Boundary element method analysis for the bioheat transfer equation.

In this paper, the boundary element method (BEM) approach is applied to solve the Pennes (1948) bioheat equation. The objective is to develop the BEM formulation and demonstrate its feasibility. The basic BEM formulations for the transient and steady-state cases are first presented. To demonstrate the usefulness of the BEM approach, numerical solutions for 2-D steady-state problems are obtained and compared to analytical solutions. Further, the BEM formulation is applied to model a conjugate problem for an artery imbedded in a perfused heated tissue. Analytical solution is possible when the conduction in the x-direction is negligible. The BEM and analytical results have very good agreement.     

The flow of a viscous liquid in a converging tube

The problem of the viscous flow of an incompressible Newtonian liquid in a converging tapered tube has been solved in spherical polar coordinates. The method of the solution involves the Stokes' stream function and a technique introduced by Stokes in the study of a sphere oscillating in a fluid. The theory for the flow in a rigid tube includes: (1) the pulsatile flow with both radial and angular velocity components; (2) the steady state flow with both radial and angular velocity components and (3) the very slow steady state flow with only a radial velocity component present. For a tapered elastic tube, the velocity of the propagated pulse wave is determined. The solution given is in terms of the elastic constants of the system and the coordinates for this type of geometry. The pulse velocity is then related to the velocity in an elastic cylindrical tube with the necessary correction terms to account for the tapered tube. Supported in part by the American Heart Association (No. 62F4EG). This work was done during the tenure of an Established Investigatorship of the American Heart Association.     

Dynamics of blood flow: modeling of Fåhraeus and Fåhraeus–Lindqvist effects using a shear-induced red blood cell migration model

Blood flow in micro capillaries of diameter approximately 15–500 μm is accompanied with a lower tube hematocrit level and lower apparent viscosity as the diameter decreases. These effects are termed the Fåhraeus and Fåhraeus–Lindqvist effects, respectively. Both effects are linked to axial accumulation of red blood cells. In the present investigation, we extend previous works using a shear-induced model for the migration of red blood cells and adopt a model for blood viscosity that accounts for the suspending medium viscosity and local hematocrit level. For fully developed hematocrit profiles (i.e., independent of axial location), the diffusion fluxes due to particle collision frequency and viscosity gradients are of equal magnitude and opposite directions. The ratio of the diffusion coefficients for the two fluxes affects both the Fåhraeus and Fåhraeus–Lindqvist effects and is found related to the capillary diameter and discharge hematocrit using a well-known data-fit correlation for apparent blood viscosity. The velocity and hematocrit profiles were determined numerically as functions of radial coordinate, tube diameter, and discharge hematocrit. The velocity profile determined numerically is consistent with the derived analytical expression and the results are in good agreement with published numerical results and experimental data for hematocrit ratio and hematocrit and velocity profiles.     

Application of boundary element method for more accurate localization of EEG dipole sources

The proposed modification of boundary element method (BEM) makes it possible to solve direct EEG problem taking into account individual head geometry. This modification decreases computational cost thus enabling practical usage of BEM for localization of EEG dipole sources. The method developed was applied for error estimation of dipole localization using spherical approximation with the aid of BESA software. It was shown that the localization error of BESA for real head geometry could reach 2 cm.     

Analysis of coupled intra- and extraluminal flows for single and multiple capillaries

Matched asymptotic expansions are used to study a model of the coupled fluid flow in the capillaries and tissue of the microcirculation. These capillaries are long, narrow cylindrical tubes embedded in a uniform tissue space. The capillary, or intraluminal, flow is assumed to be that of an incompressible Navier-Stokes fluid wherein colloids are represented as dilute solute; the extraluminal flow in the tissue is according to Darcy's law. Central to this fluid exchange is the boundary condition on the fluid radial velocity at the semipermeable wall of the capillary. This boundary condition, involving the local hydrostatic and colloidal osmotic pressures in both the capillary and the tissue, together with the radial gradient of the tissue hydrostatic pressure, couples the intra- and extraluminal flow fields. With this model we investigate the relationship between transport properties, hydrostatic pressures, and flow exchange for a single capillary, and describe the fluid transport in the tissue space produced by an array of such capillaries.     

Quantifying retinal capillary circulation using the scanning laser ophthalmoscope

The quantitative picture analysis of video fluorescein angiograms allows one to evaluate retinal hemodynamics. However, this method does not permit us to quantify the retinal capillary perfusion. Now, for the first time, we are able to quantify capillary bloodflow directly and objectively by means of scanning laser ophthalmoscopy. Even the measurement of the bloodflow velocity in all capillaries is now possible. Using digital frame-to-frame picture analysis of digital recordings, bloodflow velocities can be measured on the basis of the movement of dye boluses.     

