Localized Collocation Meshless Method for Modeling Transdermal Pharmacokinetics in Multiphase Skin Structures

The human skin has a complicated structure with many multi-scale, biophysical effects impacting the propagation of skin-injected substances, such as partitioning, metabolic reactions, adsorption and elimination. An extended version of Fick’s second law governing the process of the compound diffusion in various skin layer is employed in the current work by considering the conservation of mass of the substance and the metabolic reaction of the substance in viable skin. Additionally, a model assuming linear coupling between the substance concentrations that are bound and unbound with blood was developed. Using such a model, a set of coupled partial differential equations are derived as the governing equations for the 3D dynamic skin pharmacokinetics model. To approximate the solution of these equations, a Meshless Method is developed employing localized collocation of inverse multi-quadric radial basis functions (RBF) to ensure a smooth, accurate, well-conditioned solution of the global field variable in space, while simultaneously implementing a forward-time marching scheme to address the time-transience of the solution. To validate the Localized RBF Collocation Meshless Model (LRCMM), the 2D and 3D cases of verapamil diffusion in viable skin was investigated. A benchmark, given in the literature employs empirically derived values for the governing equation parameters, providing a point of comparison for the LRCMM. In conjunction with the benchmark, the numerical analysis simulated the verapamil skin diffusion process during 4 hours of continuous injection with an input concentration of 43 mg/ml, followed by 4 hours of diffusion without further injection. The 2D and 3D LRCMM solutions compare well versus the benchmark, demonstrating the ability of the LRCMM to accurately model the system behavior. Thus, results from this study will be further implemented in future compound permeation studies, where advantages of the LRCMM will be leveraged for the optimization of various pharmacokinetic parameters for transdermal drug delivery. This is a joint work with Anthony Khoury, Vladimir V. Golubev, and Alain J. Kassa

Transient Non-linear Heat Conduction Solution by a Dual Reciprocity Boundary Element Method with an Effective Posteriori Error Estimator

A Dual Reciprocity Boundary Element Method is formulated to solve non-linear heat conduction problems. The approach is based on using the Kirchhoff transform along with lagging of the effective non-linear thermal diffusivity. A posteriori error estimate is used to provide effective estimates of the temporal and spatial error. A numerical example is used to demonstrate the approach

Singular Superposition/Boundary Element Method for Reconstruction of Multi-dimensional Heat Flux Distributions with Application to Film Cooling Holes

A hybrid singularity superposition/boundary element-based inverse problem method for the reconstruction of multi-dimensional heat flux distributions is developed. Cauchy conditions are imposed at exposed surfaces that are readily reached for measurements while convective boundary conditions are unknown at surfaces that are not amenable to measurements such as the walls of the cooling holes. The purpose of the inverse analysis is to determine the heat flux distribution along cooling hole surfaces. This is accomplished in an iterative process by distributing a set of singularities (sinks) inside the physical boundaries of the cooling hole (usually along cooling hole centerline) with a given initial strength distribution. A forward steady-state heat conduction problem is solved using the boundary element method (BEM), and an objective function is defined to measure the difference between the heat flux measured at the exposed surfaces and the heat flux predicted by the BEM under the current strength distribution of the singularities. A Genetic Algorithm (GA) iteratively alters the strength distribution of the singularities until the measuring surfaces heat fluxes are matched, thus satisfying Cauchy conditions. The distribution of the heat flux at the walls of the cooling hole is determined in a post-processing stage after the inverse problem is solved. The advantage of this technique is to eliminate the need of meshing the surfaces of the cooling holes, which requires a large amount of effort to achieve a high quality mesh. Moreover, the use of singularity distributions significantly reduces the number of parameters sought in the inverse problem, which constitutes a tremendous advantage in solving the inverse problem, particularly in the application of film cooling holes

