Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA
097661 - BIOMATHEMATICAL MODELING
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.
Risultati di apprendimento attesi
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.
Stability in continuous media. Elements of Fourier analysis, linear stability analysis. Application of the theory to reaction-diffusion systems of equations: Turing instability and pattern formation.
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.
Arteries and their structure. Application of nonlinear elasticity to the arterial wall.
Muscles and their structure. Application of nonlinear elasticity to muscle tissue.
Porous media. Derivation of the theory and main applications to biological models.
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.
Modalità di valutazione
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 may prepare a short presentation (10 minutes approx.) based on material on models NOT covered in the Course, writing a mail to prof. Marzocchi in due time (at least 2 weeks before) for the authorization and this will be certainly a topic in the exam. The presentation will be evaluated mainly on the clarity of the exposition and it must be in English.
Keener-Sneyd, Mathematical PhysiologyM.E. Gurtin, Elements of Continuum MechanicsJ.D.Humphrey, Cardiovascular solid mechanicsY.C. Fung, BiomechanicsB. Perthame, Growth, reaction, movement and diffusion from biologyLubarda, Hoger, On the mechanics of solids with a growing mass, Fascicolo: Int. J. of Solids Struct. 39 (2002) 4627¿4664
I-S. Liu, Continuum MechanicsS. Forte, L. Preziosi, M. Vianello, Meccanica dei continui
Tipo Forma Didattica
Ore di attività svolte in aula
Ore di studio autonome
Laboratorio Di Progetto
Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua
Disponibilità di materiale didattico/slides in lingua inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese