Linda J. Hayes

A Finite Element Model of Burn Injury in Blood-Perfused Skin

The burn process resulting from the application of a hot, cylindrical source to the skin surface was modeled using the finite element technique. A rotationally symmetric 125-element mesh was defined within the tissue beneath and outlying to an applied heating disk. The disk temperature and duration of contact were varied, respectively, between 50 and 100 degrees C for up to 30 s. Natural convection with ambient air was assumed for areas of skin surface not in direct contact with the disk. The simulated thermal history was used in a damage integral model to calculate the extent and severity of injury in the radial and axial dimensions.

Analysis of tissue injury by burning: comparison of in situ and skin flap models

Abstract The transient temperature field created in skin during a surface burn is modeled using the finite element technique. The two physical systems which are simulated are in situ tissue and an experimental skin flap chamber which is implemented for burn studies. Local cumulative injury in the tissue is calculated using an Arrhenius type injury model. The analysis shows that the burn experiments conducted in the skin flap chamber at temperatures up to 70°C will produce thermal histories which are negligibly different from those for in situ tissue for insult durations of less than 15 s.

Prediction of local cooling rates and cell survival during the freezing of a cylindrical specimen.

A finite element numerical model was implemented to simulate the freezing process of an aqueous salt solution in a cylindrical container. Local cooling rates within the container were computed for several defined cooling protocols applied at the boundary. Characteristic cell survival signatures were used to predict the associated local survival rates throughout the system. These calculations show that there are two definite time domains during a typical freezing process: (1) while the surface temperature is changing and (2) after the surface temperature reaches a constant storage value. The calculations also show significant spatial variations in the local cooling rates within the container and considerable local deviation from the volumetric average survival for various simulated freezing protocols.

Prediction of transient temperature fields and cumulative tissue destruction for radio frequency heating of a tumor

A therapeutic hyperthermia protocol using a radio frequency (rf) electrode placed adjacent to a bronchial wall tumor has been modeled using the finite element technique. Variable physical properties and variable blood perfusion have been assigned to the tumor and to the surrounding normal lung tissue. The Laplace equation was solved on a curvilinear grid for a single rf source electrode to determine the steady-state electric field, which in turn governs the energy deposition function. The heat generation in the tumor and in the lung tissue is then calculated from the energy deposition profile, and the bioheat equation is solved on the same finite element mesh to determine the transient temperature history. The temperatures are displayed as isothermal contours at designated times during the protocol and as temperature histories at selected points. In addition, an Arrhenius-type injury model has been implemented to predict thermally induced damage, from which equal total amounts of energy are deposited into the tissue using a constant power density for an appropriate time or using a cyclic heating pattern. The cyclic heating pattern consisted of a series of equal duration time periods during which the rf current source is alternately turned on and off (50% duty cycle). This study illustrates how a finite element model could be used to evaluate alternative protocols for heating a tumor of a specific geometry and to evaluate thermally induced damage to surrounding normal tissue.

Research: A Mathematical Model for the Thermal Efficacy of Cooling Therapy for Burns

The finite element technique was used to model the transient thermal field in skin during a burn. The resulting temperature field was substituted into a damage integral equation to determine cumulative injury as a function of time and position in the tissue. Postburn cooling protocols were simulated to predict the thermal efficacy of cooling therapy in manipulating skin temperature to reduce injury. Protocol parameters included delay prior to initiation of cooling, cooling temperature, and duration of therapy. The analysis indicates that it is not possible by postburn water cooling to lower the tissue temperature rapidly enough to reduce the extent of burn injury. Any clinical efficacy of postburn cooling should therefore be attributed to nonthermal effects.

Implementation of Phase Change in Numerical Models of Heat Transfer

This paper investigates some of the numerical problems involved in simulating heat transfer in porous media in the presence of phase change. Applications of this type of simulation include modeling of certain metal forming processes, of biological tissues and organs during cryosurgery or cyropreservation, and of heat transfer in frozen soils subjected to transient environmental conditions. A two-dimensional finite element model was used in which the latent heat is treated directly as an energy source in the problem formulation. Several parameters addressed in this work are crucial to the successful implementation of numerical methods for nonlinear heat transport with phase change, including: the effect of nodal point spacing on the occurrence and magnitude of numerical oscillations in the temperature solution and the use of grid point spacing to control these oscillations; the limiting element size which should be used in order to insure stable temperature fields; and the effect which the range of temperatures over which latent heat is liberated has on the solution. The results indicate that numerical stability is achieved for combinations of the foregoing parameters which yield small values of the Stefan number.

An alternating-direction collocation method for finite element approximations on rectangles

Abstract An alternating-direction finite element collocation procedure is presented for parabolic boundary value problems posed on rectangular regions. Collocation offers a great savings in time as compared to standard finite element methods during the matrix formation stage of numerical process. With an alternating-direction procedure, multidimensional problems can be solved as a series of one-dimensional problems, which greatly reduces both the work and the storage requirements during the matrix solution phase of the numerical procedure.

Pulse-power integrated-decay technique for the measurement of thermal conductivity

A pulse-power integrated-decay technique for the measurement of thermal conductivity of biological tissues is presented. A self-heated thermistor probe is used to deliver heat and also to measure the temperature response. Three-dimensional finite element analyses are used in this paper to design and optimize the technique. The thermal conductivity measurements from the computer simulations were in close accordance with the experimental data. An empirical calibration process, performed in glycerol and agar-gelled water, provides accurate thermal conductivity measurements. An accuracy analysis evaluated multiple experimental protocols using three solutions of known thermal properties. The results indicate that the thermal decay technique protocol had better accuracy than the constant temperature heating techniques. In vitro measurements demonstrate the variability of tissue thermal conductivity, and the need to perform direct measurements for tissues of interest. The factors that may introduce error in the experimental data are (i) poor thermal/physical contact between the thermistor probe and tissue sample, (ii) water loss from tissue during the course of experimentation and (iii) temperature stability.

Grid orthogonalization for curvilinear alternating-direction techniques

Abstract A method is developed for an a posteriori iterative improvement to an arbitrary computational grid. Local corrections to the coordinates of the grid points are used to form a global cost function which is minimized with respect to a single parameter. The local corrections and cost function can be constructed to maximize the local smoothness and/or the local orthogonality of the grid. The advantage of this method is that it allows the user to generate an initial grid using any inexpensive method, and then the grid can be improved with respect to both orthogonality and smoothness. This technique was used to generate grids for a finite element alternating-direction method which uses curved elements. A sample transient diffusion problem was solved on a series of grids to investigate the sensitivity of the curvilinear alternating-direction method to grid orthogonalization. The initial grid was highly nonorthogonal and each grid produced by the automatic grid generation program was smoother and more orthogonal. This work shows that the adaptive grid program can be easily used to generate nearly orthogonal grids and it shows that the curvilinear alternating-direction technique is not highly sensitive to nonorthogonality of the grid. It is shown that as long as a grid is somewhat reasonable, the alternating-direction method will perform quite well.

A Vectorized Matrix-Vector Multiply and Overlapping Block Iterative Method

An overlapping block iterative method is presented for solving large systems of equations which result from finite element discretizations. This algorithm is very much in the spirit of the element-by-element techniques described by Hughes et al. and by Carey and Jiang; however, it differs from these in that the algorithm here is a block iterative method in the classical sense where the blocks or groups are the nodes in the individual elements. Since any given node may appear in several elements, this technique has been called an “overlapping” block iterative method.