__Biophysical_Bases_of_Electrotherapy

SHORTWAVE DIATHERMY 197 (b) On the basis of your classification which tissue would be associated with maximum heat production? (c) Describe the complications (as far as prediction of heat production is concerned) caused by the presence of Iymphatic and blood vessels in fatty tissue. 6 Figure 7.4 shows current pathways in a model for an arm or leg. Describe the principal factors determining the relative rate of heating of each tissue layer. 7 A patient's lower limb is enclosed in a solenoidally wound coil as shown in figure 7.6. (a) Describe the motion of polar, non-polar and ionic molecules when a high frequency alternating current flows through the coil. (b) Indicate (with a diagram) the direction of movement of molecules in the limb. 8 If the solenoidally wound coil in question 7 was replaced by a pair of capacitor plates (one above the knee, one below the sole of the foot), what would be the new directions of molecular motion? Draw a diagram to illustrate. 9 Figure 7.8 shows the electric field associated with two adjacent turns of an induction coil (a) what is the practical significance of this electric field in patient treatment? (b) how can the effects of this electric field be minimized? 10 For shortwave diathermy it is common practice to use electrodes which are somewhat larger than the structure to be treated (figure 7.9). Explain in terms of: (a) the field pattern produced SHORTWAVE DIATHERMY 198 (b) the pattern of heating of tissue. What are the advantages and disadvantages of using unequal size electrode (figure 7.9(c) )? 11 (a) It is normal practice to space electrodes as far apart as possible (figure 7.10(b)) in shortwave diathermy treatment. Why is this the case? (b) What is the practical limitation on the electrode spacing which can be used? (c) Is there any advantage in positioning one electrode close to the patient's tissue as shown in figure 7.10(c)? 12 Consider the electrode arrangements shown in figure 7.11. Explain why the field intensity is non-uniform in diagrams (a) and (c). Under what circumstances will the field intensity be uniform, as in (b)? 13 (a) Draw a diagram showing a coplanar arrangement of electrodes over tissue and the resulting field pattern. (b) What are the advantages and disadvantages of coplanar electrode arrangements? (c) What practical limit is there on the spacing of coplanar electrodes? 14 Consider the hollow dielectric between capacitor plates which is shown in figure 7.14. (a) Explain where heat production is greatest and why. b) What technique can be used to produce more uniform heating of the dielectric? Explain.

SHORTWAVE DIATHERMY 199 15 When coplanar electrodes are used for patient treatment the tissues can be considered to be in series electrically (see figure 7.15(a)). (a) what approximations are implicit in this statement? (b) draw an electrical equivalent circuit similar to that in figure 7.15(a) for the situation where the electrodes are close together. (c) how would bringing the electrodes closer together affect the relative heating rate of muscle and fatty tissue? Justify your answer. 16 (a) Explain why, in principle, it is easier to produce selective heating of muscle with an inductive coil rather than capacitor field electrodes. (b) what practical constraints limit the selective heating of muscle with an induction coil? 17 (a) Explain the meaning of each of the terms in equation 7.6. (b) The initial rate of temperature increase in fatty tissue in an experiment is found to be double that of muscle. Assume that the densities of each tissue are the same and that muscle has a 50% greater heat capacity and calculate the relative rate of heating of the tissues. 18 The relationship between heat production (∆Q) and current flow (I) in a conductor is given by Joule's law: ∆Q = V.I.∆t where V is the potential difference across the conductor and ∆t is the time interval for which current I flows. (a) Show how equation 7.3 can be obtained as an alternative form of Joule's law. (b) An electric field intensity of 100 V.m-1 in a conductor results in a current density of 50 A.m-2. Use equation 7.2 to calculate the amount of heat produced in a 30 second time interval. You may assume that the current is entirely real. SHORTWAVE DIATHERMY 200 19 (a) Explain the meaning of each of the terms in equation 7.7. (b) An electric field of intensity 200 V.m-1 in a material results in a real current density of 50 A.m-2. The mass density of the material is 900 kg.m-3 and its specific heat capacity is 4.0 kJ.kg-1.K-1. Calculate the initial rate of increase in temperature (∆T/∆t) of the material. 20 A block of conducting material is placed in an electric field. The field intensity in the material is 300 V.m-1 and the resulting real current density is 120 A.m-2. If the material has a density of 1000 kg.m-3 and it has a specific heat capacity of 3.8 kJ.kg-1.K-1, calculate the initial rate of increase in temperature of the material (using equation 7.7). 21 Equation 7.7 describes the initial rate of increase in temperature of tissue in shortwave diathermy treatment. Describe the physiological response to the initial temperature rise and the effect this has on the subsequent rate of increase of temperature (figure 7.16).

NON-DIATHERMIC FIELDS 201 8 Non-Diathermic Fields PULSED SHORTWAVE APPARATUS Most shortwave diathermy machines offer the option of pulsed or continuous output. With continuous output, tissue heating is maximized as energy is transferred continually from the apparatus to the tissue. With pulsed output, energy is delivered in brief bursts with a long off-time between the bursts, so the average energy transferred is low. Pulsed shortwave is classed as non-diathermic in that the average power dissipated in the patient's tissue is too low to produce the appreciable temperature rises associated with traditional (continuous) shortwave treatment. For this reason it is described here rather than in the previous chapter where the emphasis was on the use of electric and magnetic fields to produce deep heating. Consider an example. If a burst of high frequency AC with a duration of 1 ms is generated at a burst frequency of 50 Hz, the period of each repetition ('on' time + 'off' time) is 1000/50 = 20 ms so 'on' time is 1 ms and the 'off' time is 19 ms and consequently the average energy is 1/20th of the peak energy (figure 8.1). Figure 8.1 (a) continuous and (b) pulsed output from shortwave machines. NON-DIATHERMIC FIELDS 202 It is asserted, though it has not been demonstrated, that pulsed shortwave is clinically Figure 8.2 beneficial because there are physiological effects of a 'non-thermal' nature which are Pulsed shortwave diathermy produced by the bursts of electromagnetic energy. Figure 8.2 shows the essential features of pulsed shortwave apparatus. It is the apparatus (schematic). similar to figure 7.1 but with the addition of a gating circuit to control the output of the 27.12 MHz sinewave generator. The gating circuit switches the sinewave generator on and off at the operator-chosen frequency (50 Hz in the previous example). It also controls the burst duration (1 ms in the previous example). Some machines have a predetermined burst duration, others allow operator selection. Components and subsections within the yellow rectangle in figure 8.2 are inside the apparatus. The functions of each subsection are as follows: * The sinewave generator produces a sinusoidal AC signal at the internationally approved frequency of 27.12 MHz [see chapter 5 and chapter 7]. * The gating circuit generates rectangular pulses to control the gain of the sinewave generator. The pulse frequency can normally be adjusted in the range approximately 1 Hz to 200 Hz. * A power amplifier in the sinewave generator circuit amplifies the signal to a level suitable for driving the patient circuit. The pulse power is high but the average power is low. * The resonant circuit (the patient tuning circuit)

NON-DIATHERMIC FIELDS 203 couples energy generated by the apparatus to the patient. Its operation was described in chapter 7. Output from the apparatus is applied to the patient using electrodes or an induction coil, in the same way that conventional (continuous- mode) shortwave diathermy is applied. * A power supply is needed to convert mains supplied 50 Hz AC to DC of whatever voltage is required to power the gating circuit, sinewave generator and amplifier. Pulsed shortwave is described as 'non-diathermic', meaning that it does not produce deep heating. The rationale is as follows. Consider and compare two pulsed shortwave machines. A difference between them is in the pulse frequencies which can be selected and the pulse width. Both have a peak power output of approximately 1000 watts. Machine 1 has a pulse width of 65 microseconds and a pulse frequency selectable between 80 Hz and 600 Hz. Machine 2 has a pulse width of 400 microseconds and a frequency range from 15 Hz to 200 Hz. Suppose machine 1 was set to a frequency of 80 Hz. The pulse width is fixed, in the 5.2 W is not a high power machine, at 65 microseconds so the output is on for a total of 65 x 80 = 5200 level. Imagine shining an microseconds each second. When the intensity control is set to deliver a maximum ordinary battery-operated output of 1000 W pulses the average power output is only: torch at your skin from a short distance. The heat and light 5200 x 10-6 x 1000 W = 5.2 W energy produced by the torch 1 has little impact on your tissues. This is a tiny fraction, approximately 0.5%, of the peak power. At a frequency of 600 Hz the average power at maximum output rises to: 65 x 600 x 10-6 x 1000 W = 39 W 1 This is 39/1000 W which is still only 3.9% of the peak power. NON-DIATHERMIC FIELDS 204 The average power is low and is only a small fraction, between 0.5% and 4%, of the The calculation of average peak power per pulse. power is left as an exercise Machine 2, with a pulse width of 400 microseconds, has a frequency range of 15 Hz to for the reader. It is the same 200 Hz and a peak power output of 1000 W. Calculations similar to the previous as the previous calculation, examples show that the maximum average power varies between 6 watts (at 15 Hz) but with different frequencies and 80 watts (at 200 Hz). Again, the average power is low and is only a small fraction, and burst durations. between 0.6% and 8%, of the peak power per pulse. See chapter 10 of JF The low power levels of pulsed shortwave ensure that gross heating effects, due to an Lehmann (Ed) 'Therapeutic appreciable increase in tissue temperature, do not occur. Heat and Cold'. Williams & Wilkins (1982) for more Effects of Pulsed Shortwave Fields details of these studies. Pulsed shortwave treatment is advocated as therapeutically beneficial due to non- thermal effects. Unfortunately, the advocates seem, in the main, to be the manufacturers of the equipment, rather than independent researchers who have carried-out proper studies. The few studies which have been undertaken include human and laboratory animal observations at the tissue level. Healing of experimentally produced skin wounds and haematomas in laboratory animals has been shown, in one study, to be promoted by pulsed shortwave treatment. Another study showed that human soft tissue injuries responded more rapidly in comparison to control (untreated) groups and groups of patients receiving a similar dose (but not dose rate) of continuous shortwave treatment. Some promising results have also been obtained in studies of rate and extent of nerve regeneration in laboratory animals. These results are indicative of therapeutic benefit but, due to their small number, are by no means conclusive. Proponents of pulsed shortwave have argued that another diathermic modality, ultrasound (chapter 10, following), has been shown to be useful when applied in

NON-DIATHERMIC FIELDS 205 pulsed mode for the treatment of a number of conditions where heating as such is In the absence of such either contraindicated or of dubious value. The argument is that if ultrasound can effects, pulsing the output so used to advantage in pulsed mode, where non-thermal effects are the explanation for that tissue heating is a any therapeutic benefits, then pulsed shortwave should also be beneficial. This minimum would be of little arguments is based on two questionable premises. First that 'non-thermal' benefits therapeutic value. of ultrasound treatment actually exist and second (perhaps more importantly) that With pulsed shortwave, the non-thermal effects will also be produced by pulsed shortwave treatment. instantaneous temperature increase is high but the Biophysical Mechanisms average temperature increase is low. Although the evidence base for pulsed shortwave treatment is small, a biophysical argument can be made for possible non-thermal effects of pulsed shortwave. The response of ions, polar molecules and non-polar molecules to an applied electric or magnetic field is well understood (chapter 7) and heat production in the applied field is readily explained. What is not known is how these same molecules responding in a biological environment can produce non-thermal cellular effects of therapeutic value. When electromagnetic energy is applied in brief bursts, the ions, polar molecules and non-polar molecules will respond equally briefly, with vigorous movement during the burst and a dying-down of activity between bursts. During the bursts, we would expect considerable molecular movement which would not increase the average temperature appreciably but which would markedly increase the instantaneous temperature. One could reasonably speculate that the transient excitation might affect concentration gradients, movement of molecules across the cell membrane and changes in membrane permeability in either or both of excitable cells and non-excitable cells. There may also be transient thermal effects on the synaptic junctions of nerve cells. These ideas remain speculative in the absence of appropriate experimental studies. NON-DIATHERMIC FIELDS 206 Optimum Treatment Parameters This limitation, of course, applies to both pulsed and The operating frequency of pulsed shortwave apparatus is 27.12 MHz. This particular continuous shortwave. frequency is used because there is international agreement that it be allocated for use in diathermy and it is known to produce heating at depth. There is no evidence to suggest that 27.12 MHz is the most appropriate frequency for clinical use. Another frequency might be optimum in terms of depth efficiency of heating. As yet, only limited experimental work has been conducted on the effect of different frequencies in the high frequency range. The burst frequency range from 10 Hz to 100 Hz or more may well turn out to be a clinically useful range as it overlaps the 'biological' frequency range. That is, the range of frequencies associated with best response from excitable cells. Unfortunately, comparative studies of the effect of different pulse frequencies (and burst durations) have yet to be conducted. The value of pulsed shortwave is thus questionable. Despite the long history of commercial availability and use, the question of therapeutic benefit remains to be answered. Therapeutic benefits seem possible but have not been adequately demonstrated. LOW FREQUENCY PULSED MAGNETIC FIELDS During the 1980s, low frequency pulsed magnetic field (PMF) apparatus gained some popularity as a therapeutic modality. The apparatus consists of a signal generating circuit (a pulse or sinewave generator), an amplifier circuit, a patient circuit and a power supply (figure 8.3). Subsections within the apparatus are shown in the yellow rectangle. * The signal generator circuit produces either low frequency sinusoidal AC or DC pulses of low frequency. The selectable frequencies are in the 'biological' frequency range up to about 100 Hz.

NON-DIATHERMIC FIELDS 207 * Current from the signal generator circuit is Figure 8.3 amplified and delivered to an induction coil, Low-frequency pulsed magnetic creating an intense, pulsed magnetic field. field apparatus (schematic). * A power supply converts mains-frequency AC to the DC which is needed to power the signal generator and the amplifier circuit. The part of the patient to be treated is placed within the induction coil. Large coils are used to enclose the trunk for treatment of, for example, low back pain. Smaller coils are used to treat smaller regions such as parts of the limbs As with pulsed shortwave, the advocacy and the recommendations for treatment seemed to come from the manufacturers of the equipment, rather than independent researchers. An argument has been presented that pulsed magnetic field (PMF) therapy is of benefit for the treatment of musculo-skeletal disorders in general and bone-healing in particular. The evidence of value for the treatment of musculo-skeletal disorders is lacking. The evidence for promotion of bone-healing is more convincing. Promotion of Bone Healing In the mid 1960's it was found that hydrated living bone is a piezo-electric material: when stressed the bone becomes charged. The phenomenon is illustrated in figure 8.4. When a bending load is applied, one surface of the bone is subject to a compressive stress (the top surface in figure 8.4) and the opposite surface is subject to a stretching or tensile stress. The bone responds by becoming NON-DIATHERMIC FIELDS 208 negatively charged on the surface under compression and positively charged Figure 8.4 on the surface under tension. This observation provides the basis for an The piezoelectric effect in bone. explanation of the biological phenomenon of stress remodelling; the mechanism by which bones or areas of bone respond to stress by growing in size and load bearing capability. It was reasoned that tiny current produced as a result of the piezoelectrically induced potentials can stimulate bone growth or resorption. Subsequent experiments demonstrated that small (microampere range) currents applied by implanted electrodes could also promote bone formation. Direct current promoted bone formation near the cathode, alternating current promoted bone formation near both electrodes. Two problems associated with the use of implanted electrodes are the risk of Extensive case studies have infection and the localization of bone formation in the vicinity of the electrodes. These established the effectiveness problems are overcome by inducing current flow in tissue using a pulsed magnetic of pulsed magnetic fields for field. As the magnetic field increases and decreases, eddy currents (chapter 6) are the treatment of non-united produced in the tissue. fractures. Induced Current in Tissue When considering the effects of low frequency pulsed magnetic fields it is important to make the distinction between the voltage waveform produced by the apparatus, the current waveform in the induction coil and the current induced in the patient's tissue. If a rectangular voltage waveform is applied to an induction coil the current waveform will not be perfectly rectangular, but more rounded. This is because when the voltage suddenly changes it takes a finite time for the current through the coil and the magnetic field around the coil to change correspondingly. The larger is the inductance of the coil, the longer it takes for the change to be complete. Figure 8.5 shows the relationship between voltage and current in the coil for different values of

