The anatomy of the heart : a finite element model

Abstract

A new formalism for describing multidimensional piecewise functional fields is presented. The global or ensemble field parameters are related to the element field parameters by a general hear map. This ensemble to element parameter map contrasts with the bolean map of traditional formalisms where each element parameter is identified with one and only one ensemble field parameter. By allowing general linear maps between the ensemble field parameters and the element field parameters significant gains in describing and evaluating fields may be made. Since this mapping is Unear it can accommodate basis transformations and interelement continuity conditions manifested as linear relations or constraints between the field parameters. One is thus free to choose element bases from a wide class of functions. The general linear mag may be viewed as shifting information from the element basis functions to the ensemble to element parameter map. As a result of this considerable savings may be made In field evaluations at the element level. A new technique is presented for transforming fields from one basts representation to another. This technique uses projection operators to determine the best approximation when an exact representafion is impossible. Two techniques are presented for optimally fitting multidimensional piecewise functional fields to irregularly spaced measurements. The linear method may be used where the ensemble position assdated with each measurement does not change with the ensernbIe configuration. The linear equations resulting from this approach are the normal equations produced by the Markov estimator. Use of the nonllnsar method Is appropriate where the ensemble position associated with each measurement changes with the ensemble canflguration. fn this method a nested nonlinear optim.isation tachnIque is employed to minirnise the mean squared distance from the closest approach of the ensemble to each measurement. It Is shown that, since lines joining each measurement to its correspondhg ensemble posltbn are normal to the ensemble, determinaff on of the deflvatives of the error function with respect to the ensemble field parameters may be considerably simplified. A detailed understanding of ventricular mechanics can only' be obtained from continuum models. Any realistic model of the ventricles whlch attempts to examhe regional variations of myocardial behavior must include accurate descriptions of the large strains, irregular geometry, and complex constitutive properties of the myocardium. In continuum models to date Pttie attempt has been made to inciude a detailed representation of ventricular architecture. We apply the above field description and fitting techniques to measurements of the geometry and fibre orientation from a single dog heart to provide an optlrnal description of the ventricular anatomy. Comparisons of the relative performance of the Ifnear and nonlinear methods are provided on the ftts to the geometry measurements. On a heart measuring 60mm from base to apex mean errors of 0-658mm, 2.1 Omm, and 1.95mm are obtained on geometric fits to the epicardial, left ventricular endocardia!, and right ventricular endocardial surfaces using 52,24, and 9 fiting parameters respectively. With 12 fibre orientation fitting parameters a mean error of 14.6" is obtained.