A Highly Scalable Boundary Integral Equation and Walk-on-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data

In this paper, we study a highly scalable communication-free parallel domain boundary decomposition algorithm for the Laplace equation based on a hybrid method combining boundary integral equations and walk-on-spheres (BIE-WOS) method, which provides a numerical approximation of the Dirichlet-to-Neumann (DtN) mapping for the Laplace equation. The BIE-WOS is a local method on the boundary of the domain and does not require a structured mesh, and only needs a covering of the domain boundary by patches and a local mesh for each patch for a local BIE. A new version of the BIE-WOS method with second kind integral equations is introduced for better error controls. The effect of errors from the Feynman-Kac formula based path integral WOS method on the overall accuracy of the BIE-WOS method is analyzed for the BIEs, especially in the calculation of the right hand sides of the BIEs. For the special case of flat patches, it is shown that the second kind integral equation of BIE-WOS method can be simplified where the local BIE solutions can be given in closed forms. A key advantage of the parallel BIE-WOS method is the absence of communications during the computation of the DtN mapping on individual patches of the boundary, resulting in a complete independent computation using a large number of cluster nodes. In addition, the BIE-WOS has an intrinsic capability of fault tolerance for exascale computations. The nearly linear scalability of the parallel BIE-WOS method on a large-scale cluster with 6400 CPU cores is verified for computing the DtN mapping of exterior Laplace problems with Dirichlet data for several domains.     

Boundary Integral Modelling of Elastic Wave Propagation in Multi-Layered 2D Media with Irregular Interfaces

We present a semi-analytic method based on the propagation matrix formulation of indirect boundary element method to compute response of elastic (and acoustic) waves in multi-layered media with irregular interfaces. The method works recursively starting from the top-most free surface at which a stress-free boundary condition is applied, and the displacement-stress boundary conditions are then subsequently applied at each interface. The basic idea behind this method is the matrix formulation of the propagation matrix (PM) or more recently the reflectivity method as wide used in the geophysics community for the computation of synthetic seismograms in stratified media. The reflected and transmitted wave-fields between arbitrary shapes of layers can be computed using the indirect boundary element method (BEM, sometimes called IBEM). Like any standard BEM, the primary task of the BEM-based propagation matrix method (thereafter called PM-BEM) is the evaluation of element boundary integral of the Green's function, for which there are standard method that can be adapted. In addition, effective absorbing boundary conditions as used in the finite difference numerical method is adapted in our implementation to suppress the spurious arrivals from the artificial boundaries due to limited model space. To our knowledge, such implementation has not appeared in the literature. We present several examples in this paper to demonstrate the effectiveness of this proposed PM-BEM for modelling elastic waves in media with complex structure.     

Ablation of neural tissue by short-pulsed lasers — a technical report

Summary The basis for most laser applications in neurosurgery is the conversion of laser light into heat when the incident laser beam is absorbed by the tissue. Irradiation of neural tissue with laser light therefore leads to its thermal damage. However, due to the diffusion of heat energy into the surrounding tissue, often there is thermal damage to neural tissue outside the area of the target volume. These are the characteristics of thermal laser/tissue interaction. In this paper we discuss how we used three different short-pulsed lasers to achieve non-thermal ablation of neural tissue.Three different short-pulsed lasers were used to generate ultrashort laser pulses in the picosecond to femtosecond range. The interaction of such laser pulses with tissue was predicted to be nonthermal. The short-pulsed lasers were used for the ablation of neural tissue using an in vitro calf brain model. The histopathological examination of the lesions revealed that the neural tissue had been removed very precisely without any sign of thermal damage to the surrounding tissue.     

A Fourth-Order Kernel-Free Boundary Integral Method for Interface Problems

This paper presents a fourth-order Cartesian grid based boundary integral method (BIM) for heterogeneous interface problems in two and three dimensional space, where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions. The method reformulates the governing equation with interface conditions into boundary integral equations (BIEs) and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region. Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver. The structure of the coefficient matrix is preserved even with the existence of the interface. In the whole calculation process, analytical expressions of Green’s functions are never determined, formulated or computed. This is the novelty of the proposed kernel-free boundary integral (KFBI) method. Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.     