NON-DIATHERMIC FIELDS 209 inductance. The larger is the inductance, the longer it takes for the Figure 8.5 current to increase to maximum. Voltage and current waveforms for (a) small The rate of increase in current intensity in the coil is important because this determines the induced current in tissue. Eddy currents are and (b) large value inductances. produced in tissue as a result of a changing magnetic field (chapter 6). When the magnetic field is constant, no current will be induced. Thus when the current in the coil is changing, and only when it is changing, a current will be induced in tissue. Figure 8.6 shows the relationship between induced current and current in the coil for the two waveforms shown in figure 8.5. When the coil current suddenly starts to increase, the rapid rate of increase (figure 8.6a) results in a high induced current. The magnetic field around the coil builds-up rapidly so the induced current is large. The rate of increase then drops rapidly and the induced current drops accordingly. A current spike is induced in tissue. When the coil current suddenly decreases, a current spike of the opposite polarity is induced due to the decreasing magnetic field intensity. The more rapidly changing coil current in 8.6(a) induces large current spikes but these are of short duration as the coil current rapidly reaches a steady value. The more slowly changing coil current in 8.6(b) induces current spikes which are smaller in amplitude but of longer duration. Figure 8.6 Current in an induction coil and resulting current induced in tissue for (a) small and (b) large value inductances. NON-DIATHERMIC FIELDS 210 The size of the induced current depends on the rate of change of the Figure 8.7 magnetic field and thus on the rate of change of current in the coil. Induced current waveforms for (a) sinusoidal If a sinusoidal current is applied and (b) triangular currents in an induction coil. to an induction coil the induced current will have the same shape but be shifted in phase. This is because the rate of change of a sine waveform is another sine waveform phase- shifted by one quarter of a wavelength; in other words a cosine waveform. This is shown in figure 8.7(a). Figure 8.7(b) shows the rectangular current waveform induced when triangular waveform is passed through an induction coil. A rectangular waveform is induced because the triangular waveform is alternately increasing at a constant rate then decreasing at a constant rate. The induced current is alternately constant and positive then constant and negative. Treatment Parameters No definite statements can yet be made regarding the most appropriate current waveform for magnetic field therapy. Even in the case of bone repair, where the clinical evidence of effectiveness is substantial, there is uncertainty as to the best waveshape and

NON-DIATHERMIC FIELDS 211 frequency. Excellent results seem to have been obtained using a burst of high- frequency pulses (frequency approximately 4 kHz) in 5 millisecond bursts repeated at a frequency of 15 Hz. Similar success has been achieved using single pulses and more intense fields wit a pulse frequency of 1 Hz. Optimum treatment parameters are yet to be established. Since heating appears to play no role (the energies involved are too low) one cannot predict effectiveness on the simple basis of total energy transfer. Nor can optimum frequencies be deduced without adequate knowledge of the cellular mechanisms involved. A conclusion is that chronic non-union of fractured bone can be successfully treated with low-frequency pulsed magnetic fields but that its value for the treatment of soft tissue injury remains open to question. EXERCISES 1 Figure 8.2 shows a schematic diagram of pulsed shortwave diathermy apparatus. (a) Briefly explain the function of each subsection. (b) What range of output pulse widths and pulse frequencies are normally provided? 2 Suppose the peak power output of a pulsed shortwave machine is 900 W and the pulse width is 300 ms. (a) What is the maximum frequency which can be used if the average power output is not to exceed 50 W? (b) For a pulse frequency of 200 Hz, what is the average power output? NON-DIATHERMIC FIELDS 212 3 One justification which is often invoked for pulsed shortwave treatment is that certain biological responses are 'non-linear' and exhibit a 'threshold effect'. (a) What is meant by the terms 'non-linear' and 'threshold effect'? (b) The average power output may be increased either by increasing the pulse frequency or by increasing the peak power output. If threshold effects are important, which of these adjustments would have the greatest biological effect? 4 Suppose it was established that for pulsed shortwave therapy an optimum pulse frequency was 100 Hz and a desirable peak to average power ratio was 50 to 1. What pulse width would be necessary? 5 (a) What is a piezo-electric material? (b) It is known that direct current promotes bone formation near the cathode. If bone, in vivo, is loaded as in figure 8.4, in what region will bone formation be promoted? 6 Figure 8.3 shows a schematic diagram of low frequency pulsed magnetic field apparatus. Briefly explain the function of each subsection. 7 Consider figures 8.5 and 8.6. (a) Explain why the current waveforms in figure 8.5 are rounded versions of the voltage waveforms. (b) Explain why the induced current waveforms in tissue have the shapes shown in figure 8.6. 8 The diagrams below show current waveforms in an induction coil placed near tissue.

NON-DIATHERMIC FIELDS 213 Draw diagrams to show the waveforms of the induced current in tissue for each of waveforms (a), (b), (c) and (d). 9 Draw diagrams showing the pathways of the induced current for biological tissue placed (a) adjacent to an induction coil (b) within an induction coil.

SOUND AND ELECTROMAGNETIC WAVES 214 9 Sound and Electromagnetic Waves Many kinds of wave motion are found in nature - in this chapter we examine only two - sound waves and electromagnetic waves. The two are very different in character but share a number of common properties and it is these common properties which we first consider. We look at how waves are produced in more detail in later chapters. DIFFERENT KINDS OF WAVES Any kind of wave motion, be it the ripples on a pond, sound or light has four characteristics which are fundamentally associated with the wave. These are: the wavelength, frequency, velocity of propagation and amplitude (or size). Figure 9.1 shows one kind of wave motion; a sinusoidal oscillation travelling in a finely coiled spring. As the spring is shaken up and down, transverse oscillations are produced which travel along the spring at a characteristic velocity, v, determined by the physical properties of the spring. The wavelength, λ, is the distance between peaks of the waves, that is the distance over which the wave repeats itself. The (peak) amplitude is the maximum Figure 9.1 displacement of the spring from the mean Transverse oscillations in a spring. position. Sometimes the peak-to-peak amplitude (which is double the peak amplitude) is specified. In order to sustain the oscillations, one end of the spring must be moved up an down with an appropriate frequency, f. The velocity, SOUND AND ELECTROMAGNETIC WAVES 215 wavelength and frequency are related by the wave equation: v = f.λ .... (9.1) The wave shown in figure 9.1 is a transverse sine wave - so called because the displacement of the spring is perpendicular or transverse to the direction of propagation. When the oscillations are along the direction of propagation the wave is called longitudinal. Figure 9.2 shows a longitudinal wave generated in a spring. For a longitudinal wave, the wavelength and velocity are easy to determine. The wavelength is the distance between two regions of compression. The velocity is determined by measuring how far a region of compression moves along the spring (∆x) in a known time interval (∆t). One region of compression, moving to the right, is coloured in figure 9.2. The velocity is calculated using the formula v = ∆x/∆t. The amplitude is less apparent but, in the case of a spring, it can be determined by attaching a marker to a point on the spring and measuring how far the marker oscillates back and forth from its mean position. Sound waves are longitudinal compressional waves. By their very nature they require a material medium for their existence as they are displacements of the material medium - solid, liquid or gas - about some mean position. The human ear can detect only a Figure 9.2 restricted range of sound frequencies, Longitudinal waves in a spring. from the lowest tones of an organ, around 16 Hz, up to some 12 to 20 kHz. The upper frequency limit of audibility diminishes with age. Frequencies greater than 20 kHz are termed ultrasonic, although some animals can hear frequencies up to 100 kHz.

SOUND AND ELECTROMAGNETIC WAVES 216 Figure 9.3 is a pictorial representation of a sound wave. Figure 9.3 Lines drawn close together represent regions of high Diagramatic representation of pressure. Widely spaced lines represent regions of low pressure (rarefaction). These regions move through any a sound wave. particular medium at a fixed velocity. For example, all Figure 9.4 sound waves travel at 340 m.s-1 in air, regardless of the sound frequency. In water, the velocity of sound waves is Diagramatic representation of close to 1100 m.s-1. an electromagnetic wave. Electromagnetic waves are a very special kind of transverse wave. They consist of a transverse sinusoidal electric field together with a transverse magnetic field. Light, radio waves, microwaves and X-rays are all electromagnetic waves with different frequencies, but the same velocity. The frequency and wavelength can vary but, in a particular medium, the velocity is constant. The speed of propagation of electromagnetic waves in empty space is a universal constant on which much of the structure of modern physics is based. A convenient representation of an electromagnetic wave is shown in figure 9.4. The sinusoidal electric field E, is transverse to the direction of propagation (arrow labelled v) and also perpendicular to the magnetic field, B. The existence of electromagnetic waves was not suspected until 1864 when the Scottish scientist, James Clerk Maxwell published a theoretical paper in which their existence was predicted. The velocity predicted for these waves turned out to be extremely close to that measured experimentally for light, which led Maxwell to conclude that light itself was an electromagnetic wave. Prior to this scientist since Newton's day had puzzled over the nature SOUND AND ELECTROMAGNETIC WAVES 217 of light: whether it was corpuscular or a wave motion, and if a wave, a wave in what? The mathematics of In many way Maxwell's work formed the keystone of 19th century physics. Maxwell's equations can be Maxwell had set himself the task of generalizing all of the accumulated knowledge of found in most textbooks on electrostatics, electric current, magnetism and electromagnetism: to write a few electromagnetism and will simple laws from which everything else could be derived. He summarized his not be discussed here. findings in a set of four equations which expressed the relationship between electric Rather the focus is on the and magnetic fields. In writing the equations he noticed they had a certain symmetry implications of his equations. about them, but that the symmetry could only be made complete by assuming the existence of a hitherto unobserved experimental result: that a changing electric field gives rise to a changing magnetic field. This assumption, together with other known facts of electricity and magnetism gave rise to the four equations which bear Maxwell's name. Not only did Maxwell's equations account for all that was known of electricity and magnetism, they also made one startling prediction: whenever charges are accelerated, an electromagnetic wave is produced. It was previously known that a moving charge produces a magnetic field which disappears when the charge stop moving. The equations predict that in addition an electromagnetic wave is produced if the charge accelerates and once the wave is produced its continued existence and propagation is independent of what subsequently happens to the charge. It is not a great step from this to the conclusion that all electromagnetic waves have their origin in the accelerated motion of charges. Since Maxwell's time electromagnetic waves with frequencies ranging from 5 Hz to 1024 Hz have been produced and used. Although they are produced an detected by seemingly different means and given different names, they all have essentially the same nature. A range of frequencies of electromagnetic waves is referred-to as an electromagnetic spectrum. Figure 9.5 shows such a spectrum and its most familiar regions. Electromagnetic waves with frequencies up to 1012 Hz can be generated electrically. For example the normal AM or FM waves received by a radio are produced by

