Finite Element Framework for Computational Fluid Dynamics in FEBio

The Problem With National Institute of Standards and Technology Thermodynamics Tables in Continuum Mechanics

Thermodynamics is a fundamental topic of continuum mechanics and biomechanics, with a wide range of applications to physiological and biological processes. This study addresses two fundamental limitations of current thermodynamic treatments. First, thermodynamics tables distributed online by the U.S. National Institute of Standards and Technology (NIST) report properties of fluids as a function of absolute temperature T and absolute pressure P. These properties include mass density ρ, specific internal energy u, enthalpy h=u+P/ρ, and entropy s. However, formulations of jump conditions across phase boundaries derived from Newton's second law of motion and the first law of thermodynamics employ the gauge pressure p=P-Pr, where Pr is an arbitrarily selected referential absolute pressure. Interchanging p with P is not innocuous as it alters tabulated NIST values for u while keeping h and s unchanged. Using p for functions of state and governing equations solves the problem with using NIST entries for the specific internal energy u in standard thermodynamics tables and analyses of phase transformation in continuum mechanics. Second, constitutive models for the free energy of fluids, such as water and air, are not typically provided in standard thermodynamics treatments. This study proposes a set of constitutive models and validates them against suitably modified NIST data.

A Biphasic Fluid-Structure Interaction Model of Backflow During Infusion Into Agarose Gel

The efficacy of convection-enhanced delivery as a technique to treat disorders of the central nervous system is limited by backflow, in which the infused fluid flows backward along surface of the catheter rather than toward the targeted area. In order to improve treatment protocols, finite element models of backflow have been developed to understand the underlying physics. García et al. (2013, "Description and Validation of a Finite Element Model of Backflow During Infusion Into a Brain Tissue Phantom," ASME J. Comput. Nonlinear Dyn., 8(1), p. 011017) presented a finite element model that accounted for the flow in the annular gap that develops between the tissue and the outer surface of the catheter by using a layer of biphasic elements with a formula for the axial hydraulic conductivity to represent annular Poiseuille flow. In this study, we present a generalization of that model using fluid-FSI and biphasic-FSI elements that are recently available in febio. We demonstrate that our model of a 0.98 mm radius catheter is able to reproduce experimental backflow lengths and maximum fluid pressures for infusions into a brain tissue surrogate and that it agrees well with the previous model by García et al. (2013, "Description and Validation of a Finite Element Model of Backflow During Infusion Into a Brain Tissue Phantom," ASME J. Comput. Nonlinear Dyn., 8(1), p. 011017). The model predicts that the backflow length and the total amount of flow into the hemispherical region forward of the catheter tip is comparable for two different catheter sizes, albeit at a higher fluid pressure for the smaller catheter. This biphasic-FSI model has the potential to be extended to a stepped catheter geometry, which has been shown in experiments to be successful in controlling backflow.

Finite Element Implementation of Computational Fluid Dynamics With Reactive Neutral and Charged Solute Transport in FEBio

The objective of this study was to implement a novel fluid-solutes solver into the open-source finite element software FEBio, that extended available modeling capabilities for biological fluids and fluid-solute mixtures. Using a reactive mixture framework, this solver accommodates diffusion, convection, chemical reactions, electrical charge effects, and external body forces, without requiring stabilization methods that were deemed necessary in previous computational implementations of the convection-diffusion-reaction equation at high Peclet numbers. Verification and validation problems demonstrated the ability of this solver to produce solutions for Peclet numbers as high as 1011, spanning the range of physiological conditions for convection-dominated solute transport. This outcome was facilitated by the use of a formulation that accommodates realistic values for solvent compressibility, and by expressing the solute mass balance such that it properly captured convective transport by the solvent and produced a natural boundary condition of zero diffusive solute flux at outflow boundaries. Since this numerical scheme was not necessarily foolproof, guidelines were included to achieve better outcomes that minimize or eliminate the potential occurrence of numerical artifacts. The fluid-solutes solver presented in this study represents an important and novel advancement in the modeling capabilities for biomechanics and biophysics as it allows modeling of mechanobiological processes via the incorporation of chemical reactions involving neutral or charged solutes within dynamic fluid flow. The incorporation of charged solutes in a reactive framework represents a significant novelty of this solver. This framework also applies to a broader range of nonbiological applications.

