Penetration of external thermal perturbations into homeothermic organisms, part I (author's transl)

The general importance of the mean surface curvature for heat conduction problems is explained and a special symmetry with constant mean curvature on the isothermal surfaces is defined. The applicability for the body shapes of homeothermic organisms is demonstrated and the partial differential equation of heat conduction for this case is derived. The definition: heat release = real heat production + convective pseudoproduction eliminates the term of convective heat transfer through the blood stream and allows the reduction to a mere heat conduction problem. Formulas for the heat loss to the environment and for steady state temperature profiles are given. In case of sudden change of heat loss the partial differential equation is solved and a formula is derived, using dimensionless coordinates of time and distance. The mean surface curvature has strongest influence to the interior temperature field. The solution shows clearly the importance of thermal inertia of the homeothermic organism, for the external temperature wave penetrates into the body with a long phase displacement in time.     

An advanced computational bioheat transfer model for a human body with an embedded systemic circulation

In the present work, an elaborate one-dimensional thermofluid model for a human body is presented. By contrast to the existing pure conduction-/perfusion-based models, the proposed methodology couples the arterial fluid dynamics of a human body with a multi-segmental bioheat model of surrounding solid tissues. In the present configuration, arterial flow is included through a network of elastic vessels. More than a dozen solid segments are employed to represent the heat conduction in the surrounding tissues, and each segment is constituted by a multilayered circular cylinder. Such multi-layers allow flexible delineation of the geometry and incorporation of properties of different tissue types. The coupling of solid tissue and fluid models requires subdivision of the arterial circulation into large and small arteries. The heat exchange between tissues and arterial wall occurs by convection in large vessels and by perfusion in small arteries. The core region, including the heart, provides the inlet conditions for the fluid equations. In the proposed model, shivering, sweating, and perfusion changes constitute the basis of the thermoregulatory system. The equations governing flow and heat transfer in the circulatory system are solved using a locally conservative Galerkin approach, and the heat conduction in the surrounding tissues is solved using a standard implicit backward Euler method. To investigate the effectiveness of the proposed model, temperature field evolutions are monitored at different points of the arterial tree and in the surrounding tissue layers. To study the differences due to flow-induced convection effects on thermal balance, the results of the current model are compared against those of the widely used modelling methodologies. The results show that the convection significantly influences the temperature distribution of the solid tissues in the vicinity of the arteries. Thus, the inner convection has a more predominant role in the human body heat balance than previously thought. To demonstrate its capabilities, the proposed new model is used to study different scenarios, including thermoregulation inactivity and variation in surrounding atmospheric conditions.     

Thermal models for microwave hazards and their role in standards development

We consider the thermal response of the body to radiofrequency (RF) energy, with emphasis on partial-body exposure, to assess potential thermal hazards. The thermal analysis is based on Pennes' bioheat equation. In this model, the thermal response is governed by two time constants. One (τ₁) pertains to heat convection by blood flow and is (for physiologically normal perfusion rates) on the order of 3 min. The second (τ₂) characterizes heat conduction, and varies as the square of a distance that characterizes the spatial extent of the heating. We examine three idealized cases. The first is a region of tissue with an insulated surface, subject to irradiation with an exponentially decreasing SAR, which models a large surface area of tissue exposed to microwaves. The second is a region of tissue in contact with a hemispherical electrode that passes current into it, which models exposure from contact with a conductor. The third is a region of tissue with an insulated surface, subject to heating from a dipole located close to it. In all three cases, we estimate the maximum steady-state temperature increase as a function of the relevant electrical and thermal parameters and the thresholds for thermal hazard. We conclude that thermal models are a potentially fruitful but underutilized means of analyzing thermal hazards from RF fields. A quantitative analysis of such hazards enables the development of data-based uncertainty factors, which can replace arbitrary “safety factors” in developing exposure limits. Finally, we comment on the need to marry quantitative modeling of data and risk assessment, and to incorporate contemporary approaches to risk assessment into RF standards development. Bioelectromagnetics 20:52–63, 1999. © 1999 Wiley-Liss, Inc.     

Effect of natural convection on the current-voltage characteristic of a DC discharge in neon at intermediate pressures

The parameters of the positive column of a glow discharge in neon are calculated with allowance for the induced hydrodynamic motion. It is shown that natural convection in the pressure range of ∼0.1 atm significantly affects the profiles of the parameters of the positive column and its current-voltage characteristic. The convection arising at large deposited energies improves heat removal, due to which the temperature in the central region of the discharge becomes lower than that calculated from the heat conduction equation. As a result, the current-voltage characteristic is shifted. With allowance for convection, the current-voltage characteristic changes at currents much lower than the critical current at which a transition into the constricted state is observed. This change is uniquely related to the Rayleigh number in the discharge. Thus, a simplified analysis of thermal conduction and diffusion, even with detailed account of kinetic processes occurring in the positive column, does not allow one to accurately calculate the current-voltage characteristic and other discharge parameters at intermediate gas pressures.     

