Introduction to some topics in Mathematical Oncology Franco Flandoli, University of Pisa Stochastic Analysis and applications in Biology, Finance and Physics, Berlin 2014 Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
The field received considerable attentions in the past 10 years One of the plenary talks at ICM 2014 was on this subject (B. Perthame) Some people believe it could be a revolution in the next 10 years if properly developed (see Forbes) For people interested in stochastic systems it is an opportunity to touch an applied field of basic importance for society where stochasticity plays a role The following discussion is the outcome of an initial investigation with Michele Coghi, Mauro Maurelli, Manuela Benedetti and other young collaborators. Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Most common models Macroscopic models (tissue level) 1 microscopic models (cell level) 2 mixed or multi-scale models. 3 Remarks. Usually, in Physics, microscopic means molecular. Cell scale is in a sense a meso-scale, but I will call it microscopic. Cell motion is stochastic to a large extent. Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Objects studied by the models Macroscopic models: density of cells, oxygen concentration etc. 1 microscopic models: single cells 2 multi-scale models: single cells and oxygen concentration, etc. 3 Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Equations Macroscopic models: Fokker-Planck equations with nonlinear reaction 1 terms microscopic models: interacting particle systems (based on SDE or 2 discrete models) mixed or multi-scale models: both 3 Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Example of Macroscopic model P. Hinow, P. Gerlee, L. J. McCawley, V. Quaranta, M. Ciobanu, S. Wang, J. M. Graham, B. P. Ayati, J. Claridge, K. R. Swanson, M. Loveless, A. R. A. Anderson, A spatial model of tumor-host interaction: application of chemoterapy, Math. Biosc. Engin. 2009. Among several other models I have chosen this one due to three facts: it is a very good paper the team is a leading one in quantitative oncology (e.g. Kristine Swanson) it does not require deep biological training. This model is made of 7 coupled PDE-ODE. I will spend some time on it in order to explain the level of complexity and realism that is usually reached in such papers. Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Normoxic, hypoxic and apoptotic cells normoxic cells: healthy, proliferating tumor cells, with normal oxygen supply hypoxic cells: quiescent tumor cells, with poor oxygen supply apoptotic cells: death or programmed to death tumor cells Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
The PDE for normoxic cells ∂ N ∂ t = k 1 ∆ N ( N ( t , x ) = normoxic cell density) � �� � background diffusion div ( σ ( N ) ∇N ) + c 1 N ( V max − V ) � �� � � �� � crowding-driven diffusion proliferation − χ 1 div ( N ∇ m ) � �� � transport along ECM gradient − α N →H 1 o ≤ o H N + α H→N 1 o ≥ o H H � �� � � �� � normoxic → hypoxic hypoxic → normoxic I will come back later to the diffusion and crowding-driven diffusion. V = N + H + A + E + m = total volume occupied by cells and ECM Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
ODEs for hypoxic and apoptotic cells and for ECM d H dt = α N →H 1 o ≤ o H N − α H→N 1 o ≥ o H H � �� � � �� � normoxic → hypoxic hypoxic → normoxic − α H→A 1 o ≤ o A H ( H ( t , x ) = hypoxic cell density) � �� � hypoxic → apoptotic d A dt = α H→A 1 o ≤ o A H ( A ( t , x ) = apoptotic cell density) � �� � hypoxic → apoptotic dm dt = − β m N ( m ( t , x ) = ExtraCellular Matrix) � �� � degradation by normoxic cells Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
The endothelial cascade hypoxic cells: need more oxygen to survive. They initiate a cascade of cellular interactions. The result is angiogenesis: new vascularization to supply the tumor (microvessels branching from main vessels in the direction of the tumor). Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
The endothelial cascade hypoxic cells: need more oxygen to survive. They initiate a cascade of cellular interactions. The result is angiogenesis: new vascularization to supply the tumor (microvessels branching from main vessels in the direction of the tumor). Messanger from hypoxic cells to endothelial cells: VEGF (Vascular Endothelial Growth Factor) Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
PDEs for the endothelial cascade ∂ g ∂ t = k 4 ∆ g ( g ( t , x ) = VEGF concentration) � �� � diffusion + α H→ g H − α g →M E g � �� � � �� � production by hypoxic cells uptake by endothelial cells ∂ E ∂ t = k 2 ∆ E ( E ( t , x ) = density of endothelial ramification) � �� � diffusion − χ 2 div ( E∇ g ) + c 2 E g ( V max − V ) � �� � � �� � transport along VEGF gradient proliferation under VEGF presence Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
PDEs for oxygen concentration ∂ o ∂ t = k 3 ∆ o ( o ( t , x ) = oxygen concentration) � �� � diffusion + c 3 E ( o max − o ) − α o →N , H , E ( N + H + E ) o − γ o ���� � �� � � �� � oxygen decay production by endothelial cells uptake by all living cells Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Summary of variables N ( t , x ) = density of normoxic cells H ( t , x ) = density of hypoxic cells A ( t , x ) = density of apoptotic cells E ( t , x ) = density of endothelial cells (or density of vasculature) o ( t , x ) = oxygen concentration g ( t , x ) = angiogenic growth factor (VEGF) concentration m ( t , x ) = ECM (ExtraCellular Matrix) Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Summary of constants (1) k 1 = background random motility coefficient of normoxic cells k 2 = random motility coefficient of endothelial cells k 3 = diffusion coefficient of oxygen k 4 = diffusion coefficient of angiogenic factor χ 1 = transport coefficient of normoxic cells along ECM gradient χ 2 = transport coefficient of endothelial cells along VEGF gradient V cr = threshold for crowding-driven diffusion V max = limit to total volume of cells and ECM c 1 = proliferation rate of normoxic cells c 2 = proliferation rate of endothelial cells c 3 = production rate of oxygen Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Summary of constants (2) α N →H = decay rate from normoxic to hypoxic cells α H→N = restoration rate from hypoxic to normoxic cells α H→A = decay rate from hypoxic to apoptotic cells α H→ g = production rate of VEGF from hypoxic cells α o →N , H , E = uptake rate of oxygen from all living cells α g →E = uptake rate of VEGF from endothelial cells o max = maximum oxygen concentration o H = oxygen threshold for transition normoxic ↔ hypoxic o A = oxygen threshold for transition hypoxic ↔ apoptotic β = rate of ECM degradation γ = oxygen decay rate Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Blue: density of normoxic cells; light blue (green): extracellular matrix Dotted black: density of hypoxic cells; red: density of endothelial ramification. Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
"Although this does not give any new insight into the dynamics of tumor invasion it nevertheless gives us a reference with which we can compare the different treatment strategies we apply." Cytostatic drugs inhibit cell division or some other function of tumor or host cells (e.g. angiogenesis). Cytotoxic drugs actively kill proliferating tumor (and healthy) cells. Drugs that specifically target proliferation of the endothelial cells could be more efficient than agents which reduce chemotaxis. Stochastic Analysis and applications in Biolog Franco Flandoli, University of Pisa () Mathematical Oncology / 36
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