Application of an RBF Blending Interpolation Method To Problems With Shocks

Radial basis functions (RBF) have become an area of research in recent years, especially in the use of solving partial differential equations (PDE). Radial basis functions have an impressive capability in interpolating scattered data, even for data with discontinuities. Although, for infinitely smooth radial basis functions such as the multi-quadrics and inverse multi-quadrics, the shape parameter must be chosen properly to obtain accurate approximations while avoiding ill-conditioning of the interpolating matrices. The optimum shape parameter can vary depending on the field, such as in locations of sharp gradients or shocks. Typically, the shape parameter is chosen to maintain a high conditioning number for the interpolation matrix, rendering the RBF smooth [1–10]. However, this strategy fails for a problem with a shock or sharp discontinuity. Instead, in such cases the conditioning number must be kept small. The focus of this work is then to demonstrate the use of RBF interpolation in the approximation of sharp gradients or shocks by use of a RBF blending interpolation approach. This RBF blending interpolation approach is used to maintain the optimum shape parameter depending on the field. The approach is able to sense gradients or shocks in the field and adjust the shape parameter accordingly to keep excellent accuracy. Presented in this work, is an explanation of the RBF blending interpolation methodology and testing of the RBF blending interpolation approach by solving the Burger’s equation using the virtual finite difference method

Biological Validation of a Microgravity Analog for Bacteria and Cell Cultures

With the future of long duration spaceflight missions looking to expand from the International Space Station (ISS) to deep space, it must be ensured that all critical systems, living and non-living, are thoroughly developed before humans begin the extended voyage. As we continue to unravel the effects of microgravity on cells, more questions arise on the molecular and cellular components and processes that sense and react to lowered gravity conditions. We have developed a microgravity analog that could simulate the effects of reduced gravity on cells and that could be maintained for a period of time long enough to observe and measure a wide variety of biological responses. On this instrument, the simulation of the effects of microgravity occurs when the samples rotate perpendicular to the gravity vector, moving in a very small circular path in the media that can be calculated based on Stoke’s Law. Once this path is significantly smaller than the natural diffusive motion, the cells can be assumed to be experiencing “functional weightlessness”. Here we present a new 2D clinostat design that operates under gravity and simulated microgravity conditions, simultaneously, and that is scalable to accommodate up to forty 2-mL liquid samples. This design was originally intended for bacterial studies that require a high number of replicates during multiple timepoints and it was mathematically and biologically validated using phenotypic and transcriptional endpoints on Escherichia coli K12 cultures

An RBF Interpolation Blending Scheme for Effective Shock-Capturing

In recent years significant focus has been given to the study of Radial basis functions (RBF), especially in their use on solving partial differential equations (PDE). RBF have an impressive capability of inter- polating scattered data, even when this data presents localized discontinuities. However, for infinitely smooth RBF such as the Multiquadrics, inverse Multiquadrics, and Gaussian, the shape parameter must be chosen properly to obtain accurate approximations while avoiding ill-conditioning of the interpolating matrices. The optimum shape parameter can vary significantly depending on the field, particularly in locations of steep gradients, shocks, or discontinuities. Typically, the shape parameter is chosen to be high value to render flatter RBF therefore yielding a high condition number for the ensuing interpola- tion matrix. However, this optimization strategy fails for problems that present steep gradients, shocks or discontinuities. Instead, in such cases, the optimal interpolation occurs when the shape parameter is chosen to be low in order to render steeper RBF therefore yielding low condition number for the interpolation matrix. The focus of this work is to demonstrate the use of RBF interpolation to capture the behaviour of steep gradients and shocks by implementing a blending scheme that combines high and low shape parameters. A formulation of the RBF blending interpolation scheme along with test- ing and validation through its implementation in the solution of the Burger’s linear advection equation and compressible Euler equations using a Localized RBF Collocation Meshless Method (LRC-MM) is presented in this paper

A Coupled Localized RBF Meshless/DRBEM Formulation for Accurate Modeling of Incompressible Fluid Flows

