Showing papers by "Gerhard Holzapfel published in 2019"

On fibre dispersion modelling of soft biological tissues: a review.

Abstract: Collagen fibres within fibrous soft biological tissues such as artery walls, cartilage, myocardiums, corneas and heart valves are responsible for their anisotropic mechanical behaviour. It has recently been recognized that the dispersed orientation of these fibres has a significant effect on the mechanical response of the tissues. Modelling of the dispersed structure is important for the prediction of the stress and deformation characteristics in (patho)physiological tissues under various loading conditions. This paper provides a timely and critical review of the continuum modelling of fibre dispersion, specifically, the angular integration and the generalized structure tensor models. The models are used in representative numerical examples to fit sets of experimental data that have been obtained from mechanical tests and fibre structural information from second-harmonic imaging. In particular, patches of healthy and diseased aortic tissues are investigated, and it is shown that the predictions of the models fit very well with the data. It is straightforward to use the models described herein within a finite-element framework, which will enable more realistic (and clinically relevant) boundary-value problems to be solved. This also provides a basis for further developments of material models and points to the need for additional mechanical and microstructural data that can inform further advances in the material modelling.

Anisotropic finite strain viscoelasticity: Constitutive modeling and finite element implementation

Abstract: A new anisotropic finite strain viscoelastic model is presented, which is based on the Holzapfel type anisotropic hyperelastic strain-energy function. The anisotropic viscous part is set to be independent from the isotropic viscous part. A corresponding multiplicative decomposition of the deformation gradient is presented, and a specific definition of the anisotropic viscous fiber term. A new method to develop the evolution equations of the viscous internal variables is also provided. The time derivatives of the internal variables for the isotropic and anisotropic viscous parts are obtained from the evolution equation of the second Piola–Kirchhoff stress for the viscous part. The corresponding analytical validation of non-negative dissipation using the second law of thermodynamics is provided. The incompressible plane stress case is used to achieve an analytical solution for the proposed constitutive model. A good agreement between the finite element results and the analytical solution is obtained. Finally, some numerical simulations are presented, including the viscous hysteresis response, experimental data fitting and a relaxation test.

Multiscale modeling of fiber recruitment and damage with a discrete fiber dispersion method

Abstract: Recently, we introduced a discrete fiber dispersion model based on triangular discretization of a unit sphere with a finite number of elementary areas. Over each elementary area, we define a representative fiber direction and an elementary fiber density based on the fiber dispersion. The strain energy of fibers distributed in each elementary area is then approximated by the deformation of the representative fiber direction weighted by the corresponding elementary fiber density. A summation of fiber contributions of all elementary areas yields the resultant fiber strain energy. However, in that study we did not consider fiber recruitment, softening and damage. The goal of this study is to incorporate these important properties of collagen fibers into the constitutive model. We first define a fiber recruitment stretch at which the fiber becomes straightened. Then, we adopt the continuum damage mechanics method for modeling fiber softening and damage. We implemented the proposed model in a finite element program and verified it with three representative examples including a uniaxial extension test of a dog-bone shaped specimen up to failure. The computational solution agrees well with the experimental result. In conclusion, the proposed model is able to capture fiber recruitment, softening, and damage. Future studies with more complex boundary conditions are necessary to verify this approach.

On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

Abstract: Quasi-incompressible behavior is a desired feature in several constitutive models within the finite elasticity of solids, such as rubber-like materials and some fiber-reinforced soft biological tissues. The Q1P0 finite element formulation, derived from the three-field Hu–Washizu variational principle, has hitherto been exploited along with the augmented Lagrangian method to enforce incompressibility. This formulation typically uses the unimodular deformation gradient. However, contributions by Sansour (Eur J Mech A Solids 27:28–39, 2007) and Helfenstein et al. (Int J Solids Struct 47:2056–2061, 2010) conspicuously demonstrate an alternative concept for analyzing fiber reinforced solids, namely the use of the (unsplit) deformation gradient for the anisotropic contribution, and these authors elaborate on their proposals with analytical evidence. The present study handles the alternative concept from a purely numerical point of view, and addresses systematic comparisons with respect to the classical treatment of the Q1P0 element and its coalescence with the augmented Lagrangian method by means of representative numerical examples. The results corroborate the new concept, show its numerical efficiency and reveal a direct physical interpretation of the fiber stretches.

