Physics of Shock‐wave Lithotripsy

Andreas Neisius¹ & Pei Zhong²

¹ Department of Urology, Brüderkrankenhaus Trier, Johannes Gutenberg University, Mainz, Germany

² Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA

Introduction

Since its inception in the early 1980s shock‐wave lithotripsy (SWL) has become the treatment of choice for the majority of urinary stones because of its non‐invasiveness compared to traditional stone treatments [1–3]. In the early 1990s more than 85% of stone patients in Europe and the United States were treated with SWL [4]. Although this trend has been in decline since then because of the rapid advance in endourological technology and technique, still about 70% of kidney stone patients in 2002 and fewer than 50% in 2005 were managed by SWL [2, 4–6].

A primary factor that has contributed to this gradual decline in the clinical use of SWL is the reduced treatment efficiency, compounded by the increased retreatment rate and higher risk of tissue injury produced by the second‐ and third‐generation lithotripters, compared to the first‐generation HM3 lithotripter [7, 8]. Although SWL technology has evolved significantly over the past two decades, there is still debate regarding the physical basis and clinical benefits of technical modifications such as the tight focal width and high peak pressure fields often used in many contemporary lithotripters [9–11]. Better design as well as effective and safe use of shock‐wave lithotripters must rely on a clear and fundamental understanding about the physics of this innovative technology.

The physics and acoustical principles of SWL have been reviewed previously [12–14]. This chapter will therefore focus on the most salient features of SWL physics that are relevant to understanding the mechanisms of stone comminution and tissue injury with emphasis on contemporary technologies. The materials in this chapter are condensed in acoustics and physics content from a recent review of the topic by the senior author [15], but otherwise expanded based on the latest studies in order to provide a guideline for better lithotripter design, and effective and safe clinical use of this non‐invasive technology for stone management.

Overview of the clinical goal and physical process in SWL

The overarching goal of SWL is to pulverize kidney stones into gravels for spontaneous discharge in urine without inflicting adverse effects to surrounding tissues. As illustrated in Figure 58.1, disintegration of kidney stones by focused shock waves is a progressive process in which fragments of different size, shape, and distribution in the renal collecting system will be produced that may all influence the clinical outcome. In particular, the rate of stone comminution is not uniform throughout SWL. Stone disintegration accelerates initially within a few hundred shocks, followed by a gradually decelerating process towards the end of treatment. This critical observation suggests that different mechanisms of stone fragmentation may exist during the course of SWL, a unique feature that was not taken into account in the design of the second‐ and third‐generation lithotripters. To succeed in SWL, it is important to gain a fundamental understanding about acoustic waves (Box 58.1), the characteristics of focused shock waves produced by lithotripters (Box 58.2), and the physical basis for the optimal selection of lithotripter parameters and treatment protocol (or strategy) to maximize stone comminution while minimizing tissue injury.

Image described by caption. — **Figure 58.1** (a) Progressive comminution of a 10 mm spherical artificial stone in a membrane holder. (b) Dose‐dependence in stone comminution (solid lines) and normalized rate of stone comminution (dashed lines) produced by a contemporary electromagnetic shock‐wave lithotripter.

*Source:* from [15] with permission of Springer‐Verlag, Berlin, Heidelberg.

Box 58.1 Basics of acoustic waves [12].

An acoustic (or sound) wave is a form of mechanical vibration that propagates within a fluid (either gas or liquid). The wave locally compresses the fluid particles, i.e., the molecules are forced closer, pushing against each other. Compression in one region is accompanied by rarefaction (or tension) of the fluid particles in the adjacent region due to conservation of mass. This alternating vibration of compression and tension leads to the formation of a “wave” that travels through the fluid. The speed of wave propagation (referred to as the sound speed) is a material property of the medium, which is about 340 m/second in air, and about 1500 m/second in water and most soft tissues of the body. Notably, individual molecules of the medium do not travel with the acoustic wave; rather they just jostle adjacent molecules. Therefore, the medium serves as a carrier for the acoustic wave to propagate. This is an important physical distinction between mechanical waves (e.g., acoustic waves, water waves, elastic waves in solids) and electromagnetic waves (e.g., light waves, radio waves, X‐rays). For electromagnetic waves energy is carried by photons, which may be thought of as particles that physically travel through space without the need of a carrier medium. For example, light can travel through a vacuum; in contrast, sound cannot.