SOUND AND ELECTROMAGNETIC WAVES 218 generating an oscillating electrical current in the transmitting aerial. Electrons are accelerated back and forth along the wire and the result is that electromagnetic waves are produced, radiating from the wire. Figure 9.5 The electromagnetic spectrum. Production of current by electronic circuitry becomes increasingly difficult at higher There are limits to the speed frequencies and above 1012 Hz it is necessary to use alternative methods for with which charges can move accelerating charges and producing the waves. through electronic (silicon Infrared radiation, sometimes referred-to as 'radiant heat' is emitted by all matter. chip) circuits. This limits the This is because the atoms and molecules are continually moving. In a solid, for frequency of electromagnetic example, the atoms are constrained but are able to vibrate about their mean position. waves which can be It is this movement energy which we call the heat energy of an object. The atomic produced by such means. jiggling means that charges (negative electrons and positive nuclei) are continually accelerating, so they radiate electromagnetic waves. At normal temperatures, most of the electromagnetic radiation has frequencies in the infrared portion of the spectrum. When something is heated, the molecules within it are given more energy and they move or jiggle more vigorously. A consequence is that the electromagnetic radiation produced has a higher average frequency. For example, if a piece of metal is heated from room temperature it first emits only infrared radiation, but as the temperature is increased, the metal becomes red-hot, then white, then blue hot. This is because the SOUND AND ELECTROMAGNETIC WAVES 219 average frequency of the emitted radiation increases. Infrared, visible and Higher frequency (visible light) radiation can also be produced by movement the ultraviolet light can be outer-shell electrons of an atom - hence the different colours produced when, for produced by heating of example, different chemicals are introduced into a bunsen flame. materials and temperature If atoms are bombarded with high energy electrons, inner shell electrons can be elevation is the major factor knocked from their orbitals, producing electromagnetic waves in the ultraviolet and X- in determining the frequency ray parts of the spectrum. Higher frequency gamma and hard-X radiation can not be distribution of the waves. produced by knocking electrons from their orbitals, but are produced when atomic Mutation may result either in nuclei are split, as in a nuclear reactor or nuclear explosion. The spontaneous decay cell death, cells with of naturally occurring radioisotopes also results in production of gamma and hard-X suboptimal function or radiation, due to the massive acceleration of the fragments when the nucleus is split. daughter cells which are cancerous (replicate Electromagnetic Waves and Safety uncontrollably). Any form of electromagnetic radiation can pose a biological hazard but higher frequencies are more dangerous. A distinction is made between ionizing and non- ionizing radiation. Electromagnetic waves with frequencies somewhat higher than those of visible light constitute ionizing radiation. When these higher frequency waves are absorbed by matter, electrons are knocked from their orbitals producing ions. If the displaced electrons are involved in bonding atoms together in a molecule, the bond will be broken and the molecule will be damaged, sometimes split. Thus higher frequency electromagnetic waves can cause molecular disruption. In most instances, molecular disruption will not harm cells or tissues as the damaged molecules can be removed and replaced. But if the disrupted molecule is DNA, a mutation can result. This is why exposure to ionizing radiation is associated with cell mutation and tissue tumours (cancer). The medical use of ionizing radiation involves a risk/benefit analysis. X-radiation is very useful for diagnostic imaging but there is no 'safe' level of exposure - rather the risk is proportional to the dose. Ultraviolet radiation is useful for treating certain conditions (see chapter 11), but again there is no 'safe' level of exposure. In

SOUND AND ELECTROMAGNETIC WAVES 220 assessing what is an acceptable level of exposure or dosage, normal environmental Low levels of naturally exposure is a consideration. If the treatment does not add appreciably to the natural occuring ionizing radiation burden then it may well be considered acceptable. The question is,. what is an are ever present. Sources 'appreciable' increase and how does this weigh against the benefits of treatment? include sunlight, cosmic In contrast to ionizing radiation, non-ionizing radiation does have safe levels of radiation and naturally exposure. At frequencies less than those of visible light, the principal effect of wave occuring radioisotopes. absorption is heating, so the risks are simply those associated with temperature It is no coincidence that elevation. Thus provided the temperature increase is within the physiological range, visible light is close to the no harm will normally occur. In this sense, exposure to non-ionizing radiation is no boundary between ionizing more harmful than any other form of heating. and non-ionizing radiation. A potential hazard with exposure to non-ionizing radiation is that it could stimulate Vision relies on light sensitive cell proliferation in malignant tissue, simply as a result of heating. It is for this molecules (opsins) which reason that any form of therapeutic heating is contraindicated when tissue split in a particular way when malignancy is known or suspected. A second potential hazard is focussing of a they absorb light. The beam of electromagnetic waves, which will result in concentration of the wave energy 'damage' is, however, in a particular region, producing a local 'hot-spot'. This is discussed further in reversible. chapter 11. A general conclusion is that, provided the temperature increase in any region is below a physiologically harmful level and the tissue is non-malignant, treatment with non-ionizing radiation is quite safe. WAVE TRANSMISSION AND ABSORPTION Having talked a little about sound and electromagnetic waves and considered some (but by no means all) hazards, we now address some fundamental questions about how these waves propagate and how they interact with and are absorbed by matter. Here the emphasis is on wave transmission and absorption. Hazards associated with particular frequencies of sound and electromagnetic waves are considered in more detail in later chapters. One striking difference between sound and electromagnetic waves is that a sound SOUND AND ELECTROMAGNETIC WAVES 221 wave, being a periodic vibration of atoms or molecules, relies on a material medium Figure 9.6 for its existence and propagation. Electromagnetic waves on the other hand, require (a) translational oscillation (b) no material medium for their transmission. Thus we can see our sun, the stars and rotation and (c) Internal vibration distant galaxies but cannot (even in principle) hear them! In applications to therapy this difference need not concern us. We examine what happens to both kinds of of a diatomic molecule. wave in a material medium. From everyday experience we know that sound and light are absorbed as they pass through materials. Ordinary window glass absorbs very little visible light - though it certainly absorbs some - but absorbs ultraviolet and infrared radiation quite strongly. Sound is absorbed by brick walls. In this case we find that low frequency sound is not absorbed as readily as higher frequencies. Absorption is related to the amount of absorbing material so it must be related to the density of the absorber. But it is also related to some other property of the material - otherwise why does glass transmit light while cardboard or paper does not? Molecular Motion in Matter To gain some insight into the absorption process we consider the motion of molecules making up a material. At any temperature above absolute zero the molecules will be in a state of agitation - oscillating back and forth and rotating. In addition, for molecules of more than one atom, vibrations of atoms relative to each other is possible. Figure 9.6 illustrates some of these modes of movement. Each of these three kinds of motion has a certain average frequency associated with it. For example if we consider a rotating molecule in a liquid, then as a result of its motion and the motion of other molecules it will suffer frequent collisions. In many instances the collisions will result in a change in the frequency of rotation of the molecule. Thus when we specify an average frequency of rotation we know that at any one instant some of the molecules will be rotating with frequencies much higher than the average, and some with frequencies much lower. A molecule may have a high

