Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias

We introduce the concept of a contracting excitable medium that is capable of conducting non-linear waves of excitation that in turn initiate contraction. Furthermore, these kinematic deformations have a feedback effect on the excitation properties of the medium. Electrical characteristics resemble basic models of cardiac excitation that have been used to successfully study mechanisms of reentrant cardiac arrhythmias in electrophysiology. We present a computational framework that employs electromechanical and mechanoelectric feedback to couple a three-variable FitzHugh–Nagumo-type excitation-tension model to the non-linear stress equilibrium equations, which govern large deformation hyperelasticity. Numerically, the coupled electromechanical model combines a finite difference method approach to integrate the excitation equations, with a Galerkin finite element method to solve the equations governing tissue mechanics. We present example computations demonstrating various effects of contraction on stationary rotating spiral waves and spiral wave break. We show that tissue mechanics significantly contributes to the dynamics of electrical propagation, and that a coupled electromechanical approach should be pursued in future electrophysiological modelling studies.     

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

In this paper, we present computational techniques to investigate the effect of surface geometry on biological pattern formation. In particular, we study two-component, nonlinear reaction–diffusion (RD) systems on arbitrary surfaces. We build on standard techniques for linear and nonlinear analysis of RD systems and extend them to operate on large-scale meshes for arbitrary surfaces. In particular, we use spectral techniques for a linear stability analysis to characterise and directly compose patterns emerging from homogeneities. We develop an implementation using surface finite element methods and a numerical eigenanalysis of the Laplace–Beltrami operator on surface meshes. In addition, we describe a technique to explore solutions of the nonlinear RD equations using numerical continuation. Here, we present a multiresolution approach that allows us to trace solution branches of the nonlinear equations efficiently even for large-scale meshes. Finally, we demonstrate the working of our framework for two RD systems with applications in biological pattern formation: a Brusselator model that has been used to model pattern development on growing plant tips, and a chemotactic model for the formation of skin pigmentation patterns. While these models have been used previously on simple geometries, our framework allows us to study the impact of arbitrary geometries on emerging patterns.     

Adaptive Finite Element Analysis of the Anisotropic Biphasic Theory of Tissue-Equivalent Mechanics

The nonlinear partial differential equations of the anisotropic biphasic theory of tissue-equivalent mechanics are solved with axial symmetry by an adaptive finite element system. The adaptive procedure operates within a method-of-lines framework using finite elements in space and backward difference software in time. Spatial meshes are automatically refined, coarsened, and relocated in response to error indications and material deformation. Problems with arbitrarily complex two-dimensional regions may be addressed. With meshes graded in high-error regions, the adaptive solutions have fewer degrees of freedom than solutions with comparable accuracy obtained on fixed quasi-uniform meshes. The adaptive software is used to address problems involving an isometric cell traction assay, where a cylindrical tissue equivalent is adhered at its end to fixed circular platens; a prototypical bioartificial artery; and a novel configuration that is intended as an initial step in a study to determine bioartificial arteries having optimal collagen and cell concentrations.     

Finite element solution of the steady-state Smoluchowski equation for rate constant calculations

This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.     

A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease

We have previously described a new approach to planning treatments for cardiovascular disease, Simulation-Based Medical Planning, whereby a physician utilizes computational tools to construct and evaluate a combined anatomic/physiologic model to predict the outcome of alternative treatment plans for an individual patient. Current systems for Simulation-Based Medical Planning utilize finite element methods to solve the time-dependent, three-dimensional equations governing blood flow and provide detailed data on blood flow distribution, pressure gradients and locations of flow recirculation, low wall shear stress and high particle residence. However, these methods are computationally expensive and often require hours of time on parallel computers. This level of computation is necessary for obtaining detailed information about blood flow, but likely is unnecessary for obtaining information about mean flow rates and pressure losses. We describe, herein, a space-time finite element method for solving the one-dimensional equations of blood flow. This method is applied to compute flow rate and pressure in a single segment model, a bifurcation, an idealized model of the abdominal aorta, in three alternate treatment plans for a case of aorto-iliac occlusive disease and in a vascular bypass graft. All of these solutions were obtained in less than 5 min of computation time on a personal computer.     

