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Mathematical modeling applied to the left ventricle of heart

Background: How can mathematics help us to understand the mechanism of the cardiac motion? The best known approach is to take a mathematical model of the fibered structure, insert it into a more-or-less complex model of cardiac architecture, and then study the resulting fibers of activation that propagate through the myocardium. In our paper, we have attempted to create a novel software capable of demonstrate left ventricular (LV) model in normal hearts. Method: Echocardiography was performed on 70 healthy volunteers. Data evaluated included: velocity (radial, longitudinal, rotational and vector point), displacement (longitudinal and rotational), strain rate (longitudinal and circumferential) and strain (radial, longitudinal and circumferential) of all 16 LV myocardial segments. Using these data, force vectors of myocardial samples were estimated by MATLAB software, interfaced in the echocardiograph system. Dynamic orientation contraction (through the cardiac cycle) of every individual myocardial fiber could be created by adding together the sequential steps of the multiple fragmented sectors of that fiber. This way we attempted to mechanically illustrate the global LV model. Result: Our study shows that in normal cases myocardial fibers initiate from the posterior basal region of the heart, continues through the LV free wall, reaches the septum, loops around the apex, ascends, and ends at the superior-anterior edge of LV. Conclusion: We were able to define the whole LV myocardial model mathematically, by MATLAB software in normal subjects. This will enable physicians to diagnose and follow up many cardiac diseases when this software is interfaced within echocardiographic machines.


Background:
In a great many problems the microscopic structure of matter can be disregarded and a biological body replaced by a continuous mathematical model whose geometrical points are identified with material points of the body. The learning of such models is in the province of the mechanics of continuous media, which covers a vast range of problems in elasticity, hemodynamics, aerodynamics, plasticity, and electrodynamics. When the relative position of points in a continuous biological body is altered, we say that the body is strained. The change in the relative position of a point is a deformation, and the study of deformations is the region of the analysis of strain. In this paper, the left ventricular myocardium is taken as a biological elastic body and modeled by mathematical/imaging techniques.
Imaging cardiology is largely based on the evaluation of the tissue mechanics by mean of medical imaging techniques like echocardiography that has the advantage of a high temporal resolution and permits to differentiate the cardiac phases and produce an enormous amount of data about the cardiac structure and function. These data are mostly used for visualization purposes, although physically-based models would be required in order to improve their interpretation and eventually support diagnostic and therapeutic practice. A novel mathematical Cardiac modeling represents a challenge in this sense. The visualization of such this modeling in the heart chambers particularly in the left ventricle is, in fact, still of little usage in diagnosis.
Left ventricular twist deformation is due to the complex helical myocardial fiber architecture.
This wringing motion of the heart is the result of the clockwise rotation of the base and the counterclockwise rotation of the apex and plays an important role in ventricular performance.
Several invasive and non-invasive techniques have been used to describe and quantify this cardiac motion. Recently developed, two-dimensional speckle-tracking echocardiography (2DSTE) has proved to be a simple non-invasive technique to quantify the LV twist mechanism.
A novel technique based on three-dimensional speckle-tracking echocardiography (3DSTE) has been developed to reduce some of the shortcomings of 2DSTE.
We present a new mathematical tool by introducing the notions of 2D and 3D strains which are two by two and three by three matrices. These matrices are computable from echocardiogram data. Having used these data and some new mathematical techniques and formulas we obtained can be achieved to test the hypothesis that myofibres move on helical bands which begin from the Septal, loop around the Apex and then go to the Anterior. The results we have obtained so far confirm this hypothesis.  wall thickening , which is the effect of tangential shortening of the myocardium. Therefore, these two dependent quantities can be used as the quantitative characteristics for the local contribution of the left ventricle myocardium to global heart function. The spatial distribution of regions that contribute the most to cardiac function acts as a functional macrostructure for the myocardium. The mere existence of such a distinct structure and the knowledge of its normal morphology will facilitate a more effective modeling of left ventricle function.

Figure 2
The force vector field on a parasternal long-axis view in MATLAB software.

Result:
Assembled from data shown in (Figure 2), we can illustrate a segment of the left ventricle with tracking results from all areas through the heart wall from endocardium to epicardium ( Figure 3).  Finally, we are able to define the whole LV myocardial model mathematically, by MATLAB software in normal subjects ( Figure 5). it just works with a force field that happens to be zero everywhere except in the immediate of the vorticity region. This approach can be used for instructional purposes and diagnosis of heart ailments.

Conclusion:
In this paper, the relation of the form and function of the left ventricle (myocardium) has been studied through investigating the spatial distribution of its regional function. In this approach, we were able to define the whole LV myocardial model mathematically, by MATLAB software in normal subjects. This will enable physicians to diagnose and follow up many cardiac diseases when this software is interfaced within echocardiographic machines.