HSC dynamics : Mathematical modeling
Mathematical modeling of HSC dynamics has been the focus of a large panel of re-searchers over the last four decades, with attempts to improve the understanding of the complex mechanisms regulating HSC functions, throughout the course of normal and pathological hematopoiesis. One of the earliest mathematical models that shed some light on this process was proposed by Mackey  in 1978 inspired by the work of Lajtha , and Burns and Tannock . Mackey’s model consists of a system of two delay diﬀerential equations describing the evolution of the HSC population divided into proliferating and quiescent cells (also called resting cells). This model has been studied, analyzed and applied to hematological diseases by many authors (see, for in-stance, [9, 95, 96, 99]). For many years, only systems with discrete delay were proposed to describe HSC dynamics see, for example, [8, 77, 78]. Then, more recently, Adimy and Crauste , Adimy, Crauste, and Ruan , and Bernard, Bélair, and Mackey  proposed and analyzed modified versions of Mackey’s model  by considering a dis-tribution of cell cycle durations. In 2005, Adimy and Crauste  and Adimy, Crauste, and Pujo-Menjouet  proposed a model of HSC dynamics in which the cell cycle duration depends upon the cell maturity.
Mathematical models describing the action of growth factors on the hematopoiesis process have been proposed by Bélair et al in 1995 , and Mahaﬀy et al in 1998 . They considered an age-structured model of HSC dynamics, coupled with a dif-ferential equation to describe the action of a growth factor on the reintroduction rate from the resting phase to the proliferating one. In 2006, Adimy et al  proposed a system of three delay diﬀerential equations describing the production of blood cells under the action of growth factors assumed to act on the rate of reintroduction into the proliferating phase. Adimy and Crauste considered and analyzed two models of hematopoiesis dynamics with: the influence of growth factors on HSC apoptosis , and the action of growth factors on the apoptosis rate as well as on the reintroduction rate into the proliferating phase . Many of the aforementioned mathematical models have subsequent implications to cancer prevention, development and treatment.
Our contribution to HSC dynamics modeling
The originality of our study regarding the influence of growth factors concentrations on diﬀerentiation, proliferation rate and apoptosis lays in the proposed model itself which, to our knowledge, has never been considered in hematopoiesis dynamics beforehand.
Regarding the originality of our study relating to the influence of the total popula-tion of quiescent cells on the cell cycle duration, it lays in the refinement of the model proposed by Adimy et al. .
Immunosenescence, Cancer and Stem Cells
The immune system is not permanently fully eﬃcient. Indeed, immunosenescence is a process that reflects a gradual decrease of immune system activity with age mainly through a decreased capacity of immunosurveillance . The beginning of immunose-nescence is assumed to be associated with the beginning of thymopoiesis decline. In-deed, the thymus play a crucial role in the development of T cells but also in main-taining immune eﬃciency . Maximal activity is reached at puberty (from 10 to 19 years old according to the World Health Organization) and decrease progressively in adults . The elderly (more than 65 years old (WHO)) usually have i) a depleted population of naive T cells (the set of T lymphocytes that can respond to novel anti-gens) [92, 93], ii) a shrinking repertoire of T cell clones [55, 83, 93], iii) an increased number of naturally occurring regulatory T cells that down-regulate T cell responses [46, 98], iv) a low grade, pro-inflammatory status , and v) increased numbers of myeloid-derived suppressor cells, which are associated with impaired T-cell functioning and produce high amounts of reactive oxygen species . All these immune-associated changes can potentially promote tumor proliferation .
The Link Between Aging and Cancer : Mathematical modeling
Mathematical models have been used, since 1954, to investigate the link between aging and cancer by correlating increasing incidences of cancer with advancing age to muta-tion accumulation . More recently, genetic and epigenetic changes in stem cells have been associated with both normal aging processes and cancer risk [23, 36, 106, 111]. Normal aging is linked with lymphocytes immunosenescence leading to increased se-cretion of cytokines such as IL − 6 and T N F − α . This state of chronic immune activation has been associated with DNA-modifying events that lead to an increased risk of malignancy . Inflammatory cytokines are also important regulators of stem cell states . Mathematical models have proven useful in the study the eﬀects of stress and hormesis  on lifespan and the relationship between accelerated aging and carcinogenesis [32–34]. Stochastic models have been used to study the balance between damaged and repaired states in stressed worms. Predictions of lifespan us-ing this model matched the experimental observations . Further stochastic models have been used to examine levels of free radicals and cumulative damage to DNA, lipid structures, and proteins, leading to genetic instability and malignant transformation . Predictions from these models matched both experimental data of survival and fertility curves in Mediterranean fruit flies, and cancer incidence in rats exposed to bromode-oxyuridine [33, 34].
