On the generalization of the Weinbaum-Jiji bioheat equation to microvessels of unequal size; the relation between the near field and local average tissue temperatures

The extensive series of experiments reported in Lemons et al. [1] show that measureable local tissue temperature fluctuations are observed primarily in the vicinity of the 100-500 micron countercurrent vessels of the microcirculation and thus strongly support the basic hypothesis in the new bioheat equation of Weinbaum and Jiji [2] that these countercurrent microvessels are the principal determinants of local blood-tissue heat transfer. However, the detailed temperature profiles in the vicinity of these vessels indicate that large asymmetries in the local temperature field can result from the significant differences in size between the countercurrent artery and vein. Using the superposition techniques of Baish et al. [9], the paper first presents a solution to the classic problem of an unequal countercurrent heat exchanger with heat loss to the far field. This solution is then used to generalize the Weinbaum-Jiji bioheat equation and the conductivity tensor that appears in this equation to vessels of unequal size. An asymptotic analysis has also been developed to elucidate the relationship between the near field temperature of the artery-vein pair and the local average tissue temperature. This analysis is used to rigorously prove the closure approximation relating the local arterial-venous temperature difference and the mean tissue temperature gradient which had been derived in [2] using a more heuristic approach.     

The bleed off perfusion term in the Weinbaum-Jiji bioheat equation.

The microvascular organization and thermal equilibration of the primary and secondary arteries and veins that comprise the bleed off circulation to the muscle fibers from the parent countercurrent supply artery and veins are analyzed. The blood perfusion heat source term in the tissue energy equation is shown to be related to this vascular organization and to undergo a fundamental change in behavior as one proceeds from the more peripheral tissue, where the perfusion term is proportional to the Ta--Tv difference in the parent supply vessels, to the deeper tissue layers where the bleed off vessels themselves form a branching countercurrent system for each muscle tissue cylinder and the venous return temperature can vary between the local tissue temperature and Ta. The consequences of this change in behavior are examined for the Weinbaum-Jiji bioheat equation and a modified expression for the effective conductivity of perfused tissue is derived for countercurrent bleed off exchange.     

A new fundamental bioheat equation for muscle tissue--part II: Temperature of SAV vessels

In this study, a new theoretical framework was developed to investigate temperature variations along countercurrent SAV blood vessels from 300 to 1000 microm diameter in skeletal muscle. Vessels of this size lie outside the range of validity of the Weinbaum-Jiji bioheat equation and, heretofore, have been treated using discrete numerical methods. A new tissue cylinder surrounding these vessel pairs is defined based on vascular anatomy, Murray's law, and the assumption of uniform perfusion. The thermal interaction between the blood vessel pair and surrounding tissue is investigated for two vascular branching patterns, pure branching and pure perfusion. It is shown that temperature variations along these large vessel pairs strongly depend on the branching pattern and the local blood perfusion rate. The arterial supply temperature in different vessel generations was evaluated to estimate the arterial inlet temperature in the modified perfusion source term for the s vessels in Part I of this study. In addition, results from the current research enable one to explore the relative contribution of the SAV vessels and the s vessels to the overall thermal equilibration between blood and tissue.     

A new simplified bioheat equation for the effect of blood flow on local average tissue temperature

A new simplified three-dimensional bioheat equation is derived to describe the effect of blood flow on blood-tissue heat transfer. In two recent theoretical and experimental studies [1, 2] the authors have demonstrated that the so-called isotropic blood perfusion term in the existing bioheat equation is negligible because of the microvascular organization, and that the primary mechanism for blood-tissue energy exchange is incomplete countercurrent exchange in the thermally significant microvessels. The new theory to describe this basic mechanism shows that the vascularization of tissue causes it to behave as an anisotropic heat transfer medium. A remarkably simple expression is derived for the tensor conductivity of the tissue as a function of the local vascular geometry and flow velocity in the thermally significant countercurrent vessels. It is also shown that directed as opposed to isotropic blood perfusion between the countercurrent vessels can have a significant influence on heat transfer in regions where the countercurrent vessels are under 70-micron diameter. The new bioheat equation also describes this mechanism.     

