Frank P. T. Baaijens

Mixed finite element methods for viscoelastic flow analysis: a review

The progress made during the past decade in the application of mixed finite element methods to solve viscoelastic flow problems using differential constitutive equations is reviewed. The algorithmic developments are discussed in detail. Starting with the classical mixed formulation, the elastic viscous stress splitting (EVSS) method as well as the related discrete EVSS and the so-called EVSS-G method are discussed among others. Furthermore, stabilization techniques such as the streamline upwind Petrov‐Galerkin (SUPG) and the discontinuous Galerkin (DG) are reviewed. The performance of the numerical schemes for both smooth and non-smooth benchmark problems is discussed. Finally, the capabilities of viscoelastic flow solvers to predict experimental observations are reviewed. # 1998 Elsevier Science B.V. All rights reserved.

A fictitious domain/mortar element method for fluid–structure interaction

A new method for the computational analysis of fluid-structure interaction of a Newtonian fluid with slender bodies is developed. It combines ideas of the fictitious domain and the mortar element method by imposing continuity of the velocity field along an interface by means of Lagrange multipliers. The key advantage of the method is that it circumvents the need for complicated mesh movement strategies common in arbitrary Lagrangian-Eulerian (ALE) methods, usually used for this purpose

Differential constitutive equations for polymer melts: The extended Pom-Pom model

The Pom‐Pom model, recently introduced by McLeish and Larson @J. Rheol. 42, 81‐110~1998!#, is a breakthrough in the field of viscoelastic constitutive equations. With this model, a correct nonlinear behavior in both elongation and shear is accomplished. The original differential equations, improved with local branch-point displacement, are modified to overcome three drawbacks: solutions in steady state elongation show discontinuities, the equation for orientation is unbounded for high strain rates, the model does not have a second normal stress difference in shear. The modified extended Pom‐Pom model does not show the three problems and is easy for implementation in finite element packages, because it is written as a single equation. Quantitative agreement is shown with experimental data in uniaxial, planar, equibiaxial elongation as well as shear, reversed flow and step-strain for two commercial low density polyethylene ~LDPE! melts and one high density polyethylene ~HDPE! melt. Such a good agreement over a full range of well defined rheometric experiments, i.e., shear, including reversed flow for one LDPE melt, and different elongational flows, is exceptional.

Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour

An improved gradient-enhanced approach for softening elasto-plasticity is proposed, which in essence is fully nonlocal, i.e. an equivalent integral nonlocal format exists. The method utilises a nonlocal field variable in its constitutive framework, but in contrast to the integral models computes this nonlocal field with a gradient formulation. This formulation is considered ‘implicit’ in the sense that it strictly incorporates the higher-order gradients of the local field variable indirectly, unlike the common (explicit) gradient approaches. Furthermore, this implicit gradient formulation constitutes an additional partial differential equation (PDE) of the Helmholtz type, which is solved in a coupled fashion with the standard equilibrium condition. Such an approach is particularly advantageous since it combines the long-range interactions of an integral (nonlocal) model with the computational efficiency of a gradient formulation. Although these implicit gradient approaches have been successfully applied within damage mechanics, e.g. for quasi-brittle materials, the first attempts were deficient for plasticity. On the basis of a thorough comparison of the gradient-enhancements for plasticity and damage this paper rephrases the problem, which leads to a formulation that overcomes most reported problems. The two-dimensional finite element implementation for geometrically linear plain strain problems is presented. One- and two-dimensional numerical examples demonstrate the ability of this method to numerically model irreversible deformations, accompanied by the intense localisation of deformation and softening up to complete failure.

Tissue Engineering of Human Heart Valve Leaflets: A Novel Bioreactor for a Strain-Based Conditioning Approach

Current mechanical conditioning approaches for heart valve tissue engineering concentrate on mimicking the opening and closing behavior of the leaflets, either or not in combination with tissue straining. This study describes a novel approach by mimicking only the diastolic phase of the cardiac cycle, resulting in tissue straining. A novel, yet simplified, bioreactor system was developed for this purpose by applying a dynamic pressure difference over a closed tissue engineered valve, thereby inducing dynamic strains within the leaflets. Besides the use of dynamic strains, the developing leaflet tissues were exposed to prestrain induced by the use of a stented geometry. To demonstrate the feasibility of this strain-based conditioning approach, human heart valve leaflets were engineered and their mechanial behavior evaluated. The actual dynamic strain magnitude in the leaflets over time was estimated using numerical analyses. Preliminary results showed superior tissue formation and non-linear tissue-like mechanical properties in the strained valves when compared to non-loaded tissue strips. In conclusion, the strain-based conditioning approach, using both prestrain and dynamic strains, offers new possibilities for bioreactor design and optimization of tissue properties towards a tissue-engineered aortic human heart valve replacement.

Minimally-invasive implantation of living tissue engineered heart valves: a comprehensive approach from autologous vascular cells to stem cells.

