Acoustic Microscopy

Abstract

Biological ultrasonic microscopy, also known as biological scanning acoustic microscope, provides quantitative acoustic parameters like sound speed and characteristic acoustic impedance that are relevant to elastic properties.

Appendix

1.1.1 Derivation of Eq. (1.1)

Sound speed of the soft tissue can be assessed by either time domain or frequency domain analysis. An example of frequency domain analysis is as follows: The spectrum of the reflection from the object slice is normalized by the reference waveform. Assuming f_m as one of the minimum and maximum points in the intensity spectrum, and ϕ_m as the corresponding phase angle, the phase difference between the two reflections at the minimum point is (2n − 1)π, giving

$$ 2\pi f_{m} \times \frac{2d}{{c_{0} }} = \phi_{m} + (2n - 1)\pi , $$

(1.4)

where d, c₀, and n are the tissue thickness, sound speed of the water, and a nonnegative integer, respectively. The phase difference at the maximum point is 2nπ, giving

$$ 2\pi f_{m} \times \frac{2d}{{c_{0} }} = \phi_{m} + 2n\pi . $$

(1.5)

The phase angle ϕ_m can be expressed by

$$ 2\pi f_{m} \times 2d\left( {\frac{1}{{c_{0} }} - \frac{1}{c}} \right) = \phi_{m} , $$

(1.6)

since ϕ_m is the phase difference between the wave passed through the distance 2d with sound speed c and that passed though the corresponding distance with sound speed c₀. Equation (1.4) gives

$$ d = \frac{{c_{0} }}{{4\pi f_{m} }}\left\{ {\phi_{m} + (2n - 1)\pi } \right\} $$

(1.7)

for the minimum point. For the maximum point, Eq. (1.5) gives

$$ d = \frac{{c_{0} }}{{4\pi f_{m} }}\left( {\phi_{m} + 2n\pi } \right). $$

(1.8)

Sound speed is finally calculated as

$$ c = \left( {\frac{1}{{c_{0} }} - \frac{{\phi_{m} }}{{4\pi f_{m} d}}} \right)^{ - 1} , $$

which corresponds to Eq. (1.1).

1.1.2 Derivation of Eq. (1.2)

Hereafter, the signal component at an arbitrary frequency will be symbolized by S. Considering the reflection coefficient, the target signal S_target can be described as

$$ S_{\text{target}} = \frac{{Z_{\text{target}} - Z_{\text{sub}} }}{{Z_{\text{target}} + Z_{\text{sub}} }}S_{0} , $$

(1.9)

where S₀ is the transmitted signal and Z_target and Z_sub are the acoustic impedances of the target and substrate, respectively. On the other hand, the reference signal can be described as

$$ S_{\text{ref}} = \frac{{Z_{\text{ref}} - Z_{\text{sub}} }}{{Z_{\text{ref}} + Z_{\text{sub}} }}S_{0} , $$

(1.10)

where Z_ref is the acoustic impedance of the reference material. In case of using water as the reference, its acoustic impedance was assumed to be 1.5 × 10⁶ Ns/m³. One can measure S_target and Z_ref; however, S₀ cannot be directly measured. The acoustic impedance of the target is subsequently calculated as a solution of the simultaneous equations for Z_target and S₀, as

$$ Z_{\text{target}} = \frac{{1 + \frac{{S_{\text{target}} }}{{S_{0} }}}}{{1 - \frac{{S_{\text{target}} }}{{S_{0} }}}}Z_{\text{sub}} = \frac{{1 - \frac{{S_{\text{target}} }}{{S_{\text{ref}} }} \cdot \frac{{Z_{\text{sub}} - Z_{\text{ref}} }}{{Z_{\text{sub}} + Z_{\text{ref}} }}}}{{1 + \frac{{S_{\text{target}} }}{{S_{\text{ref}} }} \cdot \frac{{Z_{\text{sub}} - Z_{\text{ref}} }}{{Z_{\text{sub}} + Z_{\text{ref}} }}}}Z_{\text{sub}} , $$

(1.11)

assuming that S₀ is constant throughout the observation process.

1.1.3 Derivation of Eq. (1.3)

Phases of the two reflection signals (D and R) are represented as

$$ \phi_{D} = 2(\ell - z)k_{0} ,\phi_{R} = 2\left( {\ell - \frac{z}{{\cos \,\theta_{R} }}} \right)k_{0} + 2zk_{R} \tan \theta_{R} , $$

(1.12)

where wave numbers k₀ and k_R are defined as $k_{0} = 2\pi f/c_{0}$ and $k_{R} = 2\pi f/c_{R}$, respectively. Relative phase shift per unit distance of z is represented as

$$ \phi (z) = (\phi_{D} (z) - \phi_{R} (z))/z = 2\left\{ {k_{0} (1 - 1/\cos \theta_{R} + k_{R} \tan \theta_{R} } \right\}. $$

(1.13)

Two signals emphasize together when $\left( {\phi_{D} (z) - \phi_{R} (z)} \right)$ is 2π. Hence, periodical change appears on the V(z) curve.

The interval Δz between minima can be represented by

$$ \frac{2\pi }{\Delta z} = 2\left( {\frac{2\pi f}{{c_{0} }} \cdot \frac{{\cos \,\theta_{R} - 1}}{{\cos \,\theta_{R} }} + \frac{2\pi f}{{c_{R} }} \cdot \frac{{\sin \,\theta_{R} }}{{\cos \,\theta_{R} }}} \right). $$

(1.14)

Applying $\theta_{R} = \sin^{ - 1} (c_{0} /c_{R} )$ (Snell’s law),

$$ \frac{1}{\Delta z} = \frac{{2f(1 - \cos \,\theta_{R} )}}{{c_{0} }}, $$

(1.15)

thus

$$ \cos \,\theta_{R} = 1 - \frac{{c_{0} }}{2f\Delta z}. $$

(1.16)

Speed of LSAW is determined as

$$ c_{R} = c_{0} /\sin \,\theta_{R} = c_{0} \left\{ {1 - \left( {1 - \frac{{c_{0} }}{2f\Delta z}} \right)^{2} } \right\}^{ - 1/2} , $$

which corresponds to Eq. (1.3).