Biomedical Engineering Reference
In-Depth Information
6
Numerical Modeling for Simulation
and Treatment Planning of
Thermal Therapy: Ultrasound
6.1 Introduction ...............................................................................................................................95
6.2 Models of Ultrasound Propagation ........................................................................................95
Rayleigh-Sommerfeld Integral • Rectangular Radiator Method • Fast Near-Field
Method • Angular Spectrum Approach • Nonlinear Ultrasound Propagation • Software
Programs • Intensity and Power Calculations
6.3 Thermal Modeling and Treatment Planning ......................................................................105
Bioheat Transfer Model • Thermal Dose Calculations • Thermal Therapy Planning
6.4 Summary ...................................................................................................................................116
References .............................................................................................................................................116
Robert J. McGough
Michigan State University
6.1 Introduction
Rayleigh-Sommerfeld integral or with expressions that are
closely related to the Rayleigh-Sommerfeld integral, where
each approach has an associated numerical accuracy, computa-
tion time, and algorithmic complexity. These numerical mod-
els are applicable to the three transducer geometries that are
most often encountered in thermal therapy, namely the circu-
lar piston, the rectangular piston, and the spherical shell. Other
pressure simulations include nonlinear propagation effects,
especially when higher intensities are generated by a focused
ultrasound transducer or a large aperture phased array. Some
of these numerical models for linear and nonlinear ultrasound
propagation are implemented in publicly available software.
After the pressure is computed with one of these numerical
models, power depositions are also calculated for subsequent
bioheat transfer simulations.
Pressure fields generated by single and multiple element transducers
are routinely simulated during the initial design of an ultrasound
applicator and also throughout the subsequent characterization
and optimization of power depositions, temperature distributions,
and thermal doses produced by these thermal therapy applicators.
Numerical models describe the diffraction of ultrasound produced
by single transducers, fixed-phase multiple transducer configura-
tions, and ultrasound phased arrays. Nonlinear effects are also
incorporated into some of these numerical models. The resulting
power deposition then provides the input to the bioheat transfer
equation. Temperatures are computed for most applications, and
thermal doses are also calculated, especially for simulations of abla-
tion therapy. In more advanced models, the effects of intervening
tissue heating are considered, beamforming algorithms are evalu-
ated, and more complicated issues such as patient anatomy and tis-
sue inhomogeneities are also included. Patient treatment planning
then combines several of these models in an effort to optimize the
temperature distribution or thermal dose in the tumor while spar-
ing sensitive normal tissues.
6.2.1 rayleigh-Sommerfeld Integral
The Rayleigh-Sommerfeld integral is the most common model
for linear ultrasound propagation in simulations of thermal
therapy. The numerical formulas derived from the Rayleigh-
Sommerfeld integral superpose contributions from a point
source surrounded by a rigid baffle radiating into an infinite
half-space. The frequency-domain Green's function describing
the contribution from each point source is
e
−
jkR
/(2π
R
), where
the 2π in the denominator indicates a baffled point source
radiating into an infinite half-space (as opposed to 4π in the
denominator, which indicates an unbaffled source radiating in
6.2 Models of Ultrasound propagation
Simulated pressures generated by ultrasound applicators
are often computed with a linear propagation model. These
numerical models capture the effects of diffraction in the
near-field region. Pressures are typically calculated with the
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