Electrical resistance of a capillary endothelium

The electrical resistance of consecutive segments of capillaries has been determined by a method in which the microvessels were treated as a leaky, infinite cable. A two-dimensional analytical model to describe the potential field in response to intracapillary current injection was formulated. The model allowed determination of the electrical resistance from four sets of data: the capillary radius, the capillary length constant, the length constant in the mesentery perpendicular to the capillary, and the relative potential drop across the capillary wall. Of particular importance were the mesothelial membranes covering the mesenteric capillaries with resistances several times higher than that of the capillary endothelium. 27 frog mesenteric capillaries were characterized. The average resistance of the endothelium was 1.85 omega cm2, which compares well with earlier determinations of the ionic permeability of such capillaries. However, heterogeneity with respect to resistance was observed, that of 10 arterial capillaries being 3.0 omega cm2 as compared with 0.95 omega cm2 for 17 mid- and venous capillaries. The average in situ length constant was 99 micrometers for the arterial capillaries and 57 micrometers for the mid- and venous capillaries. It is likely that the ions that carry the current must move paracellularly, through junctions that are leaky to small solutes.     

IB–LBM simulation of the haemocyte dynamics in a stenotic capillary

To study the behaviour of a haemocyte when crossing a stenotic capillary, the immersed boundary–lattice Boltzmann method was used to establish a quantitative analysis model. The haemocyte was assumed to be spherical and to have an elastic cell membrane, which can be driven by blood flow to adopt a highly deformable character. In the stenotic capillary, the spherical blood cell was stressed both by the flow and the wall dimension, and the cell shape was forced to be stretched to cross the stenosis. Our simulation investigated the haemocyte crossing process in detail. The velocity and pressure were anatomised to obtain information on how blood flows through a capillary and to estimate the degree of cell damage caused by excessive pressure. Quantitative velocity analysis results demonstrated that a large haemocyte crossing a small stenosis would have a noticeable effect on blood flow, while quantitative pressure distribution analysis results indicated that the crossing process would produce a special pressure distribution in the cell interior and to some extent a sudden change between the cell interior and the surrounding plasma.     

Finite and Boundary Element Methods for Simulating Optical Properties of Plasmonic Nanostructures

In this study, a numerical investigation was done on the optical properties of silver nanostructures using the boundary element method (BEM) and finite element method (FEM). The BEM simulation was done using a freely available code called MNBEM in MATLAB with minor modifications. The FEM simulation was done by Comsol Multiphysics, a commercial software package. Silver nanostructures in the sphere, rod, and triangle geometries and the presence of different polarization angles were compared between these two methods. According to the obtained results, the absorption cross-sections for both BEM and FEM were consistent with their actual optical properties. For instance, both longitudinal and transverse resonance modes were observed in the case of nanorods, and all three in–plane dipole, in–plane quadrupole, and out–plane quadrupole plasmon resonances were observed successfully obtained for triangular nanostructures. Although both BEM and FEM results were similar to each other (from the number and position of the peaks in the final spectra), this similarity was decreased as the anisotropy was increased in the structure. For example, nearly 40 nm difference was observed between the BEM and FEM results in the triangular nanostructures, even though the trends and shape of the peaks were similar. It was revealed that specific points should be considered in the discretization process (especially the corner fillets) to close the gap in the obtained results from BEM and FEM. According to the obtained results, BEM significantly reduces the computational cost and time by discretizing only the boundary of the domain. A self-written software was developed to predict the optical cross-section of a plasmonic-ensemble consisting of spherical, rod-shaped, and triangular nanostructures, which can be used in different disciplines such as plasmon-enhanced solar cells, plasmon-enhanced photocatalysis, and plasmon-enhanced fluorescence.

Graphical Abstract

     

Computation of molecular electrostatics with boundary element methods.

In continuum approaches to molecular electrostatics, the boundary element method (BEM) can provide accurate solutions to the Poisson-Boltzmann equation. However, the numerical aspects of this method pose significant problems. We describe our approach, applying an alpha shape-based method to generate a high-quality mesh, which represents the shape and topology of the molecule precisely. We also describe an analytical method for mapping points from the planar mesh to their exact locations on the surface of the molecule. We demonstrate that derivative boundary integral formulation has numerical advantages over the nonderivative formulation: the well-conditioned influence matrix can be maintained without deterioration of the condition number when the number of the mesh elements scales up. Singular integrand kernels are characteristics of the BEM. Their accurate integration is an important issue. We describe variable transformations that allow accurate numerical integration. The latter is the only plausible integral evaluation method when using curve-shaped boundary elements.     

