Multi-scale simulations of pulmonary airflow based on a coupled 3D-1D-0D model

Pulmonary airflow simulation is a valuable tool for studying respiratory function and disease. However, the respiratory system is a complex multiscale system that involves various physical and biological processes across different spatial and temporal scales. In this study, we propose a 3D-1D-0D multiscale method for simulating pulmonary airflow, which integrates different levels of detail and complexity of the respiratory system. The method consists of three components: a 3D computational fluid dynamics model for the airflow in the trachea and bronchus, a 1D pipe model for the airflow in the terminal bronchioles, and a 0D biphasic mixture model for the airflow in the respiratory bronchioles and alveoli coupled with the lung deformation. The coupling between the different components is achieved by satisfying the mass and momentum conservation law and the pressure continuity condition at the interfaces. We demonstrate the validity and applicability of our method by comparing the results with data of previous models. We also investigate the reduction in inhaled air volume due to the pulmonary fibrosis using the developed multiscale model. Our method provides a comprehensive and realistic framework for simulating pulmonary airflow and can potentially facilitate the diagnosis and treatment of respiratory diseases.

Coupling solid and fluid stresses with brain tumour growth and white matter tract deformations in a neuroimaging-informed model

Brain tumours are among the deadliest types of cancer, since they display a strong ability to invade the surrounding tissues and an extensive resistance to common therapeutic treatments. It is therefore important to reproduce the heterogeneity of brain microstructure through mathematical and computational models, that can provide powerful instruments to investigate cancer progression. However, only a few models include a proper mechanical and constitutive description of brain tissue, which instead may be relevant to predict the progression of the pathology and to analyse the reorganization of healthy tissues occurring during tumour growth and, possibly, after surgical resection. Motivated by the need to enrich the description of brain cancer growth through mechanics, in this paper we present a mathematical multiphase model that explicitly includes brain hyperelasticity. We find that our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding healthy tissue, quantifying the displacements, deformations, and stresses induced by its proliferation. At the same time, the knowledge of the mechanical variables may be used to model the stress-induced inhibition of growth, as well as to properly modify the preferential directions of white matter tracts as a consequence of deformations caused by the tumour. Finally, the simulations of our model are implemented in a personalized framework, which allows to incorporate the realistic brain geometry, the patient-specific diffusion and permeability tensors reconstructed from imaging data and to modify them as a consequence of the mechanical deformation due to cancer growth.

Individualized body geometry correction factor (KB) for use when predicting body composition from bioimpedance spectroscopy

OBJECTIVE

Prediction of body composition from bioimpedance spectroscopy (BIS) measurements using mixture theory-based biophysical modelling invokes a factor (K_B) to account for differing body geometry (or proportions) between individuals. To date, a single constant value is commonly used. The aim of this study was to investigate variation in KB across individuals and to develop a procedure for estimating an individualized K_Bvalue.

APPROACH

Publicly available body dimension data, primarily from the garment industry, were used to calculate K_Bvalues for individuals of varying body sizes across the life-span. The 3-D surface relationship between weight, height and K_B, was determined and used to create look-up tables to enable estimation of K_Bin individuals based on height and weight. The utility of the proposed method was assessed by comparing body composition predictions from BIS using either a constant K_Bvalue or the individualized value.

RESULTS

Computed KB values were well fitted to height and weight by a 3-D surface (R2 = 0.988). Body composition was predicted more accurately compared to reference methods when using individualized K_Bthan a constant value in infants and children but improvement in prediction was less in adults particularly those with high body mass index.

SIGNIFICANCE

Prediction of body composition from BIS and mixture theory is improved by using an individualized body proportion factor in those of small body habitus, e.g. children. Improvement is small in adults or non-existent in those of large body size. Further improvements may be possible by incorporating a factor to account for trunk size, i.e., waist circumference.

Bayesian-based predictions of COVID-19 evolution in Texas using multispecies mixture-theoretic continuum models

We consider a mixture-theoretic continuum model of the spread of COVID-19 in Texas. The model consists of multiple coupled partial differential reaction-diffusion equations governing the evolution of susceptible, exposed, infectious, recovered, and deceased fractions of the total population in a given region. We consider the problem of model calibration, validation, and prediction following a Bayesian learning approach implemented in OPAL (the Occam Plausibility Algorithm). Our goal is to incorporate COVID-19 data to calibrate the model in real-time and make meaningful predictions and specify the confidence level in the prediction by quantifying the uncertainty in key quantities of interests. Our results show smaller mortality rates in Texas than what is reported in the literature. We predict 7003 deceased cases by September 1, 2020 in Texas with 95 % CI 6802-7204. The model is validated for the total deceased cases, however, is found to be invalid for the total infected cases. We discuss possible improvements of the model.

