Nonstandard Fourier Pseudospectral Time Domain (PSTD) Schemes for Partial Differential Equations

Treeby, Bradley E.; Wise, Elliott S.; Cox, Ben

doi:10.4208/cicp.oa-2017-0192

Cited by 11 publications

(6 citation statements)

References 12 publications

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“….. Note, higher-order time integration schemes are also possible (e.g., [35,31,36,37,38]). However, these are not discussed here as they do not influence the implementation of boundary conditions, which is the focus of the current work.…”

Section: The Fourier Pstd Methods For the Wave Equationmentioning

confidence: 99%

Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms

Wise,

Jaros,

Cox

et al. 2020

Preprint

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Pseudospectral time domain (PSTD) methods are widely used in many branches of acoustics for the numerical solution of the wave equation, including biomedical ultrasound and seismology. The use of the Fourier collocation spectral method in particular has many computational advantages, including a reduced number of grid points required for accurate simulations. However, the use of a discrete Fourier basis is also inherently restricted to solving problems with periodic boundary conditions. This means that waves exiting one side of the domain reappear on the opposite side. Practically, this is usually overcome by implementing a perfectly matched layer to simulate freefield conditions. However, in some cases, other boundary conditions are required, and these are not straightforward to implement. Here, a family of spectral collocation methods based on the use of a sine or cosine basis is described. These retain the computational advantages of the Fourier collocation method but instead allow homogeneous Dirichlet (sound-soft) and Neumann (sound-hard) boundary conditions to be imposed. The basis function weights are computed numerically using the discrete sine and cosine transforms, which can be implemented using O(N log N ) operations analogous to the fast Fourier transform. The different combinations of discrete symmetry give rise to sixteen possible discrete trigonometric transforms. The properties of these transforms are described, and practical details of how to implement spectral methods using discrete sine and cosine transforms are provided. The technique is then illustrated through the solution of the wave equation in a rectangular domain subject to different combinations of boundary conditions. The extension to boundaries with arbitrary real reflection coefficients or boundaries that are non-reflecting is also demonstrated using the weighted summation of the solutions with Dirichlet and Neumann boundary conditions.

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Section: The Fourier Pstd Methods For the Wave Equationmentioning

confidence: 99%

Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms

Wise,

Jaros,

Cox

et al. 2020

Preprint

Self Cite

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“…In this study, the above parameters were replicated in silico. Thermal simulations were conducted using the k-wave package [37] implementation of the Pennes bioheat equation [38] (Equation (4)):…”

Section: Thermal Simulationmentioning

confidence: 99%

Quantitative prediction of the extent of pelvic tumour ablation by magnetic resonance-guided high intensity focused ultrasound

Lam

Rivens

Giles

et al. 2021

International Journal of Hyperthermia

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Background: Patient suitability for magnetic resonance-guided high intensity focused ultrasound (MRgHIFU) therapy of pelvic tumors is currently assessed by visual estimation of the proportion of tumor that can be reached by the device's focus (coverage). Since it is important to assess whether enough energy reaches the tumor to achieve ablation, a methodology for estimating the proportion of the tumor that can be ablated (treatability) was developed. Predicted treatability was compared against clinically achieved thermal ablation. Methods: MR Dixon sequence images of five patients with recurrent gynecological tumors were acquired during their treatment. Acousto-thermal simulations were performed using k-Wave for three exposure points (the deepest and shallowest reachable focal points within the tumor, identified from tumor coverage analysis, and a point halfway in-between) per patient. Interpolation between the resulting simulated ablated tissue volumes was used to estimate the maximum treatable depth and hence, tumor treatability. Predicted treatability was compared both to predicted tumor coverage and to the clinically treated tumor volume. The intended and simulated volumes and positions of ablated tissues were compared. Results: Predicted treatability was less than coverage by 52% (range: 31-78%) of the tumor volume. Predicted and clinical treatability differed by 9% (range: 1-25%) of tumor volume. Ablated tissue volume and position varied with beam path length through tissue. Conclusion: Tumor coverage overestimated patient suitability for MRgHIFU therapy. Employing patient-specific simulations improved treatability assessment. Patient treatability assessment using simulations is feasible.

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“…The thermal simulations were conducted using the k‐Wave Toolbox. This solves Pennes bioheat transfer equation using a nonstandard pseudospectral method . The model is exact in the case of homogeneous media, and gives high accuracy for low computational cost in the case of heterogeneous material parameters.…”

Section: Treatment Simulationsmentioning

confidence: 99%

“…This solves Pennes bioheat transfer equation using a nonstandard pseudospectral method. 31 The model is exact in the case of homogeneous media, and gives high accuracy for low computational cost in the case of heterogeneous material parameters. The model accounts for heat diffusion, advective heat loss due to tissue perfusion, and heat deposition due to nonlinear ultrasound absorption.…”

Section: C Thermal Simulation Parameters and Executionmentioning

confidence: 99%

Transurethral ultrasound therapy of the prostate in the presence of calcifications: A simulation study

et al. 2018

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Prostatic calcifications limit the reach of therapeutic ultrasound treatment due to reflections and attenuation. The tissue regions behind the calcifications can possibly be treated using longer sonication durations combined with proper transducer element selection. However, caution should be taken with calcifications located close to sensitive organs such as the urethra, bladder neck, or rectal wall.

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Nonstandard Fourier Pseudospectral Time Domain (PSTD) Schemes for Partial Differential Equations

Cited by 11 publications

References 12 publications

Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms

Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms

Quantitative prediction of the extent of pelvic tumour ablation by magnetic resonance-guided high intensity focused ultrasound

Transurethral ultrasound therapy of the prostate in the presence of calcifications: A simulation study

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