# Utilizing sliding mode control for the cavitation phenomenon and using the obtaining result in modern medicine

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## Abstract

The existence of a bubble in the vicinity of an elastic boundary appears in many situations such as medical and mechanical systems. On the other hand, bubble collapse is considered a source of energy loss in most systems and caused a lot of damages to it. This research is the first attempt to prevent bubble collapse in the vicinity of an elastic boundary by using control algorithms. In this paper, first, the nonlinear dynamic model of bubble in the vicinity of an elastic wall is introduced and then rewritten into state-space form. The second part of this paper is devoted to the design of the sliding mode controller for the bubble system, where the ultrasonic pressure plays the role of control input and the output is the bubble radius. Our main objective is to design a stabilizing controller that is able to regulate the radius of the bubble to the desired radius. At first, traditional sliding mode controller is proposed. Despite the successful tracking error, the chattering problem of this method leads us to introduce the boundary layer sliding mode control. Although the chattering phenomenon has been attenuated, it increases the steady-state error. Finally, robust integral sliding mode control is suggested to minimize the steady-state error while the chattering problem is removed. Numerical simulations including the case of parametric uncertainty are also presented. The results of this study are of immediate interest for medical applications such as ultrasound imaging and also industrial applications such as designing long-lasting pumps and valves.

## Keywords

Rayleigh–Plesset equation Encapsulated microbubble Sliding mode control Nonlinear Chattering## 1 Introduction

Lord Rayleigh was the first one who analyzed the dynamic of a spherical bubble surrounded by liquid. He ignored surface viscosity, tension, compressibility and considered a bubble full of gas. Additional attempts by Plesset lead to the Rayleigh–Plesset equation, the simplest equation to describe the bubble’s radial behavior [1]. In continuation, Keller presented the Rayleigh–Plesset–Keller equation which is the time evolution of a single and spherical bubble [2].

Recently, a comprehensive investigation into the dynamic of bubble near different types of boundaries has been carried out [3, 4]. There are different versions of the Rayleigh–Plesset equation that study the radial behavior of an encapsulated microbubble under different conditions. In [5], Doinikov attempted to model an encapsulated microbubble in the vicinity of a rigid wall by a modified version of Rayleigh–Plesset equation. In [6, 7], authors explained the bubble behavior under different conditions such as near a fluid layer and elastic wall with mathematical language. However, they ignored the fact that in the actual world contrast agent microbubble oscillates in the vicinity of a boundary layer [8]. Two modified Rayleigh–Plesset equations also describe the governing dynamics and the secondary Bjerknes force between two bubbles [9].The bubble radial behavior changes significantly in the presence of a boundary layer.

In [10], the authors reported that the scattered pressure of an encapsulated microbubble was influenced by the boundary layer such as an elastic wall. Similarly in [11], experimental studies show that bubble’s proximity to the boundary layer caused noticeable changes in the radius- time curve of bubble, Lankford experimental observation also confirms that the acoustic signal of the contrast agent microbubble is reduced due to its proximity to the rigid wall [12]. Garbin observed more than 50% reduction in the oscillation amplitude of an encapsulated microbubble due to the presence of the boundary layer [13].

Recently, encapsulated microbubble is widely used in modern medicine because it can pass through the narrowest vessels in the body when it is administrated to the circulatory system. Encapsulated microbubble is considered a novel method for transporting and releasing drugs to the desired location [14, 15]. Vibration of microbubble produces a local pulsating disturbance in the surrounding liquid and this property helps to blood–brain barrier opening [16]. Encapsulated microbubbles are also employed as an ultrasound contrast agent for clinical diagnosis in cardiology to improve the detection and characterization of tumors [17]. Nonlinear oscillation is generated, when microbubble is exposed to a typical acoustic field. This emission signal contains higher harmonics which can be distinguished from the primary ultrasound [18]. This property is employed in the ultrasound radiography application ranges from stroke detection to blood volume and perfusion measurement [19, 20].

The results of this study are also useful for industrial design. Control valves and pumps are vital components in industrial applications. The useful life of industrial valves and pumps are highly affected by the phenomenon of bubble collapse [21, 22]. In [23], the authors investigate the effect of the number of trims on the performance of the globe valve and how it suppresses the cavitation problem. It should be mentioned that most of the proposed methods are not economically justified and demand to change the mechanical structure of valves and pumps. This paper suggests a novel solution to prevent from bubble collapse and efficient method to extend the life of industrial valves.

Recently, sliding mode control (SMC) was turned into a popular method by robustness and simple implementation and applied to a variety of systems [24]. The main negative aspect of SMC control is a chattering phenomenon [25]. This phenomenon is extremely damaging to the actuators of the physical system [26]. In order to prevent chattering in SMC, Slotine proposed quasi-SMC, which considered a boundary layer for the sliding manifold [27]. The main drawback of the proposed method is to increase the tracking error of the state vector, which in many cases is undesirable. In [28], integral sliding mode control (ISMC) is introduced which improves both tracking error and chattering problem.

The rest of this paper is organized as follows. Section 2 contain model description including governing equation and also control challenges. Section 3 introduces various sliding mode control schemes and checking the stability of the proposed sliding manifold via the Lyapunov theory. Simulation results and analysis of their performances including the case of uncertain initial bubble radius and liquid viscosity are also presented.

## 2 Model description

Encapsulated micro bubble parameters

Parameter | Description | Value | Unit |
---|---|---|---|

\(\rho_{L}\) | Density of liquid | 1000 | \({{\text{kg}} \mathord{\left/ {\vphantom {{\text{kg}} {{\text{m}}^{3} }}} \right. \kern-0pt} {{\text{m}}^{3} }}\) |

\(\tau \left( \varepsilon \right)\) | Liquid density gain | .622 | * |

\(\mu_{L}\) | Liquid viscosity | .001 | Pa.s |

\(\kappa_{s}\) | Shell viscosity | 0.6e − 8 | Kg/s |

\(\chi\) | Shell elasticity | 2.5 | N/m |

\(\gamma\) | Polytrophic coefficient | 1.07 | ** |

\(\alpha (R_{0} )\) | Initial surface tension | .02 | N/m |

\(P_{{}}\) | Hydrostatic pressure | 1.2 | KPa |

\(\dot{R}_{o}\) | Bubble initial velocity | 0 | m/s |

\(R_{o}\) | Initial radius | 2.0 | \(\upmu{\text{m}}\) |

\(a\) | Bubble’s van der Waals radius | \(.197R_{0}\) | \(\upmu{\text{m}}\) |

\(c\) | Sound velocity in the liquid | 1500 | m/s |

### *Remark 1*

The distance between wall and microbubble is in order of bubble size; therefore, time delays problem is negligible for sound wave propagation between wall and bubble [33].

### *Remark 2*

The term \(\rho_{L} \tau \left( \varepsilon \right)\) in Eq. (1) determines the effective density. The term \(\tau \left( \varepsilon \right)\) is a dimensionless factor that depends on the wall’s mechanical properties. In other words, the material of considered wall has direct effect on the behavior of designated microbubble.

Both experimental and theoretical results confirmed that microbubble in the vicinity of an elastic wall presents radial oscillation with lower amplitude but higher frequency than bubble surrounded by liquid. In this paper, we intend to use nonlinear control algorithm in order to take the radial oscillation of microbubble under control. In the other word, our main control objective is that bubble follow the desired radius by applying control input signal i.e. acoustic pressure wave.

This equation relies heavily on the bubble radius and its derivatives. Different harmonics in the Fourier spectrum of scatter pressure are observed due to the pulsating behavior of the designated bubble, which makes it difficult to distinguish the scattered pressure from other agents. The nonlinear control scheme presented in this paper prevents from pulsating behavior and makes it easy to distinguish the scattered pressure.

## 3 Controller design

In this section, the procedure of designing a sliding mode controller is discussed. Our main control objective is to obtain a stabilizing control input signal \(\left( {u\left( t \right)} \right)\) which is able to regulate the radius of the bubble \(\left( {x_{1} \left( t \right)} \right)\) into the desired radius \(\left( {x_{d} \left( t \right)} \right)\). The stabilization of the bubble dynamic is guaranteed by using the Lyapunov stability analysis theory.

### **Theorem 1**

*For the nonlinear system of encapsulated microbubble in the vicinity of an elastic wall*(5)

*, the tracking error*(9)

*exponentially converges to zero if the following control law holds;*

*where*\(\left( {K_{1} > 0} \right)\)

*is a design constant.*

### *Proof*

It is obvious that \(\left( {\dot{V}\left( X \right)} \right)\) is negative definite function if a positive value is selected for the gain \(\left( {K_{1} } \right)\). According to the Lyapunov theory, the sliding manifold (10) is asymptotically stable and tracking error (9) exponentially converges to zero. The proof is completed.□

It should be noted that oftentimes the parameters like liquid viscosity and initial radius are uncertain in the dynamic of microbubble in the vicinity of an elastic wall [34]. In this case, the estimated dynamic \(\left( {\hat{f}\left( X \right)} \right)\) should be replaced instead of \(\left( {f\left( X \right)} \right)\) in Eq. (13).We assume that the estimated error \(\left| {f\left( X \right) - \hat{f}\left( X \right)} \right|\) is bounded by some known function as \(F = F\left( {x,\dot{x}} \right)\), i.e. \(\left| {\hat{f}\left( X \right) - f\left( X \right)} \right| < F\).

### Theorem 2

Tracking error (9) converges to zero and the chattering phenomenon is optimally minimized if the gain \(\left( {K_{1} } \right)\) is selected as \(K_{1} = F + \eta ,\eta > 0\)

### *Proof*

Here \(u_{\rm{max} }\) is the maximum value of the control input signal.

### Theorem 3

### *Proof*

### *Remark 3*

by a simple comparison between Eqs. (23) and (32), it is understood that using integral sliding manifold leads to faster reaching time than the traditional sliding manifold.

### *Remark 4*

the thickness of the boundary layer \(\left( \phi \right)\) in the saturation function of Eq. (24) has a direct relationship with the faster reaching time (32) and attenuation of chattering magnitude but has a reverse relationship with steady-state error (9).

### **Corollary 1**

*with regards to Eq.* (22)*, in a saturated condition* \(\left( {\left| {u\left( t \right)} \right| \ge u_{\hbox{max} } } \right)\)*, the control input signal of Eq.* (26) *acts like a conventional PID controller with coefficients of* \(K_{p} = \frac{{\lambda_{1} }}{\phi },K_{I} = \frac{{\lambda_{2} }}{\phi },K_{d} = \frac{1}{\phi }\)

### *Proof*

The proof is completed.□

## 4 Results and discussion

In order to evaluate the performance of proposed methods, the nonlinear dynamics of the bubble in the vicinity of an elastic boundary (5) are tested under control input signals \(\left( {u_{Tra} \left( t \right)} \right)\) and \(\left( {u_{ISMC} \left( t \right)} \right)\). The simulations are performed by using Matlab-Simulink software with the step size \(.001\).

Comparison between controllers performance

Indices | Methods | ||
---|---|---|---|

Traditional SMC | Boundary layer SMC | Integral SMC | |

Reaching time (\(\upmu{\text{s}}\)) | .17 | .47 | .09 |

Controller output variance | .9788 | .432 | .0273 |

RMSE | .1730 | .2600 | .1382 |

## 5 Conclusion

In this paper, we have tackled the challenging problem of designing a robust integral sliding mode controller for an encapsulated microbubble in the vicinity of an elastic wall. The main contribution of this paper is to prevent nonlinear radial oscillation of encapsulated microbubble using nonlinear control methods. We start by representing the Rayleigh–Plesset-like equation into state-space form. Then three types of sliding mode controller including traditional, boundary layer and robust integral SMC are introduced. The simulation results confirmed the efficiency of integral SMC over traditional and boundary layer SMC in terms of minimizing chattering problem and improving time response performances. The work in this paper can be extended in many ways. For example, the design of a fuzzy controller for the nonlinear dynamic of encapsulated microbubble can also be tested. It is also possible to consider more complex bubble dynamics, such as two coupled bubbles or dynamic of encapsulated microbubble between two elastic walls.

## Notes

### Acknowledgements

The authors are grateful to Dr. Mahmoud Najafi for his valuable comments on the progress of this research.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Human and animal rights

Research involving no human participants and/or animals. The manuscript is processed through proper channel.

## References

- 1.Plesset MS, Prosperetti A (1977) Bubble dynamics and cavitation. Annu Rev Fluid Mech 9(1):145–185CrossRefGoogle Scholar
- 2.Franc J-P (2007) The Rayleigh–Plesset equation: a simple and powerful tool to understand various aspects of cavitation. Fluid dynamics of cavitation and cavitating turbopumps. Springer, Berlin, pp 1–41Google Scholar
- 3.Turangan C et al (2006) Experimental and numerical study of transient bubble-elastic membrane interaction. J Appl Phys 100(5):054910CrossRefGoogle Scholar
- 4.Sankin G, Zhong P (2006) Interaction between shock wave and single inertial bubbles near an elastic boundary. Phys Rev E 74(4):046304CrossRefGoogle Scholar
- 5.Doinikov AA, Zhao S, Dayton PA (2009) Modeling of the acoustic response from contrast agent microbubbles near a rigid wall. Ultrasonics 49(2):195–201CrossRefGoogle Scholar
- 6.Doinikov AA, Aired L, Bouakaz A (2011) Acoustic response from a bubble pulsating near a fluid layer of finite density and thickness. J Acoust Soc Am 129(2):616–621CrossRefGoogle Scholar
- 7.Doinikov AA, Aired L, Bouakaz A (2011) Acoustic scattering from a contrast agent microbubble near an elastic wall of finite thickness. Phys Med Biol 56(21):6951CrossRefGoogle Scholar
- 8.Caskey CF et al (2007) Direct observations of ultrasound microbubble contrast agent interaction with the microvessel wall. The Journal of the Acoustical Society of America 122(2):1191–1200CrossRefGoogle Scholar
- 9.Doinikov A (2001) Translational motion of two interacting bubbles in a strong acoustic field. Phys Rev 64:026301Google Scholar
- 10.Zhao S, Ferrara KW, Dayton PA (2005) Asymmetric oscillation of adherent targeted ultrasound contrast agents. Appl Phys Lett 87(13):134103CrossRefGoogle Scholar
- 11.Thomas D et al (2009) Single microbubble response using pulse sequences: initial results. Ultrasound Med Biol 35(1):112–119CrossRefGoogle Scholar
- 12.Lankford M et al (2006) Effect of microbubble ligation to cells on ultrasound signal enhancement: implications for targeted imaging. Invest Radiol 41(10):721–728CrossRefGoogle Scholar
- 13.Garbin V et al (2007) Changes in microbubble dynamics near a boundary revealed by combined optical micromanipulation and high-speed imaging. Appl Phys Lett 90(11):114103CrossRefGoogle Scholar
- 14.Lentacker I, De Smedt SC, Sanders NN (2009) Drug loaded microbubble design for ultrasound triggered delivery. Soft Matter 5(11):2161–2170CrossRefGoogle Scholar
- 15.Mayer CR, Bekeredjian R (2008) Ultrasonic gene and drug delivery to the cardiovascular system. Adv Drug Deliv Rev 60(10):1177–1192CrossRefGoogle Scholar
- 16.Baseri B et al (2010) Multi-modality safety assessment of blood-brain barrier opening using focused ultrasound and definity microbubbles: a short-term study. Ultrasound Med Biol 36(9):1445–1459CrossRefGoogle Scholar
- 17.Lindner JR (2004) Microbubbles in medical imaging: current applications and future directions. Nat Rev Drug Discovery 3(6):527CrossRefGoogle Scholar
- 18.Mulvagh SL et al (2000) Contrast echocardiography: current and future applications. J Am Soc Echocardiogr 13(4):331–342CrossRefGoogle Scholar
- 19.Schrope BA, Newhouse VL (1993) Second harmonic ultrasonic blood perfusion measurement. Ultrasound Med Biol 19(7):567–579CrossRefGoogle Scholar
- 20.Meyer K et al (2003) Harmonic imaging in acute stroke: detection of a cerebral perfusion deficit with ultrasound and perfusion MRI. J Neuroimaging 13(2):166–168CrossRefGoogle Scholar
- 21.Rammohan S, Saseendran S, Kumaraswamy S (2009) Effect of multi jets on cavitation performance of globe valves. Journal of fluid science and technology 4(1):128–137CrossRefGoogle Scholar
- 22.Amirante R, Distaso E, Tamburrano P (2014) Experimental and numerical analysis of cavitation in hydraulic proportional directional valves. Energy Convers Manag 87:208–219CrossRefGoogle Scholar
- 23.Yaghoubi H, Madani SAH, Alizadeh M (2018) Numerical study on cavitation in a globe control valve with different numbers of anti-cavitation trims. Journal of Central South University 25(11):2677–2687CrossRefGoogle Scholar
- 24.Gao F et al (2019) Distributed sliding mode control for formation of multiple nonlinear AVs coupled by uncertain topology. SN Applied Sciences 1(4):374CrossRefGoogle Scholar
- 25.Levant A (2010) Chattering analysis. IEEE Trans Autom Control 55(6):1380–1389MathSciNetCrossRefGoogle Scholar
- 26.Li Jn et al (2013) Chattering free sliding mode control for uncertain discrete time-delay singular systems. Asian J Control 15(1):260–269MathSciNetCrossRefGoogle Scholar
- 27.Slotine J-JE (1984) Sliding controller design for non-linear systems. Int J Control 40(2):421–434MathSciNetCrossRefGoogle Scholar
- 28.Rubagotti M et al (2011) Integral sliding mode control for nonlinear systems with matched and unmatched perturbations. IEEE Trans Autom Control 56(11):2699–2704MathSciNetCrossRefGoogle Scholar
- 29.Munson BR et al (2013) Fluid mechanics. Wiley, SingaporeGoogle Scholar
- 30.Brennen CE (2014) Cavitation and bubble dynamics. Cambridge University Press, CambridgezbMATHGoogle Scholar
- 31.Doinikov AA, Aired L, Bouakaz A (2012) Dynamics of a contrast agent microbubble attached to an elastic wall. IEEE Trans Med Imaging 31(3):654–662CrossRefGoogle Scholar
- 32.Katiyar A, Sarkar K, Jain P (2009) Effects of encapsulation elasticity on the stability of an encapsulated microbubble. J Colloid Interface Sci 336(2):519–525CrossRefGoogle Scholar
- 33.Mettin R, et al. (2000) Dynamics of delay-coupled spherical bubbles. InL AIP conference proceedings. AIPGoogle Scholar
- 34.Kang C et al (2018) Effects of initial bubble size on geometric and motion characteristics of bubble released in water. J Cent South Univ 25(12):3021–3032CrossRefGoogle Scholar
- 35.Castaños F, Fridman L (2006) Analysis and design of integral sliding manifolds for systems with unmatched perturbations. IEEE Trans Autom Control 51(5):853–858MathSciNetCrossRefGoogle Scholar