Our contribution to the modeling of the interactions between aging and cancer
The originality of our study lays in the ability of our model to predict that acute immunosuppressive infections could also impact cancer risk and in a larger extent than persistent infections. Empirical evidences of such situation are obviously harder to identify, but the impact of “common” diseases on immune system and their relation with cancer risk are worthy of investigation. Our results suggest a stronger impact of acute and repeated immune challenges after the beginning of immunosencence.
Global Asymptotic Stability of the Trivial Steady State
First, we are interested in the local asymptotic stability of the trivial steady state of (3.1.16). The characteristic equation (3.3.1) becomes, when x∗ = 0, Δ(λ) = λ + δ + β(0) − 2β(0)e−γτ0 e−λτ0 . ddλΔ(λ) = 1 + 2τ0e−γτ0 β(0)e−λτ0 > 0, Δ(0) = δ + β(0) − 2β(0)e−γτ0 = δ − 2e−γτ0 − 1 β(0), and limλ→∞ Δ(λ) = +∞. Then there exists λ0 ∈ R, which is unique, such that Δ(λ0) = 0. When (3.2.1) holds, then Δ(0) < 0, so λ0 > 0, which proves the instability of the trivial steady state. When (3.2.1) does not hold, we show that all roots λ 6= λ0 of Δ satisfy Re(λ) < λ0. Suppose that λ = ρ + iσ is a root of Δ such that λ =6 λ0. Considering the real parts of Δ = 0 we get . ρ − λ0 = 2β(0)e−γτ0 e−ρτ0 cos(στ0) − e−λ0τ0 (3.4.1) By contradiction, we suppose that ρ > λ0 then e−ρτ0 cos(στ0) − e−λ0τ0 < 0. We obtain a contradiction, thus ρ ≤ λ0. Now, if ρ = λ0, (3.4.1) implies cos(στ0) = 1, for τ0 ≥ 0. It follows that sin(στ0) = 0, considering the imaginary part of equation Δ = 0 we get σ + 2β(0)e−γτ0 e−ρτ0 sin(στ0) = 0, then σ = 0 and λ = λ0 which gives a contradiction. Therefore ρ < λ0. We can now conclude that the trivial steady state N ≡ 0 of (3.1.16) is locally asymptotically stable when 1 ln 2β(0) 1 ln 2β(0) , <τ0≤ γ β(0) + δ γ δ.
Table of contents :
List of Figures
1 General Introduction
1.1 Hematopoietic Stem Cells
1.1.1 HSC dynamics : Mathematical modeling
1.1.2 Our contribution to HSC dynamics modeling
1.2 Immunosenescence, Cancer and Stem Cells
1.2.1 The Link Between Aging and Cancer : Mathematical modeling .
1.2.2 Our contribution to the modeling of the interactions between aging and cancer
Organization of the thesis
I Hematopoietic Stem Cell Dynamics
2 Age-structured model of hematopoiesis dynamics with growth factordependent coefficients
2.2 Age-structured partial differential model
2.3 Reduction to a delay differential system
2.4 Positivity and boundedness of solutions
2.5 Existence of steady states
2.6 Global asymptotic stability of trivial steady state
2.7 Local asymptotic stability of the positive steady state
2.8 Numerical illustrations
3 Hematopoietic Stem Cells dynamics model with state dependant delay
3.1 Age-structured partial differential model
3.2 Properties of the model and existence of steady states
3.3 Linearization and Characteristic Equation
3.4 Global Asymptotic Stability of the Trivial Steady State
3.5 Transcritical Bifurcation and Hopf Bifurcation
3.6 Numerical illustrations
II Cancer cells Proliferation: Immune System Response
4 Interactions between immune challenges and cancer cells proliferation: timing does matter!
4.2 Materials and Methods
Influence of timing and duration of a single immunosuppressive challenge
Combined effect of duration and the number of immunosuppressive challenges
Influence of immune activation challenges combined with immunosenescence
Conclusion and Perspectives
Appendix A Supplementary data to Chapter 4