Cells in 3D matrices under interstitial flow: Effects of extracellular matrix alignment on cell shear stress and drag forces

Interstitial flow is an important regulator of various cell behaviors both in vitro and in vivo, yet the forces that fluid flow imposes on cells embedded in a 3D extracellular matrix (ECM), and the effects of matrix architecture on those forces, are not well understood. Here, we demonstrate how fiber alignment can affect the shear and pressure forces on the cell and ECM. Using computational fluid dynamics simulations, we show that while the solutions of the Brinkman equation accurately estimate the average fluid shear stress and the drag forces on a cell within a 3D fibrous medium, the distribution of shear stress on the cellular surface as well as the peak shear stresses remain intimately related to the pericellular fiber architecture and cannot be estimated using bulk-averaged properties. We demonstrate that perpendicular fiber alignment of the ECM yields lower shear stress and pressure forces on the cells and higher stresses on the ECM, leading to decreased permeability, while parallel fiber alignment leads to higher stresses on cells and increased permeability, as compared to a cubic lattice arrangement. The Spielman–Goren permeability relationships for fibrous media agreed well with CFD simulations of flow with explicitly considered fibers. These results suggest that the experimentally observed active remodeling of ECM fibers by fibroblasts under interstitial flow to a perpendicular alignment could serve to decrease the shear and drag forces on the cell.     

Flow-induced deformation from pressurized cavities in absorbing porous tissues

The behaviour of a cavity during an injection of fluid into biological tissue is considered. High cavity pressure drives fluid into the neighbouring tissue where it is absorbed by capillaries and lymphatics. The tissue is modelled as a nonlinear deformable porous medium with the injected fluid absorbed by the tissue at a rate proportional to the local pressure. A model with a spherical cavity in an infinite medium is used to find the pressure and displacement of the tissue as a function of time and radial distance. Analytical and numerical solutions for a step change in cavity pressure show that the flow induces a radial compression in the medium together with an annular expansion, the net result being an overall expansion of the medium. Thus any flow induced deformation of the material will aid in the absorption of fluid.     

On the electrophysiological response of bone cells using a Stokesian fluid stimulus probe for delivery of quantifiable localized picoNewton level forces

A Stokesian fluid stimulus probe (SFSP), capable of delivering quantifiable pN level hydrodynamic forces, is developed to distinguish the electrophysiological response of the cell process and cell body of osteocyte-like MLO-Y4 cells without touching the cell or its substrate. The hydrodynamic disturbance is a short lived (100 ms), constant strength pressure pulse that propagates nearly instantaneously through the medium creating a nearly spherical expanding fluid bolus surrounding a 0.8 μm micropipette tip. Laboratory model experiments show that the growth of the bolus and the pressure field can be closely modeled by quasi-steady Stokes flow through a circular orifice provided the tip Reynolds number, Re(t)<0.03. By measuring the deflection of the dendritic processes between discrete attachment sites, and applying a detailed ultrastructural model for the central actin filament bundle within the process, one is able to calculate the forces produced by the probe using elastic beam theory. One finds that forces between 1 and 2.3 pN are sufficient to initiate electrical signaling when applied to the cell process, but not the much softer cell body. Even more significantly, cellular excitation by the process only occurs when the probe is directed at discrete focal attachment sites along the cell process. This suggests that electrical signaling is initiated at discrete focal attachments along the cell process and that these sites are likely integrin-mediated complexes associated with stretch-activated ion channels though their molecular structure is unknown.     

Theory of non-Newtonian viscosity of red blood cell suspension: effect of red cell deformation

The effects of the deformation of red blood cells on non-Newtonian viscosity of a concentrated red cell suspension are investigated theoretically. To simplify the problem an elastic spherical shell filled with an incompressible Newtonian fluid is considered as a model of a normal red cell. The equation of the surface of the shell suspended in a steady simple shear flow is calculated on the assumption that the deformation from a spherical shape is very small. The relative viscosity of a concentrated suspension of such particles is obtained based on the "free surface cell" method proposed by Happel. It is shown that the relative viscosity decreases as the shear rate increases.     

Non-Newtonian flow-induced deformation from pressurized cavities in absorbing porous tissues

We investigate the behavior of a spherical cavity in a soft biological tissue modeled as a deformable porous material during an injection of non-Newtonian fluid that follows a power law model. Fluid flows into the neighboring tissue due to high cavity pressure where it is absorbed by capillaries and lymphatics at a rate proportional to the local pressure. Power law fluid pressure and displacement of a solid in the tissue are computed as function of radial distance and time. Numerical solutions indicate that shear thickening fluids exhibit less fluid pressure and induce small solid deformation as compared to shear thinning fluids. Absorption in the biological tissue increases as a consequence of flow-induced deformation for power law fluids. In most cases non-Newtonian results are compared with the viscous fluid case to magnify the differences.     

