Experimental study on damage mechanism of blood vessel by cavitation bubbles

Highlights

• •Bubble dynamics varies with the thickness of vessel model wall.
• •No micro-jet toward the vessel wall appear when bubbles are off-axis.
• •Expansion deformation amplitude of vessel wall is larger than that of contraction under the action of bubble.
• •Period of bubble oscillation under the influence of the elastic vessel was lower than that of the elastic membrane.
• •Differences in the deformation of elastic membrane and elastic blood vessels under the action of bubble.

Abstract

Ultrasound-induced cavitation in blood vessels is a common scenario in medical procedures. This paper focuses on understanding the mechanism of microscopic damage to vessel walls caused by the evolution of cavitation bubbles within the vessels. In this study, cavitation bubbles were generated using the low-voltage discharge method in 0.9% sodium chloride saline, and vessel models with wall thicknesses ranging from 0.7 mm to 2 mm were made using a 3D laminating process. The interaction between cavitation bubbles and vessel models with different wall thicknesses was observed using a combination of high-speed photography. Results show that cavitation bubble morphology and collapse time increased and then stabilized as the vessel wall thickness increased. When the cavitation bubble was located in vessel axial line, pair of opposing micro-jets were formed along the axis of the vessel, and the peak of micro-jet velocity decreased with increasing wall thickness. However, when the cavitation bubble deviated from the vessel model center, no micro-jet towards the vessel model wall was observed. Further analysis of the vessel wall deformation under varying distances from the cavitation bubble to the vessel wall revealed that the magnitude of vessel wall stretch due to the cavitation bubble expansion was greater than that of the contraction. A comparative analysis of the interaction of between the cavitation bubble and different forms of elastic membranes showed that the oscillation period of the cavitation bubble under the influence of elastic vessel model was lower than the elastic membrane. Furthermore, the degree of deformation of elastic vessel models under the expansion of the cavitation bubble was smaller than that of elastic membranes, whereas the degree of deformation of elastic vessel models in the contraction phase of the cavitation bubble was larger than that of elastic membranes. These new findings provide important theoretical insights into the microscopic mechanisms of blood vessel potential damage caused by ultrasound-induced cavitation bubble, as well as cavitation in pipelines in hydrodynamic systems.

Notations

Number	Symbols	Symbol Meaning	Unit
1	A	Cavitation bubble long axis	mm
2	B	Cavitation bubble short axis	mm
3	d	Distance between the cavitation bubble centroid and boundary	mm
4	h	Vessel wall thickness	mm
5	L	Initial inner diameter of vessel models	mm
6	$L^{\prime }$	External diameter of vessel after deformation	mm
7	$L_l^{}$	Distance of the left vessel wall from the center of the vessel after deformation	mm
8	$L_r^{}$	Distance of the right vessel wall from the center of the vessel after deformation	mm
9	P	Surrounding liquid pressure	Pa
10	$P_{\mathrm{v}}$	Saturated vapor pressure	Pa
11	R	Equivalent radius of the cavitation bubble	mm
12	Rmax	Maximum radius of the cavitation bubble	mm
13	t	Cavitation bubble collapse time	μs
14	tb	Cavitation bubble evolution time	μs
15	Tc	Rayleigh collapse time	μs
16	$t^{\ast }$	Non-dimensional cavitation bubble collapse time
17	T*	Non-dimensional cavitation bubble evolution time
18	V	Micro-jet velocity	m/s
19	Vb,l	Left surface contracting velocity of cavitation bubble	m/s
20	Vb,r	Right surface contracting velocity of cavitation bubble	m/s
21	$V_{w,l}^{}$	Velocity of left vessel wall deformation	m/s
22	$V_{w,r}^{}$	Velocity of right vessel wall deformation	m/s
23	ΔH	Deformation distance of vessels wall and elastic membrane	mm
24	Δt	Cavitation bubble contracting time from maximum radius to a minimum radius	μs
25	ΔT	Cavitation bubble oscillation time	μs
26	|ΔV|	Difference in contracting velocity of cavitation bubble wall	m/s
27	$\rho$	Density of 0.9% sodium chloride saline	kg/m3
28	γ	Dimensionless distance from the cavitation bubble to the boundary
29	δ	Non-dimensional vessel wall thickness
30	φ	Cavitation bubble morphology
31	τ	Non-dimensional collapse period
32	τ*	Non-dimensional bubble oscillation period
33	Ω	Non-dimensional deformation distance of vessel wall and elastic membrane
34	ξ	Degree of vessel wall deformation
35	${\xi }_l$	Degree of left vessel wall deformation
36	${\xi }_r$	Degree of right vessel wall deformation

1. Introduction

In the medical field, numerous new therapeutic applications of High Intensity Focused Ultrasound (HIFU) are rapidly developing. These include noninvasive ultrasound (US) surgery [1], treatment of stones disease [2], [3], [4], targeted drug delivery [5]. Research continues on the thrombolysis, stimulating the growth of capillaries after a heart attack, and others [6].

The ultrasound used for treatment has a higher time-averaged intensity, and the HIFU transducer provides ultrasound with an intensity of approximately 0.5–3 W/cm², in the surgical-based ultrasound field, even more than 10 W/cm² [5]. Among other factors, the effectiveness and safety of HIFU treatment in the clinic depends on the cavitation bubble dynamics induced by ultrasound [7], particularly when treating certain blood vessels. It is very difficult to observe and measure the evolution of cavitation bubbles and the deformation of tissues and blood vessels by experimental means. Therefore, experimental work is mainly performed in vitro usually using gel and blood vessel models to simulate tissues and blood vessels, respectively [8], [9], [10], [11]. In extracorporeal shock wave lithotripsy (SWL), the primary cause of kidney damage is the rupture of small vessels due to cavitation, including capillaries, small arteries, and small veins, which range in diameter from 5 to 100 µm [8], [12]. Recent experiments conducted by scientists on the isolated rat mesenteric vascular system have demonstrated that the growth and collapse of cavitation bubbles, as well as the impact of micro-jets, can cause significant deformation of small vessels, leading to their rupture. [13], [14], [15]. Caskey et al. [16] were the first to investigate the in vitro vascular and clinical bubble dynamics of rat cecum micro-bubbles through ultrasonic cavitation. Their study focused on gene therapy and local drug delivery, without specifying the potential mechanism by which vessel damage could occur. Freund et al. [17] and Kobayashi et al. [18] contributed to characterizing the damage mechanism by linking the deformation of the tissue boundary to the pulse generated by the water hammer shock wave released during the formation of the liquid jet. Miller et al. [19] discovered that ultrasound-induced micro-bubbles are restricted to blood vessels and identified that cavitation bubbles can not only rupture blood vessels but also impact the vessel endothelium.

The intensity of diagnostic ultrasound is low compared to the ultrasound used for treatment, with a typical diagnostic ultrasound transducer outputting about 0.0001–0.5 W/cm² [5]. Ultrasound contrast agents (UCA) are encapsulated micro-bubbles and were originally designed to image perfusion by improving the echo amplitude from the blood pool [20]. When driven by ultrasonic pulses, the UCA can generate unique echo signatures resulting from the nonlinear oscillation of bubbles [21]. Microbubbles are widely used in clinical ultrasound diagnosis because of their high compressibility compared to the surrounding tissue, thus allowing the visualization of blood vessels down to the capillary level. The applications of UCAs have recently expanded to include targeted imaging, thrombolysis, and drug and gene delivery [21], [22], [23], [24]. Efforts in modelling the dynamics of UCAs have largely been focused on using various modified Rayleigh-Plesset bubble dynamics equations, of which the cornerstone assumption is that a single UCA is surrounded by an infinite fluid and remains spherical until it collapses. Comparing the theoretical predictions with experimental results demonstrates that the Rayleigh-Plesset equation and the various modified models work remarkably well for the dynamics of cavitation in an unbounded field or in large vessels [22], [25].

