Effect of Circadian Rhythm Modulated Blood Flow on Nanoparticle based Targeted Drug Delivery in Virtual In Vivo Arterial Geometries.

Delivery of drug using nanocarriers tethered with vasculature-targeting epitopes aims to maximize the therapeutic efficacy of the drug while minimizing the drug side effects. Circadian rhythm which is governed by the central nervous system has implications for targeted drug delivery due to sleep-wake cycle changes in blood flow dynamics. This paper presents an advanced fluid dynamics modeling method that is based on viscous incompressible shear-rate fluid (blood) coupled with an advection-diffusion equation to simulate the formation of drug concentration gradients in the blood stream and buildup of concentration at the targeted site. The method is equipped with an experimentally calibrated nanoparticle-endothelial cell adhesion model that employs Robin boundary conditions to describe nanoparticle retention based on probability of adhesion, a friction model accounting for surface roughness of endothelial cell layer, and a dispersion model based on Taylor-Aris expression for effective diffusion in the boundary layer. The computational model is first experimentally validated and then tested on engineered bifurcating arterial systems where impedance boundary conditions are applied at the outflow to account for the downstream resistance at each outlet. It is then applied to a virtual geometric model of an in vivo arterial tree developed through MRI-based image processing techniques. These simulations highlight the potential of the computational model for drug transport, adhesion, and retention at multiple sites in virtual in vivo models. The model provides a virtual platform for exploring circadian rhythm modulated blood flow for targeted drug delivery while minimizing the in vivo experimentation.

Modeling of spatiotemporal dynamics of ligand-coated particle flow in targeted drug delivery processes.

Nanoparticles tethered with vasculature-binding epitopes have been used to deliver the drug into injured or diseased tissues via the bloodstream. However, the extent that blood flow dynamics affects nanoparticle retention at the target site after adhesion needs to be better understood. This knowledge gap potentially underlies significantly different therapeutic efficacies between animal models and humans. An experimentally validated mathematical model that accurately simulates the effects of blood flow on nanoparticle adhesion and retention, thus circumventing the limitations of conventional trial-and-error-based drug design in animal models, is lacking. This paper addresses this technical bottleneck and presents an integrated mathematical method that derives heavily from a unique combination of a mechanics-based dispersion model for nanoparticle transport and diffusion in the boundary layers, an asperity model to account for surface roughness of endothelium, and an experimentally calibrated stochastic nanoparticle-cell adhesion model to describe nanoparticle adhesion and subsequent retention at the target site under external flow. PLGA-b-HA nanoparticles tethered with VHSPNKK peptides that specifically bind to vascular cell adhesion molecules on the inflamed vascular wall were investigated. The computational model revealed that larger particles perform better in adhesion and retention at the endothelium for the particle sizes suitable for drug delivery applications and within physiologically relevant shear rates. The computational model corresponded closely to the in vitro experiments which demonstrates the impact that model-based simulations can have on optimizing nanocarriers in vascular microenvironments, thereby substantially reducing in vivo experimentation as well as the development costs.

A Unified Determinant-Preserving Formulation for Compressible/Incompressible Finite Viscoelasticity.

This paper presents a formulation alongside a numerical solution algorithm to describe the mechanical response of bodies made of a large class of viscoelastic materials undergoing arbitrary quasistatic finite deformations. With the objective of having a unified formulation that applies to a wide range of highly compressible, nearly incompressible, and fully incompressible soft organic materials in a numerically tractable manner, the viscoelasticity is described within a Lagrangian setting by a two-potential mixed formulation. In this formulation, the deformation field, a pressure field that ensues from a Legendre transform, and an internal variable of state Fv that describes the viscous part of the deformation are the independent fields. Consistent with the experimental evidence that viscous deformation is a volume-preserving process, the internal variable Fv is required to satisfy the constraint det Fv=1. To solve the resulting initial-boundary-value problem, a numerical solution algorithm is proposed that is based on a finite-element (FE) discretization of space and a finite-difference discretization of time. Specifically, a Variational Multiscale FE method is employed that allows for an arbitrary combination of shape functions for the deformation and pressure fields. To deal with the challenging non-convex constraint det Fv=1, a new time integration scheme is introduced that allows to convert any explicit or implicit scheme of choice into a stable scheme that preserves the constraint det Fv=1 identically. A series of test cases is presented that showcase the capabilities of the proposed formulation.

Error estimates and physics informed augmentation of neural networks for thermally coupled incompressible Navier Stokes equations.