A mathematical model of the dynamics of odontogenic cyst growth

OBJECTIVE: To formulate a mathematical model of odontogenic cyst growth and establish the dynamics of cyst enlargement and role of osmotic pressure forces throughout its growth. STUDY DESIGN: The model assumed a spherical cyst with a semipermeable lining of living cells and a core consisting of degraded cellular material, including generic osmotic material, fed by the continuous death of epithelial cells in the lining. The lining cells were assumed to have both elastic and viscous properties, reflecting the action of physical stresses by the surrounding cyst capsule, composed of fibroblasts and collagen fibers. The model couples the cyst radius and osmotic pressure differences resulting in a system of 2 nonlinear ordinary differential equations. RESULTS: The model predicts that in all parameter regimens the long-time behavior of the cyst is the same and that linear radial expansion results. CONCLUSION: In the early and intermediate stages of cystic growth, osmotic pressure differences play an important role; however, in very large cysts, this role becomes negligible, and cell birth in the lining dominates growth.     

Coupling of the Deformation of a Model Endothelial Cell and a Steady Laminar Flow

Abstract

Mechanical forces influence endothelial cell's (EC) morphology and functions. In this work it was proposed a numerical analysis of steady laminar flows near a modelled monolayer of elastic ECs in order to determine the local distributions of mechanical forces on the surface and inside the cell.

Numerical results showed that the flow induced non uniform mechanical stresses on cell surface and led to a cell deformation. These numerical results were compared with experimental measurements of the deformation of cultured human aortic endothelial cells under flow. It will be interesting to study eventual correlations between the distributions of biological receptors (cytoskeleton, adhesion molecules, etc.) and that of the non-uniform mechanical forces.     

Cell and tissue mechanics in cell migration

Migrating cells generate traction forces to counteract the movement-resisting forces arising from cell-internal stresses and matrix adhesions. In the case of collective migration in a cell colony, or in the case of 3-dimensional migration through connective tissue, movement-resisting forces arise also from external stresses. Although the deformation of a stiffer cell or matrix causes larger movement-resisting forces, at the same time a larger stiffness can also promote cell migration due to a feedback between forces, deformations, and deformation speed that is mediated by the acto-myosin contractile machinery of cells. This mechanical feedback is also important for stiffness sensing, durotaxis, plithotaxis, and collective migration in cell colonies.     

Finite element analysis of a three-dimensional open-celled model for trabecular bone

Based on a regular array of cubic unit cells, each containing a body-centered spherical void, we created an idealized three-dimensional model for both subchondral trabecular bone and a class of porous foams. By considering only face-to-face stacking of unit cells, the inherent symmetry was such that, except at the surface, the displacements and stresses within any one unit cell were representative of the entire porous structure. Using prescribed displacements the model was loaded in both uniaxial compressive strain and uniaxial shear strain. Based on the response to these loads, we found the tensor of elastic constants for an equivalent homogeneous elastic solid with cubic symmetry. We then compared the predicted modulus with our experimental values for bovine trabecular bone and literature values for an open-celled latex rubber foam.     

Effect of the stress phase angle on the strain energy density of the endothelial plasma membrane

Endothelial cells are simultaneously exposed to the mechanical forces of fluid wall shear stress (WSS) imposed by blood flow and solid circumferential stress (CS) induced by the blood vessel's elastic response to the pressure pulse. Experiments have demonstrated that these combined forces induce unique endothelial biomolecular responses that are not characteristic of either driving force alone and that the temporal phase angle between WSS and CS, referred to as the stress phase angle, modulates endothelial responses. In this article, we provide the first theoretical model to examine the combined forces of WSS and CS on a model of the endothelial cell plasma membrane. We focus on the strain energy density of the membrane that modulates the opening of ion channels that can mediate signal transduction. The model shows a significant influence of the stress phase angle on the strain energy density at the upstream and downstream ends of the cell where mechanotransduction is most likely to occur.     

