Ultrasonography

Summary

Ultrasound (US) imaging is one of the principle tools of our project. We use ultrasonic instrument to detect the exist of bubbles and the effectiveness and the sensitivity of our end product. So, it is necessary to model the our product's response in the ultrasonic field, explain the phenomenon in our experiments and design better device for our detection. In summary, there are several fundamental purposes in this part:

1. Explain why we can detect such small bubbles in the Escherichia coli.

2. Dicuss how we can get more significant signal.

3. Give some tips for ultrasonic experiment.

Basic Concepts

Basic principle of B-mode ultrasound imaging

The ultrasonic testing method we use is a pulse-echo approach with a brightness-mode (B-mode) display, so does the most primary modern medical ultrasound test ^[1] Reference 1 Chan, V., & Perlas, A. (2011). Basics of ultrasound imaging. In Atlas of ultrasound-guided procedures in interventional pain management (pp. 13-19). Springer, New York, NY. . It involves transmitting small pulses of ultrasound waves from a transducer into the determinand. As the ultrasound waves penetrate body tissues of different acoustic impedances along the path of transmission, some are reflected back to the transducer (echo signals) and some continue to penetrate deeper. The echosignals returned from many sequential coplanar pulses are processed and combined to generate an image. Thus, an ultrasound transducer works both as a speaker (generating sound waves) and a microphone (receiving sound waves). The direction of ultrasound propagation along the beam line is called the axial direction, and the direction in the image plane perpendicular to axial is called the lateral direction. Usually only a small fraction of the ultrasound pulse returns as a reflected echo after reaching a body tissue interface, while the remainder of the pulse continues along the beam line to greater tissue depths.

Wavelength and Frequency

According to the formula $c=\lambda f$ (where $c$ is sound velocity, $\lambda$ is wavelength, $f$ is frequency), the wavelength and frequency of US are inversely related, ultrasound of high frequency has a short wavelength and vice versa. The ultrasound wave frequency is determined by the transducer, and the transducers' transmitting frequency we can get range from 4MHz to 44MHz.

Resolution

Spacial resolution ^[2] Reference 2 Ng, A., & Swanevelder, J. (2011). Resolution in ultrasound imaging. Continuing Education in Anaesthesia Critical Care & Pain, 11(5), 186-192. has great influence on our detection part. The ability of an ultrasound system to distinguish between two points at a particular depth in tissue, that is to say, axial resolution and lateral resolution, is determined predominantly by the transducer (Figure 1A).

Axial (also called longitudinal) resolution is the minimum distance that can be differentiated between two reflectors located parallel to the direction of ultrasound beam. Mathematically, it is equal to half the spatial pulse length spatial pulse length Spatial pulse length is the product of the number of cycles in a pulse of ultrasound and the wavelength. . Axial resolution is high when the spatial pulse length is short.

Lateral resolution, with respect to an image containing pulses of ultrasound scanned across a plane of tissue, is the minimum distance that can be distinguished between two reflectors located perpendicular to the direction of the ultrasound beam. Lateral resolution is high when the width of the beam of ultrasound is narrow.

Temporal resolution is the time from the beginning of one frame to the next; it represents the ability of the ultrasound system to distinguish between instantaneous events of rapidly moving structures, for example, during the cardiac cycle.

Contrast resolution refers to the ability to distinguish between different echo amplitudes of adjacent structures. Contrast agents are used when conventional ultrasound imaging does not provide sufficient distinction between two kinds of tissue. The contrast enhancement phenomenon arises because the impedance for ultrasound in gas is markedly different from that for soft tissue. When such a disparity occurs, ultrasound is reflected strongly from the microbubbles, thus enhancing contrast resolution and visualization of structures of interest. In our project, we use the principle of contrast agents (microbubbles in E.coli) to detect the intestinal tumor tissue.

Interaction with tissue

As the ultrasound waves travel through the tissue, they will interact with it. The ultrasound waves can be absorbed inside the tissue and be reflected, transmitted, scattered on the surface. Reflection-mode (the mode we use) ultrasound images display the reflectivity of the object. The reflectivity depends on both the object shape and the material in a complex way.Two important types of reflections are surface reflections and volumetric scattering.

