϑ)M(θ, ϑ)n(t,r, ϑ)dϑ,

− αs(ρ(t,r))∆s(t,r) +

• γs +

1 p(θ)n(t,r, θ)dθ

• s(t,r) = 0,

− αc1(ρ(t,r))∆c1(t,r) +

• γc1 +

1 µ1(θ)n(t,r, θ)dθ

• c1(t,r) = 0,

− αc2(ρ(t,r))∆c2(t,r) +

• γc2 + µ2

1 n(t,r, θ)dθ

• c2(t,r) = 0.

(10

★

Cell diffusion, Mutations

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Diffusion depends on cell density

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The effect of the cytotoxic drug depends on the dosage µ1 = µ1(θ, c1)

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SLIDE 16

T T y Type to enter text

Linear vs. nonlinear uptake functions

★

Avoids the asymptotic delta at the boundary Linear (Lorz et al) Nonlinear (Cho-DL)

• − αs(ρ(t,r))∆s(t,r) +
• γs +

1 p(θ)n(t,r, θ)dθ

• s(t,r) = 0,

− αc1(ρ(t,r))∆c1(t,r) +

• γc1 +

1 µ1(θ)n(t,r, θ)dθ

• c1(t,r) = 0,
• 1
• p(θ) = a1θ + a2,

µ1(θ) = b1θ + b2

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p(θ) = a1 1 + a2θ5 , µ1(θ, c1) = b1 1 + θ2+0.5c1

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R(t,r, θ) = R(t,r, θ; s, c2) = p(θ) 1 + µ2c2(t,r)s(t,r).

SLIDE 17

T T y Type to enter text

Linear vs. nonlinear uptake functions

no drug

• n-off cytostatic drug

t θ

100 200 1 0.5 1 2 3 4 5

t θ

100 200 1 0.5 1 2 3

t θ

100 200 1 0.5 0.2 0.4 0.6

t θ

100 200 1 0.5 0.2 0.4 0.6

(a) (b)

Linear Nonlinear

SLIDE 18

Relapse & emerging resistance

(one figure to remember)

dosage 1

t θ

C1(t) = 2 200 400 600 1 0.5 0.05 0.1

t θ

C1(t) = 6 100 200 1 0.9 1 2

t θ

C1(t) = 8 100 200 1 0.9 0.5 1

dosage 2 dosage 1

t θ

C1(t) = 2 100 200 1 0.5 0.1 0.2 0.3

t θ

C1(t) = 6 100 200 1 0.5 0.05 0.1

t θ

C1(t) = 8 100 200 1 0.5 0.05 0.1

t θ

C1(t) = 7Cb(t) 100 200 1 0.5 0.1 0.2 0.3

t θ

C1(t) = 16Cb(t) 100 200 1 0.5 0.1 0.2 0.3

t θ

C1(t) = 21Cb(t) 100 200 1 0.5 0.1 0.2 0.3

(a) (b) (c)

Linear Nonlinear Continuous On-Off Nonlinear Continuous

SLIDE 19

Diffusion of nutrients & drug

0.5 1 1 0.5 0.5 1 1 0.5 0.5 1 1 0.5 0.5 1 1 0.5 0.5 1 1 0.5 0.5 1 1 0.5

Resistance vs. r Increased diffusion of nutrients high drug diffusion low drug diffusion no cell diffusion

★

The drug and the nutrients diffuse from the boundary (r=1)

★

With a high diffusion of the nutrient, the cancer cell growth near the center (r=0) is expedited

★

This increases the heterogeneity in particular when the drug permeability is low

SLIDE 20

Diffusion of nutrients & drug

100 200 1 0.5 0.05 0.1 0.15 100 200 1 0.5 0.04 0.08 100 200 1 0.5 0.04 0.08 100 200 1 0.5 0.1 0.2 0.3 100 200 1 0.5 0.1 0.2 0.3 100 200 1 0.5 0.1 0.2 0.3

high drug diffusion low drug diffusion Resistance vs. time Increased diffusion of nutrients no cell diffusion

★

The phenotype heterogeneity becomes relatively large when the resource is more permeable compared to the drug (upper-right corner)

SLIDE 21

Emerging tumor heterogeneity

0.2 0.4 0.6 0.8 2 4 6 8 10

θ

Q(t, θ) 0.2 0.4 0.6 0.8 2 4 6 8 10

θ r

n(t, r, θ)

¯ αs = 0.8 θ

0.5 1 1 0.5

r ¯ αs = 8 θ

0.5 1 1 0.5

¯ β [ · ]= 0, αn = 0 ¯ αn = 10−4 ¯ αn = 10−3 ¯ αn = 10−2

★

Increased cell diffusion results with a bi-modal distribution

★

Different levels of resistance in different locations

SLIDE 22

The role of mutations

t = 50 t = 100 t = 150

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15

θ

¯ αs = ¯ αc1 = 0.08 q(θ, t)

