## MPHY3893 Mathematical Methods in Medical Physics

### Contents

Course information Purpose Aims and Objectives Teaching and Exams |
Prerequisites Description Brief syllabus Recommended texts |

### Course information

This course also has a Moodle page.

Unit value Year of study Term Course organiser Second examiner |
0.5 3 Term 1 Prof Jem Hebden Dr Tom Vercauteren |

### Purpose

The component provides an essential grounding in mathematical methods for all students enrolled in the Intercalated B.Sc. in Medical Physics & Bioengineering. A significant proportion of the course covers material that in previous years was covered in group tutorials, which proved to be a very important element of the intercalated programme. The component will provide students with sufficient knowledge in mathematics in order that they are at no significant disadvantage compared to 3rd and 4th year physics and electrical engineering students who also take the other courses which form the intercalated degree.

### Aims and Objectives

The aim is to provide intercalating medical students with the most fundamental skills in mathematics and statistics which are essential for a thorough understanding of the most important physical principles and techniques in medical physics. The specific objectives are to introduce students to a series of mathematical topics necessary for their comprehension of other components of the intercalated programme, and to engender a proficiency in statistics and basic analytical techniques.

### Teaching and exams

Teaching will consist of:

- Lectures, 30 hours.
- Seminars/problem classes, 6 hours.
- Required written work 4 coursework sheets (problem sheets).
- Private reading 45 hours.

The assessment will consist of:

- 1 Unseen written examination (2.5 hours) worth 80% of the total course mark.
- 3 Problem Sheets completed during term-time worth 20% of the total course mark.

### Prerequisites

This module is for Intercalating Medical Physics students only. An A Level in Mathematics (or its equivalent) is required, although some exceptions are sometime made where students are prepared to undertake additional directed learning during the summer prior to starting the course.

**Specific knowledge
assumed:**

**Mathematics**: Familiarity with
manipulation of equations, basic calculus (differentiation and integration),
exponentials. Ability to sketch graphs.

**Physics: **None.

**Engineering:** None

**Biology: **None.

**Other: **None.

### Description

The course, taught during Term 1, is designed specifically for intercalated students in order that they gain a basic familiarity with various mathematical techniques and notation which form part of their other lecture courses. It is intended to be sensitive to the current mathematical skills and limited previous experience of medical students, including the minority without A'level Mathematics (although such students are expected to undertake some directed background reading during the summer before starting the intercalated degree). The lectures emphasise the need for an intuitive understanding of specific methods and their application rather than a rigorous training in mathematics. The course will also enable students to develop greater skills to process and manipulate data, which forms a significant part of most student projects.

### Brief Syllabus

**Part A. Mathematics and applications**

**Coordinate Geometry**. Coordinate systems, angles, solid angles.**Vectors**. Vectors, Cartesian components, multiplication of vectors. Applications in Medical Physics.**Differentiation and integration**. Limits, derivatives, differentials, partial derivatives, total differentials and chain rule, integration.**Directional derivatives**. The directional derivative. The Del operator.**Exponentials and logarithms**. The exponential function, logarithms, natural logarithms, changing the base of a logarithm.**Series and the Taylor expansion**. Sequences and series, convergence of infinite series, the Taylor series expansion, the binomial theorem.**Partial fractions**. Completing the square. Rules for decomposing complicated rational functions into a sum of simple rational functions.**Complex numbers**. Imaginary and complex numbers, Argand diagram, Euler's formula. Application of complex numbers in electronics.**The Fourier transform**. A non-mathematical description, Fourier series, The Fourier transform, Delta functions.**Convolution and deconvolution**. Convolution, Deconvolution.**Sampling and aliasing**. The Sampling theorem, Aliasing.**Differential equations**. Classification, First order equations, Second order equations. Examples.**Matrices**. Definitions and applications.**Mathematics of computed tomography**. Central slice theorem. Filtered backprojection.

**Part B. Medical Statistics.**

By the end of the course, you will

- know about the
**Gaussian**,**Poisson**and**Binomial**distributions - understand the concept of
**statistical significance** - be able to apply the
**t-test** - be able to apply the
**chi-squared test** - be able to statistically analyse
**photon counting experiments** - be able to characterise the
**precision**of a measurement - understand how
**measurement errors**propagate

### Core Texts

**Mathematics and applications**

- K. A. Stroud and Dexter J. Booth.
*Engineering Mathematics*. Palgrave. 6th Edition 2007 [ISBN 10-1-4039-4246-3] (JH) - K. A. Stroud and Dexter J. Booth.
*Advanced Engineering Mathematics*. Palgrave. 4th Edition 2003 [ISBN 1-4039-0312-3] (JH) - Alan Jeffrey.
*Essentials of Engineering Mathematics*. Chapman & Hall. 1992 [ISBN 0-412-39680-7] (JH)

**Statistics**

- T D V Swinscow (1997) Statistics at square one
- R J Barlow (1989) Statistics
*John Wiley & Sons* - G F Knoll (1992) Radiation Detection and Measurement
*John Wiley & Sons* - DM Lane (2003) Hyperstat online