Flow around cells adhered to a microvessel wall. I. Fluid stresses and forces acting on the cells

To evaluate the fluid forces acting on cells adhered to a microvessel wall, we numerically studied the flow field around adherent cells and the distribution of the stresses on their surfaces. For simplicity, the cells were modeled as rigid particles attached to a wall of a circular cylindrical tube regularly in the flow direction, in a row or two rows. It was found that not the detailed shape of the model cells but their height from the vessel wall is a key determinant of the fluid forces and torque acting on them. In both arrangements of one row and two rows, the axial spacing between neighboring adherent cells significantly affects the distributions of the stresses on them, which results in drastic variations of the fluid forces with the axial spacing and the relative positions with respect to their neighboring cells. The drag force acting on an adherent cell in the vessel was evaluated to be larger than the value in the 2D chamber flow at the same wall shear stress, mainly due to much larger variations of the pressure distribution on the cell surface in the vessel flow.     

On the infusion of a therapeutic agent into a solid tumor modeled as a poroelastic medium

The direct infusion of an agent into a solid tumor, modeled as a spherical poroelastic material with anisotropic dependence of the tumor hydraulic conductivity upon the tissue deformation, is treated both by solving the coupled fluid/elastic equations, and by expressing the solution as an asymptotic expansion in terms of a small parameter, ratio between the driving pressure force in the fluid system, and the elastic properties of the solid. Results at order one match almost perfectly the solutions of the full system over a large range of infusion pressures. Comparison with experimental results is acceptable after the hydraulic conductivity of the medium is properly calibrated. Given the uncertain estimates of some model constants, the order zero solution of the expansion, for which fluid and porous matrix are decoupled, yields acceptable values and trends for all the physical fields of interest, rendering the coupled analysis (in the limit of small displacements) of little use. When the deformation of the tissue becomes large nonlinear elasticity theory must be resorted to.     

Body forces and pressures in elastic models of the myocardium.

Tension strands are introduced to represent active myocardial fibers. They create one body force proportional to the divergence of the tension-direction vector, and a second equal to the tension divided by the radius of curvature. Explicit solutions to isotropic linearly elastic tensor equations with these body forces are found for the radially-symmetric, linearly-isotropic, elastic spherical heart with arbitrary radial body force. They confirm experiments showing supraluminal intramural pressures. Such pressures may affect coronary perfusion. A tension strand model which is a reasonable compromise between actual myofibrillar geometry and analytical simplicity is the iso-oblique, terminating, nonintersecting model. The body force from that or any other axially symmetric body force can be the forcing term for equations in which the heart is modeled as a thin, ellipsoidal, elastic membrane.     

A bioreactor test system to mimic the biological and mechanical environment of oral soft tissues and to evaluate substitutes for connective tissue grafts

Gingival cells of the oral connective tissue are exposed to complex mechanical forces during mastication, speech, tooth movement and orthodontic treatments. Especially during wound healing following surgical procedures, internal and external forces may occur, creating pressure upon the newly formed tissue. This clinical situation has to be considered when developing biomaterials to augment soft tissue in the oral cavity. In order to pre‐evaluate a collagen sponge intended to serve as a substitute for autogenous connective tissue grafts (CTGs), a dynamic bioreactor system was developed. Pressure and shear forces can be applied in this bioreactor in addition to a constant medium perfusion to cell‐material constructs. Three‐dimensional volume changes and stiffness of the matrices were analyzed. In addition, cell responses such as cell vitality and extracellular matrix (ECM) production were investigated. The number of metabolic active cells constantly increased under fully dynamic culture conditions. The sponges remained elastic even after mechanical forces were applied for 14 days. Analysis of collagen type I and fibronectin revealed a statistically significant accumulation of these ECM molecules (P < 0.05–0.001) when compared to static cultures. An increased expression of tenascin‐c, indicating tissue remodeling processes, was observed under dynamic conditions only. The results indicate that the tested in vitro cell culture system was able to mimic both the biological and mechanical environments of the clinical situation in a healing wound. Biotechnol. Bioeng. 2010;107: 1029–1039. © 2010 Wiley Periodicals, Inc.     

Geometry and mechanics of the morphogenesis of active membranes on the example of plant trichome cells of the genus <Emphasis Type="Italic">Draba</Emphasis> L.

