Heat transfer normal to paired arterioles and venules embedded in perfused tissue during hyperthermia

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.

A Three-Dimensional Finite Element Analysis of Heat Transfer in the Forearm

The finite element method was used to analyze heat transfer within a section of the forearm while exposed to different ambient conditions and with different metabolic states. The three-dimensional model accounts for the different material properties of bone, muscle and blood and incorporates a single artery-vein pair for counter-current heat exchange. The geometry of the model was developed from anatomical cross-sectional images of the forearm. The model was used to determine the effects or rest vs. exercise, free vs. forced surface convection and 0 degrees C vs. -20 degrees C external temperatures. The results of the model were compared to experimental data and the model exhibits qualitatively correct behaviour. This model can be used to study hyperthermia, burns and cryogenic freezing of tissue.

A counter current vascular network model of heat transfer in tissues

A fully conjugated blood vessel network model (FCBVNM) for calculating tissue temperatures has been developed, tested, and studied. This type of model represents a more fundamental approach to modeling temperatures in tissues than do the generally used approximate equations such as the Pennes'BHTE or effective thermal conductivity equations. As such, this type of model can be used to study many important questions at a more basic level. For example, in the particular hyperthermia application studied herein, a simple vessel network model predicts that the role of counter current veins is minimal and that their presence does not significantly affect the tissue temperature profiles: the arteries, however, removed a significant fraction of the power deposited in the tissue. These more fundamental models can also be used to check the validity of approximate equations. For example, using the present simple model, when the temperatures calculated by the FCBVNM are used for comparing predictions from two approximation equations (a simple effective thermal conductivity and a simple Pennes' bio-heat transfer equation formulation of the same problem) it is found that the Pennes' equation better approximates the FCBVNM temperatures than does the k(eff) model. These results also show that the "perfusion" value (W) in the Pennes' BHTE is not necessarily equal to the "true" tissue perfusion (P) as calculated from mass flow rate considerations, but can be greater than, equal to, or less than that value depending on (1) how many vessel levels are modeled by the BHTE, and (2) the "true" tissue perfusion magnitude. This study uses a simple, generic vessel network model to demonstrate the potential usefulness of such fully conjugated vessel network models, and the associated need for developing and applying more complicated and realistic vascular network models. As more realistic vascular models (vessel sizes, orientations, and flow rates) are developed, the predictions of the fully conjugated models should more closely model and approach the true tissue temperature distributions, thus making these fully conjugated models more accurate and valuable tools for studying tissue heat transfer processes.

Use of vascular and non-vascular models for the assessment of temperature distribution during induced hyperthermia

We developed a three-dimensional thermal model in which convective heat transfer in a human extremity is explicitly quantified by taking into account the physical details of the vascular system. The spatial pattern in the arterial, venous and tissue temperature is computed during hyperthermia treatment. As such a complex vascular model is not generally applicable, a comparative study of the results of simpler, substitutional, non-vascular concepts applied to hyperthermia was carried out. It turned out that classical bioheat approaches may lead to wrong conclusions. Satisfactory results are to be expected from an approach suggested by Wissler/Baish and Charny/Levin, and by a far simpler efficiency function (EF) concept, developed by us, that can be used as easily as the simple classical Pennes approach while avoiding its deficiencies. The EF model is used to predict whether the temperature of the entire volume of vascularized muscle tissue will be raised to a therapeutic level. With the help of the vascular model, local thermal non-uniformities near individual blood vessels are analysed. Underheating in small volumes of tissue near the vessel walls is detected. This might explain tumour regrowth following local hyperthermia treatment in the tissue adjacent to blood vessels.

A three-dimensional thermal and electromagnetic model of whole limb heating with a MAPA

Previous studies by the authors have shown that if properly implemented, the Pennes assumptions can be applied to quantify bioheat transfer during extremity heating. Given its relative numerical simplicity and its ability to predict temperatures in thermoregulated tissue, the Pennes model of bioheat transfer was utilized in a three-dimensional thermal model of limb heating. While the arterial blood temperature was assumed to be radially uniform within a cross section of the limb, axial gradients in the arterial and venous blood temperatures were computed with this three-dimensional model. A realistically shaped, three-dimensional finite element model of a tumor-bearing human lower leg was constructed and was "attached" mathematically to the whole body thermal model of man described in previous studies by the authors. The central as well as local thermoregulatory feedback control mechanisms which determine blood perfusion to the various tissues and rate of evaporation by sweating were input into the limb model. In addition, the temperature of the arterial blood which feeds into the most proximal section of the lower leg was computed by the whole body thermal model. The variations in the shape of the tissues which comprise the limb were obtained from computerized tomography scans. Axial variations in the energy deposition patterns along the length of the limb exposed to a miniannular phased array (MAPA) applicator were also input into this model of limb heating. Results indicate that proper positioning of the limb relative to the MAPA is a significant factor in determining the effectiveness of the treatment. A patient-specific hyperthermia protocol can be designed using this coupled electromagnetic and thermal model.

Comparison of numerical calculations with phantom experiments and clinical measurements

Three-dimensional models, while fundamentally desirable in hyperthermia treatment simulation, are only beginning to emerge and may take a number of years to perfect for routine clinical use. Two dimensional calculations, on the other hand, can be efficiently performed on today's inexpensive computer workstations; however, the accuracy of two-dimensional models in the pretreatment planning context is questionable. This paper investigates the ability of a general two-dimensional finite element model to predict power deposition patterns in phantoms and temperature distributions during actual clinical treatments. The experiments and simulations have been performed for an annular phased array (APA) operating at 70 MHz. Comparisons between model predictions and measurements show that quantitative agreement occurs in phantoms containing moderate complexities in heterogeneity, but that only qualitative agreement appears possible in clinical treatments. However, the results suggest that the lack of blood flow information may contribute as much, if not more, to the uncertainties in the clinical predictions than the two-dimensional nature of the model itself.