Box 58.2 Basics of lithotripter‐generated shock waves.

Several different approaches have been used to model the shock‐wave propagation and focusing in a lithotripter [19–21]. Majority of the models use the Euler equations for compressible fluids, in combination with the Tai equation of state for water shown below:

(1)

where p and ρ are the pressure and density of water immediately behind the shock front, ρ₀ is the density of water under ambient conditions, B (= 3000 bars) is a characteristic constant, and n = 7.15 [22]. It can be shown based on conservation of mass and momentum analysis that the Mach number at the shock front (M_s = c/c₀; where c is the speed of the shock front and c₀ is the sound speed in water under ambient conditions) is determined by [14]:

(2)

Considering that positive peak pressure p + in a lithotripter field is usually below 100 MPa, equations (1) and (2) indicate a density ratio below 1.05 and a Mach number less than 1.09 across the shock front of typical lithotripter‐generated shock waves (LSWs). As a LSW approaches the lithotripter focus, a Mach stem will be formed, leading to a planar shock front followed by diffracted waves originated from the aperture of the shock‐wave source [14, 23].

LSW can be significantly attenuated when propagating through interposing skin and soft tissues to reach the kidney stone. Although the pulse profile of the LSW remains similar, there is about 30% reduction in p + with a concomitant threefold increase in the rise time of the shock front and 44–67% broadening of the focal width from in vitro to in vivo [24]. Thermal effects in SWL are negligible. In clinical SWL, a pulse repetition frequency (PRF) of ≤ 2 Hz is typically utilized. However, even at a PRF of 100 Hz, the accumulated temperature increase produced by a clinically relevant shock‐wave exposure was estimated to be less than 2 °C [25]. Therefore the interaction of LSWs with biological tissues and renal calculi is primarily through nonthermal mechanical effects.

In comparison, electromagnetic (EM) and piezoelectric (PE) lithotripters have a larger dynamic range in pressure output, longer lifetime of the shock source, and more stable output than electrohydraulic (EH) lithotripters. Clinically, EM and EH lithotripters are more commonly used because of their broader focal width in combination with the higher effective acoustic pulse energy produced than PE lithotripters [9, 26]. Low energy output compounded with narrow focal width of the PE lithotripters result in longer treatment time, making them less favorable in high‐volume stone clinics [10, 27]. The primary field parameters of different types of lithotripters are summarized in Table 58.1.

Table 58.1 Characteristics of different type of shock‐wave lithotripters.

Source: from [15] with permission of Springer‐Verlag, Berlin, Heidelberg.

Lithotripter type		C (nF)	V (kV)	Focal length (cm)	Aperture angle (°)	p + (MPa), typical/max	p − (MPa)	t_r (ns)	t + (µs)	t − (µs)	Focal width (mm)	E_12mm (mJ)
EH		20–200	12–30	13–17	60–80	20/80	−8 to −10	<30 ns	1–2	4–6	8–17	20–70
EM	Acoustic lens	500–1500	8–20	15–18	50–70	50/80	−5 to −12	30–120	1–2	2–4	6–12	10–120
	Parabolic reflector		8–20		85–90	70/120	−5 to −15	50–500	≈1	2–3	4–9	20–150
	Self‐focusing		8–12		30–34	20/40	−4 to −7	70–120	2–3	4–5	15–20	N/A
PE		300–800	1–10	10–16	70–90	80/120	−9 to −10	30–250	≈1	0.5–1	2–6	N/A

General principles of shock‐wave lithotripsy

All clinical lithotripters are constructed using four primary components: (i) shock‐wave generator, (ii) focusing device, (iii) coupling medium, and (iv) stone localization system. The original Dornier HM3 lithotripter employed a water tub for coupling to ensure effective transmission of the lithotripter‐generated shock wave (LSW) without significant loss of energy at the patient’s surface (Figure 58.2a). In contrast, for operation convenience all contemporary lithotripters use dry coupling modalities with a water cushion system (such as the Siemens Lithoskop shown in Figure 58.2b). Three primary types of shock‐wave lithotripters have been developed – electrohydraulic (EH), electromagnetic (EM), and piezoelectric (PE) shock sources – with a variety of focusing systems (Figure 58.3). Stone localization in modern lithotripters is performed using isocentric C‐arm fluoroscopy with optional inline or offline ultrasound imaging [3]. The combination of the shock‐wave generator and the focusing device determines the main characteristics of the lithotripter field.

