70 C.A. Sciammarella and L. Lamberti 10.5 Computation of the Derivatives of the Displacements In [5], it is mentioned that there are four fundamental tensors that can be utilized in Continuum Mechanics to define different forms of strain tensors and the corresponding definitions are given. The tensors are denoted in the literature as [F], [G], the inverse [F] 1 and [J]. It is necessary to address a problem that occurs when it is not feasible to get all the system of tagging planes indicated in Figs. 10.1 and 10.4. ŒJ D 2 6 6 6 6 6 6 6 4 @u1 @x1 @u1 @x2 @u1 @x3 @u2 @x1 @u2 @x2 @u2 @x3 @u3 @x1 @u3 @x2 @u3 @x3 3 7 7 7 7 7 7 7 5 (10.30) Equation (10.30) gives the derivatives of the components of the displacement vector in Eulerian coordinates with respect to the Eulerian coordinates of the point. In [5], it is proven that these derivatives are related by the continuity of the medium deformation through relationships, that in the notation of Fig. 10.3c, can be written for the plane x1x2, @u2 @x1 D 1C @u1 @x1 tan™12 (10.31) A similar equation can be obtained for the plane x1-x3, @u3 @x1 D 1C @u1 @x1 tan™13 (10.32) For x2x3 the equation is, @u3 @x2 D 1C @u2 @x2 tan™23 (10.33) All these derivatives can be computed utilizing the techniques outlined in [2] by differentiation in the frequency plane [6] without unwrapping the phase information. 10.6 Determination of Displacements and Strains of the Heart As an illustration of the proposed approach to 3D displacement and strain determination, images generated by the Magnetic Image Resonance (MRI) method are utilized. MRI images are produced by “illuminating” the observed sample with an intense magnetic field. To this magnetic field, interrogating signals are applied in a temporal sequence that is required to obtain the magnetization information of the different voxels inside the observed object volume. What information is collected? Position of the voxels in the 3D space and the state of magnetization of the voxels. The state of magnetization depends on the chemical composition of the observed voxels thus though MRI it is possible to obtain information on the 3D spatial composition of the observed object. The magnetic resonance law depends on the changing spinning properties of protons in the nuclei of molecules and can only be observed in chemical elements that have different isotopes. An important property of magnetic resonance is the process of tagging the observed volume with planes that modulate periodically the intensity of the resulting image. The tagging planes are attached to the observed volume elements and create the system of reference illustrated in Figs. 10.1 and 10.2. The tagging process is a time-dependent phenomenon and fades away with time, thus limitations arise in the observation process of deformations. One should also realize that the functions of a camera and a sensor recorder in visible light are performed by hardware and software that generates an image output in levels of gray needed by the human observed to understand the content of the output of the MRI system.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==