Atherosclerosis develops at arterial sites where endothelial cells (ECs) are exposed to low time-averaged shear stress, in particular in regions of recirculating disturbed flow. To understand how hemodynamics contributes to EC dysfunction in atheroma development, an in vitro parallel plate flow chamber gasket was modified with protruding baffles to produce large recirculating flow regions. Computational fluid dynamics predicted that more than 60% of the flow surface area was below the 12 dynes/cm2 atheroprotective threshold. Bovine aortic endothelial cells (BAECs) were then seeded in the parallel plate flow chamber with either the standard laminar or the new disturbed flow gasket and exposed to flow for 36 hours. Cell morphology, nitric oxide, proliferation, permeability, and monocyte adhesion were assessed by phase contrast and confocal microscopy. BAEC exposed to 20 dynes/cm2 shear stress in the laminar flow device aligned and elongated in the flow direction while increasing nitric oxide, decreasing permeability, and maintaining low proliferation and monocyte adhesion. As expected, BAEC in the recirculating flow and low shear stress disturbed flow device regions did not elongate or align, produced less nitric oxide, and showed higher proliferation, permeability, and monocyte adhesion than cells in the laminar flow device. Surprisingly, cells in disturbed flow device regions exposed to atheroprotective shear stress did not consistently align or decrease permeability, and these cells demonstrated low nitric oxide levels. These results suggest that atheroprotective signaling may be inhibited by neighboring cells exposed to recirculating flow, highlighting the complex relationship between hemodynamics and atheroma.

An analytical theory for the unconfined creep behavior of a cylindrical inclusion (simulating a soft tissue tumor) embedded in a cylindrical background sample (simulating normal tissue) is presented and analyzed in this paper. Both the inclusion and the background are considered as fluid-filled, porous materials, each of them being characterized by a set of mechanical parameters. Specifically, in this derivation, the inclusion is assumed to have significantly higher interstitial permeability than the background. The formulations of the effective Poisson's ratio (EPR) and fluid pressure in the inclusion and in the background are derived for the case of a sample subjected to a creep compression. The developed analytical expressions are validated using finite element models (FEM). Statistical comparison between the results obtained from the developed model and the results from FEM demonstrates accuracy of the proposed theoretical model higher than 99.4%. The model presented in this paper complements the one reported in the companion paper (Part I), which refers to the case of an inclusion having less interstitial permeability than the background.

An analytical theory for the unconfined creep behavior of a cylindrical inclusion (simulating a soft tissue tumor) embedded in a cylindrical background sample (simulating normal tissue) is presented and analyzed in this paper. Both the inclusion and the background are considered as fluid-filled, porous materials, each of them being characterized by a set of mechanical properties. Specifically, in this paper, the inclusion is considered to be less permeable than the background. The cylindrical sample is compressed using a constant pressure within two frictionless plates and is allowed to expand in an unconfined way along the radial direction. Analytical expressions for the effective Poisson's ratio (EPR) and fluid pressure inside and outside the inclusion are derived and analyzed. The theoretical results are validated using finite element models (FEMs). Statistical analysis shows excellent agreement between the results obtained from the developed model and the results from FEM. Thus, the developed theoretical model can be used in medical imaging modalities such as ultrasound poroelastography to extract the mechanical parameters of tissues and/or to better understand the impact of different mechanical parameters on the estimated displacements, strains, stresses, and fluid pressure inside a tumor and in the surrounding tissue.

Nerve guidance conduits (NGCs) are tubular tissue engineering scaffolds used for nerve regeneration. The poor mechanical properties and porosity have always compromised their performances for guiding and supporting axonal growth. Therefore, in order to improve the properties of NGCs, the computational design approach was adopted to investigate the effects of different NGC structural features on their various properties, and finally, design an ideal NGC with mechanical properties matching human nerves and high porosity and permeability. Three common NGC designs, namely hollow luminal, multichannel, and microgrooved, were chosen in this study. Simulations were conducted to study the mechanical properties and permeability. The results show that pore size is the most influential structural feature for NGC tensile modulus. Multichannel NGCs have higher mechanical strength but lower permeability compared to other designs. Square pores lead to higher permeability but lower mechanical strength than circular pores. The study finally selected an optimized hollow luminal NGC with a porosity of 71% and a tensile modulus of 8 MPa to achieve multiple design requirements. The use of computational design and optimization was shown to be promising in future NGC design and nerve tissue engineering research.

