OVERVIEW

This course introduces students to the mathematical modeling of biomedical data through four main thematic areas: X-ray computed tomography, positron emission tomography, magnetic resonance imaging, and molecular interaction maps. The course provides mathematical and computational tools to understand key medical imaging techniques and to build descriptive models across different biological scales, from the organ level to the molecular scale. Lectures are held in Italian; upon request, they may be delivered in English. Laboratory sessions are an integral part of the course.

AIMS AND CONTENT

LEARNING OUTCOMES

The course intends to describe the mathematical modeling of two very important tomographic problems in biomedical field: X-ray tomography and magnetic resonance. In both cases, the objective is twofold: on the one hand, to emphasize how sophisticated mathematical formalisms are indispensable to fully understand problems of such great application value; On the other hand, to provide students with the numerical tools needed to process the images from these acquisition modes.

AIMS AND LEARNING OUTCOMES

The course offers a multiscale perspective on mathematical modeling in medicine:

• Organ scale: modeling of X-ray data via the Radon transform and magnetic resonance imaging data via the Fourier transform, with emphasis on theoretical and computational aspects related to data inversion from incomplete measurements.
• Cellular scale: modeling of positron emission tomography data through compartmental analysis, including inverse problems and issues of uniqueness and stability.
• Molecular scale: construction and mathematical analysis of molecular interaction maps derived from protein data, with a focus on the structure and properties of biological networks.

At the end of the course, students will have acquired:

• theoretical knowledge of the mathematical foundations of major medical imaging techniques;
• skills to model cellular behavior using compartmental dynamical systems;
• tools for analyzing complex biological networks at the molecular level;
• the ability to implement numerical methods for processing both simulated and real biomedical data.

PREREQUISITES

Fondamenti di calcolo numerico

TEACHING METHODS

The course consists of approximately 40 hours of lectures and 12 hours of laboratory sessions.

In the lab, students will implement the computational methodologies introduced in class, including:

• inversion of the Radon transform for the reconstruction of simulated X-ray tomography data;
• formulation and solution of the direct and inverse problems for a compartmental model applied to both simulated and real positron emission tomography data.

SYLLABUS/CONTENT

Part I – X-ray Computed Tomography

• General introduction
• Radon transform: definition, inversion formulas, uniqueness properties

Part II – Positron Emission Tomography

• General introduction
• Two related inverse problems:
  1. Imaging: inversion of the Radon transform
  2. Compartmental modeling: Gauss-Newton method

Part III – Magnetic Resonance Imaging

• General introduction
• Acquisition models and magnetic field distortions
• Fourier transform and inversion from limited data

Part IV – Molecular Interaction Maps

• Modeling of protein data
• Mathematical analysis of interaction networks

RECOMMENDED READING/BIBLIOGRAPHY

Lecture notes provided by the instructor (available on AulaWeb).

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

Classes will begin according to the academic calendar

Class schedule

APPLICATIONS OF MATHEMATICS TO MEDICINE

EXAMS

EXAM DESCRIPTION

The final assessment consists of an oral exam and the submission of laboratory assignments.

ASSESSMENT METHODS

Evaluation is based on the quality of the laboratory assignments and the oral examination. There are no midterm tests. During the oral exam, students will be asked questions covering the full syllabus, and their understanding of the subject will be assessed.

FURTHER INFORMATION

Prerequisites: the only essential prerequisites are familiarity with Hilbert spaces, the theory of bounded linear operators between such spaces, and basic notions of Fourier analysis.

Attendance: attendance is recommended.

Exam Registration: details regarding exam registration will be arranged directly with the instructor.