A nonlinear viscoelastic finite element model of polyethylene

A nonlinear viscoelastic finite element model of ultra-high molecular weight polyethylene (UHMWPE) was developed in this study. Eight cylindrical specimens were machined from ram extruded UHMWPE bar stock (GUR 1020) and tested under constant compression at 7% strain for 100 sec. The stress strain data during the initial ramp up to 7% strain was utilized to model the instantaneous stress-strain response using a Mooney-Rivlin material model. The viscoelastic behavior was modeled using the time-dependent relaxation in stress seen after the initial maximum stress was achieved using a stored energy formulation. A cylindrical model of similar dimensions was created using a finite element analysis software program. The cylinder was made up of hexahedral elements, which were given the material properties utilizing the instantaneous stress-strain curve and the energy-relaxation curve obtained from the experimental data. The cylinder was compressed between two flat rigid bodies that simulated the fixtures of the testing machine. Experimental stress-relaxation, creep and dynamic testing data were then used to validate the model. The mean error for predicted versus experimental data for stress relaxation at different strain levels was 4.2%. The mean error for the creep test was 7% and for dynamic test was 5.4%. Finally, dynamic loading in a hip arthroplasty was modeled and validated experimentally with an error of 8%. This study establishes a working finite element material model of UHMWPE that can be utilized to simulate a variety of postoperative arthroplasty conditions.     

A semi-empirical nonlinear viscoelastic model of the arterial wall

A mathematical model was developed for the characterization of quasistatic nonlinear viscoelastic behaviour of large arteries with activated smooth muscle. Arteries were considered to belong to the class of viscoelastic materials with fading memory and the first order term of the integral polynomial constitutive equations of Pipkin and Rogers satisfying the nonlinear superposition principle was modified to handle responses to increasing and decreasing loads independently. The two creep functions contained by the obtained one dimensional constitutive equation were determined experimentally from the series of creep and recovery tests of increasing amplitude performed on isolated canine iliac arteries following the activation of the vascular smooth muscle by normal dose of norepinephrine in vitro. Utilizing the constitutive equation of the arterial wall and the tabulated values of creep functions successive stress-strain hysteresis loops of various constant stress rates were simulated by digital computer. The computed hysteresis loops demonstrated the main characteristics, such as the weak and asymmetric rate-sensitivity of the experimentally observed hysteresis of arteries qualitatively well, thus allowing certain conclusions on the mechanism of quasistatic viscoelastic behaviour of vascular smooth muscle.     

Modified Bilston nonlinear viscoelastic model for finite element head injury studies

This paper proposes a modified nonlinear viscoelastic Bilston model (Bilston et al., 2001, Biorheol., 38, pp. 335-345). for the modeling of brain tissue constitutive properties. The modified model can be readily implemented in a commercial explicit finite element (FE) code, PamCrash. Critical parameters of the model have been determined through a series of rheological tests on porcine brain tissue samples and the time-temperature superposition (TTS) principle has been used to extend the frequency to a high region. Simulations by using PamCrash are compared with the test results. Through the use of the TTS principle, the mechanical and rheological behavior at high frequencies up to 10(4) rads may be obtained. This is important because the properties of the brain tissue at high frequencies and impact rates are especially relevant to studies of traumatic head injury. The averaged dynamic modulus ranges from 130 Pa to 1500 Pa and loss modulus ranges from 35 Pa to 800 Pa in the frequency regime studied (0.01 rads to 3700 rads). The errors between theoretical predictions and averaged relaxation test results are within 20% for strains up to 20%. The FEM simulation results are in good agreement with experimental results. The proposed model will be especially useful for application to FE analysis of the head under impact loads. More realistic analysis of head injury can be carried out by incorporating the nonlinear viscoelastic constitutive law for brain tissue into a commercial FE code.     

