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

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.

Pulse wave imaging of a stenotic artery model with plaque constituents of different stiffnesses: Experimental demonstration in phantoms and fluid-structure interaction simulation

Vulnerable plaques associated with softer components may rupture, releasing thrombotic emboli to smaller vessels in the brain, thus causing an ischemic stroke. Pulse Wave Imaging (PWI) is an ultrasound-based method that allows for pulse wave visualization while the regional pulse wave velocity (PWV) is mapped along the arterial wall to infer the underlying wall compliance. One potential application of PWI is the non-invasive estimation of plaque's mechanical properties for investigating its vulnerability. In this study, the accuracy of PWV estimation in stenotic vessels was investigated by computational simulation and PWI in validation phantoms to evaluate this modality for assessing future stroke risk. Polyvinyl alcohol (PVA) phantoms with plaque constituents of different stiffnesses were designed and constructed to emulate stenotic arteries in the experiment, and the novel fabrication process was described. Finite-element fluid-structure interaction simulations were performed in a stenotic phantom model that matched the geometry and parameters of the experiment in phantoms. The peak distension acceleration of the phantom wall was tracked to estimate PWV. PWVs of 2.57 ms-1, 3.41 ms-1, and 4.48 ms-1 were respectively obtained in the soft, intermediate, and stiff plaque material in phantoms during the experiment using PWI. PWVs of 2.10 ms-1, 3.33 ms-1, and 4.02 ms-1 were respectively found in the soft, intermediate, and stiff plaque material in the computational simulation. These results demonstrate that PWI can effectively distinguish the mechanical properties of plaque in phantoms as compared to computational simulation.

Beyond CFD: Emerging methodologies for predictive simulation in cardiovascular health and disease

Physics-based computational models of the cardiovascular system are increasingly used to simulate hemodynamics, tissue mechanics, and physiology in evolving healthy and diseased states. While predictive models using computational fluid dynamics (CFD) originated primarily for use in surgical planning, their application now extends well beyond this purpose. In this review, we describe an increasingly wide range of modeling applications aimed at uncovering fundamental mechanisms of disease progression and development, performing model-guided design, and generating testable hypotheses to drive targeted experiments. Increasingly, models are incorporating multiple physical processes spanning a wide range of time and length scales in the heart and vasculature. With these expanded capabilities, clinical adoption of patient-specific modeling in congenital and acquired cardiovascular disease is also increasing, impacting clinical care and treatment decisions in complex congenital heart disease, coronary artery disease, vascular surgery, pulmonary artery disease, and medical device design. In support of these efforts, we discuss recent advances in modeling methodology, which are most impactful when driven by clinical needs. We describe pivotal recent developments in image processing, fluid-structure interaction, modeling under uncertainty, and reduced order modeling to enable simulations in clinically relevant timeframes. In all these areas, we argue that traditional CFD alone is insufficient to tackle increasingly complex clinical and biological problems across scales and systems. Rather, CFD should be coupled with appropriate multiscale biological, physical, and physiological models needed to produce comprehensive, impactful models of mechanobiological systems and complex clinical scenarios. With this perspective, we finally outline open problems and future challenges in the field.

Flow-based method demonstrates improved accuracy for calculating wall shear stress in arterial flows from 4D flow MRI data

Four-dimensional flow magnetic resonance imaging (i.e., 4D flow MRI) has become a valuable tool for the in vivo assessment of blood flow within large vessels and cardiac chambers. As wall shear stress (WSS) has been correlated with the development and progression of cardiovascular disease, focus has been directed at developing techniques to quantify WSS directly from 4D flow MRI data. The goal of this study was to compare the accuracy of two such techniques - termed the velocity and flow-based methods - in the setting of simplified and complex flow scenarios. Synthetic MR data were created from exact solutions to the Navier-Stokes equations for the steady and pulsatile flow of an incompressible, Newtonian fluid through a rigid cylinder. In addition, synthetic MR data were created from the predicted velocity fields derived from a fluid-structure interaction (FSI) model of pulsatile flow through a thick-walled, multi-layered model of the carotid bifurcation. Compared to the analytical solutions for steady and pulsatile flow, the flow-based method demonstrated greater accuracy than the velocity-based method in calculating WSS across all changes in fluid velocity/flow rate, tube radius, and image signal-to-noise (p < 0.001). Furthermore, the velocity-based method was more sensitive to boundary segmentation than the flow-based method. When compared to results from the FSI model, the flow-based method demonstrated greater accuracy than the velocity-based method with average differences in time-averaged WSS of 0.31 ± 1.03 Pa and 0.45 ± 1.03 Pa, respectively (p <0.005). These results have implications on the utility, accuracy, and clinical translational of methods to determine WSS from 4D flow MRI.

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.

