1 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations Mathematical modeling and analysis of tumor-immune interactions 6 août 2019 Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019 Kevin Atsou 1 Thierry Goudon 1 Biologists : Véronique Braud 2 Fabienne Anjuere 2 1 Université Côte d’Azur, Inria (Team COFFEE), CNRS, LJAD 2 CNRS, IPMC (Institut de Pharmacologie Moléculaire et Cellulaire)
2 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations 1 tumors and efgector T-cells interactions : Biological context 2 Mathematical modeling : Earlier stages of tumor-growth 3 Numerical simulations Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019
3 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The Immune system Summary 1 tumors and efgector T-cells interactions : Biological context 2 Mathematical modeling : Earlier stages of tumor-growth 3 Numerical simulations Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019
4 often develop cancer Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon source : https ://gifer.com/en/7ftZ . 1 tumors or in the development of important role in the control Immune cells play an immunodefjcient patients tumors and efgector T-cells interactions : Biological context appearance of AIDS ; seriously With the Immunotherapy restarted toxins) ; 1891 with W. Coley (Coley’s Immunotherapy starts in The Immune system Attacks Tumor cells The Immune system Numerical simulations Mathematical modeling : Earlier stages of tumor-growth 6 août 2019
5 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The Immune system Two types of Immune system 2 2. source : https ://www.biosa.co.nz/, what is your immune system Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019
6 tumors and efgector T-cells interactions : Biological context Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon source : https ://curioussciencewriters.org/ 3. 3 = alterations Genes involved in cell cycle control are subject to the genetic Tumor cells cycle The Immune system Numerical simulations Mathematical modeling : Earlier stages of tumor-growth 6 août 2019 ⇒ cancer = ⇒ Uncontrolled Cell division
7 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The Immune system Tumor Immunity Cycle 4 4. source : http ://biocc.hrbmu.edu.cn/TIP/ Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019
8 tumors and efgector T-cells interactions : Biological context Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon angiogenesis ; Production of VEGF (Vascular Endothelial Growth Factor) for tumors ; high expression of PD-L1 (Programmed death-ligand 1 ) by high levels of suppressive cytokines ; have a strong enough efgect on Tumors. The prevalence of cancer indicates that the immune system does not Tumor has many suppressive infmuences K.Atsou - Impediments of the tumor immunity cycle The Immune system Numerical simulations Mathematical modeling : Earlier stages of tumor-growth 6 août 2019 T regulatory cells (Tregs produce TGF- β and IL-10) ;
9 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations Summary 1 tumors and efgector T-cells interactions : Biological context 2 Mathematical modeling : Earlier stages of tumor-growth 3 Numerical simulations Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019
10 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations A growth-fragmentation modeling approach : Tumor distribution cells Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019 ( t , z ) �→ n ( t , z ) (in cell n · µ m − 3 ) the size-structured tumor cells ( t , x ) �→ c ( t , x ) (in cell c · mm − 3 ) the concentration of Efgector T
11 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations Tumor growth main processes Tumor growth is splitted into two main processes : natural (microscopic) growth of the cells, cell division. 5 . source : https ://gifer.com/en/9uMp Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019
12 the cell’s volume growth Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon and the boundary condition , We can complete the equation by the initial data , Cellular division operator tumors and efgector T-cells interactions : Biological context 6 août 2019 We model the tumor growth by the following equation : Mathematical modeling : Earlier stages of tumor-growth Numerical simulations A growth-fragmentation modeling approach : Tumor ∂ tn ( t , z ) = − ∂ ∂ ∂ z ( V ( z ) n ( t , z )) + Q ( n ( t , z )) . � �� � � �� � n ( t = 0 , z ) = n 0 n ( t , 0) = 0
13 the cell’s volume growth Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon dividing . Cellular division operator tumors and efgector T-cells interactions : Biological context 6 août 2019 The cellular division Mathematical modeling : Earlier stages of tumor-growth Numerical simulations ∂ tn ( t , z ) = − ∂ ∂ ∂ z ( V ( z ) n ( t , z )) + Q ( n ( t , z )) . � �� � � �� � Q ( n ( t , z )) ? a ( z ) , the rate at which cells of size z process division . k ( z ′ | z ) the distribution of products from a cells of size z
14 Consequently, Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon (2) gain of cells with size z z The loss of cells of size z tumors and efgector T-cells interactions : Biological context 6 août 2019 (1) Mathematical modeling : Earlier stages of tumor-growth Numerical simulations The cellular division k ( z ′ | z ) must be normalized so that mass is conserved : ∫ z z ′ k ( z ′ | z ) dz ′ = z . 0 ∫ ∞ a ( z ′ ) k ( z | z ′ ) n ( t , z ′ ) dz ′ Q ( n ) = − a ( z ) n ( t , z ) + . � �� � � �� �
15 z Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon (5) The loss of cells of size z gain of cells with size z (4) We assume binary and symmetric division process . Therefore tumors and efgector T-cells interactions : Biological context gain of cells with size z (3) 6 août 2019 The loss of cells of size z Mathematical modeling : Earlier stages of tumor-growth Numerical simulations Binary and symmetric The cellular division ∫ ∞ a ( z ′ ) k ( z | z ′ ) n ( t , z ′ ) dz ′ Q ( n ) = − a ( z ) n ( t , z ) + . � �� � � �� � cells of size 2 z give birth to cells of size z . k ( z | 2 z ) = 2 δ z ′ =2 z Q ( n ( t , z )) = − a ( z ) n ( t , z ) + 2 a (2 z ) n ( t , 2 z ) d (2 z ) Q ( n ( t , z )) = 4 a (2 z ) n ( t , 2 z ) − a ( z ) n ( t , z ) � �� � � �� �
16 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations Chemotaxis phenomenon 6 T cells displacement towards the tumor microenvironment is model by chemotaxis . Atsou, Goudon Mathematical modeling and analysis of tumor-immune interactions 6 août 2019
N x 17 concentration of immune cells in Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon (6) c t x dx t c : a volume c t the time dependent tumors and efgector T-cells interactions : Biological context R x Let’s denote by the activated T cells follows the gradient of the chemical signal displacement K.A - The antigen-specifjc CD8+ efgector T cells Numerical simulations Mathematical modeling : Earlier stages of tumor-growth 6 août 2019
17 Let’s denote by Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon (6) tumors and efgector T-cells interactions : Biological context concentration of immune cells in the activated T cells follows the gradient of the chemical signal displacement K.A - The antigen-specifjc CD8+ efgector T cells Numerical simulations Mathematical modeling : Earlier stages of tumor-growth 6 août 2019 Ω = { x ∈ R N , | x | ≤ R } , c ω ( t ) the time dependent a volume ω ⊂ Ω : ∫ c ω ( t ) = c ( t , x ) dx ω
18 tumors and efgector T-cells interactions : Biological context Mathematical modeling and analysis of tumor-immune interactions Atsou, Goudon (8) volume of the tumor ; (7) number of tumor cells in the tumor ; The tumor mass : kth order moments Numerical simulations Mathematical modeling : Earlier stages of tumor-growth 6 août 2019 Let’s denote by µ 0 ( t ) (the zeroth order moment) the total ∫ ∞ µ 0 ( t ) = n ( t , z ) dz 0 Let’s denote by µ 1 ( t ) (the fjrst order moment) the total ∫ ∞ µ 1 ( t ) = zn ( t , z ) dz 0
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