Numerical differentiation is an ill-posed problem. This article demonstrates that the application of the regularization theory together with the methods of projections on convex sets of constraints improve the accuracy of the derivatives calculation. Simulation results are presented and the applications of the proposed method to edge detection are discussed.>

Detection of the left ventricular (LV) endocardial (inner) and epicardial (outer) boundaries in cardiac images, provided by fast computer tomography (cine CT), magnetic resonance (MR), or ultrasound (echocardiography), is addressed. The automatic detection of the LV boundaries is difficult due to background noise, poor contrast, and often unclear differentiation of the tissue characteristics of the ventricles, papillary muscles, and surrounding tissues. An approach to the automatic ventricular boundary detection that employs set-theoretic techniques, and is based on incorporating a priori knowledge of the heart geometry, its brightness, spatial structure, and temporal dynamics into the boundaries detection algorithm is presented. Available knowledge is interpreted as constraint sets in the functional space, and the consistent boundaries are considered to belong to the intersection of all the introduced sets, thus satisfying the a priori information. An algorithm is also suggested for the simultaneous detection of the endocardial and epicardial boundaries of the LV. The procedure is demonstrated using cine CT images of the human heart.

An integrated approach to modeling the cardiac system is described and related to physiological measurements and other model approaches. The model is composed of a number of parts that will ultimately be used as a tool to study the various functions of the cardiac system. The 3-D geometry of the LV is reconstructed by using a helical coordinate system which provides a quantitative tool for shape description and analysis. Analysis of the regular geometry may be achieved using either the analytical helical system or a finite element technique. A simplified geometry is used to study the physiological interrelationship between the global ventricular function and the transmural temporal distribution of parameters which relate to mechanics, perfusion, energetics, and temperature in the myocardium. Transmural electrical activation maps are calculated based on analytical as well as a finite-element approach, eventually to be used in combination with the mechanical model to study LV contraction.>