Structured Equations for Complex Living Systems - Modeling, Asymptotics and Numerics

Complex living systems differ from those systems whose evolution is well described by the laws of Classical Physics. In fact, they are endowed with self-organizing abilities that result from the interactions among their constituent individuals, which behave according to specific functions, strategies or traits. These functions/strategies/traits can evolve over time, as a result of adaptation to the surrounding environment, and are usually heterogeneously distributed over the individuals, so that the global features expressed by the system as a whole cannot be reduced to the superposition of the single functions/strategies/traits. Quoting Aristotle, we can say that, within these systems, "the whole is more than the sum of its parts". As a result, when we study the dynamics of complex living systems, there are new concepts that come into play, such as adaptation, herding and learning, which do not belong to the traditional vocabulary of physical sciences and make the dynamics of these systems hardly to be forecast. Moving from the above considerations, the subject of my PhD was the development and the study of structured equations for population dynamics (partial differential equations and integro-differential equations) applied to modelling the evolution of complex living systems. In particular, I designed models for multicellular systems, living species and socio-economic systems with the aim of inspecting mechanisms underlying the emergence of collective behaviors and self-organization. In the framework of structured equations, individuals belonging to a given system are divided into different populations and heterogeneously distributed characteristics are modelled by suitable independent variables, the so-called structuring variables. For each population, a function describing the distribution of the individuals over the structuring variables is introduced, which evolves through a partial differential equation, or an integro-differential equation, whose parameter functions are defined according to the phenomena under study. I decided to use such mathematical framework since it makes possible to effectively model the afore mentioned complexity aspects of living systems and provides an efficient way to reduce complexity in view of the mathematical formalization. With particular reference to multicellular systems, I focused on the design and the study of mathematical models describing the evolutionary dynamics of cancer cell populations under the selective pressures exerted by therapeutic agents and the immune system. Proliferation, mutation and competition phenomena are included in these models, which rely on the idea that the process leading to the emergence of resistance to anti-cancer therapies and immune action can be considered, at least in principles, as a Darwinian micro-evolution. It is worth noting that most of these models stem from direct collaborations with biologists and clinicians. Besides local and global existence results for the mathematical problems linked to the models, my PhD thesis presents results related to concentration phenomena arising in phenotype-structured equations and opinion-structured equations (i.e., the weak convergence of the solutions to sums of Dirac masses), and with the derivation of macroscopic models from space-velocity structured equations. From the applicative standpoint such concentration phenomena provide a possible mathematical formalization of the selection principle in evolutionary biology and the emergence of opinions; macroscopic models, instead, offer an overall view of the systems at hand. Numerical simulations are performed with the aim of illustrating, and extending, analytical results and verifying the consistency of the model with empirical dat

Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells

Accumulating evidence indicates that the interaction between epithelial and mesenchymal cells plays a pivotal role in cancer development and metastasis formation. Here we propose an integro-differential model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Our model takes into account the effects of chemotaxis, adhesive interactions between epithelial-like cells, proliferation and competition for nutrients. We present a sample of numerical results which display the emergence of spots, stripes and honeycomb patterns, depending on parameters and initial data. These simulations also suggest that epithelial-like and mesenchymal-like cells can segregate when there is little competition for nutrients. Furthermore, our computational results provide a possible explanation for how the concerted action between epithelial-cell adhesion and mesenchymal-cell spreading can precipitate the formation of ring-like structures, which resemble the fibrous capsules frequently observed in hepatic tumours.PostprintPeer reviewe

Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies

Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance gene (or genes, yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects of chemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work by demonstrating how qualitatively different actions of chemotherapeutic and cytostatic treatments may induce different levels of resistance. The mathematical interest of our study is in the formalism of constrained Hamilton-Jacobi equations in the framework of viscosity solutions. We derive the long-term temporal dynamics of the fittest traits in the regime of small mutations. In the context of adaptive cancer management, we also analyse whether an optimal drug level is better than the maximal tolerated dose

Macroscopic limit of a kinetic model describing the switch in T cell migration modes via binary interactions

Experimental results on the immune response to cancer indicate that activation of cytotoxic T lymphocytes (CTLs) through interactions with dendritic cells (DCs) can trigger a change in CTL migration patterns. In particular, while CTLs in the pre-activation state move in a non-local search pattern, the search pattern of activated CTLs is more localised. In this paper, we develop a kinetic model for such a switch in CTL migration modes. The model is formulated as a coupled system of balance equations for the one-particle distribution functions of CTLs in the pre-activation state, activated CTLs and DCs. CTL activation is modelled via binary interactions between CTLs in the pre-activation state and DCs. Moreover, cell motion is represented as a velocity-jump process, with the running time of CTLs in the pre-activation state following a long-tailed distribution, which is consistent with a L\'evy walk, and the running time of activated CTLs following a Poisson distribution, which corresponds to Brownian motion. We formally show that the macroscopic limit of the model comprises a coupled system of balance equations for the cell densities whereby activated CTL movement is described via a classical diffusion term, whilst a fractional diffusion term describes the movement of CTLs in the pre-activation state. The modelling approach presented here and its possible generalisations are expected to find applications in the study of the immune response to cancer and in other biological contexts in which switch from non-local to localised migration patterns occurs.Comment: 21 pages, 1 figur

On interfaces between cell populations with different mobilities

Partial differential equations describing the dynamics of cell population densities from a fluid mechanical perspective can model the growth of avascular tumours. In this framework, we consider a system of equations that describes the interaction between a population of dividing cells and a population of non-dividing cells. The two cell populations are characterised by different mobilities. We present the results of numerical simulations displaying two-dimensional spherical waves with sharp interfaces between dividing and non-dividing cells. Furthermore, we numerically observe how different ratios between the mobilities change the morphology of the interfaces, and lead to the emergence of finger-like patterns of invasion above a threshold. Motivated by these simulations, we study the existence of one-dimensional travelling wave solutions.PostprintPeer reviewe

Modelling coevolutionary dynamics in heterogeneous SI epidemiological systems across scales

We develop a new structured compartmental model for the coevolutionary dynamics between susceptible and infectious individuals in heterogeneous SI epidemiological systems. In this model, the susceptible compartment is structured by a continuous variable that represents the level of resistance to infection of susceptible individuals, while the infectious compartment is structured by a continuous variable that represents the viral load of infectious individuals. We first formulate an individual-based model wherein the dynamics of single individuals is described through stochastic processes, which permits a fine-grain representation of individual dynamics and captures stochastic variability in evolutionary trajectories amongst individuals. Next we formally derive the mesoscopic counterpart of this model, which consists of a system of coupled integro-differential equations for the population density functions of susceptible and infectious individuals. Then we consider an appropriately rescaled version of this system and we carry out formal asymptotic analysis to derive the corresponding macroscopic model, which comprises a system of coupled ordinary differential equations for the proportions of susceptible and infectious individuals, the mean level of resistance to infection of susceptible individuals, and the mean viral load of infectious individuals. Overall, this leads to a coherent mathematical representation of the coevolutionary dynamics between susceptible and infectious individuals across scales. We provide well-posedness results for the mesoscopic and macroscopic models, and we show that there is excellent agreement between analytical results on the long-time behaviour of the components of the solution to the macroscopic model, the results of Monte Carlo simulations of the individual-based model, and numerical solutions of the macroscopic model