A Coupled Mass Transport and Deformation Theory of Multi-constituent Tumor Growth

We develop a general class of thermodynamically consistent, continuum models based on mixture theory with phase effects that describe the behavior of a mass of multiple interacting constituents. The constituents consist of solid species undergoing large elastic deformations and compressible viscous fluids. The fundamental building blocks framing the mixture theories consist of the mass balance law of diffusing species and microscopic (cellular scale) and macroscopic (tissue scale) force balances, as well as energy balance and the entropy production inequality derived from the first and second laws of thermodynamics. A general phase-field framework is developed by closing the system through postulating constitutive equations (i.e., specific forms of free energy and rate of dissipation potentials) to depict the growth of tumors in a microenvironment. A notable feature of this theory is that it contains a unified continuum mechanics framework for addressing the interactions of multiple species evolving in both space and time and involved in biological growth of soft tissues (e.g., tumor cells and nutrients). The formulation also accounts for the regulating roles of the mechanical deformation on the growth of tumors, through a physically and mathematically consistent coupled diffusion and deformation framework. A new algorithm for numerical approximation of the proposed model using mixed finite elements is presented. The results of numerical experiments indicate that the proposed theory captures critical features of avascular tumor growth in the various microenvironment of living tissue, in agreement with the experimental studies in the literature.

Finite element analysis of biological soft tissue surrounded by a deformable membrane that controls transmembrane flow

Background

Many biological soft tissues are hydrated porous hyperelastic materials, which consist of a complex solid skeleton with fine voids and fluid filling these voids. Mechanical interactions between the solid and the fluid in hydrated porous tissues have been analyzed by finite element methods (FEMs) in which the mixture theory was introduced in various ways. Although most of the tissues are surrounded by deformable membranes that control transmembrane flows, the boundaries of the tissues have been treated as rigid and/or freely permeable in these studies. The purpose of this study was to develop a method for the analysis of hydrated porous hyperelastic tissues surrounded by deformable membranes that control transmembrane flows.

Results

For this, we developed a new nonlinear finite element formulation of the mixture theory, where the nodal unknowns were the pore water pressure and solid displacement. This method allows the control of the fluid flow rate across the membrane using Neumann boundary condition. Using the method, we conducted a compression test of the hydrated porous hyperelastic tissue, which was surrounded by a flaccid impermeable membrane, and a part of the top surface of this tissue was pushed by a platen. The simulation results showed a stress relaxation phenomenon, resulting from the interaction between the elastic deformation of the tissue, pore water pressure gradient, and the movement of fluid. The results also showed that the fluid trapped by the impermeable membrane led to the swelling of the tissue around the platen.

Conclusions

These facts suggest that our new method can be effectively used for the analysis of a large deformation of hydrated porous hyperelastic material surrounded by a deformable membrane that controls transmembrane flow, and further investigations may allow more realistic analyses of the biological soft tissues, such as brain edema, brain trauma, the flow of blood and lymph in capillaries and pitting edema.

Reactive Constrained Mixtures for Modeling the Solid Matrix of Biological Tissues

This article illustrates our approach for modeling the solid matrix of biological tissues using reactive constrained mixtures. Several examples are presented to highlight the potential benefits of this approach, showing that seemingly disparate fields of mechanics and chemical kinetics are actually closely interrelated and may be elegantly expressed in a unified framework. Thus, constrained mixture models recover classical theories for fibrous materials with bundles oriented in different directions or having different reference configurations, that produce characteristic fiber recruitment patterns under loading. Reactions that exchange mass among various constituents of a mixture may be used to describe tissue growth and remodeling, which may also alter the material's anisotropy. Similarly, reactions that describe the breaking and reforming of bonds may be used to model free energy dissipation in a viscoelastic material. Therefore, this framework is particularly well suited for modeling biological tissues.

Kinetics of charged antibiotic penetration into human intervertebral discs: A numerical study

Little quantitative information exists on the kinetics of charged antibiotic penetration into human intervertebral discs (IVD). This information is crucial for determining the dosage to use, timing of administration, and duration of treatment for infected IVDs. The objective of this study was to quantitatively analyze the transport of various charged antibiotics into human lumbar IVDs. Penetration of charged and uncharged antibiotics into a human lumbar disc was analyzed using a 3D finite element model. The valence (z) of the electrical charge of antibiotics varied from z=+2 (positively charged) to z=-2 (negatively charged). An uncharged antibiotic (z=0) was used as a control. Cases with intravenous (IV) administrations of different charged antibiotics were simulated. Our results showed that the electrical charge had great effects on kinetics of an antibiotic penetration into the IVD; with higher concentrations and uptakes for positively charged antibiotics than those for negatively charged ones. This study provides quantitative information on selecting antibiotics for treating intervertebral disc infections.

