Blood perfusion and thermal conduction effects in Gaussian beam, minimum time single-pulse thermal therapies

An Optimized Control Approach for HIFU Tissue Ablation Using PDE Constrained Optimization Method

High-intensity focused ultrasound (HIFU) is a widely used technique capable of providing noninvasive heating and ablation for a wide range of applications. However, the major challenges lie in the determination of the position and the amount of heat deposition over a target area. In order to assure that the thermal area is confined to tumor locations, an optimization method should be employed. Sequential quadratic programming and steepest gradient method with closed-form solution have been previously used to solve this kind of problem. However, these methods are complex and computationally inefficient. The goal of this article is to solve and control the solution of inverse problems with partial differential equation (PDE) constraints. Therefore, a distinguishing challenge of this technique is the handling of large numbers of optimization variables in combination with the complexities of discretized PDEs. In our method, the objective function is formulated as the square difference between the actual thermal dose and the desired one. At each iteration of the optimization procedure, we need to develop and solve the variation problem, the adjoint problem, and the gradient of the objective function. The analytical formula for the gradient is derived and calculated based on the solution of the adjoint problem. Several factors have been taken into consideration to demonstrate the robustness and efficiency of the proposed algorithm. The simulation results for all cases indicate the robustness and the computational efficiency of our proposed method compared to the steepest gradient descent method with the closed-form solution.

Non-invasive Measurement of Thermal Diffusivity Using High-Intensity Focused Ultrasound and Through-Transmission Ultrasonic Imaging

Thermal diffusivity at the site ablated by high-intensity focused ultrasound (HIFU) plays an important role in the final therapeutic outcome, as it influences the temperature's spatial and temporal distribution. Moreover, as tissue thermal diffusivity is different in tumors as compared with normal tissue, it could also potentially be used as a new source of imaging contrast. The aim of this study was to examine the feasibility of combining through-transmission ultrasonic imaging and HIFU to estimate thermal diffusivity non-invasively. The concept was initially evaluated using a computer simulation. Then it was experimentally tested on phantoms made of agar and ex vivo porcine fat. A computerized imaging system combined with a HIFU system was used to heat the phantoms to temperatures below 42°C to avoid irreversible damage. Through-transmission scanning provided the time-of-flight values in a region of interest during its cooling process. The time-of-flight values were consequently converted into mean values of speed of sound. Using the speed-of-sound profiles along with the developed model, we estimated the changes in temperature profiles over time. These changes in temperature profiles were then used to calculate the corresponding thermal diffusivity of the studied specimen. Thermal diffusivity for porcine fat was found to be lower by one order of magnitude than that obtained for agar (0.313×10(-7)m(2)/s vs. 4.83×10(-7)m(2)/s, respectively, p < 0.041). The fact that there is a substantial difference between agar and fat implies that non-invasive all-ultrasound thermal diffusivity mapping is feasible. The suggested method may particularly be suitable for breast scanning.

Effects of different parameters in the fast scanning method for HIFU treatment

PURPOSE

High-intensity focused ultrasound is a promising method for the noninvasive treatment of benign and malignant tumors. This study analyzes the effects of scanning path, applied power, and geometric characteristics of the transducer on ablation using fast scanning method, a new scanning method that uses high-intensity focused ultrasound at different blood perfusion levels.

METHODS

Two transducers, six scanning paths, and three focal patterns were used to examine the ablation results of the fast scanning method using power densities from 1.35 × 10(7) W∕m(3) to 4.5 × 10(7) W∕m(3) and blood perfusion rates from 2 × 10(-3) ml∕ml∕s to 16 × 10(-3) ml∕ml∕s. The Pennes equation was solved using the finite-difference time-domain method to simulate the heating procedure.

RESULTS

Based on the results of the fast-scanning method, the different scanning paths exhibited small effect on the total treatment time supported by both simulation and least-square fit. Similar-sized lesions can result from the five different repeated paths, whereas a random path may lead to relative large fluctuations in ablation volume because of asymmetry of the lesions. Higher power levels increase the lesion volume and decrease the treatment time required for ablating a target area using the fast scanning method, whereas increased blood perfusion has the opposite effect on ablation volume and treatment time. A symmetric lesion can be produced through fast scanning method using a 65-element and a 90-element transducer. However, lesion production using the same operation scheme differs between the two transducers.