SOUND AND ELECTROMAGNETIC WAVES 222 frequency of rotation at one instant, suffer a collision and lose some rotational energy This is analogous to the to the other molecule, thus changing to a lower rotation frequency. situation with resonant The same is true for vibration of the molecules - there is continual transfer of the circuits (chapter 2) where if vibrational energy back and forth between molecules. two circuits have the same The motion of molecules within a material is, of course, what we measure as the heat resonant frequency, energy energy of an object. As we heat up a material the energy we put in results in greater can be transferred very agitation and thus greater kinetic energy of the molecules. The extent to which energy efficiently between them. is shared between the different modes of movement will depend on whether the material is a solid, liquid or gas and how many atoms make up molecule. For example in the case of a large protein molecule with many atom and many bonds, a significant proportion of the heat energy will appear as internal vibrations of the molecule. Sound Waves in Matter What happens then to a sound (or ultrasound) wave as it travels through a medium? In generating a sound wave we are producing mechanical vibrations - an oscillating displacement of the molecules - with a specific frequency. Consider what happens when the sound frequency is the same as that of some of the molecules. The sound wave is oscillating the molecules in a particular direction in the medium (the direction of propagation) while the molecules are naturally oscillating in all directions and these directions are continually and randomly changing as a result of collisions. The tendency is for the collisions to randomize the direction of sound vibrations and so convert sound energy into heat energy. If any natural oscillation of the molecule corresponds in frequency to the sound wave then the sound will be rapidly absorbed in the medium. Even if the sound frequency differs somewhat from any average frequency of molecular movement the natural spread of oscillation frequency of the molecules will enable some energy to be absorbed. In addition if the difference in frequency of two natural modes of molecular oscillation is equal to the sound frequency, energy can be absorbed in converting one frequency of oscillation to the other. SOUND AND ELECTROMAGNETIC WAVES 223 Electromagnetic Waves in Matter What happens as an electromagnetic wave travels through a medium? Since the Figure 7.2 shows the wave consists of an alternating electric and magnetic field as in figure 9.4 we would response of ions, polar expect the effect of the wave on the medium to be similar to the field effects discussed molecules and non-polar in chapter 7. Non-polar molecules will polarize in alternate directions in the molecules to a high alternating field, polar molecules will rotate back and forth and ions will try to move in frequency alternating electric the field direction. The energy losses in these processes - discussed previously in field. chapter 7 - will result in electromagnetic energy being converted into heat energy. e, like π is an irrational number, it cannot be The absorption of electromagnetic energy as a wave travels through a medium will expressed as a whole thus depend on the frequency of the wave and the electric and magnetic properties of number or a simple fraction. the material - the dielectric constant, conductivity and permeability. Using this Its value, to an accuracy of (somewhat simplified) model we predict that biological tissues with low dielectric four significant figures, is constant and conductivity, such as fatty tissue will absorb electromagnetic energy to a 2.718. lesser extent than substances such as muscle and other tissues with a high dielectric constant and conductivity. PENETRATION DEPTH In general, for any kind of wave of a certain frequency, we find that the wave energy decreases exponentially with distance. Mathematically this is written: E = Eoe-x/δ .... (9.2) Where Eo is the original energy and E is the energy remaining after the waves have travelled a distance x through the medium. The quantity δ is called the penetration depth of the waves in the medium. It depends on the frequency of the wave and the properties of the medium through which the wave travels. The quantity e is a constant which crops-up in any mathematical description of exponential increases or decreases, in the same way that π crops-up when we are dealing with circular geometry.

SOUND AND ELECTROMAGNETIC WAVES 224 Figure 9.7 shows a graph of E against x for an exponential decrease. To see what is meant by equation 9.2 and the term 'penetration depth', try substituting different values for x into equation 9.2. * when the distance x is zero, e-x/δ is eo = 1 since any number raised to the power zero is one. Thus E is equal to Eo the original energy, as we might expect. * when x = δ, e-x/δ is e-1 = 1/e = 1/2.718 = 0.37. Equation 9.2 then becomes E = Eo x 0.37. In other words the wave energy is reduced to 37% of the incident energy at a distance x equal to δ, the penetration depth. * when x = 2δ, e-x/δ is e-2 = 1/(2.718)2 = (0.37)2. Equation 9.2 then becomes E = Eo x (0.37)2. In other words the wave energy is reduced to 14% (37% of 37%) of the incident energy. The calculations show that as the wave travels through a material the energy is Figure 9.7 progressively absorbed. At a distance δ (the 'penetration depth') the wave energy Graph showing an exponential is decreased to 37% of the original energy. At a distance 2δ the wave energy is drop in energy, E, with distance, x. reduced to 37% of 37% of the incident energy and so on. In other words the wave energy is reduced by 63% every time the wave travels a distance δ in the medium. The wave energy is never completely absorbed but is reduced by a certain fraction with every centimetre it travels through the material. Clearly we cannot specify 'depth for complete absorption' of the wave energy as this will never occur. Instead we specify the penetration depth as the depth required to absorb 63% of the incident wave energy. An example. The penetration depth, δ, of 2000 MHz microwaves in fatty tissue is SOUND AND ELECTROMAGNETIC WAVES 225 5.3 cm. Use equation 9.2 to calculate the energy remaining after travelling a distance of (a) 2 cm and (b) 10 cm through fatty tissue. (a) For a distance x of 2 cm the wave energy, E, is given by equation 9.2 as E = Eoe-2/5.3 = Eoe-0.38 = 0.69Eo so E = 0.69 Eo The energy remaining after travelling a distance of 2 cm in fatty tissue is 69% of the incident energy. (b) For a distance x of 10 cm the wave energy E is given by E = Eoe-10/5.3 = Eoe-1.9 = 0.15Eo so E = 0.15 Eo The energy remaining after travelling a distance of 10 cm in fatty tissue is 15% of the incident energy. Some authors prefer to specify a 'half-value depth' rather than a penetration depth to The half-value depth, d1/2, is describe the rate of absorption of wave energy. The relationship between half-value the thickness of material depth and penetration depth can be calculated from equation 9.2 as follows. required to absorb 50% of the incident wave energy. The half-value depth, the thickness required to reduce the wave energy by 50%, is d1/2 where Eo = Eoe-d1/2/δ 2 In other words we have substituted E = Eo/2 (50% of Eo) when x = d1/2 into equation 9.2. 1 Cancelling the Eo on each side gives 2 = e-d1/2/δ Taking logarithms to the base e we have In 1 = - d1/2 i.e. In 2 = d1/2 so δ = d1/2 2 δ δ ln2 hence δ = 1.44 d1/2

SOUND AND ELECTROMAGNETIC WAVES 226 That is, the penetration depth is obtained from the half-value depth simply by multiplying by 1.44. Ultrasound and Microwaves In the frequency range of therapeutic interest, microwave and ultrasound radiation share two common features: * their penetration depths in fatty tissue are much higher than in muscle (or other tissues with high water and ion content). * as the frequency increases the penetration depth decreases. In other words as the wavelength decreases so does the penetration depth. Table 9.1 shows values of the penetration depth, δ, for different frequencies of ultrasound and microwave radiation in different body tissues. It is clear from the table that microwaves and radiation δ (cm) δ (cm) δ (cm) ultrasound are true diathermic modalities; that is, in fat in muscle in bone the waves are able to penetrate deeply into tissue. Ultrasound 1 MHz A significant proportion of the wave energy will be Ultrasound 2 MHz 7.2 1.7 0.22 available for heating of muscle and other tissues Ultrasound 3 MHz 4.8 1.2 0.15 Iying beneath the subcutaneous fat. Microwave 1000 MHz 2.4 0.6 0.07 In considering which frequencies are most useful Microwave 2000 MHz 7.0 1.6 ) similar for diathermy we would choose a frequency which Microwave 4000 MHz 5.3 1.2 ) to gives adequate penetration of the waves. We 4.0 0.6 ) fat would not, however, aim for a maximum penetration depth since if δ is too large the waves Table 9.1 will penetrate right through the tissue with little Penetration depth, δ, for microwaves absorption and thus little heating. The choice of 1 MHz for therapeutic application of ultrasound is a and ultrasound in body tissues. good compromise between adequate penetration SOUND AND ELECTROMAGNETIC WAVES 227 and adequate heating of underlying tissue. The pattern of heating does not, however, depend solely on penetration depth - reflection of the waves plays an important role. We will discuss reflection shortly. Infrared and Higher Frequency Radiation At higher frequencies than microwaves we have infrared, visible and ultraviolet radiation (see figure 9.5). As we go to these 'optical' frequencies the penetration depth becomes dependent on frequency in a complicated way; there is no longer a smooth increase or decrease in δ with frequency. Let us consider, first, infrared radiation. The infrared region of the spectrum extends from about 3 x 1011 Hz up to 4 x 1014 Hz. Traditionally we refer to the wavelength of these radiations rather than the frequency and the unit in popular usage is the nanometre which is abbreviated nm. One nanometre is 10-9 metre. We can convert from frequency in Hz to wavelength in nanometres by using equation 9.1. Since the speed of electromagnetic waves is close to 3 x 108 metres per second in most materials the wavelength, λ, in nanometres is related to the frequency, f, by λ = v = 3 x 1017 .... (9.3) To obtain this equation we f f have substituted v = 3 x 108 m.s-1 into equation 7.1 and The infrared region of the spectrum extends from 700 nm wavelength up to about 400 multiplied by 109 to convert 000 nm. For therapeutic application, sources of infrared radiation are used which put from metres to nm. out most of their radiation at the end of the spectrum close to visible light: from about 700 nm to about 15 000 nm. This includes both the so called 'near' infrared region, from about 700 nm to 4000 nm and part of the 'far' infrared region. The far infrared region extends from 4000 nm to about 400 000 nm. The penetration depth of near infrared radiation is very small. A maximum penetration depth of a few mm is obtained at about 1200 nm wavelength, and this decreases to a fraction of a millimetre at longer wavelengths. Wavelengths longer than 3000 nm are absorbed by the moisture on the surface of the skin. You may have noticed that the red end of the visible spectrum can be transmitted through the full thickness of your

