ID:
500173
Duration (hours):
24
CFU:
3
SSD:
ANALISI MATEMATICA
Year:
2025
Overview
Date/time interval
Primo Semestre (01/10/2025 - 16/01/2026)
Syllabus
Course Objectives
The course aims to offer students the basic knowledge to understand some simple mathematical models applied in the medical field.
In the first part of the course, the fundamental tools of analytical calculation are considered, as well as several examples which emphasize their application. The second part of the course is devoted to the analysis of some differential models applied in oncology and epidemiology.
In the first part of the course, the fundamental tools of analytical calculation are considered, as well as several examples which emphasize their application. The second part of the course is devoted to the analysis of some differential models applied in oncology and epidemiology.
Course Prerequisites
The course requires the basic mathematical knowledge of the Second Degree Secondary School according to the Ministerial Decree n. 277-19 / 20 for the admission test to the Faculty of Medicine.
Teaching Methods
Frontal teaching.
Assessment Methods
Written Test.
Texts
D. Benedetto, M. Degli Esposti, C. Maffei, Matematica per le scienze della vita, Editrice Ambrosiana, 2015.
V. Villani, G. Gentili, Comprendere e interpretare fenomeni delle scienze della vita, McGraw-Hill, 2017.
E.N. Bodine, S. Lenhart, L. J. Gross, Matematica per le scienze della vita, UTET, 2017.
V. Villani, Matematica per discipline biomediche, McGraw-Hill, 2007.
E. Vitali, Lezioni introduttive sulle equazioni differenziali ordinarie, Università di Pavia, Dispensa a cura del docente.
M. Degli Esposti, Trasmissione di malattie infettive e diffusione di epidemie su network: modelli matematici, Università di Bologna, Dispensa a cura del docente.
J.F. Holland, E. Frei, Cancer medicine, BC Decker Inc., Hamilton, 2000.
V.T. Devit, T.S. Lawrence, S. A. Rosenberg, Cancer principles and practice of oncology, Wilkins, Philadephia, 2008.
L. Marconato, Principi di chemioterapia in oncologia, Poletto Editore, Milano, 2009.
I.F. Tannock, R.P. Hill, R.G. Bristow, L. Harrington, The basic science of oncology, MacGraw-Hill, 2005.
L. Norton, Conceptual and practical implications of breast tissue geometry, toward a more effective, less toxic therapy, The Oncologist, 2005.
V. Villani, G. Gentili, Comprendere e interpretare fenomeni delle scienze della vita, McGraw-Hill, 2017.
E.N. Bodine, S. Lenhart, L. J. Gross, Matematica per le scienze della vita, UTET, 2017.
V. Villani, Matematica per discipline biomediche, McGraw-Hill, 2007.
E. Vitali, Lezioni introduttive sulle equazioni differenziali ordinarie, Università di Pavia, Dispensa a cura del docente.
M. Degli Esposti, Trasmissione di malattie infettive e diffusione di epidemie su network: modelli matematici, Università di Bologna, Dispensa a cura del docente.
J.F. Holland, E. Frei, Cancer medicine, BC Decker Inc., Hamilton, 2000.
V.T. Devit, T.S. Lawrence, S. A. Rosenberg, Cancer principles and practice of oncology, Wilkins, Philadephia, 2008.
L. Marconato, Principi di chemioterapia in oncologia, Poletto Editore, Milano, 2009.
I.F. Tannock, R.P. Hill, R.G. Bristow, L. Harrington, The basic science of oncology, MacGraw-Hill, 2005.
L. Norton, Conceptual and practical implications of breast tissue geometry, toward a more effective, less toxic therapy, The Oncologist, 2005.
Contents
The course aims to offer students the basic knowledge to understand some simple mathematical models applied in the medical field.
In the first part of the course, the fundamental tools of analytical calculation are considered, as well as several examples which emphasize their application. The second part of the course is devoted to the analysis of some differential models applied in oncology and epidemiology.
Sequences. Arithmetic sequence (example: growth of a tumor mass). Geometric sequence (example: cell mitosis). Sum of the geometric sequence (example: bacterial proliferation) - (example: predict hospitalization time, note the probability of recovery and vice versa). Sequences: definition and properties. Limits and classification (several examples). The number of Napier interpreted as a limit of a suitable sequence. Theorem: if a sequence is convergent, then it is bounded. Theorem: if a sequence is monotone, then it is not indeterminate. The Fibonacci sequence (∗)
Limits. Domain. Adherent points. Exercises and examples. Continuous functions. Definition and properties. Weierstrass theorem. Examples: the body mass index, the skeletal mass index, the model of the artificial neuron. Exercises and examples.