Illustration displaying different types of shock‐wave lithotripters: electrohydraulic, electromagnetic (acoustic lens, parabolic reflector, and self-focusing), and piezoelectric (left–right). — **Figure 58.3** Different types of shock‐wave lithotripters. EH, electrohydraulic; EM, electromagnetic; PE, piezoelectric.

*Source:* from [15] with permission of Springer‐Verlag, Berlin, Heidelberg.

Shock‐wave generation

In EH lithotripters, a spherically divergent shock wave is generated by a high voltage (12–30 kV) discharge between the tips of an electrode, which vaporizes and expands supersonically the surrounding fluid. In EM lithotripters, a magnetic field is produced by applying a high voltage (8–20 kV) pulse to a coil in flat, cylindrical, or spherical configuration to inject a strong current into the coil that repels an adjacent thin metallic membrane (in which a counter current is induced) to launch a high‐amplitude acoustic wave into the surrounding fluid. In PE lithotripters, a high‐voltage (1–10 kV) pulse is applied simultaneously to a mosaic of hundreds to few thousands piezoceramic elements that are mounted on a spherical carrier to generate a strong converging acoustic wave towards a common focus. Shock waves in EM and PE lithotripters are formed by nonlinear wave propagation in the coupling medium or interposing soft tissues between the shock source and the lithotripter focus [13].

Shock‐wave focusing

Different designs of acoustic reflector and lens have been utilized for shock‐wave focusing. In EH lithotripters, the spark‐generated shock wave (at F₁) is focused by a truncated ellipsoidal reflector to the second focus (F₂) of the ellipsoid where the kidney stones are aligned under imaging guidance. In majority of EM lithotripters, either an acoustic lens or parabolic reflectors are used (Figure 58.3). In PE lithotripters, self‐focusing ensured by the geometrical configuration of the shock‐wave source is typically utilized. Self‐focusing is also employed in a recently developed EM lithotripter by Eisenmenger et al. [16].

Lithotripter‐generated shock‐wave profile and distribution

Representative shock waves measured at the focus of the three different types of lithotripters are shown in Figure 58.4. A fiber‐optic probe hydrophone (FOPH) [17] recommended for lithotripter output characterization by the IEC standard is often used for such measurements [18]. All shock waves at the lithotripter focus consist of a leading shock front with a positive peak pressure (p+) of 40–70 MPa (1 MPa is about 10 times the atmospheric pressure) and an associated compressive wave of 1–2 microseconds in duration (1 microsecond = 1 millionth of a second). This fast rise of the compressive wave is referred to as “shock.” After falling to zero there is then a region of tensile wave lasting about 3–5 microseconds in duration with a peak negative pressure (p−) about −10 MPa. The entire shock‐wave pulse duration varies from 5 to 10 microseconds. Shock waves are broadband acoustic waves with a fundamental frequency in the range of 150–800 kHz (Figure 58.4a inset) [13]. As depicted in Figure 58.4a there is a notable pressure oscillation after the tensile wave produced by EM and PE lithotripters but not in EH lithotripters. In contrary to EH lithotripters in which the rise time of the leading shock front is less than 30 ns the shock‐wave rise time is about 100 ns for EM and PE lithotripters [13]. In general, pressure amplitudes are machine‐ and output‐setting‐specific, and the values of p + can vary from 20 to 120 MPa and p − from −4 to −15 MPa (see Table 58.1).

Image described by caption and surrounding text. — **Figure 58.4** (a) Pressure waveforms produced by different types of lithotripters and (b) model calculated peak pressure distribution in an EM lithotripter. EH, electrohydraulic; EM, electromagnetic; PE, piezoelectric.

*Source:* from [15] with permission of Springer‐Verlag, Berlin, Heidelberg.

Limited aperture size of the shock source causes the pressure field to be elongated along the central axis of the lithotripter to form a so‐called “cigar‐shaped” focus (Figure 58.4b). The size of the lithotripter focal region is measured by the −6 dB focal width (i.e., the dimension in which the positive pressure is equal or above half of p+) along the longitudinal and transverse axis of the lithotripter, respectively. It should be noted that the focal width is only a measure of the tightness of the pressure concentration in a lithotripter field. In general, the focal width may not be equated to the dimension of the fragmentation zone in a lithotripter field.

Composition and physical properties of kidney stones

To understand the mechanisms of stone fragmentation some basic features of kidney stones need to be considered. Kidney stones vary substantially in their chemical composition and structure. Stones are about 95% aggregated crystalline substances by weight, which are bound by cellular matrix proteins [28, 29]. As depicted in Figure 58.5 the composition of a kidney stone can often change from the core to the shell, as revealed by micro‐computed tomography (CT) images [30]. Calcium oxalate monohydrate (COM) or dihydrate (COD) stones represent the most common kidney stones. Other prevalent compositions include calcium hydrogen phosphate dihydrate (or brushite), carbonate‐apatite (CA), magnesium ammonium phosphate hexahydrate (MAPH or struvite), uric acid (UA), and cystine. It has been observed that the fragility of kidney stones during SWL varies significantly with composition and structural heterogeneity [31–33].

Table 58.2 summarizes the acoustic and mechanical properties of kidney stones of different compositions, together with their counterpart values in artificial kidney stones that are often used in research to understand the mechanisms of stone fragmentation, and for comparisons of different lithotripters and treatment strategies.

Mechanisms of stone fragmentation in SWL

Stone fragmentation in SWL is the direct consequence of shock‐wave–stone interaction, which is illustrated in Figure 58.6 by high‐speed photoelastic and shadowgraph imaging [34]. When a LSW strikes on the stone surface, part of the wave will be reflected back into the water while the other part will be refracted (or transmitted) into the stone. In general, the wave refraction at a fluid/solid boundary will produce two types of stress wave in the solid. One is called the longitudinal (or P) wave, in which the solid particles vibrate in the same direction as the wave propagation. The other is called the transverse (or S) wave, in which the solid particles vibrate perpendicular to the direction of the wave propagation. The acoustical principles of wave reflection and refraction at fluid/solid boundaries are described in Box 58.3. In addition, cavitation – the formation, expansion, and collapse of gas/vapor bubbles in liquids – will be generated by the tensile component of the LSW, which plays an important, yet different, role in stone comminution and in tissue injury.

Box 58.3 Acoustics of wave reflection and refraction at fluid/solid boundaries.

Figure 58.7 illustrates several general scenarios of wave reflection and refraction (or transmission) at fluid/stone boundaries that are relevant to SWL. The directions of the reflected pressure wave (P_r) in the fluid, and the transmitted P and S waves into the solid are determined by the Snell’s law:

(3)

Figure 58.7 Shock‐wave reflection and refraction (transmission) at the fluid/solid boundary. Left: shock‐wave incidence on a stone of flat surfaces, (a) zero incident angle (θ_i = 0°) and (b) at an oblique incident angle (θ_i > 0°). Right: shock‐wave incidence on a stone with curved surfaces and sharp corners, (c) at an oblique incident angle (θ_i > 0°) on a curved surface and (d) wave diffraction at a sharp corner when the incident angle changes abruptly from θ_i = 0° to θ_i = 90°, leading to generation of strong shear wave (P−S) and head wave (H) near the lateral boundary of the stone. In water, P_i and P_r denote incident and reflected pressure wave, DW indicates diffracted wave. In stone, P indicates longitudinal (or P) wave, S denotes transverse (or S for shear) wave, and P−S, S−S, and S−P indicate different reflected waves (denoted by the second letter) produced by the incident wave (denoted by the first letter) at stone/water boundaries. Lines with arrowheads indicate rays along the wavefront propagation direction, and dashed lines are used to show the locations of different wavefronts in water and stone.

where θ_i, θ_r, θ_P, and θ_S are the incident and reflected angles of the pressure waves in the fluid, and refracted angles of the P and S waves in the solid; c₀, c_P, and c_S are the wave speed in water, and P and S wave speeds in the solid, respectively. The mode conversion from pressure wave in water into P and S waves in the solid is ensured by the boundary conditions (i.e., continuities of displacement and stress [force per unit area] along the surface normal direction of the boundary). In general the reflection coefficient of the potential for the pressure wave in water (R), and transmission coefficients of the potentials for the P and S waves (T_P and T_S) in the solid depend critically on θ_i and are given by: [35]

(4.1)

(4.2)

(4.3)

where ρ_s is the density of the solid, and the acoustic impedances (i.e., the product of medium density and wave speed) for the oblique incident pressure wave in water (Z₀), and refracted P and S waves in the solid (Z_P and Z_S) are given by:

(4.4)