A framework for incorporating 3D hyperelastic vascular wall models in 1D blood flow simulations

We present a novel framework for investigating the role of vascular structure on arterial haemodynamics in large vessels, with a special focus on the human common carotid artery (CCA). The analysis is carried out by adopting a three-dimensional (3D) derived, fibre-reinforced, hyperelastic structural model, which is coupled with an axisymmetric, reduced order model describing blood flow. The vessel transmural pressure and lumen area are related via a Holzapfel–Ogden type of law, and the residual stresses along the thickness and length of the vessel are also accounted for. After a structural characterization of the adopted hyperelastic model, we investigate the link underlying the vascular wall response and blood-flow dynamics by comparing the proposed framework results against a popular tube law. The comparison shows that the behaviour of the model can be captured by the simpler linear surrogate only if a representative value of compliance is applied. Sobol’s multi-variable sensitivity analysis is then carried out in order to identify the extent to which the structural parameters have an impact on the CCA haemodynamics. In this case, the local pulse wave velocity (PWV) is used as index for representing the arterial transmission capacity of blood pressure waveforms. The sensitivity analysis suggests that some geometrical factors, such as the stress-free inner radius and opening angle, play a major role on the system’s haemodynamics. Subsequently, we quantified the differences in haemodynamic variables obtained from different virtual CCAs, tube laws and flow conditions. Although each artery presents a distinct vascular response, the differences obtained across different flow regimes are not significant. As expected, the linear tube law is unable to accurately capture all the haemodynamic features characterizing the current model. The findings from the sensitivity analysis are further confirmed by investigating the axial stretching effect on the CCA fluid dynamics. This factor does not seem to alter the pressure and flow waveforms. On the contrary, it is shown that, for an axially stretched vessel, the vascular wall exhibits an attenuation in absolute distension and an increase in circumferential stress, corroborating the findings of previous studies. This analysis shows that the new model offers a good balance between computational complexity and physics captured, making it an ideal framework for studies aiming to investigate the profound link between vascular mechanobiology and blood flow.

     

Comparative study of flow in right-sided and left-sided aortas: numerical simulations in patient-based models

A right-sided aorta is a rare malformation which may be associated with other various types of congenital heart disease. We utilised haemodynamic, echocardiographic measurements, computerised tomography and image reconstruction software packages that were integrated in a computational fluid dynamics model to determine blood flow patterns in patient-based aortas. In the left-sided aorta, a systolic clockwise rotational component was present, while helical flow was depicted in the aortic arch that was converted in the descending aorta as counter-rotating vortices with accompanying retrograde flow. The right-sided configuration has not altered the orientation of the three-dimensional vortex, but intensification of polymorphic flow patterns, alterations in wall shear stress distribution and development of a lateral pressure gradient at the area of an aneurysmal anomaly was observed. Moreover, increments of Reynolds, Womersley and Dean numbers were evident. These phenomena along with the formation of the aneurysm might influence cardiovascular risk in patients with right-sided aortas.     

The importance of nonlinear tissue modelling in finite element simulations of infant head impacts

Despite recent efforts on the development of finite element (FE) head models of infants, a model capable of capturing head responses under various impact scenarios has not been reported. This is hypothesized partially attributed to the use of simplified linear elastic models for soft tissues of suture, scalp and dura. Orthotropic elastic constants are yet to be determined to incorporate the direction-specific material properties of infant cranial bone due to grain fibres radiating from the ossification centres. We report here on our efforts in advancing the above-mentioned aspects in material modelling in infant head and further incorporate them into subject-specific FE head models of a newborn, 5- and 9-month-old infant. Each model is subjected to five impact tests (forehead, occiput, vertex, right and left parietal impacts) and two compression tests. The predicted global head impact responses of the acceleration–time impact curves and the force–deflection compression curves for different age groups agree well with the experimental data reported in the literature. In particular, the newly developed Ogden hyperelastic model for suture, together with the nonlinear modelling of scalp and dura mater, enables the models to achieve more realistic impact performance compared with linear elastic models. The proposed approach for obtaining age-dependent skull bone orthotropic material constants counts both an increase in stiffness and decrease in anisotropy in the skull bone—two essential biological growth parameters during early infancy. The profound deformation of infant head causes a large stretch at the interfaces between the skull bones and the suture, suggesting that infant skull fractures are likely to initiate from the interfaces; the impact angle has a profound influence on global head impact responses and the skull injury metrics for certain impact locations, especially true for a parietal impact.     

