• Ch038: The Physics of Shock Wave Lithotripsy

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Part IV
Extracorporeal Shock Wave Lithotripsy
The Physics of Shock Wave Lithotripsy
Robin O. Cleveland, PhD
James A. McAteer, PhD
Shock wave lithotripsy (SWL) was introduced in • Characteristics of a lithotriptor shock wave pressure around –10 MPa. The amplitude of the
the 1980s for the treatment of urinary stones and • The acoustics of SWL negative pressure is always much less than the
earned near-instantaneous acceptance as a first- • Acoustics primer peak positive pressure, and the negative phase of
line treatment option.1 Since then SWL has revo- • Acoustic cavitation the waveform generally does not have a shock in
lutionized treatment in nephrolithiasis world- • The physics of clinical lithotriptors it—that is, there is no abrupt transition. The
wide, and in the United States, it has been • Shock generation and shock focusing entire 5 µs pulse is generally referred to as a
estimated that approximately 70% of kidney • Coupling the shock wave to the body shock wave, shock pulse or pressure pulse—tech-
stones are treated using SWL.2 Over the years, • The focal zone of high acoustic pressure nically, however, it is only the sharp leading tran-
lithotripsy has undergone several waves of tech- • Mechanisms of shock wave action sition that is formally a shock.
nological advancement, but with little change in • How shock wave break stones Figure 38-1B shows the amplitude spectrum
the fundamentals of shock wave generation and • Mechanisms of tissue damage of the shock pulse (that is, it displays the different
delivery. That is, lithotriptors have changed in • The evolution of the lithotriptor frequency components in the pulse). We see that
form and mode of operation from a user perspec- • Future directions in lithotripsy a lithotriptor shock wave does not have a domi-
tive—and in certain respects the changes have nant frequency or tone, but rather its energy is
been dramatic—but the lithotriptor pressure CHARACTERISTICS OF spread over a very large frequency range—this is
pulse is still essentially the same. Lithotriptors A LITHOTRIPTOR SHOCK WAVE a characteristic feature of a short pulse. It can be
produce a signature waveform, an acoustic shock seen that most of the energy in the shock wave is
wave. This pressure pulse, or shock wave, is A typical shock wave measured at the focus of a between 100 kHz and 1 MHz. This means that it
responsible for breaking stones. However, it is lithotriptor is shown in Figure 38-1A. The wave is is unlikely that a lithotriptor breaks kidney stones
also responsible for collateral tissue damage that a short pulse of about 5 µs duration.* In this by exciting its resonance—as an opera singer
in some cases can be significant.3–6 example, the wave begins with a near instanta- might shatter a crystal glass.
Lithotriptors produce a powerful acoustic neous jump to a peak positive pressure of about The waveform shown in Figure 38-1A was
field that results in two mechanical forces on 40 MPa.† This fast transition in the waveform is measured in an electrohydraulic lithotriptor. A
stones and tissue: (1) direct stress associated with referred to as a “shock.” The transition is faster description of different types of shock wave gen-
the high amplitude shock wave and (2) stresses than can be measured and is less than 5 ns in erators is given below (see “The Physics of Clini-
and microjets associated with the growth and vio- duration.‡ The pressure then falls to zero about cal Lithotripsy”). Most lithotriptors produce a
lent collapse of cavitation bubbles. Recent 1 µs later. There is then a region of negative pres- similarly shaped shock wave, but depending on
research has made significant advances in deter- sure that lasts around 3 µs and has a peak negative the machine and the setting, the peak positive
mining the mechanisms of shock wave action, but pressure typically varies between 30 and 110 MPa
the story is by no means complete. What fuels this and the negative pressure between –5 and –15
*1 microsecond (µs) is 1 millionth of a second.
effort is the realization that a totally safe, yet †1 megapascal (MPa) is about 10 atmospheres of pressure. MPa. In Figure 38-2A, we compare waveforms
effective lithotriptor has yet to be developed. ‡1 nanosecond (ns) is 1 billionth of a second. measured in an electrohydraulic lithotriptor and
Indeed, there is compelling evidence to suggest
that a recent trend toward the development of
lithotriptors that produce very high amplitude and
tightly focused shock waves has led to increased
adverse effects and higher re-treatment rates.2,7–9
A major objective within the lithotripsy com-
munity is to find ways to make SWL safer and
more efficacious. The perfect lithotriptor may not
exist, so urologists are left to determine how best
to use the machines at hand. One step toward
improving outcomes in SWL is to have a better
understanding of how current machines work.