A mixture approach to investigate interstitial growth in engineering scaffolds

Controlling biological growth within a cell-laden polymeric scaffold is a critical challenge in the tissue engineering community. Indeed, construct growth must often be balanced with scaffold degradation and is often coupled to varying degrees of deformation that originate from swelling, external forces and the effects of confinement. These factors have been shown to affect growth in many ways, but to date, our understanding is mostly qualitative. While cell sensing, molecular transport and scaffold/tissue interactions are believed to be important players, it will be critical to quantify, predict and control these effects in order to eventually optimize tissue growth in the laboratory. The aim of this paper was thus to provide a theoretical framework to better understand how the scaffold-mediated mechanisms of transport, deposition (and possibly degradation) and elasticity affect the overall growth of a tissue subjected to finite deformations. We propose a formulation in which the macroscopic evolutions in tissue size, density as well as the appearance of residual stresses can be directly related to changes in internal composition by considering three fundamental principles: mechanical equilibrium, chemical equilibrium and molecular incompressibility. The resulting model allows us to pay particular attention to features that are critical to the interaction between growth and deformation: osmotic pressure and swelling, the strain mismatch between old and newly deposited material as well as the mechano-sensitive cell-mediated production. We show that all of these phenomena may indeed strongly affect the overall growth of a construct under finite deformations.

Viscoelasticity using reactive constrained solid mixtures

This study presents a framework for viscoelasticity where the free energy density depends on the stored energy of intact strong and weak bonds, where weak bonds break and reform in response to loading. The stress is evaluated by differentiating the free energy density with respect to the deformation gradient, similar to the conventional approach for hyperelasticity. The breaking and reformation of weak bonds is treated as a reaction governed by the axiom of mass balance, where the constitutive relation for the mass supply governs the bond kinetics. The evolving mass contents of these weak bonds serve as observable state variables. Weak bonds reform in an energy-free and stress-free state, therefore their reference configuration is given by the current configuration at the time of their reformation. A principal advantage of this formulation is the availability of a strain energy density function that depends only on observable state variables, also allowing for a separation of the contributions of strong and weak bonds. The Clausius-Duhem inequality is satisfied by requiring that the net free energy from all breaking bonds must be decreasing at all times. In the limit of infinitesimal strains, linear stress-strain responses and first-order kinetics for breaking and reforming of weak bonds, the reactive framework reduces exactly to classical linear viscoelasticity. For large strains, the reactive and classical quasilinear viscoelasticity theories produce different equations, though responses to standard loading configurations behave similarly. This formulation complements existing tools for modeling the nonlinear viscoelastic response of biological soft tissues under large deformations.

A numerical study of blood flow using mixture theory

In this paper, we consider the two dimensional flow of blood in a rectangular microfluidic channel. We use Mixture Theory to treat this problem as a two-component system: One component is the red blood cells (RBCs) modeled as a generalized Reiner-Rivlin type fluid, which considers the effects of volume fraction (hematocrit) and influence of shear rate upon viscosity. The other component, plasma, is assumed to behave as a linear viscous fluid. A CFD solver based on OpenFOAM^® was developed and employed to simulate a specific problem, namely blood flow in a two dimensional micro-channel, is studied. Finally to better understand this two-component flow system and the effects of the different parameters, the equations are made dimensionless and a parametric study is performed.

Interstitial growth and remodeling of biological tissues: tissue composition as state variables

Growth and remodeling of biological tissues involves mass exchanges between soluble building blocks in the tissue's interstitial fluid and the various constituents of cells and the extracellular matrix. As the content of these various constituents evolves with growth, associated material properties, such as the elastic modulus of the extracellular matrix, may similarly evolve. Therefore, growth theories may be formulated by accounting for the evolution of tissue composition over time in response to various biological and mechanical triggers. This approach has been the foundation of classical bone remodeling theories that successfully describe Wolff's law by establishing a dependence between Young's modulus and bone apparent density and by formulating a constitutive relation between bone mass supply and the state of strain. The goal of this study is to demonstrate that adding tissue composition as state variables in the constitutive relations governing the stress-strain response and the mass supply represents a very general and straightforward method to model interstitial growth and remodeling in a wide variety of biological tissues. The foundation for this approach is rooted in the framework of mixture theory, which models the tissue as a mixture of multiple solid and fluid constituents. A further generalization is to allow each solid constituent in a constrained solid mixture to have its own reference (stress-free) configuration. Several illustrations are provided, ranging from bone remodeling to cartilage tissue engineering and cervical remodeling during pregnancy.