A revised approach for an exact analytical solution for thermal response in biological tissues significant in therapeutic treatments

The genesis of the present research paper is to develop a revised exact analytical solution of thermal profile of 1-D Pennes’ bioheat equation (PBHE) for living tissues influenced in thermal therapeutic treatments. In order to illustrate the temperature distribution in living tissue both Fourier and non-Fourier model of 1-D PBHE has been solved by ‘Separation of variables’ technique. Till date most of the research works have been carried out with the constant initial steady temperature of tissue which is not at all relevant for the biological body due to its nonhomogeneous living cells. There should be a temperature variation in the body before the therapeutic treatment. Therefore, a coupled heat transfer in skin surface before therapeutic heating must be taken account for establishment of exact temperature propagation. This approach has not yet been considered in any research work. In this work, an initial condition for solving governing differential equation of heat conduction in biological tissues has been represented as a function of spatial coordinate. In a few research work, initial temperature distribution with PBHE has been coupled in such a way that it eliminates metabolic heat generation. The study has been devoted to establish the comparison of thermal profile between present approach and published theoretical approach for particular initial and boundary conditions inflicted in this investigation. It has been studied that maximum temperature difference of existing approach for Fourier temperature distribution is 19.6% while in case of non-Fourier, it is 52.8%. We have validated our present analysis with experimental results and it has been observed that the temperature response based on the spatial dependent variable initial condition matches more accurately than other approaches.     

Numerical analysis of the cooling effect of blood over inflamed atherosclerotic plaque

Atherosclerotic plaques with high likelihood of rupture often show local temperature increase with respect to the surrounding arterial wall temperature. In this work, atherosclerotic plaque temperature was numerically determined during the different levels of blood flow reduction produced by the introduction of catheters at the vessel lumen. The temperature was calculated by solving the energy equation and the Navier-Stokes equations in 2D idealized arterial models. Arterial wall temperature depends on three basic factors: metabolic activity of the inflammatory cells embedded in the plaque, heat convection due to luminal blood flow, and heat conduction through the arterial wall and plaque. The calculations performed serve to simulate transient blood flow reduction produced by the presence of thermography catheters used to measure arterial wall temperature. The calculations estimate the spatial and temporal alterations in the cooling effect of blood flow and plaque temperature during the measurement process. The mathematical model developed provides a tool for analyzing the contribution of factors known to affect heat transfer at the plaque surface. Blood flow reduction leads to a nonuniform temperature increase ranging from 0.1 to 0.25 degrees Celsius in the plaque/lumen interface of the arterial geometries considered in this study. The temperature variation as well as the Nusselt number calculated along the plaque surface strongly depended on the arterial geometry and distribution of inflammatory cells. The calculations indicate that the minimum required time to obtain a steady temperature profile after arterial occlusion is 6 s. It was seen that in arteries with geometries involving bends, the temperature profiles appear asymmetrical and lean toward the downstream edge of the plaque.     

Galerkin's finite element-Laplace transform technique to transient heat migration in human skin and subcutaneous tissues

Galerkin's finite element-Laplace transform technique (GAFELTTE) has been used to study transient temperature distribution in human skin and subcutaneous tissues. This study incorporates heat conduction, heat carried by perfusion of blood in the capillary beds and metabolic heat generation in the tissues. Different values of various quantities have been considered in all three parts, namely epidermis, dermis and subcutaneous tissues, depending on physiological considerations. The GAFELTTE provides interface temperatures for a wide range of the values of skin surface temperatures. These values have been used to obtain temperature profiles in the region considered. Steady-state temperature distribution has been deduced from the solution obtained by GAFELTTE and has been compared with the results obtained by using different methods.     

A computer model of temperature distribution inside a lossy sphere after microwave radiation

The temperature distribution inside a lossy sphere resulting from the absorption of microwave energy was approximated by successive numerical iterations, of the thermal energy equation. Heat transfer within the sphere by conduction was considered. In the model energy was not dissipated by convection but was contained in the sphere for over 200 seconds. Exposure of a 5-cm sphere to 3,000 MHz at 30 mW/cm² for 200 seconds was calculated to produce a temperature rise of 0.56°C near the front surface.     

