Effects of MRTI sampling characteristics on estimation of HIFU SAR and tissue thermal diffusivity.

While the non-invasive and three-dimensional nature of magnetic-resonance temperature imaging (MRTI) makes it a valuable tool for high-intensity focused ultrasound (HIFU) treatments, random and systematic errors in MRTI measurements may propagate into temperature-based parameter estimates used for pretreatment planning. This study assesses the MRTI effects of zero-mean Gaussian noise (SD = 0.0-2.0 °C), temporal sampling (tacq = 1.0-8.0 s), and spatial averaging (Res = 0.5-2.0 mm isotropic) on HIFU temperature measurements and temperature-based estimates of the amplitude and full width half maximum (FWHM) of the HIFU specific absorption rate and of tissue thermal diffusivity. The ultrasound beam used in simulations and ex vivo pork loin experiments has lateral and axial FWHM dimensions of 1.4 mm and 7.9 mm respectively. For spatial averaging simulations, beams with lateral FWHM varying from 1.2-2.2 mm are also assessed. Under noisy conditions, parameter estimates are improved by fitting to data from larger voxel regions. Varying the temporal sampling results in minimal changes in measured temperatures (<2% change) and parameter estimates (<5% change). For the HIFU beams studied, a spatial resolution of 1 × 1 × 3 mm(3) or smaller is required to keep errors in temperature and all estimated parameters less than 10%. By quantifying the errors associated with these sampling characteristics, this work provides researchers with appropriate MRTI conditions for obtaining estimates of parameters essential to pretreatment modeling of HIFU thermal therapies.

Mathematical model of cycad cones' thermogenic temperature responses: inverse calorimetry to estimate metabolic heating rates.

A mathematical model based on conservation of energy has been developed and used to simulate the temperature responses of cones of the Australian cycads Macrozamia lucida and Macrozamia. macleayi during their daily thermogenic cycle. These cones generate diel midday thermogenic temperature increases as large as 12 °C above ambient during their approximately two week pollination period. The cone temperature response model is shown to accurately predict the cones' temperatures over multiple days as based on simulations of experimental results from 28 thermogenic events from 3 different cones, each simulated for either 9 or 10 sequential days. The verified model is then used as the foundation of a new, parameter estimation based technique (termed inverse calorimetry) that estimates the cones' daily metabolic heating rates from temperature measurements alone. The inverse calorimetry technique's predictions of the major features of the cones' thermogenic metabolism compare favorably with the estimates from conventional respirometry (indirect calorimetry). Because the new technique uses only temperature measurements, and does not require measurements of oxygen consumption, it provides a simple, inexpensive and portable complement to conventional respirometry for estimating metabolic heating rates. It thus provides an additional tool to facilitate field and laboratory investigations of the bio-physics of thermogenic plants.

An analytical solution for improved HIFU SAR estimation.

Accurate determination of the specific absorption rates (SARs) present during high intensity focused ultrasound (HIFU) experiments and treatments provides a solid physical basis for scientific comparison of results among HIFU studies and is necessary to validate and improve SAR predictive software, which will improve patient treatment planning, control and evaluation. This study develops and tests an analytical solution that significantly improves the accuracy of SAR values obtained from HIFU temperature data. SAR estimates are obtained by fitting the analytical temperature solution for a one-dimensional radial Gaussian heating pattern to the temperature versus time data following a step in applied power and evaluating the initial slope of the analytical solution. The analytical method is evaluated in multiple parametric simulations for which it consistently (except at high perfusions) yields maximum errors of less than 10% at the center of the focal zone compared with errors up to 90% and 55% for the commonly used linear method and an exponential method, respectively. For high perfusion, an extension of the analytical method estimates SAR with less than 10% error. The analytical method is validated experimentally by showing that the temperature elevations predicted using the analytical method's SAR values determined for the entire 3D focal region agree well with the experimental temperature elevations in a HIFU-heated tissue-mimicking phantom.

Closed-form solution for the thermal dose delivered during single pulse thermal therapies.

