In this paper, we investigate the applicability of the Bayesian approximation error approach to compensate for the discrepancy of the diffusion approximation in diffuse optical tomography close to the light sources and in weakly scattering subdomains. While the approximation error approach has earlier been shown to be a feasible approach to compensating for discretization errors, uncertain boundary data and geometry, the ability of the approach to recover from using a qualitatively incorrect physical model has not been contested. In the case of weakly scattering subdomains and close to sources, the radiative transfer equation is commonly considered to be the most accurate model for light scattering in turbid media. In this paper, we construct the approximation error statistics based on predictions of the radiative transfer and diffusion models. In addition, we investigate the combined approximation errors due to using the diffusion approximation and using a very lowdimensional approximation in the forward problem. We show that recovery is feasible in the sense that with the approximation error model the reconstructions with a low-dimensional diffusion approximation are essentially as good as with using a very high-dimensional radiative transfer model.
Measurements of temperature elevations induced by sonications in a single intact cadaver skull filled with soft-tissue mimicking phantom material were performed using magnetic resonance thermometry. The sonications were done using a clinical transcranial ultrasound therapy device operating at 230 kHz and the measurements were compared with simulations done using a model incorporating both the longitudinal and shear wave propagation. Both the measurements and simulations showed that in some situations the temperature increase could be higher in the phantom material adjacent to the skull-base than at the focus, which could lead to undesired soft-tissue damage in treatment situations. On average the measurements of the sonicated locations, as well as the comparative simulations, showed 32 ± 64% and 49 ± 32% higher temperature elevations adjacent to the skull-base than at the focus, respectively. The simulation model was used to extend the measurements by simulating multiple sonications of brain tissue in five different skulls with and without correcting the aberrations caused by the skull on the ultrasound. Without aberration correction the closest sonications to the skulls that were treatable in any brain location without undesired tissue damage were at a distance of 19.1 ± 2.6 mm. None of the sonications beyond a distance of 41.2 ± 5.3 mm were found to cause undesired tissue damage. When using the aberration correction closest treatable, safe distances for sonications were found to be 16.0 ± 1.6 and 38.8 ± 3.8 mm, respectively. New active cooling of the skull-base through the nasal cavities was introduced and the treatment area was investigated. The closest treatable distance without aberration correction reduced to 17.4 ± 1.9 mm with the new cooling method. All sonications beyond a distance of 39.7 ± 6.6 mm were found treatable. With the aberration correction no difference in the closest treatable or the safety distance was found in comparison to sonications without nasal cavity cooling. To counteract undesired skull-base heating a new anti-focus within solid media was developed along with a new regularized phasing method. Mathematical bases for both the methods and simulations utilizing them were presented. It was found that utilizing the anti-focus in solid media and regularized phasing, the fraction of temperature increase of the brain tissue at the focus and the peak temperature increase adjacent to the skull-base can be increased from 1.00 to 1.95. This improves the efficiency of the sonication by reducing the energy transfer to the skull-base.
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