SOUND AND ELECTROMAGNETIC WAVES 228 hand: this property does not extend to the infrared region of the spectrum. There is no sharp dividing Visible and ultraviolet radiation have frequencies corresponding to natural frequencies line, but the boundary associated with electrons in the outer shells of atoms. Since these electrons are the between ionizing and non- ones involved in bonding between atoms it is possible for light and ultraviolet ionizing radiation is between radiation to cause breaking of chemical bonds. the near and far ultraviolet We may summarize the absorption mechanisms for infrared, visible and ultraviolet regions of the electro- radiations as follows: magnetic spectrum. * Infrared radiation has frequencies corresponding to molecular and atomic motion and to differences in vibration frequency between two modes of motion. It can thus produce heating directly (hence the term 'radiant heat') but has a very small depth of penetration. * Visible and near ultraviolet radiation have frequencies corresponding to the difference in natural frequency between two energy states of bonding electrons in atoms. Such radiation can initiate chemical reactions and is only indirectly associated with the production of heat. * Far ultraviolet radiation, at higher frequencies than visible and near ultraviolet light, can separate electrons completely from an atom thus producing an ion. For this reason there is some risk of causing irreversible damage to biological molecules. The absorption mechanism for ultraviolet and visible light means that absorption and hence penetration depth, depends critically on frequency. Certain frequencies will be rapidly absorbed and have small penetration depths while others will not be absorbed so readily and hence have large penetration depths. Clearly ultraviolet therapy is of more value in initiating chemical change than in heating as such. Infrared radiation would be indicated when heating of superficial tissue is required. SOUND AND ELECTROMAGNETIC WAVES 229 WAVES AT BOUNDARIES The energy E of a wave is proportional to the square of So far we have discussed the absorption of a wave as it is transmitted through a the wave amplitude, a. This is medium. A knowledge of the rate of absorption of a wave in different tissues is not, written: however, sufficient to predict the amount of heating in a given tissue layer. Not all of the radiation striking a tissue interface will be transmitted, some will be reflected. In E α a2 this section we consider the factors determining the relative proportions of reflection It is also proportional to the and transmission which occur in tissues. square of the wave frequency, f, so we can also write: Energy and Impedance E α f2 First let us ask what determines the energy carried by a wave. The example of a transverse wave produced in a spring (figure 9.1) is a useful one. If the human oscillator in this figure were to shake the spring at a higher frequency this would result in more work being done and thus a greater energy in the wave. If the spring is displaced over a larger distance, resulting in a greater amplitude more energy is also produced in the wave motion. Wave energy depends on both the amplitude and frequency of the oscillations. The energy also depends on the properties of the spring itself. A very heavy spring will require more energy to move it: thus the energy depends on the mass of the spring, or for waves generally on the inertia of the medium. In the case of solids and liquids carrying sound waves the property which specifies the inertia of the medium is the density. Another property which determines the energy needed to produce oscillations in the spring is the elasticity. If waves are produced in a spring the energy needed will depend on its 'stretchiness' or elasticity. If the spring has high elastic compliance it will stretch easily and the restoring force which returns the spring to its original length is small. The two factors of elasticity and inertia together specify the impedance, Z, of a medium. In the case of sound waves in a solid or liquid the impedance is determined by the density and elasticity of the medium.

SOUND AND ELECTROMAGNETIC WAVES 230 In the case of electromagnetic waves the properties determining the impedance are the dielectric constant and conductivity. Consider, for example, an ideal dielectric. The molecules will polarize in the electric field. The electron cloud will alternate about the atomic nucleus and be drawn back to the normal position by the electrostatic attraction of negative electrons for the positive nucleus. The polarizing of the atom is analogous to stretching of a spring, and the polarizability (elasticity) is determined by the dielectric constant. For any kind of wave, the relationship between wave energy and the three quantities amplitude (a), frequency (f) and impedance (Z) is E α a2.f2.Z .... (9.4) Impedance and Reflection We now consider what happens when a wave strikes a boundary between two media. Although we have used the The example of two different springs connected together is a useful one. If the first example of two joined spring is made to oscillate with a certain frequency then if any energy is transferred to springs, the rule that waves the second spring, the frequency of the waves in each spring must be identical. This do not change frequency must be so since the springs are fastened together so that the oscillations in the when passing into a new joined ends of each spring are the same. medium applies to any kind of wave motion. How would we arrange things so as to transfer all of the wave energy from one medium to another? For maximum energy transfer the wave amplitude must be a maximum in spring 2. It cannot be larger than in spring 1 as the springs are joined. So for maximum energy transfer the wave amplitudes must be equal. The frequency is always the same in each medium, so for maximum energy transfer we require both equal amplitudes and frequencies. The wave energy, however, depends not only on frequency and amplitude but also on the impedance of the medium (equation 9.4). It follows that complete energy transfer can only occur when the impedances of each medium are the same. If a wave arrives at a boundary between two media of different impedance only part of SOUND AND ELECTROMAGNETIC WAVES 231 the wave energy can be transmitted: the rest must be Figure 9.8 reflected. Reflection from a low impedance boundary. Figure 9.8 shows two springs of different impedance connected together. Spring 2 is of lower impedance (more elastically compliant and/or lighter) than spring 1. A pulse travels along spring 1 until meeting the low impedance boundary. When the pulse hits the low impedance boundary the end of spring 2 is lifted to the same height as the incoming pulse. The energy transferred to spring 2 is given by equation 9.4 as E α a2f2Z so if the impedance, Z of spring 2 is lower than spring 1 but a and f are the same, a2f2Z is less so the transmitted energy is lower than the incident energy. The energy that is not transmitted is reflected, producing the reflected pulse in figure 9.8. Part of the original pulse is reflected and part continues in the original direction. Note that the displacement of the reflected pulse is in the same direction as the original. Figure 9.9 shows the opposite scenario, reflection at a high impedance boundary. In this case spring 2 is heavier and less compliant (has a higher impedance) than spring 1. In this case the stiffness of spring 2 prevents the spring junction from moving as high as the crest of the incoming wave. The effect is that a net downward force is exerted on spring 1 when the pulse reaches the junction. A reflected pulse is generated with the displacement downward rather than upward as in figure 9.8.

SOUND AND ELECTROMAGNETIC WAVES 232 The relationship between the mismatch in impedance and the amplitude of the reflected pulse (or wave) is given by equation 9.5. ρ = ar = Z1 - Z2 .... (9.5) ai Z1 + Z2 The reflection coefficient, ρ, is defined as the ratio of the reflected wave amplitude (ar) to the incident wave amplitude (ai) and this depends on the difference in impedance of the two media. Only when the impedances are equal (Z1 = Z2) will the reflection coefficient be zero and the amplitude of the reflected wave be zero. If there is a mismatch in impedance some wave energy will be reflected. An example. Suppose that two springs are connected together as in figure 9.8 and the impedance of spring 1 is three times the impedance of spring 2. Calculate the proportion of energy reflected at the junction. Substituting Z1 = 3Z2 into equation 9.5 the reflection coefficient is ρ = 3Z2 - Z2 = 0.50 3Z2 + Z2 This means that the reflected wave has an amplitude one Figure 9.9 half of the incident wave. Wave energy is proportional to Reflection from a high impedance boundary. the square of the amplitude, thus the fraction of energy reflected is one quarter. Although we have talked in terms of pulses or waves in a spring to illustrate the application of equation 9.5, the equation holds true for any kind of wave motion SOUND AND ELECTROMAGNETIC WAVES 233 including sound and electromagnetic waves. In the case of sound waves, Z refers to the acoustic impedance of the medium. In the case of electromagnetic waves, Z is the electrical impedance. STANDING WAVES Consider what happens if a transverse wave rather than a pulse strikes the boundary between two media. Unless the impedances of both media are identical a reflected wave will be produced travelling in the opposite direction. The two waves will add together, sometimes reinforcing, sometimes cancelling and the result is a standing wave pattern. Figure 9.10 shows the resultant waveform (in red) when two waves of equal amplitude and frequency are travelling in opposite directions The incident wave (blue) travels to the right and strikes a boundary (not shown). The wave is fully reflected, generating a wave (green) travelling in the opposite direction. The waves add together, so that what is actually observed is no longer two separate waves travelling in opposite directions but a single resultant. The resultant is a stationary wave pattern (hence the term 'standing wave'). The wave amplitude varies from instant to instant, changing from zero to maximum and back again, but the wave crests do not change position. At certain points (called nodes) the wave amplitude is always zero while at other points (the antinodes) the wave amplitude alternates rapidly between extreme values. Figure 9.10 A standing wave produced by interference of two equal size waves travelling in opposite directions.

