Accurate analytical calculation of the rate coefficient for the diffusion-controlled reactions due to hyperbolic diffusion

Using an approach based on the diffusion analog of the Cattaneo-Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski's boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski's theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice's formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper bound of the exact value. Here, the obtained formula seems to be of great significance for bridging a known gap between an analytically estimated rate coefficient on the one hand and molecular dynamics simulations together with experimentally observed results for the short times regime on the other hand. A particular emphasis has been placed on the rigorous mathematical treatment and important properties of the relevant initial boundary value problems in parabolic and hyperbolic diffusion theories.

Effects of non-Fourier bioheat transfer on bone drilling temperature in orthopedic surgery: Theoretical and in vitro experimental investigation

The mechanical drilling process is a typical step in treating bone fractures to fix broken parts with screws and plates. Drilling generates a significant amount of heat and elevates the temperature of the bone, which can cause thermal osteonecrosis and damage to the surrounding bone tissue and nerves. Thermal inertia between heat flux and temperature gradient in nonhomogeneous interior structural medium-like biological tissues is arguable. Therefore, this paper proposes an analytical model of heat propagation in bone drilling for orthopedic surgery based on the hyperbolic Pennes bioheat transfer equation (HPBTE). Drilling experiments in bovine cortical bone samples were also carried out using an infrared thermography approach to confirm the proposed analytical model. Around the drilled hole surface, thermal necrosis is spread out from 1 to 10 mm. Increased feed rate reduces necrosis penetration distance and increases intense bone necrosis. The HPBTE includes thermal relaxation time effect and internal convective function of tissue perfusion rate. As these factors are not considered in the parabolic heat transfer equation (PHTE), the results show that the HPBTE is more accurate in predicting temperature and thermal osteonecrosis than the PHTE. As a result, proposed analytical model is a handy tool for calculating temperature to avoid thermal damage while improving process efficiency. Furthermore, it has the capability of controlling the manual or robotic drilling procedure for minimally invasive operations.

Learning the Hydrophobic, Hydrophilic, and Aromatic Character of Amino Acids from Thermal Relaxation and Interfacial Thermal Conductance

In this study, the thermal relaxation of the 20 naturally occurring amino acids in water and in the protein lysozyme is investigated using transient nonequilibrium molecular dynamics simulations. By modeling the thermal relaxation process, the relaxation times of the amino acids in water occurs over a time scale covering 2-5 ps. For the hydrophobic amino acids, the relaxation time is controlled by the size of the hydrocarbon side chain, while for hydrophilic amino acids, the number of hydrogen bonds does not significantly affect the time scales of the heat dissipation. Our results show that the interfacial thermal conductance at the amino acid-water interface is in the range of 40-80 MW m^-2 K^-1. Hydrophobic and aromatic amino acids tend to have a lower interfacial thermal conductance. Notably, we show that amino acids can be correlated with their thermal relaxation times and molar masses, into simply connected phases with the same hydrophilicity, hydrophobicity, and aromaticity. The thermal relaxation slows down by a factor of up to five in the protein relative to that in water. In the case of the hydrophobic amino acids in the protein lysozyme, the slow down in the thermal relaxation relative to that in water appears to be controlled primarily by the size of the side chain.

Non-Fourier bioheat model for bone grinding with application to skull base neurosurgery

Bone grinding is used to remove the skull bone and access tumors through the nasal passage during cranial base neurosurgery. The generated heat of the spherical diamond tool propagates and could damage the nerves or coagulate the arteries blood. Little is known about the non-Fourier behavior of heat propagation during bone grinding. Therefore, this study develops an analytical model of the hyperbolic Pennes bioheat transfer equation (HPBTE) to calculate the three-dimensional temperature and necrosis in the grinding region. In vitro experimental investigations were carried out, and the contact zone temperature was measured using an infrared thermography system to validate the proposed thermal model. The results demonstrate that the HPBTE provides more reliable temperature evaluation and thermal damage than Fourier or parabolic heat transfer equation (PHTE). Due to the low thermal diffusivity of the bone, the lower grinding feed rate leads to higher temperature amplitude and a smaller radius of the affected zone in the surface and depth of the bone. Also, the intensity of bone necrosis decreases with the increase of the feed rate, and the shape of the damage zone becomes stretched. This analytical model can assess the potential risk of the surgery before clinical trials. Also, it could be used for comparing the different operating conditions to minimize bone necrosis and improve the control process in neurosurgeries.

Analysis of hyperbolic Pennes bioheat equation in perfused homogeneous biological tissue subject to the instantaneous moving heat source

The one-dimensional hyperbolic Pennes bioheat equation under instantaneous moving heat source is solved analytically based on the Eigenvalue method. Comparison with results of in vivo experiments performed earlier by other authors shows the excellent prediction of the presented closed-form solution. We present three examples for calculating the Arrhenius equation to predict the tissue thermal damage analysis with our solution, i.e., characteristics of skin, liver, and kidney are modeled by using their thermophysical properties. Furthermore, the effects of moving velocity and perfusion rate on temperature profiles and thermal tissue damage are investigated. Results illustrate that the perfusion rate plays the cooling role in the heating source moving path. Also, increasing the moving velocity leads to a decrease in absorbed heat and temperature profiles. The closed-form analytical solution could be applied to verify the numerical heating model and optimize surgery planning parameters.

Thermal field and tissue damage analysis of moving laser in cancer thermal therapy

In this paper, a closed-form analytical solution of hyperbolic Pennes bioheat equation is obtained for spatial evolution of temperature distributions during moving laser thermotherapy of the skin and kidney tissues. The three-dimensional cubic homogeneous perfused biological tissue is adopted as a media and the Gaussian distributed function in surface and exponentially distributed in depth is used for modeling of laser moving heat source. The solution procedure is Eigen value method which leads to a closed form solution. The effect of moving velocity, perfusion rate, laser intensity, absorption and scattering coefficients, and thermal relaxation time on temperature profiles and tissue thermal damage are investigated. Results are illustrated that the moving velocity and the perfusion rate of the tissues are the main important parameters in produced temperatures under moving heat source. The higher perfusion rate of kidney compared with skin may lead to lower induced temperature amplitude in moving path of laser due to the convective role of the perfusion term. Furthermore, the analytical solution can be a powerful tool for analysis and optimization of practical treatment in the clinical setting and laser procedure therapeutic applications and can be used for verification of other numerical heating models.

Numerical Analysis of Human Cancer Therapy Using Microwave Ablation

Microwave ablation is one type of hyperthermia treatment of cancer that involves heating tumor cells. This technique uses electromagnetic wave effects to kill cancer cells. A micro-coaxial antenna is introduced into the biological tissue. The radiation emitted by the antenna is absorbed by the tissue and leads to the heating of cancer cells. The diffuse increase in temperature should reach a certain value to achieve the treatment of cancer cells but it should be less than a certain other value to avoid damaging normal cells. This is why hyperthermia treatment should be carefully monitored. A numerical simulation is useful and may provide valuable information. The bio-heat equation and Maxwell’s equations are solved using the finite element method. Electro-thermal effects, temperature distribution profile, specific absorption rate (SAR), and fraction of necrotic tissue within cancer cells are analyzed. The results show that SAR and temperature distribution are strongly affected by input microwave power. High microwave power causes a high SAR value and raises the temperature above 50 °C, which may destroy healthy cells. It is revealed that with a power of 10 W, the tumor cells will be killed without damaging the surrounding tissue.