A computationally efficient formal optimization of regional myocardial contractility in a sheep with left ventricular aneurysm

Abstract

A non-invasive method for estimating regional myocardial contractility in vivo would be of great value in the design and evaluation of new surgical and medical strategies to treat and/or prevent infarction-induced heart failure. As a first step towards developing such a method, an explicit finite element (FE) model-based formal optimization of regional myocardial contractility in a sheep with left ventricular (LV) aneurysm was performed using tagged magnetic resonance (MR) images and cardiac catheterization pressures. From the tagged MR images, 3-dimensional (3D) myocardial strains, LV volumes and geometry for the animal-specific 3D FE model of the LV were calculated, while the LV pressures provided physiological loading conditions. Active material parameters (T(max_B) and T(max_R)) in the non-infarcted myocardium adjacent to the aneurysm (borderzone) and in myocardium remote from the aneurysm were estimated by minimizing the errors between FE model-predicted and measured systolic strains and LV volumes using the successive response surface method for optimization. The significant depression in optimized T(max_B) relative to T(max_R) was confirmed by direct ex vivo force measurements from skinned fiber preparations. The optimized values of T(max_B) and T(max_R) were not overly sensitive to the passive material parameters specified. The computation time of less than 5 hours associated with our proposed method for estimating regional myocardial contractility in vivo makes it a potentially very useful clinical tool.

Keywords: cardiac mechanics; finite element modeling; numerical optimization; tagged magnetic resonance imaging.

Fig. 1

3D cardiac strain analysis from in vivo tagged MR images. Endocardial and epicardial contours as well as segmented tag-lines were traced from (a) short and (b) long-axis MR images to create (c) a 3D geometry. (d) Each short axis slice was divided into 12 sectors and a 4D B-spline-based motion tracking technique was applied to the tag-line (dotted lines) deformations in order to calculate the Lagrangian Green’s strains in cylindrical coordinates. For each sector of each short axis slice, longitudinal, radial, (e) circumferential and shear strains throughout systole were determined.

Fig. 2

Creation of the FE model of the LV using geometry from in vivo tagged MR images. Endocardial and epicardial contours extracted from (Fig. 1a) short and (Fig. 1b) long-axis MR images were used to generate (a) a surface mesh with three distinct LV regions (remote, borderzone and aneurysm). The boundaries between these three LV regions are based on wall thickness. The surface meshes provide projection surfaces for (b) the volumetric mesh, which is refined into three elements transmurally. A layer of shell elements line the endocardial surface and cap off the top of the LV to form a closed volume for LV volume measurements.

Fig. 3

A graphical representation of the RSM optimization approach employed by LS-OPT. (a) For the optimization of 2 parameters, 5 experimental points including the initial guess were selected using a D-Optimal method bounded by their respective upper and lower limits. (b) FE simulations were performed at each experimental point and the strains and volumes for each point were calculated. (c) A linear response surface was fitted onto the strains and volumes using a least-square fit method. The MSE was then computed using the predicted and experimental strain and volume values. The optimum design for the resulting approximate MSE was determined by the minimization of the response surface using the leap-frog algorithm.

Fig. 4

A graphical representation of the SRSM. From the initial experimental points and parameter space bounded by the ranges of the 2 parameters, T_{max_R} and T_{max_B}, the first implementation of the RSM produced an initial response surface and its optimum. This optimum becomes the starting experimental point for the next iteration with a narrower sub-region of the parameter space. Repeating the RSM for each successive design iteration reduced and shifted the sub-regions of the parameter space until a final optimum was found. For clarity, only the first 5 iteration out of the 10 total is illustrated.

Fig. 5

A flowchart illustrating the process involved in determining the optimum myocardial material parameters from tagged MR images and LV pressures from cardiac catheterization.

Fig. 6

The narrowing of the parameter space for the systolic parameters, T_{max_R} and T_{max_B}, with each iteration resulted in a precise final converged optimum.

Fig. 7

Circumferential strains predicted from the present FE model are generally in decent agreement with the values measured in vivo from tagged MR images. Slice 1 is the most basal while slice 16 is the most apical. I is the posterior right ventricular insertion, II is the free wall, III is the anterior right ventricular insertion and IV is the septum. The area of largest discrepancy between the measured and predicted circumferential strains is at the insertion points of the right ventricle to the left ventricle since the right ventricle was not included in the model.