magnetic resonance (MR), its application to image generation, the introduction of the tagging process and finally the analysis of displacements and deformations of organs of the human body. In this paper, a comprehensive presentation of the overall subject is developed. 7.2 Fundamentals of MRI Images Generation The MRI method to visualize the internal structure of human organs is not unlike the usual methods based on the utilization of visible light to get information on displacements and their derivatives in transparent media. Both methodologies are based on the electromagnetic spectrum: that is, they have the same foundations, solution of the Maxwell equation of propagation of the electromagnetic fields. In the case of optical methods with visible light, the electric vector Eis the vector utilized to obtain displacement information and the presence of the magnetic vector B is ignored in most cases. In the MRI methodology, the vector Bplays the role of the illuminating vector since it can penetrate the human body different tissues and generate signals that can reflect the state of magnetization of a voxel located inside a volume. The electrical and the magnetic fields can be represented by a mathematical entity, a phasor: B) x; t ð Þ¼B x; t ð ÞeiQx;t ð Þ ð7:1Þ Equation (7.1) represents a propagating wavefront in space and time. B) x; t ð Þis the field amplitude at a given point and at a given time in the 3-D space, xis the position vector, B(x, t) is the amplitude of the phasor, t is the time, Q(x, t) is an angular variable (phase). The phase Q(x, t) can be expressed as: Qx; t ð Þ¼k x ot þa ð7:2Þ where: k is the wave vector, x is the position vector, ois the angular frequency, a and initial phase that depends on the selected initial conditions. In the case of the visible spectrum, the angular frequency term is ignored because the frequency of the signal is so high that in general the times required for signal sensing are very large to detect this part of the signal. Equation (7.1) represents the signal that needs to be detected. Hence, we need an illumination field so to speak: in this case, it is a strong magnetic field where the object to be studied is immersed. This magnetic field propagates plane wavefronts; this implies a very uniform field in the region of observation. The interaction of the magnetic field with human tissues depends on a phenomenon, the magnetic resonance of the protons in the nuclei of the hydrogen contained in the tissues. The response depends on the density of the present nuclei and hence will depend on the chemical composition: this makes possible to separate different types of tissues. To collect information, it is necessary to detect weak changes in the uniform magnetic field existing in the volume of the illuminated body. To achieve this objective, signals must be sent to interrogate the state of magnetization at given points of the analyzed volume. These signals generate electromagnetic waves that upon recording and processing provide local information of the magnetic state of the volume. In order to get information concerning human tissues, we need to produce an image that provides features of the observed tissues as levels of gray as it is done usually in conventional images produced with a camera and a lens. Except that in this case there is not going to be a lens or an array sensor of an electronic camera to record information in pixels and voxels. In the present case, the illumination is a magnetic field and the distinctive features come from small changes experienced by the magnetic field that have to be sensed and transformed in levels of gray at a given location. This last operation is the interface between the system generating and processing the information and the human observer that has to make inferences on the obtained information. Furthermore, if we want to measure displacements we must create a reference system embedded in the tissues and that experiences the same changes in time as the tissue due to the applied deformation between two time intervals. Let us consider these two aspects separately beginning with the generation of an image of the organ under observation. 7.3 The Image Generation We adopt a Cartesian system of coordinates (X, Y, Z) and to simplify the analysis, as it is usual in MRI literature, in-plane images are given by the coordinates (X, Y) while depth information is given by the coordinate Z (see Fig. 7.1). The information is given as a level of gray in voxels that cover the analyzed volume. It is necessary to locate the voxel position given by the coordinates (X, Y, Z). Therefore, we need to perform separate measurements that allow us to locate the 64 C.A. Sciammarella et al.
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