Recently, there has been growing attention focused on the impact of micro-bubble oscillation on small vessels [26], [27], [28], [29]. Experimental studies of the dynamics of microbubbles both in vitro and in vivo indicate that micro-bubble oscillation can enhance vascular permeability and even locally damage vasculature [30], [31], [32]. Stieger et al. [33] believed that it is essential to examine the mechanism whereby vascular permeability is enhanced and vascular injuries are produced, so that the appropriate strategies are designed to improve local drug and gene delivery efficiency and to minimize any permanent damage to capillaries. Caskey et al. [34] experimentally investigated the oscillation of micro-bubbles in microvessels with diameters similar to capillaries, and found that the expansion rate of bubbles in rigid capillary model is greatly reduced compared to the predictions of the Raligh-Plesset model for micro-bubbles in infinite liquids.

Appropriate models of vascular compliance are necessary in theoretical and experimental studies, especially for micro-bubble oscillations in ultrasound-assisted thrombolysis, as well as drug and gene delivery [22]. Khismatullin [35] suggested that blood viscosity greatly influences the oscillations of micro-bubbles in small vessels and capillaries, and therefore the effect of blood viscosity on bubble dynamics was considered in their model. Qin et al. [36] studied asymmetric oscillation of microbubbles in curved microvessels during ultrasonic lithotripsy. In their model, the characteristics of vascular compliance are the static nonlinear relationship between intracavitary pressure and the expansion ratio of vessel radius, which represents the change of vessel stiffness with intracavitary liquid pressure. The evolutionary structure of micro-bubbles is pre-assumed to be ellipsoidal, and the blood model is simplified as incompressible, viscosity fluid.

Calvisi et al. [37] and Fong et al. [38], [39] used potential flow theory and the boundary element method (BEM) to simulate the dynamics of acoustic bubbles in axisymmetric structures. Wang and Blake [40], [41] to simulate the compressible effect of liquid flow around bubbles by developing the weakly compressible theory. Tang et al. [42] measure the behavior of a vapor bubble and its induced liquid flow simultaneously during subcooled boiling using the high-speed two-phase particle image velocimetry method. Wang and Manmi [43], [44] implemented a three-dimensional boundary element model that can be used for bubble dynamics affected by acoustic waves propagating parallel to the boundary near a rigid boundary. Wang et al. [45] based on Klaseboer [46] and Turangan et al. [47] model, both blood flow inside the vessel and tissue flow outside the vessel were modeled using incompressible potential flow theory and the boundary element method (BEM), and the two fluids were considered to have different densities, assuming that the fluid outside the elastic vessel is related to elasticity. Freund and Jonathan [48] studied the effect of elastic membrane and surrounding elastic tissue on cavitation bubble by numerical simulations method, and the results showed that the effect of the elastic membrane on the growth of the cavitation bubble is small. Zhang et al. [49] established a novel theory for the dynamics of oscillating bubbles, which unifies different classical bubble equations. Supponen et al. [50] conducted a comprehensive study on micro-jets produced when cavitation bubbles collapse near different boundaries, thereby introducing an anisotropy parameter to categorize the jets as weak jets, intermediate jets, and strong jets. The scaling laws derived for these jets have significantly contributed to the unified mechanism research of cavitation bubbles in diverse application scenarios.

According to review of the above research literature, it can be found that in the medical field, both low-intensity ultrasound used for diagnosis and high-intensity focused ultrasound (HIFU) used for treatment are very likely to induce cavitation bubbles in blood vessels. The dynamics of cavitation bubbles and the deformation mechanism of the blood vessel under the influence of cavitation bubble need further researches, especially for small vessels or capillaries, where the bursting of vessels under the influence of cavitation bubbles can have a negative impact on diagnosis or treatment. For this reason, this paper focuses on the microscopic potential damage mechanism of the vessel wall by the evolution of the cavitation bubble in the vessel model. In the experiments, the cavitation bubble was generated by means of the low-voltage discharge method in 0.9% sodium chloride saline, vessel models with varying wall thicknesses were made using a 3D laminating process, and the combination of high-speed photography observed the interaction process of cavitation bubbles with blood vessels of different wall thickness. The aim of this paper is to describe the microscopic mechanism of blood vessel potential damage caused by ultrasound-induced cavitation bubbles in medicine.

2. Experimental methods

In medical field, ultrasound induced cavitation bubble are generated by the negative pressure of ultrasound waves acting on liquid, with peak ultrasound waves in the range of 0.8–7.2 MPa [14]. Ultrasound-induced cavitation bubbles in blood vessels often exist as cavitation clusters [8], [51], for the study of the damage mechanism of blood vessels under the action of cavitation bubbles, the interaction between cavitation bubbles inside the cavitation clusters makes the problem extremely complicated. Thus, the cavitation bubble in this paper was generated by means of the low-voltage discharge method to simulate a single cavitation bubble in blood vessel model, and combination of high-speed photography observed the interaction process of a cavitation bubble with the blood vessel model. The experimental set-up is shown in Fig. 1a.

Fig. 1.

Schematic diagram of experimental set-up.

The cavitation bubble was generated by means of the low-voltage discharge method in 0.9% sodium chloride saline is to convenience of observation, and the system is mainly composed of capacitor, resistor, charge–discharge switch and copper wire with a diameter of 0.1 mm was selected as the discharge electrode [47], [52], [53], [54]. In this paper, the voltage is 113 V, the experimental liquid temperature was 22 °C, the local ambient pressure was 95.5 kPa, and the radius of the cavitation bubble in the free field was 9.4 ± 0.4 mm. In this experiment, the bubbles in the elastic tube and the bubbles in the free field were induced using the same voltage. However, the bubbles inside the elastic tube are slightly smaller in size compared to the bubbles in the free field due to the constraint imposed by the elastic boundaries.

The high-speed camera system consisted of a high-speed camera (Photron Inc., Japan; Fastcam SA-Z, maximum acquisition rate: 1,000,000 fps), lens and light source. Due to the millimeter size of bubbles and vessel models wall, micro lenses (Nikon, Micro 105/2.8G) were used with the high-speed camera. In the experiment of this study, the frame rate is 180,000 fps, such a high acquisition rate resulted in a severe lack of exposure, so the LED (300 W) light source had to be used as auxiliary illumination during the shooting process, and the final image was taken with the exposure time is 3.95 μs, the resolution is 384 × 200 pixels, the pixel size is approximately 0.139 mm/pixel.

Human blood vessel wall thicknesses from 0.001 to 3 mm [55], [56], vessel diameters from 0.008 to 30 mm [57], [58], [59], elastic modulus of vessel from 0.5 to 5.5 N/mm² [60], [61], tensile strength of vessel from 1.47 to 5.07 MPa [62]. However, poor transparency of human blood vessels poses a great difficulty to the observation of the deformation of the vessel wall under the action of cavitation bubbles. Therefore, in this paper, a three-component system of Hei-Cast T0387 transparent soft rubber (T0387-A: T0387-B: T0387-C = 100:95:200) was used to created vessel models by 3D lamination process. The inner diameter of the simulated blood vessel is 16 mm and the length is 50 mm, and the vessel wall thicknesses used in experiments are 0.7 mm, 1 mm, 1.2 mm, 1.5 mm and 2 mm, respectively. The elastic modulus and tensile strength of the simulated blood vessels were measured using INSTRON 5567 (USA) and are shown in Table 1. They both fall within the respective parameter ranges of human blood vessels. In this paper, we mainly study the potential damage mechanism of the vessel wall under the influence of transient cavitation bubbles, so the osmotic effect of the vessel model is ignored [63]. 0.9% sodium chloride saline was chosen to simulate human blood medium in this study [8], [14].

Table 1.

Mechanical properties of human blood vessels and vessel models.