Physics Informed Neural Networks (PINNs) are shown to be a promising method for the approximation of partial differential equations (PDEs). PINNs approximate the PDE solution by minimizing physics-based loss functions over a given domain. Despite substantial progress in the application of PINNs to a range of problem classes, investigation of error estimation and convergence properties of PINNs, which is important for establishing the rationale behind their good empirical performance, has been lacking. This paper presents convergence analysis and error estimates of PINNs for a multi-physics problem of thermally coupled incompressible Navier-Stokes equations. Through a model problem of Beltrami flow it is shown that a small training error implies a small generalization error. Posteriori convergence rates of total error with respect to the training residual and collocation points are presented. This is of practical significance in determining appropriate number of training parameters and training residual thresholds to get good PINNs prediction of thermally coupled steady state laminar flows. These convergence rates are then generalized to different spatial geometries as well as to different flow parameters that lie in the laminar regime. A pressure stabilization term in the form of pressure Poisson equation is added to the PDE residuals for PINNs. This physics informed augmentation is shown to improve accuracy of the pressure field by an order of magnitude as compared to the case without augmentation. Results from PINNs are compared to the ones obtained from stabilized finite element method and good properties of PINNs are highlighted.

Physics-Constrained Data-Driven Variational Method for Discrepancy Modeling.

This paper presents a data-driven discrepancy modeling method that variationally embeds measured data in the modeling and analysis framework. The proposed method exploits the residual between the first-principles theory and sensor-based measurements from the dynamical system, and it augments the physics-based model with a variationally derived loss function that is comprised of this residual. The method was first developed in the context of linear elasticity (Masud and Goraya, J. Appl. Mech. 89 (11), 111001 (2022)) wherein the relation between the discrepancy model and loss terms was derived to show that the data embedding terms behave like residual-based least-squares regression functions. An interpretation of the stabilization tensor as a kernel function was formally established and its role in assimilating a-priori knowledge of the problem in the modeling method was highlighted. The present paper employs linear elastodynamics as a model problem where the Data-Driven Variational (DDV) method incorporates high-fidelity data into the forward simulations, thereby driving the problem with not only the boundary and initial conditions, but also by measurement data that is taken at only a small subset of the total domain. The effect of the loss function on the time-dependent response of the system is investigated under a variety of loading conditions and model discrepancies. The energy and Morlet wavelet analyses reveal that the problem with embedded data recovers the energy and the fundamental frequency band of the target system. Time histories of strain energy and kinetic energy of a cantilever beam undergoing damped oscillations are recovered by including known data in an undamped model to highlight the data-driven discrepancy modeling feature of the method under the combined effect of parameter and model discrepancy.

Blood-Artery Interaction in Calcified Aortas and Abdominal Aortic Aneurysms.

A stabilized FSI method is presented for coupling shear-rate dependent model of blood with finitely deforming anisotropic hyperelastic model of arteries. The blood-artery coupling conditions are weakly enforced to accommodate non-matching blood-artery meshes which provides great flexibility in independent discretization of fluid and solid subdomains in the patient-specific geometric models. The variationally derived interface coupling terms play an important role in the concurrent solution of the nonlinear mixed-field problem across non-matching discretizations. Two test cases are presented that investigate the effect of growth of aortic aneurysm on local changes in blood flow and stress concentrations in calcified arteries under pulsating flows to highlight the clinical relevance of the proposed method for cardiovascular applications.

Variational coupling of non-matching discretizations across finitely deforming fluid-structure interfaces.

This paper presents a stabilized monolithic method for coupling incompressible viscous fluids with finitely deforming elastic solids across non-matching interfacial meshes. Governing equations for the solid are written in the finite deformation Lagrangian frame using velocity field as the primary unknown, while the governing equations for the fluid are written in an Arbitrary Lagrangian-Eulerian (ALE) frame to accommodate large motions of the fluid-solid interfaces. Interface coupling terms are derived by embedding Discontinuous Galerkin (DG) ideas in the Variational Multiscale (VMS) framework and locally resolving the fine-scale variational equations from the fluid and the solid subdomains along the common interface. The unique attribute of the proposed Variational Multiscale Discontinuous Galerkin (VMDG) method is a systematic procedure for deriving analytical expression for the traction Lagrange multiplier at the fluid-solid interface. The structure of the interface stabilization tensor emerges naturally and is shown to be a function of the boundary operators that are associated with the domain interior operators from the fluid and the solid subdomains. The derived stabilization tensor possesses the features of area-averaging and stress-averaging and evolves spatially and temporally with the evolving nonlinear fields at the interface. An analysis of the mathematical properties of the interface stabilization tensor is presented and subsequently verified numerically. The method is implemented using four-node tetrahedral elements for the fluid as well as the solid subdomains while employing equal-order interpolations for the various interacting fields. Benchmark problems are presented for the verification of the method for finitely deforming fluid-structure interfaces.

Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows.