A Parallel Adaptive Treecode Algorithm for Evolution of Elastically Stressed Solids

The evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions. The governing equations are solved numerically using a boundary integral method (BIM). A critical step in applying BIM is to develop fast algorithms to reduce the arithmetic operation count of matrix-vector multiplications. In this paper, we develop a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions (2D). We present a novel source dividing strategy to parallelize the treecode. Numerical results show that the speedup factor is nearly perfect up to a moderate number of processors. This approach of parallelization can be readily implemented in other treecodes using either uniform or non-uniform point distribution. We demonstrate the effectiveness of the treecode by computing the long-time evolution of a complicated microstructure in elastic media, which would be extremely difficult with a direct summation method due to CPU time constraint. The treecode speeds up computations dramatically while fulfilling the stringent precision requirement dictated by the spectrally accurate BIM.     

Two-layer simulation model of laser-induced interstitial thermo-therapy

A two-layer model using different properties for the pathological tissue and the normal tissue was developed to describe the spatial photon, temperature and thermal damage distributions during laser-induced interstitial thermo-therapy (LITT). The photon distribution was simulated using the Monte Carlo method. The optical tissue parameters and the blood perfusion were derived based on the Arrhenius rate process formulation of thermal damage and kinetics of vasodilatation. The corresponding temperature distribution was numerically calculated using the Pennes bio-heat equation. The calculated results showed that the two-layer model predicted different results on the temperature variation and distribution, the thermal damage distribution and the thermal damage volume etc. from the one-layer model. As a more reasonable physical model, the two-layer model can be used to optimize the therapeutic parameters for improved LITT treatments.     

Picosecond optical breakdown: tissue effects and reduction of collateral damage

The effects of picosecond laser-induced optical breakdown on tissue were investigated using high-intensity 40 ps Nd:YAG laser pulses at 1.06 microns. Tissue damage was evaluated using the corneal endothelium in vitro as a model system. Systematic studies were performed to determine the scaling of the tissue damage and damage range with pulse energy. For suprathreshold lesions, the radius of the damage zone varies as the cube root of the pulse energy, in agreement with simple physical scaling laws. A minimum damage range of less than 100 microns was observed for pulse energies of 8 muJ. Damage morphology was investigated by scanning electron microscopy. Three different damage patterns were observed; cell damage, cell removal, and rupture of Descemet's membrane. Different irradiation geometries were used to study damage mediated by either the shock wave or the cavitation bubble. Comparative studies using 10 ns pulses demonstrated that picosecond pulses yielded a significant reduction in collateral tissue damage.     

A Fast Time Splitting Finite Difference Approach to Gross–Pitaevskii Equations

We propose an idea to solve the Gross–Pitaevskii equation for dark structures inside an infinite constant background density $ρ_∞$=${|ψ_∞|}^2$, without the introduction of artificial boundary conditions. We map the unbounded physical domain $\mathbb{R}^3$ into the bounded domain ${(−1,1)}^3$ and discretize the rescaled equation by equispaced 4th-order finite differences. This results in a free boundary approach, which can be solved in time by the Strang splitting method. The linear part is solved by a new, fast approximation of the action of the matrix exponential at machine precision accuracy, while the nonlinear part can be solved exactly. Numerical results confirm existing ones based on the Fourier pseudospectral method and point out some weaknesses of the latter such as the need of a quite large computational domain, and thus a consequent critical computational effort, in order to provide reliable time evolution of the vortical structures, of their reconnections, and of integral quantities like mass, energy, and momentum. The free boundary approach reproduces them correctly, also in finite subdomains, at low computational cost. We show the versatility of this method by carrying out one- and three-dimensional simulations and by using it also in the case of Bose–Einstein condensates, for which $ψ$→0 as the spatial variables tend to infinity.     

Review of Feynman's Path Integral in Quantum Statistics: From the Molecular Schrödinger Equation to Kleinert's Variational Perturbation Theory

Feynman's path integral reformulates the quantum Schrödinger differential equation to be an integral equation. It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems. In this Review, the molecular Schrödinger equation will first be introduced, together with the Born-Oppenheimer approximation that decouples electronic and internuclear motions. Some effective semiclassical potentials, e.g., centroid potential, which are all formulated in terms of Feynman's path integral, will be discussed and compared. These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger's energy eigenvalues. As a result, path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques. To complement these techniques, we will examine how Kleinert's variational perturbation (KP) theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential. To enable the powerful KP theory to be practical for many-body molecular systems, we have proposed a new path-integral method: automated integration-free path-integral (AIF-PI) method. Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method, we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions. The computational procedure of using our AIF-PI method, along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy (AMAZE), are also highlighted in this review.     