	Blood vessel model					Human blood vessel
Vessel inner diameter (mm)	16					0.008–30
Vessel wall thickness (mm)	0.7	1	1.2	1.5	2	0.001–3
Elastic modulus (N/mm2)	0.846	1.009	1.40	1.99	2.97	0.5–5.5
Tensile strength (MPa)	2.511	2.266	2.011	1.631	1.633	1.47–5.07

To ensure the accuracy and repeatability of the experimental data, each experiment under each condition in this study was repeated 5–10 times. In this paper, the ‘equivalent area method’ was used to define the radius of the cavitation bubble R ($R=\sqrt{S/\pi }$, S is the area occupied by the cavitation bubble in the image), and the maximum radius of the cavitation bubble in the first oscillation period was denoted as R_max, as shown in Fig. 1b. The non-dimensional vessel wall thickness is defined as δ, δ = h/R_max, where h is the thickness of vessel wall. The dimensionless distance is defined the central of cavitation bubble to the boundary as γ, γ = d/R_max, where d is the distance between the cavitation bubble centroid and boundary. The morphology of the cavitation bubble was defined as φ when the area of the bubble reached its maximum. Specifically, φ was calculated as the ratio of the long axis of the cavitation bubble (B) to its short axis (A), such that: φ = B/ A. Define the degree of vessel wall model deformation as ξ, $\xi =L^{\prime }/(L+2h)$, where L is the initial inner diameter of the vessel model and L' is the external diameter of vessel models after deformation; when ξ greater than 1, it means that the vessel model undergo expansion deformation, and ξ less than 1 means that the vessel models undergo contraction deformation. In Fig. 1 b, the direction perpendicular to the vessel wall is radial and parallel to the vessel wall is axial. The degree of left vessel wall deformation is defined as ${\xi }_l,{\xi }_{\mathrm{l}}=L_{\mathrm{l}}^{}/(\frac{L}{2}+h)$, where $L_l^{}$ is the distance of the left vessel wall from the center of the vessel after deformation. The degree of right vessel wall deformation is defined as ${\xi }_r,{\xi }_{\mathrm{r}}=L_{\mathrm{r}}^{}/(\frac{L}{2}+h)$, where $L_r^{}$ is the distance of the right vessel wall from the center of the vessel after deformation. When the ${\xi }_l$ and ${\xi }_r$ are greater than 1, which means the vessel wall is deformed by expansion, and less than 1, which means the vessel wall is deformed by contraction.

3. Effect of vessel wall thickness on the evolution of the cavitation bubble in vessel axial line

In this part, these effects of vessel wall thickness on the morphology, collapse time and micro-jets of the cavitation bubble located in the central of vessel were studied.

3.1. Effect of vessel wall thickness on morphology of cavitation bubble

Fig. 2 shows high-speed photographic images of the evolution of cavitation bubbles located in vessel model axial line (d = 0.5L) under different conditions of vessel models wall thickness, including 0.7 mm, 1 mm, 1.2 mm, 1.5 mm, and 2 mm.

Fig. 2.

Morphology of cavitation bubble in vessel models with different wall thicknesses.

In Fig. 2a, h = 0.7 mm, R_max = 8.09 mm, L = 16 mm, d = 0.5L. From Fig. 2a, it can be seen that the expansion phase of the bubble begins as a sphere, with the vessel wall expanding at the same time. In Fig. 2a₃, the vessel wall expands to its maximum extent, resulting in an increase in the vessel diameter by 4.02 mm, while the bubble continues to expand, causing it to evolve into an ellipsoidal shape when constrained within the vessel model as shown in Fig. 2a₄, and the long axis B of the ellipsoidal cavitation bubble was 18.33 mm and the short axis A was 15.31 mm at this time. Then the cavitation bubble entered into the contraction and collapse stage, the top and bottom sides of the ellipsoidal cavitation bubble gradually contracted from a circular arc to a sharp one, as is shown in Fig. 2a₅. While the cavitation bubble is bound radially by the vessel wall during further contraction, the cavitation bubble is replenished axially by the surrounding liquid in time and depressed toward the center of the cavitation bubble to form a pair of opposing micro-jets, as shown by the red arrow in Fig. 2a₇. The contraction of the cavitation bubble to minimum volume at t = 1517 μs, and the vessel wall contraction also reaches to the maximum, resulting in a decrease in the vessel diameter by 2.68 mm. In Fig. 2 b, h = 1 mm, R_max = 8.34 mm, d = 0.5L. The process of the cavitation bubble expansion and collapse in this experiment is similar to that in Fig. 2 a, the difference is the degree of deformation of the vessel model and the morphology of the cavitation bubble in this experiment. The diameter of the vessel model increased by 3.67 mm at the moment of vessel wall expansion reaches to the maximum. The diameter of the vessel decreased by 2.58 mm at the moment of cavitation bubble contraction reaches to the minimum volume. The difference is also reflected in the fact that the ellipsoidal cavitation bubble becomes flatter and longer when the cavitation bubble reaches its maximum radius. At this time, the long axis of the cavitation bubble was 19.17 mm and the short axis was 15.45 mm, as is shown in Fig. 2b₄. In the experiment of vessel wall thickness was further increased to 1.2 mm, R_max = 7.5 mm, The diameter of the maximum expansion deformation of the vessel increased by 2.3 mm and the diameter of the maximum contraction deformation of the vessel decreased by 2.17 mm. The long axis of cavitation bubble is 16.83 mm and the short axis is 16.31 mm at the moment of maximum cavitation bubble radius. In Fig. 2d, the thickness of vessel wall is 1.5 mm, and R_max = 7.74 mm, the vessel diameter increased by 2.11 mm when the wall expanded to the maximum, the vessel diameter decreased by 2.06 mm when the wall contracted to the maximum. The long axis of cavitation bubble is 17.39 mm and the short axis is 13.79 mm at the moment of maximum cavitation bubble radius. When the vessel wall thickness is 2 mm, again after the maximum vessel wall deformation, the cavitation bubble continues to expand and develop into an ellipsoidal shape and forms an upward and downward opposing micro-jet during the collapse phase. However, compared with the vessel wall thickness of 0.7 mm, the vessel wall deformation is further reduced, and the vessel model diameter at the time of maximum expansion deformation of vessel wall increased by only 1.39 mm, while the diameter of vessel maximum contraction deformation decreased by only 1.11 mm. From the time interval in Fig. 2, it can be seen that the time interval from the onset of micro-jets to the first collapse is about 84 μs in the experimental group with 0.7 mm vessel wall thickness, while in the experimental group with 2 mm vessel wall thickness, the time interval from the onset of micro-jets to the first collapse is 334 μs. Indicating that as the thickness of vessel wall increases, the time interval from the onset of micro-jets to reach the first collapse also increases.

The analysis of the effect of vessel model thickness changes on the evolution of the cavitation bubble in Fig. 2 shows that the vessel model forces the cavitation bubble to evolve from spherical to ellipsoidal shape, and the ellipsoidal shape also changes with the vessel wall thickness changes. To further quantify the effect of vessel wall thickness on the extent of the cavitation bubble morphology, the ratio of long axis to the short axis of the cavitation bubble, φ, was used to define the cavitation bubble morphology. The experiments were repeated for vessels with wall thicknesses of 0.7 mm, 1 mm, 1.2 mm, 1.5 mm, and 2 mm, respectively, and the magnitude of the cavitation bubble morphology φ when the cavitation bubble radius R reached the maximum in each group was counted, as shown in Fig. 3. When the vessel wall thickness was 0.7 mm, the cavitation bubble morphology φ = 1.196 ± 0.037. When the vessel wall thickness increased to 1 mm. the cavitation bubble morphology φ also increased to 1.251 ± 0.065. When the vessel wall continued to increase to 1.2 mm, the increase in the cavitation bubble morphology was smaller compared to that under the influence of 1 mm vessel wall thickness, at which time φ = 1.264 ± 0.037. When the vessel wall thickness continues to increase to 1.5 mm, the cavitation bubble morphology φ = 1.266 ± 0.039. When the vessel wall thickness is 2 mm, the cavitation bubble morphology φ = 1.290 ± 0.043. It can be seen that the cavitation bubble morphology φ increases first and then stabilizes with the increase of the vessel wall thickness of the vessel models, especially when the wall thickness reaches 1.2 mm, the variation of cavitation bubble morphology φ was small. It indicates that the cavitation bubble morphology φ gradually reach a limiting value with the increase of the vessel wall thickness δ. Moreover, as the vessel wall thickness increases from 0.7 mm to 2 mm, the maximum increase of the morphology φ (1.006 ± 0.008) compared to that of a single cavitation bubble in the free field (red dot shown in Fig. 3) is about 30%.

