Development and Application of Low-Order Computational Models of the Left Ventricle

Abstract

Current methods of echocardiography provide detailed information about heart wall motion during the cardiac cycle. However, the ability to interpret these data in terms of cardiac muscle properties is limited. An eventual goal of the present work is to develop theoretical models for left ventricle (LV) dynamics that allow rapid estimation of myocardial stiffness and contractility from echocardiogram data. To reduce the computational complexity of this inverse problem, we represent LV deformation in terms of a limited number of modes. In the first part of this work, we present a low-order theoretical model of the LV. The LV shape is assumed to be axisymmetric and described using a small number of parameters, allowing rapid computational simulations of LV dynamics. The LV is represented as a thick-walled prolate spheroid containing helical muscle fibers with nonlinear passive and time-dependent active contractile properties. The displacement field during the cardiac cycle is described by three time-dependent parameters, using a family of volume-preserving mappings based on prolate spheroidal coordinates. Stress equilibrium is imposed in weak form and the resulting force balance equations are coupled to a lumped-parameter model of the circulation. This leads to a system of differential-algebraic equations, whose numerical solution yields predictions of LV pressure and volume, together with spatial distributions of stresses and strains throughout the cardiac cycle. In the second part, we derive a generalized low-order (GLO) left ventricle model. The GLO model extends the axisymmetric framework to a more general set of volume-conserving deformations. The myocardial boundaries are defined by prolate spheroidal spline functions in the reference configuration. The three parameter family of mappings used to define displacements in the axisymmetric model is generalized to a seven parameter family. Non-axisymmetric modes of deformation, such as those resulting from differences between the septal and lateral walls, are included. As in the axisymmetric model, the mechanical equilibrium equations are imposed in weak form, leading to a system of differential-algebraic equations that describe LV dynamics. In the third part, we use the GLO LV model to estimate the myocardial stiffness parameters in a mouse model. We assume the "Guccione law," a four parameter transversely isotropic material law, for the elastic properties of the myocardium. The stiffness parameters are estimated using two objective functions: a displacement objective and an aggregated objective. We find that the GLO model is capable of identifying material parameters to accurately fit four selected aggregated measurements: volume, work, short axis diameter, and long axis length. The results of the GLO model agree well with the findings in [45], where the Guccione material law parameters were identified for the same data set using a finite element model. However, as in many FEM studies, multiple combinations of the stiffness parameters lead to equally good fits to the data, implying uncertainty in parameter identification. We find that only a combined parameter representing the overall stiffness of the LV is identified robustly. In the fourth part, we develop a multigrid approach to deformable image registration of the LV. We apply this method to estimate LV displacements from 3D echocardiogram data. To improve temporal continuity and reduce registration errors, this technique uses three stages to register LV deformations. In the first stage, the seven parameter family of deformations developed for the GLO LV model are used to register the majority of LV motion. In the second stage, the low-order deformations are smoothed. In the third stage, these low-order estimates of LV motion are refined by registering local deformations with increasing resolution. The local displacements are described by a multi-resolution set of prolate spheroidal splines. The result of this multi-stage and multi-grid procedure is a smooth approximation of LV deformation over the full cardiac cycle. In the final part, we couple the GLO LV model to important components of the circulatory system. We represent the left atrium using a spherical model that is constructed analogously to the LV. The aorta is described by a 1D wave propagation model. The mitral and aortic valves are formulated as both time-delay and variable resistance valves. The circulation loop is closed using lumped parameter models. We explore the coupled circulatory system with several sets of simulations. In the first, we find that the closed circulation loop is necessary to accurately represent the preload and afterload systems of the LV if LV function is altered. In the second, we show that interactions between the right and left ventricles have a pronounced effect during diastole. In the third, we demonstrate that, according to the coupled mathematical system, loss of atrial systole results in elevated left atrial pressure. In the fourth set of simulations, we illustrate the effect of using different valve models on the filling and emptying characteristics of the LV. In the final simulation set, we show that early wave reflections from the arterial tree can inhibit LV function. We find that a 25% increase in LV contractility is necessary to overcome elevated systolic pressures resulting from early wave reflections.