Boundary of the trapping region in the problem of disturbance of collisionless plasma by an absorbing sphere

The structure of the plasma disturbance near a spherical charged body is analyzed with allowance for the boundedness of the region of finite particle motion. The significance of the outer radius of the spherical region in which trapped particles can exist is demonstrated. The radius of the trapping sphere is determined by the boundary condition imposed on Poisson’s equation. The important role played by the boundary of the trapping region in the general analysis and solution of the problem is illustrated by simple examples.     

Dynamical clustering of red blood cells in capillary vessels

We have modeled the dynamics of a 3-D system consisting of red blood cells (RBCs), plasma and capillary walls using a discrete-particle approach. The blood cells and capillary walls are composed of a mesh of particles interacting with harmonic forces between nearest neighbors. We employ classical mechanics to mimic the elastic properties of RBCs with a biconcave disk composed of a mesh of spring-like particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, translational and angular momenta. Realistic behavior of blood cells is modeled by considering RBCs and plasma flowing through capillaries of various shapes. Three types of vessels are employed: a pipe with a choking point, a curved vessel and bifurcating capillaries. There is a strong tendency to produce RBC clusters in capillaries. The choking points and other irregularities in geometry influence both the flow and RBC shapes, considerably increasing the clotting effect. We also discuss other clotting factors coming from the physical properties of blood, such as the viscosity of the plasma and the elasticity of the RBCs. Modeling has been carried out with adequate resolution by using 1 to 10 million particles. Discrete particle simulations open a new pathway for modeling the dynamics of complex, viscoelastic fluids at the microscale, where both liquid and solid phases are treated with discrete particles. Figure A snapshot from fluid particle simulation of RBCs flowing along a curved capillary. The red color corresponds to the highest velocity. We can observe aggregation of RBCs at places with the most stagnant plasma flow.     

Velocity profiles in central airways with endotracheal intubation: a model study

Steady inspiratory velocity profiles were measured at two flow rates in a 3:1 scale model of the human central airways in the presence of five modes of endotracheal intubation. The presence of an orifice or a short endotracheal tube had no significant effect on the velocity profiles distal to the carina. Long endotracheal tubes change the profiles in both main bronchi. A significant peak occurred in the frontal plane near the walls, and the maximum velocity in the airway was almost identical to the endotracheal tube center-line velocity. The flow impinging on the medial wall of the main bronchus was redirected up around the anterior and posterior walls yielding bipeak velocity profiles in the sagittal plane. A tube placed eccentrically in the trachea over the right main bronchus did not alter the velocity profiles in the left main bronchus, suggesting a redirection of flow over the carina into the left lung. An endobronchial tube at the mouth of the right main bronchus did change the shape of the velocity profiles in the left main bronchus. In the left upper lobar bronchus the presence of trachea intubation had no effect on the velocity profiles. However, in the right upper lobar bronchus, the long endotracheal tube flattened the velocity profiles from the strongly skewed ones seen in the absence of the endotracheal inserts. These results not only are relevant to distribution of ventilation and aerosol particle deposition, but also have strong implications in intrapulmonary gas mixing, especially when high-frequency low tidal-volume ventilation is involved.     

Mathematical analysis of oxygen concentration in a two dimensional array of capillaries

 An approach is presented for modeling transport and exchange in skeletal muscle that can be used to analyze vascular beds consisting of a large number of interacting capillaries. First the oxygen concentration is determined in a functional unit consisting of a single capillary surrounded by a region of tissue in which a flux is prescribed on the outer boundary of the region. This flux, which is a result of the interaction among all of the capillaries comprising the vascular bed, is then found by matching the concentration along the borders between adjacent units. This leads to a system of ordinary differential equations for the oxygen concentration in the capillaries coupled with a system of algebraic equations for the fluxes. The method is illustrated by obtaining the oxygen concentration within an array of capillaries for the case when each capillary has a different initial concentration and for the case when each capillary has a different flow rate. Received: 12 June 2001 / Revised version: 18 April 2002 / Published online: 17 January 2003 Key words or phrases: Skeletal muscle – Transport – Microcirculation     