Fig. 3.

Relationship between cavitation bubble morphology and vessel wall thickness.

The above analysis suggests that the cavitation bubble morphology φ is related to the dimensionless vessel wall thickness δ. Therefore, based on the experimental results, the exponential function relationship between φ and δ is obtained by the Levenberg-Marquardt optimization algorithm analyzing [64]:

$$\phi =1.55-\frac{0.099}{0.187+\delta }(\delta >0)$$ (1)

The adjusted correlation coefficient of the curve fitted by Eq. (1) is 0.94. The trend of the curve fitted by Fig. 3 shows that φ increases first and then stabilizes with the increase of δ, indicating that there is a limit value of the cavitation bubble morphology φ, which is about 1.55. When the thickness of the vessel wall is extremely large, it is similar to the collapse of the cavitation bubble in the rigid tube, at which time the cavitation bubble morphology φ is near the limit value of 1.55; when the vessel wall thickness tends to 0, the wall binding effect on the cavitation bubble is very small, which is similarity to the spherical collapse of the cavitation bubble in the free field. Unfortunately, the vessel model with wall thickness less than 0.7 mm could not be achieved due to the experimental conditions.

The reason for the change from spherical to ellipsoidal shape during the evolution of the cavitation bubble is that the radius of the cavitation bubble is small at the initial stage of expansion, and the action of the surrounding fluid on the cavitation bubble in the radial direction is basically the same, resulting in a spherical expansion of the cavitation bubble. When the radius of the cavitation bubble increases to a certain degree, the expansion of the vessel wall in the radial direction is smaller than the expansion of the cavitation bubble in the radial direction, and the axial direction of the vessel has no restraining effect on the cavitation bubble, which eventually leads to the cavitation bubble expanding along the vessel axial direction to form an ellipsoidal shape.

3.2. Effect of vessel wall thickness on collapse time of cavitation bubble

To compare the difference in cavitation bubble collapse time by different vessel model wall thicknesses, the cavitation bubble radius R and cavitation bubble collapse time t were non-dimensionalized in this study using Eqs. (2), (3) [65], [66]:

$$t^{\ast }=t/T_c$$ (2)

$$T_c=0.915R_{\mathrm{m}\mathrm{a}\mathrm{x}}{\left(\frac{\rho }{P-P_{\mathrm{v}}}\right)}^{0.5}$$ (3)

where $T_c$ is the Rayleigh collapse time, P is the surrounding liquid pressure, ρ is the 0.9% sodium chloride saline density of 1030 kg/m³, $P_{\mathrm{v}}$ is the saturated vapor pressure of the cavitation bubble, and since $P_{\mathrm{v}}$ is very small compared to P, it can be neglected [66].

Fig. 4 gives the variation pattern of the dimensionless cavitation bubble radius R/R_max with the dimensionless collapse time $t^{\ast }$ in the vessel models with different wall thicknesses and compares it with the Rayleigh collapse time and single cavitation bubble collapse time in the free field. As shown in Fig. 4, when the cavitation bubble is positioned in the center of the vessel model, it is evident that the collapse time is significantly longer when the bubble is affected by a larger wall thickness compared to a smaller wall thickness. Furthermore, the collapse time of the cavitation bubble obtained under experimental conditions involving vessel wall thicknesses of 0.7 mm, 1 mm, and 1.2 mm was found to be smaller than that of both the Rayleigh collapse time and the free field cavitation bubble collapse time. As the wall thickness decreases, there is a greater decrease in the collapse time of the cavitation bubble. Specifically, under experimental conditions involving vessel wall thicknesses of 1.5 mm and 2 mm, the collapse time of the cavitation bubble is larger than both the free field cavitation bubble collapse time and the Rayleigh collapse time. Moreover, the cavitation bubble collapse time is prolonged with the increase of wall thickness. In Fig. 4, it can also be derived that the surface contracting speed of the cavitation bubble at the last moment of collapse decreases with wall thickness increases. It can be observed that when the wall thickness is 2 mm, the cavitation bubble surface contracting speed is the smallest.

Fig. 4.

Relationship between collapse time and bubble radius.

Fig. 4 shows that the cavitation bubble collapse time increases as the increase of vessel wall thickness, in order to further statistical the effect of dimensionless wall thickness on the cavitation bubble collapse period. In this paper, the cavitation bubble non-dimension collapse period τ was calculated using Eq. (4):

$$\tau =\frac{\mathrm{\Delta }t}{T_c}$$ (4)

where $\mathrm{\Delta }t$ refers to the time taken for process the cavitation bubble from its maximum to minimum radius during collapse.

The experiments were conducted under varying conditions of wall thickness, and the cavitation bubble collapse time $\mathrm{\Delta }t$ was measured for each group. This data was used to obtain the non-dimensional collapse period τ as a function of the dimensionless vessel wall thickness δ, as shown in Fig. 5. Based on the data presented in Fig. 5, we can determine that the non-dimensional collapse period of the cavitation bubble for a vessel wall thickness of 0.7 mm is 0.927 ± 0.056. For a wall thickness of 1 mm, the non-dimensional collapse period of the cavitation bubble increases to 0.943 ± 0.051. When the wall thickness is further increased to 1.2 mm, the non-dimensional collapse period of the cavitation bubble increases to 0.977 ± 0.059. However, these are still smaller than the Rayleigh collapse period (τ = 1); when the wall thickness is 1.5 mm, the non-dimension collapse period of the cavitation bubble is 1.030 ± 0.074, which is greater than the Rayleigh collapse period; as the wall thickness continues to increase to 2 mm, the non-dimension collapse period of the cavitation bubble continues to increase to 1.098 ± 0.074. It can be obtained in this paper under the influence of five vessel wall thicknesses, the cavitation bubble collapse period increases with the increase of vessel wall thickness, and the cavitation bubble collapse period increases about 18.5%, which indicates that the contracting speed of the bubble surface will be reduced with the increase of vessel wall thickness.

Fig. 5.

Effect of vessel wall thickness on cavitation bubble collapse period.

Based on the regression analysis of the above experimental data, the curve relationship between the standardized collapse period τ and δ is obtained as shown in Fig. 5, and the correlation coefficient of the fitted curve is 0.99. From the fitted results, we can obtain that when the vessel wall thickness is less than 0.7 mm, the smaller the vessel wall thickness is, the larger the cavitation bubble collapse period is. When the vessel wall thickness is infinitely small, the rupture of the vascular wall may occur during the cavitation bubble expansion phase, at this time the cavitation bubble collapse is similar to the collapse in the free field (red dot in Fig. 5). When the vessel wall thickness is greater than 0.7 mm, the cavitation bubble collapse period tends to increase and then stabilize with increasing vessel wall thickness, eventually converging to 2.83 (red dashed line τ = 2.83 in the Fig. 5). When the vessel wall thickness of the vessel model is infinite, it is similar to the collapse of the cavitation bubble under the influence of the rigid tube. It can also be found by Fig. 5 that the cavitation bubble collapse period is greater than the Rayleigh collapse time when the dimensionless vessel wall thickness δ is greater than 0.165 (gray dashed line τ = 1 in the Fig. 5), indicating that the cavitation bubble collapse period can be prolonged when the dimensionless vessel wall thickness δ is greater than 0.165, and the simulated vessels can shorten the cavitation bubble collapse period collapse time when δ is less than 0.165.