Theory on thermal probe arrays for the distinction between the convective and the perfusive modalities of heat transfer in living tissues

Recent suggestions for an improved model of heat transfer in living tissues emphasize the existence of a convective mode due to flowing blood in addition to, or even instead of, the perfusive mode, as proposed in Pennes' "classic" bioheat equation. In view of these suggestions, it might be beneficial to develop a technique that will enable one to distinguish between these two modes of bioheat transfer. To this end, a concept that utilizes a multiprobe array of thermistors in conjunction with a revised bioheat transfer equation has been derived to distinguish between, and to quantify the perfusive and convective contribution of blood to heat transfer in living tissues. The array consists of two or more temperature sensors one of which also serves to locally insert a short pulse of heat into the tissue prior to the temperature measurements. A theoretical analysis shows that such a concept is feasible. The construction of the system involves the selection of several important design parameters, i.e., the distance between the probes, the heating power, and the pulse duration. The choice of these parameters is based on computer simulations of the actual experiment.     

A parametric study of thermal therapy of skin tissue

A thermal therapy for cancer in skin tissue is numerically investigated using three bioheat conduction models, namely Pennes, thermal wave and dual-phase lag models. A laser is applied at the surface of the skin for cancer ablation, and the temperature and thermal damage distributions are predicted using the three bioheat models and two different modeling approaches of the laser effect. The first one is a prescribed surface heat flux, in which the tissue is assumed to be highly absorbent, while the second approach is a volumetric heat source, which is reasonable if the scattering and absorption skin effects are of similar magnitude. The finite volume method is applied to solve the governing bioheat equation. A parametric study is carried out to ascertain the effects of the thermophysical properties of the cancer on the thermal damage. The temperature distributions predicted by the three models exhibit significant differences, even though the temperature distributions are similar when the laser is turned off. The type of bioheat model has more influence on the predicted thermal damage than the type of modeling approach used for the laser. The phase lags of heat flux and temperature gradient have an important influence on the results, as well as the thermal conductivity of the cancer. In contrast, the uncertainty in the specific heat and blood perfusion rate has a minor influence on the thermal damage.     

Peripheral tissue freezing in cryosurgery

The recently formulated bioheat equation of Weinbaum and Jiji which accounts for the vascular ultrastructure and blood perfusion was applied to the freezing of peripheral tissue. Using quasi-steady approximation the temperature distribution in the two-phase tissue and the motion of the frozen front were determined. Results are in good agreement with Pennes' bioheat equation.     

Laser-induced hyperthermia of nanoshell mediated vascularized tissue – A numerical study

Laser-induced hyperthermia treatment of tumor in a 2-D axisymmetric tissue embedded with moderate size (100–150 µm) blood vessels is studied. Laser absorption is enhanced by embedding gold–silica nanoshells in the tumor. Heat transfer in the tissue is modeled using Weinbaum–Jiji bioheat transfer equation. With laser irradiation, the volumetric radiation is accounted in the governing bioheat equation. Radiative information needed in the bioheat equation is calculated using the discrete ordinate method, and the coupled bioheat-radiation equation is solved using the finite volume method. Effects of power density, laser exposure time, beam radius, diameter of blood vessel and volume fractions of nanoshells on temperature spread in the tissue are analyzed.     

Dimensionless solutions and general characteristics of bioheat transfer during thermal therapy

The derivation and application of the general characteristics of bioheat transfer for medical applications are shown in this paper. Two general bioheat transfer characteristics are derived from solutions of one-dimensional Pennes’ bioheat transfer equation: steady-state thermal penetration depth, which is the deepest depth where the heat effect reaches; and time to reach steady-state, which represents the amount of time necessary for temperature distribution to converge to a steady-state. All results are described by dimensionless form; therefore, these results provide information on temperature distribution in biological tissue for various thermal therapies by transforming to dimension form.     

Heat transfer analysis of skin during thermal therapy using thermal wave equation

Specifying exact geometry of vessel network and its effect on temperature distribution in living tissues is one of the most complicated problems of the bioheat field. In this paper, the effects of blood vessels on temperature distribution in a skin tissue subjected to various thermal therapy conditions are investigated. Present model consists of counter-current multilevel vessel network embedded in a three-dimensional triple-layered skin structure. Branching angles of vessels are calculated using the physiological principle of minimum work. Length and diameter ratios are specified using length doubling rule and Cube law, respectively. By solving continuity, momentum and energy equations for blood flow and Pennes and modified Pennes bioheat equations for the tissue, temperature distributions in the tissue are measured. Effects of considering modified Pennes bioheat equation are investigated, comprehensively. It is also observed that blood has an impressive role in temperature distribution of the tissue, especially at high temperatures. The effects of different parameters such as boundary conditions, relaxation time, thermal properties of skin, metabolism and pulse heat flux on temperature distribution are investigated. Tremendous effect of boundary condition type at the lower boundary is noted. It seems that neither insulation nor constant temperature at this boundary can completely describe the real physical phenomena. It is expected that real temperature at the lower levels is somewhat between two predicted values. The effect of temperature on the thermal properties of skin tissue is considered. It is shown that considering temperature dependent values for thermal conductivity is important in the temperature distribution estimation of skin tissue; however, the effect of temperature dependent values for specific heat capacity is negligible. It is seen that considering modified Pennes equation in processes with high heat flux during low times is significant.     