Surface reflecting happens when ultrasound waves encounter Large planar surface (relative to wavelength λ) boundary between two materials of different acoustic impedances.(Figure 1B ^[3] Reference 3 Chapter, U. Reflection-mode ultrasound imaging. )

Figure 1B--Note:
$\bullet$ $p$ is the pressure (force per unit area) [ $kg/(m\cdot s^2)$ ]
$\bullet$ $Z=\rho_0 c$ is characteristic impedance (for a plane harmonic wave) [ $kg/(m^2\cdot s)$ ]
$\bullet$ $\rho_0$ is density [ $g/m^3$ ]
$\bullet$ $c$ is wave velocity [ $m/s$ ]

Boundary conditions:
$\bullet$ Equilibrium total pressure at boundary: $p_{ref}+p_{inc}=p_{trn}$ (Pressure must be continuous at the surface)
$\bullet$ Snell's Law: $\sin \theta_{inc}/\sin \theta_{trn} = c_1/c_2$
$\bullet$ Angle of reflection: $\theta_{ref}=-\theta_{inc}$

The pressure reflectivity at the surface is $$R = \frac{{{{\rm{p}}_{ref}}}}{{{p_{inc}}}} = \frac{{{Z_2}\cos {\theta _{inc}} - {Z_1}\cos {\theta _{trn}}}}{{{Z_2}\cos {\theta _{inc}} + {Z_1}\cos {\theta _{trn}}}}$$

When interface parallel to wavefront, $\theta_{inc}=\theta_{ref}=\theta_{trn}=0$. Thus the reflectivity or pressure coefficient for waves at normal incidence to surface is: $$R = {R_{12}} = - {R_{21}} = \frac{{{p_{ref}}}}{{{p_{inc}}}} = \frac{{{Z_2} - {Z_1}}}{{{Z_1} + {Z_2}}}$$

If $Z_1$ is approximate to $Z_2$, $$R\approx \frac{{\Delta Z}}{{2{Z_0}}}$$ where $Z_0$ denotes the typical acoustic impedance of soft tissue. $-1\leq R\leq1$.

The intensity of an ultrasonic wave is $I=p^2/(2Z)$. Thus the reflected and transmitted intensities are $$\frac{{{I_{ref}}}}{{{I_{inc}}}} = {\left( {\frac{{{Z_2} - {Z_1}}}{{{Z_2} + {Z_1}}}} \right)^2}{\rm{ , }}\frac{{{I_{trn}}}}{{{I_{inc}}}} = \frac{{4{Z_1}{Z_2}}}{{{{\left( {{Z_1} + {Z_2}} \right)}^2}}}$$

On a microscopic level (less than or comparable to an ultrasonic wavelength), mechanical inhomogeneities inherent in tissue will scatter sound. $\eta$ = Backscatter coefficient = Backscatter cross section/unit volume.

According to the relationship of ultrasound wavelength and the object's size, the interaction can be divided into three types:

If wavelength is much larger than the size ($\lambda \gg R\;$), reflection, transmission should be considered first;

If wavelength is much smaller than the size ($\lambda \ll R\;$ ), Rayleigh scattering should be considered first;

If wavelength is approximate to the size ($\lambda \approx R\;$ ), diffraction theory (similar to the light) should be considered first.

Bubbles' Model 1

To obtain the accurate formula for bubbles' response we must specialize our calculations for objects of relatively simple shape. In this model ^[4] Reference 4 Morse, P. M., & Ingard, K. U. (1986). Theoretical acoustics. Princeton university press. , we suppose the micro gas bubble as the rigid, motionless sphere of radius R, centered at the origin. (Figure 2A)

According to the TEM figure in the literature ^[5] Reference 5 Bourdeau, R. W., Lee-Gosselin, A., Lakshmanan, A., Farhadi, A., Kumar, S. R., Nety, S. P., & Shapiro, M. G. (2018). Acoustic reporter genes for noninvasive imaging of microorganisms in mammalian hosts. Nature, 553(7686), 86. , we suppose $R = 100nm = 1 \times {10^{ - 7}}m$. We fix the bacteria in agarose gel, and the bubbles are in the bacterial cytoplasm, therefore it is reasonable to suppose the environmental sound velocity is average sound velocity of biological soft tissue ($1540m/s\;$ ). According to the transducer we used, ultrasound frequency $f = 21MHz$, ultrasound wavelength $\lambda=c/f=73.3\mu m$. It is obvious that the size of bubbles is far less than the spacial resolution. However, why we can get the signal of bubbles?