The increase in the cavitation bubble collapse period with vessel wall thickness can be explained from two perspectives. On the one hand, it is because the smaller the vessel wall thickness of the cavitation bubble contracting collapse phase, the larger the contracting deformation (as is shown in. Fig. 2a₉–e₉), resulting in a smaller suppression of the cavitation bubble collapse period by the smaller the wall thickness. On the other hand, in terms of the effect of rigid and elastic wall on the collapse period of the cavitation bubble, the cavitation bubble collapse period is prolonged under the influence of rigid wall [67] and elastic wall [9]; polyacrylamide PAA with 50% water content) compared to the Rayleigh collapse period. The free liquid surface boundary shortens the non-dimension collapse period of the cavitation bubble compared to the Rayleigh collapse period [67]. From Table 1, it is obtained that the larger the vessel wall thickness of the vessel in this paper, the larger the elastic modulus is, so it leads to longer cavitation bubble collapse period; while the smaller the wall thickness, the smaller the elastic modulus is, so it leads to shorter cavitation bubble collapse period.

3.3. Effect of vessel wall thickness on micro-jet velocity of cavitation bubble in vessel axial line

The experiments in Fig. 2 showed that the collapse of the cavitation bubble located in the center of vessel model resulted in the form opposing micro-jets along the vessel axis, and the effect of the vessel wall thickness on the micro-jet is given in this section. Fig. 6 a shows the spherical collapse morphology of a single cavitation bubble in the free field. Fig. 6 b - e show the cavitation bubble micro-jets under the influence of the vessel wall thickness of 0.7 mm, 1 mm, 1.2 mm and 1.5 mm, respectively. Fig. 6 f shows the cavitation bubble micro-jets under the influence of rigid tube. It can be observed from Fig. 6 that the micro-jets under the influence of various wall thicknesses all appear as opposing jets in the axial direction of vessel model, and the morphology of the jets shows a narrow front end and a wide tail end. This phenomenon of micro-jets caused by bubble collapse inside the pipeline has also been observed in the studies conducted by Ren et al. [68] and Wang et al. [45].

Fig. 6.

Pair of opposing micro-jets inside vessel model.

To investigate the relationship between micro-jet velocity magnitude and vessel wall thickness, this paper selects the maximum velocity during the development of micro-jets for each tested wall thickness. For each type of vessel wall thickness, the experiments were repeated 5–10 times to obtain a statistical sample. Based on these experimental samples, Fig. 7 illustrates the variation of peak velocity of micro-jet with dimensionless vessel wall thickness. It should be noted that the red dot in Fig. 7 is the surface contraction velocity during the collapse of a single cavitation bubble in the free field (corresponding to the experiment in Fig. 6a), which has a value of about 43.91 m/s. Fig. 7 shows that the peak velocity of micro-jet decreases as the dimensionless vessel wall thickness increases. When the wall thickness of vessel model is 0.7 mm, the peak velocity of micro-jet reaches its maximum value. The upward and downward micro-jet velocities are approximately 84.4 m/s and 85.7 m/s, respectively. The peak velocity of micro-jet is minimum when the vessel model wall thickness is 2 mm, and the upward micro-jet velocity is about 48.1 m/s and the downward micro-jet velocity is about 53.6 m/s. Moreover, it can be seen from Fig. 7 that the peak velocities of the opposing micro-jets are almost equal for each wall thickness condition, and the micro-jet velocities are greater than the surface velocity of bubble in the free field. Through their study on the dynamics of bubbles inside a funnel-shaped tube, Ren et al. [68] firstly discovered that when a bubble collapses in the cylindrical section, it generates micro-jets that develop away from one side of the funnel. The velocity of these micro-jets is approximately around 8 m/s, while the jet velocity in a rigid tube in Fig. 6f is about 8.09 m/s. This indicates that the elasticity variation of the tube wall has a significant impact on the velocity of micro-jets caused by bubble collapse.

Fig. 7.

Influence of vessel wall thickness on the micro-jet velocity.

The formation of opposing micro-jets along the vessel axial direction can be attributed to the radial constraint of the cavitation bubble by the vessel wall and the unrestricted free space in the axial direction. During the contraction of the cavitation bubble, the space released by the bubble is replenished by the axial liquid in the vessel model, which results in the formation of a high-velocity jet along the vessel model axial line [45]. The slowdown of the axial micro-jet velocity with increasing vessel wall thickness can be explained by the results in Fig. 5. Fig. 5 shows that the period of cavitation bubble collapse increases with increasing vessel wall thickness or elastic modulus. This result indicates that the increase in vessel wall thickness or elastic modulus leads to a slower shrinkage process of the cavitation bubble (as shown in Fig. 4). Secondly, the larger the vessel wall thickness, the smaller the vessel wall contraction deformation (Fig. 2a₉–e₉), which further slows down the bubble collapse process and leads to a decrease in the micro-jet velocity.

4. Deformation of vessel under the influence of cavitation bubble

From the above analysis, it can be seen that different thicknesses of vessel models affect the evolution of the cavitation bubble to different degrees, but the vessel wall deformation produced by the cavitation bubble needs further analysis. Therefore, the following contents will be analyzed for the deformation pattern produced by the vessel wall under the action of the cavitation bubble.

4.1. Deformation of vessel wall under effect of cavitation bubble in vessel axial line

In Fig. 2, the evolution of the cavitation bubble in the vessel model axial line and the deformation of the vessel wall are presented. The images in Fig. 2a₃–e₃ depict the moments of maximum expansion deformation of the vessel wall for different wall thicknesses of 0.7 mm, 1 mm, 1.2 mm, 1.5 mm, and 2 mm, respectively. As the wall thickness of the vessel model increases, its expansion deformation decreases, as can be seen from high-speed photographic images. Fig. 8 shows the variation pattern of the maximum deformation degree ξ with the dimensionless thickness δ of the vessel model wall. Here, ξ > 1 indicates the expansion deformation of the vessel model wall, where a larger value represents a greater degree of expansion deformation. From Fig. 8, it can be observed that the range of expansion deformation for vessel wall thicknesses of 0.7 mm, 1 mm, 1.2 mm, 1.5 mm, and 2 mm are 1.14–1.22, 1.16–1.21, 1.13–1.17, 1.08–1.13, and 1.07–1.13, respectively. These results indicate that the degree of expansion deformation ξ tends to decrease with an increase in the dimensionless vessel wall thickness δ. Moreover, Table 1 reveals that the tensile strength is smaller and the elastic modulus is larger for greater vessel wall thicknesses, implying that thicker vessel walls are harder and less deformable.

Fig. 8.

Variation of vessel model deformation with wall thickness.

Further analysis of Fig. 2a₉–e₉, the moment of minimum volume of cavitation bubble contraction, and also the moment of maximum deformation of the vessel wall contraction. The smaller value of the degree of shrinkage deformation ξ indicates the larger degree of shrinkage deformation. The variation pattern of the maximum degree of shrinkage deformation with the increase of the vessel wall thickness is given in the Fig. 8. The results show that as the dimensionless vessel wall thickness increases, the ξ value gradually increases from about 0.85 to 0.97, and the degree of contraction deformation gradually decreases. It is worth noting in Fig. 8 that the degree of expansion deformation of the vessel wall is greater than the degree of shrinkage deformation.

4.2. Deformation of vessel wall under effect of off-axis cavitation bubble

Fig. 9a–e shows the evolution of the cavitation bubbles and the vessel models deformation process at the distance of 0.5L, 0.375L, 0.25L, 0.125L and 0L from the center of cavitation bubble to the right side of the vessel wall for a vessel wall thickness of 0.7 mm, respectively.

Fig. 9.

Deformation of vessel during changes in bubble-wall distance.