CONCLUSIONS

Unlike traditional scanning methods, fast scanning method produces a planned lesion regardless of scanning path, as long as the path consists of repeated subsequences. This attribute makes fast scanning method an easy-operation scheme that produces relatively large symmetric lesions in homogeneous tissues. Applied power is the most important factor; however, high blood perfusion levels can limit or even hinder the full ablation of the target area. Therefore, tissue perfusion and transducer type should be given special attention to ensure the success and safety of ablation treatment.

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.

In vivo characterization of tissue thermal properties of the kidney during local hyperthermia induced by MR-guided high-intensity focused ultrasound

The purpose of this study was to evaluate quantitatively in vivo the tissue thermal properties during high-intensity focused ultrasound (HIFU) heating. For this purpose, a total of 52 localized sonications were performed in the kidneys of six pigs with HIFU monitored in real time by volumetric MR thermometry. The kidney perfusion was modified by modulation of the flow in the aorta by insertion of an inflatable angioplasty balloon. The resulting temperature data were analyzed using the bio-heat transfer model in order to validate the model under in vivo conditions and to estimate quantitatively the absorption (α), thermal diffusivity (D) and perfusion (w(b)) of renal tissue. An excellent correspondence was observed between the bio-heat transfer model and the experimental data. The absorption and thermal diffusivity were independent of the flow, with mean values (± standard deviation) of 20.7 ± 5.1 mm(3) K J(-1) and 0.23 ± 0.11 mm(2) s(-1), respectively, whereas the perfusion decreased significantly by 84% (p < 0.01) with arterial flow (mean values of w(b) of 0.06 ± 0.02 and 0.008 ± 0.007 mL(-1) mL s(-1)), as predicted by the model. The quantitative analysis of the volumetric temperature distribution during nondestructive HIFU sonication allows the determination of the thermal parameters, and may therefore improve the quality of the planning of noninvasive therapy with MR-guided HIFU.

Mathematical formulation and analysis of the nonlinear system reconstruction of the online image-guided adaptive control of hyperthermia

PURPOSE

A nonlinear system reconstruction can theoretically provide timely system reconstruction when designing a real-time image-guided adaptive control for multisource heating for hyperthermia. This clinical need motivates an analysis of the essential mathematical characteristics and constraints of such an approach.

METHODS

The implicit function theorem (IFT), the Karush-Kuhn-Tucker (KKT) necessary condition of optimality, and the Tikhonov-Phillips regularization (TPR) were used to analyze and determine the requirements of the optimal system reconstruction. Two mutually exclusive generic approaches were analyzed to reconstruct the physical system: The traditional full reconstruction and the recently suggested partial reconstruction. Rigorous mathematical analysis based on IFT, KKT, and TPR was provided for all four possible nonlinear reconstructions: (1) Nonlinear noiseless full reconstruction, (2) nonlinear noisy full reconstruction, (3) nonlinear noiseless partial reconstruction, and (4) nonlinear noisy partial reconstruction, when a class of nonlinear formulations of system reconstruction is employed.

RESULTS

Effective numerical algorithms for solving each of the aforementioned four nonlinear reconstructions were introduced and formal derivations and analyses were provided. The analyses revealed the necessity of adding regularization when partial reconstruction is used. Regularization provides the theoretical support for one to uniquely reconstruct the optimal system. It also helps alleviate the negative influences of unavoidable measurement noise. Both theoretical analysis and numerical examples showed the importance of having a good initial guess for accomplishing nonlinear system reconstruction.