A One-dimensional Finite Element Method for Simulation-based Medical Planning for Cardiovascular Disease

We have previously described a new approach to planning treatments for cardiovascular disease, Simulation-Based Medical Planning, whereby a physician utilizes computational tools to construct and evaluate a combined anatomic/physiologic model to predict the outcome of alternative treatment plans for an individual patient. Current systems for Simulation-Based Medical Planning utilize finite element methods to solve the time-dependent, three-dimensional equations governing blood flow and provide detailed data on blood flow distribution, pressure gradients and locations of flow recirculation, low wall shear stress and high particle residence. However, these methods are computationally expensive and often require hours of time on parallel computers. This level of computation is necessary for obtaining detailed information about blood flow, but likely is unnecessary for obtaining information about mean flow rates and pressure losses. We describe, herein, a space-time finite element method for solving the one-dimensional equations of blood flow. This method is applied to compute flow rate and pressure in a single segment model, a bifurcation, an idealized model of the abdominal aorta, in three alternate treatment plans for a case of aorto-iliac occlusive disease and in a vascular bypass graft. All of these solutions were obtained in less than 5 min of computation time on a personal computer.     

Adaptive Finite Element Analysis of the Anisotropic Biphasic Theory of Tissue-Equivalent Mechanics

Abstract

The nonlinear partial differential equations of the anisotropic biphasic theory of tissue-equivalent mechanics are solved with axial symmetry by an adaptive finite element system. The adaptive procedure operates within a method-of-lines framework using finite elements in space and backward difference software in time. Spatial meshes are automatically refined, coarsened, and relocated in response to error indications and material deformation. Problems with arbitrarily complex two-dimensional regions may be addressed. With meshes graded in high-error regions, the adaptive solutions have fewer degrees of freedom than solutions with comparable accuracy obtained on fixed quasi-uniform meshes. The adaptive software is used to address problems involving an isometric cell traction assay, where a cylindrical tissue equivalent is adhered at its end to fixed circular platens; a prototypical bioartificial artery; and a novel configuration that is intended as an initial step in a study to determine bioartificial arteries having optimal collagen and cell concentrations.     

The mesh-matching algorithm: an automatic 3D mesh generator for finite element structures

Several authors have employed finite element analysis for stress and strain analysis in orthopaedic biomechanics. Unfortunately, the definition of three-dimensional models is time consuming (mainly because of the manual 3D meshing process) and consequently the number of analyses to be performed is limited. The authors have investigated a new patient-specific method allowing automatically 3D mesh generation for structures as complex as bone for example. This method, called the mesh-matching (M-M) algorithm, generated automatically customized 3D meshes of anatomical structures from an already existing model. The M-M algorithm has been used to generate FE models of 10 proximal human femora from an initial one which had been experimentally validated. The automatically generated meshes seemed to demonstrate satisfying results.     

Smooth surface meshing for automated finite element model generation from 3D image data

Finite element (FE) modelling based on data from three-dimensional high-resolution computed tomography (CT) imaging systems provides a non-invasive method to assess structural mechanics. Automated mesh generation from these voxel based image data can be achieved by direct conversion to hexahedron elements, however these model representations have jagged edges. This paper proposes an automated method to generate smoothed FE meshes from voxel-based image data. Mesh fairing processes are utilized that allow constraints that control the smoothing process, and are computationally efficient. Surfaces of the mesh on the exterior, as well as interfaces between two tissues, can be smoothed by varying fairing parameters and constraint criteria. The method was tested on a variety of real and simulated three-dimensional data sets, resulting in both hexahedron and tetrahedron meshes. It was shown that the fairing process is linearly related to the number of smoothing iterations, and that peak stresses are reduced in FE simulations of the smoothed models. Although developed for micro-CT data sets, this fast and reliable mesh smoothing method could be applied to any three-dimensional image data where node and element connectivity have been defined.     

Numerical study to establish relationship between coagulation volume and target tip temperature during temperature-controlled radiofrequency ablation

The present study aims at proposing a relationship between the coagulation volume and the target tip temperature in different tissues (viz., liver, lung, kidney, and breast) during temperature-controlled radiofrequency ablation (RFA). A 20-min RFA has been modelled using commercially available monopolar multi-tine electrode subjected to different target tip temperatures that varied from 70°C to 100°C with an increment of 10°C. A closed-loop feedback proportional-integral-derivative (PID) controller has been employed within the finite element model to perform temperature-controlled RFA. The coagulation necrosis has been attained by solving the coupled electric field distribution, the Pennes bioheat and the first-order Arrhenius rate equations within the three-dimensional finite element model of different tissues. The computational study considers temperature-dependent electrical and thermal conductivities along with the non-linear piecewise model of blood perfusion. The comparison between coagulation volume obtained from the numerical and in vitro experimental studies has been done to evaluate the aptness of the numerical models. In the present study, a total of 20 numerical simulations have been performed along with 12 experiments on tissue-mimicking phantom gel using RFA device. The study revealed a strong dependence of the coagulation volume on the pre-set target tip temperature and ablation time during RFA application. Further, the effect of target tip temperature on the applied input voltage has been studied in different tissues. Based on the results attained from the numerical study, statistical correlations between the coagulation volume and treatment time have been developed at different target tip temperatures for each tissue.     

