Breast cancer and its environment

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Table of contents

1 Breast cancer and its environment 
1.1 Breast cancer : a brief overview
1.1.1 Some statistics in France
1.1.2 Location of breast cancers
1.1.3 Different types of breast cancers
1.1.4 Therapeutics for breast cancer
1.2 Interactions between the tumour and the adipose tissue
1.2.1 Short description of the adipose tissue
1.2.2 Mutualistic interactions
1.3 The key role of adipocytes
1.3.1 In cell proliferation
1.3.2 In invasive phenotype
1.3.3 In resistance to therapy
2 Experimental setting and mathematical modelling 
2.1 Experimental setting
2.1.1 Principle and cell lines
2.1.2 Estimating cell proliferation and distribution in phenotypes
2.2 Mathematical modelling
2.2.1 Underlying principles
2.2.2 Typical equation
2.2.3 Equation for the interaction between adipocytes and cancer cells
3 Single integro-differential equation 
3.1 Existence and uniqueness
3.1.1 A priori bounds
3.1.2 Proof of existence and uniqueness
3.2 Asymptotics
3.2.1 Convergence for
3.2.2 Convergence for n
4 System of two integro-differential equations 
4.1 Mutualistic 2 2 Lotka-Volterra systems
4.1.1 Possible blow-up
4.1.2 Convergence in the case (24)
4.2 Existence and uniqueness
4.2.1 Regularity and non blow-up assumptions
4.2.2 A priori bounds for 1 and 2
4.2.3 Existence and uniqueness
4.3 Asymptotics
4.3.1 Convergence for 1 and 2
5 Parametrization of the model 
5.1 With explicit formulas
5.1.1 Main ideas and model simplification
5.1.2 Explicit formulas
5.2 By means of numerical simulations
5.2.1 Numerical scheme
5.2.2 Examples of results
A BV functions on R+ 30
B Handling positive and negative parts for ODEs 
C Computations for the explicit solution to the equation on R 
References