Wensheng Shen

Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue

A mathematical model describing the thermomechanical interactions in biological bodies at high temperature is proposed by treating the soft tissue in biological bodies as a thermoporoelastic media. The heat transfer and elastic deformation in soft tissues are examined based on the Pennes bioheat transfer equation and the modified Duhamel-Neuman equations. The three-dimensional governing equations based on the proposed model is discretized using a 19-point finite-difference scheme. The resulting large sparse linear system is solved by a preconditioned Krylov subspace method. Numerical simulations show that the proposed model is valid under our test conditions and the proposed numerical techniques are efficient.

AN EXPLICIT TVD SCHEME FOR HYPERBOLIC HEAT CONDUCTION IN COMPLEX GEOMETRY

Two-dimensional hyperbolic heat conduction problems of complex geometry are investigated numerically. A second-order total variation diminishing (TVD) scheme is introduced and its application to the hyperbolic heat conduction is developed in detail using the knowledge of characteristics. In current work primitive variables, rather than characteristic variables, are used as the dependent variables. The governing equations of two-dimensional heat conduction are transformed from the physical coordinates to the computational coordinates, so that the hyperbolic heat conduction problems of irregular geometry can be solved numerically by the present TVD scheme. Three examples with different geometry are used to verify the accuracy of the present numerical scheme. Results show the explicit TVD scheme can predict the thermal wave without oscillation.

A Computational Model of FGF-2 Binding and HSPG Regulation Under Flow

A novel convection--diffusion--reaction model is developed to simulate fibroblast growth factor (FGF-2) binding to cell surface receptors (FGFRs) and heparan sulfate proteoglycans (HSPGs) under flow conditions within a cylindrical-shaped vessel or capillary. The model consists of a set of coupled nonlinear partial differential equations (PDEs) and a set of coupled nonlinear ordinary differential equations (ODEs). The time-dependent PDE system is discretized and solved by a second-order implicit Euler scheme using the finite volume method. The ODE system is solved by a stiff ODE solver VODE using backward differencing formulation (BDF). The transient solution of FGF-2, FGFR, HSPG, and their bound complexes for three different flow rates are computed and presented. Simulation results indicate that the model can predict growth factor transport and binding to receptors with/without the presence of heparan sulfate, as well as the effect of flow rate on growth factor-receptor binding. Our computational model may provide a useful means to investigate the impact of fluid flow on growth factor dynamics, and ultimately, signaling within the circulation.

Skin Thermal Injury Prediction with Strain Energy

A three-dimensional model is presented for the quantitative prediction of skin injury resulting from certain thermal exposure on the surface. The model is based on the skin damage equation proposed by Henriques and Moritz for the process of protein denaturation. Different from the standard Arrhenius model for protein damage rate, in which the activation energy includes chemical reaction only, strain energy of tissue due to thermal stress is also considered in the current model. Skin thermal response is modeled using the bioheat transfer equation by including water diffusion on the skin surface, and the corresponding thermal stress is predicted using the modified Duhamel-Neuman equation. Strain energy is then obtained by the stress-strain relation. The extent of burn injury is computed from the transient temperature solution and the effect of strain energy on skin damage is investigated. The time-dependent partial differential equations (PDEs) are discretized using Crank-Nicholson finite difference scheme and the resulting sparse linear systems are solved iteratively.

Three-dimensional model on thermal response of skin subject to laser heating

A three-dimensional (3D) multilayer model based on the skin physical structure is developed to investigate the transient thermal response of human skin subject to laser heating. The temperature distribution of the skin is modeled by the bioheat transfer equation, and the influence of laser heating is expressed as a source term where the strength of the source is a product of a Gaussian shaped incident irradiance, an exponentially shaped axial attenuation, and a time function. The water evaporation and diffusion is included in the model by adding two terms regarding the heat loss due to the evaporation and diffusion, where the rate of water evaporation is determined based on the theory of laminar boundary layer. Cryogen spray cooling (CSC) in laser therapy is studied, as well as its effect on the skin thermal response. The time-dependent equation is discretized using the finite difference method with the Crank–Nicholson scheme and the stability of the numerical method is analyzed. The large sparse linear system resulted from discretizing the governing partial differential equation is solved by a GMRES solver and the expected simulation results are obtained.

