Self-sustained oscillations in blood flow through a honeycomb capillary network.

Numerical simulations of unsteady blood flow through a honeycomb network originating at multiple inlets and terminating at multiple outlets are presented and discussed under the assumption that blood behaves as a continuum with variable constitution. Unlike a tree network, the honeycomb network exhibits both diverging and converging bifurcations between branching capillary segments. Numerical results based on a finite difference method demonstrate that as in the case of tree networks considered in previous studies, the cell partitioning law at diverging bifurcations is an important parameter in both steady and unsteady flow. Specifically, a steady flow may spontaneously develop self-sustained oscillations at critical conditions by way of a Hopf bifurcation. Contrary to tree-like networks comprised entirely of diverging bifurcations, the critical parameters for instability in honeycomb networks depend weakly on the system size. The blockage of one or more network segments due to the presence of large cells or the occurrence of capillary constriction may cause flow reversal or trigger a transition to unsteady flow.

A floating prolate spheroid.

The equilibrium position of a spherical or prolate spheroidal particle resembling a needle floating at the interface between two immiscible fluids is discussed. A three-dimensional meniscus attached to an a priori unknown contact line at a specified contact angle is established around the particle, imparting to the particle a capillary force due to surface tension that is balanced by the buoyancy force and the particle weight. An accurate numerical solution for a floating sphere is obtained by solving a boundary-value problem, and the results are compared favorably with an approximate solution where the effect of the particle surface curvature is ignored and the elevation of the contact line is computed using an analytical solution for the meniscus attached to an inclined flat plate. The approximate formulation is applied locally around the nearly planar elliptical contact line of a prolate spheroid to derive a nonlinear algebraic equation governing the position of the particle center and the mean elevation of the contact line. The effect of the fluid and particle densities, contact angle, and capillary length is discussed, and the shape of the contact line is reconstructed and displayed from the local solution.

On the shape of a hydrostatic meniscus attached to a corrugated plate or wavy cylinder.

The shape of a hydrostatic meniscus attached at a fixed contact angle to a vertical plate or circular cylinder with periodic corrugations is studied by analytical and numerical methods, and the effect of wall irregularities on the shape of the contact line and vertical component of the capillary force is discussed. An asymptotic analysis for a plate with small-amplitude sinusoidal corrugations is carried out to first order with respect to the corrugation amplitude, and a boundary-value problem is formulated and solved by a shooting method to determine the meniscus shape and elevation of the contact line. The meniscus attached to a corrugated plate with rounded corners produced by a Schwarz-Christoffel mapping function for a triangular wave is considered by numerical methods. The Laplace-Young equation determining the meniscus shape is solved in orthogonal curvilinear coordinates generated by conformal mapping using a finite-difference method. The numerical results are successfully compared with the predictions of the perturbation expansion for small amplitudes and discussed with reference to the rise of a meniscus inside a dihedral angle for large amplitudes. A companion asymptotic analysis is presented for a meniscus outside a vertical circular cylinder with small-amplitude sinusoidal corrugations. The analytical predictions are successfully compared with numerical solutions of the Laplace-Young equation for a meniscus outside an elliptical cylinder with aspect ratio near unity, regarded as a deformed circle.

Numerical Simulation of Unsteady Blood Flow through Capillary Networks.

A numerical method is implemented for computing unsteady blood flow through a branching capillary network. The evolution of the discharge hematocrit along each capillary segment is computed by integrating in time a one-dimensional convection equation using a finite-difference method. The convection velocity is determined by the local and instantaneous effective capillary blood viscosity, while the tube to discharge hematocrit ratio is deduced from available correlations. Boundary conditions for the discharge hematocrit at divergent bifurcations arise from the partitioning law proposed by Klitzman and Johnson involving a dimensionless exponent, q≥1. When q=1, the cells are partitioned in proportion to the flow rate; as q tends to infinity, the cells are channeled into the branch with the highest flow rate. Simulations are performed for a tree-like, perfectly symmetric or randomly perturbed capillary network with m generations. When the tree involves more than a few generations, a supercritical Hopf bifurcation occurs at a critical value of q, yielding spontaneous self-sustained oscillations in the absence of external forcing. A phase diagram in the m-q plane is presented to establish conditions for unsteady flow, and the effect of various geometrical and physical parameters is examined. For a given network tree order, m, oscillations can be induced for a sufficiently high value of q by increasing the apparent intrinsic viscosity, decreasing the ratio of the vessel diameter from one generation to the next, or by decreasing the diameter of the terminal vessels. With other parameters fixed, oscillations are inhibited by increasing m. The results of the continuum model are in excellent agreement with the predictions of a discrete model where the motion of individual cells is followed from inlet to outlet.

