Kevin D. Costa

Finite element stress analysis of left ventricular mechanics in the beating dog heart

A three-dimensional finite element model was used to explore whether or not transmural distributions of end-diastolic and end-systolic fiber stress are uniform from the apex to the base of the canine left ventricular wall. An elastance model for active fiber stress was incorporated in an axisymmetric model that accurately represented the geometry and fiber angle distribution of the anterior free wall. The nonlinear constitutive equation for the resting myocardium was transversely isotropic with respect to the local fiber axis. Transmural distributions of end-diastolic fiber stress became increasingly nonuniform from midventricle toward the apex or the base. At a typical diastolic left ventricular pressure (1 kPa), the differences between largest and smallest fiber stresses were only 0.5 kPa near midventricle, compared with 4.6 kPa at the apex, and 3.3 kPa at the base. Transmural fiber stress differences at end-systole (14 kPa) were relatively small in regions from the base to the midventricle (13-22 kPa), but were larger between midventricle and the apex (30-43 kPa). All six three-dimensional end-diastolic strain components were within or very close to one standard deviation of published measurements through the midanterior left ventricular free wall of the passive canine heart [Omens et al., Am. J. Physiol. 261, H918-H928 (1991)]. End-systolic in-plane normal and shear strains also agreed closely with published experimental measurements in the beating dog heart [Waldman et al., Circ. Res. 63, 550-562 (1988)]. The results indicate that, unlike in the midventricle region that has been studied most fully, there may be significant regional nonhomogeneity of fiber stress in the normal left ventricle associated with regional variations in shape and fiber angle.

Analysis of Indentation: Implications for Measuring Mechanical Properties With Atomic Force Microscopy

Indentation using the atomic force microscope (AFM) has potential to measure detailed micromechanical properties of soft biological samples. However, interpretation of the results is complicated by the tapered shape of the AFM probe tip, and its small size relative to the depth of indentation. Finite element models (FEMs) were used to examine effects of indentation depth, tip geometry, and material nonlinearity and heterogeneity on the finite indentation response. Widely applied infinitesimal strain models agreed with FEM results for linear elastic materials, but yielded substantial errors in the estimated properties for nonlinear elastic materials. By accounting for the indenter geometry to compute an apparent elastic modulus as a function of indentation depth, nonlinearity and heterogeneity of material properties may be identified. Furthermore, combined finite indentation and biaxial stretch may reveal the specific functional form of the constitutive law--a requirement for quantitative estimates of material constants to be extracted from AFM indentation data.

Enhanced cell sorting and manipulation with combined optical tweezer and microfluidic chip technologies.

Sorting (or isolation) and manipulation of rare cells with high recovery rate and purity are of critical importance to a wide range of physiological applications. In the current paper, we report on a generic single cell manipulation tool that integrates optical tweezers and microfluidic chip technologies for handling small cell population sorting with high accuracy. The laminar flow nature of microfluidics enables the targeted cells to be focused on a desired area for cell isolation. To recognize the target cells, we develop an image processing methodology with a recognition capability of multiple features, e.g., cell size and fluorescence label. The target cells can be moved precisely by optical tweezers to the desired destination in a noninvasive manner. The unique advantages of this sorter are its high recovery rate and purity in small cell population sorting. The design is based on dynamic fluid and dynamic light pattern, in which single as well as multiple laser traps are employed for cell transportation, and a recognition capability of multiple cell features. Experiments of sorting yeast cells and human embryonic stem cells are performed to demonstrate the effectiveness of the proposed cell sorting approach.

Single-cell elastography: probing for disease with the atomic force microscope.

