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The physics of the oscillating bubble made simple
European Journal of Radiology 41 (2002) 176– 178 www.elsevier.com/locate/ejrad
The physics of the oscillating bubble made simple P. Dawson UCL Hospitals, The Middlesex Hospital, Mortimer Street, London W 1N 8AA, UK
Abstract The physics of bubbles and of their oscillations is extremely complex and attempts at its mathematical description are generally inaccessible. Yet some idea, in broad descriptive terms at least, is very helpful in understanding bubble phenomena of clinical interest. A brief attempt to provide such a description is given in this article. © 2002 Published by Elsevier Science Ireland Ltd.
1. Natural oscillations Many physical objects will oscillate or vibrate if disturbed from equilibrium, a pendulum given a push, a glass flicked with a finger, for example. The oscillations have fixed frequencies determined by the physical characteristics of the material and its geometry. Such objects, or systems, may oscillate in more than one way. London’s new and famous (infamous?) ‘‘Millenium’’ bridge, for example, worryingly oscillates both ‘up and down’ and from ‘side to side’. Which ‘modes’ of oscillation occur depend on the type of disturbance – the direction of the original kick so to speak. Although rather difficult to visualise, systems may oscillate simultaneously in more than one mode. For a bridge it is not too hard to imagine simultaneous side to side and up and down oscillations. The result may look very complex but may be ‘resolved’ into two such independent component modes of oscillation. The road to breaking down a complex repeating (periodic) vibration into simpler components is a road that, ultimately, leads to Fourier analysis and Transforms but we may go no further here than a simple statement that a system capable of vibrating in several different ways may adopt one, several or all of these simultaneously according to the disturbance. A spherical gas bubble in liquid may seem to have only one choice, a symmetrical increase and decrease in radius (spherical oscillations), but in fact such a bubble may adopt much more complex periodic motions involving a variety of shape
Tel.: +44-171-6368333; fax: + 44-171-3809068. E-mail address: [email protected] (P. Dawson).
distortions. Again, note that these may occur all together or in combinations. An oscillating system in air or fluid will convey the periodic disturbance to the surrounding medium and a sound wave will move through this medium away from it in all directions. If the frequency is within the appropriate range this will be heard as sound. An oscillating bubble in fluid, water, blood will emit sound, at the frequency of its own oscillation. Conversely, sound waves (periodic mechanical vibrations in the surrounding medium) may set bubbles oscillating (see below).
2. Driven oscillating systems A system capable of oscillation may be given a single brief disturbance – a push on the pendulum, a tap on the glass, for example. In this case the oscillations engendered will soon die away because of frictional forces (‘damping’). In fact the surrounding medium conducts much of the energy away as sound, of course. However, a system may be disturbed repeatedly not by a single disturbance but repeated disturbances at a steady frequency. What happens then? All depends on the relationship between the disturbing frequency and the system’s natural frequency. Imagine two tuning forks one of 256Hz(Hz) and one of 300Hz side by side in air. If the 256Hz fork is struck it oscillates at its natural frequency and sends out sound waves of this frequency which are heard as ‘middle C’ and which also fall on the other fork, subjecting it to repeated disturbances. What will happen? Something very interesting and not intuitively obvious. Firstly the second fork will begin to vibrate at its own natural frequency of 300Hz,
0720-048X/02/$ - see front matter © 2002 Published by Elsevier Science Ireland Ltd. PII: S 0 7 2 0 - 0 4 8 X ( 0 2 ) 0 0 0 2 3 - 2
P. Dawson / European Journal of Radiology 41 (2002) 176–178
not at the ‘driving’ frequency of 256Hz. However, this vibration will die away quite quickly. For this reason it is called a ‘transient’, a term we will return to shortly. Then the fork will settle down to oscillating at the frequency of the first fork. It will have been ‘forced’ into this (for it) unnatural vibration frequency. How vigorously it oscillates – how big an amplitude – will depend on how close the frequencies are. The closer they are the bigger the amplitude. When the frequencies are identical the amplitudes may become very big. Indeed, theory indicates that if there were no damping forces the amplitude would be infinite. This is resonance. With identical tuning forks the phenomenon is impressive. The first fork may be taken away after a few seconds and the second fork will still be readily heard. Resonance may be understood in a very simple way. If you push a pendulum repeatedly exactly at each time it comes to the top of its swing on one side, i.e. your pushes are at the same frequency as its natural swing frequency, its amplitude will steadily increase.