This study provides a closed form, analytical expression for the thermal dose delivered by a single heating pulse. The solution is derived using the effective cooling method and the non-linear Sapareto-Dewey equation to determine the thermal dose delivered by the time-temperature history of a treatment. The analytical solutions are used to determine the optimal treatment conditions, i.e. those that exactly deliver the desired thermal dose at a specified time. For purposes of illustration, this study focuses on a 'conservative' clinical approach in which the desired thermal dose is delivered at the end of the 'cool down' period. The analytical results show that, after a clinical strategy has been chosen (e.g. conservative, aggressive or intermediate), the user can only specify two free variables for such an optimal treatment. Results are presented which suggest that a practical approach would be to specify both (1) the desired thermal dose to be delivered to the target (the clinically relevant outcome) and (2) the peak temperature to be reached (a measurable, clinically useful, patient dependent response variable that can be employed in feedback control systems); and then determine the associated, optimal heating magnitude and duration that need to be used to reach that dose and temperature. The results also reveal that, with a given patient condition and power deposition distribution (together specifying an effective cooling time constant for the treatment) and a specified thermal dose, there is a maximum allowable peak temperature that, if exceeded, will result in 'over-dosing' the heated tissue. The results also show that avoiding such non-optimal 'over-dosing' will be difficult in most high temperature therapies since, when high temperatures are produced in tissues, the temperature decay must be very fast in order to avoid over-dosing during the cooling period. Such rapid cooling can only occur if short effective cooling time constants are present-either as a result of large tissue blood flows in the patient or due to large conduction effects induced by the use of highly localized power deposition sources.

Improved accuracy and consistency in T1 measurement of flowing blood by using inversion recovery GE-EPI.

Problems associated with techniques currently used to measure the T1 of flowing blood are evaluated and a method to improve the consistency and repeatability of measurements is presented. Similar to some currently used techniques, the pulse sequence employs a nonselective adiabatic inversion pulse followed by a series of ECG-gated gradient echo EPI (echo planar imaging) images to obtain images where the blood (fluid) signal exhibits a T1-dependent inversion recovery signal from which the spin lattice relaxation constant (T1) of the flowing fluid can be measured. The new method combines curve fitting with a measure of the curve null point to acquire more accurate and consistent T1 values. Simulation and experimental results show that this combined fitting-nulling method is more stable and consistent in measuring the T1 of flowing fluid. The feasibility of temperature measurement of a flowing fluid based on the temperature dependence of the T1 of water protons is shown in this paper. ECG gating is used to reduce the effects of cyclic intensity changes for measurement of T1 in pulsatile flowing blood.

A thermo-pharmacokinetic model of tissue temperature oscillations during localized heating.

Thermally-induced large blood flow increases and oscillations have been experimentally observed in both muscle and prostate tissues. However, the bio-physical/-chemical mechanisms underlying these phenomena remain undiscovered. To study the basic nature of these coupled thermal-mass transport processes, this study combines a compartmental vasodilator pharmacokinetics model with a bio-heat-transfer temperature model. The resulting simulated temperature responses to different applied power levels closely match both the overall behaviour and the fine structure of the complex temperature responses observed in vivo. This suggests that the coupled thermo-pharmacokinetic model captures the essence of the links between tissue temperature and blood flow oscillations and of the role of the important vaso-active substances. Thus, it appears that such thermo-pharmacokinetic models can provide a basis for helping to understand and quantify the fundamental bio-physical/-chemical processes that couple the transient tissue temperature distributions to blood flow oscillations. Such combined models allow investigators to directly predict tissue blood flow responses to applied power and avoid the need to make ad hoc assumptions regulating the blood flow rates present in heated tissues.

Optimal power deposition patterns for ideal high temperature therapy/hyperthermia treatments.