Theoretical analysis of thermal damage in biological tissues caused by laser irradiation

A bioheat transfer approach is proposed to study thermal damage in biological tissues caused by laser radiation. The laser light propagation in the tissue is first solved by using a robust seven-flux model in cylindrical coordinate system. The resulting spatial distribution of the absorbed laser energy is incorporated into the bioheat transfer equation for solving temperature response. Thermal damage to the tissue is assessed by the extent of denatured protein using a rate process equation. It is found that for the tissue studied, a significant protein denaturation process would take place when temperature exceeds about 53 degrees C. The effects of laser power, exposure time and beam size as well as the tissue absorption and scattering coefficients on the thermal damage process are examined and discussed. The laser conditions that cause irreversible damage to the tissue are also identified.     

Thermal detection of embedded tumors using infrared imaging

Breast cancer is the most common cancer among women. Thermography, also known as thermal or infrared imaging, is a procedure to determine if an abnormality is present in the breast tissue temperature distribution. This abnormality in temperature distribution might indicate the presence of an embedded tumor. Although thermography is currently used to indicate the presence of an abnormality, there are no standard procedures to interpret these and determine the location of an embedded tumor. This research is a first step towards this direction. It explores the relationship between the characteristics (location and power) of an embedded heat source and the resulting temperature distribution on the surface. Experiments were conducted using a resistance heater that was embedded in agar in order to simulate the heat produced by a tumor in the biological tissue. The resulting temperature distribution on the surface was imaged using an infrared camera. In order to estimate the location and heat generation rate of the source from these temperature distributions, a genetic algorithm was used as the estimation method. The genetic algorithm utilizes a finite difference scheme for the direct solution of the Pennes bioheat equation. It was determined that a genetic algorithm based approach is well suited for the estimation problem since both the depth and the heat generation rate of the heat source were accurately predicted.     

Transport lattice models of heat transport in skin with spatially heterogeneous,temperature-dependent perfusion

Background  

Investigation of bioheat transfer problems requires the evaluation of temporal and spatial distributions of temperature. This class of problems has been traditionally addressed using the Pennes bioheat equation. Transport of heat by conduction, and by temperature-dependent, spatially heterogeneous blood perfusion is modeled here using a transport lattice approach.     

Specific sequences determine the stability and cooperativity of folding of the C-terminal half of tropomyosin

Tropomyosin is a flexible 410 A coiled-coil protein in which the relative stabilities of specific regions may be important for its proper function in the control of muscle contraction. In addition, tropomyosin can be used as a simple model of natural occurrence to understand the inter- and intramolecular interactions that govern the stability of coiled-coils. We have produced eight recombinant tropomyosin fragments (Tm(143-284(5OHW),) Tm(189-284(5OHW)), Tm(189-284), Tm(220-284(5OHW)), Tm(220-284), Tm(143-235), Tm(167-260), and Tm(143-260)) and one synthetic peptide (Ac-Tm(215-235)) to investigate the relative conformational stability of different regions derived from the C-terminal region of the protein, which is known to interact with the troponin complex. Analytical ultracentrifugation experiments show that the fragments that include the last 24 residues of the molecule (Tm(143-284(5OHW)), Tm(189-284(5OHW)), Tm(220-284(5OHW)), Tm(220-284)) are completely dimerized at 10 microm dimer (50 mm phosphate, 100 mm NaCl, 1.0 mm dithiothreitol, and 0.5 mm EDTA, 10 degrees C), whereas fragments that lack the native C terminus (Tm(143-235),Tm(167-260), and Tm(143-260)) are in a monomer-dimer equilibrium under these conditions. The presence of trifluoroethanol resulted in a reduction in the [theta](222)/[theta](208) circular dichroism ratio in all of the fragments and induced stable trimer formation only in those containing residues 261-284. Urea denaturation monitored by circular dichroism and fluorescence revealed that residues 261-284 of tropomyosin are very important for the stability of the C-terminal half of the molecule as a whole. Furthermore, the absence of this region greatly increases the cooperativity of urea-induced unfolding. Temperature and urea denaturation experiments show that Tm(143-235) is less stable than other fragments of the same size. We have identified a number of factors that may contribute to this particular instability, including an interhelix repulsion between g and e' positions of the heptad repeat, a charged residue at the hydrophobic coiled-coil interface, and a greater fraction of beta-branched residues located at d positions.     

