PDEs and evolutionary biology€¦ · PDEs and evolutionary biology come together to ght cancer 1 Tommaso Lorenzi Sorbonne Universit es, UPMC Univ Paris 06, UMR 7598, Laboratoire

PDEs and evolutionary biologycome together to fight cancer 1

Tommaso Lorenzi

Sorbonne Universites, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions

INRIA-Paris-Rocquencourt, EPC MAMBA

Mathematiques en mouvement 2014Universite Paris 1 - Pantheon-Sorbonne, 28th May 2014

1supported by the Fondation Sciences Mathematiques de Paris

Research framework

Structured PDEs and

integro-‐differen4al equa4ons

Structured Popula4ons

Evolu4onary Biology

Study evolu+onary dynamics in cancer cell popula+ons

• with L.Almeida, R.Chisholm, J.Clairambault, A.Escargueil, A.Lorz,B.Perthame

• A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame,Effects of space structure and combination therapies on phenotypicheterogeneity and drug resistance in solid tumors, under review

T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 2 / 27

Research framework

Structured PDEs and

integro-‐differen4al equa4ons

Structured Popula4ons

Evolu4onary Biology

Study evolu+onary dynamics in cancer cell popula+ons

• with L.Almeida, R.Chisholm, J.Clairambault, A.Escargueil, A.Lorz,B.Perthame

• A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame,Effects of space structure and combination therapies on phenotypicheterogeneity and drug resistance in solid tumors, under review


Outline

Key Biological Ideas

Mathematical Formalization

Study of Cell Environmental Adaptation and Phenotypic Heterogeneity

Study of Optimized Therapeutic Protocols


Outline






Outline






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Outline






Cancer as an evolutionary and ecological process


Resistance to cytotoxic drugs: evolutionary explanation

Exposure to cytotoxic drugs

z

• Cytotoxic drug treatments can create a population bottleneck ⇒selection for the few clones that may randomly posses a mutationthat confers resistance to the drug.

• The subsequent population will no longer respond to treatment ⇒resistance to cytotoxic drugs.


Resistance to cytotoxic drugs: a possible mechanism


A possible way to obstacle the emergence of resistance

• Cytotoxic drugs = killing cells ⇒ emergence of resistance.

• Cytostatic drugs = slowing down cell proliferation ⇒ reducing thegrowth of resistant clones.

• Cytotoxic drugs + cytostatic drugs ⇒ maintaining the persistence ofsensitive tumor cells, which are fitter than the resistant cells in lowdrug pressure conditions ⇒ adaptive therapies.


Intratumor heterogeneity: clinical implications

• Heterogeneous cancer cell populations are more likely to harbor cellswith a mutation that confers resistance.

• Heterogeneity in treatment response is partly due to heterogeneity inthe tumor microenvironment.


Intratumor heterogeneity: evolutionary explanation

• The concentration of abiotic factors (e.g., nutrients and drugs) insidea solid tumor varies in space ⇒ gradients of abiotic factors.

• Gradients of abiotic factors ⇒ distinct ecological niches inside thesame solid tumor.

• Distinct ecological niches inside the same solid tumor ⇒ ecologicalopportunities for diversification in cancer cell populations.


Outline






Aim

Use PDEs for structured populations as in-silico laboratories to

1. answer the following questions• can we explain intra-tumor heterogeneity in terms of cell adaptation to

local conditions?

• is it possible to overcome the emergence of resistance and favor theeradication of cancer cells by using combination therapies?

2. support• the comprehension of mechanisms underlying cancer cell evolution,

• the design of hypothesis-driven experiments and optimized anti-cancerprotocols.


Moving toward a mathematical formalization• Radially symmetric tumor spheroid exposed to cytotoxic and

cytostatic drugs ⇒ one cell population structured by y ∈ [0, 1] andx ∈ [0, 1].

source: http://www.oncolab.unimi.it

• Variable y : normalized linear distance from the center of the spheroid.• Variable x : expression level of a resistance phenotype influencing the

cellular birth and death rates, and the sensitivity to cytotoxictherapies.


The model

∂tn(t, y , x) =[

cytostatic therapy︷︸︸︷1

1 + µ2c2(t, y)

proliferation︷︸︸︷p(x)

s(t, y)

−

death︷︸︸︷d%(t, y)

−

cytotoxic therapy︷︸︸︷µ1(x)c1(t, y)

]n(t, y , x)

,

−σs ∆s(t, y)︸︷︷︸diffusion

+[

γs︸︷︷︸degradation

+

∫p(x)n(t, y , x)dx︸︷︷︸

consumption

]s(t, y) = 0,

−σc ∆c1(t, y)︸︷︷︸diffusion

+[

γc︸︷︷︸degradation

+

∫µ1(x)n(t, y , x)dx︸︷︷︸

consumption

]c1(t, y) = 0,

− σc ∆c2(t, y) +[γc + µ2

∫n(t, y , x)dx

]c2(t, y) = 0.