2.
3.
4.
5.
3. Non-linear oscillations The concept described by this term is basically very simple. A system will oscillate when disturbed around its resting equilibrium position because there are ‘restoring forces’ of some kind. A spring, for example, will resist both stretching and compressing. If the restoring force is simply proportional to the magnitude of the displacement from equilibrium (as in a spring, in which, if the stretch or compression is doubled the resisting force is doubled) the system is said to be ‘linear’. Systems (which in practice is, strictly, all systems) where this is not so are said to be ‘non-linear’. The simple spring is very close to linear for small displacements from equilibrium but not so for large displacements. It is a complex matter but the mathematics predicts elegantly what happens in practice in a system which diverges from simple linearity: such a system may oscillate not only at a fundamental frequency but also at multiples of it. The 2x multiple frequency is termed the 2nd Harmonic (1st overtone in musical terminology). For small amplitudes of radial oscillation, bubbles are almost linear systems, though never quite so since, for any given change in radius from the equilibrium, the restoring force is greater if the change is a decrease rather than if it is an increase. There is an intrinsic asymmetry and non-linearity. To focus on a gas bubble in fluid we may summarise as follows: 1. The bubble will be capable of oscillating in a variety of ways in which there are periodic variations in its shape. Some or all of these ‘modes’ may occur simultaneously, producing a very complex summation periodic change. The simplest of these is the
6.
7.
177
‘spherical’ mode of symmetric expression and compression. There is non linearity in the bubble and multiples of the fundamental frequency(ies) may additionally occur. This is more likely if the amplitude of the oscillation(s) is larger. A bubble may be caused to oscillate in some mode either by a single disturbance impulse or by continuous driving. A sound wave, a periodic mechanical disturbance of the surrounding medium, will cause the bubble to oscillate. Initially and very briefly a bubble driven by a periodic disturbance will oscillate at its own natural frequency(ies) but soon will settle down to oscillate at the driving frequency(ies). The brief period of oscillation at its own frequency is called a ‘transient’. This term when used in other ways is inappropriate as will be discussed below. The closer the driving frequency(ies) to the natural frequency(ies) of the oscillation of the bubble the better will be the ‘coupling’ and the greater the amplitude of oscillation. When the frequencies are equal there is resonance and only damping forces prevent oscillation being infinite in amplitude. Clearly, this would imply disruption of the system. In the case of a gas bubble of a few microns diameter in water the natural frequency of spherical oscillation is 1-2MHz, very close to that of the transducer frequency typically used in clinical work so there is a near resonance and large amplitude oscillations. This is the more so when insonating power levels are high. Bubbles insonated in these circumstances will be non-linear systems and will oscillate in harmonics. This is the basis of 2nd Harmonic Imaging where insonation is at a frequency close to the natural frequency of the bubble and the bubble oscillates at f, 2f, etc. The system may be set to detect only the 2f oscillations with consequent elimination of background clutter at the f frequency [1–3].
4. Bubble collapse/cavitation This is a very complex subject indeed, having its origins historically in Lord Raleigh’s musings on why kettles ‘sing’ and his being asked to solve the British Admiralty’s problems with mysteriously damaged ship’s propellers [4,5]. Concentrating on the single ultrasound bubble oscillating in a driving acoustic field we may say the following. A driven bubble may oscillate over a long period of time and the situation is sometimes described, somewhat confusingly, as ‘stable (non-inertial) cavitation’. On the other hand a situation may develop which is described as ‘unstable’ (inertial) cavitation [5,6]. Here the bubble expands and then rapidly
P. Dawson / European Journal of Radiology 41 (2002) 176–178
178
5. Disappearing bubble contrast agents
Fig. 1. Temperature during adiabatic compression.