Predicting effects of blood flow rate and size of vessels in a vasculature on hyperthermia treatments using computer simulation

Background

Pennes Bio Heat Transfer Equation (PBHTE) has been widely used to approximate the overall temperature distribution in tissue using a perfusion parameter term in the equation during hyperthermia treatment. In the similar modeling, effective thermal conductivity (K_eff) model uses thermal conductivity as a parameter to predict temperatures. However the equations do not describe the thermal contribution of blood vessels. A countercurrent vascular network model which represents a more fundamental approach to modeling temperatures in tissue than do the generally used approximate equations such as the Pennes BHTE or effective thermal conductivity equations was presented in 1996. This type of model is capable of calculating the blood temperature in vessels and describing a vasculature in the tissue regions.

Methods

In this paper, a countercurrent blood vessel network (CBVN) model for calculating tissue temperatures has been developed for studying hyperthermia cancer treatment. We use a systematic approach to reveal the impact of a vasculature of blood vessels against a single vessel which most studies have presented. A vasculature illustrates branching vessels at the periphery of the tumor volume. The general trends present in this vascular model are similar to those shown for physiological systems in Green and Whitmore. The 3-D temperature distributions are obtained by solving the conduction equation in the tissue and the convective energy equation with specified Nusselt number in the vessels.

Results

This paper investigates effects of size of blood vessels in the CBVN model on total absorbed power in the treated region and blood flow rates (or perfusion rate) in the CBVN on temperature distributions during hyperthermia cancer treatment. Also, the same optimized power distribution during hyperthermia treatment is used to illustrate the differences between PBHTE and CBVN models. K_eff (effective thermal conductivity model) delivers the same difference as compared to the CBVN model. The optimization used here is adjusting power based on the local temperature in the treated region in an attempt to reach the ideal therapeutic temperature of 43°C. The scheme can be used (or adapted) in a non-invasive power supply application such as high-intensity focused ultrasound (HIFU). Results show that, for low perfusion rates in CBVN model vessels, impacts on tissue temperature becomes insignificant. Uniform temperature in the treated region is obtained.

Conclusion

Therefore, any method that could decrease or prevent blood flow rates into the tumorous region is recommended as a pre-process to hyperthermia cancer treatment. Second, the size of vessels in vasculatures does not significantly affect on total power consumption during hyperthermia therapy when the total blood flow rate is constant. It is about 0.8% decreasing in total optimized absorbed power in the heated region as γ (the ratio of diameters of successive vessel generations) increases from 0.6 to 0.7, or from 0.7 to 0.8, or from 0.8 to 0.9. Last, in hyperthermia treatments, when the heated region consists of thermally significant vessels, much of absorbed power is required to heat the region and (provided that finer spatial power deposition exists) to heat vessels which could lead to higher blood temperatures than tissue temperatures when modeled them using PBHTE.     

An inverse problem in estimating the laser irradiance and thermal damage in laser-irradiated biological tissue with a dual-phase-lag model

The aim of this study is to solve an inverse heat conduction problem to estimate the unknown time-dependent laser irradiance and thermal damage in laser-irradiated biological tissue from the temperature measurements taken within the tissue. The dual-phase-lag model is considered in the formulation of heat conduction equation. The inverse algorithm used in the study is based on the conjugate gradient method and the discrepancy principle. The effect of measurement errors and measurement locations on the estimation accuracy is also investigated. Two different examples of laser irradiance are discussed. Results show that the unknown laser irradiance and thermal damage can be predicted precisely by using the present approach for the test cases considered in this study.     

Heat transfer in perfused biological tissue—II. The “macroscopic” temperature distribution in tissue

In Part I, the general solution of the heat transport equation was given for a medium of constant thermal properties, containing a general convection fieldv(r,t). Starting from this general solution, an average or “macroscopic” temperature distribution is calculated in this paper, since in most physiological problems only an average or “macroscopic” temperature is accessible to the measurement. It is shown that the averaged temperature distribution is influenced alone by the order of the local symmetries of the convection field. These local symmetries can be deduced from anatomical data on the vascular architecture. In order to perform the averaging process, a detailed discussion, and an estimate of the correlation between the microscopic temperature variation and the convection field is carried out. Hereby, great emphasis is put on the experimental consequences of these results. As an illustrative example, the calculation of the macroscopic temperature distribution in a medium traversed by parallel capillaries is outlined.     