The atomic force microscope (AFM) is emerging as a powerful tool in cell biology. Originally developed for high-resolution imaging purposes, the AFM also has unique capabilities as a nano-indenter to probe the dynamic viscoelastic material properties of living cells in culture. In particular, AFM elastography combines imaging and indentation modalities to map the spatial distribution of cell mechanical properties, which in turn reflect the structure and function of the underlying cytoskeleton. Such measurements have contributed to our understanding of cell mechanics and cell biology and appear to be sensitive to the presence of disease in individual cells. This chapter provides a background on the principles and practice of AFM elastography and reviews the literature comparing cell mechanics in normal and diseased states, making a case for the use of such measurements as disease markers. Emphasis is placed on the need for more comprehensive and detailed quantification of cell biomechanical properties beyond the current standard methods of analysis. A number of technical and practical hurdles have yet to be overcome before the method can be of clinical use. However, the future holds great promise for AFM elastography of living cells to provide novel biomechanical markers that will enhance the detection, diagnosis, and treatment of disease.

A three-dimensional finite element method for large elastic deformations of ventricular myocardium : II-prolate spheroidal coordinates

A three-dimensional finite element method for nonlinear finite elasticity is presented using prolate spheroidal coordinates. For a thick-walled ellipsoidal model of passive anisotropic left ventricle, a high-order (cubic Hermite) mesh with 3 elements gave accurate continuous stresses and strains, with a 69 percent savings in degrees of freedom (dof) versus a 70-element standard low-order model. A custom mixed-order model offered 55 percent savings in dof and 39 percent savings in solution time compared with the low-order model. A nonsymmetric 3D model of the passive canine LV was solved using 16 high-order elements. Continuous nonhomogeneous stresses and strains were obtained within 1 hour on a laboratory workstation, with an estimated solution time of less than 4 hours to model end-systole. This method represents the first practical opportunity to solve large-scale anatomically detailed models for cardiac stress analysis.

Osteoblast elastic modulus measured by atomic force microscopy is substrate dependent.

The actin and microtubule cytoskeleton have been found to contribute to the elastic modulus of cells, which may be modulated by adhesion to extracellular matrix (ECM) proteins and subsequent alterations in the cytoskeleton. In this study, the apparent elastic modulus (Eapp) of osteoblast-like MC3T3-E1 cells adhered to fibronectin (FN), vitronectin (VN), type I collagen (COLI), fetal bovine serum (FBS), or poly-l-lysine (PLL), and bare glass were determined using an atomic force microscope (AFM). The Eapp of osteoblasts adhered to ECM proteins (FN, VN, COLI, and FBS) that bind cells via integrins were higher compared to cells on glass and PLL, which adhere cells through nonspecific binding. Also, osteoblasts adhered to FN, VN, COLI, and FBS had F-actin stress fiber formation, while osteoblasts on glass and PLL showed few F-actin fibers. Disruption of the actin cytoskeleton decreased Eapp of osteoblasts plated on FN to the level of osteoblasts plated on glass, while microtubule disruption had no significant effect. This suggests that the elevated modulus of osteoblasts adhered to FN was due to remodeling of the actin cytoskeleton upon adhesion to ECM proteins. Modulation of cell stiffness upon adhesion to various substrates may influence mechanosignal transduction in osteoblasts.

Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system.

Electrical and mechanical anisotropy arise from matrix and cellular alignment in native myocardium. Generation of anisotropy in engineered heart tissue will be required to match native properties and will provide immediate opportunities to investigate the genesis and structural determinants of functional anisotropy. We investigated the influence of geometry and boundary conditions on fibroblast alignment in thin collagen gels. Consistent with previous reports, we found that human dermal fibroblasts align parallel to free edges in partially constrained gels; in contrast to at least one report, fibroblasts in fully constrained gels remained randomly aligned independent of geometry. These experiments allowed us to distinguish between two possible mechanisms for such alignment. Mean orientations that followed the shape of the free edges and stronger alignment nearest the free edges in gels with a variety of geometries suggested that cells align parallel to a local free boundary rather than to local lines of tension. These findings focus attention on the presence of voids and free surfaces such as the endocardium and epicardium, cleavage planes, and blood vessels in governing cell and fiber alignment in developing and remodeling myocardium, myocardial scar tissue, and engineered heart constructs.

