Endocavitary thermal therapy by MRI-guided phased-array contact ultrasound: Experimental and numerical studies on the multi-input single-output PID temperature controller’s convergence and stability
Illustration of the endocavitary ultrasound device. (a) Diagram of the active surface which is a 64-element cylindrical phased array. (b) Picture of the device active head. (c) MR image of the active head equipped with the cooling balloon (shown FOV is , using the T2w-TSE sequence of Table I). (d) Thermal pattern in the longitudinal plane shown at different time points for an elementary sonication (quasiplanar beam mode). The same linear color scale was used from to temperature elevation. FOV is for each map.
Numerical simulation in 3D of the acoustic field. (a) Symmetries are exploited to reduce the computing time (mirror plane and translation along ). Sampling step of the observed volume was half wavelength (isotropic grid). Complex Rayleigh–Sommerfeld integral had to be explicitly computed for the source surface corresponding only to the red band (4-element width and half-lambda height). [(b) and (c)] Magnitude of the acoustic pressure generated by a set of eight adjacent transducers, shown in the midplane of the phased-array, for quasiplanar wave (b) and, respectively, convergent beam (c) (aperture ratio in the circular dimension equal to 0.37). The tissue attenuation was considered equal to 5.4 Np/m MHz. The two maps are normalized to the same maximum magnitude, which was obtained for the convergent beam. FOV is 53.8 mm. [(d) and (e)] −3 dB isosurface of acoustic intensity. Same parameters were used as for (b) and (c). Tick is 5 mm on each axis.
Illustration of the thermal pattern generated by the ultrasound applicator in the transverse plane. FOV is for each image. The first row corresponds to a quasiplanar acoustic beam (40 W electric power), while 1D convergent focusing has been used for the second row (geometrical focus at 20 mm distance from the cylindrical surface, 24 W electric power). In both cases, the sonication duration was 90 s. Images (a), (b), (c), (e), (f), and (g) are temperature maps, with linear color scale, taken at 50%, 75%, and 100%, respectively, of the sonication time base. Images (d) and (h) are cumulated thermal dose maps at the end of sonications (logarithmic color scale was used with 0 dB corresponding to the lethal threshold of 240 CEM43). Note that the 10 mm thick nonabsorbent water layer traversed by the beam before reaching the tissue.
Illustration of the device capability for fast electronic rotation. FOV is square for each image. (a) Diagram of the sonication paradigm with interleaved switching of two individual beams (eight adjacent transducers per beam), asynchronous with respect to MR data acquisition. The duty cycle was 50% per beam, with unbalanced power level. (b), (c), and (d) are temperature mapping frames obtained during two-beam sonication at 45° relative angle, as recorded at half duration, full duration, and 30 s postsonication, respectively. (e), (f), and (g) are cumulated thermal dose maps for two-beam sonication with relative beam angles of 22.5°, 45° and 78.5°, respectively. A logarithmic color scale was used with 0 dB corresponding to the lethal threshold of 240 CEM43. Note the reflection of the quasiplanar beam on the cylinder wall. The 80 mm diameter holder was used in this experiment.
Compartments of the 2D thermal model, segmented on T2w-TSE transverse image. The central circle indicates the applicator of therapeutic ultrasound (UsTx). The first compartment (C1) is the tip-cooling balloon with circulating degassed water. The temperature is considered invariant inside C1, i.e., a constant room-temperature reservoir. The second compartment (C2) is the biological tissue that is ultrasound-absorbent and heat-conductive medium. The last compartment (C3) is stagnant degassed saline serum, at room temperature, which fills the sample holder. Note that C3 was only added for the in vitro configuration without direct correspondence in vivo. The binary mask in Eqs. (2) and (5) is set to 1 over C2 and set to 0 over C1 (and also over C3, if considered).
Temperature dependence of nonlinear parameters simulating in vivo phenomena. Dynamic changes are considered for the heat deposition rate (as ultrasound absorption increases with temperature above ) and for the perfusion cooling rate of Eq. (6). Plots correspond to the normalized parameters with respect to the initial situation at baseline temperature.
(a) Typical performance of the temperature controller with parameters correctly adjusted (physical values of and , respectively); ex vivo experiment. Eight adjacent transducers were operated with one-dimensional convergent focusing of the beam . The target temperature elevation at the control point (8 mm deep in the tissue) was set here arbitrarily to . The experimental temperature (○ symbols) and the automatically adjusted power (red × symbols) are plotted against time. (b) The same experiment has been simulated using Eq. (6) when considering static parameter (ultrasound attenuation independent of temperature), , no perfusion (ex vivo), and no measurement noise (theoretical performance). Consider that a static attenuation is justified here, as the initial absolute baseline is the room temperature.
Results from 15 ex vivo experiments with active temperature control are shown, where the values of the -couple were deliberately changed. Except for the modification of these parameters into the automatic interface, the experimental setup was identical in all cases, and at least 30 min waiting time was allowed from one heating procedure to the next one. The effective diffusivity parameter is varied along the horizontal axis, while the ultrasound absorption parameter is varied along the vertical axis (values are given relative to the correct ones). The average value (AV) and the standard deviation (SD) of the temperature for the flattop target region are indicated on each plot.
Evolution of the integral term [i.e., the last term of Eq. (4)] during the HICU sonication, shown for five different combinations chosen from Fig. 8. When couple is set to true values, i.e., (100%, 100%), the integral term is close to 0. When the -couple is set to an erroneous combination, but still within the convergence domain, the integral term initially drifts to a maximum (positive or negative) value and further attenuates. In the extreme case of very large errors of the couple , the feedback loop undergoes an unstable regime.
Result of numerical simulations considering a static-tuning PID controller in situation of nonlinear thermal response of tissue. Dynamic changes in the heat deposition rate and blood perfusion term were considered according to Fig. 6. The left column is the theoretical performance without measurement noise, while white noise was added to the simulated temperature for the right column. Temperature elevation at control location (, , and ) is indicated with black symbols and dynamically adjusted power is indicated with red × symbols. Tuning of the controller’s parameters is indicated on each plot as ratios (%).
Acquisition parameters for MR sequences, as used in this study for device positioning (T2w-TSE) and for fast MR thermometry (PRFS-EPI).
Top-down priority of the parallel threads running under the real-time application kernel, and the assigned tasks per thread.
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