On the anatomical definition of arterial networks in blood flow simulations: comparison of detailed and simplified models

Biomechanics and Modeling in Mechanobiology - The goal of this work is to assess the impact of vascular anatomy definition degree in the predictions of blood flow models of the arterial network. To...     

Inlet boundary conditions for blood flow simulations in truncated arterial networks

In the context of patient-specific cardiovascular applications, hemodynamics models (going from 3D to 0D) are often limited to a part of the arterial tree. This restriction implies the set up of artificial interfaces with the remaining parts of the cardiovascular system. In particular, the inlet boundary condition is crucial: it supplies the impulsion to the system and receives the reflected backward waves created by the distal network. Some aspects of this boundary condition need to be properly defined such as the treatment of backward waves (reflected or absorbed) and the value of the imposed hemodynamic wave (total or forward component). Most authors prescribe as inlet boundary condition (BC) the total measured variable (pressure, velocity or flow rate) in a reflective way. We show that with this type of inlet boundary condition, the model does not produce physiological waveforms. We suggest instead to prescribe only the forward component of the prescribed variable in an absorbing way. In this way, the computed reflected waves superpose with the prescribed forward waves to produce the total wave at the inlet. In this work, different inlet boundary conditions are implemented and compared for a 1D blood flow model. We test our boundary conditions on a truncated arterial model presented in the literature as well as on a patient-specific lower-limb model of a femoral bypass. We show that with this new boundary condition, a much better fitting is observed on the shape and intensity of the simulated pressure and velocity waves.     

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.     

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.     

An efficient full space-time discretization method for subject-specific hemodynamic simulations of cerebral arterial blood flow with distensible wall mechanics

A computationally inexpensive mathematical solution approach using orthogonal collocations for space discretization with temporal Fourier series is proposed to compute subject-specific blood flow in distensible vessels of large cerebral arterial networks. Several models of wall biomechanics were considered to assess their impact on hemodynamic predictions. Simulations were validated against in vivo blood flow measurements in six human subjects. The average root-mean-square relative differences were found to be less than 4.3% for all subjects with a linear elastic wall model. This discrepancy decreased further in a viscoelastic Kelvin-Voigt biomechanical wall. The results provide support for the use of collocation-Fourier series approach to predict clinically relevant blood flow distribution and collateral blood supply in large portions of the cerebral circulation at reasonable computational costs. It thus opens the possibility of performing computationally inexpensive subject-specific simulations that are robust and fast enough to predict clinical results in real time on the same day.     

A novel,FFT-based one-dimensional blood flow solution method for arterial network

In the present work, we propose an FFT-based method for solving blood flow equations in an arterial network with variable properties and geometrical changes. An essential advantage of this approach is in correctly accounting for the vessel skin friction through the use of Womersley solution. To incorporate nonlinear effects, a novel approximation method is proposed to enable calculation of nonlinear corrections. Unlike similar methods available in the literature, the set of algebraic equations required for every harmonic is constructed automatically. The result is a generalized, robust and fast method to accurately capture the increasing pulse wave velocity downstream as well as steepening of the pulse front. The proposed method is shown to be appropriate for incorporating correct convection and diffusion coefficients. We show that the proposed method is fast and accurate and it can be an effective tool for 1D modelling of blood flow in human arterial networks.