Hydrostatic meniscus between two eccentric circular cylinders.

A numerical method is implemented for computing the shape of a three-dimensional hydrostatic meniscus extending between two arbitrary closed contact lines under the restriction that the projections of the contact lines in a horizontal plane are eccentric circles. In a physical realization, the contact lines are attached to vertical circular cylinders, spherical particles or containers. The Laplace-Young equation determining the meniscus shape is solved in bipolar coordinates generated by conformal mapping using a finite-difference method, and the capillary force and torque exerted on the cylinders are evaluated. Numerical results are presented for a meniscus extending between two circular horizontal contact lines. The horizontal component of the capillary force at each contact line is found to increase monotonically with the cylinder center offset, favoring the concentric configuration.

Numerical simulation of blood and interstitial flow through a solid tumor.

A theoretical framework is presented for describing blood flow through the irregular vasculature of a solid tumor. The tumor capillary bed is modeled as a capillary tree of bifurcating segments whose geometrical construction involves deterministic and random parameters. Blood flow along the individual capillaries accounts for plasma leakage through the capillary walls due to the transmural pressure according to Sterling's law. The extravasation flow into the interstitium is described by Darcy's law for a biological porous medium. The pressure field developing in the interstitium is computed by solving Laplace's equation subject to derived boundary conditions at the capillary vessel walls. Given the arterial, venous, and tumor surface pressures, the problem is formulated as a coupled system of integral and differential equations arising from the interstitium and capillary flow transport equations. Numerical discretization yields a system of linear algebraic equations for the interstitial and capillary segment pressures whose solution is found by iterative methods. Results of numerical computations document the effect of the interstitial hydraulic and vascular permeability on the fractional plasma leakage. Given the material properties, the fractional leakage reaches a maximum at a particular grade of the bifurcating vascular tree.

Numerical simulation of blood flow through microvascular capillary networks.

A numerical method is implemented for computing blood flow through a branching microvascular capillary network. The simulations follow the motion of individual red blood cells as they enter the network from an arterial entrance point with a specified tube hematocrit, while simultaneously updating the nodal capillary pressures. Poiseuille's law is used to describe flow in the capillary segments with an effective viscosity that depends on the number of cells residing inside each segment. The relative apparent viscosity is available from previous computational studies of individual red blood cell motion. Simulations are performed for a tree-like capillary network consisting of bifurcating segments. The results reveal that the probability of directional cell motion at a bifurcation (phase separation) may have an important effect on the statistical measures of the cell residence time and scattering of the tube hematocrit across the network. Blood cells act as regulators of the flow rate through the network branches by increasing the effective viscosity when the flow rate is high and decreasing the effective viscosity when the flow rate is low. Comparison with simulations based on conventional models of blood flow regarded as a continuum indicates that the latter underestimates the variance of the hematocrit across the vascular tree.

Numerical simulation of cell motion in tube flow.

A theoretical model is presented for describing the motion of a deformable cell encapsulating a Newtonian fluid and enclosed by an elastic membrane in tube flow. In the mathematical formulation, the interior and exterior hydrodynamics are coupled with the membrane mechanics by means of surface equilibrium equations, and the problem is formulated as a system of integral equations for the interfacial velocity, the disturbance tube-wall traction, and the pressure difference across the two ends to the tube due to the presence of the cell. Numerical solutions obtained by a boundary-element method are presented for flow in a cylindrical tube with a circular cross-section, cytoplasm viscosity equal to the ambient fluid viscosity, and cells positioned sufficiently far from the tube wall so that strong lubrication forces do not arise. In the numerical simulations, cells with spherical, oblate ellipsoidal, and biconcave unstressed shapes enclosed by membranes that obey a neo-Hookean constitutive equation are considered. Spherical cells are found to slowly migrate toward the tube centerline at a rate that depends on the mean flow velocity, whereas oblate and biconcave cells are found to develop parachute and slipper-like shapes, respectively, from axisymmetric and more general initial orientations.

Resting shape and spontaneous membrane curvature of red blood cells.