The model

∂tn(t, y , x) =[


1 + µ2c2(t, y)

proliferation︷︸︸︷p(x)s(t, y)−

death︷︸︸︷d%(t, y)

−


]n(t, y , x),


+[


+

∫p(x)n(t, y , x)dx︸︷︷︸

consumption

]s(t, y) = 0

,


+[


+

∫µ1(x)n(t, y , x)dx︸︷︷︸

consumption

]c1(t, y) = 0,

− σc ∆c2(t, y) +[γc + µ2

∫n(t, y , x)dx

]c2(t, y) = 0.


The model

∂tn(t, y , x) =[


1 + µ2c2(t, y)


death︷︸︸︷d%(t, y) −


]n(t, y , x),


+[


+

∫p(x)n(t, y , x)dx︸︷︷︸

consumption

]s(t, y) = 0,


+[


+

∫µ1(x)n(t, y , x)dx︸︷︷︸

consumption

]c1(t, y) = 0

,

− σc ∆c2(t, y) +[γc + µ2

∫n(t, y , x)dx

]c2(t, y) = 0.


The model

∂tn(t, y , x) =[ cytostatic therapy︷︸︸︷

1

1 + µ2c2(t, y)


death︷︸︸︷d%(t, y) −


]n(t, y , x),


+[


+

∫p(x)n(t, y , x)dx︸︷︷︸

consumption

]s(t, y) = 0,


+[


+

∫µ1(x)n(t, y , x)dx︸︷︷︸

consumption

]c1(t, y) = 0,

− σc ∆c2(t, y) +[γc + µ2

∫n(t, y , x)dx

]c2(t, y) = 0.


The model

• Radial symmetry ⇒ zero Neumann conditions at y = 0:

∂yc1,2(t, y = 0) = 0, ∂y s(t, y = 0) = 0.

• Supply of nutrients and drugs ⇒ Dirichlet boundary conditions aty = 1:

c1,2(t, y = 1) = C1,2(t), s(t, y = 1) = S .


Outline






Initial conditions


Cell dynamics without therapies


Cell dynamics in presence of cytostatic drugs only

0 T/2 T0

Ca

C1,2

(t)

t


Cell dynamics in presence of cytotoxic drugs only

0 T/2 T0

Ca

C1,2

(t)

t


Outline






Cell dynamics in the presence of cytotoxic drugs only

0 T/2 T0

Ca

C1,2

(t)

t0 T/2 T

0

Cb

C1,2

(t)

t

0 T/2 T0

0.005

0.01Cytotoxic drugs (constant infusion)

t0 T/2 T

0

0.05

0.1

Cytotoxic drugs (bang−bang infusion)

t


Cell dynamics in the presence of cytostatic drugs only

0 T/2 T0

Ca

C1,2

(t)

t0 T/2 T

0

Cb

C1,2

(t)

t

0 T/2 T0

0.005

0.01Cytostatic drugs (constant infusion)

t0 T/2 T

0

0.05

0.1

Cytostatic drugs (bang−bang infusion)

t


Cell dynamics in the presence of cytotoxic and cytostaticdrugs (same infusion protocols)

0 T/2 T0

Ca

C1,2

(t)

t0 T/2 T

0

Cb

C1,2

(t)

t

0 T/2 T0

0.005

0.01Cytotoxic and cytostatic drugs (constant infusion)

t0 T/2 T

0

0.05

0.1Cytotoxic and cytostatic drugs (bang−bang infusion)

t


Cell dynamics in the presence of cytotoxic and cytostaticdrugs (mixed infusion protocols)

0 T/2 T0

Cd

Ce

t

C1,2

(t)

0 T/2 T0

0.05

0.1Cytotoxic (constant infusion) and cytostatic drugs (bang−bang infusion)

t0 T/2 T

0

0.005

0.01Cytotoxic (bang−bang infusion) and cytostatic drugs (constant infusion)

t


PDEs and evolutionary biologycome together to fight cancer 1

Tommaso Lorenzi

Sorbonne Universites, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions

INRIA-Paris-Rocquencourt, EPC MAMBA

Mathematiques en mouvement 2014Universite Paris 1 - Pantheon-Sorbonne, 28th May 2014

1supported by the Fondation Sciences Mathematiques de Paris

Documents

PDEs and evolutionary biology€¦ · PDEs and evolutionary biology come together to ght cancer 1 Tommaso Lorenzi Sorbonne Universit es, UPMC Univ Paris 06, UMR 7598, Laboratoire