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15

θ

q(θ, t)

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15

θ

q(θ, t) θ θ

0.5 0.5 0.4 0.8

A continuous process: asymmetric Gaussian mutation kernel

★

Increased variance

★

Regularized distribution

★

Concentration near the point where the maximum growth rate is achieved

★

The phenotype distribution is shifted towards the higher resistance levels due to the asymmetry in the kernel

SLIDE 23

The role of mutations

θ θ θ

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15

θ

¯ αs = 0.8, ¯ αc1 = 0.08 q(θ, t)

0.2 0.4 0.6 0.8 1

0.05

0.1 0.15

θ

q(θ, t)

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15

θ

q(θ, t)

θ θ

0.5 0.5 0.4 0.8

A jump process: mutations occurring at certain resistance levels

★

Dashed line - without mutations. Solid lines - with mutations.

★

Piecewise-linear mutation kernel

★

Cancer cells accumulate before crossing the bottleneck trait values

M(θ, θ) = nD

i=1 K(θ)(θ − θi−1) χ(θ,θ) Ωi×Ωi ,

θ ≥ θ, 0,

• therwise,

SLIDE 24

The full 2D problem (Cho-DL JTB 2018)

p(t, x) = k k − 1ρk−1(t, x), k > 1

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G(t, x, θ) = g(t, x, θ)h(p, g), h(p, g) = 1 − H(p − ˜ p)H(g)

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R(t, x, θ; s, c2) = φ(θ) 1 + µ2c2(t, x)s(t, x), C(t, x, θ; c1) = µ1(θ, c1)c1(t, x)

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g(t, x, θ) = R(t, x, θ) − D(t, x, θ) − C(t, x, θ)

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✓ ˜ p = k k − 1 ◆

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Cancer cells Pressure Effective growth Cytostatic drug Cytotoxic drug Homeostatic pressure

Follows Perthame, Quirós, Vázquez, Arch. Rat. Mech., 2014

∂tn(t, x, θ) = G(t, x, θ)n(t, x, θ) + νn∆n(t, x, θ) + νpr · (n(t, x, θ)rp(t, x))

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SLIDE 25

The nutrients and the drugs

∈

νs

s (t,

x

) =

• γs +

1

0 ϕ(θ)

n

(t,

x,

θ)

dθ

• s

(t,

x

)

,

νc

1

c

1

(t,

x

) =

• γc

1 +

1

0 µ1

(θ)

n

(t,

x,

θ)

dθ

• c

1

(t,

x

)

,

νc

2

c

2

(t,

x

) =

• γc

2 + µ2

1 n

(t,

x,

θ)

dθ

• c

2

(t,

x

)

. s

(t,

x

)

|

∂0 = S

, c

i

(t,

x

)

|

∂0 =

C

i

, i

=1 , 2

. Cytotoxic drug Cytostatic drug Nutrients Boundary conditions

(following Peng, Trucu, Lin, Thompson, Chaplain, BMB 2016)

SLIDE 26

Is this a good model for tumor growth? The emergence of a necrotic core

★

The tumor at t=20 for different motility constants

★

The population increases up to and becomes restricted by the homeostatic pressure after which it starts to expand in space

Increased velocity Increased diffusion

(a) (νn, νp) = (10−6, 10−5) (b) (νn, νp) = (10−5, 10−6) (c) (νn, νp) = (10−4, 10−6)

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ρ = 1

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• v =
• 2

νp

G

(θ) ¯

p + νn ∞

r(t)

n

′

r

s ds + ∞

r(t)

n

r

ds,

The velocity of the boundary (radially symmetric tumor):

∂tn(t, x, θ) = G(t, x, θ)n(t, x, θ) + νn∆n(t, x, θ) + νpr · (n(t, x, θ), rp(t, x))

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SLIDE 27

Relapse with resistant colonies

★

The evolution of the mean phenotype. A higher dosage results with a later relapse but with a more resistant tumor

★

The cross section of a radially symmetric tumor with different drug

• dosages. A higher dosage (cytotoxic

drug) results with a late relapse

SLIDE 28

Estimating the time of relapse

★

The time it takes the tumor density to reach trelapse = tK + 1 p 2νpG(θ)˜ p r ρT

M

π − r0 ! , tK ∼ 1 R(θ) − C(θ)

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ρT

M

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★

The time of relapse for different values of the initial density and the drug dosages

★

The estimate provides a lower bound for the time of relapse c1 = 3, . . . , 8

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SLIDE 29

Continuous drug vs. on-off

★

Comparing constant and on-off cytotoxic drug administration with the same total drug dosage.

★

Shorter and stronger bursts, lead to the emergence of bigger tumors during the off times, expressing more resistant traits.

E [ n

(t,

x,

θ)] .