OBJECTIVES The aim of this study was to demonstrate the feasibility of combining the novel heart valve replacement technologies of: 1) tissue engineering; and 2) minimally-invasive implantation based on autologous cells and composite self-expandable biodegradable biomaterials. BACKGROUND Minimally-invasive valve replacement procedures are rapidly evolving as alternative treatment option for patients with valvular heart disease. However, currently used valve substitutes are bioprosthetic and as such have limited durability. To overcome this limitation, tissue engineering technologies provide living autologous valve replacements with regeneration and growth potential. METHODS Trileaflet heart valves fabricated from biodegradable synthetic scaffolds, integrated in self-expanding stents and seeded with autologous vascular or stem cells (bone marrow and peripheral blood), were generated in vitro using dynamic bioreactors. Subsequently, the tissue engineered heart valves (TEHV) were minimally-invasively implanted as pulmonary valve replacements in sheep. In vivo functionality was assessed by echocardiography and angiography up to 8 weeks. The tissue composition of explanted TEHV and corresponding control valves was analyzed. RESULTS The transapical implantations were successful in all animals. The TEHV demonstrated in vivo functionality with mobile but thickened leaflets. Histology revealed layered neotissues with endothelialized surfaces. Quantitative extracellular matrix analysis at 8 weeks showed higher values for deoxyribonucleic acid, collagen, and glycosaminoglycans compared to native valves. Mechanical profiles demonstrated sufficient tissue strength, but less pliability independent of the cell source. CONCLUSIONS This study demonstrates the principal feasibility of merging tissue engineering and minimally-invasive valve replacement technologies. Using adult stem cells is successful, enabling minimally-invasive cell harvest. Thus, this new technology may enable a valid alternative to current bioprosthetic devices.

Autologous Human Tissue-Engineered Heart Valves Prospects for Systemic Application

Background— Tissue engineering represents a promising approach for the development of living heart valve replacements. In vivo animal studies of tissue-engineered autologous heart valves have focused on pulmonary valve replacements, leaving the challenge to tissue engineer heart valves suitable for systemic application using human cells. Methods and Results— Tissue-engineered human heart valves were analyzed up to 4 weeks and conditioning using bioreactors was compared with static culturing. Tissue formation and mechanical properties increased with time and when using conditioning. Organization of the tissue, in terms of anisotropic properties, increased when conditioning was dynamic in nature. Exposure of the valves to physiological aortic valve flow demonstrated proper opening motion. Closure dynamics were suboptimal, most likely caused by the lower degree of anisotropy when compared with native aortic valve leaflets. Conclusions— This study presents autologous tissue-engineered heart valves based on human saphenous vein cells and a rapid degrading synthetic scaffold. Tissue properties and mechanical behavior might allow for use as living aortic valve replacements.

A structural constitutive model for collagenous cardiovascular tissues incorporating the angular fiber distribution

Accurate constitutive models are required to gain further insight into the mechanical behavior of cardiovascular tissues. In this study, a structural constitutive framework for cardiovascular tissues is introduced that accounts for the angular distribution of collagen fibers. To demonstrate its capabilities, the model is applied to study the biaxial behavior of the arterial wall and the aortic valve. The pressure-radius relationships of the arterial wall accurately describe experimentally observed sigma-shaped curves. In addition, the nonlinear and anisotropic mechanical properties of the aortic valve can be analyzed with the proposed model. We expect that the current model offers strong possibilities to further investigate the complex mechanical behavior of cardiovascular tissues, including their response to mechanical stimuli.

Viscoelastic flow past a confined cylinder of a low-density polyethylene melt

The capabilities of the exponential version of the Phan-Thien-Tanner (PTT) model and the Giesekus model to predict stress fields for the viscoelastic flow of a low density polyethylene melt around a confined cylinder are investigated. Computations are based on a newly developed version of the discontinuous Galerkin method. This method gives convergent results up to a Deborah number of 2.5 for the falling sphere in a tube benchmark problem. Moreover, the specific implicit-explicit implementation allows the efficient resolution of problems with multiple relaxation times which are mandatory for polymer melts. Experimentally, stress fields are related to birefringence distributions by means of the stress optical rule. Three different fits, of equal quality, to available viscometric shear data are used: two for the PTT model and one for the Giesekus model. Comparison of computed and measured fringes reveals that neither of the models is capable of describing the full birefringence pattern sufficiently well. In particular it appears difficult to predict both the birefringent tail at the wake of the cylinder that is dominated by elongational effects and the fringe pattern between cylinder and the walls where a combined shear-elongational flow is present.

TISSUE ENGINEERING OF HEART VALVES: ADVANCES AND CURRENT CHALLENGES

It is estimated that the number of patients requiring heart valve replacement will triple over the next five decades. None of the current replacement valves can fully restore native valve function because they lack growth and remodeling capabilities. Heart valve tissue engineering is a promising technology to overcome these limitations. Various approaches are being employed, either aimed at development of the valve substitute in vitro or at the use of the regenerative potential of the body (in situ) for the tissue culture phase. This review provides an overview of the progress within both the in vitro and in situ tissue engineering approaches for trileaflet heart valve tissue engineering. Current challenges with these approaches are discussed, focusing in particular on the use of synthetic scaffold materials.