Diverging Polymer Acoustic Lens Design for High-Resolution Row-Column Array Ultrasound Transducers.

Spherical diverging acoustic lenses mounted on flat 2-D row-column-addressed (RCA) ultrasound transducers have shown the potential to extend the field of view (FOV) from a rectilinear to a curvilinear volume region and, thereby, enable 3-D imaging of large organs. Such lenses are usually designed for small aperture low-frequency transducers, which have limited resolution. Moreover, they are made of off-the-shelf pieces of materials, which leaves no room for optimization. We hypothesize that acoustic lenses can be designed to fit high-resolution transducers, and they can be fabricated in a fast, cost-effective, and flexible manner using a combination of 3-D printing and casting or computer numerical control (CNC) machining techniques. These lenses should increase the FOV of the array while preserving the image quality. In this work, such lenses are made in concave, convex, and compound spherical shapes and from thermoplastics and thermosetting polymers. Polymethylpentene (TPX), polystyrene (PS), polypropylene (PP), polymethyl methacrylate (PMMA), polydimethylsiloxane (PDMS), and room-temperature-vulcanizing (RTV) silicone diverging lenses have been fabricated and mounted on a 128 + 128 6-MHz RCA transducer. The performances of the lenses have been assessed and compared in terms of FOV, signal-to-noise ratio (SNR), bandwidth, and potential artifacts. The largest FOV (24.0.) is obtained with a 42.64-mm radius PMMA-RTV compound lens, which maintains a decent fractional bandwidth (53%) and SNR at 6 MHz (.9.1-dB amplitude drop compared with the unlensed transducer). The simple PMMA TPX, PS, PP, PDMS, and RTV lenses provide an FOV of 12.2°, 6.3°, 8.1°, 11.7°, 0.6°, and 10.4°; a fractional bandwidth of 97%, 46%, 103%, 46%, 97%, 53%, and 49%; and an amplitude drop of -5.2, -4.4, -2.8, -15.4, -6.0, and -1.8 dB, respectively. This work demonstrates that thermoplastics are suitable materials for fabricating low-attenuation convex diverging lenses for large-aperture, high-frequency 2-D transducers. This is highly desired to achieve high-resolution volumetric imaging of large organs.

Contrast-enhanced ultrasound imaging using capacitive micromachined ultrasonic transducers.

Capacitive micromachined ultrasonic transducers (CMUTs) have a nonlinear relationship between the applied voltage and the emitted signal, which is detrimental to conventional contrast enhanced ultrasound (CEUS) techniques. Instead, a three-pulse amplitude modulation (AM) sequence has been proposed, which is not adversely affected by the nonlinearly emitted harmonics. In this paper, this is shown theoretically, and the performance of the sequence is verified using a 4.8 MHz linear capacitive micromachined ultrasonic transducer (CMUT) array, and a comparable lead zirconate titanate (PZT) array, across 6-60 V applied alternating current (AC) voltage. CEUS images of the contrast agent SonoVue flowing through a 3D printed hydrogel phantom showed an average enhancement in contrast-to-tissue ratio (CTR) between B-mode and CEUS images of 49.9 and 37.4 dB for the PZT array and CMUT, respectively. Furthermore, hydrophone recordings of the emitted signals showed that the nonlinear emissions from the CMUT did not significantly degrade the cancellation in the compounded AM signal, leaving an average of 2% of the emitted power between 26 and 60 V of AC. Thus, it is demonstrated that CMUTs are capable of CEUS imaging independent of the applied excitation voltage when using a three-pulse AM sequence.

Anatomic and Functional Imaging Using Row-Column Arrays.

Row-column (RC) arrays have the potential to yield full 3-D ultrasound imaging with a greatly reduced number of elements compared to fully populated arrays. They, however, have several challenges due to their special geometry. This review article summarizes the current literature for RC imaging and demonstrates that full anatomic and functional imaging can attain a high quality using synthetic aperture (SA) sequences and modified delay-and-sum beamforming. Resolution can approach the diffraction limit with an isotropic resolution of half a wavelength with low sidelobe levels, and the field of view can be expanded by using convex or lensed RC probes. GPU beamforming allows for three orthogonal planes to be beamformed at 30 Hz, providing near real-time imaging ideal for positioning the probe and improving the operator's workflow. Functional imaging is also attainable using transverse oscillation and dedicated SA sequence for tensor velocity imaging for revealing the full 3-D velocity vector as a function of spatial position and time for both blood velocity and tissue motion estimation. Using RC arrays with commercial contrast agents can reveal super-resolution imaging (SRI) with isotropic resolution below [Formula: see text]. RC arrays can, thus, yield full 3-D imaging at high resolution, contrast, and volumetric rates for both anatomic and functional imaging with the same number of receive channels as current commercial 1-D arrays.