If it were possible to achieve, an ideal high temperature therapy or hyperthermia treatment would involve a single heating session and yield a desired thermal dose distribution in the tumour that would be attained in the shortest possible treatment time without heating critical normal tissues excessively. Simultaneously achieving all of these goals is impossible in practice, thus requiring trade-offs that allow clinicians to approach more closely some of these ideal goals at the expense of others. To study the basic nature of a subset of these trade-offs, the present simulation study looked at a simple, ideal case in which the tumour is heated by a single, optimized (with respect to space) power pulse, with no power deposition in the normal tissue. Results were obtained for two different clinical strategies (i.e. trade-off approaches), including: (1) an 'aggressive' approach, wherein the desired, uniform thermal dose is completely delivered to the tumour during the power-on period. This approach gives the clinician the satisfaction of knowing that the tumour was treated completely while power was being delivered, and yields the shortest attainable tumour dose delivery time. However, that benefit is attained at the cost of both 'overdosing' the tumour during the subsequent cool down period and, paradoxically, requiring a longer, overall treatment time. Here, the treatment time is considered as that time interval from the initiation of the heating pulse to the time at which the entire tumour has decayed to a specified 'safe' temperature--below 43 degrees C for our calculations. And, (2) a 'conservative' approach is considered, wherein the desired uniform dose is attained at the post-heating time at which the complete tumour cools back down to 'basal' conditions, taken as 4 h in this study. This conservative approach requires less applied power and energy and avoids the 'overdosing' problem, but at the cost of having a tumour dose delivery time that can be significantly longer than the heating pulse duration. This approach can require that clinicians wait a significant time after the power has been turned off before being able to confirm that the desired tumour thermal dose was reached. The present findings show that: (1) for both clinical strategies, an optimal power deposition shape (with respect to position in the tumour) can always be found that provides the desired uniform thermal dose in the tumour, regardless of the heating pulse duration chosen or the tumour perfusion pattern; and (2) shorter heating pulses are preferable to longer ones in that they require less total energy, take less total time to treat the patients, and have optimal power deposition patterns less influenced by perfusion. On the other hand, shorter pulses always require higher temperatures, and for the 'aggressive' clinical approach, they give significantly larger excess thermal doses in the tumour. The aggressive approach always requires longer treatment times than comparable conservative treatments. The optimal power patterns for both strategies involve a high-power density at the tumour boundary, which frequently creates a 'thermal wave' that contributes significantly to the final thermal dose distribution attained.

Hybrid finite element-finite difference method for thermal analysis of blood vessels.

A hybrid finite-difference/finite-element technique for the thermal analysis of blood vessels embedded in perfused tissue has been developed and evaluated. This method provides efficient and accurate solutions to the conjugated heat transfer problem of convection by blood coupled to conduction in the tissue. The technique uses a previously developed 3D automatic meshing method for creating a finite element mesh in the tissue surrounding the vessels, coupled iteratively with a 1-D marching finite difference method for the interior of the vessels. This hybrid technique retains the flexibility and ease of automated finite-element meshing techniques for modelling the complex geometry of blood vessels and irregularly shaped tissues, and speeds the solution time by using a simple finite-difference method to calculate the bulk mean temperatures within all blood vessels. The use of the 1D finite-difference technique in the blood vessels also eliminates the large computer memory requirements needed to accurately solve large vessel network problems when fine FE meshes are used in the interior of vessels. The accuracy of the hybrid technique has been verified against previously verified numerical solutions. In summary, the hybrid technique combines the accuracy and flexibility found in automated finite-element techniques, with the speed and reduction of computational memory requirements associated with the 1D finite-difference technique, something which has not been done before. This method, thus, has the potential to provide accurate, flexible and relatively fast solutions for the thermal analysis of coupled perfusion/blood vessel problems, and large vessel network problems.

An accurate, convective energy equation based automated meshing technique for analysis of blood vessels and tissues.