Derivatives. Geometric definition and interpretation. Left and right derivative. Classification of non-differentiable points: angular points, cusps, inflections with vertical tangents. Derivative of the product, of the quotient and of the compound function. Increasing and decreasing d functions: the link with the value of the derivative. Theorem: if a function is differentiable, then it is continuous. Higher-order derivatives. Concavity and convexity. Points of inflection. Taylor's polynomial: definition and application to the calculation of limits. Exercises and examples.
Integrals. Application and graphic interpretation. Example: absorption of a medicine during chemotherapy. Definition of primitive function. Fundamental theorem of integral calculus. Integrals of elementary functions. Integrals by parts. Integrals by substitution. Integrals of fractioned polynomial functions (case ∆> 0). Examples: the Poiseuille model to determine the speed of blood and the flow rate of an artery (∗). Exercises.
ab.
Curve. Norm of a vector. Tangent vector. Closed curves. Length of a curve. Exercises and examples.
First order ordinary differential equations. Autonomous equations. Equilibrium points and their classification.
Population dynamics model. Maltus model (∗). Example: the Louisiana shrimp invasion. Badger Maltus model (∗). Maltus model with injection or withdrawal (∗). Verhulst model (∗). Richards model (∗). Lotka-Volterra model (∗).
Tumor growth model. Gompertz model (melanoma) (∗). Simulation of a radiotherapy. Bertanlaffy model(∗): spherical approximation of tumor cells. Fister-Panetta model (∗); Skipper's hypothesis (murine leukemia); Holford-Shiner hypothesis (enzyme deficiency); Norton-Simon hypothesis (Hodgkin's lymphomas - acute leukemia of fibroblasts). Torqui model derivation of the dynamical system.
In the first part of the course, the fundamental tools of analytical calculation are considered, as well as several examples which emphasize their application. The second part of the course is devoted to the analysis of some differential models applied in oncology and epidemiology.
Sequences. Arithmetic sequence (example: growth of a tumor mass). Geometric sequence (example: cell mitosis). Sum of the geometric sequence (example: bacterial proliferation) - (example: predict hospitalization time, note the probability of recovery and vice versa). Sequences: definition and properties. Limits and classification (several examples). The number of Napier interpreted as a limit of a suitable sequence. Theorem: if a sequence is convergent, then it is bounded. Theorem: if a sequence is monotone, then it is not indeterminate. The Fibonacci sequence (∗)
Limits. Domain. Adherent points. Exercises and examples. Continuous functions. Definition and properties. Weierstrass theorem. Examples: the body mass index, the skeletal mass index, the model of the artificial neuron. Exercises and examples.
Derivatives. Geometric definition and interpretation. Left and right derivative. Classification of non-differentiable points: angular points, cusps, inflections with vertical tangents. Derivative of the product, of the quotient and of the compound function. Increasing and decreasing d functions: the link with the value of the derivative. Theorem: if a function is differentiable, then it is continuous. Higher-order derivatives. Concavity and convexity. Points of inflection. Taylor's polynomial: definition and application to the calculation of limits. Exercises and examples.
Integrals. Application and graphic interpretation. Example: absorption of a medicine during chemotherapy. Definition of primitive function. Fundamental theorem of integral calculus. Integrals of elementary functions. Integrals by parts. Integrals by substitution. Integrals of fractioned polynomial functions (case ∆> 0). Examples: the Poiseuille model to determine the speed of blood and the flow rate of an artery (∗). Exercises.
ab.
Curve. Norm of a vector. Tangent vector. Closed curves. Length of a curve. Exercises and examples.
First order ordinary differential equations. Autonomous equations. Equilibrium points and their classification.
Population dynamics model. Maltus model (∗). Example: the Louisiana shrimp invasion. Badger Maltus model (∗). Maltus model with injection or withdrawal (∗). Verhulst model (∗). Richards model (∗). Lotka-Volterra model (∗).
Tumor growth model. Gompertz model (melanoma) (∗). Simulation of a radiotherapy. Bertanlaffy model(∗): spherical approximation of tumor cells. Fister-Panetta model (∗); Skipper's hypothesis (murine leukemia); Holford-Shiner hypothesis (enzyme deficiency); Norton-Simon hypothesis (Hodgkin's lymphomas - acute leukemia of fibroblasts). Torqui model derivation of the dynamical system.
Course Language
Italian
Degrees
Degrees
MEDICINE AND SURGERY (IN ENGLISH LANGUAGE)
Single-cycle Master’s Degree (6 Years)
6 years
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People
People
Teaching staff
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