Viewed in renal physiology as a refined filtration device, the glomerulus filters large volumes of blood plasma while keeping proteins within blood circulation. Effects of macromolecule size and macromolecule hydrodynamic interaction with the nanostructure of the cellular layers of the glomerular capillary wall on the glomerular size selectivity are investigated through a mathematical simulation based on an ultrastructural model. The epithelial slit, a planar arrangement of fibers connecting the epithelial podocytes, is represented as a row of parallel cylinders with nonuniform spacing between adjacent fibers. The mean and standard deviation of gap half-width between its fibers are based on values recently reported from electron microscopy. The glomerular basement membrane (GBM) is represented as a fibrous medium containing fibers of two different sizes: the size of type IV collagens and that of glycosaminoglycans (GAGs). The endothelial cell layer is modeled as a layer full of fenestrae that are much larger than solute size and filled with GAGs. The calculated total sieving coefficient agrees well with the sieving coefficients of ficolls obtained from in vivo urinalysis in humans, whereas the computed glomerular hydraulic permeability also falls within the range estimated from human glomerular filtration rate (GFR). Our result indicates that the endothelial cell layer and GBM significantly contribute to solute and fluid restriction of the glomerular barrier, whereas, based on the structure of the epithelial slit obtained from electron microscopy, the contribution of the epithelial slit could be smaller than previously believed.

In this study, statistical models are developed for modeling uncertain heterogeneous permeability and porosity in tumors, and the resulting uncertainties in pressure and velocity fields during an intratumoral injection are quantified using a nonintrusive spectral uncertainty quantification (UQ) method. Specifically, the uncertain permeability is modeled as a log-Gaussian random field, represented using a truncated Karhunen–Lòeve (KL) expansion, and the uncertain porosity is modeled as a log-normal random variable. The efficacy of the developed statistical models is validated by simulating the concentration fields with permeability and porosity of different uncertainty levels. The irregularity in the concentration field bears reasonable visual agreement with that in MicroCT images from experiments. The pressure and velocity fields are represented using polynomial chaos (PC) expansions to enable efficient computation of their statistical properties. The coefficients in the PC expansion are computed using a nonintrusive spectral projection method with the Smolyak sparse quadrature. The developed UQ approach is then used to quantify the uncertainties in the random pressure and velocity fields. A global sensitivity analysis is also performed to assess the contribution of individual KL modes of the log-permeability field to the total variance of the pressure field. It is demonstrated that the developed UQ approach can effectively quantify the flow uncertainties induced by uncertain material properties of the tumor.

An existing axisymmetric fluid/structure-interaction (FSI) model of the spinal cord, pia mater, subarachnoid space, and dura mater in the presence of syringomyelia and subarachnoid-space stenosis was modified to include porous solids. This allowed investigation of a hypothesis for syrinx fluid ingress from cerebrospinal fluid (CSF). Gross model deformation was unchanged by the addition of porosity, but pressure oscillated more in the syrinx and the subarachnoid space below the stenosis. The poroelastic model still exhibited elevated mean pressure in the subarachnoid space below the stenosis and in the syrinx. With realistic cord permeability, there was slight oscillatory shunt flow bypassing the stenosis via the porous tissue over the syrinx. Weak steady streaming flow occurred in a circuit involving craniocaudal flow through the stenosis and back via the syrinx. Mean syrinx volume was scarcely altered when the adjacent stenosis bisected the syrinx, but increased slightly when the syrinx was predominantly located caudal to the stenosis. The fluid content of the tissues over the syrinx oscillated, absorbing most of the radial flow seeping from the subarachnoid space so that it did not reach the syrinx. To a lesser extent, this cyclic swelling in a boundary layer of cord tissue just below the pia occurred all along the cord, representing a mechanism for exchange of interstitial fluid (ISF) and cerebrospinal fluid which could explain recent tracer findings without invoking perivascular conduits. The model demonstrates that syrinx volume increase is possible when there is subarachnoid-space stenosis and the cord and pia are permeable.

