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.     

Requirements for mesh resolution in 3D computational hemodynamics

Computational techniques are widely used for studying large artery hemodynamics. Current trends favor analyzing flow in more anatomically realistic arteries. A significant obstacle to such analyses is generation of computational meshes that accurately resolve both the complex geometry and the physiologically relevant flow features. Here we examine, for a single arterial geometry, how velocity and wall shear stress patterns depend on mesh characteristics. A well-validated Navier-Stokes solver was used to simulate flow in an anatomically realistic human right coronary artery (RCA) using unstructured high-order tetrahedral finite element meshes. Velocities, wall shear stresses (WSS), and wall shear stress gradients were computed on a conventional "high-resolution" mesh series (60,000 to 160,000 velocity nodes) generated with a commercial meshing package. Similar calculations were then performed in a series of meshes generated through an adaptive mesh refinement (AMR) methodology. Mesh-independent velocity fields were not very difficult to obtain for both the conventional and adaptive mesh series. However, wall shear stress fields, and, in particular, wall shear stress gradient fields, were much more difficult to accurately resolve. The conventional (nonadaptive) mesh series did not show a consistent trend towards mesh-independence of WSS results. For the adaptive series, it required approximately 190,000 velocity nodes to reach an r.m.s. error in normalized WSS of less than 10 percent. Achieving mesh-independence in computed WSS fields requires a surprisingly large number of nodes, and is best approached through a systematic solution-adaptive mesh refinement technique. Calculations of WSS, and particularly WSS gradients, show appreciable errors even on meshes that appear to produce mesh-independent velocity fields.     

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.     

Absorbing phenomena and escaping time for Muller's ratchet in adaptive landscape

Background

The accumulation of deleterious mutations of a population directly contributes to the fate as to how long the population would exist, a process often described as Muller's ratchet with the absorbing phenomenon. The key to understand this absorbing phenomenon is to characterize the decaying time of the fittest class of the population. Adaptive landscape introduced by Wright, a re-emerging powerful concept in systems biology, is used as a tool to describe biological processes. To our knowledge, the dynamical behaviors for Muller's ratchet over the full parameter regimes are not studied from the point of the adaptive landscape. And the characterization of the absorbing phenomenon is not yet quantitatively obtained without extraneous assumptions as well.

Methods

We describe how Muller's ratchet can be mapped to the classical Wright-Fisher process in both discrete and continuous manners. Furthermore, we construct the adaptive landscape for the system analytically from the general diffusion equation. The constructed adaptive landscape is independent of the existence and normalization of the stationary distribution. We derive the formula of the single click time in finite and infinite potential barrier for all parameters regimes by mean first passage time.

Results

We describe the dynamical behavior of the population exposed to Muller's ratchet in all parameters regimes by adaptive landscape. The adaptive landscape has rich structures such as finite and infinite potential, real and imaginary fixed points. We give the formula about the single click time with finite and infinite potential. And we find the single click time increases with selection rates and population size increasing, decreases with mutation rates increasing. These results provide a new understanding of infinite potential. We analytically demonstrate the adaptive and unadaptive states for the whole parameters regimes. Interesting issues about the parameters regions with the imaginary fixed points is demonstrated. Most importantly, we find that the absorbing phenomenon is characterized by the adaptive landscape and the single click time without any extraneous assumptions. These results suggest a graphical and quantitative framework to study the absorbing phenomenon.

     

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.     

A generalized finite difference method for modeling cardiac electrical activation on arbitrary, irregular computational meshes

A generalized finite difference (GFD) method is presented that can be used to solve the bi-domain equations modeling cardiac electrical activity. Classical finite difference methods have been applied by many researchers to the bi-domain equations. However, these methods suffer from the limitation of requiring computational meshes that are structured and orthogonal. Finite element or finite volume methods enable the bi-domain equations to be solved on unstructured meshes, although implementations of such methods do not always cater for meshes with varying element topology. The GFD method solves the bi-domain equations on arbitrary and irregular computational meshes without any need to specify element basis functions. The method is useful as it can be easily applied to activation problems using existing meshes that have originally been created for use by finite element or finite difference methods. In addition, the GFD method employs an innovative approach to enforcing nodal and non-nodal boundary conditions. The GFD method performs effectively for a range of two and three-dimensional test problems and when computing bi-domain electrical activation moving through a fully anisotropic three-dimensional model of canine ventricles.     