contracts to a small fraction of its original radius/ volume. Such a bubble may rebound to grow again and collapse again or it may collapse and disintegrate into smaller bubbles (some of which may grow larger). Very high pressures and temperatures are reached during this extreme compression or ‘collapse’ phase and physical injury to either ship propellers or human tissues is possible. An idea of the temperature changes may be obtained by a simple calculation and the results are rather startling. If it is assumed that the process of compression of a bubble is so rapid that there is no time for heat exchange between the gas in the bubble and its surroundings the process is said to be adiabatic. For an ideal gas the temperature and volume during such an adiabatic compression is:TV k − 1 =K, a constant, when k is a parameter dependent on the gas molecules. For a diatomic gas such as Oxygen or Nitrogen, it is 1.4.
so T= T0 = T0
V0 V
r0 r
k−1
=T0
r0 r
3(k − 1)
1.2
,
since Vhr 3
The temperature in a compressed bubble for increasing compression ratios is shown in Fig. 1. If T0 is body temperature, i.e. 310K, and if the radius of the bubble is rapidly reduced to, say, 1/10th of its resting radius the temperature will rise to 5000K. This is the basis of the phenomenon of sonoluminescence in which oscillating bubbles glow with a ‘black body’ spectral output [7]. Such temperatures may be injurious to tissues and may be associated with the generation of free radicals.
Insonated bubbles disappear rapidly as do bubbles ‘trapped’ in the normal liver [8–10]. The disappearance of the latter is associated with the emission of a flash of broad spectrum noise. This is not what is actually ‘seen’ in the ultrasound display; it is, rather, the loss of correlation which is reflected in Colour Doppler mode display as a brief flash of colour. This is sometimes described as ‘transient scattering’, an entirely inappropriate term as the precise use of the terms ‘transient’ in oscillating systems has been described earlier. The alternative terms ‘stimulated acoustic emission’ or ‘sonascintography’ seem no better, however, given that all ‘acoustic emission’ by bubbles is ‘stimulated’ and given that the nuclear medicine association of ‘scintography’. It is not yet proven that the bubbles are ‘collapsing’ in the sense described above but this seems probable. Recent attempts to detect the generation of free radicals during these events with Levovist have so far proved unconvincing but are continuing [11]. The paper by te Haar examines possible harmful effects of bubbles.
References [1] Burns PN, Powers JE, Fritzsch T. Harmonic imaging: a new imaging and Doppler method for contrast enhanced ultrasound. Radiology 1992;185:142 – 3. [2] Schwarz KQ, Chen X, Steinmetz S, Phillips D. Harmonic Imaging with Levovist. J Am Soc Echocardiol 1997;10:1 – 10. [3] Kono Y, Moriasu F, Mine Y. Grey-scale second harmonic imaging of the liver with galactose-based microbubbles. Invest Radiol 1997;32:120 – 90. [4] Rayleigh Lord. On the pressure developed in a liquid during the collapse of a spherical cavity. Philosophical Magazine 1917;34:94 – 8. [5] Leighton TG. The Acoustic Bubble. Academic Press 1994. [6] The Textbook of Contrast Media. Eds: Dawson, Cosgrove, Grainger. ISIS Medical Media 1999. Ch 37, page 487. [7] Walton AJ, Reynolds GT. Sonoluminescence. Advances in Physics 1984;33:595 – 660. [8] Blomley MJK, Albrecht T, Cosgrove DO, et al. Stimulated acoustic emission (sono-scintigraphy) imaging with Levovist. Ultrasound Med Biol 1997;23(S1):33. [9] Blomley MJK, Albrecht T, Cosgrove DO, et al. Stimulated acoustic emission with the echo-enhancing agent Levovist: a reproducible effect with potential clinical utility. Radiology 1997;205(P):278. [10] Kamiyama K, Moriasu F, Kono Y, et al. Investigation of the ‘flash echo’ signal associated with a US contrast agent. Radiology 1996;201(P):158. [11] Dawson and Blomley (2001), unpublished data.