The microvasculature is an extensive, heterogeneous, and complex system that plays a critical role in human physiology and disease. It nourishes almost all living human cells and maintains a local microenvironment that is vital for tissue and organ function. Operating under a state of continuous flow, with an intricate architecture despite its small caliber, and subject to a multitude of biophysical and biochemical stimuli, the microvasculature can be a complex subject to study in the laboratory setting. Engineered microvessels provide an ideal platform that recapitulates essential elements of in vivo physiology and allows study of the microvasculature in a precise and reproducible way. Here, we review relevant structural and functional vascular biology, discuss different methods to engineer microvessels, and explore the applications of this exciting tool for the study of human disease.

The biological response of living arteries to mechanical forces is an important component of the atherosclerotic process and is responsible, at least in part, for the well-recognized spatial variation in atherosusceptibility in man. Experiments to elucidate this response often generate maps of force and response variables over the arterial surface, from which the force–response relationship is sought. Rowland et al. discussed several statistical approaches to the spatial autocorrelation that confounds the analysis of such maps and applied them to maps of hemodynamic stress and vascular response obtained by averaging these variables in multiple animals. Here, we point out an alternative approach, in which discrete surface regions are defined by the hemodynamic stress levels they experience, and the stress and response in each animal are treated separately. This approach, applied properly, is insensitive to autocorrelation and less sensitive to the effect of confounding hemodynamic variables. The analysis suggests an inverse relation between permeability and shear that differs from that in Rowland et al. Possible sources of this difference are suggested.

This paper reviews and further develops pore-scale computational flow modeling techniques used for creeping flow through orthotropic fiber bundles used in blood oxygenators. Porous model significantly reduces geometrical complexity by taking a homogenization approach to model the fiber bundles. This significantly simplifies meshing and can avoid large time-consuming simulations. Analytical relationships between permeability and porosity exist for Newtonian flow through regular arrangements of fibers and are commonly used in macroscale porous models by introducing a Darcy viscous term in the flow momentum equations. To this extent, verification of analytical Newtonian permeability–porosity relationships has been conducted for parallel and transverse flow through square and staggered arrangements of fibers. Similar procedures are then used to determine the permeability–porosity relationship for non-Newtonian blood. The results demonstrate that modeling non-Newtonian shear-thinning fluids in porous media can be performed via a generalized Darcy equation with a porous medium viscosity decomposed into a constant term and a directional expression through least squares fitting. This concept is then investigated for various non-Newtonian blood viscosity models. The proposed methodology is conducted with two different porous model approaches, homogeneous and heterogeneous, and validated against a high-fidelity model. The results of the heterogeneous porous model approach yield improved pressure and velocity distribution which highlights the importance of wall effects.

Assessing the anatomical correlation of atherosclerosis with biomechanical localizing factors is hindered by spatial autocorrelation (SA), wherein neighboring arterial regions tend to have similar properties rather than being independent, and by the use of aggregated data, which artificially inflates correlation coefficients. Resampling data at lower resolution or reducing degrees-of-freedom in significance tests negated effects of SA but only in artificial situations where it occurred at a single length scale. Using Fourier or wavelet transforms to generate autocorrelation-preserving surrogate datasets, and thus to compute the null distribution, avoided this problem. Bootstrap methods additionally circumvented the errors caused by aggregating data. The bootstrap technique showed that wall shear stress (WSS) was significantly correlated with atherosclerotic lesion frequency and endothelial nuclear elongation, but not with the permeability of the arterial wall to albumin, in immature rabbits.

