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COURSE: MATHEMATICAL METHODS FOR BIOLOGICAL AND MEDICAL ENGINEERING
TRANSLATION: METODOS MATEMATICOS PARA LA INGENIERIA BIOLOGICA Y MEDICA
COURSE CODE: IBM3103
CREDITS: 10 UC / 6 SCT
MODULES: 03
PREREQUISITE COURSES: --
CONNECTOR:
RESTRICTIONS: Curriculum 040201 o 040301 o 040401 o 140601 o 148101 o 120401 o 120501 o 120801 0 180001 0 181001
CHARACTER: Minimum
TYPE: Lecture
QUALIFICATION: Standard
DISCIPLINE: Engineering, Biology and Medicine


me. DESCRIPTION

The aim of this graduate-level course is to teach students the mathematical concepts and computational tools that are commonly used in modeling and simulating physical, biological, and medical phenomena. The course will focus on using numerical simulations and experiments to illustrate teh application of teh mathematical concepts being presented. Intuition, algorithms and computational issues will be emphasized over abstract results.
The main objective of teh course is for students to learn the essential mathematical and computational tools needed to undertake research in biological and medical engineering, and to be able to develop new mathematical models or computational methods tailored to their specific research areas in biological and medical engineering.


II. OBJECTIVES

1. Identify the elementary mathematical models underlying some common biological or medical phenomena.
2. Implement teh essential numerical methods required to perform a numerical simulation of an elementary mathematical model of biological or medical phenomena.
3. Evaluate the benefits and drawbacks of mathematical models of biological or medical phenomena through numerical simulations.
4. Propose new mathematical models or modifications of common mathematical models for biological or medical phenomena.


III. CONTENTS

1. Introduction to Scientific Computing
1.1. Introduction to the Python language for Scientific Computing
1.2. Elements of Scientific Computing.
1.3. Elements of Numerical Linear Algebra.
2. Ordinary Differential Equations (ODEs)
2.1. Time-varying biological and medical phenomena and the intuition behind an ODE
2.2. Definition, geometric intuition, existence and uniqueness of solutions
2.3. Equilibria and stability
2.4. Linear systems of ODEs
2.5. Numerical solution to ODEs
3. Applied Harmonic Analysis
3.1. The importance of acquiring biological and medical signals
3.2. Linear Time Invariant (LTI) systems as a modeling tool
3.3. Action of an LTI system on complex exponentials, and the Fourier transform
3.4. The transfer function of an LTI system, harmonic analysis and synthesis
3.5. The discrete Fourier transform (DFT) and the fast Fourier transform (FFT)
3.6. Sampling of signals, bandlimited signals, and the Shannon-Whittaker theorem
3.7. Elements of Time Series analysis
4. Optimization
4.1. Biological and medical phenomena as an energy-minimizing process
4.2. Unconstrained optimization: necessary and sufficient conditions and first- and second-order optimality conditions
4.3. Elements of line search and trust-region algorithms
4.4. Linearly constrained problems: necessary and sufficient conditions and the Karush-Kuhn-Tucker (KKT) conditions.
4.5. Elements of non-linearly constrained optimization and numerical solvers.
5. Probabilistic Models
5.1. The random nature of some biological and medical phenomena
5.2. Elements of probability theory, probability measure, random variables, and probability distributions.
5.3. Expectation and variance, statistical independence, covariance and correlation, and conditional expectation
5.4. The Law of Large Numbers (LLN) and the Central Limit Theorem (CLT)
5.5. Random walks
5.6. Markov chains
5.7. Renewal processes


IV. METHODOLOGY

Lectures
Computational workshops


V. EVALUATION

Homework (60%)
Project (40%)


VI. BIBLIOGRAPHY
Mandatory:
? Ascher, U. & Greif, C. (2011) A First Course in Numerical Methods. Society for Applied and Computational Mathematics: PA.

Complementary:
? Durret, R. (2010) Probability: Theory and Examples, 4th Edition. Cambridge University Press: Cambridge.
? Holmes, M. H. (2016) Introduction to Scientific Computing and Data Analysis. Springer International Publishing: Switzerland.
? Ingalls, B. (2013) Mathematical Modelling in Systems Biology: An Introduction, MIT Press: MA.
? Strang, G. (2007) Computational Science and Engineering, Wellesley-Cambridge Press: MA.
? Tveito, A. et al. (2010) Elements of Scientific Computing, Springer-Verlag, Berlin.