Robert J. McGough

NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION

In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Rapid calculations of time-harmonic nearfield pressures produced by rectangular pistons

A rapid method for calculating the nearfield pressure distribution generated by a rectangular piston is derived for time-harmonic excitations. This rapid approach improves the numerical performance relative to the impulse response with an equivalent integral expression that removes the numerical singularities caused by inverse trigonometric functions. The resulting errors are demonstrated in pressure field calculations using the time-harmonic impulse response solution for a rectangular source 5 wavelengths wide by 7.5 wavelengths high. Simulations using this source geometry show that the rapid method eliminates the singularities introduced by the impulse response. The results of pressure field computations are then evaluated in terms of relative errors and computational speeds. The results show that, when the same number of Gauss abscissas are applied to both approaches for time-harmonic pressure field calculations, the rapid method is consistently faster than the impulse response, and the rapid method consistently produces smaller maximum errors than the impulse response. For specified maximum error values of 10% and 1%, the rapid method is 2.6 times faster than the impulse response for pressure field calculations performed on a 61 by 101 point grid. The rapid approach achieves even greater reductions in the computation time for smaller errors and larger grids.

Treatment planning for hyperthermia with ultrasound phased arrays

Treatment planning for ultrasound phased arrays suggests a strategy for hyperthermia therapy which satisfies therapeutic conditions at the target and spares other sensitive anatomical structures. To predict both desirable and harmful interactions between ultrasound and important structures such as the tumor, bones, and air pockets, a hyperthermia treatment planning system has been developed for ultrasound phased arrays. This collection of treatment planning routines consists of geometric and thermal optimization procedures specific to ultrasound phased arrays, where geometric treatment planning, combined with thermal treatment planning and three-dimensional visualization, provides essential information for the optimization of individual patient treatments. A patient image data set for cancer of the prostate, a difficult target situated in the midst of multiple pelvic bone obstructions, illustrates the geometric treatment planning algorithm and other tools for treatment analysis. The results indicate that the analysis of complex three-dimensional relationships between the applicator, anatomical structures, and incident fields provides an important means of predicting treatment limiting conditions, thereby allowing the hyperthermia applicator to electronically adapt to individual patients and specific sites.

Limits on focused ultrasound for deep hyperthermia

It becomes increasingly difficult for ultrasound to noninvasively heat larger tumor volumes at depth without overheating intervening normal tissues. For a given acoustic window, there exists a maximum tumor volume at a given depth which can be treated effectively without causing “hot spots” in the normal tissues, even if “ideal” applicators and power deposition techniques are applied. In this paper, the upper bound on the radii of treatable spherical volumes under given physical, anatomic, and physiological constraints is determined. Parametric studies are performed to determine relationships between the upper bound radius and constraints such as acoustic window size, tumor depth, attenuation rate, the tolerable temperature in normal tissues, and blood perfusion rate

Evaluation of the angular spectrum approach for simulations of near-field pressures

The implementation of the angular spectrum approach based on the two-dimensional fast Fourier transform is evaluated for near-field pressure simulations of square ultrasound transducers, where the three-dimensional pressure field is calculated from the normal velocity distribution on the transducer surface. The pressure field is propagated in the spatial frequency domain with the spatial propagator or the spectral propagator. The spatial propagator yields accurate results in the central portion of the computational grid while significant errors are produced near the edge due to the finite extent of the window applied to the spatial propagator. Likewise, the spectral propagator is inherently undersampled in the spatial frequency domain, and this causes high frequency errors in the computed pressure field. This aliasing problem is alleviated with angular restriction. The results show that, in nonattenuating media, the spatial propagator achieves smaller errors than the spectral propagator after the region of interest is truncated to exclude the windowing error. For pressure calculations in attenuating media or with apodized pistons as sources, the spatial and spectral propagator achieve similar accuracies. In all simulations, the angular spectrum calculations with the spatial propagator take more time than calculations with the spectral propagator.

An efficient grid sectoring method for calculations of the near-field pressure generated by a circular piston.