Computational modeling of progressive damage and rupture in fibrous biological tissues: application to aortic dissection

Abstract: This study analyzes the lethal clinical condition of aortic dissections from a numerical point of view. On the basis of previous contributions by Gultekin et al. (Comput Methods Appl Mech Eng 312:542-566, 2016 and 331:23-52, 2018), we apply a holistic geometrical approach to fracture, namely the crack phase-field, which inherits the intrinsic features of gradient damage and variational fracture mechanics. The continuum framework captures anisotropy, is thermodynamically consistent and is based on finite strains. The balance of linear momentum and the crack evolution equation govern the coupled mechanical and phase-field problem. The solution scheme features the robust one-pass operator-splitting algorithm upon temporal and spatial discretizations. Based on experimental data of diseased human thoracic aortic samples, the elastic material parameters are identified followed by a sensitivity analysis of the anisotropic phase-field model. Finally, we simulate an incipient propagation of an aortic dissection within a multi-layered segment of a thoracic aorta that involves a prescribed initial tear. The finite element results demonstrate a severe damage zone around the initial tear and exhibit a rather helical crack pattern, which aligns with the fiber orientation. It is hoped that the current contribution can provide some directions for further investigations of this disease.

Anisotropic residual stresses in arteries.

Abstract: The paper provides a deepened insight into the role of anisotropy in the analysis of residual stresses in arteries. Residual deformations are modelled following Holzapfel and Ogden (Holzapfel and Ogden 2010, J. R. Soc. Interface 7, 787-799. ( doi:10.1098/rsif.2009.0357 )), which is based on extensive experimental data on human abdominal aortas (Holzapfel et al. 2007, Ann. Biomed. Eng. 35, 530-545. ( doi:10.1007/s10439-006-9252-z )) and accounts for both circumferential and axial residual deformations of the individual layers of arteries-intima, media and adventitia. Each layer exhibits distinctive nonlinear and anisotropic mechanical behaviour originating from its unique microstructure; therefore, we use the most general form of strain-energy function (Holzapfel et al. 2015, J. R. Soc. Interface 12, 20150188. ( doi:10.1098/rsif.2015.0188 )) to derive residual stresses for each layer individually. Finally, the systematic experimental data (Niestrawska et al. 2016, J. R. Soc. Interface 13, 20160620. ( doi:10.1098/rsif.2016.0620 )) on both mechanical and structural properties of the different layers of the human abdominal aorta facilitate our discussion on (i) the importance of anisotropy in modelling residual stresses; (ii) the variability of residual stresses within the same class of tissue, the abdominal aorta; (iii) the limitations of conventional opening angle method to account for complex residual deformations; and (iv) the effect of residual stresses on the loaded configuration of the aorta mimicking in vivo conditions.

Cholesterol Deficiency Causes Impaired Osmotic Stability of Cultured Red Blood Cells.

Abstract: Ex vivo generation of red blood cells (cRBCs) is an attractive tool in basic research and for replacing blood components donated by volunteers. As a prerequisite for the survival of cRBCs during storage as well as in the circulation, the quality of the membrane is of utmost importance. Besides the cytoskeleton and embedded proteins, the lipid bilayer is critical for membrane integrity. Although cRBCs suffer from increased fragility, studies investigating the lipid content of their membrane are still lacking. We investigated the membrane lipid profile of cRBCs from CD34+ human stem and progenitor cells compared to native red blood cells (nRBCs) and native reticulocytes (nRETs). Ex vivo erythropoiesis was performed in a well-established liquid assay. cRBCs showed a maturation grade between nRETs and nRBCs. High-resolution mass spectrometry analysis for cholesterol and the major phospholipid classes, phosphatidylcholine, phosphatidylethanolamine, phosphatidylinositol, phosphatidylserine, sphingomyelin and lysophosphatidylcholin, demonstrated severe cholesterol deficiency in cRBCs. Although cRBCs showed normal deformability capacity, they suffered from increased hemolysis due to minimal changes in the osmotic conditions. After additional lipid supplementation, especially cholesterol during culturing, the cholesterol content of cRBCs increased to a subnormal amount. Concurrently, the osmotic resistance recovered completely and became comparable to that of nRETs. Minor differences in the amount of phospholipids in cRBCs compared to native cells could mainly be attributed to the ongoing membrane remodeling process from the reticulocyte to the erythrocyte stage. Obtained results demonstrate severe cholesterol deficiency as a reason for enhanced fragility of cRBCs. Therefore, the supplementation of lipids, especially cholesterol during ex vivo erythropoiesis may overcome this limitation and strengthens the survival of cRBCs ex vivo and in vivo.