The Effects of leaflet material properties on the simulated function of regurgitant mitral valves

Advances in three-dimensional imaging provide the ability to construct and analyze finite element (FE) models to evaluate the biomechanical behavior and function of atrioventricular valves. However, while obtaining patient-specific valve geometry is now possible, non-invasive measurement of patient-specific leaflet material properties remains nearly impossible. Both valve geometry and tissue properties play a significant role in governing valve dynamics, leading to the central question of whether clinically relevant insights can be attained from FE analysis of atrioventricular valves without precise knowledge of tissue properties. As such we investigated 1) the influence of tissue extensibility and 2) the effects of constitutive model parameters and leaflet thickness on simulated valve function and mechanics. We compared metrics of valve function (e.g., leaflet coaptation and regurgitant orifice area) and mechanics (e.g., stress and strain) across one normal and three regurgitant mitral valve (MV) models with common mechanisms of regurgitation (annular dilation, leaflet prolapse, leaflet tethering) of both moderate and severe degree. We developed a novel fully-automated approach to accurately quantify regurgitant orifice areas of complex valve geometries. We found that the relative ordering of the mechanical and functional metrics was maintained across a group of valves using material properties up to 15% softer than the representative adult mitral constitutive model. Our findings suggest that FE simulations can be used to qualitatively compare how differences and alterations in valve structure affect relative atrioventricular valve function even in populations where material properties are not precisely known.

Adaptive Wall Shear Stress Imaging in Phantoms, Simulations and In Vivo

WSS measurement is challenging since it requires sensitive flow measurements at a distance close to the wall. The aim of this study is to develop an ultrasound imaging technique which combines vector flow imaging with an unsupervised data clustering approach that automatically detects the region close to the wall with optimally linear flow profile, to provide direct and robust WSS estimation. The proposed technique was evaluated in phantoms, mimicking normal and atherosclerotic vessels, and spatially registered Fluid Structure Interaction (FSI) simulations. A relative error of 6.7% and 19.8% was obtained for peak systolic (WSSPS) and end diastolic (WSSED) WSS in the straight phantom, while in the stenotic phantom, a good similarity was found between measured and simulated WSS distribution, with a correlation coefficient, R, of 0.89 and 0.85 for WSSPS and WSSED, respectively. Moreover, the feasibility of the technique to detect pre-clinical atherosclerosis was tested in an atherosclerotic swine model. Six swines were fed atherogenic diet, while their left carotid artery was ligated in order to disturb flow patterns. Ligated arterial segments that were exposed to low WSSPS and WSS characterized by high frequency oscillations at baseline, developed either moderately or highly stenotic plaques (p < 0.05). Finally, feasibility of the technique was demonstrated in normal and atherosclerotic human subjects. Atherosclerotic carotid arteries with low stenosis had lower WSSPS as compared to control subjects (p < 0.01), while in one subject with high stenosis, elevated WSS was found on an arterial segment, which coincided with plaque rupture site, as determined through histological examination.

Cerebral hemodynamics during atrial fibrillation: Computational fluid dynamics analysis of lenticulostriate arteries using 7 T high-resolution magnetic resonance imaging

