Une approche stochastique à la modélisation de l’immunothérapie contre le cancer Loren Coquille - Travail en collaboration avec Martina Baar, Anton Bovier, Hannah Mayer (IAM Bonn) Michael Hölzel, Meri Rogava, Thomas Tüting (UniKlinik Bonn) Institut Fourier – Grenoble Séminaire commun IF/LJK - 17 septembre 2015 Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 1 / 45
Plan Biological motivations 1 Adaptative dynamics 2 The model State of the art Only switches : Relapse caused by stochastic fluctuations 3 Therapy with 1 type of T-cell Biological parameters Therapy with 2 types of T-cells Only mutations : Early mutation induced by the therapy 4 Mutations and switches : Polymorphic Evolution Sequence 5 Conclusion 6 Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 2 / 45
Biological motivations Plan Biological motivations 1 Adaptative dynamics 2 The model State of the art Only switches : Relapse caused by stochastic fluctuations 3 Therapy with 1 type of T-cell Biological parameters Therapy with 2 types of T-cells Only mutations : Early mutation induced by the therapy 4 Mutations and switches : Polymorphic Evolution Sequence 5 Conclusion 6 Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 3 / 45
Biological motivations Experiment on melanoma (UniKlinik Bonn) Injection of T-cells able to kill a specific type of melanoma. The treatment induces an inflammation , to which the melanoma react by changing their phenotype (markers disappear on their surface, " switch "). The T-cells cannot kill them any more, the tumor continues to grow. Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 4 / 45
Biological motivations Without therapy : exponential growth of the tumor. With therapy : relapse after 140 days. With therapy and restimulation : late relapse. Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 5 / 45
Adaptative dynamics Plan Biological motivations 1 Adaptative dynamics 2 The model State of the art Only switches : Relapse caused by stochastic fluctuations 3 Therapy with 1 type of T-cell Biological parameters Therapy with 2 types of T-cells Only mutations : Early mutation induced by the therapy 4 Mutations and switches : Polymorphic Evolution Sequence 5 Conclusion 6 Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 6 / 45
Adaptative dynamics The model Individual-based model Cancer cells (melanoma): each cell is characterized by a genotype and a phenotype. Each can reproduce, die, mutate (reproduction with genotypic change) or switch (phenotypic change, without reproduction) at prescribed rates. Immune cells (T-cells): Each cell can reproduce, die, or kill a cancer cell of prescribed type (which produces a chemical messenger) at prescribed rates. Chemical messenger (TNF − α ): Each particle can die at a prescribed rate. Its presence influences the ability of a fixed type of cancer cell to switch. Trait space and measure : X = G × P ⊔ Z ⊔ W = { g 1 , . . . , g |G| } × { p 1 , . . . , p |P| } ⊔ { z 1 , . . . , z |Z| } ⊔ w n = ( n ( g 1 , p 1 ) , . . . , n ( g |G| , p |P| ) , n z 1 , . . . , n z |Z| , n w ) Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 7 / 45
Adaptative dynamics The model Example for 2 types of melanoma and 1 type of T-cell The stochastic model converges, in the limit of large populations, towards the solution this dynamical system with logistic , predator-prey , switch : � � ˙ = n x b x − d x − c xx · n x − c xy · n y + s · n y − s w · n w n x − t xz · n z x n x n x � � ˙ = n y b y − d y − c yy · n y − c yx · n x − s · n y + s w · n w n x n y ˙ = − d zx · n z x + b zx · n z x n x n z x ˙ = − d w · n w + ℓ x · t xz · n x n z x n w Event Rates for x Rates for y for z for w (Re)production b x b y b zx n x Natural death d x + c xx n x + c xy n y d y + c yy n x + c yx n y d zx d w Therapy death t xz n z x 0 Switch s w n w s Deterministically, a number ℓ w of TNF- α particles are produced when z x kills x . Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 8 / 45
Adaptative dynamics State of the art State of the art for the BPDL model In general X continuous. Measure ν t = � N t i = 1 δ x i . Markov process on the space of positive measures. Event Rate Clonal reproduction ( 1 − p ( x )) · b ( x ) Reproduction with mutation m ( x , dy ) · p ( x ) · b ( x ) � Death d ( x ) + X c ( x , y ) ν ( dy ) Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 9 / 45