3D printed calibration micro-phantoms for super-resolution ultrasound imaging validation.

This study evaluates the use of 3D printed phantoms for 3D super-resolution ultrasound imaging (SRI) algorithm calibration. The main benefit of the presented method is the ability to do absolute 3D micro-positioning of sub-wavelength sized ultrasound scatterers in a material having a speed of sound comparable to that of tissue. Stereolithography is used for 3D printing soft material calibration micro-phantoms containing eight randomly placed scatterers of nominal size 205 µm × 205 µm × 200 µm. The backscattered pressure spatial distribution is evaluated to show similar distributions from micro-bubbles as the 3D printed scatterers. The printed structures are found through optical validation to expand linearly in all three dimensions by 2.6% after printing. SRI algorithm calibration is demonstrated by imaging a phantom using a λ/2 pitch 3 MHz 62+62 row-column addressed (RCA) ultrasound probe. The printed scatterers will act as point targets, as their dimensions are below the diffraction limit of the ultrasound system used. Two sets of 640 volumes containing the phantom features are imaged, with an intervolume uni-axial movement of the phantom of 12.5 µm, to emulate a flow velocity of 2 mm/s at a frame rate of 160 Hz. The ultrasound signal is passed to a super-resolution pipeline to localise the positions of the scatterers and track them across the 640 volumes. After compensating for the phantom expansion, a scaling of 0.989 is found between the distance between the eight scatterers calculated from the ultrasound data and the designed distances. The standard deviation of the variation in the scatterer positions along each track is used as an estimate of the precision of the super-resolution algorithm, and is expected to be between the two limiting estimates of (σ̃x,σ̃y,σ̃z) = (22.7 µm, 27.6 µm, 9.7 µm) and (σ̃x,σ̃y,σ̃z) = (18.7 µm, 19.3 µm, 8.9 µm). In conclusion, this study demonstrates the use of 3D printed phantoms for determining the accuracy and precision of volumetric super-resolution algorithms.

Detection and Localization of Ultrasound Scatterers Using Convolutional Neural Networks.

Delay-and-sum (DAS) beamforming is unable to identify individual scatterers when their density is so high that their point spread functions overlap. This paper proposes a convolutional neural network (CNN)-based method to detect and localize high-density scatterers, some of which are closer than the resolution limit of delay-and-sum (DAS) beamforming. A CNN was designed to take radio frequency channel data and return non-overlapping Gaussian confidence maps. The scatterer positions were estimated from the confidence maps by identifying local maxima. On simulated test sets, the CNN method with three plane waves achieved a precision of 1.00 and a recall of 0.91. Localization uncertainties after excluding outliers were ±46 [Formula: see text] (outlier ratio: 4%) laterally and ±26 [Formula: see text] (outlier ratio: 1%) axially. To evaluate the proposed method on measured data, two phantoms containing cavities were 3-D printed and imaged. For the phantom study, the training data were modified according to the physical properties of the phantoms and a new CNN was trained. On an uniformly spaced scatterer phantom, a precision of 0.98 and a recall of 1.00 were achieved with the localization uncertainties of ±101 [Formula: see text] (outlier ratio: 1%) laterally and ±37 [Formula: see text] (outlier ratio: 1%) axially. On a randomly spaced scatterer phantom, a precision of 0.59 and a recall of 0.63 were achieved. The localization uncertainties were ±132 [Formula: see text] (outlier ratio: 0%) laterally and ±44 [Formula: see text] with a bias of 22 [Formula: see text] (outlier ratio: 0%) axially. This method can potentially be extended to detect highly concentrated microbubbles in order to shorten data acquisition times of super-resolution ultrasound imaging.

Three-Dimensional Super-Resolution Imaging Using a Row-Column Array.

