Finite element modeling and experimental verification of lower extremity shape change under load.

Prediction and measurement of residuum shape change inside the prosthesis under various loading conditions is important for prosthesis design and evaluation. Residual limb surface measurements with the prosthesis in situ were used for construction of a finite element model (FEM). These surface measurements were obtained from volumetric computed tomography. A new experimental method for modeling the shape of the in situ lower residual limb was developed based on spiral X-ray computed tomography (SXCT) imaging. The p-version of the finite element method was used for estimating the material properties from known load and displacement data. A homogeneous, isotropic, linear constitutive model with accommodation of the constitutive soft and hard tissues of the residuum was evaluated with static axial loading applied to the in situ prosthesis and compared with experimental results obtained in a human volunteer. Two FEMs were created for similar coronal cross sections of the below knee residuum under two loading conditions. Agreement between observed (from SXCT) and predicted (from FEA) residual limb shape changes inside the prosthesis were maximized with a single modulus of elasticity for the residuum soft tissue of 0.06 MPa, consistent with previously published results. This methodology provides a framework to predict and objectively evaluate FEMs and determine residuum material properties by inverse methods.

Myocardial material property determination in the in vivo heart using magnetic resonance imaging.

OBJECTIVES: To determine nonlinear material properties of passive, diastolic myocardium using magnetic resonance imaging (MRI) tissue-tagging, finite element analysis (FEA) and nonlinear optimization. BACKGROUND: Alterations in the diastolic material properties of myocardium may pre-date the onset of or exist exclusive of systolic ventricular dysfunction in disease states such as hypertrophy and heart failure. Accordingly, significant effort has been expended recently to characterize the material properties of myocardium in diastole. The present study defines a new technique for determining material properties of passive myocardium using finite element (FE) models of the heart, MRI tissue-tagging and nonlinear optimization. This material parameter estimation algorithm is employed to estimate nonlinear material parameter sin the in vivo canine heart and provides the necessary framework to study the full complexities of myocardial material behavior in health and disease. METHODS AND RESULTS: Material parameters for a proposed exponential strain energy function were determined by minimizing the least squares difference between FE model-predicted and MRI-measured diastolic strains. Six mongrel dogs underwent MRI imaging with radiofrequency (RF) tissue-tagging. Two-dimensional diastolic strains were measured from the deformations of the MRI tag lines. Finite element models were constructed from early diastolic images and were loaded with the mean early to late left ventricular and right ventricular diastolic change in pressure measured at the time of imaging. A nonlinear optimization algorithm was employed to solve the least squares objective function for hte material parameters. Average material parameters for the six dogs were E = 28,722 +/- 15984 dynes/cm2 and c = 0.00182 +/- 0.00232 cm2/dyne. CONCLUSION: This parameter estimation algorithm provides the necessary framework for estimating the nonlinear, anisotropic and non-homogeneous material properties of passive myocardium in health and disease in the in vivo beating heart.

Spline surface interpolation for calculating 3-D ventricular strains from MRI tissue tagging.

A method is developed and validated for approximating continuous smooth distributions of finite strains in the ventricles from the deformations of magnetic resonance imaging (MRI) tissue tagging "tag lines" or "tag surfaces." Tag lines and intersections of orthogonal tag lines are determined using a semiautomated algorithm. Three-dimensional (3-D) reconstruction of the displacement field on tag surfaces is performed using two orthogonal sets of MRI images and employing spline surface interpolation. The 3-D regional ventricular wall strains are computed from an initial reference image to a deformed image in diastole or systole by defining a mapping or transformation of space between the two states. The resultant mapping is termed the measurement analysis solution and is defined by determining a set of coefficients for the approximating functions that best fit the measured tag surface displacements. Validation of the method is performed by simulating tag line or surface deformations with a finite element (FE) elasticity solution of the heart and incorporating the measured root-mean-square (rms) errors of tag line detection into the simulations. The FE-computed strains are compared with strains calculated by the proposed procedure. The average difference between two-dimensional (2-D) FE-computed strains and strains calculated by the measurement analysis was 0.022 +/- 0.009 or 14.2 +/- 3.6% of the average FE elasticity strain solution. The 3-D displacement reconstruction errors averaged 0.087 +/- 0.002 mm or 2.4 +/- 0.1% of the average FE solution, and 3-D strain fitting errors averaged 0.024 +/- 0.011 or 15.9 +/- 2.8% of the average 3-D FE elasticity solution. When the rms errors in tag line detection were included in the 2-D simulations, the agreement between FE solution and fitted solution was 24.7% for the 2-D simulations and 19.2% for the 3-D simulations. We conclude that the 3-D displacements of MRI tag lines may be reconstructed accurately; however, the strain solution magnifies the small errors in locating tag lines and reconstructing 3-D displacements.