The physics of the oscillating bubble made simple P. Dawson UCL Hospitals, The Middlesex Hospital, Mortimer Street, London W 1N 8AA, UK
Abstract The physics of bubbles and of their oscillations is extremely complex and attempts at its mathematical description are generally inaccessible. Yet some idea, in broad descriptive terms at least, is very helpful in understanding bubble phenomena of clinical interest. A brief attempt to provide such a description is given in this article. © 2002 Published by Elsevier Science Ireland Ltd.
1. Natural oscillations Many physical objects will oscillate or vibrate if disturbed from equilibrium, a pendulum given a push, a glass flicked with a finger, for example. The oscillations have fixed frequencies determined by the physical characteristics of the material and its geometry. Such objects, or systems, may oscillate in more than one way. London’s new and famous (infamous?) ‘‘Millenium’’ bridge, for example, worryingly oscillates both ‘up and down’ and from ‘side to side’. Which ‘modes’ of oscillation occur depend on the type of disturbance – the direction of the original kick so to speak. Although rather difficult to visualise, systems may oscillate simultaneously in more than one mode. For a bridge it is not too hard to imagine simultaneous side to side and up and down oscillations. The result may look very complex but may be ‘resolved’ into two such independent component modes of oscillation. The road to breaking down a complex repeating (periodic) vibration into simpler components is a road that, ultimately, leads to Fourier analysis and Transforms but we may go no further here than a simple statement that a system capable of vibrating in several different ways may adopt one, several or all of these simultaneously according to the disturbance. A spherical gas bubble in liquid may seem to have only one choice, a symmetrical increase and decrease in radius (spherical oscillations), but in fact such a bubble may adopt much more complex periodic motions involving a variety of shape
Tel.: +44-171-6368333; fax: + 44-171-3809068. E-mail address: [email protected] (P. Dawson).
distortions. Again, note that these may occur all together or in combinations. An oscillating system in air or fluid will convey the periodic disturbance to the surrounding medium and a sound wave will move through this medium away from it in all directions. If the frequency is within the appropriate range this will be heard as sound. An oscillating bubble in fluid, water, blood will emit sound, at the frequency of its own oscillation. Conversely, sound waves (periodic mechanical vibrations in the surrounding medium) may set bubbles oscillating (see below).
2. Driven oscillating systems A system capable of oscillation may be given a single brief disturbance – a push on the pendulum, a tap on the glass, for example. In this case the oscillations engendered will soon die away because of frictional forces (‘damping’). In fact the surrounding medium conducts much of the energy away as sound, of course. However, a system may be disturbed repeatedly not by a single disturbance but repeated disturbances at a steady frequency. What happens then? All depends on the relationship between the disturbing frequency and the system’s natural frequency. Imagine two tuning forks one of 256Hz(Hz) and one of 300Hz side by side in air. If the 256Hz fork is struck it oscillates at its natural frequency and sends out sound waves of this frequency which are heard as ‘middle C’ and which also fall on the other fork, subjecting it to repeated disturbances. What will happen? Something very interesting and not intuitively obvious. Firstly the second fork will begin to vibrate at its own natural frequency of 300Hz,
0720-048X/02/$ - see front matter © 2002 Published by Elsevier Science Ireland Ltd. PII: S 0 7 2 0 - 0 4 8 X ( 0 2 ) 0 0 0 2 3 - 2
P. Dawson / European Journal of Radiology 41 (2002) 176–178
not at the ‘driving’ frequency of 256Hz. However, this vibration will die away quite quickly. For this reason it is called a ‘transient’, a term we will return to shortly. Then the fork will settle down to oscillating at the frequency of the first fork. It will have been ‘forced’ into this (for it) unnatural vibration frequency. How vigorously it oscillates – how big an amplitude – will depend on how close the frequencies are. The closer they are the bigger the amplitude. When the frequencies are identical the amplitudes may become very big. Indeed, theory indicates that if there were no damping forces the amplitude would be infinite. This is resonance. With identical tuning forks the phenomenon is impressive. The first fork may be taken away after a few seconds and the second fork will still be readily heard. Resonance may be understood in a very simple way. If you push a pendulum repeatedly exactly at each time it comes to the top of its swing on one side, i.e. your pushes are at the same frequency as its natural swing frequency, its amplitude will steadily increase.