$\lambda \ll R$, therefore we should consider Rayleigh scattering first. The expression for a plane wave traveling to the right along the polar axis is $${p_p} = A{e^{ik(r\cos \theta - ct)}} = A\sum\limits_{m = 0}^\infty {(2m + 1){i^m}{P_m}(\cos \theta ){j_m}(kr){e^{ - 2\pi ivt}}}$$

where $A = \sqrt {{\rho _0}cI}$, $P_m$ is m-order Lagrangian function, $j_m$ is m-order the real part of the ball Bessel function. The expression for the wave scattered from a sphere of radius R whose center is the polar origin is $${p_s} = - A\sum\limits_{m = 0}^\infty {(2m + 1){i^{m + 1}}{e^{ - i{\delta _m}}}} \sin {\delta _m}{P_m}(\cos \theta ) \times [{j_m}(kr) + i{n_m}(kr)]{e^{ - 2\pi ivt}}$$

where the angles $\delta _m$ is in connection with the radiation from a sphere. The intensity of the scattered wave and the total power scattered are $${I_S} \simeq \frac{{{a^2}I}}{{{r^2}}}\frac{1}{{{k^2}{a^2}}}\sum\limits_{m,n = 0}^\infty {(2m + 1)(2n + 1) \times \sin {\delta _m}} \sin {\delta _n}\cos ({\delta _m} - {\delta _n}){P_m}(\cos \theta ){P_n}(\cos \theta )$$ $${I_S} \simeq \left\{ \begin{array}{l} \frac{{16{\pi ^4}{v^4}{R^6}I}}{{9{c^4}{r^2}}}{(1 - 3\cos \theta )^2}{\rm{ }}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;ka \ll 1{\rm{ }}\\ I\left[ {\frac{{{a^2}}}{{4{r^2}}} + \frac{{{a^2}}}{{4{r^2}}}{{\cot }^2}(\frac{\theta }{2}){J_1}^2(kR\sin \theta )} \right]{\rm{ }}\;\;\;\;\;\;\;\;ka \gg 1 \end{array} \right.$$ $$\prod _S = 2\pi {R^2}I\frac{2}{{{k^2}{R^2}}}\sum\limits_{m = 0}^\infty {(2m + 1){{\sin }^2}{\delta _m} \simeq \left\{ \begin{array}{l} \frac{{256{\pi ^5}{R^6}}}{{9{\lambda ^4}}}I{\rm{ }}\;\;\;\;\;\;(\lambda \gg 2\pi R)\\ 2\pi {R^2}I{\rm{ }}\;\;\;\;\;\;\;\;\;(\lambda \ll {\rm{2}}\pi {\rm{R})} \end{array} \right.} $$ where $k$ is wave number,$k=2\pi /\lambda$ .

Under our conditions, $k=2\pi/\lambda=8.57 \times {10^4} m^{-1}$

$kR\ll 1$ and $\lambda \gg 2\pi R$ , therefore $$\frac{{{I_S}}}{I} \simeq \frac{{16{\pi ^4}{v^4}{R^6}}}{{9{c^4}{r^2}}}{(1 - 3\cos \theta )^2}$$ $$\frac{{\prod_S}}{I}= \frac{256\pi^5 R^6}{9\lambda^4} $$ $$\frac{{\prod_S}}{R^2I}= {\frac{16}{9}}{(kR)}^2 $$

Figure 2C-D shows the polar curve of the scattered intensity as a function of the angle of scattering $\theta$, for different values of $kR$, a curve $\frac{{\prod_S}}{R^2I}$ as a function of $kR$.

The maximum of $\frac{{{I_S}}}{I}$ at the surface of the ball is $5.99\times 10^{-10}\;$, the total power scattered ratio $\frac{{\prod_S}}{I}$ is about $4.7\times 10^{-24} m^2 $. Therefore, according to this model, the scattered sound of a single bubble is so small that we can't detect it. But with a large amout of bubbles gathering, the scattered ultrasound waves superposing, the transducer may perceive the sound waves and we can get the signal of bubbles.

According to the formula, the scattering intensity is in inverse proportion to the fourth power of the sound velocity and the square of the distance from the center of the sphere, in proportional to the fourth power of the wave frequency and six power of the object's size. Therefore, if we want to get higher signal, we should use higher frequency ultrasound and try to enlarge the size of the object.

Model 2

Based on the discussion above, we can solve the scattering ultrasound wave produced by the bubbles. In that model, we suppose the bubble as rigid, motionless sphere, ignoring the bubble's material composition and deformation,that is to say, a gas-bubble and a metal ball of the same size will get the same results using the modeling 1. However, as we all known, gas get much stronger sound wave reflection and scattering than solids. In this model, we want to discuss the difference if we take the physical properties of gas in to consideration.