A boundary-value problem is formulated describing the biconcave resting shape of normal red blood cells, based on local constitutive equations for the membrane tensions and bending moments. The fundamental physical assumption is that curvature-dependent anisotropic membrane stress resultants accompanied by isotropic bending moments arise from isotropic tensions developing in each leaflet of the lipid bilayer, while the cytoskeleton is unstressed in the resting configuration. Families of equilibrium resting shapes parametrized by the spontaneous bilayer curvature and cell sphericity compare favourably with the average shape of normal red blood cells. The successful comparison supports Helfrich's notion of a non-zero spontaneous curvature whose magnitude is nearly equal to the negative of the equivalent cell radius defined with respect to the membrane surface area. The structure of the solution space suggests a minimum spontaneous curvature below which the cell sphericity is lower than that of the red blood cell, independent of the transmural pressure. The computed cell shapes also compare favourably with the shapes of swollen red blood cells, though for a different value of the spontaneous curvature. The dependence of the spontaneous curvature on the cell volume is attributed to in-plane elastic tensions developing due to the deformation of the cytoskeleton. An alternative formulation based on a non-local model for the monolayer tensions is found to be incapable of predicting non-spherical shapes.

Numerical simulation of the flow-induced deformation of red blood cells.

A theoretical model is presented for describing the flow-induced deformation of red blood cells. The cells are modeled as deformable liquid capsules enclosed by a membrane that is nearly incompressible and exhibits elastic response to shearing and bending deformation. In the mathematical formulation, the hydrodynamics is coupled with the membrane mechanics by means of surface equilibrium equations expressed in global Cartesian coordinates. Numerical simulations are carried out to investigate the deformation of a cell in simple shear flow, in the physiological range of physical properties and flow conditions. The results show that the cell performs flipping motion accompanied by periodic deformation in which the cross section of the membrane in the plane that is perpendicular to the vorticity of the shear flow alternates between the nearly biconcave resting shape and a reverse S shape. The period of the overall rotation is in good agreement with the experimental observations of Goldsmith and Marlow for red blood cells suspended in plasma. Parametric investigations reveal that, in the range of shear rates considered, membrane compressibility has a secondary influence on the cell deformation and on the effective viscosity of a dilute suspension. The numerical results illustrate in quantitative terms the distribution of the membrane tensions developing due to the flow-induced deformation, and show that the membrane is subjected to stretching and compression in the course of the rotation.

Film flow of a suspension down an inclined plane.

A method is developed for simulating the film flow of a suspension of rigid particles with arbitrary shapes down an inclined plane in the limit of vanishing Reynolds number. The problem is formulated in terms of a system of integral equations of the first and second kind for the free-surface velocity and the traction distribution along the particle surfaces involving the a priori unknown particle linear velocity of translation and angular velocity of rotation about designated centres. The problem statement is completed by introducing scalar constraints that specify the force and torque exerted on the individual particles. A boundary-element method is implemented for solving the governing equations for the case of a two-dimensional periodic suspension. The system of linear equations arising from numerical discretization is solved using a preconditioner based on a particle-cluster iterative method recently developed by Pozrikidis (2000 Engng Analysis Bound. Elem. 25, 19-30). Numerical investigations show that the generalized minimal residual (GMRES) method with this preconditioner is significantly more efficient than the plain GMRES method used routinely in boundary-element implementations. Extensive numerical simulations for solitary particles and random suspensions illustrate the effect of the particle shape, size and aspect ratio in semi-finite shear flow, and the effect of free-surface deformability in film flow.

A model of fluid flow in solid tumors.

Solid tumors consist of a porous interstitium and a neoplastic vasculature composed of a network of capillaries with highly permeable walls. Blood flows across the vasculature from the arterial entrance point to the venous exit point, and enters the tumor by convective and diffusive extravasation through the permeable capillary walls. In this paper, an integrated theoretical model of the flow through the tumor is developed. The flow through the interstitium is described by Darcy's law for an isotropic porous medium, the flow along the capillaries is described by Poiseuille's law, and the extravasation flux is described by Starling's law involving the pressure on either side of the capillaries. Given the arterial, the venous, and the ambient pressure, the problem is formulated in terms of a coupled system of integral and differential equations for the vascular and interstitial pressures. The overall hydrodynamics is described in terms of hydraulic conductivity coefficients for the arterial and venous flow rates whose functional form provides an explanation for the singular behavior of the vascular resistance observed in experiments. Numerical solutions are computed for an idealized case where the vasculature is modeled as a single tube, and charts of the hydraulic conductivities are presented for a broad range of tissue and capillary wall conductivities. The results in the physiological range of conditions are found to be in good agreement with laboratory observations. It is shown that the assumption of uniform interstitial pressure is not generally appropriate, and predictions of the extravasation rate based on it may carry a significant amount of error.