Lecturer: Piero Colli Franzone  

Course name: Biomathematics
Course code: 500703
Degree course: Bioingegneria
Disciplinary field of science: MAT/08
The course relates to:
University credits: ECTS 6
Course website: n.d.

Specific course objectives

The course proposes an introduction to the mathematical modeling and simulation of physiological systems in biological sciences ( cellular physiology dynamics of excitable cells) providing the main analytical and numerical methods for the investigation of the mathematical models and the interpretation of the simulated results.

Course programme

The course proposes an introduction to the mathematical modeling and simulation of some physiological systems: enzyme kinetics, dynamics of excitable cells, reaction-diffusion systems, bioelectric cardiac processes.

Models of cellular physiology
Regular and singular perturbation theory. Mass action law, biochemical and enzymatic reactions, enzyme kinetics and quasi-steady approximation, cooperative and inhibition phenomena

Population dynamics
Malthusian and logistic models; difference equations in one dimension, stability and biforcations. Interaction in biological systems: pray-predator Lotka-Volterra model, more realistic pray-predator models, models of interaction of two populations, competition and cooperative effects.

Cellular electrophysiology
Nernst potential, electro-diffusion models, approximate current-voltage relationships, dynamics of the ionic currents, ion channels with multiple subunits, voltage-clamping, Hodgkin-Huxley formalism, generation of the action potential of the giant axon of the squid. Approximate two variable excitable models (FitzHugh-Nagumo, Morris-Lecar models): threshold effect and limit cycles.

Introduction to bifurcation involving fixed points and limit cycles in biological systems

Introduction to propagation and reentry in excitable one dimensional ring
Homogenization a 1D arrangement of excitable cells, cable model, threshold, action potential duration, propagation of the excitation.

Introduction to reaction-diffusion systems
Conservation laws; diffusion equation, initial and boundary conditions. Travelling wave solutions; numerical approximation of nonlinear parabolic equations.

Computational Electrocardiology
Anisotropic bidomain model, excitation wavefront propagation, macroscopic structure of the bioelectric cardiac sources, extracellular potential and electrograms.

Course entry requirements

Basic mathematical courses of the "laurea triennale" or " undergraduate degree" and or "bachelor degree".

Course structure and teaching

Lectures (hours/year in lecture theatre): 36
Practical class (hours/year in lecture theatre): 20
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

F. Britton. Essential Mathematical Biology. Springer-Verlag, Heidelberg, 2003.

J.P. Keneer, J. Sneyd. Mathematical Physiology. Springer-Verlag, New York, 1998.

J.D. Murray. Mathematical Biology I : An Introduction, II : Spatial Models and Biomedical Applications. Springer-Verlag, New York, 2002.

Testing and exams

Oral examination with discussion and interpretation of the models simulations developed in the computer laboratory.