Atrial fibrillation (AF) is the most common cardiac arrhythmia, inducing irregular and faster heart beating. Aside from disabling symptoms-such as palpitations, chest discomfort, and reduced exercise capacity-there is growing evidence that AF increases the risk of dementia and cognitive decline, even in the absence of clinical strokes. Among the possible mechanisms, the alteration of deep cerebral hemodynamics during AF is one of the most fascinating and least investigated hypotheses. Lenticulostriate arteries (LSAs)-small perforating arteries perpendicularly departing from the anterior and middle cerebral arteries and supplying blood flow to basal ganglia-are especially involved in silent strokes and cerebral small vessel diseases, which are considered among the main vascular drivers of dementia. We propose for the first time a computational fluid dynamics analysis to investigate the AF effects on the LSAs hemodynamics by using 7 T high-resolution magnetic resonance imaging (MRI). We explored different heart rates (HRs)-from 50 to 130 bpm-in sinus rhythm and AF, exploiting MRI data from a healthy young male and internal carotid artery data from validated 0D cardiovascular-cerebral modeling as inflow condition. Our results reveal that AF induces a marked reduction of wall shear stress and flow velocity fields. This study suggests that AF at higher HR leads to a more hazardous hemodynamic scenario by increasing the atheromatosis and thrombogenesis risks in the LSAs region.

Generation and evaluation of input values for computational analysis of transport processes within tissue cultures

Techniques for tissue culture have seen significant advances during the last decades and novel 3D cell culture systems have become available. To control their high complexity, experimental techniques and their Digital Twins (modelling and computational tools) are combined to link different variables to process conditions and critical process parameters. This allows a rapid evaluation of the expected product quality. However, the use of mathematical simulation and Digital Twins is critically dependent on the precise description of the problem and correct input parameters. Errors here can lead to dramatically wrong conclusions. The intention of this review is to provide an overview of the state-of-the-art and remaining challenges with respect to generating input values for computational analysis of mass and momentum transport processes within tissue cultures. It gives an overview on relevant aspects of transport processes in tissue cultures as well as modelling and computational tools to tackle these problems. Further focus is on techniques used for the determination of cell-specific parameters and characterization of culture systems, including sensors for on-line determination of relevant parameters. In conclusion, tissue culture techniques are well-established, and modelling tools are technically mature. New sensor technologies are on the way, especially for organ chips. The greatest remaining challenge seems to be the proper addressing and handling of input parameters required for mathematical models. Following Good Modelling Practice approaches when setting up and validating computational models is, therefore, essential to get to better estimations of the interesting complex processes inside organotypic tissue cultures in the future.

Patient-Specific Inverse Modeling of In Vivo Cardiovascular Mechanics with Medical Image-Derived Kinematics as Input Data: Concepts, Methods, and Applications

Inverse modeling approaches in cardiovascular medicine are a collection of methodologies that can provide non-invasive patient-specific estimations of tissue properties, mechanical loads, and other mechanics-based risk factors using medical imaging as inputs. Its incorporation into clinical practice has the potential to improve diagnosis and treatment planning with low associated risks and costs. These methods have become available for medical applications mainly due to the continuing development of image-based kinematic techniques, the maturity of the associated theories describing cardiovascular function, and recent progress in computer science, modeling, and simulation engineering. Inverse method applications are multidisciplinary, requiring tailored solutions to the available clinical data, pathology of interest, and available computational resources. Herein, we review biomechanical modeling and simulation principles, methods of solving inverse problems, and techniques for image-based kinematic analysis. In the final section, the major advances in inverse modeling of human cardiovascular mechanics since its early development in the early 2000s are reviewed with emphasis on method-specific descriptions, results, and conclusions. We draw selected studies on healthy and diseased hearts, aortas, and pulmonary arteries achieved through the incorporation of tissue mechanics, hemodynamics, and fluid-structure interaction methods paired with patient-specific data acquired with medical imaging in inverse modeling approaches.