In order to understand how interstitial fluid pressure and flow affect cell behavior, many studies use microfluidic approaches to apply externally controlled pressures to the boundary of a cell-containing gel. It is generally assumed that the resulting interstitial pressure distribution quickly reaches a steady-state, but this assumption has not been rigorously tested. Here, we demonstrate experimentally and computationally that the interstitial fluid pressure within an extracellular matrix gel in a microfluidic device can, in some cases, react with a long time delay to external loading. Remarkably, the source of this delay is the slight (∼100 nm in the cases examined here) distension of the walls of the device under pressure. Finite-element models show that the dynamics of interstitial pressure can be described as an instantaneous jump, followed by axial and transverse diffusion, until the steady pressure distribution is reached. The dynamics follow scaling laws that enable estimation of a gel's poroelastic constants from time-resolved measurements of interstitial fluid pressure.

In this paper, a quantitative interpretation for atomic force microscopy-based dynamic nanoindentation (AFM-DN) tests on the superficial layers of bovine articular cartilage (AC) is provided. The relevant constitutive parameters of the tissue are estimated by fitting experimental results with a finite element model in the frequency domain. Such model comprises a poroelastic stress–strain relationship for a fibril reinforced tissue constitution, assuming a continuous distribution of the collagen network orientations. The identification procedure was first validated using a simplified transversely isotropic constitutive relationship; then, the experimental data were manually fitted by using the continuous distribution fibril model. Tissue permeability is derived from the maximum value of the phase shift between the input harmonic loading and the harmonic tissue response. Tissue parameters related to the stiffness are obtained from the frequency response of the experimental storage modulus and phase shift. With this procedure, an axial to transverse stiffness ratio (anisotropy ratio) of about 0.15 is estimated.

Hydrated soft tissues, such as articular cartilage, are often modeled as biphasic systems with individually incompressible solid and fluid phases, and biphasic models are employed to fit experimental data in order to determine the mechanical and hydraulic properties of the tissues. Two of the most common experimental setups are confined and unconfined compression. Analytical solutions exist for the unconfined case with the linear, isotropic, homogeneous model of articular cartilage, and for the confined case with the non-linear, isotropic, homogeneous model. The aim of this contribution is to provide an easily implementable numerical tool to determine a solution to the governing differential equations of (homogeneous and isotropic) unconfined and (inhomogeneous and isotropic) confined compression under large deformations. The large-deformation governing equations are reduced to equivalent diffusive equations, which are then solved by means of finite difference (FD) methods. The solution strategy proposed here could be used to generate benchmark tests for validating complex user-defined material models within finite element (FE) implementations, and for determining the tissue's mechanical and hydraulic properties from experimental data.

Recent advances in modulating collagen building blocks enable the design and control of the microstructure and functional properties of collagen matrices for tissue engineering and regenerative medicine. However, this is typically achieved by iterative experimentations and that process can be substantially shortened by computational predictions. Computational efforts to correlate the microstructure of fibrous and/or nonfibrous scaffolds to their functionality such as mechanical or transport properties have been reported, but the predictability is still significantly limited due to the intrinsic complexity of fibrous/nonfibrous networks. In this study, a new computational method is developed to predict two transport properties, permeability and diffusivity, based on a microstructural parameter, the specific number of interfibril branching points (or branching points). This method consists of the reconstruction of a three-dimensional (3D) fibrous matrix structure based on branching points and the computation of fluid velocity and solute displacement to predict permeability and diffusivity. The computational results are compared with experimental measurements of collagen gels. The computed permeability was slightly lower than the measured experimental values, but diffusivity agreed well. The results are further discussed by comparing them with empirical correlations in the literature for the implication for predictive engineering of collagen matrices for tissue engineering applications.

