Acoustic microscopy was first developed in 1972 by Lemons and Quate at Stanford University for cellular imaging with a nearly optical resolution [5, 10, 11]. Since then, it has been used for mapping the elasticity of biological samples [4, 12, 13]. Biomechanical properties (e.g., thickness, longitudinal sound speed, density, acoustic impedance, elasticity, attenuation) of cells and tissues can be determined using the time-resolved technique. This technique relies on echoes reflected from the sample and the interface between the sample and its substrate. Fig. (1) illustrates the functioning concept of time-resolved acoustic microscopy. A reference signal (t₀ ) is acquired from the reflected echo when the transducer is at focus. The arrival time of the echo reflected off the cell surface is defined as t₁ while the arrival time of the echo reflected from the cell/substrate interface is defined as t₂.

Fig. (1). Principles of time-resolved scanning acoustic microscopy. The schematic diagram illustrates the working principle of time-resolved scanning acoustic microscopy. Echoes originating from the surface of the sample, the substrate and the specimen/substrate interface are resolved in the time domain for both the quantification of mechanoelastic properties and image generation.

Since the arrival times originating from the surface of the sample and the substrate/sample interface are only separated by a few nanoseconds (in the ultra-high frequency regime), it is of paramount importance to be able to separate the echo signals in the time domain. This is achieved by emitting sufficiently short pulses so that the reflected signals do not overlap [1]. The mechanical and elastic properties can be determined using time-resolved techniques [1]. We define τ₁ to be the time delay between the surface and the reference echoes (τ₁ = t₀ – t₁), τ₂ the time delay between the interface and the reference echoes (τ₂ = t₀ – t₂), and τ₁₂ the time delay between the interface and the surface echoes (τ₁₂ = t₂ – t₁). By applying a simple mathematical optics formula, the cell thickness, d, at any scanning position coordinates (x, y) can be computed using Eq. (1), when the sound velocity in the coupling medium, c₀, is known. Following the calculation of the cell thickness at given coordinates, the longitudinal sound velocity, c, can then be determined using Eq. (2) as follows [1]:

(1)

(2)

The acoustic impedance of the sample can be derived based on the amplitude of the reference signal reflected from the coupling medium and the substrate interface, A₀; the amplitude of the substrate/cell echo signal, A₁; and the acoustic impedance of the coupling medium, Z₀ from Eq. (3) [1]:

(3)

It is noted that the acoustic impedance of the coupling medium needs to be known a priori. Typically, cell and tissue media or buffer solutions used for coupling purposes exhibit very close viscoelastic properties to water at given temperatures, therefore this requirement does not pose any technical limitations.

Once the ultrasound velocity and acoustic impedance of the sample have been determined, the density at any given coordinate position (x, y) can be obtained by applying Eq. (4) [1]:

(4)

Briggs et al. established the mathematical expression for obtaining the acoustic attenuation [1]. The attenuation in the coupling medium α₀, the amplitude of the reflected signal from the cell surface A₂, the amplitudes A₁ and A₀ (see above), and the impedance of the substrate Z_s are all used for calculating the attenuation coefficient (Fig. 2) in Eq. (5):

Fig. (2). Quantitative methods in time-resolved acoustic microscopy. (A) Radio-frequency signal reflected from a glass substrate (A0, t0) at the focal point without any interfering samples; (B) reflected signal from the substrate/sample interface (A1, t1) and the cell surface (A2, t2), respectively, of an MCF-7 cell attached on a glass substrate.

(5)

The contrast in acoustic images of biological specimens can be created using the difference in attenuation rather than differences in the reflection coefficient because their acoustic impedances are close to the acoustic impedance of the coupling medium. It has been shown that the difference in attenuation can be enhanced by using highly reflective materials [14]. For example, silica glass was used as a substrate when operating SAM at frequencies in the range of 200 - 600 MHz. On the other hand, sapphire was used as a substrate when operating SAM at a frequency of 1 GHz or higher [13, 15]. The corresponding contrast mechanism varies with the V(z) distribution. Additionally, some information about the adhesive condition between the cells and the substrate can be explored by using the scanning reflection acoustic microscope [13, 15]. The V(z) distribution can be explained by Eqs. (6) and (7):

(6)

(7)

where u is the acoustic field; P is the pupil function of the lens; R is the reflectance function; k is the wave number in the coupling medium; and f is the focal length. By substituting r = fsinθ into Eqs. (6) and (7), we obtain expressions in terms of the half aperture angle of the lens as shown in Eqs. (8) and (9):

(8)

(9)

where θ is half the aperture angle of the acoustic lens.

By replacing k_z = k cosϑ into Eqs. (8) and (9), we obtain Eqs. (10), (11) and (12) as follows:

(10)

(11)

(12)

Furthermore, Eq. (10) yields the following expression:

(13)

where, F^-1{V(z))} is the inverse Fourier transform.

