Automated adaptive analysis of the biphasic equations for soft tissue mechanics using a posteriori error indicators

Abstract

SUMMARYHydrated soft tissues of the human musculoskeletal system, such as articular cartilage in diarthrodial joints or the annulus fibrosis of the intervertebral disk, are accurately represented by a biphasic continuum model consisting of an incompressible solid phase (collagen and proteoglycan) and incompressible, inviscid fluid (interstitial fluid) and derive from the continuum theory of mixtures. These tissues exhibit a viscoelastic-type response which is caused by the diffusive drag of the fluid phase as it flows past the solid phase. In this study an automated, adaptive, finite element solution of the governing biphasic equations is presented. The finite element formulation is based on a mixed-penalty approach in which the penalty form of the continuity equation for the mixture is included in the weak form. Pressure, solid and fluid velocities are interpolated independently, and the coefficients of the C -' pressure field are eliminated at the element level. The resulting matrix form is a system of first order differential equations which is solved via standard finite difference methods. Mesh generation and updating, including both refinement and coarsening, for the two-dimensional examples examined in this study are performed using Finite Quadtree. The adaptive analysis is based on an error indicator which is the L2 norm of the difference between the finite element solution and a projected finite element solution. Total stress, calculated as the sum of the solid and fluid phase stresses, is used in the error indicator. Rezoning is accomplished by transferring the finite element solution for the primary variables onto the locally updated mesh using a projected field. These projected values allow the finite difference algorithm to proceed in time using the updated mesh. The accuracy and effectiveness of this adaptive finite element analysis is demonstrated using two-dimensional axisymmetric problems corresponding to the unconfined compression of a cylindrical disk of soft tissue and the indentation of a thin sheet of soft tissue. The method is shown to effectively capture the steep gradients in both problems and to produce solutions in good agreement with independent, converged, numerical solutions.

Advances in Cardiac Signal Processing

“…Triangular interpolation approximates the RR interval distribution by a linear function and the baseline width of this approximation triangle is used as a measure of the heart rate variability index [109,110]. The triangular index is a measure, where the length of RR intervals serves as the x-axis of the plot and the number of each RR interval length serves as the y-axis.…”

“…[96,97], numerical methods [98,99], finite element analysis [91,101,102] and error analysis. Also pointed out are errors during measurement of strain distribution affect stresses and myocardial properties [107][108][109]. An augmented Lagrangian formulation to enforce the saturation and incompressibility constraint-related to soft biological tissue has been used by Levenston et al [106].…”

A mixed finite element formulation of triphasic mechano-electrochemical theory for charged, hydrated biological soft tissues

“…With the progress of biphasic "nite element method, a variety of problems such as friction e!ect on the uncon"ned compression [27], indentation analysis under "nite deformation [28] and non-isotropic analysis [29] have been solved. Some challenging topics such as automated-mesh-adaptivity [30] and joint contact problems [31,32] are being addressed. Recently, some investigators began to describe the contributions of ions using "nite element method [11, 33}35].…”

FEBio: History and Advances

“…FE implementations of biphasic theory for cartilage were presented by Spilker and coworkers, for small strains (17) and finite deformation of the solid matrix (18), using penalty methods to enforce incompressibility. They considered various refinements to their approach (19), including hybrid, mixed-penalty and velocity-pressure formulations (20; 21) as well as tissue anisotropy (22) and explicit modeling of interstitial fluid viscosity (23). u-p (displacement-pressure) formulations of the biphasic mixture theory were presented by Oomens et al (13) and Wayne et al (24), and a biphasic formulation accommodating solid matrix viscoelasticity was developed by Ehlers et al (25).…”