     

Numerical simulations of the blood flow in the patient-specific arterial cerebral circle region

The Cerebral Circle Region, also known as the Circle of Willis (CoW), is a loop of arteries that form arterial connections between supply arteries to distribute blood throughout the cerebral mass. Among the population, only 25% to 50% have a complete system of arteries forming the CoW. 3D time-varying simulations for three different patient-specific artery anatomies of CoW were performed in order to gain a better insight into the phenomena existing in the cerebral blood flow. The models reconstructed on the basis of computer tomography images start from the aorta and include the largest arteries that supply the CoW and the arteries of CoW. Velocity values measured during the ultrasound examination have been compared with the results of simulations. It is shown that the flow in the right anterior artery in some cases may be supplied from the left internal carotid artery via the anterior communicating artery. The investigations conducted show that the computational fluid dynamic tool, which provides high resolution in both time and space domains, can be used to support physicians in diagnosing patients of different ages and various anatomical arterial structures.     

Search for critical loading condition of the spine--a meta analysis of a nonlinear viscoelastic finite element model

The relative vulnerability of spinal motion segments to different loading combinations remains unknown. The meta-analysis described here using the results of a validated L2-L3 nonlinear viscoelastic finite element model was designed to investigate the critical loading and its effect on the internal mechanics of the human lumbar spine. A Box-Behnken experimental design was used to design the magnitude of seven independent variables associated with loads, rotations and velocity of motion. Subsequently, an optimization method was used to find the primary and secondary variables that influence spine mechanical output related to facet forces, disc pressure, ligament forces, annulus matrix compressive/shear stresses and anulus fibers strain. The mechanical responses with respect to the two most-relevant variables were then regressed linearly using the response surface quadratic model. Axial force and sagittal rotation were identified as the most-relevant variables for mechanical responses. The procedure developed can be used to find the critical loading for finite element models with multi input variables. The derived meta-models can be used to predict the risk associated with various loading parameters and in setting safer load limits.     

A comparative computational study of blood flow through prosthetic heart valves using the finite element method

The steady flow of blood through three common types of prosthetic heart valves was simulated numerically using the finite element method. The velocity, pressure and stress fields were obtained for the disk-type, tilting-disk and ball-type prosthetic heart valves in aortic position, for increasing Reynolds numbers up to 900, 1500 and 2000 respectively. Computer graphics of calculated velocities are presented, showing in detail the accelerated flow, recirculation and stagnation areas developed in the prosthesis. Maximum wall shear stresses were found at 0.5, 1.4, and 1.2 diameters from the sewing ring downstream for the disk, tilting-disk and ball valves being the values of 55, 18 and 33 dyn cm-2 respectively. In the vicinity of the occluder, maximum shear stresses of 38, 30 and 47 dyn cm-2 respectively were computed. The flow characteristics and performance for each valve are compared, the results are presented in terms of energy loss and maximum shear stress. The velocity and stress fields are compared with in vitro evaluations found in the literature.     

Comparative analysis of bone remodelling models with respect to computerised tomography-based finite element models of bone

Subject-specific finite element models are an extensively used tool for the numerical analysis of the biomechanical behaviour of human bones. However, bone modelling is not an easy task due to the complex behaviour of bone tissue, involving non-homogeneous and anisotropic mechanical properties. Moreover, bone is a living tissue and therefore its microstructure and mechanical properties evolve with time in a known process called bone remodelling. This phenomenon has been widely studied, many being the numerical models that have been formulated to predict density distribution and its evolution in several bones. The aim of the present study is to assess the capability of a bone remodelling model to predict the bone density distribution of different types of human bone (femur, tibia and mandible) comparing the obtained results with the bone density estimated by means of computerised tomography. Good accuracy was observed for the bone remodelling predictions including the thickness of the cortical layer.     

Feasibility of coronary blood flow simulations using mid-fidelity numeric and geometric models

Biomechanics and Modeling in Mechanobiology - The fractional flow reserve index (FFR) is currently used as a gold standard to quantify coronary stenosis’s functional relevance. Due to its...