A 3-D super-resolution (SR) pipeline based on data from a row-column (RC) array is presented. The 3-MHz RC array contains 62 rows and 62 columns with a half wavelength pitch. A synthetic aperture (SA) pulse inversion sequence with 32 positive and 32 negative row emissions is used for acquiring volumetric data using the SARUS research ultrasound scanner. Data received on the 62 columns are beamformed on a GPU for a maximum volume rate of 156 Hz when the pulse repetition frequency is 10 kHz. Simulated and 3-D printed point and flow microphantoms are used for investigating the approach. The flow microphantom contains a 100- [Formula: see text] radius tube injected with the contrast agent SonoVue. The 3-D processing pipeline uses the volumetric envelope data to find the bubble's positions from their interpolated maximum signal and yields a high resolution in all three coordinates. For the point microphantom, the standard deviation on the position is (20.7, 19.8, 9.1) [Formula: see text]. The precision estimated for the flow phantom is below [Formula: see text] in all three coordinates, making it possible to locate structures on the order of a capillary in all three dimensions. The RC imaging sequence's point spread function has a size of 0.58 × 1.05 × 0.31 mm3 ( 1.17λ×2.12λ×0.63λ ), so the possible volume resolution is 28900 times smaller than for SA RC B-mode imaging.

Imaging Performance for Two Row-Column Arrays.

This study evaluates the volumetric imaging performance of two prototyped 62 + 62 row-column-addressed (RCA) 2-D array transducer probes using three synthetic aperture imaging (SAI) emission sequences and two different beamformers. The probes are fabricated using capacitive micromachined ultrasonic transducer (CMUT) and piezoelectric transducer (PZT) technology. Both have integrated apodization to reduce ghost echoes and are designed with similar acoustical features, i.e., 3-MHz center frequency, λ /2 pitch, and [Formula: see text] active footprint. Raw RF data are obtained using an experimental research ultrasound scanner, SARUS. The SAI sequences are designed for imaging down to 14 cm at a volume rate of 88 Hz. Two beamforming methods, spatial matched filtering and row-column adapted delay-and-sum, are used for beamforming the RF data. The imaging quality is investigated through simulations and phantom measurements. Both probes on average have similar lateral full-width at half-maximum (FWHM) values, but the PZT probe has 20% smaller cystic resolution values and 70% larger contrast-to-noise ratio (CNR) compared to the capacitive micromachined ultrasonic transducer (CMUT) probe. The CMUT probe can penetrate down to 15 cm, and the PZT probe down to 30 cm. The CMUT probe has 17% smaller axial FWHM values. The matched filter focusing shows an improved B-mode image for measurements on a cyst phantom with an improved speckle pattern and better visualization of deeper lying cysts. The results of this study demonstrate the potentials of RCA 2-D arrays against fully addressed 2-D arrays, which are low channel count (e.g., 124 instead of 3844), low acoustic intensity mechanical index (MI ≤ 0.88 and spatial-peak-temporal-average intensity [Formula: see text]), and high penetration depth (down to 30 cm), which makes 3-D imaging at high volume rates possible with equipment in the price range of conventional 2-D imaging.

CMUT Electrode Resistance Design: Modelling and Experimental Verification by a Row-Column Array.

This paper addresses the importance of having control over the resistivity of the electrodes for capacitive micromachined ultrasonic transducers (CMUT) devices. The electrode resistivity can vary depending on the fabrication technology used, and resistivity control becomes especially important in the cases where metal electrodes can not be used. This raises the question: When is the resistivity of an electrode sufficiently low? To answer this question we have developed a simple design criterion. The criterion describes the attenuation of AC signals along a CMUT element. It is shown that the non-dimensional product between angular excitation frequency, resistance, and capacitance ωRC of an element has to be smaller than 0.35 to ensure an AC potential drop along the element of less than 1%. The optimal magnitude and directionality of the transmit pressure will be achieved if CMUT elements are designed according to the developed criteria. Hence, the model can be used to estimate device parameters that will ensure the CMUT is suitable for generating ultrasound images. An example is given where the model is used to predict the required electrode thickness for structured electrodes made of Gold, Aluminium, and Indium-Tin-Oxide, respectively. To verify the model, two Row-Column addressed (RCA) CMUT transducers were used to illustrate the effect of high and low electrode resistivity. One transducer had a sufficient electrode resistivity, and the other had an insufficient electrode resistivity. The RCA CMUT transducers were fabricated using fusion bonding, where the top electrode is made of aluminium and the bottom electrode is made of doped silicon. The resistivity of the aluminium top electrode is 2×10-6 Ωcm for both transducers, whereas the resistivity for the bottom electrode is 0.1 Ωcm for the first transducer and 0.005 Ωcm for the second transducer. The transducer with low resistivity emits pressure uniformly along both the rows and columns, whereas the transmit pressure field from the other transducer has a uniformly distributed pressure field along the rows, but a decreasing pressure field along the columns due to the high resistivity in the bottom electrode. The pressure drop, along the columns is frequency dependent and has been observed to be 63%, 74%, and 82% for the excitation frequencies 2 MHz, 4.5 MHz, and 7 MHz, respectively.