CONCLUSIONS

Regularization is mandatory for partial reconstruction to make it well posed. The Tikhonov-Phillips regularized Gauss-Newton algorithm has nice theoretical performance for partial reconstruction of systems with and without noise. The Levenberg-Marquardt algorithm is a more robust algorithmic option compared to the Gauss-Newton algorithm for nonlinear full reconstruction. A severe limitation of nonlinear reconstruction is the time consuming calculations required for the derivatives of temperatures to unknowns. Developing a method of model reduction or implementing a parallel algorithm can resolve this. The results provided herein are applicable to hyperthermia with blood perfusion nonlinearly depending on temperature and in the presence of thermally significant blood vessels.

Effective learning strategies for real-time image-guided adaptive control of multiple-source hyperthermia applicators

PURPOSE

This paper investigates overall theoretical requirements for reducing the times required for the iterative learning of a real-time image-guided adaptive control routine for multiple-source heat applicators, as used in hyperthermia and thermal ablative therapy for cancer.

METHODS

Methods for partial reconstruction of the physical system with and without model reduction to find solutions within a clinically practical timeframe were analyzed. A mathematical analysis based on the Fredholm alternative theorem (FAT) was used to compactly analyze the existence and uniqueness of the optimal heating vector under two fundamental situations: (1) noiseless partial reconstruction and (2) noisy partial reconstruction. These results were coupled with a method for further acceleration of the solution using virtual source (VS) model reduction. The matrix approximation theorem (MAT) was used to choose the optimal vectors spanning the reduced-order subspace to reduce the time for system reconstruction and to determine the associated approximation error. Numerical simulations of the adaptive control of hyperthermia using VS were also performed to test the predictions derived from the theoretical analysis. A thigh sarcoma patient model surrounded by a ten-antenna phased-array applicator was retained for this purpose. The impacts of the convective cooling from blood flow and the presence of sudden increase of perfusion in muscle and tumor were also simulated.

RESULTS

By FAT, partial system reconstruction directly conducted in the full space of the physical variables such as phases and magnitudes of the heat sources cannot guarantee reconstructing the optimal system to determine the global optimal setting of the heat sources. A remedy for this limitation is to conduct the partial reconstruction within a reduced-order subspace spanned by the first few maximum eigenvectors of the true system matrix. By MAT, this VS subspace is the optimal one when the goal is to maximize the average tumor temperature. When more than 6 sources present, the steps required for a nonlinear learning scheme is theoretically fewer than that of a linear one, however, finite number of iterative corrections is necessary for a single learning step of a nonlinear algorithm. Thus, the actual computational workload for a nonlinear algorithm is not necessarily less than that required by a linear algorithm.

CONCLUSIONS

Based on the analysis presented herein, obtaining a unique global optimal heating vector for a multiple-source applicator within the constraints of real-time clinical hyperthermia treatments and thermal ablative therapies appears attainable using partial reconstruction with minimum norm least-squares method with supplemental equations. One way to supplement equations is the inclusion of a method of model reduction.

Real-time MRI-guided hyperthermia treatment using a fast adaptive algorithm

Magnetic resonance (MR) imaging is promising for monitoring and guiding hyperthermia treatments. The goal of this work is to investigate the stability of an algorithm for online MR thermal image guided steering and focusing of heat into the target volume. The control platform comprised a four-antenna mini-annular phased array (MAPA) applicator operating at 140 MHz (used for extremity sarcoma heating) and a GE Signa Excite 1.5 T MR system, both of which were driven by a control workstation. MR proton resonance frequency shift images acquired during heating were used to iteratively update a model of the heated object, starting with an initial finite element computed model estimate. At each iterative step, the current model was used to compute a focusing vector, which was then used to drive the next iteration, until convergence. Perturbation of the driving vector was used to prevent the process from stalling away from the desired focus. Experimental validation of the performance of the automatic treatment platform was conducted with two cylindrical phantom studies, one homogeneous and one muscle equivalent with tumor tissue (conductivity 50% higher) inserted, with initial focal spots being intentionally rotated 90 degrees and 50 degrees away from the desired focus, mimicking initial setup errors in applicator rotation. The integrated MR-HT treatment platform steered the focus of heating into the desired target volume in two quite different phantom tissue loads which model expected patient treatment configurations. For the homogeneous phantom test where the target was intentionally offset by 90 degrees rotation of the applicator, convergence to the proper phase focus in the target occurred after 16 iterations of the algorithm. For the more realistic test with a muscle equivalent phantom with tumor inserted with 50 degrees applicator displacement, only two iterations were necessary to steer the focus into the tumor target. Convergence improved the heating efficacy (the ratio of integral temperature in the tumor to integral temperature in normal tissue) by up to six-fold, compared to the first iteration. The integrated MR-HT treatment algorithm successfully steered the focus of heating into the desired target volume for both the simple homogeneous and the more challenging muscle equivalent phantom with tumor insert models of human extremity sarcomas after 16 and 2 iterations, correspondingly. The adaptive method for MR thermal image guided focal steering shows promise when tested in phantom experiments on a four-antenna phased array applicator.