When the cavitation bubble is located in vessel model axial line (d = 0.5L), the evolution of the cavitation bubble and the deformation of the vessel model wall is consistent with Fig. 2. Fig. 9b is the cavitation bubble moving toward the right side of the vessel wall (d = 0.375 L, R_max = 7.82 mm). Fig. 9b₃ shows the moment of maximum deformation of the vessel wall, and the expansion degree is larger than that in Fig. 9a₃. With the development of the cavitation bubble, the shape of the cavitation bubble is no longer an ellipsoidal, as shown in Fig. 9b₅, but the right side of the cavitation bubble is close to the right vessel wall, and the left side is an ellipsoidal arc. No micro-jets towards or away from the vessel wall were observed during the collapse of the cavitation bubble. Fig. 9 c shows high-speed photographic images of the cavitation bubble further closer to the right vessel wall, where the center of the cavitation bubble is at a distance d = 0.25 L from the right vessel wall and R_max = 8.4 mm. Fig. 9c₃ shows the moment of maximum vessel wall deformation, where the right vessel wall expansion is further increase compared to Fig. 9a₃ and b₃. The micro-jets toward or away from the vessel wall was also not observed in the contraction phase of the cavitation bubble. In Fig. 9d, d = 0.125 L, R_max = 7.83 mm. Fig. 9d₃ is the moment of maximum vessel wall deformation, at this time the right vessel wall swells and bulges further, and the cavitation bubble is no longer spherical during the expansion phase because the right side of the cavitation bubble is close to the vessel wall. There is no appearance of micro-jets in all directions in the vessel model.

Fig. 9e shows the center of the cavitation bubble attached to the right vessel wall (d = 0L, R_max = 8.3 mm). Compared with other groups of experiments, Fig. 9e₃ displays the largest expansion deformation of the right vessel wall, with the cavitation bubble becoming hemispherical during the expansion phase. The distribution of cavitation bubble along the axial direction of the vessel model is wider during the contraction phase, as shown in Fig. 9e₆. When the cavitation bubble contracted to the moment of minimum volume, the contraction deformation of the right vessel wall was the largest compared to the other groups of experiment.

By gradually moving the cavitation bubble to the side of the vessel wall, it was found that the opposing micro-jets along the vessel axial direction appeared only when the cavitation bubble was located in vessel model axial line. Once the off-axis cavitation bubble in vessel model, no micro-jets were observed in the radial direction of the vessel model, and no obvious bulging of the vessel wall at the late stage of cavitation bubble collapse was observed. In the late stage of the collapse of the cavitation bubble near the elastic wall, the micro-jet will penetration the interior of the elastic wall, and this penetration phenomenon can be seen in the experimental images listed in the literature [47], [69]. It can be seen that the cavitation bubble inside the elastic vessel and near the elastic wall show two completely different results in terms of collapse morphology. This will be further discussed in Section 5.2.

Fig. 10 shows the radial contraction velocity difference |ΔV| = |V_b,l − V_b,r| with time for the surfaces near the side of the vessel model wall and away from the side of the vessel wall, respectively. In Fig. 10, the velocity difference |ΔV| of surface contraction is obviously large under the influence of rigid walls, and caused the non-spherical collapse of the cavitation bubble [45], [53], [54], [69], [70]. For the cavitation bubble deviated central vessel model, the surface contraction velocity difference |ΔV| is all lower or close to the velocity difference of the cavitation bubble surface contraction in the free field (theoretically, the velocity difference |ΔV| between the contraction of both sides of the surface of a single cavitation bubble in the free field is 0, but due to the difficulty of achieving a completely spherical Rayleigh cavitation bubble in the experiment, a smaller velocity difference |ΔV| exists when both sides of the surface of a single cavitation bubble in the free field contract). It can be seen that the contraction deformation of the vessel wall occurs along with the contraction of cavitation bubble, which makes it difficult for the bubble to form a velocity difference along radial direction of the vessel model. As a result, it is difficult for the formation of micro-jets in the radial direction of the vessel model.

Fig. 10.

Radial contraction velocity difference of cavitation bubble surface with time (Thickness of vessel wall is 0.7 mm).

Based on the analysis of Fig. 9, it is concluded that altering the distance between the cavitation bubble center and the vessel wall causes variations in the morphology of the expansion and collapse phases. Two phenomena were observed: firstly, no micro-jets were observed in the radial direction of vessel model when the cavitation bubble deviated from the center of the vessel model; secondly, the change of the bubble-wall distance affects the degree of vessel wall deformation. Fig. 11 depicts how the degree of deformation of the vessel wall (with ${\xi }_l$ and ${\xi }_r$ indicate the left and right wall deformations, respectively) changes with the dimensionless distance γ.

Fig. 11.

Effect of bubble-wall distance on the degree of vessel model deformation.

Fig. 11a shows the expansion-deformation relationship between ${\xi }_l$ and the dimensionless distance γ. These findings reveal that the degree of expansion deformation in the left vessel wall ${\xi }_l$ increases with an increase in the dimensionless distance γ, suggesting that proximity of the cavitation bubble to the left vessel wall results in greater expansion deformation of the left vessel wall. When the cavitation bubble center is at a certain distance from the vessel wall, the degree of deformation ${\xi }_l$ in the left wall of vessel model decreases as the vessel wall thickness increases. The variation range of ${\xi }_l$ is between 1.05 and 1.2, indicating that the deformation of the left vessel wall is smaller with increasing vessel wall thickness. This result is consistent with the relationship of small expansion deformation for large vessel wall thickness when the cavitation bubble is located in the vessel model axial line in Fig. 8. Fig. 11b shows the variation of expansion deformation ${\xi }_r$ dimensionless distance γ. It can be seen that as the dimensionless distance γ increases, the degree of expansion deformation of the right vessel wall ${\xi }_r$ decreases, and the variation range of ${\xi }_r$ is 1.10–1.47. In case of the center of the cavitation bubble is close to the right vessel wall, the expansion deformation of the right vessel wall is the largest. It is worth noting that the deformation of the right vessel wall is larger than the left vessel wall. When the center of the cavitation bubble is at the same position from the wall, the deformation degree ${\xi }_r$ of the right wall decreases as the vessel wall thickness increases, which further indicates that the thinner the wall becomes, the greater the deformation of the vessel wall.

Fig. 11c illustrates the contraction deformation relationship between ${\xi }_l$ and the dimensionless distance γ of the cavitation bubble-vessel wall. The results demonstrate that as γ increases, the degree of contraction deformation in the vessel wall ${\xi }_l$ decreases. The variation range of ${\xi }_l$ is between 0.86 and 0.98, which suggests that the farther the cavitation bubble center is from the right vessel wall, the greater the contraction deformation of the left vessel wall.

Fig. 11d is the contraction deformation relationship of ${\xi }_r$ with the dimensionless distance γ of the cavitation bubble-vessel wall. It can be seen that the contraction degree ${\xi }_r$ of the right vessel wall increases with the increase of the dimensionless distance γ. The variation range of ${\xi }_r$ is 0.75–0.90, and the contraction degree of the right vessel wall is larger than that of the left vessel wall.

By comparing Fig. 11a with c, and Fig. 11d with e, it can be found that the degree of expansion deformation is greater than that of contraction deformation for different vessel wall thicknesses at the same dimensionless distance γ. When d = 0 L and the vessel wall thickness is 0.7 mm, the difference between the degree of expansion and contraction deformation is the largest, which is about 30%; when the vessel wall thickness is 2 mm, the difference between the degrees of expansion and contraction difference is 4%. This indicates that vessel potential damage caused by expansion stretching is more significant than that caused by contraction stretching.

