We developed a mathematical representation of ventricular geometry and muscle fiber organization using three-dimensional finite elements referred to a prolate spheroid coordinate system. Within elements, fields are approximated using basis functions with associated parameters defined at the element nodes. Four parameters per node are used to describe ventricular geometry. The radial coordinate is interpolated using cubic Hermite basis functions that preserve slope continuity, while the angular coordinates are interpolated linearly. Two further nodal parameters describe the orientation of myocardial fibers. The orientation of fibers within coordinate planes bounded by epicardial and endocardial surfaces is interpolated linearly, with transmural variation given by cubic Hermite basis functions. Left and right ventricular geometry and myocardial fiber orientations were characterized for a canine heart arrested in diastole and fixed at zero transmural pressure. The geometry was represented by a 24-element ensemble with 41 nodes. Nodal parameters fitted using least squares provided a realistic description of ventricular epicardial [root mean square (RMS) error less than 0.9 mm] and endocardial (RMS error less than 2.6 mm) surfaces. Measured fiber fields were also fitted (RMS error less than 17 degrees) with a 60-element, 99-node mesh obtained by subdividing the 24-element mesh. These methods provide a compact and accurate anatomic description of the ventricles suitable for use in finite element stress analysis, simulation of cardiac electrical activation, and other cardiac field modeling problems.

Mathematical model of geometry and fibrous structure of the heart.

Three-dimensional (3-D) imaging of the heart is a rapidly developing area of research in medical imaging. Advances in hardware and methods for fast spatio-temporal cardiac imaging are extending the frontiers of clinical diagnosis and research on cardiovascular diseases. In the last few years, many approaches have been proposed to analyze images and extract parameters of cardiac shape and function from a variety of cardiac imaging modalities. In particular, techniques based on spatio-temporal geometric models have received considerable attention. This paper surveys the literature of two decades of research on cardiac modeling. The contribution of the paper is three-fold: (1) to serve as a tutorial of the field for both clinicians and technologists, (2) to provide an extensive account of modeling techniques in a comprehensive and systematic manner, and (3) to critically review these approaches in terms of their performance and degree of clinical evaluation with respect to the final goal of cardiac functional analysis. From this review it is concluded that whereas 3-D model-based approaches have the capability to improve the diagnostic value of cardiac images, issues as robustness, 3-D interaction, computational complexity and clinical validation still require significant attention.

http://www.iacl.ece.jhu.edu/~prince/smia/2009/papers2009/FraxTMI01-Frangi-Cardiac-Function-Review.pdf

Three-dimensional modeling for functional analysis of cardiac images, a review

The analysis of cardiac function: a continuum approach.

The normal right ventricular pressure-volume relation was studied by recording biplane right ventriculograms with simultaneous high fidelity pressure recordings in 10 adults found to have normal coronary arteries and haemodynamic function at diagnostic cardiac catheterisation. Right ventricular volume was measured frame by frame from digitised ventriculograms by a modification of Simpson's rule. The accuracy of this method was tested in a study of 22 human and animal right ventricular casts. There was excellent agreement between calculated volumes and those measured by fluid displacement. The derived regression equations were used to correct right ventricular volumes calculated from in vivo studies. The mean (SD) end diastolic volume index for the group was 62 (13) ml/m2, the stroke volume index was 43 (8) ml/m2, and the ejection fraction was 62 (6)%. Right ventricular pressure-volume loops were generated by combining simultaneous volume and pressure curves. The normal right ventricular pressure-volume loop was triangular, departing significantly from the square or rectangle of the normal left ventricular pressure-volume loop. Ejection from the right ventricle began early during the pressure rise and continued as right ventricular pressure fell. As a result phases of isovolumic contraction and relaxation were difficult to define. These observations show that normal right ventricular pressure-volume relations differ considerably from those of the normal left ventricle, presumably reflecting the different loading conditions of the two ventricles.

https://heart.bmj.com/content/heartjnl/59/1/23.full.pdf

Characterisation of the normal right ventricular pressure-volume relation by biplane angiography and simultaneous micromanometer pressure measurements.

A model of the left ventricle which combines a spheroidal geometry with a spatial fiber angle distribution is presented. The mechanics of each muscle fiber is described by its passive stress-strain relationship, active stress-strain relationship, and an activation function (half a sinusoid) which represents the time-dependent degree of activation of the fiber. A stress-strain rate relationship which characterizes the muscle fibers is used to calculate the mechanics of left ventricular contraction during ejection. Furthermore, a radial electrical signal propagation from the endocardium to the epicardium is used here as a first approximation to the actual depolarization sequence. The model is used to describe the process of contraction throughout the systole. The different calculated parameters and indices of left ventricular function are presented and discussed for different preloading, afterloading and contractility conditions. The maximum elastance is found to be an optimal macroscale parameter of contractility, as it is completely preload and afterload independent, and is a good reflection of the active microscale sarcomere stress-strain relationship.

C. A. Vinson

Papers

Effect of myocardial fibre architecture on the behaviour of the human left ventricle in diastole.

Computer modeling of the human left ventricle.

Geometric Modelling of the Human Left Venticle