A Hybrid Biphasic Mixture Formulation for Modeling Dynamics in Porous Deformable Biological Tissues

The primary aim of this study is to establish the theoretical foundations for a solid-fluid biphasic mixture domain that can accommodate inertial effects and a viscous interstitial fluid, which can interface with a dynamic viscous fluid domain. Most mixture formulations consist of constituents that are either all intrinsically incompressible or compressible, thereby introducing inherent limitations. In particular, mixtures with intrinsically incompressible constituents can only model wave propagation in the porous solid matrix, whereas those with compressible constituents require internal variables, and related evolution equations, to distinguish the compressibility of the solid and fluid under hydrostatic pressure. In this study, we propose a hybrid framework for a biphasic mixture where the skeleton of the porous solid is intrinsically incompressible but the interstitial fluid is compressible. We define a state variable as a measure of the fluid volumetric strain. Within an isothermal framework, the Clausius-Duhem inequality shows that a function of state arises for the fluid pressure as a function of this strain measure. We derive jump conditions across hybrid biphasic interfaces, which are suitable for modeling hydrated biological tissues. We then illustrate this framework using confined compression and dilatational wave propagation analyses. The governing equations for this hybrid biphasic framework reduce to those of the classical biphasic theory whenever the bulk modulus of the fluid is set to infinity and inertia terms and viscous fluid effects are neglected. The availability of this novel framework facilitates the implementation of finite element solvers for fluid-structure interactions at interfaces between viscous fluids and porous-deformable biphasic domains, which can include fluid exchanges across those interfaces.

A Hybrid Reactive Multiphasic Mixture With a Compressible Fluid Solvent

Mixture theory is a general framework that has been used to model mixtures of solid, fluid, and solute constituents, leading to significant advances in modeling the mechanics of biological tissues and cells. Though versatile and applicable to a wide range of problems in biomechanics and biophysics, standard multiphasic mixture frameworks incorporate neither dynamics of viscous fluids nor fluid compressibility, both of which facilitate the finite element implementation of computational fluid dynamics solvers. This study formulates governing equations for reactive multiphasic mixtures where the interstitial fluid has a solvent which is viscous and compressible. This hybrid reactive multiphasic framework uses state variables that include the deformation gradient of the porous solid matrix, the volumetric strain and rate of deformation of the solvent, the solute concentrations, and the relative velocities between the various constituents. Unlike standard formulations which employ a Lagrange multiplier to model fluid pressure, this framework requires the formulation of a function of state for the pressure, which depends on solvent volumetric strain and solute concentrations. Under isothermal conditions the formulation shows that the solvent volumetric strain remains continuous across interfaces between hybrid multiphasic domains. Apart from the Lagrange multiplier-state function distinction for the fluid pressure, and the ability to accommodate viscous fluid dynamics, this hybrid multiphasic framework remains fully consistent with standard multiphasic formulations previously employed in biomechanics. With these additional features, the hybrid multiphasic mixture theory makes it possible to address a wider range of problems that are important in biomechanics and mechanobiology.

Effect of Subject-Specific, Spatially Reduced, and Idealized Boundary Conditions on the Predicted Hemodynamic Environment in the Murine Aorta

Mouse models of atherosclerosis have become effective resources to study atherogenesis, including the relationship between hemodynamics and lesion development. Computational methods aid the prediction of the in vivo hemodynamic environment in the mouse vasculature, but careful selection of inflow and outflow boundary conditions (BCs) is warranted to promote model accuracy. Herein, we investigated the impact of animal-specific versus reduced/idealized flow boundary conditions on predicted blood flow patterns in the mouse thoracic aorta. Blood velocities were measured in the aortic root, arch branch vessel, and descending aorta in ApoE^-/- mice using phase-contrast MRI. Computational geometries were derived from micro-CT imaging and combinations of high-fidelity or reduced/idealized MR-derived BCs were applied to predict the bulk flow field and hemodynamic metrics (e.g., wall shear stress, WSS; cross-flow index, CFI). Results demonstrate that pressure-free outlet BCs significantly overestimate outlet flow rates as compared to measured values. When compared to models that incorporate 3-component inlet velocity data [[Formula: see text]] and time-varying outlet mass flow splits [[Formula: see text]] (i.e., high-fidelity model), neglecting in-plane inlet velocity components (i.e., [Formula: see text])) leads to errors in WSS and CFI values ranging from 10 to 30% across the model domain whereas the application of a steady outlet mass flow splits results in negligible differences in these hemodynamics metrics. This investigation highlights that 3-component inlet velocity data and at least steady mass flow splits are required for accurate predictions of flow patterns in the mouse thoracic aorta.