The deformation patterns of blood vessels under the influence of cavitation bubbles indicate that the main cause of damage to blood vessels is the expansion and contraction of the cavitation bubble. Miao et al. (2008) [56] conducted numerical simulations to investigate the evolution of cavitation bubbles in blood vessels and discovered that during the deformation of blood vessels, the radial tensile stress was greater than the axial tensile stress. Furthermore, the magnitude of tensile stress was proportionate to the degree of deformation [71], [72]. Combined with the experimental results in this paper, the expansion stretching of the vessel model under the action of the cavitation bubble are greater than the contraction stretching. It is inferred that under the influence of cavitation bubbles, the degree of potential damage caused by expansion is greater than that caused by contraction. As the vessel wall thickness increases, both the degree of expansion deformation and contraction deformation decrease. This suggests that cavitation bubble is more likely to cause injury in vessels with thinner walls than in those with thicker walls. This observation is consistent with the numerical simulation results in the Ye and Bull [73].

4.3. Deformation velocity of vessel walls under effect of off-axis cavitation bubble

To compare the differences in deformation velocity of vessel model walls with varying thicknesses under the influence of cavitation bubbles, this section first analyzes the relationship between vessel wall deformation velocity and dimensionless evolution time of the cavitation bubble ($T^{\ast }=t_{\mathrm{b}}/{2T}_c$, where t_b is the cavitation bubble evolution time), under varying distances between the cavitation bubble and the vessel wall at a wall thickness of 0.7 mm. Fig. 12 a shows the variation pattern of the deformation velocity $V_{w,l}^{}$ of the left vessel wall with the dimensionless evolution time T*. When the distance between the cavitation bubble and the vessel wall is certain, the left wall expansion velocity $V_{w,l}^{}$ tends to increase and then decrease as T* increases, and the left vessel wall expansion velocity $V_{w,l}^{}$ is the smallest when the left vessel wall expansion deformation of is the largest; the contraction velocity $V_{w,l}^{}$ of the left vessel wall increases as T* increases, and the contraction velocity $V_{w,l}^{}$ is maximum when the cavitation bubble volume is minimum. When γ decreases, the expansion velocity and contraction velocity of the left vessel wall decrease. Fig. 12b shows the variation relationship of the right vessel wall velocity $V_{w,r}^{}$ with the dimensionless evolution time T* of the cavitation bubble. When γ is certain, it can be seen that when the cavitation bubble is near the center of the vessel model (d = 0.5 L, d = 0.375 L), the right vessel wall expansion velocity $V_{w,r}^{}$ increases and then decreases with increasing T*. However, in cases of d = 0.25 L, d = 0.125 L and d = 0 L, the right vessel wall expansion velocity $V_{w,r}^{}$ decreases as T* increases, and the smaller γ is, the more severe the vessel wall expansion is. The contraction velocity $V_{w,r}^{}$ of the right vessel wall increases slowly as T* increases and gradually increases as γ decreases. Comparing Fig. 12a and b, it can be seen that when the cavitation bubble in vessel model axial line, the deformation velocity of the left and right vessel wall is basically the same, and as the cavitation bubble approaches the right vessel wall, the deformation velocity of the right vessel wall is larger than that of the left vessel wall.

Fig. 12.

Deformation velocity of vessel wall with time (Vessel wall thickness is 0.7 mm).

Fig. 12 is the relationship between the vessel wall deformation velocity and time for a vessel wall thickness of 0.7 mm. The maximum velocity of expansion $V_{w,l}^{\prime }$ and the maximum velocity of contraction $V_{w,r}^{\prime }$ with the dimensionless distance γ of the cavitation bubble-vessel wall for each group of experiments is counted, as shown in Fig. 13.

Fig. 13.

Effect of bubble-wall distance on the peak of deformation velocity of vessel walls.

Fig. 13a shows the variation pattern of the maximum velocity $V_{w,l}^{\prime }$ of the left vessel wall expansion with the cavitation bubble-vessel wall dimensionless distance γ. It can be seen from the Fig. 13 that the maximum velocity $V_{w,l}^{\prime }$ of the left vessel wall expansion at each vessel wall thickness increases with the increase of the dimensionless distance γ. When the center of the cavitation bubble is at the same position from the vessel wall, the maximum velocity $V_{w,l}^{\prime }$ of the left vessel wall expansion decreases as the vessel wall thickness increases, and the variation range of $V_{w,l}^{\prime }$ is 1–3.5 m/s. Fig. 13b shows the variation pattern of the maximum velocity $V_{w,r}^{\prime }$ of the right vessel wall expansion with the cavitation bubble-vessel wall dimensionless distance γ. It can be seen that the maximum velocity $V_{w,r}^{\prime }$ of the right vessel wall expansion decreases with the increase of the dimensionless distance γ. The variation range of $V_{w,r}^{\prime }$ is 2–11 m/s.

Fig. 13c shows the variation pattern of the maximum velocity $V_{w,l}^{\prime }$ of the left vessel wall contraction with the cavitation bubble-vessel wall dimensionless distance γ. It can be seen from Fig. 13c that the left vessel wall contraction maximum velocity $V_{w,l}^{\prime }$ decreases as the dimensionless distance γ increases, and the variation range of $V_{w,l}^{\prime }$ is 1–4 m/s. Fig. 13d is the variation pattern of the right vessel contraction maximum velocity $V_{w,r}^{\prime }$ with dimensionless distance γ. It can be seen that the $V_{w,r}^{\prime }$ increases with the increase of the dimensionless distance γ. The variation range of $V_{w,r}^{\prime }$ is 2–6 m/s. Comparing Fig. 13c and d, it is found that the contraction maximum velocity of the right vessel wall is larger than that of the left vessel wall when the cavitation bubble is deviated the center of vessel model.

By comparing Figs. 11 and 13, it is found that the degree of wall deformation and the velocity of deformation of the vessel wall have similar trends. Table 1 shows that the elastic modulus of vessel models increases as wall thicknesses increase. The thicker vessels deform less under the action of the cavitation bubble, resulting in a slower deformation velocity, and conversely, the thinner vessels deform more and deform faster accordingly. Combined with the deformation degree of the cavitation bubble in Section 3.1, the thicker vessels have a stronger radial inhibition effect on the cavitation bubble.

5. Interaction of between cavitation bubble and different forms of elastic membrane

5.1. Differences in the period of cavitation bubble under the influence of blood vessels and elastic membranes

Fig. 14 shows the variation pattern of the dimensionless oscillation period of cavitation bubble with the dimensionless distance γ with different vessel wall thicknesses. Notably, the cavitation bubble oscillation period τ* (${\tau }^{\ast }=\frac{\mathrm{\Delta }T}{{2\ast T}_c}$) is calculated using the time from its initiation to the first collapse (ΔT), divided by twice the Rayleigh collapse time. Previous studies have reported that the cavitation bubble oscillation period near rigid walls [67], elastic walls (such as polyacrylamide with 50% PAA content) [10]; elastic membranes, and free liquid surfaces [67] are different. These values are also included in Fig. 14 to enable comparison with the oscillation period observed under elastic vessel action.

Fig. 14.

Comparison of dimensionless period of cavitation bubble near different boundaries.

Although the above results were obtained by different experimental methods, the trend of the cavitation bubble oscillation period τ* can be divided into two parts: one is the dimensionless cavitation bubble oscillation period τ* > 1 under the influence of rigid and elastic walls, and the other is the dimensionless cavitation bubble oscillation period τ* < 1 under the influence of free liquid surface. The τ* under the influence of rigid wall, elastic wall and elastic membrane decreases with the increase of γ, while the τ* under the influence of free surface boundary increases with the increase of γ. Interestingly, in the vessel model, the dimensionless cavitation bubble oscillation period τ* tends to decrease and then increase as the dimensionless bubble-wall distance γ increases, and the dimensionless cavitation bubble oscillation period τ* increases with the increase of the vessel wall thickness. In the experiments of Brujan et al. [10], the material of the elastic wall is 50% PAA, and its elastic modulus is 2.03 MPa, while the elastic modulus of vessel models with wall thickness less than 1.5 mm used in this paper are all lower than 2 MPa, so the range of τ* are all smaller than the τ* under the influence of 50% PAA boundary. On the contrary, the elastic modulus of vessel models with 2 mm wall thickness is larger than the elastic modulus of 50% PAA. PAA elastic modulus, so there exists a dimensionless oscillation period τ* greater than the τ* under the influence of elastic boundaries in the literature and less than the period τ* under the influence of rigid walls. In addition, it can be seen from Fig. 14 that the oscillation period τ* of the cavitation bubble under the action of the elastic membrane is larger than that of the action of the elastic vessel model with the same thickness, and the trend of the oscillation period τ* of the cavitation bubble under the action of the elastic membrane with the bubble-wall distance is consistent with the oscillation period τ* of the cavitation bubble under the action of the elastic wall.