This paper presents a stabilized formulation for the generalized Navier-Stokes equations for weak enforcement of essential boundary conditions. The non-Newtonian behavior of blood is modeled via shear-rate dependent constitutive equations. The boundary terms for weak enforcement of Dirichlet boundary conditions are derived via locally resolving the fine-scale variational equation facilitated by the Variational Multiscale (VMS) framework. The proposed method reproduces the consistency and stabilization terms that are present in the Nitsche type approaches. In addition, for the shear-rate fluids, two more boundary terms appear. One of these terms is the viscosity-derivative term and is a function of the shear-rate, while the other term is a zeroth-order term. These terms play an important role in attaining optimal convergence rates for the velocity and pressure fields in the norms considered. A most significant contribution is the form of the stabilization tensors that are also variationally derived. Employing edge functions the edge stabilization tensor is numerically evaluated, and it adaptively adjusts itself to the magnitude of the boundary residual. The resulting formulation is variationally consistent and the weakly imposed no-slip boundary condition leads to higher accuracy of the spatial gradients for coarse boundary-layer meshes when compared with the traditional strongly imposed boundary conditions. This feature of the present approach will be of significance in imposing interfacial continuity conditions across non-matching discretizations in blood-artery interaction problems. A set of test cases is presented to investigate the mathematical attributes of the method and a patient-specific case is presented to show its clinical relevance.

A MULTISCALE VISION-ILLUSTRATIVE APPLICATIONS FROM BIOLOGY TO ENGINEERING.

Modeling and simulation have quickly become equivalent pillars of research along with traditional theory and experimentation. The growing realization that most complex phenomena of interest span many orders of spatial and temporal scales has led to an exponential rise in the development and application of multiscale modeling and simulation over the past two decades. In this perspective, the associate editors of the International Journal for Multiscale Computational Engineering and their co-workers illustrate current applications in their respective fields spanning biomolecular structure and dynamics, civil engineering and materials science, computational mechanics, aerospace and mechanical engineering, and more. Such applications are highly tailored, exploit the latest and ever-evolving advances in both computer hardware and software, and contribute significantly to science, technology, and medical challenges in the 21st century.

A Variational Multiscale method with immersed boundary conditions for incompressible flows.

This paper presents a new stabilized form of incompressible Navier-Stokes equations for weak enforcement of Dirichlet boundary conditions at immersed boundaries. The boundary terms are derived via the Variational Multiscale (VMS) method which involves solving the fine-scale variational problem locally within a narrow band along the boundary. The fine-scale model is then variationally embedded into the coarse-scale form that yields a stabilized method which is free of user defined parameters. The derived boundary terms weakly enforce the Dirichlet boundary conditions along the immersed boundaries that may not align with the inter-element edges in the mesh. A unique feature of this rigorous derivation is that the structure of the stabilization tensor which emerges is naturally endowed with the mathematical attributes of area-averaging and stress-averaging. The method is implemented using 4-node quadrilateral and 8-node hexahedral elements. A set of 2D and 3D benchmark problems is presented that investigate the mathematical attributes of the method. These test cases show that the proposed method is mathematically robust as well as computationally stable and accurate for modeling boundary layers around immersed objects in the fluid domain.

Hemodynamic analysis in an idealized artery tree: differences in wall shear stress between Newtonian and non-Newtonian blood models.

Development of many conditions and disorders, such as atherosclerosis and stroke, are dependent upon hemodynamic forces. To accurately predict and prevent these conditions and disorders hemodynamic forces must be properly mapped. Here we compare a shear-rate dependent fluid (SDF) constitutive model, based on the works by Yasuda et al in 1981, against a Newtonian model of blood. We verify our stabilized finite element numerical method with the benchmark lid-driven cavity flow problem. Numerical simulations show that the Newtonian model gives similar velocity profiles in the 2-dimensional cavity given different height and width dimensions, given the same Reynolds number. Conversely, the SDF model gave dissimilar velocity profiles, differing from the Newtonian velocity profiles by up to 25% in velocity magnitudes. This difference can affect estimation in platelet distribution within blood vessels or magnetic nanoparticle delivery. Wall shear stress (WSS) is an important quantity involved in vascular remodeling through integrin and adhesion molecule mechanotransduction. The SDF model gave a 7.3-fold greater WSS than the Newtonian model at the top of the 3-dimensional cavity. The SDF model gave a 37.7-fold greater WSS than the Newtonian model at artery walls located immediately after bifurcations in the idealized femoral artery tree. The pressure drop across arteries reveals arterial sections highly resistive to flow which correlates with stenosis formation. Numerical simulations give the pressure drop across the idealized femoral artery tree with the SDF model which is approximately 2.3-fold higher than with the Newtonian model. In atherosclerotic lesion models, the SDF model gives over 1 Pa higher WSS than the Newtonian model, a difference correlated with over twice as many adherent monocytes to endothelial cells from the Newtonian model compared to the SDF model.