Control time reduction using virtual source projection for treating a leg sarcoma with nonlinear perfusion

PURPOSE

Blood perfusion is a well-known factor that complicates accurate control of heating during hyperthermia treatments of cancer. Since blood perfusion varies as a function of time, temperature and location, determination of appropriate power deposition pattern from multiple antenna array Hyperthermia systems and heterogeneous tissues is a difficult control problem. Therefore, we investigate the applicability of a real-time eigenvalue model reduction (virtual source - VS) reduced-order controller for hyperthermic treatments of tissue with nonlinearly varying perfusion.

METHODS

We impose a piecewise linear approximation to a set of heat pulses, each consisting of a 1-min heat-up, followed by a 2-min cool-down. The controller is designed for feedback from magnetic resonance temperature images (MRTI) obtained after each iteration of heat pulses to adjust the projected optimal setting of antenna phase and magnitude for selective tumor heating. Simulated temperature patterns with additive Gaussian noise with a standard deviation of 1.0°C and zero mean were used as a surrogate for MRTI. Robustness tests were conducted numerically for a patient's right leg placed at the middle of a water bolus surrounded by a 10-antenna applicator driven at 150 MHz. Robustness tests included added discrepancies in perfusion, electrical and thermal properties, and patient model simplifications.

RESULTS

The controller improved selective tumor heating after an average of 4-9 iterative adjustments of power and phase, and fulfilled satisfactory therapeutic outcomes with approximately 75% of tumor volumes heated to temperatures >43°C while maintaining about 93% of healthy tissue volume < 41°C. Adequate sarcoma heating was realized by using only 2 to 3 VSs rather than a much larger number of control signals for all 10 antennas, which reduced the convergence time to only 4 to 9% of the original value.

CONCLUSIONS

Using a piecewise linear approximation to a set of heat pulses in a VS reduced-order controller, the proposed algorithm greatly improves the efficiency of hyperthermic treatment of leg sarcomas while accommodating practical nonlinear variation of tissue properties such as perfusion.

Noninvasive measurement of local thermal diffusivity using backscattered ultrasound and focused ultrasound heating

Previously, noninvasive methods of estimating local tissue thermal and acoustic properties using backscattered ultrasound have been proposed in the literature. In this article, a noninvasive method of estimating local thermal diffusivity in situ during focused ultrasound heating using beamformed acoustic backscatter data and applying novel signal processing techniques is developed. A high intensity focused ultrasound (HIFU) transducer operating at subablative intensities is employed to create a brief local temperature rise of no more than 10 degrees C. Beamformed radio-frequency (RF) data are collected during heating and cooling using a clinical ultrasound scanner. Measurements of the time-varying "acoustic strain", that is, spatiotemporal variations in the RF echo shifts induced by the temperature related sound speed changes, are related to a solution of the heat transfer equation to estimate the thermal diffusivity in the heated zone. Numerical simulations and experiments performed in vitro in tissue mimicking phantoms and excised turkey breast muscle tissue demonstrate agreement between the ultrasound derived thermal diffusivity estimates and independent estimates made by a traditional hot-wire technique. The new noninvasive ultrasonic method has potential applications in thermal therapy planning and monitoring, physiological monitoring and as a means of noninvasive tissue characterization.