Integral Equation Method for a Non-Selfadjoint Steklov Eigenvalue Problem

We propose a numerical method for a non-selfadjoint Steklov eigenvalue problem of the Helmholtz equation. The problem is formulated using boundary integrals. The Nyström method is employed to discretize the integral operators, which leads to a non-Hermitian generalized matrix eigenvalue problems. The spectral indicator method (SIM) is then applied to calculate the (complex) eigenvalues. The convergence is proved using the spectral approximation theory for (non-selfadjoint) compact operators. Numerical examples are presented for validation.     

Hollow-waveguide-based nanosecond, near-infrared pulsed laser ablation of tissue

BACKGROUND AND OBJECTIVE: Short-pulse solid-state lasers have recently received much attention as new coherent light sources for medical applications, but steady transmission of their high-energy output pulses through a solid quartz fiber is difficult because of the onset of laser-induced breakdown. We previously demonstrated that hollow waveguides could be used to deliver nanosecond laser pulses for tissue ablation. The aim of this study was to determine the optimum laser pulse energy and range of defocused distance for obtaining a deep and sharp ablation channel in myocardial tissue with laser pulses transmitted through a hollow waveguide. STUDY DESIGN/MATERIALS AND METHODS: Cyclic-olefin-polymer-coated silver hollow waveguides of 1 mm in inner diameter and 1 m in length were used. A vacuum-cored scheme was applied to the waveguides to suppress laser-induced air breakdown. Porcine myocardial tissue was irradiated with 300 laser pulses that were delivered through the waveguide in vitro at various laser energy levels and defocused distances, and depths and diameters of channels were measured. Histological analysis of the ablated tissues was also performed. RESULTS: At an ablation energy of approximately 60 mJ/pulse, deep (>4.5 mm) and sharp (depth-to-diameter ratio of > 6) channels were created in tissue in the range of defocused distances of -4 approximately + 0.5 mm. Under these conditions, waveguide bending did not cause a remarkable change in ablation characteristics. Histological analysis of ablated tissue showed limited thermal damage but suggested a certain extent of mechanical effects in the tissue. CONCLUSION: With near-infrared, nanosecond laser pulses delivered through a cyclic-olefin-polymer-coated silver hollow waveguide, efficient and sharp ablation of myocardial tissue can be achieved, suggesting the usefulness of the hollow waveguide as a new flexible delivery system for high-intensity laser pulses.     

Fast Multipole Accelerated Boundary Integral Equation Method for Evaluating the Stress Field Associated with Dislocations in a Finite Medium

In this paper, we develop an efficient numerical method based on the boundary integral equation formulation and new version of fast multipole method to solve the boundary value problem for the stress field associated with dislocations in a finite medium. Numerical examples are presented to examine the influence from material boundaries on dislocations.     

A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs

This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations (PDEs), including both boundary value and interface problems. The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations. Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution, which satisfies an equivalent discrete interface problem, while the interface problem is solved by a fast solver in the Cartesian grid. The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain. This paper gives implementation details for a second-order version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems. The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface are of different types.     

Study of mechanical and thermal damage in brain tissue after ablation by Erbium-YAG laser

This work studied the ablation of bovine brain tissue by free-running Erbium-YAG laser pulses. Single-shot interactions were investigated by means of an ultra-fast imaging technique. Thin sections of the treated tissue were processed for histochemical analysis of enzyme activity to assess the extent of thermal/mechanical damage. Thereafter, a scanning beam technique was employed to deliver multiple pulses over a definite region of tissue. An analytical balance was used to measure the removed mass in order to calculate the ablation efficiency. The present quantity has been compared to the amount of the tissue damaged, as assessed by the histochemical analysis. The present work shows that the interaction of the Erbium-YAG laser pulses with a soft tissue may cause a large amount of mechanical damage, while thermal damage is restricted within a thin layer around the ablation crater. A precise control of fluence and operating conditions prevents overwhelming side-effects, and possibly allows the use of the Erbium-YAG laser for the ablation of brain and other soft tissues.     

