Peter M. Pinsky

Computational modeling of mechanical anisotropy in the cornea and sclera

Purpose: To determine the biomechanical deformation of the cornea resulting from tissue cutting and removal by use of a new computational model and to investigate the effect of mechanical anisotrophy resulting from the fibrillar architecture. Setting: Department of Mechanical Engineering, Stanford University, Stanford, California, USA. Methods: A mathematical model for a typical lamella that explicitly accounts for the strain energy of the collagen fibrils, extrafibrillar matrix, and proteoglycan cross‐linking was developed. A stromal model was then obtained by generalized averaging of the lamella properties through the stromal thickness, taking into account the preferred orientations of the collagen fibrils, which were obtained from x‐ray scattering data. Results: The model was used to predict astigmatism induced by a tunnel incision in the sclera, such as is used for cataract extraction and intraocular lens implantation. The amount of induced cylinder was in good agreement with published clinical data. Results show it is important for the model to incorporate preexisting corneal physiological stress caused by intraocular pressure. Conclusions: The mathematical model described appears to provide a framework for further development, capturing the essential features of mechanical anisotropy of the cornea. The tunnel incision simulation indicated the importance of the anisotropy in this case.

A MICROSTRUCTURALLY-BASED FINITE ELEMENT MODEL OF THE INCISED HUMAN CORNEA

A mechanical model of the human cornea is proposed and employed in a finite element formulation for simulating the effects of surgical procedures, such as radial keratotomy, on the cornea. The model assumes that the structural behavior of the cornea is governed by the properties of the stroma. Arguments based on the microstructural organization and properties of the stroma lead to the conclusion that the human cornea exhibits flexural and shear rigidities which are negligible compared to its membrane rigidity. Accordingly, it is proposed that to a first approximation, the structural behavior of the cornea is that of a thick membrane shell. The tensile forces in the cornea are resisted by very fine collagen fibrils embedded in the ground substance of the stromal lamellae. When the collagen fibrils are cut, as in radial keratotomy, it is argued that they become relaxed since there is negligible transfer of load between adjacent fibrils due to the low shear modulus of the ground substance. The forces in the cornea are then resisted only by the remaining uncut fibrils. The cutting of fibrils induces an anisotropy and inhomogeneity in the membrane rigidity. By assuming a uniform angular distribution of stromal lamellae through the corneal thickness, geometric arguments lead to a quantitative representation for the anisotropy and inhomogeneity. All material behavior is assumed to be in the linear elastic regime and with no time-dependency. The resulting constitutive model for the incised cornea has been employed in a geometrically non-linear finite element membrane shell formulation for small strains with moderate rotations. A number of numerical examples are presented to illustrate the effectiveness of the proposed constitutive model and finite element formulation. The dependence of the outcome of radial keratotomy, measured in terms of the immediate postoperative shift in corneal power, on a number of important factors is investigated. These factors include the value of the elastic moduli of the stromal lamellae (dependent on the patients age), the incision depth, the optic zone size, the number of incisions and their positions, and the intraocular pressure. Results have also been compared with expected surgical corrections predicted by three expert surgeons and show an excellent correspondence.

Complex wavenumber Fourier analysis of the p-version finite element method

High-order finite element discretizations of the reduced wave equation have frequency bands where the solutions are harmonic decaying waves. In these so called “stopping” bands, the solutions are not purely propagating (real wavenumbers) but are attenuated (complex wavenumbers). In this paper we extend the standard dispersion analysis technique to include complex wavenumbers. We then use this complex Fourier analysis technique to examine the dispersion and attenuation characteristics of the p-version finite element method. Practical guidelines are reported for phase and amplitude accuracy in terms of the spectral order and the number of elements per wavelength.

Recent developments in finite element methods for structural acoustics

SummaryThe study of structural acoustics involves modeling acoustic radiation and scattering, primarily in exterior regions, coupled with elastic and structural wave propagation. This paper reviews recent progress in finite element analysis that renders computation a practical tool for solving problems of structural acoustics. The cost-effectiveness of finite element methods is composed of several ingredients. Boundary-value problems in unbounded domains are inappropriate for direct discretization. Employing DtN methodology yields an equivalent problem that is suitable for finite element analysis by posing impedance relations at an artificial exterior boundary. Vell-posedness of the resulting continuous formulations is discussed, leading to simple guidelines for practical implementation and verifying that DtN boundary conditions provide a suitable basis for computation.Approximation by Galerkin finite element methods results in spurious dispersion, degrading with reduced wave resolution. Accuracy is improved by Galerkin/least-squares and related technologies on the basis of detailed examinations of discrete errors in simplified settings, relaxing wave-resolution requirements. This methodology is applied to time-harmonic problems of acoustics and coupled problems of structural acoustics. Space-time finite methods based on time-discontinuous Galerkin/least-squares are derived for transient problems of structural acoustics. Numerical results validate the superior performance of Galerkin/least-squares finite elements for problems of structural acoustics.A comparative study of the cost of computation demonstrates that Galerkon/least-squares finite element methods are economically competitive with boundary element methods, the prevailing numerical approach to exterior problems of acoustics. Efficient iterative methods are derived for solving the large-scale matrix problems that arise in structural acoustics computation of realistics configuration at high wavenumbers. An a posteriori error estimator and adaptive strategy are developed for time-harmonic acoustic problems and the role of adaptivity in reducing the cost of computation is addressed.

