An initial geometry was chosen to be close to the current treatment probe in use for the Ablatherm^TM machine [7] (50-mm spherical cap truncated at 31 mm with a central opening for lodging an imaging transducer). The array configurations were then constructed as described in [11]. Two array configurations were analyzed: a concentric ring array with independent sectional elements and a circular-element array with a pseudorandom pattern (Figure 1).

A simulation environment was created from full body histological images taken from the Visible Human Male Server Project (VHSMP, Ecole Polytechnique Federale de Lausanne, Switzerland) [12]. One hundred transverse images with a resolution of 0.2 × 0.2 × 1 mm were analyzed using an in-house developed interface in order to obtain a 3D structure of the prostate and neighboring structures (bladder, rectum, muscular tissue, ductus deferens, seminal vesicles, rectal wall, rectal muscle and nerve bundle).

A parametric study was performed to determine the optimal focal length (L), operating frequency (f), element size (a) and central opening (r) for a device capable of treating different locations within the prostate without secondary lesions while minimizing the number of elements.

Using an in-house interface and the images from the VHSMP, the device was positioned at the center of the rectum and the maximum and minimum distances necessary to achieve a treatment throughout the organ were defined. Figure 2 shows the closest and farthest distances from the centre of the device needed to treat the prostate at different anatomical locations. This determined the minimal and maximal focusing locations as 31 and 65 mm.

The acoustic field was simulated in a model of the pelvic region. The propagation media considered for the simulation was: coolant liquid (water) between the transducer ant the tissues, rectum and then soft tissue. The geometry of the coolant was modeled as an elliptical cylinder with a major diameter of 39 mm and a minor

diameter of 26 mm [7]. The rectal wall was modeled as an elliptical cylinder with a major diameter of 45 mm and a minor diameter of 32 mm [7] and a thickness of 6 mm (obtained from the VHSMP images).

The acoustic field simulation was performed using the Rayleigh-Sommerfeld diffraction integral given by [13]

where ν_s is the particle velocity at surface dS, R = R₁ + R₂ + R₃, R is the distance sound propagates with R₁ through the coolant liquid, R₂ through the rectal wall, and R₃ through the prostatic tissue, α is the attenuation factor obtained as R₁α₁ + R₂α₂ + R₃α₃, dS is the surface area, k = 2π/λ, λ is the wavelength and Z is the acoustic impedance of water.

The focal length was ranged from 45 to 60 mm by with 1-mm steps and the element size from 1λ to 8λ by steps of 0.05λ. The operating frequency was varied from 2.25 to 3 MHz by steps of 0.25 MHz. The size of the central opening was varied from a radius of 1 to 11 mm by steps of 2 mm. For each combination of parameters we evaluated the degree of focusing by the ratio [14].

where q₂ and q₁ are the acoustic pressure at the secondary and primary lobe, respectively. The goal was set for a configuration with the fewest elements and value of η ≤ 0.5. The acoustic attenuation was 2.9 × 10⁻⁴ Np/m/MHz for the coolant, 4.1 Np/m/MHz for the rectal wall and 9 Np/m/MHz for the prostate [7]. Simulations were performed using in-house developed and optimized code using two Tesla C1060 (NVIDIA, Santa Clara, CA, USA) graphics processors.

For the optimal configuration obtained from the parametric study, the steering capabilities were evaluated to ensure that tumors at different locations within the prostate could be treated. We first determined the maximum lateral steering possible at the anterior and posterior of the prostate with the device located in front of the apex, center and base of the prostate (Figure 2). Then the maximum vertical steering obtained at the center of the prostate was evaluated. The value of η (Equation (2)) was used to evaluate the degree of focusing. Focusing was tested at off-axis distances from 1 mm to 16 mm by steps of 0.25 mm.