In the ultrasonic field, the gas-bubbles will deforms under the action of sound pressure. Their scattering feature like cross-section change. A linear model for ultrasound scattering from gas-bubbles is described by Medwin ^[6] Reference 6 Medwin, H. (1977). Counting bubbles acoustically: a review. Ultrasonics, 15(1), 7-13. , and the effect of encapsulating the gas in a shell was studied by Fox and Herzfield ^[7] Reference 7 Fox, F. E., & Herzfeld, K. F. (1954). Gas bubbles with organic skin as cavitation nuclei. The Journal of the Acoustical Society of America, 26(6), 984-989. and de Jong and Hoff ^[8] Reference 8 de Jong, N., Hoff, L., Skotland, T., & Bom, N. (1992). Absorption and scatter of encapsulated gas filled microspheres: theoretical considerations and some measurements. Ultrasonics, 30(2), 95-103. ^{, [9]} Reference 9 de Jong, N., & Hoff, L. (1993). Ultrasound scattering properties of Albunex microspheres. Ultrasonics, 31(3), 175-181. . In these models,the acoustic scattering cross-section $\sigma_S$ of a gas-bubble as function of radius and frequency ^[10] Reference 10 Hoff, L., Sontum, P. C., & Hoff, B. (1996, November). Acoustic properties of shell-encapsulated, gas-filled ultrasound contrast agents. In Ultrasonics Symposium, 1996. Proceedings., 1996 IEEE (Vol. 2, pp. 1441-1444). IEEE. is: $${\sigma _S} = \frac{{4\pi {R^2}}}{{{{({{({\omega _0}/\omega )}^2} - 1)}^2} + {\delta ^2}}}$$

where $\sigma_S=$ scattering cross-section, $\omega=$ angular frequency of the incoming sound, $\omega_0=$ resonance frequency, $\delta=$ damping constant, $R=$ is particle radius. The Kelvin-Voigt model for visco-elastic solids is used to describe the bulk modulus. In frequency domain, it takes the form $$K_C=K+i\omega \mu_K$$ where $K$ and $\mu_K$ are properties of the particle material, with values determined by experiments. Replacing the bulk modulus of air in Medwin's model with K, the resonance frequency $\omega_0\;$ and damping coefficient of the particle is: $${\omega_0}^2=\frac{3K}{R^2\rho}$$ $$\delta=\frac{3\mu_K}{\omega R^2\rho}+kR+\delta_v$$ where $\rho=$ density of the surrounding material, $\delta_v=$ damping due to viscosity in the surrounding liquid.

In our project, bacteria used for ultrasonic bubbles detecting are fixed in the agarose gel. Considering the similar acoustic properties of different material composing the environment surrounding the bubbles (Figure 3A), we simplify the model of the gas-bubble and suppose it is surrounded by gel (Figure 3B).

Although parameters such as elastic modulus can only be obtained accurately by experiment, we can estimate them by experience and existing literature ^[11] Reference 11 Watase, M., & Nishinari, K. (1980). Rheological properties of agarose-gelatin gels. Rheologica Acta, 19(2), 220-225. .

$\omega_0\sim 10^5 P $
$\rho\sim 10^3 kg/{m^3}$
$\omega\sim 10^8 Hz \sim 10^2 MHz$
$\sigma_S\sim {\omega}^2\times 10^{-30} \sim f^2 \times 10^{-28} m^2\cdot s^2 $

It shows that the scattering cross-section is approximately proportional to the square of the ultrasonic frequency. If the frequency of ultrasound wave is $21MHz$, $$\frac{\prod_S}{I} = \sigma_S\sim 10^{-14} m^2$$

Comparing to the result $\frac{\prod_S}{I}\sim 5\times10^{-24} m^2 $ in the model 1, the result in model 2 is about $10^9$ times that of model 1. The ultrasonic imaging will confirm our conclusion (Figure 3C-E). According to this conclusion, the higher the ultrasonic frequency, the stronger the signal may be. Therefore, we should better use higher frequency to do the experiment. However, ultrasonic transmission capability is negatively correlated with frequency. Ultra-high frequency ultrasound cannot meet our requirements for depth of detection. In our team, the available ultrasonic frequencies are 4MHz-10MHz, 11.2MHz, 21MHz, 44MHz. we eventually chose 21MHz.

Discussion

We raised three questions in the summary of this part model and we'll answer them now.