Air movement and heat loss from sheep. III. Components of insulation in a controlled environment

The rate of sensible heat loss from a Clun Forest ewe was studied at several fleece depths in a temperature-controlled chamber. A simple resistance analogue was used to describe the heat flow from different body regions. Heat loss from the trunk depends largely on the mean fleece depth l. The fleece resistance was about 1.5 s cm-1 per centimetre depth. Heat transfer through the fleece was accounted for by molecular conduction, thermal radiation and free convection. The fleece conductivity -kb attributed to free convection depends on the mean temperature difference (-Tst---Tct) across the fleece according to the relation -kb = 8.0 (-Tst---Tct)0.53. Estimates of the sensible heat flux from the trunk at environmental temperatures, Ta, between 0 and 30 degrees C range from about 8 W (l = 7.0 cm, Ta = 30 degrees C) to about 160 W (l = 0.1 cm, Ta = 0 degrees C). In contrast, the sensible heat loss from the legs depends mainly on the local tissue resistance. For environmental temperatures between 0 and 30 degrees C, the calculated tissue resistance for this region of the body varied from about 8 to 1 s cm-1. The corresponding heat loss from the legs was between 10 and 20 W, compared with between 3 and 7 W from the head. The fastest heat loss from the legs occurred at an environmental temperature of about 12 degrees C. Although the proportion of the heat loss from the extremities depends on environmental temperature, the total heat loss (sensible or latent) was closely related to the mean skin temperature of the trunk.     

Temperature evolution in tissues embedded with large blood vessels during photo-thermal heating

During laser-assisted photo-thermal therapy, the temperature of the heated tissue region must rise to the therapeutic value (e.g., 43 °C) for complete ablation of the target cells. Large blood vessels (larger than 500 micron in diameter) at or near the irradiated tissues have a considerable impact on the transient temperature distribution in the tissue. In this study, the cooling effects of large blood vessels on temperature distribution in tissues during laser irradiation are predicted using finite element based simulation. A uniform flow is assumed at the entrance and three-dimensional conjugate heat transfer equations in the tissue region and the blood region are simultaneously solved for different vascular models. A volumetric heat source term based on Beer–Lambert law is introduced into the energy equation to account for laser heating. The heating pattern is taken to depend on the absorption and scattering coefficients of the tissue medium. Experiments are also conducted on tissue mimics in the presence and absence of simulated blood vessels to validate the numerical model. The coupled heat transfer between thermally significant blood vessels and their surrounding tissue for three different tissue-vascular networks are analyzed keeping the laser irradiation constant. A surface temperature map is obtained for different vascular models and for the bare tissue (without blood vessels). The transient temperature distribution is seen to differ according to the nature of the vascular network, blood vessel size, flow rate, laser spot size, laser power and tissue blood perfusion rate. The simulations suggest that the blood flow through large blood vessels in the vicinity of the photothermally heated tissue can lead to inefficient heating of the target.     

Effect of tissue and atmosphere's parameters on human eye temperature distribution

A three dimensional finite element method analysis was employed to investigate the effect of tissue and atmosphere parameters namely, ambient temperature, ambient convection coefficient, local blood temperature, and blood convection coefficient upon temperature distribution of human eyes. As a matter of simplification, only eye ball and skull bone are considered as the system of eye modeling. Decreasing the local blood temperature and keeping it cool is one of the most important ways to control bleeding during surgeries. By lower temperature of body organs such as the eye, the need for oxygenated blood is reduced, allowing for an extension in time for surgery. With this in mind, this study is done to see which one of parameters, such as ambient temperature, ambient convection coefficient, local blood temperature, and blood convection coefficient, has an effective role in decreasing the temperature of the eye. To this end, 3 different paths were employed to find out about the temperature distribution through the eye. The analysis of the three paths demonstrates the interaction of ambient and blood temperature in modeling temperature changes in specific locations of the eye. These data will be important in applications such as eye surgery, relaxation, and sleep therapy.     

Measuring the effects of heat wave episodes on the human body’s thermal balance

During the peak of an extensive heat wave episode on 23–25 July 2007, simultaneous thermophysiological measurements were made in two non-acclimated healthy adults of different sex in a suburban area of Greater Athens, Greece. Based on experimental measurements of mean skin temperature and metabolic heat production, heat fluxes to and from the human body were calculated, and the biometeorological index heat load (HL) produced was determined according to the heat balance equation. Comparing experimental values with those derived from theoretical estimates revealed a great heat stress for both individuals, especially the male, while theoretical values underestimated heat stress. The study also revealed that thermophysiological factors, such as mean skin temperature and metabolic heat production, play an important role in determining heat fluxes patterns in the heat balance equation. The theoretical values of mean skin temperature as derived from an empirical equation may not be appropriate to describe the changes that take place in a non-acclimated individual. Furthermore, the changes in metabolic heat production were significant even for standard activity.     

