## Year 2 in L'Aquila

## Cancer Modelling and Simulation

**#ABOUT** Year 2 in L'Aquila**;**

## Cancer Modelling and Simulation

The group at L’Aquila features an important research stream on these subjects, see the contribution by M. Di Francesco in the mathematical theory of chemotaxis modelling and biological aggregation phenomena, as well as the results obtained by D. Donatelli in mathematical fluid dynamics and biofluid dynamics and cancer modelling and simulation.

The course “Advanced analysis” (taught by C. Lattanzio) provides advanced modelling tools proper for mathematical analysis. The course “Mathematical biofluid dynamics” (taught by D. Donatelli) tackles modern fluid-dynamical modelling techniques in cancer modelling, specifically cells-ECM interaction models. “Biomathematics”, taught by C. Pignotti, deals with cell population models and chemotaxis modelling, with applications to cancer modelling. The course “Systems Biology” covers deterministic and stochastic modelling and control of gene transcription networks and enzymatic reactions with application to cancer drug response. The Course “Cancer Genetics and Biology for Mathematical Modelling” is jointly taught by Dr. Alessandra Tessitore and Dr. Daria Capece (formerly at Imperial College, London), who are members of the Department of Biotechnological and Applied Clinical Sciences at UAQ.

Students of this specialization branch will have the opportunity to prepare their MSc thesis in collaboration with the staff from external institutions such as CSCAMM at the University of Maryland, WWU Muenster in Germany, KAUST, University of Oxford, Imperial College London. They may also spend the thesis semester in one of the pharmaceutical companies of the Capitank Consortium carrying out R&D projects on cancer genetics, evolution and treatment.

March 2025

September 2026

🎓 Graduation

**#Year 2 in L'Aquila** EMJMD InterMaths Study Track

Cancer Modelling and Simulation**;**

ECTS Credits:
9 **|** Semester:
1 **|** Year:
2 **|** Campus:
University of L'Aquila **|** Language:
English

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.

The course aims at providing advanced mathematical notions used in the field of (applied) mathematical analysis and their applications to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.

Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematics was verified at the time of the planning of the master program.

LEARNING OUTCOMES.

At the end of the course, the student should:

1. know the advanced mathematical notions used in the field of (applied) mathematical analysis, as measure theory, Sobolev Spaces, distributions, and their applications to the theory of linear and non-linear partial differential equations;

2. understand and be able to explain thesis and proofs in advanced mathematical analysis;

3. have strengthened the logic and computational skills;

4. be able to read and understand other mathematical texts on related topics.

Content:

Distributions. Locally integrable functions. The space of test function D(Ω). Distributions associated to locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions. Fourier transform and tempered distributions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).

Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and embedding theorem for p < n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich–Kondrachov theorem, Poincaré inequalities. Embedding theorem for p = n. Characterization of the dual space H-1.

Second order parabolic equations. Definition of parabolic operator. Weak solutions for linear parabolic equations. Existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions.

First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem.

Measures. System of sets, Positive Measures, Outer Measures, Construction of Measures, Signed Measures, Borel and Radon Measures

Integration. Measurable Functions, Simple Functions, Convergence Almost Everywhere, Integral of Measurable Functions, Convergences of Integrals, Fubini-Tonelli Theorems.

Diﬀerentiation. The Radon-Nikodym Theorem, Diﬀerentiation on Euclidean space, Diﬀerentiation of the real line.

Radon measures and continuous functions. Spaces of continuous functions, Riesz Theorem.

Pre-requisites:

Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

Reading list:

- L. Ambrosio, G. Da Prato, A. Mennucci. Introduction to measure theory and integration. Edizioni della Normale.

- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.

- V.I. Bogachev. Measure theory, Volume I, Springer.

- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer.

- P. Cannarsa, T. D’Aprile. Introduction to Measure Theory and Functional Analysis. Springer.

- C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, Springer.

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.

- L. Evans, R. Gariepy. Measure Theory and Fine Properties of Functions, CRC Press. Revised Edition.

- G.B. Folland. Real analysis: Modern techniques and their applications. New York Wiley

- G. Gilardi. Analisi 3. McGraw–Hill.

- L. Grafakos, Classical Fourier Analysis. Springer.

- V.S. Vladimirov. Equations of Mathematical Physics. Marcel Dekker, Inc

ECTS Credits:
6 **|** Semester:
1 **|** Year:
2 **|** Campus:
University of L'Aquila **|** Language:
English

Unit Coordinator: Donatella Donatelli

Aims:

Learning Objectives:

The aim of the course is to give an overview of fluid dynamics from a mathematical viewpoint, and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for the analysis of other partial differential equations.

Learning Outcomes:

On successful completion of this course, the student should:

- understand the basic principles governing the dynamics of non-viscous fluids;

- be able to derive and deduce the consequences of the equation of conservation of mass;

- be able to apply Bernoulli's theorem and the momentum integral to simple problems including river flows;

- understand the concept of vorticity and the conditions in which it may be assumed to be zero;

- calculate velocity fields and forces on bodies for simple steady and unsteady flows derived from potentials;

- demonstrate skill in mathematical reasoning and ability to conceive proofs for fluid dynamics equations.

- demonstrate capacity for reading and understand other texts on related topics.

Content:

CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical Modelling of Continuum Media (3 ECTS)

- Derivation of the governing equations: Euler and Navier-Stokes

- Eulerian and Lagrangian description of fluid motion; examples of fluid flows

- Fluidi di tipo Poiseulle e Couette

- Vorticity equation in 2D and 3D

CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS),

- Dimensional analysis: Reynolds number, Mach Number, Frohde number.

- From compressible to incompressible models

- Existence of solutions for viscid and inviscid fluids

- Fluid dynamic modeling in various fields: mixture of fluids, combustion, astrophysics, geophysical fluids (atmosphere, ocean)

CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS)

- Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.

Pre-requisites:

PREREQUISITES for Mathematical Modelling of Continuum Media:

Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.

PREREQUISITES for Mathematical fluid and biofluid dynamics, Mathematical fluid dynamics, Modelling and analysis of fluids and biofluids:

Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations, Sobolev spaces.

Reading list:

- Alexandre Chorin, Jerrold E. Marsden, A Mathematical Introduction to Fluid Mechanics. Springer.

- Roger M. Temam, Alain M. Miranville, Mathematical Modeling in Continum Mechanics. Cambridge University Press.

- Franck Boyer, Pierre Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Springer-Verlag Italia.

- Andrea Bertozzi, Andrew Majda, Vorticity and Incompressible Flow. Cambridge University Press.

**#Final Semester** Dissertation;

The thesis topic can be proposed by the track coordinator or by the student. In any case, the local coordinator has the responsibility to provide an advisor. The student’s taste and expectations are met whenever possible. The student must write a short thesis project, with the help of her/his advisor, to be submitted to the Executive Committee, which has to approve the thesis project before its formal start. The thesis topic will preferably deal with a problem proposed by a private company, if possible chosen among the Consortium Industrial Partners.

The final master’s degree examination, while respecting the local regulations, will consist as a general rule in two parts: an oral examination on the topic of the thesis, and the defense of the thesis.

In this examination the candidate will be required to demonstrate good knowledge of his/her specialization track and a capability for working independently and solving problems on experimental, numerical, technological, design or modelling applications. The semester may include an internship within a collaborating company or institution. In this case, a tutor from the involved partner and an academic supervisor will be appointed.

Students who have satisfied all the requirements of the degree programme will be awarded a **Joint Master Degree in Interdisciplinary Mathematics** by the Universities where the student has **spent at least one semester**.