Finite Element Implementation of Biphasic-Fluid Structure Interactions in febio

In biomechanics, solid-fluid mixtures have commonly been used to model the response of hydrated biological tissues. In cartilage mechanics, this type of mixture, where the fluid and solid constituents are both assumed to be intrinsically incompressible, is often called a biphasic material. Various physiological processes involve the interaction of a viscous fluid with a porous-hydrated tissue, as encountered in synovial joint lubrication, cardiovascular mechanics, and respiratory mechanics. The objective of this study was to implement a finite element solver in the open-source software febio that models dynamic interactions between a viscous fluid and a biphasic domain, accommodating finite deformations of both domains as well as fluid exchanges between them. For compatibility with our recent implementation of solvers for computational fluid dynamics (CFD) and fluid-structure interactions (FSI), where the fluid is slightly compressible, this study employs a novel hybrid biphasic formulation where the porous skeleton is intrinsically incompressible but the fluid is also slightly compressible. The resulting biphasic-FSI (BFSI) implementation is verified against published analytical and numerical benchmark problems, as well as novel analytical solutions derived for the purposes of this study. An illustration of this BFSI solver is presented for two-dimensional (2D) airflow through a simulated face mask under five cycles of breathing, showing that masks significantly reduce air dispersion compared to the no-mask control analysis. In addition, we model three-dimensional (3D) blood flow in a bifurcated carotid artery assuming porous arterial walls and verify that mass is conserved across all fluid-permeable boundaries. The successful formulation and implementation of this BFSI solver offers enhanced multiphysics modeling capabilities that are accessible via an open-source software platform.

Pulse Wave Imaging Coupled With Vector Flow Mapping: A Phantom, Simulation, and In Vivo Study

Pulse wave imaging (PWI) is an ultrasound imaging modality that estimates the wall stiffness of an imaged arterial segment by tracking the pulse wave propagation. The aim of the present study is to integrate PWI with vector flow imaging, enabling simultaneous and co-localized mapping of vessel wall mechanical properties and 2-D flow patterns. Two vector flow imaging techniques were implemented using the PWI acquisition sequence: 1) multiangle vector Doppler and 2) a cross-correlation-based vector flow imaging (CC VFI) method. The two vector flow imaging techniques were evaluated in vitro using a vessel phantom with an embedded plaque, along with spatially registered fluid structure interaction (FSI) simulations with the same geometry and inlet flow as the phantom setup. The flow magnitude and vector direction obtained through simulations and phantom experiments were compared in a prestenotic and stenotic segment of the phantom and at five different time frames. In most comparisons, CC VFI provided significantly lower bias or precision than the vector Doppler method ( ) indicating better performance. In addition, the proposed technique was applied to the carotid arteries of nonatherosclerotic subjects of different ages to investigate the relationship between PWI-derived compliance of the arterial wall and flow velocity in vivo. Spearman's rank-order test revealed positive correlation between compliance and peak flow velocity magnitude ( rs = 0.90 and ), while significantly lower compliance ( ) and lower peak flow velocity magnitude ( ) were determined in older (54-73 y.o.) compared with young (24-32 y.o.) subjects. Finally, initial feasibility was shown in an atherosclerotic common carotid artery in vivo. The proposed imaging modality successfully provided information on blood flow patterns and arterial wall stiffness and is expected to provide additional insight in studying carotid artery biomechanics, as well as aid in carotid artery disease diagnosis and monitoring.

Febio finite element models of the human cervical spine

Finite element (FE) analysis has proven to be useful when studying the biomechanics of the cervical spine. Although many FE studies of the cervical spine have been published, they typically develop their models using commercial software, making the sharing of models between researchers difficult. They also often model only one part of the cervical spine. The goal of this study was to develop and evaluate three FE models of the adult cervical spine using open-source software and to freely provide these models to the scientific community. The models were created from computed tomography scans of 26-, 59-, and 64-year old female subjects. These models were evaluated against previously published experimental and FE data. Despite the fact that all three models were assigned identical material properties and boundary conditions, there was notable variation in their biomechanical behavior. It was therefore apparent that these differences were the result of morphological differences between the models.

