Incorporating an immersed boundary method to study thermal effects of vascular systems during tissue cryo-freezing

In this paper, the three-dimensional thermal effects of a clinically-extracted vascular tissue undergoing cryo-freezing are numerically investigated. Based on the measured experimental temperature field, the numerical results of the Pennes bioheat model combined with the boundary condition-enforced immersed boundary method (IBM) agreed well with experimental data with a maximum temperature discrepancy of 2.9 °C. For simulating the temperature profile of a tumor sited in a dominantly vascularized tissue, our model is able to capture with ease the thermal effects at specified junctions of the blood vessels. The vascular complexity and the ice-ball shape irregularity which cannot be easily quantified via clinical experiments are also analyzed and compared for both two-dimensional and three-dimensional settings with different vessel configurations and developments. For the three-dimensional numerical simulations, a n-furcated liver vessels model from a three-dimensional segmented volume using hole-making and subdivision methods is applied. A specific study revealed that the structure and complexity of the vascular network can markedly affect the tissue's freezing configuration with increasing ice-ball irregularity for greater blood vessel complexity.     

A novel method for measuring the thermal conductivity of organisms

Based on the theories of tissue optics and artificial neural network, the relationship between the optical properties and biological parameters was studied, and a new experimental method was derived. The properties of the organism were obtained indirectly by a black-box model derived by self-study of the artificial neural network between optical parameters and thermo-physical properties without using the heat transfer equation. In this method, the energy of light in diffuse radiation, diffuse transmission and collimated transmission was absorbed by a dual-integrating sphere experimental system of a spectrometer, and the spectrogram of the energy was obtained. Combining these spectral data of the energy, the diffuse-reflecting power, the diffuse transmissivity and the collimated transmissivity were calculated. The calculated results were taken as the input parameters of a black-box model. The experimental results show that there are apparent differences between the spectrogram of the energy on the diffuse radiation, the diffuse transmission and the collimated transmission of different matters, while there is a little difference in the same matter. Each spectrogram has its own characteristic. The values of the four thermal properties including the density, the constant pressure specific heat, the thermal diffusivity and the viscosity were calculated using the black-box model. Compared with the real values the calculated one has an average relative error between −5% and 5%. The conductivity of the tongue is 0.68 W/(m K) that calculated from the value of the density, the constant pressure specific heat and the thermal diffusivity. The results also show that there is a little difference on the conductivities in the longitudinal cross-section and the transverse section, but the effect of temperature on the conductivity of the tongue is not apparent. The difference implies the anisotropy of the properties of the organism, which cannot be easily obtained by a conventional experimental method.     

Numerical investigation of thermal response of laser-irradiated biological tissue phantoms embedded with gold nanoshells

The work presented in this paper focuses on numerically investigating the thermal response of gold nanoshells-embedded biological tissue phantoms with potential applications into photo-thermal therapy wherein the interest is in destroying the cancerous cells with minimum damage to the surrounding healthy cells. The tissue phantom has been irradiated with a pico-second laser. Radiative transfer equation (RTE) has been employed to model the light-tissue interaction using discrete ordinate method (DOM). For determining the temperature distribution inside the tissue phantom, the RTE has been solved in combination with a generalized non-Fourier heat conduction model namely the dual phase lag bio-heat transfer model. The numerical code comprising the coupled RTE-bio-heat transfer equation, developed as a part of the current work, has been benchmarked against the experimental as well as the numerical results available in the literature. It has been demonstrated that the temperature of the optical inhomogeneity inside the biological tissue phantom embedded with gold nanoshells is relatively higher than that of the baseline case (no nanoshells) for the same laser power and operation time. The study clearly underlines the impact of nanoshell concentration and its size on the thermal response of the biological tissue sample. The comparative study concerned with the size and concentration of nanoshells showed that 60 nm nanoshells with concentration of 5×10¹⁵ mm⁻³ result into the temperature levels that are optimum for the irreversible destruction of cancer infected cells in the context of photo-thermal therapy. To the best of the knowledge of the authors, the present study is one of the first attempts to quantify the influence of gold nanoshells on the temperature distributions inside the biological tissue phantoms upon laser irradiation using the dual phase lag heat conduction model.     