Curvilinear 3-D Imaging Using Row-Column-Addressed 2-D Arrays With a Diverging Lens: Phantom Study.

A double-curved diverging lens over the flat row-column-addressed (RCA) 2-D array can extend its inherent rectilinear 3-D imaging field of view (FOV) to a curvilinear volume region, which is necessary for applications such as abdominal and cardiac imaging. Two concave lenses with radii of 12.7 and 25.4 mm were manufactured using RTV664 silicone. The diverging properties of the lenses were evaluated based on simulations and measurements on several phantoms. The measured FOV for both lenses in contact with tissue mimicking phantom was less than 15% different from the theoretical predictions, i.e., a curvilinear FOV of and for the 12.7- and 25.4-mm radii lenses. A synthetic aperture imaging sequence with single-element transmissions was designed for imaging down to 140 mm at a volume rate of 88 Hz. The performance was evaluated in terms of signal-to-noise ratio, FOV, and full-width at half-maximum (FWHM) of a focused beam. The penetration depths in a tissue mimicking phantom with 0.5-dB/(cm MHz) attenuation were 100 and 125 mm for the lenses with radii of 12.7 and 25.4 mm. The azimuth, elevation, and radial FWHM at 43-mm depth were (5.8, 5.8, 1) and (6, 6, 1) . The results of this study confirm that the proposed lens approach is an effective method for increasing the FOV, when imaging with RCA 2-D arrays.

Increasing the field-of-view of row-column-addressed ultrasound transducers: implementation of a diverging compound lens.

The purpose of this work is to investigate compound lenses for row-column-addressed (RCA) ultrasound transducers for increasing the field-of-view (FOV) to a curvilinear volume region, while retaining a flat sole to avoid trapping air between the transducer sole and the patient, which would otherwise lead to unwanted reflections. The primary motivation behind this research is to develop a RCA ultrasound transducer for abdominal or cardiac imaging, where a curvilinear volume region is a necessity. RCA transducers provide 3-D ultrasound imaging with fewer channels than fully-addressed 2-D arrays (2N instead of N2), but they have inherently limited FOV. By increasing the RCA FOV, these transducers can be used for the same applications as fully-addressed transducers while retaining the same price range as conventional 2-D imaging due to the lower channel count. Analytical and finite element method (FEM) models were employed to evaluate design options. Composite materials were developed by loading polymers with inorganic powders to satisfy the corresponding speed of sound and specific acoustical impedance requirements. A Bi2O3 powder with a density of 8.9g/cm3 was used to decrease the speed of sound of a room temperature vulcanizing (RTV) silicone, RTV615, from 1.03mm/µs to 0.792mm/µs. Using micro-balloons in RTV615 and a urethane, Hapflex 541, their speeds of sound were increased from 1.03mm/µs to 1.50mm/µs and from 1.52mm/µs to 1.93mm/µs, respectively. A diverging add-on lens was fabricated of a Bi2O3 loaded RTV615 and an unloaded Hapflex 541. The lens was tested using a RCA probe, and a FOV of 32.2° was measured from water tank tests, while the FEM model yielded 33.4°. A wire phantom with 0.15mm diameter wires was imaged at 3MHz down to a depth of 14cm using a synthetic aperture imaging sequence with single element transmissions. The beamformed image showed that wires outside the array footprint were visible, demonstrating the increased FOV.

Curvilinear 3-D Imaging Using Row-Column-Addressed 2-D Arrays With a Diverging Lens: Feasibility Study.