SOUND AND ELECTROMAGNETIC WAVES 234 In figure 9.10(a) the incident and reflected waves are out of phase by one half of a You should find that the wavelength. In this case the two waves exactly cancel and the resultant has zero resultant is the same as amplitude. An instant later (figure 9.10(b)) the incident wave has moved 1/8th figures 9.10(b) and 9.10(a) wavelength to the right and the reflected wave 1/8th wavelength to the left. Now the respectively. waves are only 1/4 of a wavelength out of phase and the resultant is non-zero. In figure 9.10(c) the waves have moved further: now they are in phase and the resultant has a maximum amplitude. See if you can construct the resultant waveform at two later times when the incident and reflected waves have progressed a further 1/8th wavelength then 1/4 wavelength. For waves travelling at high velocity, the variation from (a) to (d) in figure 9.10 would occur in a tiny fraction of a second and the resulting variation in amplitude would be so fast as to be seen as a blur. This is illustrated in figure 9.11. Notice that in figure 9.11 the nodes and antinodes are readily discerned. The nodes are one half of a wavelength apart (as are the antinodes). One wavelength is one sinewave cycle, which is two of the 'beats' in figure 9.11. So half a wavelength is the distance between two antinodes or two nodes. An everyday example of standing wave production is seen Figure 9.11 with stretched wires or strings (for example guitar strings) The (blurred) standing wave pattern which which, when plucked, resonate and produce standing waves at any frequency for which the string length is a would be seen when the incident and multiple of half a wavelength. The mismatch in impedance reflected waves travel at high velocity. at each end of the string results in almost complete reflection and superposition of the waves results in a standing wave. If a wave is not fully reflected at a boundary (ρ < 1 in equation 9.5) the incident and reflected waves have different amplitudes and the resultant will be a combination of a standing wave and a travelling wave. This is the more SOUND AND ELECTROMAGNETIC WAVES 235 usual case with reflection. Figure 9.12 It is a useful exercise to draw two waves as in figure 9.10 Effect on the standing wave pattern of but with unequal amplitude and see what effect this has on unequal size incident and reflected waves. the resultant. What is produced is an amplitude modulated oscillation with maxima and minima but no true nodes Figure 9.13 (figure 9.12). Reflected and refracted waves at a boundary. REFLECTION AND REFRACTION We have seen that a mismatch in impedance results in reflection of waves at a boundary. A difference in impedance also results in the phenomenon of refraction. When a beam of waves is incident on a boundary at a certain angle (i in figure 9.13) the reflected wave will leave the boundary at the same angle. i' in figure 9.13 is the same size as i. The transmitted wave will be refracted: that is, its direction of propagation will change. The angle of refraction, r, will not be equal to the angle of incidence, i. The laws of reflection and refraction arise in most discussions of how light behaves, but these laws are not restricted to optics: they apply equally to any kind of wave motion. All that is required for refraction to occur is that the wave have a different velocity in the two media. The wave velocity is in turn determined by the impedance of the medium. Thus refraction of light occurs when a beam passes from air to glass because of the different velocity of

SOUND AND ELECTROMAGNETIC WAVES 236 light (and, more fundamentally, electrical impedance) of the two media. Figure 9.14 Reflection of a beam. To see how the laws arise, consider a beam incident on a boundary as shown in figure 9.14. First we look at the reflected wave. For simplicity, we consider a beam of width AB chosen so that AC is exactly one wavelength and we assume that the wave- crests are synchronized (the results are perfectly general, but the maths is more complicated when the waves are not synchronized and the distances do not match). Waves will be reflected at B while those at A on the same wavefront still have to travel a distance AC before being reflected. This will take a time t where AC = v1t During this time waves reflected at B will have travelled a distance BD where BD = v2t and the new wavefront is DC. Clearly distance AC is equal to distance BD: this is because the velocities of the incident and reflected waves are equal. If the angle of incidence is i then angle ACB is (90-i) - from simple geometric considerations - thus angle ABC is i and sin i = AC .... (9.6) BC In triangle DBC angle DBC is (90-i') and angle DCB is thus i' and i' is given by sin i' = BD .... (9.7) BC Since we know that AC and BD are equal, equations 9.6 and 9.7 together give sin i = sin i' thus i = i' and we have the 'Law of Reflection': angle of incidence = angle of reflection SOUND AND ELECTROMAGNETIC WAVES 237 Now consider figure 9.15 where the refracted beam is Figure 9.15 shown. Refraction of a beam. Waves entering medium 2 at B will travel a distance BD where BD = v2t .... (9.8) in the same time it takes for waves at A to travel the distance AC where AC = v1t .... (9.9) In this case v1 is not necessarily equal to v2 so distance BD is not equal to distance AC. Angle ABC is equal to i (as in the previous example with reflected waves). Hence AC sin i = BC .... (9.10) Similarly, angle BCD is equal to r and sin r = BD .... (9.11) BC Dividing equation 9.10 by equation 9.11 gives: sin i = AC sin r BD and substituting equations 9.8 and 9.9 this becomes sin i = v1 ..... (9.12) sin r v2 Which is the 'Law of Refraction' for waves at a boundary.

SOUND AND ELECTROMAGNETIC WAVES 238 Since v1 is not necessarily equal to v2, sin i is not equal to sin r and so the angle of incidence is not equal to the angle of refraction. The angles of incidence and refraction depend on the relative velocity of the waves in each medium. Equation 9.12 is a less familiar form of the law of refraction. It is more common in the The law of refraction is often case of light to define an 'index of refraction'. This is simply the ratio of the velocity of written: light in a vacuum to its velocity in the medium. Equation 9.12 then has v1 and v2 replaced by n1 and n2, the refractive indices of each medium. The refractive index is sin i = n1 dictated by the wave velocity in the medium. sin r n2 For light waves in air their velocity, v1, is always greater than the velocity, v2, in a where n1 and n2 are the so- denser medium (glass or whatever). Consequently the angle of incidence is always called refractive indices of greater than the angle of refraction. each medium. An example. The velocity of sound in air is 340 m.s-1 and in water is close to 1500 m.s-1 (see table When sound waves enter a 10.1 in the next chapter). Use equation 9.12 to calculate the angle of refraction when denser medium, they travel sound waves in air are incident upon water at an angle of 6o. faster. Electromagnetic waves are slowed in a Substituting v1 = 340 m.s-1 and v2 = 1500 m.s-1 into equation 9.12 we have denser medium. sin i = 340 = 0.23 ..... (9.13) sin r 1500 that is sln i 0.23 sin r = In this example i = 6o so we have sin r = sin 6o = 0.1045 = 0 4548 0.23 0.23 and the angle of refraction, r, is calculated to be 27o. SOUND AND ELECTROMAGNETIC WAVES 239 Critical Angle angle of angle of incidence refraction From the previous discussion it is apparent that waves are refracted at a boundary when the wave velocity is different in each medium. The relationship between incident i r and refracted angle is given by equation 9.12. Consider again the example of sound waves in air incident upon a boundary with 3o 13o water. Equation 9.13 relates the incident and refracted angle in this case. If this 6o 27o equation is used to calculate r for different values of i, a table similar to table 9.2 is 9o 44o produced. 12o 67o The results show a smooth increase in r as i increases in the range 0o to 13o. The 13o 90o value i = 13o is called the critical angle for the air/water system. At this angle of incidence the angle of refraction, r, is 90o. In other words the refracted wave travels Table 9.2 along the air/water boundary. For angles of incidence greater than 13o there is no real Angle of incidence and solution to equation 9.13. Experimentally what we observe is that total reflection refraction for sound waves occurs; that is, no refracted wave is produced. The critical angle is the largest incident at an air-water interface. angle for which a refracted wave exists. Although we have used the air/water system as an example, the general conclusions apply to any pair of materials where the wave velocity in medium 2 is greater than in medium 1. In this circumstance, the angle of refraction is greater than the angle of incidence and at a critical angle of incidence the refracted angle will be 90o. For angles of incidence greater than the critical angle, total reflection occurs. The actual value of the critical angle for a given pair of materials is calculated using equation 9.12. An example. The velocity of sound in muscle tissue is 1550 m.s-1 and in bone is 2800 m.s-1 (table 10.1 following). Calculate the critical angle for sound waves incident upon a muscle/bone boundary.