Thus, the goal of this chapter is to introduce the
basic physical concepts that underlie the mecha-
nisms of shock wave action in SWL. Our aim is
to give the background necessary to appreciate
how the design features of a lithotriptor can affect A B
its function. We also present a synopsis of current
Figure 38-1 A, A pressure waveform measured at the focus of an electrohydraulic lithotriptor (Dornier HM3). B, The
theories of shock wave action in stone breakage Fourier transform of the waveform in A showing how the energy is distributed as a function of frequency. (Both axes are
and tissue damage, and we summarize recent shown on a log scale.) The peak of the amplitude response is around 300 kHz, which corresponds to the 4 µs duration. The
developments in lithotriptor technology. The energy between 1 MHz and 20 MHz can be attributed to the shock in the waveform. The steeper drop-off of energy for
main topics to be covered are as follows: frequencies above 20 MHz is because that was the limit of the hydrophone for measuring the rise-time of the shock wave.
318 PART IV / Extracorporeal Shock Wave Lithotripsy
the molecules are compressed). For the case
where the object moves away from the fluid, there
is a resulting rarefaction of the molecules (that is,
the moving object leaves a partial vacuum). In this
case, the neighboring molecules will move to fill
the void, leaving a new region of rarefaction. This
continues one region to the next, and the rarefac-
tional disturbance propagates through the
medium as a tensile acoustic wave. In most cases,
a tensile wave propagates just like a compressive
wave and with the same sound speed.
Typical acoustic sources, such as audio
speakers, vibrate backwards and forwards. This
A B produces alternating compression and rarefaction
Figure 38-2 A, Focal waveforms measured in the Dornier HM3 at 24 kV and the Storz SLX at energy level 9. B, Com- waves that are referred to as the compressive
parison at lower settings—in this case the amplitudes are about the same, but the SLX waveform has not formed a shock. phase and tensile phase of the acoustic wave.
Often the waveform is sinusoidal in nature. Note,
however, that the majority of acoustic waves,
including the acoustic pulses generated in
lithotripsy, are not sinusoidal in form. For small-
amplitude waves (linear acoustics), every point of
an electromagnetic lithotriptor. It can be seen that cules in that region, in turn, push against the mol- the waveform moves at the same speed: the sound
the basic shape of both waveforms is very similar, ecules next to them. This relieves the compres- speed c0. This is a material property, and for
consisting of a shock front, compressive phase, sion in the first region but leads to a new com- water and tissue, it is about 1,500 m/s. We will
and tensile tail. For the settings chosen here, the pressed region. The molecules in the second see later for large amplitude (nonlinear) acoustic
main difference is the amplitude. Figure 38-2B region then start to compress the next adjacent waves, such as shock waves, that the sound speed
shows waveforms measured at lower power set- region, and so on; it is thus that a “wave” of com- is slightly changed by the presence of the wave.
tings of both machines, and again, the waveforms pression travels through the fluid. This is an The waveform shown in Figure 38-1 displays
are similar, but the amplitudes are different. “acoustic wave,” and the speed of wave propaga- the pressure pulse as a function of time at a given
Thus, lithotriptor shock waves show a unique tion (called the sound-speed) is a material prop- point in space. This is typically how acoustic
form that contains a high amplitude, compressive erty of the medium. Note that individual mole- waves are measured; for example, a microphone
phase of extremely rapid transition and short cules do not travel with the acoustic wave; rather, will record how pressure varies in time at one
duration followed by a trailing tensile phase. The they just jostle their adjacent neighbors. There- point in space. Acoustic waves, however, also
features of this waveform are similar regardless fore, for an acoustic wave to propagate, there vary in space and it is often useful to think of the
of the type of lithotriptor, but there are consider- must be a medium present that can support the wave in terms of its spatial extent. The relation-
able differences in the amplitude and spatial vibrations. This is an important physical differ- ship between the temporal separation of points on
extent of the acoustic output. It is likely that the ence between classical waves (eg, acoustic an acoustic wave (Δt) and the spatial separation
amplitude and size of the focal zone of different waves, seismic waves, water waves) and electro- of the points (Δx) is related by:
lithotriptors affects their performance. magnetic waves (eg, light, radio waves, x-rays).