A novel approach to improve the efficacy of tumour ablation during cryosurgery

Freezing tumours and ablating it using cryosurgery is becoming a popular surgical procedure for treatment of carcinomas. In order to improve the efficiency of the cryosurgical procedure different approaches have been implemented till now, e.g., injecting high thermal conductivity fluid inside the tumour, low latent heat fluids inside the tumour prior to cryosurgery etc. These techniques improve the cryosurgical process to some extent but lack in minimising the damage to the surrounding healthy tissues. In this study, a novel concept is proposed which advocates the use of solutions with specific thermophysical properties around the interface of tumour. Numerical modelling has been done to determine the location of the ice fronts in the presence of this solution around the boundary of the tumour. It is noticed that in the presence of solution layer, owing to its distinct thermophysical properties like low thermal conductivity, not only the cellular destruction is enhanced but also the damage to the surrounding healthy tissue is minimised. Further, results indicate that this strategy leads to a faster ablation rate reducing the surgical time immensely. Also, an optimal offset, the minimum distance between the tip of cryoprobe and the boundary of the tumour, is identified for a given tumour radius with a given active length which gives maximum tumour necrosis in less time. This optimal offset which has been identified for each case will help the surgeons in proper planning of cryosurgery and improving the effectiveness of this technique greatly, making it a better treatment modality than its counterparts in many ways. It is also observed that for a 2 mm increase in activelength of the cryoprobe, the decrease in optimal offset is approximately 1 mm, i.e. optimal offset decreases linearly with an increase in the activelength for a given radius of the tumour. Also, for tumour with different radii, ranging between 10 mm to 15 mm, with same active length, the time taken for complete ablation by the larger tumour is nearly 2.7 times the time taken by the smaller one for every 2.5 mm increase in the tumour radius.     

Mathematical modeling of thermal physiological responses of clothed infants

The objective of this study is to develop a mathematical model for simulating the thermal physiological responses of clothed infants. By modifying and integrating Gagge's two-node model and Stolwijk's multi-node model, and coupled with the model of dynamic couple heat and moisture transfer in functional clothing, a new seven-node thermoregulation model for closed infants was developed. A series of preliminary numerical test were carried out for naked and clothed infant. The prediction of the new model was compared with the published data, the comparison results demonstrate that the model has good potential predictability.     

The effects of thermal and mechanical material properties on tumorous tissue during hyperthermia treatment

Although there have been numerous reports in several articles about the viscoelastic properties of biological tissues, no effort has been made to investigate the combined thermal and mechanical behavior of the viscoelastic tissue. At present, the model of thermo-viscoelasticity theory with variable thermal conductivity and rheological properties of the volume is considered to investigate bio-thermo-mechanics behavior in living tissue within the context of the Lord-Shulman theory. The model is applied to a limited thickness, cancerous layer problem. The problem was solved analytically in the transformed domain using Laplace transform as a tool. The exact solution is obtained in the context of transformation Laplace. Numerical results are given and illustrated graphically for the distributions of temperature, displacement, and stress. Some correlations are produced with the results obtained for the absence of the thermal relaxation parameter. The effects of variable thermal and volume materials properties, blood perfusion rate on the behavior of various fields are examined.     

Relations between heat transfer in perfused biological tissue and the local symmetry components of the vascular system

The relation between perfusion and heat transfer in tissue can be formulated in terms of local symmetry components of the vascular system. It is shown that the order of symmetry (dipole, quadrupole etc. symmetry) gives the order of magnitude of perfusion heat transfer. A unidirectional flow component results in a D'Arcy like bulk flow. A non unidirectional flow component contributes to heat transfer at least by a second order term. It acts like an additional effective heat diffusivity. Comparison with experiments confirm the theoretical results.Supported by Deutsche Forschungsgemeinshcaft.     

Heat transfer modeling of the tongue

The heat transfer mechanism of tongue was investigated on the basis of experimental and theoretical research. Firstly, the relationship between tongue temperature and blood perfusion was obtained from animal experiment that mainly carried out on porcine tongue, subordinate on human tongue. Secondly, a one-dimensional variable coefficients second-order inhomogeneous heat transfer equation is developed by simplifying tongue as fin cube and the analytical solution is got. The results show that the change regulations of temperature by blood perfusion rate are the same in human and porcine tongue, and also, there is a good agreement between calculation and experimental results. When checking the model with corresponding properties of human tongue, the result is also satisfied. In conclusion, predicting temperature distribution of tongue is feasible with the fin cube model.     

Study of thermal behavior of a biological tissue: An equivalence of Pennes bioheat equation and Wulff continuum model

Equivalence of Pennes bioheat equation (PBHE) and Wulff continuum model (WCM) is established for a 1-D planar tissue. The derived condition of equivalence is specific to tissue without metabolic heat generation. Mathematical analysis is carried out to relate blood perfusion rate and local mean blood velocity that are needed in the analysis of WCM. It is found that the local mean blood velocity in a tissue is a direct function of square root of blood perfusion rate. This functional dependence is also established numerically by having same solution obtained from PBHE and WCM. Analysis is also done to check how closely the derived relation can be used for practical cases of breast tissue with and without a tumor. Blood velocity is a very important physiological quantity. Its measurement is a difficult process and requires a state-of-the-art technique. The proposed relation allows its computation merely from the knowledge of blood perfusion rate.     