肝细胞永生化策略研究进展

正常体细胞在细胞分裂过程中由于无法维持端粒的长度和染色体的稳定而衰老、死亡。建立基因稳定、无致瘤性的永生化细胞是永生化研究的目的所在。利用Cre-loxP系统构建的可恢复性永生化细胞可用于以细胞为基础的临床治疗。分化良好并具有肝脏正常合成代谢功能的永生化肝细胞在生物医学研究,尤其是在肝细胞移植、生物人工肝支持治疗以及药理学和毒理学研究方面具有广泛的应用前景。     

Patient-specific modelling of the foot: automated hexahedral meshing of the bones

Subject-specific finite element modelling is a powerful tool for carrying out controlled investigations of the effects of geometric and material property differences on performance and injury risk. Unfortunately, the creation of suitable meshes for these models is a challenging and time-intensive task. This paper presents an automated method of generating fully hexahedral meshes of the bones of the feet which requires only surface representations as inputs. The method is outlined and example meshes, using two human feet and the foot of a Japanese macaque, are given to demonstrate its flexibility. Mesh quality is also evaluated for the calcaneus, first metatarsal, navicular and talus. Streamlining the generation of finite element meshes of the foot will ease investigations into the patient-specific biomechanics of injury.     

The steady-state properties of an ion exchange membrane with mobile sites

A study of the properties of the steady states of a system composed of two solutions separated by an ion exchange membrane having mobile sites is presented. It is assumed that the membrane is impermeable to coions; the solutions contain no more than two species of counterions, both of the same valence; and no flow of bulk solution occurs. Assuming that all ions are completely dissociated, behave ideally, and have constant mobilities throughout the membrane, explicit expressions are derived for the steady states of the electric current, individual fluxes, and concentration profiles as functions of the compositions of the solutions and of the difference of electric potential between them. The derived expressions are compared with those for an ion exchange membrane having fixed sites; and it is found that the expressions of certain quantities, such as the difference of electric potential between the two solutions for zero current or the ratio of the fluxes of the counterions as functions of the external parameters of the system, are the same for both types of membranes. On the other hand, differences in the behavior of the two types of membranes are found from other expressions-for example, the current-voltage relationship. In the mobile site ion exchanger the current asymptotically approaches finite limiting values for high positive and negative voltages while in the fixed site ion exchanger it is the conductance which approaches finite limiting values.     

A novel design of bioartificial kidneys with improved cell performance and haemocompatibility

Treatment with bioartificial kidneys had beneficial effects in animal experiments and improved survival of critically ill patients with acute kidney injury in a Phase II clinical trial. However, a Phase II b clinical trial failed. This and other results suggested various problems with the current design of bioartificial kidneys. We propose a novel design to improve various properties of device, including haemocompatibility and cell performance. An important feature of the novel design is confinement of the blood to the lumina of the hollow fibre membranes. This avoids exposure of the blood to the non‐haemocompatible outer surfaces of hollow fibre membranes, which usually occurs in bioartificial kidneys. We use these outer surfaces as substrate for cell growth. Our results show that commercial hollow fibre membranes can be directly applied in the bioreactor when human primary renal proximal tubular cells are grown in this configuration, and no coatings are required for the formation of robust and functional renal epithelia. Furthermore, we demonstrate that the bioreactor unit produces significant amounts of interleukins. This result helps to understand the immunomodulatory effects of bioartificial kidneys, which have been observed previously. The novel bioartificial kidney design outlined here and the results obtained would be expected to improve the safety and performance of bioartificial kidneys and to contribute to a better understanding of their effects.     