For electromagnetic waves, energy is carried by Δx = Δt c0 (Equation 38-1)
AN ACOUSTICS PRIMER FOR SWL photons, which may be thought of as particles
that physically travel through space; thus a Recall that, in water or tissue, the sound
WHAT IS AN ACOUSTIC WAVE? An acoustic medium is not needed for the signal to be trans- speed is c0 = 1,500 m/s = 1.5 mm/µs and there-
wave, or sound wave, is created whenever an ferred. Therefore, light can travel through a vac- fore, for the shock wave shown in Figure 38-1,
object moves within a fluid (a fluid can be either uum, but sound cannot. the positive part of the wave—a portion 1 µs long
a gas or liquid). In Figure 38-3, we show that, as in time—will have a spatial extent in water of
an object moves, it locally compresses the fluid SOUND WAVES HAVE COMPRESSIVE AND TENSILE 1.5 mm. For a sinusoidal wave the spatial extent
that surrounds it—that is, the molecules are PHASES The explanation above describes the of one cycle of the wave is called the wavelength.
forced closer together. The compressed mole- compressive phase of a sound wave (that is, where
When a sound wave propagates, it affects the
density, pressure, and particle velocity of the
fluid particles. The impact on the density occurs
because, as molecules are compressed together,
the local density (ρ) will increase and in regions
of rarefraction the density will decrease. For an
acoustic wave it is convenient to write the total
density as:
ρ = ρ0+ ρa (Equation 38-2)
A B C D where ρ0 is the ambient density of the
Figure 38-3 Illustration showing a molecular view point of a sound wave. A, Medium is at rest. B, A piston pushes all medium (in the absence of sound) and ρa is the
the molecules out of the left side, resulting in a localized region of compression at the face of the piston (dark region). C, variation in the density due to the acoustic wave.
The neighboring molecules are compressed and the compression region moves away from the piston. D, The wave con- The pressure in the fluid can similarly be
tinues to moved away from the piston. The molecules at the piston return to their ambient state. written as the sum of two terms:
Chapter 38 / The Physics of Shock Wave Lithotripsy 319
p = p0 + pa (Equation 38-3) often referred to as Rayls—after the eminent energy that passes through that area can then be
nineteeth century acoustician Lord Rayleigh— calculated as:
where p0 is the ambient pressure (in the although the Rayl is not an international standard.
absence of sound) and pa, the acoustic pressure, Therefore, for a progressive acoustic wave,
(Equation 38-9)
is the fluctuation due to the sound wave. For the pressure, density and particle velocity are
most fluids, acoustic pressure and density are not independent, but are linearly related to
directly related by an “equation of state” which each other: where the double integral indicates a surface
takes the form: integral over the area A. The unit for energy is
pa = ua Z0 = ρa c02 (Equation 38-6) joules (J). The energy, E, will depend on both the
pa = ρa c0 2 (Equation 38-4) size of the area A and how the intensity varies
where the coefficients are material proper- across the area. The focal acoustic pulse energy
That is, where the wave is compressed, the ties. It follows that regions of high pressure are is calculated using the area in the focal plane,
pressure will be positive, and where the fluid is also compressed, and high particle velocity where the pressure is greater than half the maxi-
rarefied, the pressure will be negative. Physically, (away from source) and regions of low pressure mum pressure (this is equivalent to the focal
pressure represents a force per unit area and has are rarefied and have a negative particle velocity zone, see below). Energy can also be calculated
units of pascals (Pa). One pascal is quite a small (towards the source). As the acoustic wave trav- over different areas, for example, the projected
pressure, and atmospheric pressure at sea level is els, the fluctuations in density, pressure, and par- area of a stone or the area where the peak pres-
approximately 100,000 Pa. In biomedical ultra- ticle velocity all move together (ie, “in phase”). sure is above 5 MPa.11
sound, acoustic pressure is normally measured in Therefore, in a fluid with known material prop- Another acoustic property used in the litera-
megapascals (MPa).§ erties, if one property of an acoustic wave (such ture is the power per unit area, or the intensity I.