The temperature increase obtained with the optimal configuration was estimated using the Bio-Heat Transfer Equation (BHTE) [15] and the thermal dose concept [16] at different relevant treatment locations. These locations were chosen using clinical MR images and pathology from 37 patients reported by Coakley et al. [17]. Virtual tumors with a size equivalent to the minimum (0.02 cc), average (0.79 cc) and maximum (3.7 cc) volumes reported in [17] were modeled as ellipsoids and located close the prostatic capsule. The BHTE was then used to calculate the temperature obtained with the optimal configuration focused at the centre and edges of the target virtual tumor [15] by

where ρ_t, c_t, and K_t are the density, specific heat and thermal conductivity of the tissue, respectively, T_P is the temperature a point P(x, y, z) and time t, ω_b, c_b and T_b are the perfusion rate, specific heat and temperature of blood, respectively, E₀ and E are the acoustic energy entering and exiting a volume V and χ is the acoustic absorption coefficient.

The BHTE was implemented using the numerical Finite Difference Time Difference technique [18] with boundary conditions at the borders of the tissue volume set to body temperature (37˚C) and thermo-mechanical properties of tissues were defined in [8]. The effect of this cooling was simulated as a boundary condition using a Newtonian flux convention [19] and a temperature of 13˚C. Tissues that reached a thermal dose of 240 minutes or greater were considered as ablated tissues [20]. If any of the tested sites within each virtual tumor formed a secondary lesion (outside of the target tumor), it was considered a failed treatment.

The optimal configuration was a concentric ring array with a focal length of 68 mm, an operating frequency of 2.75 MHz, a diameter of 50 mm, a truncated width of 31 mm, a 9-mm in radium central opening and an element width of 2.05λ for a total of 761 elements.

Table 1 shows the steering capabilities reached by the optimal concentric ring and random arrays. The random array was capable of steering farther in the lateral direction when focusing close to the transducer but this deteriorated when the focusing was done at deeper locations. Overall, the deeper the focal region the more difficult it was to achieve steering and the random array was unable to steer at the deeper locations.

Adequate focusing and lesion formation was possible at all limits of the virtual tumors using the concentric ring device, except for the two largest located at the prostate peripheral zone (located closest to the device).

Figure 3 shows the simulated lesions (thermal dose above 14,400 s) obtained when electronically steering the optimal configuration to reach the edges of the largest tumor that was virtually defined.

When treating tumors in the peripheral zone, secondary lesions were observed when targeting the posterior limit of the tumors (closest to the device). These secondary lesions appeared deeper than the targeted focus at approximately twice the focal length (towards the anterior region of the prostate). They were caused by grating lobes produced when the dynamic focus was located out of the steering range of the elements on the outer rings of

Table 1. Steering capabilities for optimal devices.

the device (Table 1) [14].

Table 2 shows the results for thermal simulations at all the defined targets. The size of the target tumors at peripheral and transitional prostate zones were defined based on MR images [17]. The size of the calculated thermal lesion when targeting the tumors varied depending on the location and the size of the target tumor since the maximum steering required was different in order to cover the entire tumor. Lesions were generally smaller for medium-sized tumors since to reach these tumors the device does not require as much steering and we are closer to the natural focus of the device. The lesions were generally larger for targets in the transitional region. The acoustic power required to obtain a lesion also increases with target lesions that require steering (larger tumors located outside of the central axis of the device).

For the peripheral zone targets where the secondary lesions appeared two solutions were proposed to avoid them: partial excitation of the elements and mechanical device rotation. By exciting only the elements closest to the center of the device we could decrease the phase difference between elements and eliminate the secondary lesion while minimizing the acoustic intensity at the surface. A series of simulations was performed with 50% to 75% of the elements excited and it was found that secondary lesions were eliminated with 56% of active elements when targeting a large peripheral tumor (3700 mm³) and with 71% when targeting the medium peripheral tumor (790 mm³). The device could also be rotated on the left-right direction by ±π/8 in order to eliminate the secondary lesions.

The overall maximum acoustic intensity at the surface of the transducer for treatment was 22.74 W/cm². This maximum occurred when focusing at the location the lateral edge of large tumors in the peripheral zone with only 56% of the elements active. This intensity was high as a result of the reduced surface area caused by elements being off.