The reflectance function R plays an important role in the V(z) response and the corresponding contrast formation across the cell. Normal cells grown under physiological conditions on a given substrate can be described by the reflectance function as determined by the layered media (e.g., coupling medium, cell, substrate) shown in Fig. (3A). Conversely, when a cell is compromised (e.g., apoptosis, necrosis), the reflectance function is described as the layered media consisting of the coupling medium, the cell itself, the fluid, and the substrate (Fig. 3B) [16].

Fig. (3). Schematic diagram of the cell-layered structure model for different conditions. (A) healthy cell; (B) injured cell.

Biological specimens can be considered as thin films with near zero shear modulus. Fig. (4) illustrates the layered structure model comprised of the cell specimen, the coupling medium and the substrate. Here, the biological cell is assumed to have zero shear velocity. Thus, the layered structure can be treated as a liquid-liquid-solid system.

Fig. (4). Layered model. The model depicts the cell/substrate structure. The cell or tissue samples are positioned between the coupling medium and the substrate. The coupling medium is typically a buffer solution with similar acoustic properties to the ones of distilled water at a given temperature.

Coupling medium (layer I):

(14)

(15)

Biological specimen (layer II):

(16)

(17)

Substrate (layer III):

(18)

(19)

The superscript “+” indicates wave propagation in the positive direction along the z-axis; the superscript “–“ indicates wave propagation in the negative direction along the z axis; the Roman numerals I, II, and III indicate media layers of the propagating wave. Φ represents the potential of the longitudinal wave and Ψ represents the potential of the shear wave; A is the amplitude of the potential of the longitudinal wave; B is the amplitude of the potential of the shear wave; k is the wave number; and ω is the angular frequency.

By applying Snell’s law, following expressions are obtained:

(20)

(21)

By substituting Eqs. (20) and (21) into Eqs. (14) through (19), we obtain the following expressions describing the layered structure of the model as illustrated in Fig. (4) for layer I, layer II and layer III respectively.

For the coupling medium (layer I) the following expressions are valid as shown in Eqs. (22) and (23):

(22)

(23)

For the biological cell or tissue (layer II) following expressions as shown in Eqs. (24) and (25):

(24)

(25)

Finally, the substrate layer (layer III) is described by the following expressions as shown in Eqs. (26) and (27):

(26)

(27)

The particle velocity along the z-axis and the continuity of stress are the boundary conditions in this layered structure are expressed in Eqs. (28) and (29):

(28)

(29)

where σ_z is the normal stress along the z-direction; τ_xz is the horizontal shear stress; and υ_z is the particle velocity component along the z direction.

The particle velocity and stress for each layer in the model can be calculated from Eqs. (22) through (27) and subsequently substituted into Eqs. (28) and (29) respectively to satisfy the boundary conditions. By solving boundary equations, the set of the first order of simultaneous equations relating to the potential can be obtained as follows:

(30)

(31)

(32)

whereas λ and µ are Lame’s constants. Then, by solving Eqs. (30), (31) and (32), the reflectance function can be obtained by Eqs. (33) and (34):

(33)

(34)

The V(z) distribution can be simulated to obtain the longitudinal velocity in biological cells once the reflectance function has been determined by using Eqs. (24) and (25).

The reflectance function for biological cells and tissues and their substrate system is calculated by employing the layer model described in the previous section. Here the biological specimen was treated as a thin film with near zero shear modulus attached on a substrate. The biological specimens used in the calculations were kidney tissue, HeLa and MCF-7 cancer cells; the substrates we used for our calculations were both fused quartz and glass substrates.

Once the calculations of reflectance function for the biological specimen and the substrate were completed, we then simulated the V(z) distributions for the system. The V(z) distributions were simulated using MATLAB by assigning parameters for the acoustics lens, the biological specimen and the substrate. For example, the thickness of an average HeLa cell is about 5 -12 µm, while its longitudinal velocity is about 1534 m/s. The glass substrate’s shear wave velocity is approximately 5845 m/s. These values are in good agreement with previously published studies [17, 18].

Here we calculated and plotted the V(z) distributions that were described in the previous section. Fig. (5) illustrates the schematic diagram for calculating the V(z) distribution by means of the Fourier Angular Spectrum method. Fig. (6) shows the schematic diagram for determining the V(z) distribution by means of the Ray theory method.

Fig. (5). Schematic diagram for the V(z) calculations using the angular spectrum technique. The Fourier Angular Spectrum method is employed for the V(z) calculations. FFT indicates the Fast Fourier Transform. IFFT is the inverse Fast Fourier Transform. The schematic shows the transducer and buffer rod. The coupling medium between the cell specimen and the transducer is shown between layers “2” and “3”.

Fig. (6). Schematic diagram for the V(z) calculations using the Ray theory method. The ultrasound transducer is positioned at the focus point above the substrate and the sample. The buffer rod is depicted in the above schematic. Examples of specular (yellow) and non-specular reflections (blue) are shown.

MATLAB scripts were developed for performing all simulations and for obtaining the local thickness, sound velocity, and attenuation of MCF-7 and HeLa cells. Furthermore, for the calculations we employed the quantitative formulas reported in detail by Briggs et al. [1, 4] and described above.

