Magnetic Resonance Imaging

Table of Content

1. Introduction3

2. Bloch equation under RF perturbations10

2.1. Bloch equation in the rotating frame10

2.2. Solving Bloch equation without RF perturbation13

2.3. Overview of the method for solving Bloch equation16

3. Analysis of the behaviour of the sequence 18

3.1. Mathematical framework19

3.2. Existence and uniqueness results for the sequence 21

3.3. Asymptotic behaviour of the sequence 24

4. Numerical approximation of the sequence 26

4.1. The algorithm26

4.2. An overview of accuracy and efficiency of the method30

4.3. Description of the magnetisation vector evolution33

4.4. matlab program35

5. Conclusion36

References36

Appendix41

1. Introduction

Magnetic Resonance Imaging (MRI) is a method that generates exquisite images of the soft tissue anatomy of the human body. The principle of MRI is to record the variations of the nuclear magnetisation of the biological tissues by using different kinds of magnetic fields. A static magnetic field is used to generate a macroscopic nuclear magnetisation in the body to be imaged; typically has a strength around 1 T. This equilibrium magnetisation is aligned with . To shift the magnetisation vector from its equilibrium position, a radio-frequency magnetic field (RF field) is applied at a very characteristic pulsation determined by the Larmor relation (1)

where ? is a constant called the gyro-magnetic ratio ( by Tesla for proton). In MRI this phenomenon is known as the resonance process. The position of the magnetisation vector at the end of the resonance process is determined by the duration of the RF field. Typically this duration is chosen so that the angle between the initial position and the resulting one is p/2 or p. When the RF field is stopped, the magnetisation tends to return to its equilibrium position in a process called the relaxation. During the relaxation process the magnetisation creates an induced electric signal in an antenna set in a plane perpendicular to . This signal is acquired for subsequent processing and gives rise to the image. Moreover, magnetic field gradients (static magnetic field aligned with with a linear varying intensity in one fixed direction) are applied during the imaging process to set up a spatial correspondence between position in the body and position in the image through a frequency encoding of the MRI signal.

Any perturbation of the magnetic fields involved in MRI can disturb the imaging process(Balac 2001).

The result is a local deformation of the image (called an artifact) that may render the image inaccurate and useless for medical diagnostics. Sources of perturbation of the magnetic fields are various and can be classified in two groups: the one connected to the static magnetic field and the one connected to the RF field. Moreover one can distinguish between defects that are properties of the MRI device (e.g. nonuniformity of the magnetic fields over the whole imaging area) and perturbations of the magnetic fields due to the patient himself. Since they are fixed properties of the MRI device, the first one can be handled with efficiency, either by the use of additional hardware components or by taking into account their effect in the reconstruction algorithm. It is much more difficult to deal ...