Jeffrey P. Laible

A poroelastic-swelling finite element model with application to the intervertebral disc.

The swelling process that occurs in soft tissue is incorporated into a poroelastic finite element model. The model is applied to a spinal segment consisting of two vertebrae and a single intervertebral disc. The theory is an extension of the poroelastic theory developed by Biot7and the model is an adaptation of an axisymmetric poroelastic finite element model of the intervertebral disc by Simon.45,46The model is completely three-dimensional although the results presented here assume symmetry about the sagittal plane. The theory is presented in two stages. First the development of the poroelastic theory. Following this, the effects of swelling caused by osmotic pressure are developed and expressed as a modification of the constitutive law and initial stresses. In the case of the disc, this pressure is produced mainly by the fixed negative charges on the proteoglycans within the disc.In this development we assume that the number of fixed charges remains constant over time and that the distribution of mobile lons has reached equilibrium. The variations over time in osmotic pressure, and thus in swelling effects are therefore only dependent on the initial state and the change by changes in mobile ion concentrations will be the subject of a future paper.47The results reported in this article illustrate the dramatic effect of swelling on the load carrying mechanisms in the disc. The authors believe it is likely that this will have important useful implications for our understanding not only of normal disc function, but also of abnormal function, such as disc degeneration, herniation, and others.

Influence of fixed charge density magnitude and distribution on the intervertebral disc: applications of a poroelastic and chemical electric (PEACE) model.

A 3-dimensional formulation for a poroelastic and chemical electric (PEACE) model is presented and applied to an intervertebral disc slice in a 1-dimensional validation problem and a 2-dimensional plane stress problem. The model was used to investigate the influence of fixed charge density magnitude and distribution on this slice of disc material. Results indicated that the mechanical, chemical, and electrical behaviors were all strongly influenced by the amount as well as the distribution of fixed charges in the matrix. Without any other changes in material properties, alterations in the fixed charge density (proteoglycan content) from a healthy to a degenerated distribution will cause an increase in solid matrix stresses and can affect whether the tissue imbibes or exudes fluid under different loading conditions. Disc tissue with a degenerated fixed charge density distribution exhibited greater solid matrix stresses and decreased streaming potential, all of which have implications for disc nutrition, disc biomechanics, and tissue remodeling. It was also seen that application of an electrical potential across the disc can induce fluid transport.

Measurement of a spinal motion segment stiffness matrix

The six-degrees-of-freedom elastic behavior of spinal motion segments can be approximated by a stiffness matrix. A method is described to measure this stiffness matrix directly with the motion segment held under physiological conditions of axial preload and in an isotonic fluid bath by measuring the forces and moments associated with each of the six orthogonal translations and rotations. The stiffness matrix was obtained from the load-displacement measurements by linear least squares assuming a symmetric matrix. Results from a pig lumbar spinal motion segment in an isotonic bath, with and without a 500 N axial preload, showed a large stiffening effect with axial preload.

Incorporation of spinal flexibility measurements into finite element analysis.

This technical note demonstrates two methods of incorporating the experimental stiffness of spinal motion segments into a finite element analysis of the spine. The first method is to incorporate the experimental data directly as a stiffness matrix. The second method approximates the experimental data as a beam element.

A comparison of exact and approximate adjoint sensitivities in fluorescence tomography

Many approaches to fluorescence tomography utilize some form of regularized nonlinear least-squares algorithm for data inversion, thus requiring repeated computation of the Jacobian sensitivity matrix relating changes in observable quantities, such as emission fluence, to changes in underlying optical parameters, such as fluorescence absorption. An exact adjoint formulation of these sensitivities comprises three terms, reflecting the individual contributions of 1) sensitivities of diffusion and decay coefficients at the emission wavelength, 2) sensitivities of diffusion and decay coefficients at the excitation wavelength, and 3) sensitivity of the emission source term. Simplifying linearity assumptions are computationally attractive in that they cause the first and second terms to drop out of the formulation. The relative importance of the three terms is thus explored in order to determine the extent to which these approximations introduce error. Computational experiments show that, while the third term of the sensitivity matrix has the largest magnitude, the second term becomes increasingly significant as target fluorophore concentration or volume increases. Image reconstructions from experimental data confirm that neglecting the second term results in overestimation of sensitivities and consequently overestimation of the value and volume of the fluorescent target, whereas contributions of the first term are so low that they are probably not worth the additional computational costs.

Limitation of Finite Element Analysis of Poroelastic Behavior of Biological Tissues Undergoing Rapid Loading

The finite element method is used in biomechanics to provide numerical solutions to simulations of structures having complex geometry and spatially differing material properties. Time-varying load deformation behaviors can result from solid viscoelasticity as well as viscous fluid flow through porous materials. Finite element poroelastic analysis of rapidly loaded slow-draining materials may be ill-conditioned, but this problem is not widely known in the biomechanics field. It appears as instabilities in the calculation of interstitial fluid pressures, especially near boundaries and between different materials. Accurate solutions can require impractical compromises between mesh size and time steps. This article investigates the constraints imposed by this problem on tissues representative of the intervertebral disc, subjected to moderate physiological rates of deformation. Two test cylindrical structures were found to require over 104 linear displacement-constant pressure elements to avoid serious oscillations in calculated fluid pressure. Fewer Taylor–Hood (quadratic displacement–linear pressure elements) were required, but with complementary increases in computational costs. The Vermeer–Verruijt criterion for 1D mesh size provided guidelines for 3D mesh sizes for given time steps. Pressure instabilities may impose limitations on the use of the finite element method for simulating fluid transport behaviors of biological soft tissues at moderately rapid physiological loading rates.