Alpha-helix to random-coil transitions of two-chain coiled coils: the use of physical models in treating thermal denaturation equilibria of isolated subsequences of alpha alpha-tropomyosin

Two extant models of thermal folding/unfolding equilibria in two-chain, alpha-helical coiled coils are tested by comparison with experimental results on excised, isolated subsequences of rabbit alpha alpha-tropomyosin (Tm). These substances are designated iTmj where i and j are, respectively, the residue numbers (in the 284-residue parent chain) of the N- and C-terminal residues of the subsequence. One model postulates that a coiled coil consists of segments, each denaturing in an all-or-none manner, like small globular proteins. Thus this model yields a small number of populated molecular species. In an extant calorimetry study of 11Tm127 and of 190Tm284, each required only two all-or-none-segments, and their enthalpies and transition temperatures were assigned. These assignments are shown here to yield the concentration of all molecular species, and therefore the helix content, as a function of temperature. Such calculations for 190Tm284 are in tolerable agreement with CD experiments, but those for 11Tm127 are in gross disagreement. Thus, either the model itself or the calorimetric assignment is faculty. In the second model, all conformational states are counted and weighted, as in the Zimm-Bragg theory for single-chain polypeptides. This theory has been extended (by Skolnick) to two-chain coiled coils and is here used to fit CD data for 11Tm127, 142Tm281, and 190Tm284. The fit is tolerable for 11Tm127, good for 142Tm281, and quantitative for 190Tm284. Thus this comparison does not falsify this second model. The helix-helix interaction free energy, obtainable from the fit, shows nonadditivity when isolated subsequences are compared with the parent. This suggests that removal of a region from a long coiled coil allows energetically substantial adjustments in side-chain packing in the helix-helix interface. Thus, the helix-helix interaction in long coiled coils is characteristic of a global free energy minimum and not just of the regional constellation of side chains.     

Transient bioheat simulation of the laser-tissue interaction in human skin using hybrid finite element formulation

This paper presents a hybrid finite element model for describing quantitatively the thermal responses of skin tissue under laser irradiation. The model is based on the boundary integral-based finite element method and the Pennes bioheat transfer equation. In this study, temporal discretization of the bioheat system is first performed and leads to the well-known modified Helmholtz equation. A radial basis function approach and the boundary integral based finite element method are employed to obtain particular and homogeneous solutions of the laser-tissue interaction problem. In the boundary integral based finite element formulation, two independent fields are assumed: intra-element field and frame field. The intra-element field is approximated through a linear combination of fundamental solutions at a number of source points outside the element domain. The frame temperature field is expressed in terms of nodal temperature and the corresponding shape function. Numerical examples are considered to verify and assess the proposed numerical model. Sensitivity analysis is performed to explore the thermal effects of various control parameters on tissue temperature and to identify the degree of burn injury due to laser heating.     

Application of polyelectrolyte theories for analysis of DNA melting in the presence of Na+ and Mg2+ ions.

Numerical calculations, using Poisson-Boltzmann (PB) and counterion condensation (CC) polyelectrolyte theories, of the electrostatic free energy difference, DeltaGel, between single-stranded (coil) and double-helical DNA have been performed for solutions of NaDNA + NaCl with and without added MgCl2. Calculations have been made for conditions relevant to systems where experimental values of helix coil transition temperature (Tm) and other thermodynamic quantities have been measured. Comparison with experimental data has been possible by invoking values of Tm for solutions containing NaCl salt only. Resulting theoretical values of enthalpy, entropy, and heat capacity (for NaCl salt-containing solutions) and of Tm as a function of NaCl concentration in NaCl + MgCl2 solutions have thus been obtained. Qualitative and, to a large extent, quantitative reproduction of the experimental Tm, DeltaHm, DeltaSm, and DeltaCp values have been found from the results of polyelectrolyte theories. However, the quantitative resemblance of experimental data is considerably better for PB theory as compared to the CC model. Furthermore, some rather implausible qualitative conclusions are obtained within the CC results for DNA melting in NaCl + MgCl2 solutions. Our results argue in favor of the Poisson-Boltzmann theory, as compared to the counterion condensation theory.     

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.     

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.