An inverse approach to determining myocardial material properties.

Passive myocardial material properties have been measured previously by subjecting test samples of myocardium to in vitro load-deformation analysis or, in the intact heart, by pressure-volume relationships. A new method for determining passive material properties, described in this paper, couples a p-version finite element model of the heart, a nonlinear optimization algorithm and a dense set of transmural measured strains that could be obtained in the intact heart by magnetic resonance imaging (MRI) radiofrequency tissue tagging. Unknown material parameters for a nonlinear, nonhomogeneous material law are determined by solving an inverse boundary value problem. An objective function relating the least-squares difference of model-predicted and measured strains is minimized with respect to the unknown material parameters using a novel optimization algorithm that utilizes forward finite element solutions to calculate derivatives of model-predicted strains with respect to the material parameters. Test cases incorporating several salient features of the inverse material identification problem for the heart are formulated to test the performance of the inverse algorithm in typical experimental conditions. Known true material parameters can be determined to within a small tolerance and random noise is shown not to affect the stability of the inverse solution appreciably. On the basis of these validation experiments, we conclude that the inverse material identification problem for the heart can be extended to solve for unknown material parameters that describe in vivo myocardial material behavior.

An experimental method for evaluating constitutive models of myocardium in in vivo hearts.

A new experimental method for the evaluation of myocardial constitutive models combines magnetic resonance (MR) radiofrequency (RF) tissue-tagging techniques with iterative two-dimensional (2-D) nonlinear finite element (FE) analysis. For demonstration, a nonlinear isotropic constitutive model for passive diastolic expansion in the in vivo canine heart is evaluated. A 2-D early diastolic FE mesh was constructed with loading parameters for the ventricular chambers taken from mean early diastolic-to-late diastolic pressure changes measured during MR imaging. FE solution was performed for regional, intramyocardial ventricular wall strains using small-strain, small-displacement theory. Corresponding regional ventricular wall strains were computed independently using MR images that incorporated RF tissue tagging. Two unknown parameters were determined for an exponential strain energy function that maximized agreement between observed (from MR) and predicted (from FE analysis) regional wall strains. Extension of this methodology will provide a framework in which to evaluate the quality of myocardial constitutive models of arbitrary complexity on a regional basis.

Ventricular interaction in the pathologic heart. A model based study.

The effects of direct ventricular interaction and interaction mediated by the pericardium on the diastolic left ventricle (LV) were quantified using idealized models of five pathologic conditions. Two-dimensional (2D) mathematical models were constructed in long and short axis views of four pathologic LV conditions and the normal heart (NL): dilated cardiomyopathy (DCM), concentric LV hypertrophy (HYP), chronic anterior-apical infarction in a normal shaped LV (CAINL), and CAI in a dilated LV (CAID). To assess the effects of RV pressure increase on the LV mechanical state, RV pressure was systematically increased for several LV pressures and changes in the LV diastolic pressure-area relationships, and LV free wall and septal principal stresses and strains were quantified. At higher RV pressures, with pericardial effects included in the models, the pressure-area relationship was similar for all models, indicating that, at these higher pressures, the effects of RV and pericardial pressures are more important than global LV shape, wall thickness, or material properties in determining the pressure-area relationship. There were significant differences among models in the changes in LV free wall and septal stress and strain after an increase in RV pressure. These models may be of use in predicting interaction in the corresponding clinical state.