An automated three-element meshing method for generating finite element based models for the accurate thermal analysis of blood vessels imbedded in tissue has been developed and evaluated. The meshing method places eight noded hexahedral elements inside the vessels where advective flows exist, and four noded tetrahedral elements in the surrounding tissue. The higher order hexahedrals are used where advective flow fields occur, since high accuracy is required and effective upwinding algorithms exist. Tetrahedral elements are placed in the remaining tissue region, since they are computationally more efficient and existing automatic tetrahedral mesh generators can be used. Five noded pyramid elements connect the hexahedrals and tetrahedrals. A convective energy equation (CEE) based finite element algorithm solves for the temperature distributions in the flowing blood, while a finite element formulation of a generalized conduction equation is used in the surrounding tissue. Use of the CEE allows accurate solutions to be obtained without the necessity of assuming ad hoc values for heat transfer coefficients. Comparisons of the predictions of the three-element model to analytical solutions show that the three-element model accurately simulates temperature fields. Energy balance checks show that the three-element model has small, acceptable errors. In summary, this method provides an accurate, automatic finite element gridding procedure for thermal analysis of irregularly shaped tissue regions that contain important blood vessels. At present, the models so generated are relatively large (in order to obtain accurate results) and are, thus, best used for providing accurate reference values for checking other approximate formulations to complicated, conjugated blood heat transfer problems.

Conditions for equivalency of countercurrent vessel heat transfer formulations.

Previous models of countercurrent blood vessel heat transfer have used one of two, different, equally valid but previously unreconciled formulations, based either on: (1) the difference between the arterial and venous vessels' average wall temperatures, or (2) the difference between those vessels' blood bulk fluid temperatures. This paper shows that these two formulations are only equivalent when the four, previously undefined, "convective heat transfer coefficients" that are used in the bulk temperature difference formulation (two coefficients each for the artery and vein) have very specific, problem-dependent relationships to the standard convective heat transfer coefficients. (The average wall temperature formulation uses those standard coefficients correctly.) The correct values of these bulk temperature difference formulation "convective heat transfer coefficients" are shown to be either: (1) specific functions of (a) the tissue conduction resistances, (b) the standard convective heat transfer coefficients, and (c) the independently specified bulk arterial, bulk venous and tissue temperatures, or (2) arbitrary, user defined values. Thus, they are generally not equivalent to the standard convective heat transfer coefficients that are regularly used, and must change values depending on the blood and tissue temperatures. This dependence can significantly limit the convenience and usefulness of the bulk temperature difference formulations.

Engineering aspects of hyperthermia therapy.

The continuing accrual of positive results in clinical cancer trials of adjunctive, synergistic hyperthermia therapy remains a strong motivation for the development of improved hyperthermia equipment and software. Indeed, the lack of needed engineering tools can be viewed as the major stumbling block to hyperthermia's effective clinical implementation. Developing clinically effective systems will be difficult, however, because (a) it requires solving several complex engineering problems, for which (b) setting appropriate design and evaluation goals is currently difficult owing to a lack of critical biological, physiological, and clinical knowledge, two tasks which must (c) be accomplished within a complicated social/political structure.

Reduced-order modeling for hyperthermia: an extended balanced-realization-based approach.

Accurate thermal models are needed in hyperthermia cancer treatments for such tasks as actuator and sensor placement design, parameter estimation, and feedback temperature control. The complexity of the human body produces full-order models which are too large for effective execution of these tasks, making use of reduced-order models necessary. However, standard balanced-realization (SBR)-based model reduction techniques require a priori knowledge of the particular placement of actuators and sensors for model reduction. Since placement design is intractable (computationally) on the full-order models, SBR techniques must use ad hoc placements. To alleviate this problem, an extended balanced-realization (EBR)-based model-order reduction approach is presented. The new technique allows model order reduction to be performed over all possible placement designs and does not require ad hoc placement designs. It is shown that models obtained using the EBR method are more robust to intratreatment changes in the placement of the applied power field than those models obtained using the SBR method.

Optimal actuator placement for large scale systems: a reduced-order modelling approach.

A recently developed, extended balanced realization, reduced order modelling technique for large-scale distributed systems is applied to the problem of optimal actuator placement for hyperthermia treatments. Extended balanced realization develops low-order models whose state reconstructions are robust to actuator and sensor placement changes, and hence can be effectively used to find optimal placements in a computationally efficient manner. This optimization approach has been tested on simulations of a scanned focused, ultrasound hyperthermia system and found to be robust and accurate over a wide range of models, and the savings in computational costs were found to be significant.

