An inverse material identification algorithm for determining in vivo myocardial material properties

Abstract

A material identification algorithm is described for determining the iri vivo material properties of the diastolic myocardium. A nonlinear optimization algorithm is used to solve a least squares objective function. The objective function relates the least squares difference of model-predicted displacements obtained from a finite element (FE) solution to measured displacements. obtained in the irt vivo case from magnetic resonance imaging (MRI) radiofrequency (RF) tissue-tagging. The algorithm is validated using a simple analytic test case by examining the effects of noise in the measured data and numerical error in the FE solution. Nonhomogeneous, linearly elastic and isotropic material parameters are determined for a normal adult mongrel dog. INTRODUCTION Continuum mechanical models of the heart provide important insight into the relationship between the microscopic structure and function of ventricular muscle and global ventricular function. The solution of boundary value problems encountered in deformation and stress analysis of the heart requires the application of physical laws as goveming differential equations when the geomey, boundary conditions and material properties are known. The material properties are the least well characterized input to the forward boundary value problem. Previous investigators have examined isolated samples of myocardium in the laboratory using loaddeformation testing. These analyses have been successful in delineating the anisotropic, nonlinear and non-homogeneous nature of passive myocardial material behavior. While the iri virro approach to material characterization has been successful in delineating the general form of the constitutive law, technical difficulties and natural objections to extrapolating iri vriru tests on small samples to the irr vivo beating heart, limit its applicability. An alternative approach, taken in this paper, dctermines unknown material parameters in the irt vivo heart for a proposed constitutive law. The material identification algorithm uses a customized nonlinear optimization algorithm to solve an inverse boundary value problem. The inverse problem is posed as a least squares minimization problem. The difference between model-predicted and mcasured displacements are minimized with respect to unknown malerial parameters. METHODS Statement of the problem. Unknown material parameters for a proposed constitutive law are determined by minimizing the least squares difference between modelpredicted and measured displacements with respect to material parameters: where it,, are model displacements. i,, are measured dispacements and & is the vector of unknown material parameters. The model displacements are determined from the finite element matrix equations. K(&)u = f , using the p version FE method [ l l where f is the load vector and the parameters are contained in the finite element stiffness matrix. K. The objective function. S, is minimized by solving the JS equations = 0. JP, Solution of the inverse problem. The set of equations is solved by applying a modified LevenbergMarquardt optimization algorithm (21. The objective function, S, is approximated by a second order Taylor series expansion. Approximations are made for the matrix of first and second order partial derivatives in the Taylor series 0 is expansion and the solution of the set of equations = reduced to solving the linear system of equations: