# Biomechanical Sensing Using Gas Bubbles Oscillations in Liquids and Adjacent Technologies: Theory and Practical Applications

^{1}

^{2}

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*Biosensors*)

## Abstract

**:**

## 1. Introduction and Motivation

## 2. Physics of Acoustically Driven Bubble Oscillations

#### 2.1. Acoustically Driven Oscillations of a Single Bubble in Unbounded Liquid

#### Extensions of Rayleigh-Plesset Equation

#### 2.2. Single Bubble Oscillating near a Boundary

#### 2.2.1. Bubble Oscillating near a Solid Wall

#### 2.2.2. Bubble Oscillating near an Elastic Wall

#### 2.3. Interaction of Oscillating Bubbles in a Bubble Cluster

## 3. Determining Mechanical Properties of Cells, Bacteria and Biological Yissues

#### 3.1. Brillouin Light Scattering Spectroscopy

#### 3.2. Scanning Acoustic Microscopy

#### 3.3. Deformation of Cells and Bacteria

#### 3.3.1. Mechanical Resonance Properties of Cells And Bacteria

#### 3.3.2. Deformation of Cells and Bacteria by Bubbles

## 4. Acoustic Frequency Combs

#### 4.1. Physical Principles of Operation of Bubble-Based Acoustic Frequency Combs

#### 4.2. Spectrally Wide Acoustic Frequency Combs

#### 4.3. Application of Bubble-Based AFCs in Biosensing

## 5. Applications of Gas Bubbles in Photoacoustic and Acousto-Optical Biosensors

#### Acousto-Optical Sensors Using Bubbles

## 6. Gas Bubble Sensors and Artificial Intelligence Algorithms

## 7. Bubble Generation

## 8. Conclusions and Outlook

## Funding

## Conflicts of Interest

## Abbreviations

atomic force microscopy | AFM |

acoustic frequency comb | AFC |

artificial intelligence | AI |

blood-brain barrier | BBB |

Brillouin light scattering | BLS |

in vitro fertilisation | IVF |

Keller-Miksis | KM |

mean squared error | MSE |

machine learning | ML |

optical frequency comb | OFC |

partial differential equation | PDE |

physics-informed neural network | PINN |

Rayleigh-Plesset | RP |

scanning acoustic microscopy | SAM |

signal-to-noise | SNR |

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**Figure 1.**Illustration of bubble oscillation in the bulk of water and near a surface such as the wall of a blood vessel discussed in Section 1. Similar behaviour is observed when a bubble interacts with the wall of an individual biological cell (e.g., [32]). (

**a**) Schematic radius-vs-time diagram for a bubble oscillating in the bulk of water. Bubble shapes at different times are shown above the curve during a single oscillation cycle. The pressure inside a bubble is high at the beginning and at the end of the oscillation cycle and is low in the middle. (

**b**) Illustration of the principle of drug delivery through the blood-brain barrier (BBB) using bubbles trapped inside a small blood vessel of the brain. Bubble oscillations are driven by an external ultrasound wave transducer and the drug (small spheres in the picture) pass through the wall of a blood vessel when the bubble either reaches its maximum radius or collapses and forms a water jet. Reproduced from [10] under the terms of a Creative Commons license. (

**c**) Representative computational axisymmetric profiles of a microbubble during its expansion and collapse near a solid surface. Parameter ${T}_{a}$ is the period of the sinusoidal acoustic pressure wave with $f=1.5$ MHz. The pressure wave incident along the z-axis towards the surface has the amplitude of 200 kPa. Reproduced from [33] under the terms of a Creative Commons license. (

**d**) Experimental observation of a bubble collapse. The first frame shows the bubble at its maximum radius. The bubble shrinks and moves towards the boundary in the second frame and then it collapses forming a jet directed towards the surface. (

**e**) Experimental observation of bubble streams that originate mostly from the surface of a transducer and move towards the focal point near the top of the left image. A schematic of a streamer trace (the right image) shows how bubbles emerge from the motes–the source of the streamers. Reproduced with permission of The Royal Society (UK) from [34]. Copyright 2015.

**Figure 2.**(

**a**) Schematic of the operating principle of gas bubble-based sensors for probing mechanical properties of an elastic wall (e.g., a wall of a blood vessel wall or a biological cell). In a practical realisation, microbubble oscillations are driven by ultrasound waves. The resonance frequency shift $\Delta f$ caused by the approach of the bubble to a wall is detected using a technically simple integrated electronic circuit. At the post-processing step, the measured value of $\Delta f$ is correlated with the values of material parameters that characterise elastic properties of the wall as discussed in Section 2. (

**b**) Illustration of complementary biochemical sensing properties of gas bubble sensors shown in panel (

**a**). The resonance frequency f of a microbubble (stage (i)) first increases when its surface is functionalised using a biorecognition ligand (e.g., an antibody, stage (ii)). It increases further when the bubble covered by an antibody captures analyte (stage (iii)), thereby resulting in a modification $\Delta f$ of the resonant frequency that is detected using the same approach as in panel (

**a**).

**Figure 3.**Schematic of a spherical bubble in the bulk of liquid. Parameters used to derive the RP equation are defined in the text.

**Figure 4.**A portion of a spherical bubble surface with physical parameters used to derive the RP equation.

**Figure 5.**Schematic view of a bubble near a solid wall. The bubble on the right is the ‘mirror’ image of the real bubble.

**Figure 7.**(

**a**,

**b**) Phase-contrast microscopy and (

**c**,

**d**) co-registered BLS spectrospcopy-based images of a mouse fibroblast cell before (

**a**,

**c**) and after (

**b**,

**d**) a hyperosmotic shock. The scale bars of the phase contrast images are 10 $\mathsf{\mu}$m. The false colour map of the BLS-based images encode the measured Brillouin frequency shifts in the range from 7.6 to 8.1 GHz. Reproduced from [139] under the terms of a Creative Commons Attribution (CC-BY) License. (

**e**) Representative anti-Stokes Brillouin peaks of the BLS spectra obtained for a healthy tissue (black curve), normal non-regressing melanoma (red curve) and regressing melanoma (blue curve). (

**f**) Close-up view of the spectra shown in panel (

**e**): dots correspond to the raw experimental data and lines show the respective Lorentzian function fit of the raw data. Reproduced from [23]. Copyright 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement.

**Figure 8.**(

**a**) Illustration of the operating principles of SAM. The acoustic echo signal with amplitude ${A}_{1}$ originating from the surface of the sample arrives at time ${t}_{1}$. The echo signal with amplitude ${A}_{2}$ is reflected from the interface between the sample and its substrate and is received at time ${t}_{2}$. From the arrival time of each maximum and their respective amplitudes, the elastic and mechanical parameters of the sample can be obtained. (

**b**) Image of MCF-7 breast cancer cells obtained using SAM at 1 GHz acoustic frequency. (

**c**) Acoustic wave attenuation distribution across a cell. (

**d**) Optical micrograph of a cell. (

**e**) Fluorescence-based micrograph of the same MCF-7 cell, where the stained cell nucleus overlaps with the darker area in the image in panel (

**c**). The scale bar in all panels corresponds to 15 $\mathsf{\mu}$m. Reproduced from [142] under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0).

**Figure 9.**Cross-sections of theoretical 3D shapes assumed by a liquid drop at ${T}_{l}=0$ (dashed curves), ${T}_{l}=1$ (solid curves) and ${T}_{l}=1/2$ (dotted curve), where ${T}_{l}={\omega}_{l}t$ is given in the units of $\pi $ radians and ${T}_{l}=1$ corresponds to a half of a drop oscillation period corresponding to mode number l.

**Figure 10.**Illustration of the interaction of an oscillating bubble with a biological cell in the ultrasound acoustic pressure field. (

**a**) Since a cell typically has a very small acoustic cross-section, the effect of a plain acoustic pressure wave on it at the frequency of its fundamental shape resonance is negligibly small. (

**b**) Using a microbubble, one can efficiently excite the shape oscillations of the cell since the bubble acts as a point-like source of acoustic pressure waves. As a result, the cell can be deformed and its mechanical properties can be measured in a non-contact manner.

**Figure 11.**(

**Right**) Sketch of an experimental setup for oscillating bubble-based measurements with in vitro porcine eye lenses. Microbubbles are generated across the equatorial plane of a lens using laser-induced optical breakdown. Piezoelectric transducers irradiate bubbles with a MHz-range ultrasound and receive pressure waves scattered by them thereby allowing one to measure both the displacement of bubbles with respect to their initial positions and their temporal responses. (

**Left**) Typical experimental bubble displacement curve. Along with the exponential time constant of the bubble temporal responses, knowledge of the maximum displacement enables determining mechanical properties of the lens tissues. Reproduced from [167]. Copyright 2007, with permission from Elsevier.

**Figure 12.**(

**a**) Measured acoustic response of a bubble. The time between the vertical dashed lines is $\Delta T=1/{f}_{nat}\approx 0.6$ ms, where ${f}_{nat}$ is the natural frequency of the bubble oscillations (see [184] for details). The insets show a closeup of the waveforms and demonstrate the amplitude modulation. (

**b**) Experimental AFC spectra obtained using gas bubbles in water insonated with an ${f}_{0}=24.6$ kHz sinusoidal signal of increasing pressure amplitude $\alpha $ =1.15, 3.75, 4, 4.2, 4.3, 7.5 and 11.5 kPa. The scattered pressure values (in dB) are given along the vertical axis with a vertical offset of 30 dB between the spectra. Reproduced from [184] published by Springer Nature under the terms of the Creative Commons CC BY license.

**Figure 13.**Columns (from left to right) show the AFC spectra produced by individual bubbles within clusters consisting of two, three and four bubbles with the equilibrium radii ${R}_{n0}=1.95/n$ mm, where n is the bubble index in the cluster. The number of panels in each column corresponds to the total number of bubbles. The red dashed lines in each panel show the spectra of individual non-interacting stationary bubbles with identical equilibrium radii. Computational parameters are given in [86]. Reproduced from [86]. Copyright 2021 by the American Physical Society.

**Figure 14.**(

**a**) Sketch of an OFC-based spectroscopy technique. The OFC as a broadband light source interrogates an absorbing sample and a spectrometer analyses the transmission spectrum revealing the sample’s molecular composition. (

**b**) Sketch of a bubble-generated AFC-based spectroscopy technique implemented as an extension of mechanical property measurements of a biological cell illustrated in Figure 10b. AFC as a broadband acoustic pressure wave source interrogates a mechanically oscillating cell and an ultrasound spectrometer analyses the transmission spectrum that provides information about mechanical properties of the cell and its individual organelles.

**Figure 15.**Diagram of a step-by-step remote activation of photoacoustic contrast agents via two physical mechanisms: vaporisation of liquid nanodroplets (steps 2 and 3) and thermal expansion caused by plasmonic nanoantennas (steps 4 and 5). The formed microbubble (step 6) provides contrast for ultrasound imaging. Reproduced from [194] with permission of Springer Nature. Copyright 2012.

**Figure 16.**(

**a**) Illustration of the operating principles of a nanoantenna-laden microbubble contrast agent. Superior photoacoustic properties are achieved owing to high optical absorption of nanoantennas embedded into the microbubble shell. These ultrasmall nanoantennas do not affect acoustic properties of a microbubble allowing it to serve as an efficient contrast agent for ultrasound waves. (

**b**) Amplitude of the photoacoustic response obtained at various fluences of a laser light for the same nanoantenna-laden microbubble contrast agents and free-standing nanoantennas. The vertical double arrow indicates the enhancement of a photoacoustic signal due to the integration of nanoantennas with a microbubble. Reproduced from [195] with permission of Royal Society of Chemistry. Copyright 2013.

**Figure 17.**(

**a**) Schematic view of a bubble-on-fibre system for detecting acoustic pressure waves. The system consists of a conventional single-mode optical fibre used to deliver both heating and interrogation light beams to a bubble. A thin gold film is used to absorb light and produce Joule heat needed to generate a microbubble. (

**b**) Temporal microbubble growth at heating light powers of 150 and 200 mW. The inset shows snapshots of a growing microbubble. (

**c**) Bubble decay after the heating light was turned off. The green curve with squares shows the behaviour of a bubble that reached radius of 101 $\mathsf{\mu}$m. The other two curves correspond to smaller bubbles with radii of 79 $\mathsf{\mu}$m and 52 $\mathsf{\mu}$m, respectively. Reproduced from [212]. Copyright 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement.

**Figure 18.**(

**a**) Schematic diagram of a bubble-on-fibre system employing an optical fibre with the tip covered by a thin metal film patterned with nanoscale through holes. Such a film can be used (i) to generate a bubble by laser heating, (ii) to create conditions for extraordinary optical transmission (see the main text) and (iii) to host bubbles inside the holes. (

**b**) Simulated optical response of a single water-filled hole in a metal film containing a bubble. The intensity of light interacting with the hole-bubble system becomes modulated due to bubble oscillations caused by an ultrasound pressure wave. Reproduced from [218]. Copyright 2017 by the American Physical Society.

**Figure 19.**Frequency spectrum of a single bubble with a 0.85 mm radius in water obtained using an optical interferometry technique. The spectrum reveals the presence of (non-spherical) shape oscillation modes in the frequency range from approximately 100 Hz to 1.5 kHz and of a volume oscillation (spherical) mode at approximately 3.55 kHz. The inset shows schematically the detection principle, where a laser beam focused on a bubble undergoes reflections at the curved interfaces (points S1 and S2) generating an interference pattern in the direction opposite to that of the incident laser beam. Reproduced with permission from [221]. Copyright 2022 Cambridge University Press.

**Figure 20.**Schematic of PINN for solving forward problems with conventional (the left dotted box) and physics-informed (the right dotted box) neural networks with trainable weights w and biases b and a nonlinear activation function $\sigma $. PINNs integrate the measured and observed information with that produced using physical models by embedding the corresponding PDEs into the loss function of a neural network at a training stage. A set of sampled inputs $({x}_{i},{t}_{i})$ is passed through the network, then the Jacobian of the neural network’s output is computed with respect to these inputs, and finally, the residual of PDEs is computed and added as an extra term in the loss function. The network is trained by minimising the loss via a gradient-based optimisation method until it is smaller than a set threshold $\u03f5$.

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**MDPI and ACS Style**

Maksymov, I.S.; Huy Nguyen, B.Q.; Suslov, S.A.
Biomechanical Sensing Using Gas Bubbles Oscillations in Liquids and Adjacent Technologies: Theory and Practical Applications. *Biosensors* **2022**, *12*, 624.
https://doi.org/10.3390/bios12080624

**AMA Style**

Maksymov IS, Huy Nguyen BQ, Suslov SA.
Biomechanical Sensing Using Gas Bubbles Oscillations in Liquids and Adjacent Technologies: Theory and Practical Applications. *Biosensors*. 2022; 12(8):624.
https://doi.org/10.3390/bios12080624

**Chicago/Turabian Style**

Maksymov, Ivan S., Bui Quoc Huy Nguyen, and Sergey A. Suslov.
2022. "Biomechanical Sensing Using Gas Bubbles Oscillations in Liquids and Adjacent Technologies: Theory and Practical Applications" *Biosensors* 12, no. 8: 624.
https://doi.org/10.3390/bios12080624