CODE 42916 2024/2025 6 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA MAT/08 Italian (English on demand) GENOVA 2° Semester AULAWEB

## OVERVIEW

The credits for the course Application of Mathematics to Medicine (AMM, code 42916) are 7. The course is held during the first semester of the 1°, 2° LM years. On request of one student, the lectures and teaching activities will be delivered in English, otherwise in Italian.

## 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

This course aims to describe the mathematical modeling of three medical imaging problems: the X-ray tomography, the Positron Emission Tomography and the Magnetic Resonance Imaging. The scope of the course is two-fold: on one hand, we want to highlight how sophisticated mathematics is needed for the comprehension of problems with high practical significance; on the other hand, we want to equip the students with the numerical analysis tools required for the processing of the data acquired with these three modalities.

After these lectures, students will know the main mathematical properties of two important medical imaging modalities, some crucial issues concerned with mathematical models of cellular respiration, and the way an in-silico model of the cancer cell can be constructed.

### PREREQUISITES

Fondamenti di Calcolo Numerico

### SYLLABUS/CONTENT

Part I: X-ray tomography (overview); Radon transform, formulas for the inversion of the Radon transform (as back projection and filtered back projection), issues of uniqueness.

Part II: positron emission tomography (overview); on the two inverse problems related to positron emission tomography: an imaging problem (inversion of the Radon transform) and a compartment alone (Gauss-Newton optimization scheme)

Part III: magnetic resonance imaging (overview); models for data acquisition and magnetic field distortion, Fourier transform, inversion of the Fourier transform from undersampled data.

Professor’s lecture notes

## LESSONS

### LESSONS START

The class will start according to the academic calendar.

### Class schedule

APPLICATIONS OF MATHEMATICS TO MEDICINE

## EXAMS

Oral Exam

### ASSESSMENT METHODS

Oral exam. We will ask students about topics discussed during the lectures and students will be assessed on the base of their knowledge of the content of the topics described during the course

### FURTHER INFORMATION

The prerequisites are: Hilbert spaces, continuous linear operators between Hilbert spaces, Fourier analysis