Velocity-pressure coupling schemes for the solution of incompressible fluid flow problems in Computational Fluid Dynamics (CFD) rely on the formulation of Poisson-like equations through projection methods. The solution of these Poisson-like equations represent the pressure correction and the velocity correction to ensure proper satisfaction of the conservation of mass equation at each step of a time-marching scheme or at each level of an iteration process. Inaccurate solutions of these Poisson-like equations result in meaningless instantaneous or intermediate approximations that do not represent the proper time-accurate behavior of the flow. The fact that these equations must be solved to convergence at every step of the overall solution process introduces a major bottleneck for the efficiency of the method. We present a formulation that achieves high levels of accuracy and efficiency by properly solving the Poisson equations at each step of the solution process by formulating a Localized RBF Collocation Meshless Method (LRC-MM) solution approach for the approximation of the diffusive and convective derivatives while employing the same framework to implement a Dual-Reciprocity Boundary Element Method (DR-BEM) for the solution of the ensuing Poisson equations. The same boundary discretization and point distribution employed in the LRC-MM is used for the DR-BEM. The methodology is implemented and tested in the solution of a backward-facing step problem

A New Control System for Left Ventricular Assist Devices Based on Patient-specific Physiological Demand

A left ventricular assist device (LVAD) is a mechanical pump that helps patients with a heart failure (HF) condition. This pump works in parallel to the ailing heart and provides a continuous flow from the weak left ventricle to the ascending aorta. The current supplied to the pump motor controls the flow of blood. A new feedback control system is developed to automatically adjust the pump motor current to provide the blood flow required by the level of activity of the patient. The systemic vascular resistance (RS) is the only undeterministic variable parameter in a patient-specific model and also a key value that expresses the level of activity of the patient. The rest of the parameters are constants for a patient-specific model. To determine the level of activity of the patient, an inverse problem approach is followed. The output data (pump flow) are observed and using an optimized search technique, the best model to describe such output is selected. Furthermore, the estimated RS is used in another patient-specific cardiovascular model that assumes a healthy heart, to determine the blood flow demand. Once the physiological demand is established, the current supplied to the pump motor of the LVAD can be adjusted to achieve the desired blood flow through the cardiovascular system. This process can be performed automatically in a real-time basis using information that is readily available and thus rendering a high degree of applicability. Results from simulated data show that the feedback control system is fast and very stable

Meshless 2D direct numerical simulation and heat transfer in a backward-facing step with heat conduction in the step

A meshless direct pressure-velocity coupling procedure is presented to perform Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) of turbulent incompressible flows in regular and irregular geometries. The proposed method is a combination of several efficient techniques found in different Computational Fluid Dynamic (CFD) procedures. With this new procedure, preliminary calculations with 2D steady state flows show that viscous effects become negligible faster that ever predicted numerically. The fundamental idea of this method lays on several important inconsistencies found in three of the most popular techniques used in CFD, segregated procedures, as well as in other formulations. The inconsistencies found become important in elliptic flows and they might lead to some wrong solutions. Preliminary calculations done in 2D laminar flows, suggest that the numerical diffusion and interpolation error are much important at low speeds, mainly when both, viscous and inertia forces are present. With this competitive and efficient procedure, the solution of the 2D Direct Numerical Simulation of turbulent flow with heat transfer on a backward-facing step is presented. The thermal energy is going to be transferred to the fluid through conduction on the step, with both constant temperature and heat flux conditions in the back wall of the step. The variation of the local Nusselt Number through the wall will be studied and its corresponding effect in the energy transfer to the fluid

Left Ventricular Assist Devices: Engineering Design Considerations

Patients with end-stage congestive heart failure awaiting heart transplantation often wait long periods of time (300 days or more on the average) before a suitable donor heart becomes available. The medical community has placed increased emphasis on the use of Left Ventricular Assist Devices or LVADs that can substitute for, or enhance, the function of the natural heart while the patient is waiting for the heart transplant (Poirier, 1997; Frazier & Myers, 1999). Essentially, a rotary LVAD is a pump that operates continuously directing blood from the left ventricle into the aorta by avoiding the aortic valve. Generally speaking, the goal of the LVAD is to assist the native heart in pumping blood through the circulatory system so as to provide the patient with as close to a normal lifestyle as possible until a donor heart becomes available or, in some cases, until the patient’s heart recovers. In many situations, this means allowing the patient to return home and/or to the workforce