Constructing a double-curved row-column-addressed (RCA) 2-D array or applying a diverging lens over the flat RCA 2-D array can extend the imaging field-of-view (FOV) to a curvilinear volume without increasing the aperture size, which is necessary for applications, such as abdominal and cardiac imaging. Extended FOV and low channel count of double-curved RCA 2-D arrays make 3-D imaging possible with equipment in the price range of conventional 2-D imaging. This paper proposes a delay-and-sum beamformation scheme specific to double-curved RCA 2-D arrays and validates its focusing ability based on simulations. A synthetic aperture imaging sequence with single element transmissions is designed for imaging down to 14 cm at a volume rate of 88 Hz. Using a diverging lens with an f-number of -1 circumscribing the underlying RCA array, the imaging quality of a double-curved λ/2 -pitch 3-MHz 62 + 62 RCA 2-D array is investigated as a function of depth within a curvilinear FOV of 60 °×60° . The simulated double-curved 2-D array exhibits the same full-width-at-half-maximum values for a point scatterer within its curvilinear FOV at a fixed radial distance compared with a flat 2-D array within its rectilinear FOV. The results of this paper demonstrate that the proposed beamforming approach is accurate for achieving correct time-of-flight calculations, and hence avoids geometrical distortions.

3-D Vector Flow Estimation With Row-Column-Addressed Arrays.

Simulation and experimental results from 3-D vector flow estimations for a 62 + 62 2-D row-column (RC) array with integrated apodization are presented. A method for implementing a 3-D transverse oscillation (TO) velocity estimator on a 3-MHz RC array is developed and validated. First, a parametric simulation study is conducted, where flow direction, ensemble length, number of pulse cycles, steering angles, transmit/receive apodization, and TO apodization profiles and spacing are varied, to find the optimal parameter configuration. The performance of the estimator is evaluated with respect to relative mean bias ~B and mean standard deviation ~σ . Second, the optimal parameter configuration is implemented on the prototype RC probe connected to the experimental ultrasound scanner SARUS. Results from measurements conducted in a flow-rig system containing a constant laminar flow and a straight-vessel phantom with a pulsating flow are presented. Both an M-mode and a steered transmit sequence are applied. The 3-D vector flow is estimated in the flow rig for four representative flow directions. In the setup with 90° beam-to-flow angle, the relative mean bias across the entire velocity profile is (-4.7, -0.9, 0.4)% with a relative standard deviation of (8.7, 5.1, 0.8)% for ( vx, vy, vz ). The estimated peak velocity is 48.5 ± 3 cm/s giving a -3% bias. The out-of-plane velocity component perpendicular to the cross section is used to estimate volumetric flow rates in the flow rig at a 90° beam-to-flow angle. The estimated mean flow rate in this setup is 91.2 ± 3.1 L/h corresponding to a bias of -11.1%. In a pulsating flow setup, flow rate measured during five cycles is 2.3 ± 0.1 mL/stroke giving a negative 9.7% bias. It is concluded that accurate 3-D vector flow estimation can be obtained using a 2-D RC-addressed array.

Electrostatic and Small-Signal Analysis of CMUTs With Circular and Square Anisotropic Plates.

Traditionally, capacitive micromachined ultrasonic transducers (CMUTs) are modeled using the isotropic plate equation, and this leads to deviations between analytical calculations and finite element modeling (FEM). In this paper, the deflection is calculated for both circular and square plates using the full anisotropic plate equation. It is shown that the anisotropic calculations match excellently with FEM, whereas an isotropic approach causes up to 10% deviations in deflection. For circular plates, an exact solution can be found. For square plates using the Galerkin method, and utilizing the symmetry of the silicon crystal, a compact and accurate expression for the deflection can be obtained. The deviation from FEM in center deflection is <0.1%. The theory of multilayer plates is also applied to the CMUT. The deflection of a square plate was measured on fabricated CMUTs using a white light interferometer. Fitting the plate parameter for the anisotropic calculated deflection to the measurement, a deviation of 0.07% is seen. Electrostatic and small-signal dynamic analysis are performed using energy considerations including anisotropy. The stable position, effective spring constant, pullin distance, and pull-in voltage are found for both circular and square anisotropic plates, and the pressure dependence is included by comparison with the corresponding analysis for a parallel plate. Measurements on fabricated devices with both circular and square plates subjected to increasing bias voltage are performed, and it is observed that the models including anisotropic effects are within the uncertainty interval of the measurements. Finally, a lumped element small-signal model for both circular and square anisotropic plates is derived to describe the dynamics of the CMUT.

Acoustical cross-talk in row-column addressed 2-D transducer arrays for ultrasound imaging.

The acoustical cross-talk in row-column addressed 2-D transducer arrays for volumetric ultrasound imaging is investigated. Experimental results from a 2.7 MHz, λ/2-pitch capacitive micromachined ultrasonic transducer (CMUT) array with 62 rows and 62 columns are presented and analyzed in the frequency-wavenumber domain. The sources of cross-talk are identified and predicted theoretically. The nearest neighbor cross-talk is -23.9±3.7 dB when the array is used as a 1-D array with the rows functioning as both transmitters and receivers. In the row-column configuration, with the columns transmitting and the rows receiving, the cross-talk is reduced to -40.2±3.5 dB.