Liver cirrhosis represents the end-stage of different liver disorders, progressively affecting hepatic architecture, hemodynamics, and function. Morphologically, cirrhosis is characterized by diffuse fibrosis, the conversion of normal liver architecture into structurally abnormal regenerative nodules and the formation of an abundant vascular network. To date, the vascular remodeling and altered hemodynamics due to cirrhosis are still poorly understood, even though they seem to play a pivotal role in cirrhogenesis. This study aims to determine the perfusion characteristics of the cirrhotic circulation using a multilevel modeling approach including computational fluid dynamics (CFD) simulations. Vascular corrosion casting and multilevel micro-CT imaging of a single human cirrhotic liver generated detailed datasets of the hepatic circulation, including typical pathological characteristics of cirrhosis such as shunt vessels and dilated sinusoids. Image processing resulted in anatomically correct 3D reconstructions of the microvasculature up to a diameter of about 500 μm. Subsequently, two cubic samples (150 × 150 × 150 μm3) were virtually dissected from vascularized zones in between regenerative nodules and applied for CFD simulations to study the altered cirrhotic microperfusion and permeability. Additionally, a conceptual 3D model of the cirrhotic macrocirculation was developed to reveal the hemodynamic impact of regenerative nodules. Our results illustrate that the cirrhotic microcirculation is characterized by an anisotropic permeability showing the highest value in the direction parallel to the central vein (kd,zz = 1.68 × 10−13 m2 and kd,zz = 7.79 × 10−13 m2 for sample 1 and 2, respectively) and lower values in the circumferential (kd,ϑϑ = 5.78 × 10−14 m2 and kd,ϑϑ = 5.65 × 10−13 m2 for sample 1 and 2, respectively) and radial (kd,rr = 9.87 × 10−14 m2 and kd,rr = 5.13 × 10−13 m2 for sample 1 and 2, respectively) direction. Overall, the observed permeabilities are markedly higher compared to a normal liver, implying a locally decreased intrahepatic vascular resistance (IVR) probably due to local compensation mechanisms (dilated sinusoids and shunt vessels). These counteract the IVR increase caused by the presence of regenerative nodules and dynamic contraction mechanisms (e.g., stellate cells, NO-concentration, etc.). Our conceptual 3D model of the cirrhotic macrocirculation indicates that regenerative nodules severely increase the IVR beyond about 65 vol. % of regenerative nodules. Numerical modeling allows quantifying perfusion characteristics of the cirrhotic macro- and microcirculation, i.e., the effect of regenerative nodules and compensation mechanisms such as dilated sinusoids and shunt vessels. Future research will focus on the development of models to study time-dependent degenerative adaptation of the cirrhotic macro- and microcirculation.

Knowledge is limited about fluid flow in tissues containing engineered microvessels, which can be substantially different in topology than native capillary networks. A need exists for a computational model that allows for flow through tissues dense in nonpercolating and possibly nonperfusable microvessels to be efficiently evaluated. A finite difference (FD) model based on Poiseuille flow through a distribution of straight tubes acting as point sources and sinks, and Darcy flow through the interstitium, was developed to describe fluid flow through a tissue containing engineered microvessels. Accuracy of the FD model was assessed by comparison to a finite element (FE) model for the case of a single tube. Because the case of interest is a tissue with microvessels aligned with the flow, accuracy was also assessed in depth for a corresponding 2D FD model. The potential utility of the 2D FD model was then explored by correlating metrics of flow through the model tissue to microvessel morphometric properties. The results indicate that the model can predict the density of perfused microvessels based on parameters that can be easily measured.

Mechanical modeling of the deformation of a liquid-filled spherical microcapsule indented by a sharp truncated-cone indenter was proposed, in which membrane permeability was taken into account. The change in the internal volume of the microcapsule due to fluid permeation was calculated on the basis of Kedem and Katchalsky equations (1958, “Thermodynamic Analysis of the Permeability of Biological Membranes to Non-electrolytes,” Biochim. Biophys. Acta, 27, pp. 229–246). The membrane hydraulic permeability, membrane initial stretch, and effective osmotic pressure difference across the membrane of an alginate–poly(l)lysine–alginate (APA) microcapsule were identified by fitting calculated and measured force–displacement curves. The difference between deformed shapes with and without membrane permeability was shown, suggesting the spatial resolution of image analysis performed to measure the membrane permeability from the volume loss. The influences of changes in permeability, initial stretch, and a parameter β, used for determining the effective osmotic pressure difference, on the force–displacement relationship were examined, and mechanisms causing changes in the force–displacement relationship were discussed.

