The role of mathematics in biological modelling has considerably increased during the last twenty years and represents today a significant fraction of all research projects in applied mathematics. On the other side, the knowledge of the mechanical and, more generally, of the mathematical aspects of biological processes like growth, remodelling, cell motility and evolution has become enormous.
Therefore, the course aims to give a solid mathematical background in the field of biomathematics and a quite up-to-date knowledge of the main biomathematical models widely used in biology, recalling the most important mathematical tools needed for their setting and comprehension.
The mathematical problems arising in these models, although they represent the translation of well-known physical laws into the realm of Biology, are often very difficult to tackle without an appropriate numerical setting and approximation. Therefore, the course wants to use some tools to show the numerical predictions arising from the models and their correspondence with the experimental observations.

The expected skills focus on the ability of the student in creating models fitting applied and experimentally verifiable situations in biology. The student must know the most important models in biomathematics treated in the course and be able to describe then critically rather than knowing formulas and expressions by heart.
The most important skill, however, is the capability of the student of drawing rigorous conclusions from the models using the existing literature in the topic and his mathematical background.
The student should then be able to read a good scientific paper or a part of a good book on biomathematical modelling and recognize first the parts where physical or biological laws are set, then the approximations used and finally the conclusions, focusing on possible drawbacks or inconsistencies with the observed experiments.
Finally, it is expected that the student acquires a fairly good vocabulary of the technical terms involved in the shown models in Biology, as well as the corresponding vocabulary of terms arising from mathematics.

Elements of dynamical systems: phase space, equilibrium and stability, bifurcations and limit cycles, homoclines and heteroclines.

Application of the theory to examples in chemical kinetics and population dynamics. Recalls on conservation laws. Effective potential and ionic currents: cellular excitation.

The model by Hodgkin and Huxley and the model by FitzHugh and Nagumo. Examples and computer simulations.

Reaction-diffusion equations. Qualitative analysis: traveling fronts, traveling pulses, spiral waves. Application of the theory to the propagation of electrical signals in cells. Examples and computer simulations. Elements of Fourier analysis, linear stability analysis. Application of the theory to reaction-diffusion systems of equations: Turing instability and pattern formation.

Formal multiscale homogenization for linear reaction diffusion equations. Two scales expansion. Periodic
and non-periodic microstructure. Homogenization of porous media flow. 

Mixture theory. Reynolds theorem in time and space. Volume fraction, saturation constraint, partial stress tensor, boundary
conditions. Porosity and permeability, Darcy equation. 

Tumor growth modeling. Linear diffusion-reaction equation and drug delivery in tissues. Growth of spheroidal tumor in vitro in its
avascular phase: the Greenspan model. Multiphase models: sharp versus diffuse interface aprroaches.

Elements of continuum mechanics. Recalls of the theory. Balance of mass, growth. Equations of balance of momentum in weak form and in the presence of growth and/or remodelling. Energy balance and entropy imbalance, Clausius-Duhem inequality and consequences for models. Nonlinear elasticity. Frame-indifference and consequences. Isotropic materials. Hyperelasticity and applications to biological materials. Transverse isotropy and orthotropy. Examples and computer simulations.

Application of nonlinear elasticity to living tissues.

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Calculus in one and several variables. Multiple integrals, vector fields in R^n, divergence theorem. Basic elements of Partial Differential Equations theory. Basics of electromagnetism.

The exam will be oral and written on the blackboard. The questions during the oral exam will concern the Theory and the Esercitazioni. As for the Theory, a good knowledge of the part of general theory is required, while for the models part a deep knowledge of the finest algebraic passages is not necessary. The questions regarding the Esercitazioni will concern examples seen during the course and some calculations made in order to explain them. The evaluation will focus on mathematical rigour, knowledge appropriateness of language,

The Student shall prepare a short presentation (10 minutes approx.) based on material on models NOT covered in the Course, writing a mail to prof. Ciarletta in due time (at least 2 weeks before), choosing a topic of interest. The presentation will be evaluated mainly on the clarity of the exposition and it must be in English.