System of two integro-differential equations 

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In cell proliferation

When cultured in a medium containing soluble factors emitted by adipocytes, cancer cells are significantly more proliferative, a conclusion that does not hold if the factors are emitted by fibroblasts [7]. Similar conclusions are obtained in terms of cell proliferation in the case of cancer cells cultured with adipocytes in a three-dimensional collagen gel matrix [10].

In invasive phenotype

Epithelial to mesenchymal transition (EMT) is a reversible process by which epithelial cells partly lose their adherent properties (and their polarity), and thus become more motile. This transition is known to occur in cancer cells, and is strongly related to invasive properties of the cancer, and metastases. Put in contact with adipocytes through soluble factors only, cancer cells undergo the EMT [5], proving that adipocytes are linked to cancer invasion. Together with the change of adipocytes from mature adipocytes (A) to CAAs, this shows that crosstalk between both populations change their phenotype, a situation summarized in Fig. 3.

Estimating cell proliferation and distribution in phenotypes

The number of viable cells will be estimated by cell count, using trypan blue exclusion (trypan blue colors dead cells) every 24 hours.
To quantify the phenotype distribution of the adipocytes, two methods are possible (and can be combined).
Estimating the level of expression of adiponectin.
Estimating the size of the adipocytes.
To quantify the distribution of cancer cells in phenotype, three methods are possible (and can be combined).
Estimating the level of expression of E-cadherin.
Estimating the adhesion properties of the cancer cells.
these properties will be investigated through dispase disaggregation assay (DDA) following the protocol schematized in Fig. 5. Loosely speaking, independent monolayers of cancer cells will be grown until confluency. Cell monolayers will be detached from the culture dishes through incubation with a proper metalloproteinase dispase. Released monolayers will be disrupted by standardized pipetting, and a count of floating cell clusters will be performed. A higher number of floating clusters (i.e., a higher cell dissociation percentage) will indicate a lower cell-cell adhesion strength.

Equation for the interaction between adipocytes and cancer cells

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To model the experiments of co-culture between cancer cells and adipocytes, we are interested in the behaviour of the two population densities nC(t; x) (cancer cells) and nA(t; y) (adipocytes). x and y stand for traits that characterize the phenotype of the individuals: for cancer cells, x ranges from 0 for an epithelial phenotype E to 1 for a mesenchymal phenotype M. x can be linked to the concentration of a protein through a proper normalization:
for adipocytes, y ranges from 0 for mature adipocytes A to 1 for cancer-associated adipocytes CAA. y can be linked to the concentration of a protein through a proper normalization.

Table of contents :

1 Breast cancer and its environment 4
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 9
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 

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