A thermal model of the human body exposed to an electromagnetic field

The human body was modeled by numerical procedures to determine the thermal response under varied electromagnetic (EM) exposures. The basic approach taken was to modify the heat transfer equations for man in air to account for thermal loading due to the energy absorbed from the EM field. The human body was represented in an EM model by a large number of small cubical cells of tissue, and the energy density was determined for each cell. This information was then analyzed by a thermal response model consisting of a series of two-dimensional transient conduction equations with internal heat generation due to metabolism, internal convective heat transfer due to blood flow, external interaction by convection and radiation, and cooling of the skin by sweating and evaporation. This model simulated the human body by a series of cylindrical segments. The local temperature at 61 discrete locations as well as the thermoregulatory responses of vasodilatation and sweating were computed for a number of EM field intensities and two frequencies, one near whole-body resonance.     

Dual phase lag bio-heat transfer during cryosurgery of lung cancer: Comparison of three heat transfer models

The Pennes bio-heat model is based on Fourier's law of heat conduction, which assumed that a thermal signal propagate with infinite speed. This gives contradiction in physical situation. Also, the hyperbolic bio-heat model considers the micro scale response in time, but it does not explain the micro scale response in space. Therefore, to consider the thermal behaviour which is not captured by the Fourier's law and to take into account the microstructural effect in space, a dual phase lag (DPL) bio-heat conduction model would be advantageous. In this paper, a two dimensional DPL model is proposed to study the phase change heat transfer process during cryosurgery of lung cancer. The governing equations are solved numerically by enthalpy based finite difference method. The non-ideal behaviour of tissue and heat source terms, metabolism and blood perfusion are also considered. This study is made to examine the effects of phase lags in heat flux and temperature gradient on interface positions and temperature distribution during freezing process. A comparative study of DPL, parabolic and hyperbolic conduction models is thoroughly investigated. It is found that the phase lags of temperature gradient and heat flux have significant effect on interface positions and temperature distribution.     

Thermal energy conduction in a honey bee comb due to cell-heating bees

Theoretical analysis and numerical calculations are performed to characterize the unsteady two-dimensional conduction of thermal energy in an idealized honey bee comb. The situation explored corresponds to a comb containing a number of brood cells occupied by pupae. These cells are surrounded by other cells containing pollen which, in turn, are surrounded (above) by cells containing honey and (below) by vacant cells containing air. Up to five vacant cells in the brood region can be occupied by cell-heating bees which, through the isometrical contraction of their flight muscles, can generate sufficient energy to raise their body temperatures by a few degrees. In this way, the cell-heating bees alter the heat flux and temperature distributions in the brood region so as to maintain conditions that benefit the pupae. The calculations show that the number of cell-heating bees significantly affects the magnitude, time rate of change, and spatial distribution of temperature throughout the comb. They also reveal a vertically aligned asymmetry in the spatial distribution of temperature that is due to the large heat capacity and thermal conductivity of honey relative to air, whereby air-filled cells experience larger temperature increases than honey-filled cells. Analysis shows that convection and radiation represent negligible modes of thermal energy transfer at all levels in the problem considered. Also, because of its small thickness, the wax wall of a comb cell simultaneously presents negligible resistance to conduction heat transfer normal to it and very large resistance along it. As a consequence the walls of a cell play no thermal role, but simply serve as mechanical supports for the materials they contain.     

Numerical modelling of conductive and convective heat transfers in retinal laser applications

The control of the temperature increase is an important issue in retinal laser treatments. Within the fundus of the eye heat, generated by absorption of light, is transmitted by diffusion in the retinal pigment epithelium and in the choroid and lost by convection due to the choroidal blood flow. The temperature can be spatially and temporally determined by solving the heat equation. In a former analytical model this was achieved by assuming uniform convection for the whole fundus of the eye. A numerical method avoiding this unrealistic assumption by considering convective heat transfer only in the choroid is used here to solve the heat equation. Numerical results are compared with experimental results obtained by using a novel method of noninvasive optoacoustic retinal temperature measurements in rabbits. Assuming global convection the perfusion coefficient was evaluated to 0.07 s^?1, whereas a value of 0.32 s^?1 – much closer to values found in the literature (between 0.28 and 0.30 s^?1) – was obtained when choroidal convection was assumed, showing the advantage of the numerical method. The modelling of retinal laser treatment is thus improved and could be considered in the future to optimize treatments by calculating retinal temperature increases under various tissues and laser properties. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)