A multiscale finite element method for the Helmholtz equation

It is well known that when the standard Galerkin method is applied to the Helmholtz equation it exhibits an error in the wavenumber and the solution does not, therefore, preserve the phase characteristics of the exact solution. Improvements on the Galerkin method, including Galerkin least-squares (GLS) methods, have been proposed. However, these approaches rely on a dispersion analysis of the underlying difference stencils in order to reduce error in the solution. In this paper we propose a multiscale finite element for the Helmholtz equation. The method employs a multiscale variational formulation which leads to a subgrid model in which subgrid scales are incorporated analytically through appropriate Greens functions. It is shown that entirely new and accurate methods emerge and that GLS methods can be obtained as special cases of the more general subgrid model.

Depth-Dependent Transverse Shear Properties of the Human Corneal Stroma

PURPOSE To measure the transverse shear modulus of the human corneal stroma and its profile through the depth by mechanical testing, and to assess the validity of the hypothesis that the shear modulus will be greater in the anterior third due to increased interweaving of lamellae. METHODS Torsional rheometry was used to measure the transverse shear properties of 6 mm diameter buttons of matched human cadaver cornea pairs. One cornea from each pair was cut into thirds through the thickness with a femtosecond laser and each stromal third was tested individually. The remaining intact corneas were tested to measure full stroma shear modulus. The shear modulus from a 1% shear strain oscillatory test was measured at various levels of axial compression for all samples. RESULTS After controlling for axial compression, the transverse shear moduli of isolated anterior layers were significantly higher than central and posterior layers. Mean modulus values at 0% axial strain were 7.71 ± 6.34 kPa in the anterior, 1.99 ± 0.45 kPa in the center, 1.31 ± 1.01 kPa in the posterior, and 9.48 ± 2.92 kPa for full thickness samples. A mean equilibrium compressive modulus of 38.7 ± 8.6 kPa at 0% axial strain was calculated from axial compression measured during the shear tests. CONCLUSIONS Transverse shear moduli are two to three orders of magnitude lower than tensile moduli reported in the literature. The profile of shear moduli through the depth displayed a significant increase from posterior to anterior. This gradient supports the hypothesis and corresponds to the gradient of interwoven lamellae seen in imaging of stromal cross-sections.

A residual‐based finite element method for the Helmholtz equation

A new residual-based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter-element boundaries. The inclusion of residuals on inter-element boundaries distinguishes this method from the well-known Galerkin least-squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright

Nonlinear dynamic modeling of micromachined microwave switches

Nonlinear dynamic lumped models of micromachined microwave switches have been formulated and successfully applied to analyses of transient characteristics and geometrical scaling. Parameter extraction through electrical measurements is summarized. The results are compared to transient quasi-2D simulations.

The role of 3-D collagen organization in stromal elasticity: a model based on X-ray diffraction data and second harmonic-generated images

Examining the cross-section of the human cornea with second harmonic-generated (SHG) imaging shows that many lamellae do not lie parallel to the cornea’s anterior surface but have inclined trajectories that take them through the corneal thickness with a depth-dependent distribution. A continuum mechanics-based model of stromal elasticity is developed based on orientation information extracted and synthesized from both X-ray scattering studies and SHG imaging. The model describes the effects of inclined lamella orientation by introducing a probability function that varies with depth through the stroma, which characterizes the range and distribution of lamellae at inclined angles. When combined with the preferred lamellar orientations found from X-ray scattering experiments, a fully 3-D representation of lamella orientation is achieved. Stromal elasticity is calculated by a weighted average of individual lamella properties based on the spatially varying 3-D orientation distribution. The model is calibrated with in vitro torsional shear experiments and in vivo indentation data and then validated with an in vitro inflation study. A quantitative explanation of the experimentally measured depth dependence of mechanical properties emerges from the model. The significance of the 3-D lamella orientation in the mechanics of the human cornea is demonstrated by investigating and contrasting the effects of previous modeling assumptions made on lamella orientation.

An application of shape optimization in the solution of inverse acoustic scattering problems

We consider the problem of determining the shape of an object immersed in an acoustic medium from measurements obtained at a distance from the object. We recast this problem as a shape optimization problem where we search for the domain that minimizes a cost function that quantifies the difference between the measured and expected signals. The measured and expected signals are assumed to satisfy a boundary-value problem given by the Helmholtz equation with the Sommerfeld condition imposed at infinity. Gradient-based algorithms are used to solve this optimization problem. At every step of the algorithm the derivative of the cost function with respect to the parameters that describe the shape of the object is calculated. We develop an efficient method based on the adjoint equations to calculate the derivative and show how this method is implemented in a finite element setting. The predominant cost of each step of the algorithm is equal to one forward solution and one adjoint solution and therefore is independent of the number of parameters used to describe the shape of the object. Numerical examples showing the efficacy of the proposed methodology are presented.