Ex vivo biomechanical testing of growth plate samples provides essential information about its structural and physiological characteristics. Experimental limitations include the preservation of the samples since working with fresh tissues involves significant time and transportation costs. Little information is available on the storage of growth plate explants. The aim of this study was to determine storage conditions that could preserve growth plate biomechanical properties. Porcine ulnar growth plate explants (n = 5 per condition) were stored at either 4 °C for periods of 1, 2, 3, and 6 days or frozen at −20 °C with slow or rapid sample thawing. Samples were tested using stress relaxation tests under unconfined compression to assess five biomechanical parameters. The maximum compressive stress (σmax) and the equilibrium stress (σeq) were directly extracted from the experimental curves, while the fibril-network reinforced biphasic model was used to obtain the matrix modulus (Em), the fibril modulus (Ef), and the permeability (k). No significant changes were observed in σeq and Em in any of the tested storage conditions. Significant decreases and increases, respectively, were observed in σmax and k in the growth plate samples refrigerated for more than 48 h and in the frozen samples, when compared with the fresh samples. The fibril modulus Ef of all stored samples was significantly reduced compared to the fresh samples. These results indicate that the storage of growth plates in a humid chamber at 4 °C for a maximum of 48 h is the condition that minimizes the effects on the measured biomechanical parameters, with only Ef significantly reduced. Refrigerating growth plate explants for less than 48 h maintains their maximal stress, equilibrium stress, matrix modulus, and permeability. However, cold storage at 4 °C for more than 48 h and freezing storage at −20 °C significantly alter the biomechanical response of growth plate samples. Appropriate growth plate sample storage will be beneficial to save time and reduce transportation costs to pick up fresh samples.

Development of an optimal systemic drug delivery strategy to the brain will require noninvasive or minimally invasive methods to quantify the permeability of the cerebral microvessel wall or blood-brain barrier (BBB) to various therapeutic agents and to measure their transport in the brain tissue. To address this problem, we used laser-scanning multiphoton microscopy to determine BBB permeability to solutes (P) and effective solute diffusion coefficients (Deff) in rat brain tissue 100–250 μm below the pia mater. The cerebral microcirculation was observed through a section of frontoparietal bone thinned with a microgrinder. Sodium fluorescein, fluorescein isothiocyanate (FITC)-dextrans, or Alexa Fluor 488-immunoglobulin G (IgG) in 1% bovine serum albumin (BSA) mammalian Ringer's solution was injected into the cerebral circulation via the ipsilateral carotid artery by a syringe pump at a constant rate of ∼3 ml/min. P and Deff were determined from the rate of tissue solute accumulation and the radial concentration gradient around individual microvessels in the brain tissue. The mean apparent permeability P values for sodium fluorescein (molecular weight (MW) 376 Da), dextran-4k, -20k, -40k, -70k, and IgG (MW ∼160 kDa) were 14.6, 6.2, 1.8, 1.4, 1.3, and 0.54 × 10−7 cm/s, respectively. These P values were not significantly different from those of rat pial microvessels for the same-sized solutes (Yuan et al., 2009, “Non-Invasive Measurement of Solute Permeability in Cerebral Microvessels of the Rat,” Microvasc. Res., 77(2), pp. 166–73), except for the small solute sodium fluorescein, suggesting that pial microvessels can be a good model for studying BBB transport of relatively large solutes. The mean Deff values were 33.2, 4.4, 1.3, 0.89, 0.59, and 0.47 × 10−7 cm2/s, respectively, for sodium fluorescein, dextran-4k, -20k, -40k, -70k, and IgG. The corresponding mean ratio of Deff to the free diffusion coefficient Dfree, Deff/Dfree, were 0.46, 0.19, 0.12, 0.12, 0.11, and 0.11 for these solutes. While there is a significant difference in Deff/Dfree between small (e.g., sodium fluorescein) and larger solutes, there is no significant difference in Deff/Dfree between solutes with molecular weights from 20,000 to 160,000 Da, suggesting that the relative resistance of the brain tissue to macromolecular solutes is similar over a wide size range. The quantitative transport parameters measured from this study can be used to develop better strategies for brain drug delivery.