# Master

Lectures for master students

### Tomographical Methods in Medicine

#### Lecturer

Prof. L. Klinkenbusch

#### Target Group

Undergraduates and Graduates in Engineering Sciences and others

#### Contents

**Introduction:**

- Overview and Classification of Imaging Processes
- History of Imaging Processes

**Impedance-Tomography**

- Principle of Impedance-Tomography
- Fundamental Equations
- Electrodes, Current Source, Measure Amplifier
- Image Reconstruction

**Computertomography (CT)**

- Principle of CT
- Fundamentals of Signal Processing (more dimensional Fourier Transformation, Convolution, Correlation, Modulation-Transfer-Function (MTF), Sampling Theorem, Noise)
- Digital Image Processing
- Radon-Transformation
- X-ray Detectors
- CT-Reconstruction Method
- CT-Artefacts

**Magnetic Resonance Tomography (MRT)**

- Principle of MRT
- Physical Fundamental Principles (magnetic Spin Top, Precession, Nuclear Spin (MRI), Spin-Lattice and Spin-Spin-Relaxation, Bloch's Equations, Spin-Echoes)
- Tomographic Methods (Selective Excitation, Frequency- and Phase Encoding, Scanning Method)
- Structure of a MR Scanner
- Functional MRT (fMRT)

#### Literature

... will be given during the lecture.

#### Note

For the treatment of the numerical exercises Laptops will be provided.

### Fields and Waves in Biological Systems

#### Lecturer

Prof. L. Klinkenbusch

#### Target Group

Undergraduates and Graduates in Engineering Sciences and others

#### Contents

**Fundamentals**(Maxwell's equations, Diffusion, Nernst-Planck equation)**Cell models**(Diffusion and ionic currents in the membrane, ionic membrane-channels, natrium-potassium pump, Nernst potentials, equivalent electric circuits, Hodgkin-Huxley model for excitable cell membranes)**Neuronal cells**(Function, cable equation, wave propagation along axons)**Electromagnetics and Heart**(Function and anatomy, electrical system of the heart, electrocardiography, magnetocardiography, pacemaker)**Electromagnetics and Brain**(Function and anatomy, electroencephalography, magnetoencephalography, deep brain stimulation)**Electromagnetic waves in biological systems**(Light perception, microwave effects in biological tissue, hyperthermia therapy, magnetic resonance tomography)

#### Literature

... will be given during the lecture.

#### Note

For the treatment of the numerical exercises Laptops will be provided.

### Computational Electromagnetics

#### Lecturer

Prof. L. Klinkenbusch

#### Target Group

Undergraduates and Graduates in Engineering Sciences and others

#### Description

Most practical problems in electromagnetics, e.g. the radiation
caused by a mobile phone near a human head or shielding of an electronic
circuit by a slotted metallic box, cannot be solved purely by means of
analytical methods. In many of such cases, numerical methods in
electromagnetics can be applied in an efficient way to come to a
satisfactory solution.

The course deals with two of the most successful numerical methods in electromagnetics: The first one is the *Finite-Difference Time-Domain*
(FDTD) method, which represents a finite difference formulation
perfectly adapted to the time domain Maxwell's equations. A functioning
FORTRAN source code will be explained and delivered to the students.

As a second topic the *Finite-Element Method*
(FEM) is treated, which is an excellent tool for solving full wave
frequency-domain problems especially since the introduction of the *vector edge elements.*

#### Contents

#### **Maxwell's equations:**

Integral Form, Differential Form, Properties of Matter: Non-Linear, Inhomogenenous, Anisotropic, Dispersive

#### **Finite-Difference Method:**

1-Dimensional Example: Wave Equation, The Finite-Difference Time-Domain Method (FDTD): Stability, Numerical Dispersion, Simulation of Scattering Problems, PML: Perfectly Matched Layer, Efficent Modelling of Special Structures (Sub-celling, Sub-gridding), Numerical treatment of dispersive media, Discussion of a 3-D FDTD FORTRAN Source Code

Example: Demonstration of the FDTD-method

#### **Finite-Element Method:**

1-Dimensional Example: Poisson Equation (Definition of Finite Elements, Galerkin Method, Introduction into Variational Methods, Suitable Functionals, Ritz Method), 2-Dimensional Problems (Discretization, Triangular Scalar and Vector Edge Elements, Suitable Functionals, Typical Problems), 3-Dimensional Problems (Discretization, Tetrahedral Scalar and Vector Edge Elements, Suitable Functionals, Typical Problems)

#### **Near-To-Far-Field-Transformation (NFT):**

Free-space Green's Dyadic Function, Efficient NFT in the Frequency Domain and in the Time Domain

**Iterative solution of a system of linear equations: **

Splitting methods, Conjugate-gradient methods

#### Manuscript

A printed manuscript including a list of references will be distributed during the course.

#### Note

For the solution of numerical exercises Laptops will be provided.

### Mathematical Methods in Field Theory

#### Lecturer

Prof. Dr.-Ing. L. Klinkenbusch

#### Target Group

Undergraduates and Graduates in Engineering Sciences and others

#### Description

The course deals with those mathematical techniques, which are helpful and essential for understanding and solving scalar (e.g. acoustical) and vector (e.g. electromagnetic) field problems. The methods may be important also for the numerical treatment of field problems, which are mosty given by means of differential and/or integral equations. Because of its strong relationship to concepts in system theory, the course may be interesting also for students with other main topics, e.g. communications engineering.

#### Contents

**Mathematical Fundamentals:**

Dirac's "δ-Function", δ-convergent sequences, orthonormalized function systems, Sturm-Liouville theory (of ordinary second-order differential equations)

#### **Green's Functions:**

Definition, properties, representations, solution of boundary-value problems, Boundary Value Problem of the 1st and of the 2nd kind.

#### **Helmholtz equation and Laplace equation:**

Separation in plane-polar coordinates, separation in spherical coordinates, Green's function of the free space.

#### **Multipole analysis of electromagnetic fields:**

Maxwell's
equations, Helmholtz equation, spherical- multipole analysis, multipole
expansion of a plane electromagnetic wave, diffraction by a sphere
(Theory of *Gustav Mie*)

#### Manuscript/ Literature

will be provided during the course.