3-D imaging using row-column-addressed arrays with integrated apodization - part i: apodization design and line element beamforming.

This paper investigates the effect of transducerintegrated apodization in row-column-addressed arrays and presents a beamforming approach specific for such arrays. Row-column addressing 2-D arrays greatly reduces the number of active channels needed to acquire a 3-D volume. A disadvantage of row-column-addressed arrays is an apparent ghost effect in the point spread function caused by edge waves. This paper investigates the origin of the edge waves and the effect of introducing an integrated apodization to reduce the ghost echoes. The performance of a λ/2-pitch 5-MHz 128 + 128 row-column-addressed array with different apodizations is simulated. A Hann apodization is shown to decrease imaging performance away from the center axis of the array because of a decrease in main lobe amplitude. Instead, a static roll-off apodization region located at the ends of the line elements is proposed. In simulations, the peak ghost echo intensity of a scatterer at (x,y, z) = (8, 3, 30) mm was decreased by 43 dB by integrating roll-off apodization into the array. The main lobe was unaffected by the apodization. Simulations of a 3-mm-diameter anechoic blood vessel at 30 mm depth showed that applying the transducer-integrated apodization increased the apparent diameter of the vessel from 2.0 mm to 2.4 mm, corresponding to an increase from 67% to 80% of the true vessel diameter. The line element beamforming approach is shown to be essential for achieving correct time-of-flight calculations, and hence avoid geometrical distortions. In Part II of this work, experimental results from a capacitive micromachined ultrasonic transducer with integrated roll-off apodization are given to validate the effect of integrating apodization into the line elements.

3-D imaging using row-column-addressed arrays with integrated apodization- part ii: transducer fabrication and experimental results.

This paper demonstrates the fabrication, characterization, and experimental imaging results of a 62+62 element λ/2-pitch row-column-addressed capacitive micromachined ultrasonic transducer (CMUT) array with integrated apodization. A new fabrication process was used to manufacture a 26.3 mm by 26.3 mm array using five lithography steps. The array includes an integrated apodization, presented in detail in Part I of this paper, which is designed to reduce the amplitude of the ghost echoes that are otherwise prominent for row-column-addressed arrays. Custom front-end electronics were produced with the capability of transmitting and receiving on all elements, and the option of disabling the integrated apodization. The center frequency and -6-dB fractional bandwidth of the array elements were 2.77 ± 0.26 MHz and 102 ± 10%, respectively. The surface transmit pressure at 2.5 MHz was 590 ± 73 kPa, and the sensitivity was 0.299 ± 0.090 V/Pa. The nearest neighbor crosstalk level was -23.9 ± 3.7 dB, while the transmit-to-receive-elements crosstalk level was -40.2 ± 3.5 dB. Imaging of a 0.3-mm-diameter steel wire using synthetic transmit focusing with 62 single-element emissions demonstrated axial and lateral FWHMs of 0.71 mm and 1.79 mm (f-number: 1.4), respectively, compared with simulated axial and lateral FWHMs of 0.69 mm and 1.76 mm. The dominant ghost echo was reduced by 15.8 dB in measurements using the integrated apodization compared with the disabled configuration. The effect was reproduced in simulations, showing a ghost echo reduction of 18.9 dB.

Modal radiation patterns of baffled circular plates and membranes.

The far field velocity potential and radiation pattern of baffled circular plates and membranes are found analytically using the full set of modal velocity profiles derived from the corresponding equation of motion. The derivation is valid for a plate or membrane subjected to an external excitation force, which is used as a sound receiver in any medium or as a sound transmitter in a gaseous medium. A general, concise expression is given for the radiation pattern of any mode of the membrane and the plate with arbitrary boundary conditions. Specific solutions are given for the four special cases of a plate with clamped, simply supported, and free edge boundary conditions as well as for the membrane. For all non-axisymmetric modes, the velocity potential along the axis of the radiator is found to be strictly zero. In the long wavelength limit, the radiation pattern of all axisymmetric modes approaches that of a monopole, while the non-axisymmetric modes exhibit multipole behavior. Numerical results are also given, demonstrating the implications of having non-axisymmetric excitation using both a point excitation with varying eccentricity and a homogeneous excitation acting on half of the circular radiator.