Development and estimation of a novel cryoprobe utilizing the Peltier effect for precise and safe cryosurgery

We have developed a novel cryoprobe for skin cryosurgery utilizing the Peltier effect. The four most important parameters for necrotizing tissue efficiently are the cooling rate, end temperature, hold time and thawing rate. In cryosurgery for small skin diseases such as flecks or early carcinoma, it is also important to control the thickness of the frozen region precisely to prevent necrotizing healthy tissue. To satisfy these exacting conditions, we have developed a novel cryoprobe to which a Peltier module was attached. The cryoprobe makes it possible to control heat transfer to skin surface precisely using a proportional-integral-derivative (PID) controller, and because it uses the Peltier effect, the cryoprobe does not need to move during the operation. We also developed a numerical simulation method that allows us to predict the frozen region and the temperature profile during cryosurgery.We tested the performance of our Peltier cryoprobe by cooling agar, and the results show that the cryoprobe has sufficient cooling performance for cryosurgery, because it can apply a cooling rate of more than 250 °C/min until the temperature reaches −40 °C. We also used a numerical simulation to reconstruct the supercooling phenomenon and examine the immediate progress of the frozen region with ice nucleation. The calculated frozen region was compared with the experimentally measured frozen region observed by an interferometer, and the calculation results showed good agreement. The results of numerical simulation confirmed that the frozen region could be predicted accurately with a margin of error as small as 150 μm during use of the cryoprobe in cryosurgery. The numerical simulation also showed that the cryoprobe can control freezing to a depth as shallow as 300 μm.     

Key issues in bioheat transfer simulations for the application of cryosurgery planning

The bioheat transfer simulation is undoubtedly the foundation for developing computerized tools for cryosurgery planning and analysis. While a large variety of techniques for bioheat transfer simulations are available in the literature of the past several decades, it is only their integration with clinical criteria and constraints which can make computerized planning a practical reality. This brief communication outlines (in the opinion of this author) the key issues that must be addressed in the application of bioheat transfer to cryosurgery planning and analysis, while drawing attention to recent and relevant publications in other journals, with reference to the most recent publication on the topic in the Journal of Cryobiology [Z. Magalov, A. Shitzer, D. Degani, Isothermal volume contours generated in a freezing gel by embedded cryo-needles with applications to cryo-surgery, Cryobiology 55 (2) (2007) 127–137].     

Numerical simulation of selective freezing of target biological tissues following injection of solutions with specific thermal properties

Recently, we proposed a method for controlling the extent of freezing during cryosurgery by percutaneously injecting some solutions with particular thermal properties into the target tissues. In order to better understand the mechanism of the enhancement of freezing by these injections, a new numerical algorithm was developed to simulate the corresponding heat transfer process that is involved. The three-dimensional phase change processes in biological tissues subjected to cryoprobe freezing, with or without injection, were compared numerically. Two specific cases were investigated to illustrate the selective freezing method: the injection of solutions with high thermal conductivity; the injection of solutions with low latent heat. It was found that the localized injection of such solutions could significantly enhance the freezing effect and decrease the lowest temperature in the target tissues. The result also suggests that the injection of these solutions may be a feasible and flexible way to control the size of the ice ball and its direction of growth during cryosurgery, which will help to optimize the treatment process.     

A general model for the propagation of uncertainty in measurements into heat transfer simulations and its application to cryosurgery

This report presents a technique for estimating the propagation of uncertainty in measurements into mathematical simulations of heat transfer. The motivation for this report is to show the dramatic uncertainty associated with estimating the value of the so-called "lethal temperature," even in a case where a perfect correlation appears to exist between histo-pathologic observations and a corresponding heat transfer simulation. Although the example presented in this report relates to cryosurgery, the technique proposed in this report is rather general and can be applied to any heat transfer problem. The uncertainty analysis presented in this report can be considered as an extension of the well-known concept of the rule of the square root of the sum of the square errors. A comparison of the new technique with the worst case scenario concept is also presented. In conclusion, it is recommended that the proposed technique be routinely applied when presenting simulated results, whether as a part of a theoretical study, or in comparison with experimental data.     

Prediction of the extent of thermal damage in the cornea during conductive keratoplasty

Conductive keratoplasty (CK) is a sub-ablative thermal therapy used to treat hyperopia and presbyopia. In this study, a 3-D finite element model of the cornea was developed to predict the transient temperature distributions and the resultant thermal damage fields in the cornea during simulated CK procedures. The model incorporated collagen denaturation and vaporization of water as well as ablation-induced thermal/electrical contact loss. The effects of the radiofrequency power applied on the electrode and the duration of the treatment on the extent of thermal damage in the tissue were examined and compared with the previously published experimental results on human cornea. It was shown that with clinical settings (60% power and 0.6 s treatment duration), the temperature maximum near the tip of the radiofrequency probe exceeded the temperature for vaporization. The results also indicated that the increase in the treatment duration had a much more significant effect on the size of the thermally modified region than increased radiofrequency power (as verified by the experimental results). The model predictions matched the experimental results well and showed the feasibility of using simulations to optimize thermal treatment of the cornea.     