1. The signal in ultrasonic imaging is produced by lots of bubbles' scattering ultrasound wave. The size of gas-bubbles is much smaller than the spacial resolution, so we can't tell them apart. What we detect is the bubbles' groups.

2. Combining the two models above, to get the higher signal, we should use transducer with higher frequency. The 21MHz frequency is the best choice for our team. And due to the resonance characteristics of the bubble, when the ultrasonic energy is sufficiently large, the bubble may be broken. We can use this feature to tell if the signal is generated by bubbles.

3.
$\bullet $Gas has very unique acoustic properties, we should put coupling agent on the surface of the detected object.
$\bullet $For experimental device, the less the interface, the better.
$\bullet $Gas can produce a lot of noise, so try to be free of gas for all detected parts.

Reference

[1] Chan, V., Perlas, A., & Narouze, S. N. (2011). Atlas of ultrasound-guided procedures in interventional pain management. Chan V, Perlas A. Basics of Ultrasound Imaging. New York: Springer Science Business Media, 13-19.

[2] Ng, A., & Swanevelder, J. (2011). Resolution in ultrasound imaging. Continuing Education in Anaesthesia Critical Care & Pain, 11(5), 186-192.

[3] Chapter, U. Reflection-mode ultrasound imaging.

[4] Morse, P. M., & Ingard, K. U. (1986). Theoretical acoustics. Princeton university press.

[5] Bourdeau, R. W., Lee-Gosselin, A., Lakshmanan, A., Farhadi, A., Kumar, S. R., Nety, S. P., & Shapiro, M. G. (2018). Acoustic reporter genes for noninvasive imaging of microorganisms in mammalian hosts. Nature, 553(7686), 86.

[6] Medwin, H. (1977). Counting bubbles acoustically: a review. Ultrasonics, 15(1), 7-13.

[7] Fox, F. E., & Herzfeld, K. F. (1954). Gas bubbles with organic skin as cavitation nuclei. The Journal of the Acoustical Society of America, 26(6), 984-989.

[8] de Jong, N., Hoff, L., Skotland, T., & Bom, N. (1992). Absorption and scatter of encapsulated gas filled microspheres: theoretical considerations and some measurements. Ultrasonics, 30(2), 95-103.

[9] de Jong, N., & Hoff, L. (1993). Ultrasound scattering properties of Albunex microspheres. Ultrasonics, 31(3), 175-181.

[10] Hoff, L., Sontum, P. C., & Hoff, B. (1996, November). Acoustic properties of shell-encapsulated, gas-filled ultrasound contrast agents. In Ultrasonics Symposium, 1996. Proceedings., 1996 IEEE (Vol. 2, pp. 1441-1444). IEEE.

[11] Watase, M., & Nishinari, K. (1980). Rheological properties of agarose-gelatin gels. Rheologica Acta, 19(2), 220-225.

$$\frac{{d[{C_{in}}]}}{{dt}} = {k_{diff}}[{C_{out}} - {C_{in}}] + {k_{diss}}[{C_{in}}] \times [pBAD] - \gamma [{C_{in}}]$$ $$\frac{{d[{C_{in}}pBAD]}}{{dt}} = {k_{ass}}[{C_{in}}] \times [pBAD] - {k_{diss}}[{C_{in}}pBAD]$$ $$\frac{{d[pBAD]}}{{dt}} = {k_{diss}}[pBAD] - {k_{ass}}[{C_{in}}] \times [pBAD] - {K_{\deg }}[pBAD]$$ $$\frac{{d[Attenuator]}}{{dt}} = {k_{trans}}[Attenuator\_mRNA] - \gamma [Attenuator$$ $$\frac{{d[X174E\_mRNA]}}{{dt}}={k_{atten}}^{[Attenuator]}\times k_{maxprodRNA}(\frac{{[C_{in}pBAD]^n}}{{(k_{\frac{1}{2}maxprodRNA})^n+[C_{in}pBAD]^n}})-k_{deg}[X174E\_mRNA]$$ $$\frac{{d[Attenuator\_mRNA]}}{{dt}} = k_{maxprodRNA}(\frac{{[C_{in}pBAD]^n}}{{(k_{\frac{1}{2}maxprodRNA})^n+[C_{in}pBAD]^n}})-k_{deg}[Attenuator\_mRNA]$$ $$\frac{{X174E}}{{dt}}=k_{trans}[X174EmRNA]-\gamma [X174E]$$