Typically, when mechanics is applied to biology, it is named biomechanics; we use cardio-mechanics to mean that specific branch of mechanics which has been successfully applied to the analysis and investigation of the cardiovascular system, whose center is the heart pump. Discuss the contribution of cardio-mechanics to clinics is beyond our aims (refer to [7] for extended references); we only aim to shortly discuss two different approaches of mechanics to clinics, with specific reference to the heart pump.

The first approach is based on the continuum analysis of the heart, aimed to model the cardiac activity. It starts from the construction of anatomical models of the heart and of constitutive model describing the passive and active material response of cardiac tissues. Electromechanical interaction are sometimes accounted for, if one has interest to also investigate cardiac electrophysiology. Typically, these models are implemented within a finite element code. Critical points are the constitutive models of the tissue, being cardiac tissue highly non homogeneous and contracting, due to an electrophysiological stimulus, and anatomical data about muscle architecture, which strongly influences the mechanical performances of the heart [14, 15, 27]. Once the model is complete, specific cardiac diseases can be included within the modeling, and the consequences on the heart activity studied. Typical examples are given by the investigations on the heart remodeling due to left anterior descending artery occlusion, to ventricular hypertrophy, to aortic stenosis, well discussed in literature [10, 18, 24]. Likewise, the model can be improved to account consequences of an infarct in different places of the left ventricular walls, with the aim of studying the ability of the pump to get on or not its work [13].

Here, we are more interested in the second approach, which starts from an analysis of real data extracted from the heart through appropriate tools such as Magnetic Risonance Image (MRI) and 3DSTE. Of course, such data are only concerning heart kinematics, and say nothing about stresses within heart walls. However, as it is recently shown, an accurate and careful analysis of real data allows to deeply investigate on heart. Let us note that the interest in in-vivo myocardial deformation dates back to ; in [26], the normal in-vivo three-dimensional finite strains were studied in dogs, through the application of appropriate markers whose coordinated were followed along the cardiac cycle through high-speed biplane cineradiography. Of course, the analysis was highly invasive. The recent techniques of visualization realized by 3DSTE make possible follow the coordinates of natural markers, automatically identified by the device, appropriately supported by an operator.

A typical example of the outcomes of a deep analysis on 3DSTE real data comes from PSLs, which are the core of this contribution. In mechanics, it is well-known that stresses and strains within a body are limited above and below by their principal counterparts; this allows for the discussion and verification of the mechanical state of that body. Moreover, the principal stress and strain lines (which are the same only when special symmetry conditions are verified) determine the directions where the largest strains and/or stresses are to be expected. Due to these characteristics, the mechanics of fiber-reinforced bodies are often based on the detection of the principal strain lines and, wherever needed, fiber architecture is conceived in order to make the fiber lines coincide with the PSLs. Fibers make a tissue highly anisotropic; hence, principal strain and stress lines may be distinct. Whereas principal strains can be measured starting with the analysis of tissue motion, being only dependent on the three-dimensional strain state of the tissue, principal stresses can only be inferred. Thus, the PSL have a predominant role where the analysis of the mechanics of a body is concerned, and can reveal which are the lines where largest strains are expected, and how they change when diseases occur.

Key point is the evaluation at any place within the body of the nonlinear strain tensor , whose eigenpairs (eigenvalues-eigenvectors) deliver principal strains and PSLs, respectively. Fixed a body, identified with the region of the three-dimensional Euclidean space it occupies at a time t _o denoted as reference configuration of the body, we are interested in following the motion of the body at any time , with the time interval identifying the duration of a human cardiac cycle (hence, different from subject to subject, as it is discussed later). The displacement field , that is a map from , delivers at any time and for any point the position of that point at that time: . Strains are related to displacement gradients within the body; precisely, it can be shown as, introduced the deformation gradient , the nonlinear Cauchy-Green strain tensor is
being the identity tensor in . In general, is a three-dimensional tensor, describing the strain state at any point y and time t of the body. If there is within the body a distinguished surface , whose unit normal field is described by the unit vector field , the corresponding surface strain tensor can be obtained through a preliminary projection of onto that surface. The projector , leads to the following definition:
It is expected that will represent a plane strain state, hence, that it will have a zero eigenvalue corresponding to the eigenvector . The primary strain lines on the surface will be the streamlines of the eigenvector , which lies on the surface and corresponds to the smallest non-zero eigenvalues; the secondary strain lines are the streamlines corresponding to the eigenvector . Of course, when the strain tensor is ab initio evaluated from surface deformation gradients , it naturally arises as a plane tensor.

Speckle tracking echocardiography (STE) is an application of pattern-matching technology to ultrasound cine data and is based on the tracking of the ‘speckles’ in a 2D plane or in a 3D volume (2DSTE and 3DSTE, respectively). Speckles are disturbances in ultrasounds caused by reflections in the ultrasound beam: each structure in the body has a unique speckle pattern that moves with tissue (Fig. 2, left panel). A square or cubic template image is created using a local myocardial region in the starting frame of the image data. The size of the template image is around 1 cm² in 2D or 1 cm³ in 3D. In the successive frame, the algorithm identifies the local speckle pattern that most closely matches the template (see [29] for further details). A displacement vector is created using the location of the template and the matching image in the subsequent frame. Multiple templates can be used to observe displacements of the entire myocardium. By using hundreds of these samples in a single image, it is possible to provide regional information on the displacement of the LV walls, and thus, other parameters such as strain, rotation, twist and torsion can be derived.

Echocardiographic examinations were performed with an Aplio-Artida ultrasound system (Toshiba Medical Systems Co, Tochigi, Japan). Full-volume ECG-gated 3D data sets were acquired from apical positions using a 1–4 MHz 3D matrix array transducer to visualize the entire LV in a volumetric image. To obtain these 3D data sets, four or six sectors were scanned from consecutive cardiac cycles and combined to provide a larger pyramidal volume covering the entire LV. The final LV geometry was reconstructed by starting from a set of 6 homologous landmarks (see Fig. 2), manually detected by the operator for all subjects under study. The manual detection for a given set of landmarks is crucial because it allows recording spatial coordinates in perfectly comparable anatomical structures of different subjects (following a homology principle). The results of our 3DSTE system is a time-sequence of shapes, each constituted by 1297 landmarks, assumed to be homologous, for both the epicardial and endocardial surfaces, positioned along 36 horizontal circles, each comprised of 36 landmarks, plus the apex (see Fig. 3).