Adaptative dynamics State of the art State of the art for the BPDL model In general X continuous. Measure ν t = 1 � N t i = 1 δ x i . K Markov process on the space of positive measures. Event Rate Clonal reproduction ( 1 − µ p ( x )) · b ( x ) Reproduction with mutation m ( x , dy ) · µ p ( x ) · b ( x ) c ( x , y ) � Death d ( x ) + ν ( dy ) X K Limit of large populations and rare mutations K → ∞ µ → 0 Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 9 / 45
Adaptative dynamics State of the art Scalings and time scales K → ∞ , µ fixed, T < ∞ : Law of large numbers, deterministic limit [ Fournier, Méléard, 2004 ] K → ∞ , µ → 0, T < ∞ : Law of large numbers, deterministic limit without mutations. K → ∞ , µ → 0, T ∼ log ( 1 /µ ) : Deterministic jump process [ Bovier, Wang, 2012 ] 1 1 ( K , µ ) → ( ∞ , 0 ) t.q. µ K ≫ log K , T ∼ µ K : Random jump process [ Champagnat, Méléard, 2009 , 2010 ] Trait Substitution Sequence Polymorphic Evolution Sequence Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 10 / 45
Adaptative dynamics State of the art Scalings and time scales K → ∞ , µ fixed, T < ∞ : Law of large numbers, deterministic limit [ Fournier, Méléard, 2004 ] limit dynamical systems (with switch) are not classified K → ∞ , µ → 0, T < ∞ : Law of large numbers, deterministic limit without mutations. K → ∞ , µ → 0, T ∼ log ( 1 /µ ) : Deterministic jump process [ Bovier, Wang, 2012 ] 1 1 ( K , µ ) → ( ∞ , 0 ) t.q. µ K ≫ log K , T ∼ µ K : Random jump process [ Champagnat, Méléard, 2009 , 2010 ] Trait Substitution Sequence Polymorphic Evolution Sequence Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 10 / 45
Only switches : Relapse caused by stochastic fluctuations Plan Biological motivations 1 Adaptative dynamics 2 The model State of the art Only switches : Relapse caused by stochastic fluctuations 3 Therapy with 1 type of T-cell Biological parameters Therapy with 2 types of T-cells Only mutations : Early mutation induced by the therapy 4 Mutations and switches : Polymorphic Evolution Sequence 5 Conclusion 6 Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 11 / 45
Only switches : Relapse caused by stochastic fluctuations Therapy with 1 type of T-cell Solution of the determinisitic system Legend : Melanoma x , melanoma y , T-cells, TNF- α 4 3 2 1 0 2 4 6 8 10 Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 12 / 45
Only switches : Relapse caused by stochastic fluctuations Therapy with 1 type of T-cell 3 fixed points With reasonable parameters we have : n y 4 Pxy000 Pxy z x 0w 3 2 P00000 1 Pxy z x z y w Pxy0 z y w n x 2 4 6 8 Pxyz is stable. Pxy 0 is stable on the invariant sub-space { n z x = 0 } . Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 13 / 45
Only switches : Relapse caused by stochastic fluctuations Therapy with 1 type of T-cell Relapse towards Pxyz , ( K = 200) Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 14 / 45
Only switches : Relapse caused by stochastic fluctuations Therapy with 1 type of T-cell Relapse towards Pxy 0 due to the death of z x Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 15 / 45
Only switches : Relapse caused by stochastic fluctuations Biological parameters Adjustment of parameters : data Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 16 / 45
Only switches : Relapse caused by stochastic fluctuations Biological parameters Adjustment of parameters : simulations ( K = 10 5 ) Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 17 / 45
Only switches : Relapse caused by stochastic fluctuations Therapy with 2 types of T-cells Therapy with 1 types of T-cells � � ˙ = n x b x − d x − c xx · n x − c xy · n y − t xz · n z x n x + s · n y − s w · n w n x n x � � ˙ = n y b y − d y − c yy · n y − c yx · n x − s · n y + s w · n w n x n y ˙ = − d zx · n z x + b zx · n z x n x n z x ˙ = − d w · n w + ℓ x · t xz · n x n z x n w Event Rates for x Rates for y Reproduction b x b y Natural death d x + c xx n x + c xy n y d y + c yy n x + c yx n y Death due to therapy t xz n z x 0 Switch s w n w s Loren Coquille (IF-Grenoble) LJK - 17 septembre 2015 18 / 45
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