Effects of scattering on ultrasound wave transmission through bioinspired scaffolds

Enhancing tissue growth in scaffolds using ultrasound waves while maintaining the structural integrity of the scaffolds is a challenging problem. Previous studies have primarily focused on the effect of ultrasound waves directly on the tissue, but how the ultrasound wave interacts with the scaffold needs to be further understood, which will have a significant effect on the response of tissue to mechanical stimulation. In this study we investigate how ultrasound wave transmission differs between scaffolds with uniform pore shapes (triangle, square, rectangle, hexagon) and a bioinspired scaffold with higher structural integrity that is inspired from the atomic structure of hydroxyapatite which is a primary component of bone. We use finite element method and ultrasound experiments on 3D-printed scaffolds composed of Acrylonitrile butadiene styrene (ABS) with constant porosity to predict the effect of pore shape and wave signal frequency in the range of 1-20 MHz on acoustic wave scattering and transmission. We find that the pore shape of the scaffold affects the magnitude of ultrasound transmission even when porosity is constant, and that the bioinspired scaffolds can allow as much as 67% more wave transmission compared to scaffolds with rectangular or square pore shapes at 1 MHz frequency. Triangular and hexagonal pores are also found to produce more nonuniform transmitted wavefronts compared to the square and rectangular pores. Peak density is defined as the number of local extrema of the transmitted wave frequency power spectrum and measures the uniformity of the transmitted wave. We find that a higher peak density value for the bioinspired scaffold due to its nonsymmetric structure further produces more nonuniform wave scattering. The results of this study are important for designing bioinspired tissue scaffold geometries to control ultrasound wave penetration and to enhance mechanical stimulation for tissue growth and will also aid in the ultrasonic characterization of porous structures based on changes in pore geometry.

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.

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.

Residual stress and osmotic swelling of the periodontal ligament

Osmotic swelling and residual stress are increasingly recognized as important factors in soft tissue biomechanics. Little attention has been given to residual stress in periodontal ligament (PDL) biomechanics despite its rapid growth and remodeling potential. Those tissues that bear compressive loads, e.g., articular cartilage, intervertebral disk, have received much attention related to their capacities for osmotic swelling. To understand residual stress and osmotic swelling in the PDL, it must be asked (1) to what extent, if any, does the PDL exhibit residual stress and osmotic swelling, and (2) if so, whether residual stress and osmotic swelling are mechanically significant to the PDL's stress/strain behavior under external loading. Here, we incrementally built a series of computer models that were fit to uniaxial loading, osmotic swelling and residual stretch data. The models were validated with in vitro shear tests and in vivo tooth-tipping data. Residual stress and osmotic swelling models were used to analyze tension and compression stress (principal stress) effects in PDL specimens under external loads. Shear-to-failure experiments under osmotic conditions were performed and modeled to determine differences in mechanics and failure of swollen periodontal ligament. Significantly higher failure shear stresses in swollen PDL suggested that osmotic swelling reduced tension and thus had a strengthening effect. The in vivo model's first and third principal stresses were both higher with residual stress and osmotic swelling, but smooth stress gradients prevailed throughout the three-dimensional PDL anatomy. The addition of PDL stresses from residual stress and osmotic swelling represents a unique concept in dental biomechanics.

Modeling Pulse Wave Propagation Through a Stenotic Artery With Fluid Structure Interaction: A Validation Study Using Ultrasound Pulse Wave Imaging

Pulse wave imaging (PWI) is an ultrasound-based method that allows spatiotemporal mapping of the arterial pulse wave propagation, from which the local pulse wave velocity (PWV) can be derived. Recent reports indicate that PWI can help the assessment of atherosclerotic plaque composition and mechanical properties. However, the effect of the atherosclerotic plaque's geometry and mechanics on the arterial wall distension and local PWV remains unclear. In this study, we investigated the accuracy of a finite element (FE) fluid-structure interaction (FSI) approach to predict the velocity of a pulse wave propagating through a stenotic artery with an asymmetrical plaque, as quantified with PWI method. Experiments were designed to compare FE-FSI modeling of the pulse wave propagation through a stenotic artery against PWI obtained with manufactured phantom arteries made of polyvinyl alcohol (PVA) material. FSI-generated spatiotemporal maps were used to estimate PWV at the plaque region and compared it to the experimental results. Velocity of the pulse wave propagation and magnitude of the wall distension were correctly predicted with the FE analysis. In addition, findings indicate that a plaque with a high degree of stenosis (>70%) attenuates the propagation of the pulse pressure wave. Results of this study support the validity of the FE-FSI methods to investigate the effect of arterial wall structural and mechanical properties on the pulse wave propagation. This modeling method can help to guide the optimization of PWI to characterize plaque properties and substantiate clinical findings.