# Bio-medicine

The main objective is to contribute to the development of a mathematical theory of life sciences. The research will then deal with the application of mathematical models and methods to describe the behavior of biological systems from both the qualitative and the quantitative viewpoint. These range from the behavior of single cells to that of cell aggregates and tissues, from the interaction between cells to that with the environment, from tumor growth to tissue remodeling and regeneration.

For this purpose models operating at different spatial scales are used, e.g., individual cell-based models, population models, kinetic theories, continuum mechanics and multiphase models. Particular attention is paid to the development of multiscale and multi-level models and methods.

- CELL AND TISSUE BIOMECHANICS

The main objective is the analysis of the mechanical aspects of cell aggregates and fiber-reinforced biological tissues, such as articular cartilages. Particular emphasis is given to the formulation of mathematical models capable of describing: (a) the growth of cell aggregates, (b) the structural transformation and the variation of the mechanical properties of tissues, (c) the remodeling of the basement membrane, and (d) the migration of cells. The research aims are to interpret and reproduce experimental results obtained in dedicated experiments.

Particular importance is given to the formulation of multi-scale models and multi-level coupling between discrete models and continuous models.

- CELLULAR MODELS POTTS

CPM is a stochastic Monte Carlo method, in which the evolution of the biological system of interest is guided by a principle of energy minimization. CPMs represent any biological element (e.g., cells or fibers of the extracellular matrix) at the mesoscopic/cellular scale as a discrete spatially extended object. Biological elements (e.g., ions, molecules, or genes) at the microscopic/subcellular scale are described by reaction-diffusion equations, which give the CPM a hybrid characterization.

The energy of the system under consideration contains energetic terms which model the action of forces and define the biophysical characteristics of discrete objects and their mutual interactions. The energy minimization is implemented through repeated probabilistic updates of the configurations of the system.

- HYBRID AND MULTI-LEVEL MODELS

Mathematical models studying biomedical systems require the development of multiscale models and methods because every phenomenon, even if described at the macroscopic scale, depends on what happens at the microscopic scale. The mathematical models have then to consider many interrelated phenomena that occur at very different spatial and temporal scales. For example, the behavior of a cell, or of a cell aggregate, or of a tissue is determined by subcellular dynamics.

The mathematical models that operate on different scales are conceived to transfer information from the smallest scale to the largest one and viceversa, or to interface mathematical models which naturally operate at different scales, thus building hybrid models that combine and exploit the advantages brought by different approaches.

- KINETIC MODELS

Modeling of systems consisting of a large number of interacting individuals, seen as active particles, that is, agents whose microscopic state also includes a variable which describes the individual capacity of each subject to express a specific function/activity.

The mathematical method used is that of the kinetic theory, developed from the analysis of models describing the interactions between living systems or groups of subjects characterized by social dynamics.

This scale of observation allows to model complex phenomena, which for example is the growth and progression of tumor cells. The succession of mutations, one after the other, allows pre-neoplastic cells to progress to tumor formation. The selection of the mutations according to the ability to facilitate this progression follows the principles similar to those of the evolution of the species: Darwinian evolution of cancer cells.

- POPULATION AND EVOLUTIONARY DYNAMICS

The main focus of this research line is the mathematical modeling of biological systems, as for instance multicellular systems from an evolutionary viewpoint. The cellular competition for resources and survival is studied as a Darwinian selection type between several populations, which leads to the selection of the more fitting populations in a given biological context. Applications related to biology and medicine, such as the tumor cells dynamics, the role of specific therapeutic agents, the action of the immune system, the competition tumor cells - immune system, cell motility are developed. In general, aspects of complexity in the biological sciences are object of study and modeling.

The mathematical formalism is that of population dynamics and of structured populations and specifically integro-differential equations, in which the entities that play the game (normally cells) are characterized by a microscopic state relative to their phenotype, or other variables which characterize particular biological functions.

In addition to the modeling activity, a qualitative analysis of the computational and mathematical problems generated by the application of these models is performed, aimed at highlighting adherence to the reality of the proposed model, exploring possible emergent behaviors.