The stages of the early morphogenesis of simple (unbranched) and complex (branched) unicellular trichomes are studied in two species of the genus Draba—D. sibirica (Pall.) Thell. and D. daurica DC. The geometry of morphogenesis is estimated by analyzing intraindividual variation of quantitative morphological characteristics of the developing leaf blade and peduncle trichomes. The surface of all types of trichome cells first acquires a spherical shape, followed by a U-shaped configuration with cylindrical proximal and spherical distal regions. In the development of complex trichomes, the area of the distal zone grows at a higher rate, which leads to separation of its volume into individual spherical regions, the morphogenesis of which repeats the early morphogenetic stages of the overall trichome cell, forming simple (unbranched) or complex (branched) trichome rays. As a rule, the lateral polarity of a trichome cell coincides with the proximodistal polarity of the leaf. Quantitative morphological data make it possible to infer an algorithm of the changes in shape common for all trichome cells, namely, the growth cycle comprising alternation of the phases of increase and decrease in the curvature of the outer cell surface. This surface is an active membrane expanded by the internal pressure and concurrently capable of actively increasing its area by incorporation of new structural elements. A distinctive feature of the proposed model is the geometrical inhomogeneity of the surface movement, changing the radius of curvature and creating internal (active) mechanical stresses in this membrane. A decrease in the ratio of the membrane surface area to the volume deprives the spatially homogeneous shape of its stability; correspondingly, the transition from elastic resistance to internal pressure to active resistance with the help of curvature differentiation becomes more energetically favorable. The source for growth and morphogenesis of the active membrane is alternation of the phases of local curvature leveling, which “charges” the membrane with active mechanical stresses and “discharge” of these stresses, leading to differentiation of the membrane’s local curvatures.     

Aggregation and disaggregation of red blood cells

The aggregation of red blood cells may be analyzed as an interaction of an adhesive surface energy and the elastic stored energy which results from deformation of the cell. The adhesive surface energy is the work required to separate a unit adhered area and is the resultant of adhesive forces due to the bridging molecules that bind the cells together and the electrostatic repulsion due to surface charge. The elastic strain energy in the case of the red blood is associated with the membrane elasticity only since the interior of the cell is liquid. The membrane elasticity is due both to bending stiffness and shear. The area expansion is small and may be neglected. These assumptions allow realistic computation of red cell shapes in rouleaux. The disaggregation of rouleaux requires an external force which must overcome the adhesive energy and also supply additional elastic energy of deformation. Depending on the geometry, the initial effect of elastic energy may tend to aid disaggregation. In a shear flow, the stresses on a suspended rouleau alternately tend to compress and to disaggregate the cells if they are free to rotate. This introduces a time dependence so that viscous effects due to the viscosity of the cell membrane, the cell cytoplasm and the external fluid may play a role in determining whether disaggregation proceeds to completion or not.     

Bending and puncturing the influenza lipid envelope

Lysosomes, enveloped viruses, as well as synaptic and secretory vesicles are all examples of natural nanocontainers (diameter ≈ 100 nm) which specifically rely on their lipid bilayer to protect and exchange their contents with the cell. We have applied methods primarily based on atomic force microscopy and finite element modeling that allow precise investigation of the mechanical properties of the influenza virus lipid envelope. The mechanical properties of small, spherical vesicles made from PR8 influenza lipids were probed by an atomic force microscopy tip applying forces up to 0.2 nN, which led to an elastic deformation up to 20%, on average. The liposome deformation was modeled using finite element methods to extract the lipid bilayer elastic properties. We found that influenza liposomes were softer than what would be expected for a gel phase bilayer and highly deformable: Consistent with previous suggestion that influenza lipids do not undergo a major phase transition, we observe that the stiffness of influenza liposomes increases gradually and weakly (within one order of magnitude) with temperature. Surprisingly, influenza liposomes were, in most cases, able to withstand wall-to-wall deformation, and forces >1 nN were generally required to puncture the influenza envelope, which is similar to viral protein shells. Hence, the choice of a highly flexible lipid envelope may provide as efficient a protection for a viral genome as a stiff protein shell.     

Fluid-Flow Induced Wall Shear Stress and Epithelial Ovarian Cancer Peritoneal Spreading

Epithelial ovarian cancer (EOC) is usually discovered after extensive metastasis have developed in the peritoneal cavity. The ovarian surface is exposed to peritoneal fluid pressures and shear forces due to the continuous peristaltic motions of the gastro-intestinal system, creating a mechanical micro-environment for the cells. An in vitro experimental model was developed to expose EOC cells to steady fluid flow induced wall shear stresses (WSS). The EOC cells were cultured from OVCAR-3 cell line on denuded amniotic membranes in special wells. Wall shear stresses of 0.5, 1.0 and 1.5 dyne/cm² were applied on the surface of the cells under conditions that mimic the physiological environment, followed by fluorescent stains of actin and β-tubulin fibers. The cytoskeleton response to WSS included cell elongation, stress fibers formation and generation of microtubules. More cytoskeletal components were produced by the cells and arranged in a denser and more organized structure within the cytoplasm. This suggests that WSS may have a significant role in the mechanical regulation of EOC peritoneal spreading.