A new method to investigate in vivo knee behavior using a finite element model of the lower limb

Several finite element models have been developed for estimating the mechanical response of joint internal structures, where direct or indirect in vivo measurement is difficult or impossible. The quality of the predictions made by those models is largely dependent on the quality of the experimental data (e.g. load/displacement) used to drive them. Also numerical problems have been described in the literature when using implicit finite element techniques to simulate problems that involve contacts and large displacements. In this study, a unique strategy was developed combining high accuracy in vivo three-dimensional kinematics and a lower limb finite element model based on explicit finite element techniques. The method presents an analytical technique applied to a dynamic loading condition (impact during hopping on one leg). The validation of the lower limb model focused on the response of the whole model and the knee joint in particular to the imposed 3D femoral in vivo kinematics and ground reaction forces. The approach outlined in this study introduces a generic tool for the study of in vivo knee joint behavior.     

Anisotropic adaptive finite element method for modelling blood flow

In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier-Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field. Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.     

Graded meshes in bio-thermal problems with transmission-line modeling method

In this study, the transmission-line modeling (TLM) applied to bio-thermal problems was improved by incorporating several novel computational techniques, which include application of graded meshes which resulted in 9 times faster in computational time and uses only a fraction (16%) of the computational resources used by regular meshes in analyzing heat flow through heterogeneous media. Graded meshes, unlike regular meshes, allow heat sources to be modeled in all segments of the mesh. A new boundary condition that considers thermal properties and thus resulting in a more realistic modeling of complex problems is introduced. Also, a new way of calculating an error parameter is introduced. The calculated temperatures between nodes were compared against the results obtained from the literature and agreed within less than 1% difference. It is reasonable, therefore, to conclude that the improved TLM model described herein has great potential in heat transfer of biological systems.     

Anisotropic adaptive finite element method for modelling blood flow

In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier–Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field.

Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.     

Analysis of pulsatile blood flow: A carotid siphon model

Numerical results for axial and secondary flow velocity and pressure in a three-dimensional model of the human carotid siphon have been calculated; the investigations were carried out under physiologically relevant pulsatile flow conditoins. Time-dependent, three-dimensional Navier-Stokes equations were solved numerically by using a special finite element method. The results of the computer simulation presented here concentrate on the secondary motion effect during the pulsatile flow cycle in multiple three-dimensional curvatures.     

Trabecular bone strains around a dental implant and associated micromotions—A micro-CT-based three-dimensional finite element study

The first objective of this computational study was to assess the strain magnitude and distribution within the three-dimensional (3D) trabecular bone structure around an osseointegrated dental implant loaded axially. The second objective was to investigate the relative micromotions between the implant and the surrounding bone. The work hypothesis adopted was that these virtual measurements would be a useful indicator of bone adaptation (resorption, homeostasis, formation).In order to reach these objectives, a μCT-based finite element model of an oral implant implanted into a Berkshire pig mandible was developed along with a robust software methodology. The finite element mesh of the 3D trabecular bone architecture was generated from the segmentation of μCT scans. The implant was meshed independently from its CAD file obtained from the manufacturer. The meshes of the implant and the bone sample were registered together in an integrated software environment. A series of non-linear contact finite element (FE) analyses considering an axial load applied to the top of the implant in combination with three sets of mechanical properties for the trabecular bone tissue was devised. Complex strain distribution patterns are reported and discussed. It was found that considering the Young’s modulus of the trabecular bone tissue to be 5, 10 and 15 GPa resulted in maximum peri-implant bone microstrains of about 3000, 2100 and 1400. These results indicate that, for the three sets of mechanical properties considered, the magnitude of maximum strain lies within an homeostatic range known to be sufficient to maintain/form bone. The corresponding micro-motions of the implant with respect to the bone microstructure were shown to be sufficiently low to prevent fibrous tissue formation and to favour long-term osseointegration.