A comparison of reduced-order modelling techniques for application in hyperthermia control and estimation.

Reduced-order modelling techniques can make important contributions in the control and state estimation of large systems. In hyperthermia, reduced-order modelling can provide a useful tool by which a large thermal model can be reduced to the most significant subset of its full-order modes, making real-time control and estimation possible. Two such reduction methods, one based on modal decomposition and the other on balanced realization, are compared in the context of simulated hyperthermia heat transfer problems. The results show that the modal decomposition reduction method has three significant advantages over that of balanced realization. First, modal decomposition reduced models result in less error, when compared to the full-order model, than balanced realization reduced models of similar order in problems with low or moderate advective heat transfer. Second, because the balanced realization based methods require a priori knowledge of the sensor and actuator placements, the reduced-order model is not robust to changes in sensor or actuator locations, a limitation not present in modal decomposition. Third, the modal decomposition transformation is less demanding computationally. On the other hand, in thermal problems dominated by advective heat transfer, numerical instabilities make modal decomposition based reduction problematic. Modal decomposition methods are therefore recommended for reduction of models in which advection is not dominant and research continues into methods to render balanced realization based reduction more suitable for real-time clinical hyperthermia control and estimation.

A generic tissue convective energy balance equation: Part I--theory and derivation.

A new equation for calculating temperatures in living tissues, the tissue convective energy balance equation (TCEBE), is derived using only a few assumptions. The resulting equation is basic, general and applicable to any tissue. The (unsolved) TCEBE is used: (a) to relate both Pennes' BHTE perfusion-related parameter (W) and the effective thermal conductivity equation's perfusion-related parameter (keff) to the true capillary perfusion Pcap, and (b) to show that both W and keff are defined, nonphysiological variables, which are only related to Pcap in a problem-dependent manner. Finally, the derivation of the relationship between W and Pcap provides a complete derivation of Pennes' BHTE, something that has not been previously done.

Cerebral bloodflow in and around spontaneous malignant gliomas.

Knowledge of the cerebral bloodflow (CBF) in and around malignant gliomas is of crucial importance in developing strategies for hyperthermia-, radiation-, and chemo-therapy of these difficult to cure lesions. To gather data regarding this important physiological variable, the perfusion distributions of 26 patients who had either a glioblastoma multiforme or an anaplastic astrocytoma were determined using stable xenon computed tomography (XeCT). Perfusion values were determined for each of the following anatomical regions: low density tumour core, the enhancing active ring of the tumour, the low density peripheral region of edema, an ipsilateral region of normal brain adjacent to the tumour, and a region of remote normal tissue on the contralateral side of the brain. A multiple regression analysis of the logs of the CBF values was used to analyse: (1) the differences in blood perfusion between the anatomical regions; and (2) the association of blood perfusion with various patient and tumour characteristics. Statistically significant differences in perfusion values were found between all of the anatomically outlined regions with the exceptions that the active tumour and edematous regions do not differ significantly from the ipsilateral normal brain tissue. The ipsilateral normal brain tissue adjacent to the tumour was found to have a relative perfusion (relative to the contra-lateral normal brain tissue perfusion) of 0.84, the edematous tissue had a relative perfusion of 0.52, the active tumour 0.78, and the core 0.39. Significant blood flow was present in the low density tumour core, contradicting the frequent assumption that there is zero or minimal blood flow in such regions. Multiple regression analysis was used to look for other variables that might be associated with blood flow after adjusting for the differences between anatomical regions. This analysis found a significant negative correlation between tumour blood-flow and tumour volume. It also estimated that blood flow in GMB tumours was approximately 67% of that in lower grade tumours. Variables that were found not to be significantly correlated with blood flow were: patient sex, multiple lobe involvement, hemisphere involved, treatment status (initial vs recurrent disease), Karnofsky performance status, age and, lobe involved.