2.
3.
4.
5.
3. Non-linear oscillations The concept described by this term is basically very simple. A system will oscillate when disturbed around its resting equilibrium position because there are ‘restoring forces’ of some kind. A spring, for example, will resist both stretching and compressing. If the restoring force is simply proportional to the magnitude of the displacement from equilibrium (as in a spring, in which, if the stretch or compression is doubled the resisting force is doubled) the system is said to be ‘linear’. Systems (which in practice is, strictly, all systems) where this is not so are said to be ‘non-linear’. The simple spring is very close to linear for small displacements from equilibrium but not so for large displacements. It is a complex matter but the mathematics predicts elegantly what happens in practice in a system which diverges from simple linearity: such a system may oscillate not only at a fundamental frequency but also at multiples of it. The 2x multiple frequency is termed the 2nd Harmonic (1st overtone in musical terminology). For small amplitudes of radial oscillation, bubbles are almost linear systems, though never quite so since, for any given change in radius from the equilibrium, the restoring force is greater if the change is a decrease rather than if it is an increase. There is an intrinsic asymmetry and non-linearity. To focus on a gas bubble in fluid we may summarise as follows: 1. The bubble will be capable of oscillating in a variety of ways in which there are periodic variations in its shape. Some or all of these ‘modes’ may occur simultaneously, producing a very complex summation periodic change. The simplest of these is the
6.
7.
177
‘spherical’ mode of symmetric expression and compression. There is non linearity in the bubble and multiples of the fundamental frequency(ies) may additionally occur. This is more likely if the amplitude of the oscillation(s) is larger. A bubble may be caused to oscillate in some mode either by a single disturbance impulse or by continuous driving. A sound wave, a periodic mechanical disturbance of the surrounding medium, will cause the bubble to oscillate. Initially and very briefly a bubble driven by a periodic disturbance will oscillate at its own natural frequency(ies) but soon will settle down to oscillate at the driving frequency(ies). The brief period of oscillation at its own frequency is called a ‘transient’. This term when used in other ways is inappropriate as will be discussed below. The closer the driving frequency(ies) to the natural frequency(ies) of the oscillation of the bubble the better will be the ‘coupling’ and the greater the amplitude of oscillation. When the frequencies are equal there is resonance and only damping forces prevent oscillation being infinite in amplitude. Clearly, this would imply disruption of the system. In the case of a gas bubble of a few microns diameter in water the natural frequency of spherical oscillation is 1-2MHz, very close to that of the transducer frequency typically used in clinical work so there is a near resonance and large amplitude oscillations. This is the more so when insonating power levels are high. Bubbles insonated in these circumstances will be non-linear systems and will oscillate in harmonics. This is the basis of 2nd Harmonic Imaging where insonation is at a frequency close to the natural frequency of the bubble and the bubble oscillates at f, 2f, etc. The system may be set to detect only the 2f oscillations with consequent elimination of background clutter at the f frequency [1–3].
4. Bubble collapse/cavitation This is a very complex subject indeed, having its origins historically in Lord Raleigh’s musings on why kettles ‘sing’ and his being asked to solve the British Admiralty’s problems with mysteriously damaged ship’s propellers [4,5]. Concentrating on the single ultrasound bubble oscillating in a driving acoustic field we may say the following. A driven bubble may oscillate over a long period of time and the situation is sometimes described, somewhat confusingly, as ‘stable (non-inertial) cavitation’. On the other hand a situation may develop which is described as ‘unstable’ (inertial) cavitation [5,6]. Here the bubble expands and then rapidly
P. Dawson / European Journal of Radiology 41 (2002) 176–178
178
5. Disappearing bubble contrast agents
Fig. 1. Temperature during adiabatic compression.