An analytical expression is derived for time-harmonic calculations of the near-field pressure produced by a circular piston. The near-field pressure is described by an efficient integral that eliminates redundant calculations and subtracts the singularity, which in turn reduces the computation time and the peak numerical error. The resulting single integral expression is then combined with an approach that divides the computational grid into sectors that are separated by straight lines. The integral is computed with Gauss quadrature in each sector, and the number of Gauss abscissas in each sector is determined by a linear mapping function that prevents large errors from occurring in the axial region. By dividing the near-field region into 10 sectors, the raw computation time is reduced by nearly a factor of 2 for each expression evaluated in this grid. The grid sectoring approach is most effective when the computation time is reduced without increasing the peak error, and this is consistently accomplished with the efficient integral formulation. Of the four single integral expressions evaluated with grid sectoring, the efficient formulation that eliminates redundant calculations and subtracts the singularity demonstrates the smallest computation time for a specified value of the maximum error.

An RF phased array applicator designed for hyperthermia breast cancer treatments

An RF phased array applicator has been constructed for hyperthermia treatments in the intact breast. This RF phased array consists of four antennas mounted on a Lexan water tank, and geometric focusing is employed so that each antenna points in the direction of the intended target. The operating frequency for this phased array is 140 MHz. The RF array has been characterized both by electric field measurements in a water tank and by electric field simulations using the finite-element method. The finite-element simulations are performed with HFSS software, where the mesh defined for finite-element calculations includes the geometry of the tank enclosure and four end-loaded dipole antennas. The material properties of the water tank enclosure and the antennas are also included in each simulation. The results of the finite-element simulations are compared to the measured values for this configuration, and the results, which include the effects of amplitude shading and phase shifting, show that the electric field predicted by finite-element simulations is similar to the measured field. Simulations also show that the contributions from standing waves are significant, which is consistent with measurement results. Simulated electric field and bio-heat transfer results are also computed within a simple 3D breast model. Temperature simulations show that, although peak temperatures are generated outside the simulated tumour target, this RF phased array applicator is an effective device for regional hyperthermia in the intact breast.

Fractal ladder models and power law wave equations.

The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100 MHz. To describe this power law behavior in soft tissue, a hierarchical fractal network model is proposed. The viscoelastic and self-similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite-ladder topology involving alternating springs and dashpots. In the low-frequency limit, this ladder network yields a stress-strain constitutive equation with a time-fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, a fractional partial differential equation that describes power law attenuation is derived. The resulting attenuation coefficient is a power law with exponent ranging between 1 and 2, while the phase velocity is in agreement with the Kramers-Kronig relations. The fractal ladder model is compared to published attenuation coefficient data, thus providing equivalent lumped parameters.

Optimal simulations of ultrasonic fields produced by large thermal therapy arrays using the angular spectrum approach

The angular spectrum approach is evaluated for the simulation of focused ultrasound fields produced by large thermal therapy arrays. For an input pressure or normal particle velocity distribution in a plane, the angular spectrum approach rapidly computes the output pressure field in a three dimensional volume. To determine the optimal combination of simulation parameters for angular spectrum calculations, the effect of the size, location, and the numerical accuracy of the input plane on the computed output pressure is evaluated. Simulation results demonstrate that angular spectrum calculations performed with an input pressure plane are more accurate than calculations with an input velocity plane. Results also indicate that when the input pressure plane is slightly larger than the array aperture and is located approximately one wavelength from the array, angular spectrum simulations have very small numerical errors for two dimensional planar arrays. Furthermore, the root mean squared error from angular spectrum simulations asymptotically approaches a nonzero lower limit as the error in the input plane decreases. Overall, the angular spectrum approach is an accurate and robust method for thermal therapy simulations of large ultrasound phased arrays when the input pressure plane is computed with the fast nearfield method and an optimal combination of input parameters.

Direct computation of ultrasound phased-array driving signals from a specified temperature distribution for hyperthermia

A method that obtains ultrasound hyperthermia applicator phased-array element driving signals from a desired temperature distribution is presented. The approach combines a technique which computes array element driving signals from focal point locations and intensities with a technique which calculates focal point locations and power deposition values from temperature requirements. Temperature specifications appear as upper and lower bounds within the tumor volume, and a focal point placement algorithm chooses focal patterns capable of achieving the temperature range objective. The linear algebraic structure of the method allows rapid calculation of both the phased-array driving signals and an approximate temperature field response. Computer simulations verify the method with a spherical section array (SSA) for a variety of temperature specifications and blood perfusion values. The scheme applies to any phased-array geometry.<<ETX>>