A Numerical Study of Pulsatile Flow Through a Hollow Fiber Cartridge: Growth Factor-Receptor Binding and Dissociation Analysis

This paper presents a numerical solution to describe growth factor-receptor binding under flow through hollow fibers of a bioreactor. The multi-physics of fluid flow, the kinetics of fibroblast growth factor (FGF-2) binding to its receptor (FGFR) and heparan sulfate proteoglycan (HSPG) and FGF-2 mass transport is modeled by a set of coupled nonlinear partial differential equations (PDEs) and coupled nonlinear ordinary differential equations (ODEs). A finite volume method is used to discretize the PDEs. The ODEs are solved by a stiff ODE solver CVODE. Overall, second order accuracy in time and space is achieved with the second order implicit Euler scheme. In order to obtain a reasonable accuracy of the binding and dissociation from cells, a uniform mesh is used. To handle pulsatile flow, several assumptions are made including neglecting any entrance effects and an analytical solution for axial velocity within the fibers is obtained. Qualitative and quantitative analysis are presented. Computational results and experimental measurements are compared and observed to agree quite well, indicating that the simulation model and methods could be used as a complementary and even predictable tool for the study of biochemical reactions in a similar flow environment.

Relaxation method for unsteady convection-diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection-diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection-diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicit-explicit (IMEX) Runge-Kutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection-diffusion equations with both smooth structures and discontinuities.

Two-Dimensional Hyperbolic Heat Conduction with Temperature-Dependent Properties

Introduction T HE phenomena of non-Fourier heat conduction are observed in many industrial applications, such as laser heating, cryogenic engineering, and nanotechnology. Various conduction models have been proposed to explain the non-Fourier conductive heat-transfer behavior in a very short period of time. These include the macrohyperbolic model1 and Tzou’s dual-phase model.2 The purpose of the present work is to present a numerical solution to the macrohyperbolic heat-conduction (HHC) model in temperature-dependent materials. Both the analytical3 and numerical4−8 methods have been used in solving HHC equation over the years. Glass et al.4 studied the effects of temperature-dependent thermal conductivity on the thermal wave propagation by using the MacCormack’s predictorcorrector scheme. Kar et al.5 solved a nonlinear HHC equation both analytically and numerically by using the Kirchhoff transformation to linearize the nonlinear terms. However, only one-dimensional problems were considered in their works.4,5 Present numerical approach employs the Roe–Sweby’s total-variation-diminishing (TVD)9 scheme to solve two-dimensional HHC equations. This scheme was used in a previous study for HHC in composite media.7 The present work investigates the effects of temperature-dependent properties on the thermal wave propagation in a homogeneous medium.

Hyperbolic Heat Conduction in Composite Materials

A hyperbolic heat conduction (HHC) equation has been proposed to replace Fourier heat conduction equation in cases heat transfer takes place in a very short period of time or at extremely low temperature. There is a growing interest in the investigation of HHC problem in recent years, but to the author’s knowledge, HHC in composite media in multidimension has not been studied up to date. This paper presents a numerical method to solve two-dimensional HHC in composite materials. The numerical method used in the paper is free of oscillation in predicting the solution of hyperbolic systems encountering discontinuities. The key to solve HHC in composite media is to keep both temperature and heat flux continuous at the interfaces. The HHC equation is non-dimensionalized in a way, so that the HHC problem in composite media can be solved in nondimensional form. Nomenclature

Parallel simulation of multiple proteins through a bioreactor coupled with biochemical reactions

This paper presents a parallel numerical solution to investigate multiple growth factors competitive binding within a bioreactor, an in vitro flow cell culture system. Since we assume all the species have the same flow, thus the multi-physics of fluid flow is modeled by the same incompressible Navier-Stokes equations. The kinetics of biochemical reactions happens in the fluid and on the cell surfaces as well, thus they are modeled by two separate sets of coupled nonlinear ordinary differential equations (ODEs). The mass transport of different species in the fluid is modeled by a distinctive set of coupled nonlinear partial differential equations (PDEs) for each of them. To solve this computational intensive system efficiently, a novel parallel algorithm is devised, in which all the numerical computations are solved in parallel, including parallel discretization of those mass transport equations PDEs and parallel linear system solver. A novel parallel strongly implicit procedure (SIP) solver is designed. For solving binding equations ODEs in the whole domain efficiently, a parallel scheme combined with a sequential CVODE solver is used for the purpose of high performance and simplicity. Overall, our parallel algorithms show good performance and stability. Preliminary simulation results are obtained. We have found that heparin or possibly other solution binding agents can effectively prevent fibroblast growth factor-2 (FGF-2) capture under flow, but only at high concentrations, and FGF-2 cross regulating receptor binding agents, such as heparin-binding epidermal growth factor-like growth factor (HB-EGF) or possibly other proteoglycan-competitors, has little effect on FGF-2 capture in single pass flow even at high concentration. Further experiments need to be conducted to verify the predictions of our parallel simulation system. This parallel modeling system can be used to any biochemical reaction analysis in a similar flow environment.