5.2. Differences in the deformation of blood vessel model and elastic membrane

Fig. 15a–e shows the evolution of the cavitation bubble in the vessel with wall thicknesses of 0.7 mm, 1 mm, 1.2 mm, 1.5 mm and 2 mm at the cavitation bubble -vessel wall distance d = 0.375L. Fig. 15a, the expansion phase of the cavitation bubble is accompanied by the vessel wall expansion deformation, whereas the contraction phase is characterized by a “mushroom-like” morphology of the cavitation bubble (Fig. 15a₄), and contraction deformation of the vessel wall. When the vessel wall thickness increases, the expansion and contraction of the cavitation bubble in each experiment is basically the same as Fig. 15a. Similarly, the deformation of the vessel wall is also basically the same. The difference is that as the vessel wall thickness increases, the deformation of the wall decreases.

Fig. 15.

Evolutionary morphology of cavitation bubble in vessel in case of difference thickness of vessel wall.

Fig. 16a–e shows the evolution of the cavitation bubble near the elastic membrane with the same bubble-wall distance and wall thicknesses of 0.7 mm, 1 mm, 1.2 mm, 1.5 mm and 2 mm, respectively. It can be seen that the cavitation bubble shows a spherical expansion during the expansion process, accompanied by the expansion of the elastic membrane (the red dashed line is the initial position of the membrane). The cavitation bubble shows a non-spherical contraction during the shrinkage process, but no “mushroom-like” morphology appears, as in Fig. 16a₄–e₄. Compared with the interaction between the cavitation bubble and the vessel wall in Fig. 15, it is interesting to note that micro-jets are formed towards the elastic membrane before the cavitation bubble shrinks to its minimum volume, as shown in Fig. 16a₅–g₅, and the elastic membrane shrinks with minimal deformation when the cavitation bubble collapses to its minimum volume (as shown in Fig. 16a₆–e₆).

Fig. 16.

Evolutionary morphology of cavitation bubble in vessel in case of difference thickness of elastic membrane.

The degree of deformation of the vessel models caused by the expansion and contraction of the cavitation bubble is different from that observed in the elastic membrane, as seen in Figs. 15 and 16. To further investigate this difference, Fig. 17 shows the deformation distance Ω of the right wall of the vessel model with varying thicknesses under the condition of different bubble-wall distance. Here, Ω = ΔH/h, where ΔH is the change distance of expansion or contraction deformation of vessels wall and elastic membranes, and Ω > 0 indicates expansion deformation of the wall, while Ω < 0 indicates contraction deformation of the wall, and the deformation of the elastic membrane of the same material as the vessel models. From Fig. 17a and b, the expansion deformation of vessel models and membranes increases with decreasing γ, and the expansion deformation of both blood vessel model and membrane decreases with increasing thickness; comparing the expansion deformation of vessel models and elastic membrane, it is found that the expansion deformation of vessel models is smaller than that of elastic membranes under the same bubble-wall condition. The vessel models undergo contraction deformation during the cavitation bubble contraction phase, but the elastic membrane almost does not undergo contraction deformation during the cavitation bubble contraction process as shown in Fig. 17b.

Fig. 17.

Deformation difference between elastic vessels and elastic membranes.

The analysis in Section 3.1 revealed that the cavitation bubbles are constrained radially by the blood vessel model, leading to their development into an elliptical shape along the axis of the vessel model. However, for the elastic membrane, since the cavitation bubble on the side close to the elastic membrane was strongly inhibited and the cavitation bubble in other directions was not affected by the boundary, the elastic membrane was deformed in compliance with the development of the cavitation bubble, which eventually led to the deformation of the elastic membrane under the effect of cavitation bubble expansion more than the deformation of the simulated blood vessel wall.

The reason why the vessel model contraction deformation is greater than the elastic membrane contraction deformation may be that during the contraction process of cavitation bubbles, they are radially constrained by the vessel model, and when cavitation bubble collapse, the water cannot quickly replenish the space released by the contraction of the bubbles, resulting in the contraction of the vessel wall. While in the vicinity of the elastic membrane, the cavitation bubble on the side away from the elastic membrane is not affected by the boundary when the cavitation bubble contracts, and the surrounding liquid can rapidly replenish the space released by the cavitation bubble contraction, resulting in a very small contraction deformation of the elastic membrane during the collapse of the cavitation bubble (significantly smaller than the contraction deformation of the vessel model wall).

According to the differences in the effects of vessel models and elastic membranes on the morphology of cavitation bubble collapse and the differences in the deformation of vessel models, and elastic membranes under the action of cavitation bubble. The result shows that a material model with similar morphology should be used for the study of the damage mechanism of blood vessels or tissues under the action of cavitation bubbles.

6. Conclusion

To investigate the mechanism of interaction between cavitation bubbles and blood vessels induced by ultrasonic waves in the medical field, this study utilized a 3D lamination process to create blood vessel models with varying wall thicknesses. The discharge method was used to induce single cavitation bubbles in 0.9% sodium chloride saline. High-speed photography technology was employed to observe the interaction between the single cavitation bubble and vessel wall. The study analyzed the effect of vessel wall thickness on the dynamics of the cavitation bubble and the deformation pattern of the vessel under the action of the cavitation bubble, and the following conclusions were obtained:

• (1)The study found that changes in vessel wall thickness can influence the morphology of the cavitation bubble, collapse time, and collapsed micro-jets. As vessel wall thickness increases, the morphology and collapse time of the cavitation bubble located in the center of vessel exhibited an increasing and then stabilizing pattern. The cavitation bubble located in the center vessel produced pair of opposing micro-jets that moved along the vessel axis, and the velocity of the micro-jet decreased as the vessel wall thickness increased. However, when the cavitation bubble was not located in the central vessel center, the micro-jet did not appear towards the vessel wall in the late stage of the cavitation bubble collapse.
• (2)Through analyzing the vessel wall deformation under various bubble-wall distance conditions, it is found that the stretching degree of the vessel due to cavitation bubble expansion was greater than that of contraction, under the same vessel wall thickness. Furthermore, the deformation amplitude and deformation velocity of the vessel wall under the influence of cavitation bubble gradually decreased as the vessel wall thickness or the vessel elastic modulus increased. Thus, these new findings suggest that the potential damage caused by the expansion stretching of the vessel wall during cavitation bubble action is greater than the contraction stretching.
• (3)By comparing the interaction of the cavitation bubble with the elastic vessel and the elastic membrane under the same texture and thickness conditions, the period of cavitation bubble oscillation under the influence of the elastic vessel was lower than that the elastic membrane. Furthermore, under the same bubble-wall distance conditions, when the cavitation bubble expanded, the deformation of the elastic vessel was smaller than that of the elastic membrane. However, when the cavitation bubble contracted and collapsed, the deformation of the elastic vessel was larger than that of the elastic membrane.

These new findings have important theoretical implications for the potential damage mechanisms of ultrasound-induced cavitation bubbles to blood vessels in the medicine field [45], as well as the cavitation erosion mechanisms within pipelines in the hydrodynamic field [68]. However, in the human body, blood vessels do not exist independently, but are instead wrapped by other tissues, and the mechanical properties of vascular-tissue combinations can vary widely [15]. Therefore, future research should aim to explore the mechanism of deformation and damage to blood vessels under different tissue wrapping conditions caused by cavitation bubble.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.