Mechanism underlying mechanical dysfunction in the border zone of left ventricular aneurysm: a finite element model study

BACKGROUND The global left ventricular dysfunction characteristic of left ventricular aneurysm is associated with muscle fiber stretching in the adjacent noninfarcted (border zone) region during isovolumic systole. The mechanism of this regional dysfunction is poorly understood. METHODS An anteroapical transmural myocardial infarct was created by coronary arterial ligation in an adult Dorset sheep and was allowed to mature into left ventricular aneurysm for 10 weeks. The animal was imaged subsequently using magnetic resonance imaging with simultaneous recording of intraventricular pressures. A realistic mathematical model of the three-dimensional ovine left ventricle with an anteroapical aneurysm was constructed from multiple short-axis and long-axis magnetic resonance imaging slices at the beginning of diastolic filling. RESULTS Three model simulations are presented: (1) normal border zone contractility and normal aneurysmal material properties; (2) greatly reduced border zone contractility (by 50%) and normal aneurysmal material properties; and (3) greatly reduced border zone contractility (by 50%) and stiffened aneurysmal material properties (by 1000%). Only the latter two simulations were able to reproduce experimentally observed stretching of border zone fibers during isovolumic systole. CONCLUSIONS The mechanism underlying mechanical dysfunction in the border zone region of left ventricular aneurysm is primarily the result of myocardial contractile dysfunction rather than increased wall stress in this region.

Theoretical Quality Assessment of Myocardial Elastography with In Vivo Validation

Myocardial elastography (ME), a radio frequency (RF)-based speckle tracking technique with one-dimensional (1-D) cross correlation and novel recorrelation methods in a 2-D search was proposed to estimate and fully image 2-1) transmural deformation field and to detect abnormal cardiac function. A theoretical framework was first developed in order to evaluate the performance of 2-D myocardial elastography based on a previously developed 3-D finite-element model of the canine left ventricle. A normal (control) and an ischemic (left-circumflex, LCx) model, which more completely represented myocardial deformation than a kinematic model, were considered. A 2-D convolu-tional image formation model was first used to generate RF signals for quality assessment of ME in the normal and ischemic cases. A 3-D image formation model was further developed to investigate the effect of the out-of-plane motion on the 2-D, in-plane motion estimation. Both orthogonal, in-plane displacement components (i.e., lateral and axial) between consecutive RF frames were iteratively estimated. All the estimated incremental 2-D displacements from end-diastole (ED) to end-systole (ES) were then accumulated to acquire the cumulative 2-D displacements, which were further used to calculate the cumulative 2-D systolic finite strains. Furthermore, the cumulative systolic radial and circumferential strains, which were angle-and frame-rate independent, were obtained from the 2-D finite-strain components and imaged in full view to detect the ischemic region. We also explored the theoretical understanding of the limitations of our technique for the accurate depiction of disease and validated it in vivo against tagged magnetic resonance imaging (tMRI) in the case of a normal human myocardium in a 2-D short-axis (SA) echocardiographic view. The theoretical framework succeeded in demonstrating that the 2-D myocardial elastography technique was a reliable tool for the complete estimation and depiction of the in-plane myocardial deformation field as well as for accurate identification of pathological mechanical function using established finite-element, left-ventricular canine models. In a preliminary study, the 2-D myocardial elastography was shown capable of imaging myocardial deformation comparable to equivalent tMRI estimates in a clinical setting.

A Three-Dimensional Finite Element Method for Large Elastic Deformations of Ventricular Myocardium: I—Cylindrical and Spherical Polar Coordinates

A three-dimensional Galerkin finite element method was developed for large deformations of ventricular myocardium and other incompressible, nonlinear elastic, anisotropic materials. Cylindrical and spherical elements were used to solve axisymmetric problems with r.m.s. errors typically less than 2 percent. Isochoric interpolation and pressure boundary constraint equations enhanced low-order curvilinear elements under special circumstances (69 percent savings in degrees of freedom, 78 percent savings in solution time for inflation of a thick-walled cylinder). Generalized tensor products of linear Lagrange and cubic Hermite polynomials permitted custom elements with improved performance, including 52 percent savings in degrees of freedom and 66 percent savings in solution time for compression of a circular disk. Such computational efficiencies become significant for large scale problems such as modeling the heart.