The transducer simulations were performed and developed using the Field II ultrasound simulation program [21, 22]. Field II originally developed by J. A. Jensen employs the concept of spatial impulse response developed by Tupholme [23] and Stepanishen [24, 25]. The spatial impulse response can be viewed as the impulse response for the linear system at a specific point in space and it was used to explain how the transducer transmits sound in space. The pressure produced by the transducer can be described by the spatial impulse response and can be determined by using the Rayleigh integral in Eq. (37):

(37)

is the position of the field point in space; c is the speed of sound and S is the surface area of the transducer. The emitted pressure field can be expressed by Eq. (38) as follows:

(38)

where v_n(t) is the surface velocity of the transducer and ρ is the density of the coupling medium. Due to the linear acoustics that were employed, apodization of the transducer surface can be included in the simulations as can the responses from transducer elements in the case of an array. Furthermore, the scattered field can be determined from the spatial impulse response. The signal received from the transducer can be calculated from Eq. (39):

(39)

*_r indicates spatial convolution and *_t denotes temporal convolution; v_pe is the pulse-echo impulse including the transducer excitation and the electro-mechanical impulse response during the pulse’s emission and reception. The inhomogeneities in the tissue caused by density and perturbations of the propagation velocity are taken into account by f_m; h_pe represents the pulse-echo spatial impulse response, which shows the relation between the transducer geometry and the spatial extent of the scattered field; v_pe, f_m and h_pe can be expressed by Eqs. (40), (41) and (42) respectively as follows:

(40)

(41)

(42)

Once the spatial impulse response for the transmitting and the receiving transducer is determined and then convolved with the impulse response of the transducer, the received response can be computed. By adding the response from a collection of scatterers, a single RF line in an image can be computed. The scattering strength of the collection of scatterers is calculated from the density and speed of sound perturbations in the tissue. In the case of homogeneous tissues, it can be generated from a collection of randomly placed scatterers, in which a scattering strength was assigned by a Gaussian distribution. The variance of the Gaussian distribution is calculated from the backscattering cross-section of the specific tissue [21]. In order to test the imaging quality of the transducers, point targets and cyst phantoms were used is the transducer simulations.

Fig. (8) depicts the V(z) distributions for fused quartz for simulations based on the ray theory and the angular spectrum technique. The amplitudes were normalized prior to plotting. Subsequently, they were plotted as a function of the focal zone, z.

Fig. (8). (z) curve simulations of fused quartz substrates. (A) V(z) curve simulations of fused quartz substrates based on ray theory; (B) V(z) curve simulations of fused quartz substrates using wave theory or the angular spectrum technique.

Fig. (9) shows the V(z) curve simulations for a thin kidney tissue mounted on fused quartz substrate and for HeLa cells mounted on sapphire substrate, respectively. It shows that the V(z) distributions have a distinct pattern and period. The periods of V(z) curve for the kidney mounted on the fused quartz are different from those of the fused quartz substrate. The periods of the V(z) curve for HeLa cells attached on the sapphire substrate were also different from those of the kidney tissue. This demonstrates that the surface acoustic velocity of a thin biological specimen can also be measured with the V(z) curve technique.

Fig. (9). (z) curve simulations for biological specimens mounted on fused quartz and sapphire substrates. (A) V(z) curve for a kidney-derived cell mounted on a fused quartz substrate. The thickness of the cell was found to be 3 μm; (B) V(z) curve for a HeLa cell adherent on a sapphire substrate. The thickness of the cell was determined to be 5.27 μm.

The acoustic images of HeLa and MCF-7 cells were created using custom MATLAB scripts. Representative acoustic images from HeLa and MCF-7 cells are shown in Fig. (10).

Fig. (10). Acoustic microscopy of adherent cells under in vitro conditions. (A) A HeLa cell mounted on a glass substrate. The scale bar corresponds to 20 μm; (B) a single MCF-7 cell as imaged by SAM at a center frequency of 860 MHz.

The biomechanical properties of HeLa and MCF-7 cells (thickness, sound velocity, acoustic impedance, density and acoustic attenuation) were determined using the time-resolved technique as described above. In brief, the RF signals collected from HeLa and MCF-7 cells were analyzed with custom developed MATLAB scripts. These images were acquired at a center frequency of 860 MHz. The mechanical properties of HeLa and MCF-7 cells cultured in vitro are listed in Table 1.

Transducer simulations were performed for generating C-scan images of MCF-7 cells in a variety of phased array settings with varying number of elements. Representative C-scans are shown in Fig. (11).

Fig. (11). Array transducer simulations at 100 MHz and 600 MHz. (A) Simulated C-scan image of MCF-7 cells with a 256-element phased array transducer at a center frequency of 600 MHz; (B) simulated C-scan image of MCF-7 cells with a 20×20 element 2D fully populated array transducer at a center frequency of 100 MHz; (C) 3D image of MCF-7 cells with a 256-element phased array transducer at a center frequency of 600 MHz; (D) simulated 3D image of MCF-7 cells with a 20×20 elements 2D fully populated array transducer at a center frequency of 100 MHz.