Adaptive properties of living beings: proposal for a generic mechanism. (Self-programming machines III)

Living systems are capable to have appropriate responses to unpredictable environment. This kind of self-organization seems to operate as a self-programming machine, i.e. an organization able to modify itself. Until now the models of self-organization of living beings proposed are functions solutions of differential systems or transition functions of automata. These functions are fixed and these models are therefore unable to modify their organization. On the other hand, computer science propose a lot of models having the properties of adaptive systems of living beings, but all these models depend on the comparison between a goal and the results and ingenious choices of parameters by programmers, whereas there are no programmer's intention nor choice in the living systems. From two best known examples of adaptive systems of living beings, nervous system and immune system that have in common that the external signals modify the rewriting of their organization and therefore work as self-organizing machines, we devised machines with a finite set of inputs, based upon a recurrence, are able to rewrite their organization (Self-programming machines or m(sp)) whenever external conditions vary and have striking properties of adaptation. M(sp) have similar properties whatever the operation defining the recurrence maybe. These results bring us to make the following statement: adaptive properties of living systems can be explained by their ability to rewrite their organization whenever external conditions vary under the only assumption that the rewriting mechanism be a deterministic constant recurrence in a finite state set.     

Characterization of the three-compartment gel-entrapment porcine hepatocyte bioartificial liver

A hybrid bioartificial liver device supporting a large mass of cells expressing differentiated hepatocyte metabolic capabilities is necessary for the successful treatment of fulminant hepatic failure. The three-compartment gel-entrapment porcine hepatocyte bioartificial liver was designed to provide "bridge" support to transplantation or until native liver recovery is achieved for patients with acute liver failure. The device is an automated mammalian cell culture system supporting 6-7 × 109 porcine hepatocytes entrapped in a collagen matrix and inoculated into the capillary lumen spaces of two 100 kDa molecular mass cut-off hollow fiber bioreactors. Gel contraction recreates a small lumen space within the hollow fiber which allows for the delivery of a nutrient medium. This configuration supported hepatocyte viability and differentiated phenotype as measured by albumin synthesis, ureagenesis, oxygen consumption, and vital dye staining during both cell culture and ex vivo application. The hollow fiber membrane was also shown to isolate the cells from xenogenic immunoglobulin attack. The gel-entrapment bioartificial liver maintained a large mass of functional hepatocytes by providing a three-dimensional cell culture matrix, by delivering basal nutrients through lumen media perfusion, and by preventing rejection of the xenocytes. These features make this device a favorable candidate for the treatment of clinical fulminant hepatic failure.     

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.     

Bioartificial kidney. II. A convective flow model of a hollow fiber bioartificial renal tubule

An existing model for volume transport in the rat proximal tubule was modified and applied to a bioartificial renal tubule. The predicted volume transport agrees well with experimental data. The volume transport model was coupled to the analytic solutions of flow in the bioartificial renal tubule bioreactor, operated in the open-shell mode with perfusion in both the lumen and surrounding shell. The results suggest that the performance of a multifiber bioreactor can be improved by controlling shell inlet conditions and fiber spacing. These results have important implications for the design and analysis not only for the bioartificial renal tubule bioreactor but also for the general case of hollow fiber bioreactors operated in the open-shell mode with perfusion in both the lumen and surrounding shell.     

Continuum model of cell adhesion and migration

The motility of cells crawling on a substratum has its origin in a thin cell organ called lamella. We present a 2-dimensional continuum model for the lamella dynamics of a slowly migrating cell, such as a human keratinocyte. The central components of the model are the dynamics of a viscous cytoskeleton capable to produce contractile and swelling stresses, and the formation of adhesive bonds in the plasma cell membrane between the lamella cytoskeleton and adhesion sites at the substratum. We will demonstrate that a simple mechanistic model, neglecting the complicated signaling pathways and regulation processes of a living cell, is able to capture the most prominent aspects of the lamella dynamics, such as quasi-periodic protrusions and retractions of the moving tip, retrograde flow of the cytoskeleton and the related accumulation of focal adhesion complexes in the leading edge of a migrating cell. The developed modeling framework consists of a nonlinearly coupled system of hyperbolic, parabolic and ordinary differential equations for the various molecular concentrations, two elliptic equations for cytoskeleton velocity and hydrodynamic pressure in a highly viscous two-phase flow, with appropriate boundary conditions including equalities and inequalities at the moving boundary. In order to analyse this hybrid continuum model by numerical simulations for different biophysical scenarios, we use suitable finite element and finite volume schemes on a fixed triangulation in combination with an adaptive level set method describing the free boundary dynamics.     