Fast temperature optimization of multi-source hyperthermia applicators with reduced-order modeling of 'virtual sources'

The goal of this work is to build the foundation for facilitating real-time magnetic resonance image guided patient treatment for heating systems with a large number of physical sources (e.g. antennas). Achieving this goal requires knowledge of how the temperature distribution will be affected by changing each source individually, which requires time expenditure on the order of the square of the number of sources. To reduce computation time, we propose a model reduction approach that combines a smaller number of predefined source configurations (fewer than the number of actual sources) that are most likely to heat tumor. The source configurations consist of magnitude and phase source excitation values for each actual source and may be computed from a CT scan based plan or a simplified generic model of the corresponding patient anatomy. Each pre-calculated source configuration is considered a 'virtual source'. We assume that the actual best source settings can be represented effectively as weighted combinations of the virtual sources. In the context of optimization, each source configuration is treated equivalently to one physical source. This model reduction approach is tested on a patient upper-leg tumor model (with and without temperature-dependent perfusion), heated using a 140 MHz ten-antenna cylindrical mini-annular phased array. Numerical simulations demonstrate that using only a few pre-defined source configurations can achieve temperature distributions that are comparable to those from full optimizations using all physical sources. The method yields close to optimal temperature distributions when using source configurations determined from a simplified model of the tumor, even when tumor position is erroneously assumed to be approximately 2.0 cm away from the actual position as often happens in practical clinical application of pre-treatment planning. The method also appears to be robust under conditions of changing, nonlinear, temperature-dependent perfusion. The proposed approach of using virtual sources reduces the number of variables that must be optimized to achieve a tumor-focused temperature distribution, thereby reducing the calculation time required in real-time control applications to about 1/3 to 1/4 of that required for full optimization.

Online feedback focusing algorithm for hyperthermia cancer treatment

PURPOSE

Magnetic resonance (MR) imaging is increasingly being utilized to visualize the 3D temperature distribution in patients during treatment with hyperthermia or thermal ablation therapy. The goal of this work is to lay the foundation for improving the localization of heat in tumors with an online focusing algorithm that uses MR images as feedback to iteratively steer and focus heat into the target.

METHODS

The algorithm iteratively updates the model that quantifies the relationship between the source (antenna) settings and resulting tissue temperature distribution. At each step in the iterative process, optimal settings of power and relative phase of each antenna are computed to maximize averaged tumor temperature in the model. The MR-measured thermal distribution is then used to update/correct the model. This iterative procedure is repeated until convergence, i.e. until the model prediction and MR thermal image are in agreement. A human thigh tumor model heated in a 140 MHz four-antenna cylindrical mini-annular phased array is used for numerical validation of the proposed algorithm. Numerically simulated temperatures are used during the iterative process as surrogates for MR thermal images. Gaussian white noise with a standard deviation of 0.3 degrees C and zero mean is added to simulate MRI measurement uncertainty. The algorithm is validated for cases where the source settings for the first iteration are based on erroneous models: (1) tissue property variability, (2) patient position mismatch, (3) a simple idealized patient model built from CT-based actual geometry, and (4) antenna excitation uncertainty due to load dependent impedance mismatch and antenna cross-coupling. Choices of starting heating vector are also validated.

RESULTS

The algorithm successfully steers and focuses a tumor when there is no antenna excitation uncertainty. Temperature is raised to > or = 43 degrees C for more than about 90% of tumor volume, accompanied by less than about 20% of normal tissue volume being raised to a temperature > or = 41 degrees C. However, when there is antenna excitation uncertainty, about 40% to 80% of normal tissue volume is raised to a temperature > or = 41 degrees C. No significant tumor heating improvement is observed in all simulations after about 25 iteration steps.

CONCLUSIONS

A feedback control algorithm is presented and shown to be successful in iteratively improving the focus of tissue heating within a four-antenna cylindrical phased array hyperthermia applicator. This algorithm appears to be robust in the presence of errors in assumed tissue properties, including realistic deviations of tissue properties and patient position in applicator. Only moderate robustness was achieved in the presence of misaligned applicator/tumor positioning and antenna excitation errors resulting from load mismatch or antenna cross coupling.