A Non-linear Viscoelastic Model of the Incudostapedial Joint

The ossicular joints of the middle ear can significantly affect middle-ear function, particularly under conditions such as high-intensity sound pressures or high quasi-static pressures. Experimental investigations of the mechanical behaviour of the human incudostapedial joint have shown strong non-linearity and asymmetry in tension and compression tests, but some previous finite-element models of the joint have had difficulty replicating such behaviour. In this paper, we present a finite-element model of the joint that can match the asymmetry and non-linearity well without using different model structures or parameters in tension and compression. The model includes some of the detailed structures of the joint seen in histological sections. The material properties are found from the literature when available, but some parameters are calculated by fitting the model to experimental data from tension, compression and relaxation tests. The model can predict the hysteresis loops of loading and unloading curves. A sensitivity analysis for various parameters shows that the geometrical parameters have substantial effects on the joint mechanical behaviour. While the joint capsule affects the tension curve more, the cartilage layers affect the compression curve more.

A Formulation for Fluid Structure-Interactions in FEBio Using Mixture Theory

Many physiological systems involve strong interactions between fluids and solids, posing a signicant challenge when modeling biomechanics. The objective of this study was to implement a fluid-structure interaction (FSI) solver in the free, open-source finite element code FEBio (febio.org), that combined the existing solid mechanics and rigid body dynamics solver with a recently-developed computational fluid dynamics (CFD) solver. A novel Galerkin-based finite element FSI formulation was introduced based on mixture theory, where the FSI domain was described as a mixture of fluid and solid constituents that have distinct motions. The mesh was defined on the solid domain, specialized to have zero mass, negligible stiffness and zero frictional interactions with the fluid, whereas the fluid was modeled as isothermal and compressible. The mixture framework provided the foundation for evaluating material time derivatives in a material frame for the solid and in a spatial frame for the fluid. Similar to our recently reported CFD solver, our FSI formulation did not require stabilization methods to achieve good convergence, producing a compact set of equations and code implementation. The code was successfully verified against benchmark problems and an analytical solution for squeeze-film lubrication. It was validated against experimental measurements of the flow rate in a peristaltic pump, and illustrated using non-Newtonian blood flow through a bifurcated carotid artery with a thick arterial wall. The successful formulation and implementation of this FSI solver enhances the multiphysics modeling capabilities in FEBio relevant to the biomechanics and biophysics communities.

A Plugin Framework for Extending the Simulation Capabilities of FEBio

The FEBio software suite is a set of software tools for nonlinear finite element analysis in biomechanics and biophysics. FEBio employs mixture theory to account for the multiconstituent nature of biological materials, integrating the field equations for irreversible thermodynamics, solid mechanics, fluid mechanics, mass transport with reactive species, and electrokinetics. This communication describes the development and application of a new "plugin" framework for FEBio. Plugins are dynamically linked libraries that allow users to add new features and to couple FEBio with other domain-specific software applications without modifying the source code directly. The governing equations and simulation capabilities of FEBio are reviewed. The implementation, structure, use, and application of the plugin framework are detailed. Several example plugins are described in detail to illustrate how plugins enrich, extend, and leverage existing capabilities in FEBio, including applications to deformable image registration, constitutive modeling of biological tissues, coupling to an external software package that simulates angiogenesis using a discrete computational model, and a nonlinear reaction-diffusion solver. The plugin feature facilitates dissemination of new simulation methods, reproduction of published results, and coupling of FEBio with other domain-specific simulation approaches such as compartmental modeling, agent-based modeling, and rigid-body dynamics. We anticipate that the new plugin framework will greatly expand the range of applications for the FEBio software suite and thus its impact.