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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
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 2 / 27
Outline
Key Biological Ideas
Mathematical Formalization
Study of Cell Environmental Adaptation and Phenotypic Heterogeneity
Study of Optimized Therapeutic Protocols
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 3 / 27
Outline
Key Biological Ideas
Mathematical Formalization
Study of Cell Environmental Adaptation and Phenotypic Heterogeneity
Study of Optimized Therapeutic Protocols
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 3 / 27
Outline
Key Biological Ideas
Mathematical Formalization
Study of Cell Environmental Adaptation and Phenotypic Heterogeneity
Study of Optimized Therapeutic Protocols
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 3 / 27
Outline
Key Biological Ideas
Mathematical Formalization
Study of Cell Environmental Adaptation and Phenotypic Heterogeneity
Study of Optimized Therapeutic Protocols
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 3 / 27
Outline
Key Biological Ideas
Mathematical Formalization
Study of Cell Environmental Adaptation and Phenotypic Heterogeneity
Study of Optimized Therapeutic Protocols
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 4 / 27
Cancer as an evolutionary and ecological process
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 5 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 6 / 27
Resistance to cytotoxic drugs: a possible mechanism
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 7 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 8 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 9 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 10 / 27
Outline
Key Biological Ideas
Mathematical Formalization
Study of Cell Environmental Adaptation and Phenotypic Heterogeneity
Study of Optimized Therapeutic Protocols
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 11 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 12 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 13 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 14 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 14 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 14 / 27
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.
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 14 / 27
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 .
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 15 / 27
Outline
Key Biological Ideas
Mathematical Formalization
Study of Cell Environmental Adaptation and Phenotypic Heterogeneity
Study of Optimized Therapeutic Protocols
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 16 / 27
Initial conditions
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 17 / 27
Cell dynamics without therapies
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 18 / 27
Cell dynamics in presence of cytostatic drugs only
0 T/2 T0
Ca
C1,2
(t)
t
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 19 / 27
Cell dynamics in presence of cytotoxic drugs only
0 T/2 T0
Ca
C1,2
(t)
t
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 20 / 27
Outline
Key Biological Ideas
Mathematical Formalization
Study of Cell Environmental Adaptation and Phenotypic Heterogeneity
Study of Optimized Therapeutic Protocols
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 21 / 27
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
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 22 / 27
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
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 23 / 27
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
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 24 / 27
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
T. Lorenzi (LJLL - UPMC & MAMBA - INRIA) May 28th, 2014 25 / 27
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