Molecular investigation of genetic assimilation during the rapid adaptive radiations of East African cichlid fishes

Adaptive radiations are characterized by adaptive diversification intertwined with rapid speciation within a lineage resulting in many ecologically specialized, phenotypically diverse species. It has been proposed that adaptive radiations can originate from ancestral lineages with pronounced phenotypic plasticity in adaptive traits, facilitating ecologically driven phenotypic diversification that is ultimately fixed through genetic assimilation of gene regulatory regions. This study aimed to investigate how phenotypic plasticity is reflected in gene expression patterns in the trophic apparatus of several lineages of East African cichlid fishes, and whether the observed patterns support genetic assimilation. This investigation used a split brood experimental design to compare adaptive plasticity in species from within and outside of adaptive radiations. The plastic response was induced in the crushing pharyngeal jaws through feeding individuals either a hard or soft diet. We find that nonradiating, basal lineages show higher levels of adaptive morphological plasticity than the derived, radiated lineages, suggesting that these differences have become partially genetically fixed during the formation of the adaptive radiations. Two candidate genes that may have undergone genetic assimilation, gif and alas1, were identified, in addition to alterations in the wiring of LPJ patterning networks. Taken together, our results suggest that genetic assimilation may have dampened the inducibility of plasticity related genes during the adaptive radiations of East African cichlids, flattening the reaction norms and canalizing their feeding phenotypes, driving adaptation to progressively more narrow ecological niches.     

Maintenance of Acidic Lateral Intercellular Spaces by Endogenous Fixed Buffers in MDCK Cell Epithelium

The lateral intercellular spaces (LIS) of MDCK cell epithelia grown on permeable supports are about 0.4 pH units acidic to the bathing solutions, presumably because of buffering by the fixed negative charges on the surface of the lateral cell membranes. To test the hypothesis that fixed buffers are responsible for the acidity, a theoretical and experimental approach was developed for the determination of the concentration and pK of the fixed buffer constituted by the glycocalyx. The pH of the solution in the LIS was measured by ratiometric fluorescence microscopy while the buffer concentration or composition of the bathing solutions was altered. In addition, the divalent cation Sr²⁺ was added to the perfusion solutions to displace protons from the fixed buffer sites for the determination of the fixed buffer properties. We conclude that the LIS contain 3.7 mm of pK 6.2 fixed buffer and that this buffer is responsible for the acidic microenvironment in the LIS. Received: 9 April 1998/Revised: 28 July 1998     

ImageParser: a tool for finite element generation from three-dimensional medical images

Background

The finite element method (FEM) is a powerful mathematical tool to simulate and visualize the mechanical deformation of tissues and organs during medical examinations or interventions. It is yet a challenge to build up an FEM mesh directly from a volumetric image partially because the regions (or structures) of interest (ROIs) may be irregular and fuzzy.

Methods

A software package, ImageParser, is developed to generate an FEM mesh from 3-D tomographic medical images. This software uses a semi-automatic method to detect ROIs from the context of image including neighboring tissues and organs, completes segmentation of different tissues, and meshes the organ into elements.

Results

The ImageParser is shown to build up an FEM model for simulating the mechanical responses of the breast based on 3-D CT images. The breast is compressed by two plate paddles under an overall displacement as large as 20% of the initial distance between the paddles. The strain and tangential Young's modulus distributions are specified for the biomechanical analysis of breast tissues.

Conclusion

The ImageParser can successfully exact the geometry of ROIs from a complex medical image and generate the FEM mesh with customer-defined segmentation information.