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

TEACHING METHODS

Traditional lectures + 1 lab

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.

RECOMMENDED READING/BIBLIOGRAPHY

Professor’s lecture notes

TEACHERS AND EXAM BOARD

Exam Board

MICHELE PIANA (President)

VALENTINA CANDIANI

CRISTINA CAMPI (President Substitute)

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

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

Exam schedule

FURTHER INFORMATION

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