=

• 1

0 θ n

(t,

x,

θ)

dθ

• 1

0 n

(t,

x,

θ)

dθ

Schedule Normalized # of cells Mean phenotype

SLIDE 30

Tumor dynamics and evolution of traits

★

Density and mean resistance level

★

Nutrients and drug are diffused from the top

★

Without the drug sensitive cells grow faster

★

As drug dosage increases, resistant cells become dominant

c=0 c=2 c=4 c=8 Initial Tumor

SLIDE 31

Tumor dynamics and evolution of traits

★

Density and mean resistance level

★

Nutrients and drug are diffused from the top

★

Without the drug sensitive cells grow faster

★

As drug dosage increases, resistant cells become dominant

c=0 c=2 c=4 c=8 Initial Tumor

SLIDE 32

Emergence of heterogeneity

★

The initial configuration is a uniform phenotype distribution

★

(a) density; (b) mean resistance level. t=40.

★

(c) uniform diffusion of nutrients and drug; (d) diffusion from the top.

SLIDE 33

Tumor evolution in a highly heterogeneous nutrients environment

Collective Migration during Cancer Progression

Alexander, S., et al. (2008). Histochemistry and Cell Biology

★

No drugs

★

The environment (following Peng, Trucu, Lin, Thompson, Chaplain, BMB 2016)

★

Irregular boundary. Regular necrotic core

SLIDE 34

MDR & continuous level of resistance

(Cho-DL, AMM 2018)

∂tnP (t, θ) = ((1 − w)R(t, θ) − D − CP (t, θ) − q) nP (t, θ) +pnQ(t, θ) + w Z

Γ

M(θ, ϑ)R(t, ϑ)nP (t, ϑ)dϑ, ∂tnQ(t, θ) = qnP (t, θ) + (−p − DQ − CQ(t, θ)) nQ(t, θ).

le θ = (θ1, ..., θM)

CP (t, θ) = Φ(C1, ..., CM) = 1 − Y

i

(1 − Ci),

R(t, θ) = ϕ(θ)s0(t) 1 + Φ(C1, ..., CM).

r Ci(t, θ) = µi(θ)ci(t),

Φ(C1, ..., CM) = X

i

Ci.

Multidrug resistance Proliferating cells Quiescent cells Cytotoxic drugs Cytostatic drugs Can be easily changed

SLIDE 35

Competition between healthy cells & cancer cells. (Cho-DL, submitted 2018)

∂tnh(t, θ) = ⇥rh(θ) − dh(θ)ρh(t) − µh(θ)c1(t)⇤ nh − ahc(θ)ρc(t)nh + νh∆θnh, ∂tnc(t, θ) = ⇥rc(θ) − dc(θ)ρc(t) − µc(θ)c1(t) −ϕc(θ)c2(t)⇤ nc − ach(θ)ρh(t)nc + νc∆θnc.

Supercompetitor Normal cell Cancer cell death

Low competition High competition

SLIDE 36

Examples

★

Evolution of the cancer cell distribution in the trait space

★

top: quadratic; bottom: linear

★

Combination therapy. 1st drug at t=10. 2nd drug at t=20

Cytotoxic

• Alt. a = 0.2

Targeted

• Alt. a = 0.8

model 1

0.5 1 1 0.5

Cytotoxic to Targeted

50 100 150 0.5 1 10 20 30 40 50 0.5 1 10 20 30 40 50 0.5 1 10 20 30 40 50 0.5 1 1 0.5

Targeted to Cytotoxic

50 100 150 0.5 1 10 20 30 40 50 0.5 1 10 20 30 40 50 0.5 1 10 20 30 40 50

model 2

0.5 1 1 0.5

Cytotoxic to Targeted

50 100 150 200 0.5 1 0.002 0.004 0.006 0.008 0.01 0.5 1 0.5 1 1.5 2 10-7 0.5 1 0.2 0.4 0.6 0.8 1 10-6 0.5 1 1 0.5

Targeted to Cytotoxic

50 100 150 200 0.5 1 0.5 1 1.5 2 10-3 0.5 1 0.5 1 1.5 2 10-7 0.5 1 0.2 0.4 0.6 0.8 1 10-5

★

Evolution of cancer cells & healthy cells

★

Single drug therapy & alternating therapy

★

Drug diffuses from the right

SLIDE 37

Conclusion

★

Medical

• Quantitative approach to therapy
• Vision: the “science fiction”

★

Math

• New challenges
• Can potentially be useful

t

dosage 2 dosage 1

t θ

C1(t) = 2 100 200 1 0.5 0.1 0.2 0.3

t θ

C1(t) = 6 100 200 1 0.5 0.05 0.1

t θ

C1(t) = 8 100 200 1 0.5 0.05 0.1

t θ

C1(t) = 7Cb(t) 100 200 1 0.5 0.1 0.2 0.3

t θ

C1(t) = 16Cb(t) 100 200 1 0.5 0.1 0.2 0.3

t θ

C1(t) = 21Cb(t) 100 200 1 0.5 0.1 0.2 0.3

(b) (c)

The figure to remember