An investigation of the glycosaminoglycan contribution to biaxial mechanical behaviours of porcine atrioventricular heart valve leaflets

Abstract: The atrioventricular heart valve (AHV) leaflets have a complex microstructure composed of four distinct layers: atrialis, ventricularis, fibrosa and spongiosa. Specifically, the spongiosa layer is primarily proteoglycans and glycosaminoglycans (GAGs). Quantification of the GAGs' mechanical contribution to the overall leaflet function has been of recent focus for aortic valve leaflets, but this characterization has not been reported for the AHV leaflets. This study seeks to expand current GAG literature through novel mechanical characterizations of GAGs in AHV leaflets. For this characterization, mitral and tricuspid valve anterior leaflets (MVAL and TVAL, respectively) were: (i) tested by biaxial mechanical loading at varying loading ratios and by stress-relaxation procedures, (ii) enzymatically treated for removal of the GAGs and (iii) biaxially mechanically tested again under the same protocols as in step (i). Removal of the GAG contents from the leaflet was conducted using a 100 min enzyme treatment to achieve approximate 74.87% and 61.24% reductions of all GAGs from the MVAL and TVAL, respectively. Our main findings demonstrated that biaxial mechanical testing yielded a statistically significant difference in tissue extensibility after GAG removal and that stress-relaxation testing revealed a statistically significant smaller stress decay of the enzyme-treated tissue than untreated tissues. These novel findings illustrate the importance of GAGs in AHV leaflet behaviour, which can be employed to better inform heart valve therapeutics and computational models.

On the importance of modeling balloon folding, pleating, and stent crimping: An FE study comparing experimental inflation tests.

Abstract: Finite element (FE)-based studies of preoperative processes such as folding, pleating, and stent crimping with a comparison with experimental inflation tests are not yet available. Therefore, a novel workflow is presented in which residual stresses of balloon folding and pleating, as well as stent crimping, and the geometries of all contact partners were ultimately implemented in an FE code to simulate stent expansion by using an implicit solver. The numerical results demonstrate that the incorporation of residual stresses and strains experienced during the production step significantly increased the accuracy of the subsequent simulations, especially of the stent expansion model. During the preoperative processes, stresses inside the membrane and the stent material also reached a rather high level. Hence, there can be no presumption that balloon catheters or stents are undamaged before the actual surgery. The implementation of the realistic geometry, in particular the balloon tapers, and the blades of the process devices improved the simulation of the expansion mechanisms, such as dogboning, concave bending, or overexpansion of stent cells. This study shows that implicit solvers are able to precisely simulate the mentioned preoperative processes and the stent expansion procedure without a preceding manipulation of the simulation time or physical mass.

Micromechanically-motivated analysis of fibrous tissue.

Abstract: Collagen fibers are the main load bearing component in fibrous tissues. Systematic analyses of their structure and orientation are thus crucial for the development of material models that enable to predict the mechanical tissue response. To this end, biaxial tests at different stretch ratios were performed on two tissue samples of the medial layer extracted from a human aorta. The tissues were loaded in the circumferential and axial directions simultaneously. We develop here a micromechanical model which is based on structural parameters of collagen fibers that were extracted from second-harmonic generation images of the two samples. The tissue is modeled as a periodic six-layered laminate in which the individual layers are treated as periodic fibrous structures with one family of fibers. We make use of the Hill-Mandel theory in the context of periodic homogenization to determine the overall mechanical tissue response. Both the analytical and numerical models are able to capture the overall mechanical response of the two tissue samples using a straightforward representation of the tissue structure together with a limited set of material parameters. Up to 10% of strains the model captures the almost linear response of both tissue samples. Beyond that stretch level the stiffening of the tissues becomes more evident, especially in the circumferential direction. In cases where the axial stretch is larger than the circumferential stretch the predictions are somewhat stiffer, while a very good agreement is obtained when the circumferential stretch is dominant. The stiffening of one tissue sample was substantially larger than the other, implying that higher-order stiffening mechanisms may kick in at larger strains. Our sensitivity analyses reveal that the parameters of the material model and the fiber dispersion have a minor effect on the tissue response. The novel modeling approach has the potential to reduce the need of time-consuming experimental data of the mechanical behavior of fibrous tissues.

Numerical simulation of the viral entry into a cell driven by the receptor diffusion

Abstract: This study focuses on the receptor driven endocytosis typical of viral entry into a cell. A locally increased density of receptors at the time of contact between the cell and the virus is necessary in this case. The virus is considered as a substrate with fixed receptors on its surface, while the receptors of the host cell are free to move over its membrane, allowing a local change in their concentration. In the contact zone the membrane inflects and forms an envelope around the virus. The created vesicle imports its cargo into the cell. The described process is simulated by the diffusion equation accompanied by two boundary conditions. The first boundary condition states that the conservation of binders expressed as the local rate of change of density has to be equal to the negative of the local flux divergence. The second boundary condition represents the energy balance condition with contributions due to the binding of receptors, the free energy of the membrane, its curvature and the kinetic energy due to the motion of the front. The described moving boundary problem in terms of the binder density and the velocity of the adhesion front is well posed and relies on biomechanically motivated assumptions. The problem is numerically solved by using the finite difference method, and the illustrative examples have been chosen to show the influence of the mobility of the receptors and of their initial densities on the velocity of the process.