contracts to a small fraction of its original radius/ volume. Such a bubble may rebound to grow again and collapse again or it may collapse and disintegrate into smaller bubbles (some of which may grow larger). Very high pressures and temperatures are reached during this extreme compression or ‘collapse’ phase and physical injury to either ship propellers or human tissues is possible. An idea of the temperature changes may be obtained by a simple calculation and the results are rather startling. If it is assumed that the process of compression of a bubble is so rapid that there is no time for heat exchange between the gas in the bubble and its surroundings the process is said to be adiabatic. For an ideal gas the temperature and volume during such an adiabatic compression is:TV k − 1 =K, a constant, when k is a parameter dependent on the gas molecules. For a diatomic gas such as Oxygen or Nitrogen, it is 1.4.
so T= T0 = T0
V0 V
r0 r
k−1
=T0
r0 r
3(k − 1)
1.2
,
since Vhr 3
The temperature in a compressed bubble for increasing compression ratios is shown in Fig. 1. If T0 is body temperature, i.e. 310K, and if the radius of the bubble is rapidly reduced to, say, 1/10th of its resting radius the temperature will rise to 5000K. This is the basis of the phenomenon of sonoluminescence in which oscillating bubbles glow with a ‘black body’ spectral output [7]. Such temperatures may be injurious to tissues and may be associated with the generation of free radicals.
Insonated bubbles disappear rapidly as do bubbles ‘trapped’ in the normal liver [8–10]. The disappearance of the latter is associated with the emission of a flash of broad spectrum noise. This is not what is actually ‘seen’ in the ultrasound display; it is, rather, the loss of correlation which is reflected in Colour Doppler mode display as a brief flash of colour. This is sometimes described as ‘transient scattering’, an entirely inappropriate term as the precise use of the terms ‘transient’ in oscillating systems has been described earlier. The alternative terms ‘stimulated acoustic emission’ or ‘sonascintography’ seem no better, however, given that all ‘acoustic emission’ by bubbles is ‘stimulated’ and given that the nuclear medicine association of ‘scintography’. It is not yet proven that the bubbles are ‘collapsing’ in the sense described above but this seems probable. Recent attempts to detect the generation of free radicals during these events with Levovist have so far proved unconvincing but are continuing [11]. The paper by te Haar examines possible harmful effects of bubbles.
References [1] Burns PN, Powers JE, Fritzsch T. Harmonic imaging: a new imaging and Doppler method for contrast enhanced ultrasound. Radiology 1992;185:142 – 3. [2] Schwarz KQ, Chen X, Steinmetz S, Phillips D. Harmonic Imaging with Levovist. J Am Soc Echocardiol 1997;10:1 – 10. [3] Kono Y, Moriasu F, Mine Y. Grey-scale second harmonic imaging of the liver with galactose-based microbubbles. Invest Radiol 1997;32:120 – 90. [4] Rayleigh Lord. On the pressure developed in a liquid during the collapse of a spherical cavity. Philosophical Magazine 1917;34:94 – 8. [5] Leighton TG. The Acoustic Bubble. Academic Press 1994. [6] The Textbook of Contrast Media. Eds: Dawson, Cosgrove, Grainger. ISIS Medical Media 1999. Ch 37, page 487. [7] Walton AJ, Reynolds GT. Sonoluminescence. Advances in Physics 1984;33:595 – 660. [8] Blomley MJK, Albrecht T, Cosgrove DO, et al. Stimulated acoustic emission (sono-scintigraphy) imaging with Levovist. Ultrasound Med Biol 1997;23(S1):33. [9] Blomley MJK, Albrecht T, Cosgrove DO, et al. Stimulated acoustic emission with the echo-enhancing agent Levovist: a reproducible effect with potential clinical utility. Radiology 1997;205(P):278. [10] Kamiyama K, Moriasu F, Kono Y, et al. Investigation of the ‘flash echo’ signal associated with a US contrast agent. Radiology 1996;201(P):158. [11] Dawson and Blomley (2001), unpublished data.