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.

Patterns of changes of tumour temperatures during clinical hyperthermia: implications for treatment planning, evaluation and control.

The patterns of changes in tumour temperatures were studied at selected times throughout 104 hyperthermia sessions. Temperature change patterns were analysed in the context of the known patterns of change of the applied power. First, of 69 extracranial treatments analysed, 74% indicated relatively flat temperatures at constant applied power during a major portion of the treatment, thereby indicating that during that time there were no major changes in any of the physical or physiological tissue parameters which contribute to the ability of the tumour tissue to remove energy (Pattern 1). Second, after reaching an initial steady state, approximately 14% of these extracranial treatments showed either steadily decreasing temperatures at constant power, or constant temperatures at steadily increasing applied power, thereby indicating that the tumour's ability to remove energy was steadily increasing in time following the initial steady state (Pattern 2). Finally, after reaching an initial steady state, the remaining 12% of these treatments showed a pronounced decrease in temperature occurring about 10-20 min into the treatment followed by increasing temperatures or levelling off of temperatures at a higher value than the temperature minimum that had occurred, all at constant applied power (Pattern 3). Of 35 brain treatments analysed, 80% followed Pattern 1, 14% followed Pattern 2, and 6% followed Pattern 3. Intratumoral heterogeneity was evident in some cases with approximately 44% of all treatments having at least one individual temperature sensor change in a manner that did not follow the average direction of change when all sensors were combined. For seven patients with permanent probes, the patterns of change presented in the first treatments were also observed during six out of seven of the second treatments. In addition, three out of the five patients who had an evaluable third treatment showed a pattern of change during that third treatment that was similar to the pattern observed in both treatment one and treatment two.

Feasibility of using neural networks to estimate minimum tumour temperature and perfusion values.

We examine the ability of neural networks to estimate the tissue perfusion values present and the minimum temperature in numerically calculated (Pennes, Bioheat Transfer Equation) steady-state hyperthermia temperature fields based on a limited number of measured temperatures within this field A hierarchical system of neural networks consisting of a first layer of pattern recognizing neural networks and a second layer of hypersurface reconstructing neural networks is shown to be capable of estimating these variables within a selected error tolerance. The results indicate that estimating the minimum tumour temperature directly with the system of neural networks may be more effective than using the indirect method of numerically recreating a temperature field with perfusion estimates and then obtaining the minimum tumour temperature from the estimated temperature field. Additional results indicate that if the locations of the measured temperatures within the temperature field are selected appropriately, the hierarchical system of neural networks can tolerate a moderate level of model mismatch. This model mismatch can come from errors in modelling the tumour boundaries, the sensor locations, or the magnitude of the power deposition. This paper is not intended to assess or demonstrate clinical applicability but to be a first step in investigating the feasibility of neural networks for parameter estimation related to hyperthermia studies.

Simulation of bidirectional ultrasound hyperthermia treatments of neck tumours.

The temperature distributions produced in neck tumours by using either a single, scanned transducer (a unidirectional scan) or two separate transducers whose axis are perpendicular (a bidirectional scan) were simulated. The three-dimensional neck model included separate anatomical regions for the normal neck muscle tissue, the tumour, the spinal column and the trachea (no large blood vessels). The effects of variations in the transducer frequency and f number, the tumour size and location, and the normal and tumour blood perfusion rates were studies. The best simulated temperature distributions were produced by bidirectionally scanned, 2 MHz, f number 2.0 ultrasound transducers whose powers were modulated as a function of position. The simulated temperature distributions from such modulated bidirectional scans were significantly better than those of both unidirectional and unmodulated bidirectional scans. The 1-MHz transducers generally produced hot spots at the tissue-spine and/or tissue-trachea interface. The 3-MHz transducers eliminated those deep hot spots but created other hot spots close to the skin surface, and did not adequately heat the deeper regions of the tumour. These results from the simplified computer simulations may be used to guide the construction of improved ultrasound hyperthermia systems for the treatment of neck tumours.