Optimal boundary impedance for the minimization of reflection: (I) Asymptotic solutions by the geometrical optics method

In the design of surfaces which absorb waves, the impedance boundary condition is used as an effective means of diminishing the reflection. In this paper, we use the geometrical optics method to approximate the optimal impedance value which minimizes the reflected field for the scalar wave equation with a monochromatic source. Our treatment yields good results for optimal impedance in the asymptoticity region of the geometrical optics solution.     

Inclusion of memory effects in a dynamic model of electroporation in biological tissues

This article presents experimental and computational results of electroporation in rat liver. The experiments were performed using different forms of electrodes and waveforms of applied electric pulses. For the numerical simulation, the electroporation model proposed by Ramos and Weinert in a previous publication was used. Dynamic adjustments were used for obtaining a good modeling of the electric current. A single set of model parameters was obtained to fit the simulated current response for different waveforms and electrodes. These parameters were obtained with the use of a genetic algorithm that minimized the error between the simulated and experimental currents. The electroporation model with dynamic adjustment proved to be an appropriate simulation tool to predict the tissue conductivity during stimulation by intense electrical fields.     

Atherosclerotic tissue analysis by time-resolved XeCl excimer laser reflectometry

The temporal modification of XeCl laser pulses reflected from human aorta tissue immersed in saline has been studied. Dynamic tissue reflectivity of both normal and atherosclerotic tissues has been examined for various incident pulse fluences between 0.7 and 6.5 J/cm². Changes in reflected pulse duration are observed for fluences at or above 2.6 J/cm² with normal tissue targets and 3.0 J/cm² with calcified plaque. Such reflected pulse analysis may prove useful in identifying tissue targets for ablation during laser angioplasty. © 1993 Wiley-Liss, Inc.

1 This article is a US Government work and, such, is in the public domain in the United States of America.

     

A Well-Conditioned Hypersingular Boundary Element Method for Electrostatic Potentials in the Presence of Inhomogeneities within Layered Media

In this paper, we will present a high-order, well-conditioned boundary element method (BEM) based on Müller's hypersingular second kind integral equation formulation to accurately compute electrostatic potentials in the presence of inhomogeneity embedded within layered media. We consider two types of inhomogeneities: the first one is a simple model of an ion channel which consists of a finite height cylindrical cavity embedded in a layered electrolytes/membrane environment, and the second one is a Janus particle made of two different semi-spherical dielectric materials. Both types of inhomogeneities have relevant applications in biology and colloidal material, respectively. The proposed BEM gives$\mathcal{O}$(1) condition numbers, allowing fast convergence of iterative solvers compared to previous work using first kind of integral equations. We also show that the second order basis converges faster and is more accurate than the first order basis for the BEM.     

Picosecond Laser Ablation of Dentine in Endodontics

. The interaction of picosecond laser radiation with human dental tissue was investigated in this study, in order to determine the ablation rates and the surface characteristics of the dentine by using scanning electron microscopy (SEM). Dentine ablation was performed by using tooth sections of different thicknesses (0.5–2.0 mm). Dental tissue samples were irradiated in air with the fundamental wavelength and first harmonic of a regenerative amplifier Nd:YAG laser system, at 1064 nm and 532 nm, respectively, with a pulse duration of 100 ps and a pulse repetition rate of 10 Hz. The results showed very clean craters surrounded by minimum melting of the surface of dentine when the 1064 nm pulses were used. In contrast, when the first harmonic 532 nm pulses were used, the SEM examinations revealed cracks and melting of dentine with irregular surface modification. Consequently, it seems that cleaning and shaping of the root canal walls during endodontic therapy with the picosecond Nd:YAG laser application may be possible in the future. The, as yet unexplored, field of the picosecond laser interaction with hard dental tissue is expected to be a potential alternative for powerful laser processing of biomedical structures. Paper received 24 February 1998; accepted following revision 20 November 1998.     

A Fast Accurate Boundary Integral Method for Potentials on Closely Packed Cells

Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other. We present a boundary integral method in two dimensions which is specially designed to maintain second order accuracy even if boundaries are arbitrarily close. The method uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy. For boundaries with many components we use the fast multipole method for efficient summation. We compute electric potentials on a domain with cells whose conductivity differs from that of the surrounding medium. We first solve an integral equation for a source term on the cell interfaces and then find values of the potential near the interfaces via integrals. Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region. A number of examples are presented. We demonstrate that increased refinement is not needed to maintain accuracy as interfaces become very close.