Solution of the shallow water equations by least squares collocation

A method of solution of the shallow water equations in their primitive form that yield solutions free of numerical oscillations, is presented. The method is based on the least squares collocation concept. The focus is on the accuracy of the solution procedure. Comparison of the numerical solution with analytic solutions for problems of irregular geometry are presented. Key features are the ability of the method to solve the equations in an irregular domain with a completely orthogonal computational mesh and the inherent nondissipative noise control of the temporal and spatial approximations.

A finite element/finite difference wave model for depth varying nearly horizontal flow

Abstract A numerical solution for the depth varying nearly horizontal flow equations is developed using the Galerkin finite element method and a wave equation type formulation. This approach allows for the simulation of non-steady circulations in shallow bodies of water which are free of numerical oscillation in both the stage solution (i.e. no 2 Δx waves) and in the velocity field. The solution variables for the velocity field are the surface and near bottom velocities and the surface and near bottom gradients (i.e. ∂u ∂z ) multiplied by the depth for each of the cartesian directions. Hermitian polynomials are used to define the vertical structure of the flow while the nine node isoparametric quadrilateral is used for horizontal discretization. The solution produces u ( x , y , z , t ), v ( x , y , z , t ), ζ ( x , y , t ) by an explicit time marching scheme for the wave equation and an implicit scheme for the momentum equation. Only a two-dimension surface grid is required and the option exists in the present program to use finite elements, finite differences, or a mixed formulation. The utility of the model is demonstrated by numerical examples for which laboratory results exist and for an actual body of water for which current studies are presently being pursued.

Spatially Resolved Streaming Potentials of Human Intervertebral Disk Motion Segments Under Dynamic Axial Compression

Intervertebral disk degeneration results in alterations in the mechanical, chemical, and electrical properties of the disk tissue. The purpose of this study is to record spatially resolved streaming potential measurements across intervertebral disks exposed to cyclic compressive loading. We hypothesize that the streaming potential profile across the disk will vary with radial position and frequency and is proportional to applied load amplitude, according to the presumed fluid-solid relative velocity and measured glycosaminoglycan content. Needle electrodes were fabricated using a linear array of AgAgCl micro-electrodes and inserted into human motion segments in the midline from anterior to posterior. They were connected to an amplifier to measure electrode potentials relative to the saline bath ground. Motion segments were loaded in axial compression under a preload of 500 N, sinusoidal amplitudes of +/-200 N and +/-400 N, and frequencies of 0.01 Hz, 0.1 Hz, and 1 Hz. Streaming potential data were normalized by applied force amplitude, and also compared with paired experimental measurements of glycosaminoglycans in each disk. Normalized streaming potentials varied significantly with sagittal position and there was a significant location difference at the different frequencies. Normalized streaming potential was largest in the central nucleus region at frequencies of 0.1 Hz and 1.0 Hz with values of approximately 3.5 microVN. Under 0.01 Hz loading, normalized streaming potential was largest in the outer annulus regions with a maximum value of 3.0 microVN. Correlations between streaming potential and glycosaminoglycan content were significant, with R(2) ranging from 0.5 to 0.8. Phasic relationships between applied force and electrical potential did not differ significantly by disk region or frequency, although the largest phase angles were observed at the outermost electrodes. Normalized streaming potentials were associated with glycosaminoglycan content, fluid, and ion transport. Results suggested that at higher frequencies the transport of water and ions in the central nucleus region may be larger, while at lower frequencies there is enhanced transport near the periphery of the annulus. This study provides data that will be helpful to validate multiphasic models of the disk.

A filtered solution of the primitive shallow-water equations

Abstract It is well documented that the direct solution of the primitive equation form (PE) of the shallow-water equations by conventional Galerkin finite elements fails to control numerical noise. While some alternative numerical techniques applied directly to the primitive equations have been successfully developed, a reformulation of the fundamental equations into the generalized wave continuity equation form (GWCE) provides an alternative noise-free procedure that has been used in numerous field applications. The selection of penalty parameter, ‘G’ for GWCE must, however, be carefully adjusted to achieve both mass balance and noise control. In this paper, a numerical method for solving the primitive equation with a filter is presented. It is shown that a filtered primitive equation method (FPE) controls numerical noise and is mass conservative. Since the FPE method derives its basis from observations about the behavior of GWCE, we present comparisons of the solution of the primitive equations via FPE with the solution of the wave equation via the GWCE form.