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Table of contents
Acknowledgements
I Introduction
I.1 Modeling cell dynamics
I.1.1 Cell average models (used in Chapter 2 and 3)
a) Exponentially stable state analysis: general theory
b) Stable state analysis: illustration of a failure case of the general theory
I.1.2 Structuring physiological variables at the population level (used in Chapter 3 to 5)
a) The McKendrick–VonFoerster and renewal equations
b) The Lotka integral equation
c) Stable state analysis
I.1.3 Cell-based level (used in Chapter 2)
a) Link with the continuous time Markov chains
b) Simulation of the Poisson process
c) The branching property
I.1.4 Structuring physiological variables at the individual level (used in Chapter 3)
a) The moment-generating functions
b) The Poisson point measure
I.1.5 Data and model matching (used in Chapter 2, 3 and 5)
I.2 Introduction to the early development of the ovarian follicle
I.2.1 The dynamics of follicle growth
a) The ovarian reserve of primordial follicles
b) Folliculogenesis stages
I.2.2 Dataset presentation
a) Follicle types in early development
b) Kinetics information: dataset building of Chapter 3 and Chapter 4
c) Dataset building of Chapter 2
I.3 Contributions and outline of the dissertation
II Modeling the ovarian follicle activation
II.1 Model definition
II.2 Model analysis
II.2.1 Analysis of the deterministic model
II.2.2 Analysis of the extinction of the precursor cell population
a) Analytical expressions in the linear case
b) Upper bound of the stochastic model
c) Numerical scheme for the mean extinction time and mean number of proliferative cells at the extinction time
II.3 Parameter calibration
II.3.1 Dataset description
II.3.2 Likelihood method
II.3.3 Fitting results
a) Two-event submodels
b) Three-event submodels and complete model
c) Comparison of models
II.3.4 Model prediction
a) Distribution of the initial condition
b) Proliferative cell proportion: reconstruction of time
c) Mean extinction time, mean number of cells at the extinction time and mean number of division events before extinction
d) Biological interpretation
II.4 Conclusion
II.5 Appendix
II.5.1 MLE parameter sets
IIIModeling the compact growth phase
III.1 Starting point
III.1.1 The [Clément-Michel-Monniaux-Stiehl] (CMMS) model [1]
III.1.2 From the [CMMS] model to our model
III.2 Introduction
III.3 Model description and main results
III.3.1 Model description
III.3.2 Hypotheses
III.3.3 Notation
III.3.4 Main results
a) Eigenproblem approach
b) Renewal equation approach
c) Calibration
III.4 Theoretical proof and illustrations
III.4.1 Eigenproblem
III.4.2 Asymptotic study for the deterministic formalism
III.4.3 Asymptotic study of the martingale problem
III.4.4 Asymptotic study of the renewal equations
III.4.5 Numerical illustration
III.5 Parameter calibration
III.5.1 Structural identifiability
III.5.2 Biological application
a) Biological background
b) Dataset description
c) Parameter estimation
III.6 Conclusion
III.7 Complements on the dataset treatment (unpublished)
IVModeling the compact growth phase: complementary works
IV.1 Back on the [CMMS] model
IV.2 Mechanical spatial-structured model
a) Free boundary problem: Hele-Shaw model
b) Link with the Keller-Segel model: from incompressible to compressible model
V Inverse problem for a structured cell population dynamics model
V.1 Model and discretized solutions
V.1.1 Formal solutions of the direct problem
V.1.2 Numerical scheme to simulate the direct problem
V.2 Analysis of the inverse problem (IP) for continuous initial conditions
V.2.1 Single layer case
a) Well-posedness of the inverse problem (IP)
b) Numerical procedure for non-synchronized populations
V.2.2 Multi-layer case
V.3 Analysis of the inverse problem (IP) for Dirac measure initial conditions
V.4 Conclusion
VI Discussion and perspectives
A Appendix
A.1 Additionnal materials of Chapter 3
A.1.1 Supplemental proofs: deterministic model
A.1.2 Supplemental proofs: Stochastic model
A.1.3 Supplemental proofs: Moment study
a) Stochastic simulation procedures
b) Deterministic simulation protocol
A.1.4 Construction of Figure III.5
A.1.5 Parameter estimation procedure
A.2 Additionnal materials of Chapter 5
A.2.1 Numerical scheme
A.2.2 Complement of proofs
A.2.3 Complementary illustrations of Algorithm
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