By way of example, the amplitude of the as the acoustic pressure) is measured, then Equa- Power is energy per unit time, and so the intensity
pressure from a diagnostic ultrasound scanner is tion 38-5 can be used to determine the other is the energy density divided by the time over
about 2 MPa at the focus. Typically, values for acoustic properties. which the integration was done (Equation 38-8),
ambient density and sound speed in tissue are which is normally the pulse length Tp:
ρ0 = 1,000 kg/m3 and c0 = 1,540 m/s, and so this WAVE INTENSITY OR ENERGY A propagating
corresponds to a relative density perturbation of acoustic wave carries energy. The amount of
(Equation 38-10)
ρa / ρ0 = 0.0009. For lithotripsy, peak pressures acoustic energy per unit area is called the energy
can be upwards of 100 MPa, which results in ρa / flux, energy density, energy flux density, or the
ρ0 = 0.04. Therefore, the density disturbances pulse intensity integral. The IEC standard10|| calls Intensity has units of watts per square meter
associated with acoustic waves in medical this the “pulse intensity integral (energy den- (W/m2) but it is more common in biomedical
devices—even the very strong waves that are pro- sity)” and it can be calculated by the following ultrasound to use centimeters (W/cm2).
duced in lithotripsy, actually result in very weak integral: For a sinusoidal pressure wave the integral
(less than 5%) compression of the fluid. can be calculated exactly and the intensity is:
(Equation 38-7)
(Equation 38-11)
The case shown in Figure 38-3, where the com- where the integration is done over the dura-
pression wave moves in one direction, is referred tion of the pulse. This is the acoustic equivalent to
to as a progressive wave. In contrast, when there the expression from physics “work equals force where p is the peak pressure of the sinusoidal
are sound waves traveling in different directions, times distance,” where acoustic pressure is the wave. If one substitutes the impedance for water
this is referred to as a compound wave, which will force per unit area and the time integral of the or tissue (Z0=1.5 MRayls) the relationship can be

not be considered here. For a progressive wave, velocity gives the distance. expressed as p = √3 I where p is in atmospheres
ˆ ˆ
the molecules in the compressed region also have The units for the pulse intensity integral (PII) of pressure and I is in W/cm2. For pulsed pressure
a small net velocity away from the source. The net are joules per square meter (J/m2). For a progres- waves, such as in lithotripsy, a simple expression
velocity of the molecules in a region of space is sive wave, we know that the particle velocity is does not exist for the intensity, as even small
referred to as the particle velocity (ua) and for a related to the acoustic pressure ua = pa/Z0 and changes in the pulse shape can have a significant
progressive acoustic wave it can be expressed as: therefore: effect on the integration used to calculate PII.
ua = pa/ρ0 c0 (Equation 38-5) (Equation 38-8) REFLECTION AND TRANSMISSION OF SOUND
WAVES When an acoustic wave encounters a
Using the example of a 100 MPa shock wave, medium with a different impedance, then part of
in which case, one only need measure the pres-
the instantaneous particle velocity at the peak is the wave will continue to propagate into the new
sure of the wave to determine PII. Note that to cal-
about 67 m/s. We will see below that the particle medium (the transmitted wave) and part of the
culate the integral, one needs to be able to accu-
velocity is needed in order to determine the wave will be reflected back into the original
rately measure the entire pressure-versus-time
energy in an acoustic wave. It has also been sug- medium (the reflected wave). In the case of nor-
waveform so that the integration can be done. The
gested that the particle velocity within a biologi- mal incidence, where the propagation direction
duration of a lithotripter pulse, for which this
cal target may produce sufficient strain to dam- of the shock wave is perpendicular to the sur-
integral needs to be evaluated, is defined as the
age the cells. face, the amplitude of the transmitted and
time from when the absolute value of the pressure
reflected waves depend only on the change in
first exceeds 10% of the peak pressure until the
ACOUSTIC IMPEDANCE The density and sound impedance between the two media, what is
last time is exceeds 10% of the peak pressure.
speed of a material (Equation 38-4) determine its referred to as the impedance mismatch. In terms
To determine the energy in an acoustic wave,
specific acoustic impedance (Z0 = ρ0 c0). This of acoustic pressure the transmission and reflec-
a specific area, A, has to be chosen, and the
term is often shortened to acoustic impedance or tion coefficients are:
just impedance. The impedance of tissue and
water is about 1.5 × 106 kg m–2 s–1. The units are
||Thisstandard describes how pressure measurements should be
taken on a lithotriptor to ensure accurate results and fair compar- (Equation 38-12)
isons across devices. The definition of terms used here is taken
§1 MPa is one million pascals, or approximately 10 atmospheres. from the IEC standard.