Engineering of human hepatic tissue with functional vascular networks

Although absolute organ shortage highlights the needs of alternative organ sources for regenerative medicine, the generation of a three-dimensional (3D) and complex vital organ, such as well-vascularized liver, remains a challenge. To this end, tissue engineering holds great promise; however, this approach is significantly limited by the failure of early vascularization in vivo after implantation. Here, we established a stable 3D in vitro pre-vascularization platform to generate human hepatic tissue after implantation in vivo. Human fetal liver cells (hFLCs) were mixed with human umbilical vein endothelial cells (HUVECs) and mesenchymal stem cells (hMSCs) and were implanted into a collagen/fibronectin matrix composite that was used as a 3-D carrier. After a couple of days, the fluorescent HUVECs developed premature vascular networks in vitro, which were stabilized by hMSCs. The establishment of functional vessels inside the pre-vascularized constructs was proven using dextran infusion studies after implantation under a transparency cranial window. Furthermore, dynamic morphological changes during embryonic liver cell maturation were intravitaly quantified with high-resolution confocal microscope analysis. The engineered human hepatic tissue demonstrated multiple liver-specific features, both structural and functional. Our new techniques discussed here can be implemented in future clinical uses and industrial uses, such as drug testing.     

Engineering of human hepatic tissue with functional vascular networks

Although absolute organ shortage highlights the needs of alternative organ sources for regenerative medicine, the generation of a three-dimensional (3D) and complex vital organ, such as well-vascularized liver, remains a challenge. To this end, tissue engineering holds great promise; however, this approach is significantly limited by the failure of early vascularization in vivo after implantation. Here, we established a stable 3D in vitro pre-vascularization platform to generate human hepatic tissue after implantation in vivo. Human fetal liver cells (hFLCs) were mixed with human umbilical vein endothelial cells (HUVECs) and mesenchymal stem cells (hMSCs) and were implanted into a collagen/fibronectin matrix composite that was used as a 3-D carrier. After a couple of days, the fluorescent HUVECs developed premature vascular networks in vitro, which were stabilized by hMSCs. The establishment of functional vessels inside the pre-vascularized constructs was proven using dextran infusion studies after implantation under a transparency cranial window. Furthermore, dynamic morphological changes during embryonic liver cell maturation were intravitaly quantified with high-resolution confocal microscope analysis. The engineered human hepatic tissue demonstrated multiple liver-specific features, both structural and functional. Our new techniques discussed here can be implemented in future clinical uses and industrial uses, such as drug testing.     

Computerized planning of prostate cryosurgery using variable cryoprobe insertion depth

The current study presents a computerized planning scheme for prostate cryosurgery using a variable insertion depth strategy. This study is a part of an ongoing effort to develop computerized tools for cryosurgery. Based on typical clinical practices, previous automated planning schemes have required that all cryoprobes be aligned at a single insertion depth. The current study investigates the benefit of removing this constraint, in comparison with results based on uniform insertion depth planning as well as the so-called “pullback procedure”. Planning is based on the so-called “bubble-packing method”, and its quality is evaluated with bioheat transfer simulations. This study is based on five 3D prostate models, reconstructed from ultrasound imaging, and cryoprobe active length in the range of 15-35 mm. The variable insertion depth technique is found to consistently provide superior results when compared to the other placement methods. Furthermore, it is shown that both the optimal active length and the optimal number of cryoprobes vary among prostate models, based on the size and shape of the target region. Due to its low computational cost, the new scheme can be used to determine the optimal cryoprobe layout for a given prostate model in real time.     

The stability of predator-prey systems subject to the Allee effects

In recent years, many theoreticians and experimentalists have concentrated on the processes that affect the stability of predator-prey systems. But few papers have addressed the Allee effect with focus on the their stability. In this paper, we select two classical models describing predator-prey systems and introduce the Allee effects into the dynamics of both the predator and prey populations in these models, respectively. By combining mathematical analysis with numerical simulation, we have shown that the Allee effect may be a destabilizing force in predator-prey systems: the equilibrium point of the system could be changed from stable to unstable or otherwise, the system, even when it is stable, will take much longer time to reach the stable state. We also conclude that the equilibrium of the prey population will be enlarged due to the Allee effect of the predator, but the Allee effects of the prey may decrease the equilibrium value of the predator, or that of both the predator and prey. It should also be pointed out that the impact of the Allee effects of predator and prey due to different mechanisms on different predator-prey systems could also vary.