Force acting on spheres adhered to a vessel wall

To evaluate the force and torque acting on leukocytes attached to the vessel wall, we numerically study the flow field around the leukocytes by using rigid spherical particles adhered to the wall of a circular cylindrical tube as a model of adherent leukocytes. The adherent particles are assumed to be placed regularly in the flow direction with equal spacings, in one row or two rows. The flow field of the suspending fluid is analyzed by a finite element method applied to the Stokes equations, and the drag force and torque acting on each particle, as well as the apparent viscosity, are evaluated as a function of the particle to tube diameter ratio and the particle arrangements. For two-row arrangements of adhered particles where neighboring particles are placed alternately on opposite sides of the vessel, the drag and the torque exerted on each particle are higher than those for single-row arrangements, for constant particle to tube diameter ratio and axial spacing between neighboring particles. This is enhanced for Larger particles and smaller axial spacings. The apparent viscosity of the flow through vessels with adhered particles is found to be significantly higher than that without adhered particles or when the particles are freely floating through the vessels.     

Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model

A computational model for retinal hemodynamics accounting for ocular curvature is presented. The model combines (i) a hierarchical Darcy model for the flow through small arterioles, capillaries and small venules in the retinal tissue, where blood vessels of different size are comprised in different hierarchical levels of a porous medium; and (ii) a one-dimensional network model for the blood flow through retinal arterioles and venules of larger size. The non-planar ocular shape is included by (i) defining the hierarchical Darcy flow model on a two-dimensional curved surface embedded in the three-dimensional space; and (ii) mapping the simplified one-dimensional network model onto the curved surface. The model is solved numerically using a finite element method in which spatial domain and hierarchical levels are discretized separately. For the finite element method, we use an exterior calculus-based implementation which permits an easier treatment of non-planar domains. Numerical solutions are verified against suitably constructed analytical solutions. Numerical experiments are performed to investigate how retinal hemodynamics is influenced by the ocular shape (sphere, oblate spheroid, prolate spheroid and barrel are compared) and vascular architecture (four vascular arcs and a branching vascular tree are compared). The model predictions show that changes in ocular shape induce non-uniform alterations of blood pressure and velocity in the retina. In particular, we found that (i) the temporal region is affected the least by changes in ocular shape, and (ii) the barrel shape departs the most from the hemispherical reference geometry in terms of associated pressure and velocity distributions in the retinal microvasculature. These results support the clinical hypothesis that alterations in ocular shape, such as those occurring in myopic eyes, might be associated with pathological alterations in retinal hemodynamics.     

Application of Different Variants of the BEM in Numerical Modeling of Bioheat Transfer Problems

Heat transfer processes proceeding in the living organisms are described by the different mathematical models. In particular, the typical continuous model of bioheat transfer bases on the most popular Pennes equation, but the Cattaneo-Vernotte equation and the dual phase lag equation are also used. It should be pointed out that in parallel are also examined the vascular models, and then for the large blood vessels and tissue domain the energy equations are formulated separately. In the paper the different variants of the boundary element method as a tool of numerical solution of bioheat transfer problems are discussed. For the steady state problems and the vascular models the classical BEM algorithm and also the multiple reciprocity BEM are presented. For the transient problems connected with the heating of tissue, the various tissue models are considered for which the 1st scheme of the BEM, the BEM using discretization in time and the general BEM are applied. Examples of computations illustrate the possibilities of practical applications of boundary element method in the scope of bioheat transfer problems.     

Rotational diffusion coefficients of a small,spherical subunit flexibly tethered to a larger sphere

The rotational diffusion coefficients of a small spherical particle, which is flexibly anchored to the surface of a much larger sphere, are calculated using the hydrodynamic theory of segmentally flexible particles. The model is intended for representing the rotational mobility of a small residue or chromophore in the surface of a globular macromolecule. The coefficients are found to be essentially independent, or to vary slowly with the relative dispositions of the spheres. They are also insensitive to the size ratio when this ratio is high enough. These findings support the use of an approximative treatment proposed by Wegener in which the small conformation dependence is averaged out. The resulting averages are tentatively used in the Lipari-Szabo model for restricted rotational diffusion in a cone. It is concluded that the rotational relaxation of the small sphere has three components: (i) a torsional rotation with the same diffusion coefficient as the free sphere; (ii) a perpendicular wobbling with a diffusion coefficient several (five in a typical case) times smaller; and (iii) an overall rotation of the whole macromolecule, that will appear in a much longer time scale if the two spheres have quite distinct sizes.