320 PART IV / Extracorporeal Shock Wave Lithotripsy
wave a pure tissue path to the kidney, is on the extreme acoustic pressures delivered to a very
(Equation 38-13) flank of the patient (delineated by the ribs, spine narrow focal zone.
and pelvic bone). For a focused acoustic source that generates a
There is a different set of coefficients for the sinusoidal waveform, such as an ultrasound
intensity or energy, called the intensity transmis- FOCUSING AND DIFFRACTION OF SOUND In transducer, there are analytical expressions for
sion and reflection coefficients: lithotripsy, focusing of the shock waves is used the size of the focal zone. The critical parameters
to concentrate the acoustic energy onto the stone are the wavelength of the sound wave (λ) and the
while reducing the impact on the surrounding half angle of the aperture:
tissue as much as possible. Lithotriptors achieve
focusing by various means, including the use of
(Equation 38-16)
reflectors, acoustic lenses, and spherically
(Equation 38-14) curved sources. Regardless of the method used,
the physics that describes the focusing of the where D is the diameter of the source and F
waves is similar for all these cases. An ideal the focal length. The formulae for the length (LFZ)
focus would be the case where all energy is and the diameter (DFZ) of the focal zone are:
localized to an infinitesimally small region in
(Equation 38-15) space. However, the physics of wave propagation
does not allow the energy to be focused to an (Equation 38-17)
arbitrarily small volume due to a process called
diffraction. This means that, even though the
In Figure 38-4, we show the intensity trans- and
acoustic pressure may be greatest at one point in
mission coefficient for an acoustic wave going space, there is a finite region or volume of sur-
from water to another medium with different rounding space that is also at high amplitude. (Equation 38-18)
impedance. We indicate typical values for tis- This is called the focal zone. For a theoretically
sue, kidney stones, bone, and air. One can see optimal focusing arrangement, where sound can
that the transmission from water to tissue is very come in from all angles, diffraction puts a bound Note that the focal length F is the distance
efficient. The water-to-stone transmission is on the size of the focal zone of about one wave- from the mouth of the therapy head to the focus
also relatively high, with 75 to 95% of the length. For the realistic focusing arrangements (where the stone should be placed). The focal
energy transmitted into the kidney stones. But a used in lithotripsy, where the sound only comes length should not be confused with the length of
water-air interface has an extremely small coef- from one direction, the focal zone can be from a the focal zone LFZ which is the region around the
ficient, and less than 0.1% of the energy of an few millimeters to tens of millimeters in size. focus where the pressure is high.
acoustic wave in water will pass into air—that For a pulsed waveform, as is generated in
is, 99.9% is reflected. This is why shock wave FOCAL ZONE The focal zone of a lithotriptor lithotripsy, there are no explicit formulae for the
generators in lithotripsy are water-filled, why (equivalent terms include focal region, hot spot, size of the focal volume because the size depends
immersion of the patient in water gives the most focal spot, focal volume, zone of high pressure) is on the waveform shape. But the focal region of a
efficient coupling of shock waves to the body, normally ellipsoidal in shape with its longest lithotriptor can be estimated using the formulae
and why in dry lithotriptors, great care must be dimension along the axis of the shock wave. To for the focal region of a sine wave. Figure 38-6A
taken to eliminate air pockets between the shock demonstrate this, Figure 38-5 shows the pre- shows how the focal zone gets shorter and nar-
head and the body. This is also one reason why dicted peak pressure of the focal zone in an rower as the diameter of the source aperture is
stones are not targeted for treatment through unmodified Dornier HM3 lithotriptor.12 The increased. Figure 38-6B shows how the focal
lung or segments of gas-filled bowel. Indeed the length and diameter of the focal zone depends on zone gets shorter and narrower as the focal length
best acoustic window, which allows the shock the diameter of the source, the focal length of the of the source (source-to-target distance) is
source, and the frequency content of the wave- decreased. Therefore, to make a small focal zone,
form. The dimension of the focal zone is thus one a shock source with a large diameter aperture and
characteristic of any given lithotriptor that is short focal length would be desired. However, the
determined by design features. Different size of the acoustic window in the flank and the

Use: 0.1034