SOUND AND ELECTROMAGNETIC WAVES 240 Using equation 9.12 we have sin i = 1550 = 0.55 sin r 2800 At the critical angle the refracted angle is 90o so sin r = 1.00. We thus have sin i = 0.55 which gives i = 34o. Hence the critical angle for the muscle/bone boundary is 34o. EXERCISES 1 (a) Figure 9.1 shows transverse oscillations induced in a spring. The oscillations are observed to travel along the spring at a velocity of 2.5 m.s-1. If the end of the spring is vibrated with a frequency of 2 Hz what will be the wavelength of the oscillations? (b) When the spring shown in figure 9.2 is vibrated back and forth at a frequency of 3 Hz regions of compression separated by a distance of 48 cm are produced. What is the velocity of longitudinal waves in the spring? 2 The frequencies of some of the waves used in therapy are: ultrasound: 1 MHz, microwave: 2450 MHz, infrared: 3 x 1011 to 4 x 1014 Hz, ultraviolet :0.8 x 1015 to 1.6 x 1015 Hz (a) Given that the speed of light is 3 x 108 m.s-1 and that of sound 340 m.s-1, calculate the wavelength (or wavelength range) of these waves. (b) For each of these waves, indicate whether the wavelength concerned is closest in size to a house. a human limb. a tissue layer. a cell a protein molecule. (c) What would be the wavelength of electromagnetic radiation of the same frequency as the ultrasound? SOUND AND ELECTROMAGNETIC WAVES 241 3 (a) Describe the similarities and differences between sound and electromagnetic waves. (b) Give four examples of electromagnetic waves. What are their differences and similarities? (c) On what basis do we distinguished sound and ultrasound? 4 The number of possible modes of vibration of a molecule depends on the number of atoms in the molecule. For example the diatomic molecule in figure 9.6 has three possible modes of vibration. How many modes of vibration has: (a) a single atom? (b) a triatomic molecule? What are the implications of this for absorption of, say, sound energy in liquids made up of diatomic compared with polyatomic molecules? 5 Sound waves in liquid A are absorbed more rapidly than in liquid B. What conclusions can you draw regarding the frequencies of molecular oscillation in each liquid? 6 Consider an electromagnetic wave travelling through a material. (a) Describe the effect of the wave on each of polar molecules, non-polar molecules and ions in the material. (b) If the proportion of ions in the material was increased what effect would this have on the rate of absorption of wave energy? Explain. (c) If the proportion of non-polar molecules was increased what effect would this have on the rate of absorption of wave energy? Explain. 7 The penetration depth of 1 MHz ultrasound in muscle tissue is 1.7 cm. Using equation 9.2 construct a table showing fraction of energy remaining (E/Eo) at different depths in the muscle. Use a range of values of depth from 0 to 5 cm.

SOUND AND ELECTROMAGNETIC WAVES 242 Plot a graph of E/Eo vs depth and determine the depth at which the energy is reduced to: (a) 75% (b) 50% (c) 37% (d) 25% (e) 10% 8 The penetration depth of 1 MHz ultrasound in fatty tissue is 7.2 cm. Use equation 9.2 to construct a table showing the fraction of energy remaining (E/Eo) at different depths in the tissue. A suitable range of values of depth is from 0 to 5 cm. Plot a graph of E/Eo versus depth. (a) Compare your graph with that obtained in question 7. What conclusions can you draw about the relative 'absorbing power' of fatty tissue compared with muscle for 1 MHz ultrasound? (b) Use your graph to determine the thickness of fatty tissue required to reduce the wave energy to 75% of the original value. (c) What thickness of fatty tissue is required to absorb 10% of the incident energy? 9 The penetration depth of 4000 MHz microwaves in muscle tissue is 0.6 cm (table 9.1). Use equation 9.2 to calculate the thickness of muscle tissue required to absorb: (a) 10% (b) 50% (c) 90% of the incident wave energy. SOUND AND ELECTROMAGNETIC WAVES 243 10 (a) Electromagnetic waves with wavelengths in the range 700 nm to 400 nm are termed 'near infrared'. What is the frequency range of near infrared radiation? (See equation 9.3). (b) The portion of the far infrared spectrum used in therapy ranges from 4000 nm to 15 000 nm in wavelength. What is the frequency range of these waves? 11 What are the factors which determine: (a) the impedance of a spring? (b) the acoustic impedance of a material? (c) the impedance of a material to electromagnetic waves (the electrical impedance)? 12 Two springs are connected together as in figure 9.8. A pulse travels along spring 1. After reflection the reflected pulse has an amplitude one fifth of the incident amplitude. (a) What is the reflection coefficient of the boundary? (b) If the impedance of spring 1 is 3 x 103 kg.m-2.s-1 what is the impedance of spring 2? 13 Two springs are connected together as in figure 9.8. The impedance of spring 1 is one quarter of the impedance of spring 2. (a) Calculate the reflection coefficient of the boundary. (b) What is the significance of the negative value for the reflection coefficient? (c) What is the fraction of energy reflected at the spring junction? (d) What is the fraction of energy transmitted from spring 1 to spring 2? 14 Calculate the reflection coefficient for two media in contact when: (a) the impedances differ by a factor of 100.

SOUND AND ELECTROMAGNETIC WAVES 244 (b) the impedances differ by a factor of 10. (c) the impedances differ by a factor of 2. (d) the impedances are equal. 15 For each of the cases considered in question 14 above calculate the percentage of the original energy which will be: (a) reflected (b) transmitted at the boundary between the media. 16 Figure 9.10 shows the standing wave pattern produced by two waves of equal amplitude travelling in opposite directions. The time interval between each diagram is the same. Construct (graphically) the standing wave pattern at four successive (equal) time intervals. 17 Figure 9.10 shows the standing wave pattern produced when incident and reflected waves are of equal amplitude. Construct (graphically) the corresponding pattern produced when the reflected wave is only 2/3 the amplitude of the incident wave. 18 Ultrasound waves travelling through water strikes fatty tissue at an angle of incidence of 45o. The velocity of sound in water is 1500 m.s-1 and in fatty tissue is 1450 m.s-1. Use equation 9.12 to calculate the angle of refraction of the waves. 19 Ultrasound waves travelling through muscle strike an interface with bone. The transmitted wave has an angle of refraction of 80o. Given that the velocity of sound in muscle is 1550 m.s-1 and in bone is 2800 m.s-1, calculate the angle of incidence of the ultrasound waves. SOUND AND ELECTROMAGNETIC WAVES 245 20 Microwaves strike a tissue with an angle of incidence of 20o. If the ratio: speed of light in air speed of light in tissue is 1.3, calculate the angle of refraction of the waves. 21 Light incident on tissue at an angle of 40o has an angle of refraction of 29o. If the speed of light in air is 3.0 x 108 m.s-1, calculate the speed of light in the tissue (using equation 9.12). 22 (a) What is meant by the term 'critical angle' for a wave at a boundary? (b) Draw diagrams showing the incident, reflected and transmitted waves when the angle of incidence is (i) zero, (ii) less than the critical angle, (iii) equal to the critical angle and (iv) greater than the critical angle. 23 (a) The velocity of sound in fatty tissue is 1450 m.s-1 and in muscle, 1550 m.s- 1. Calculate the critical angle for the fat/muscle interface. (b) The critical angle for an air/water interface is 13o. Given that the velocity of sound in air is 340 m.s-1, calculate the velocity of sound in water. 24 Consider three parallel layers, fat/muscle/bone, on top of each other. A sound wave travelling through fatty tissue is incident upon the fat/muscle interface at an angle of 20o. Given: vfatty tissue = 1450 m.s-1 vmuscle = 1550 m.s-1 vbone = 2800 m.s-1

SOUND AND ELECTROMAGNETIC WAVES 246 Calculate: (a) the angle of refraction at the fat/muscle interface, (b) the angle of incidence at the muscle/bone interface. Would any wave energy be transmitted at the muscle/bone interface?

THERAPEUTIC WAVES: ULTRASOUND 247 10 Therapeutic Waves: Ultrasound Their popularity may be because ultrasound units are Of the diathermic modalities commonly used in therapy, ultrasound is the most relatively cheap, simple to popular. This is not because ultrasound is necessarily the most depth-effective. use, compact and portable. While ultrasound is a deep-heating modality and more depth-efficient than superficial modalities such as hot packs, infrared lamps or lasers, the depth effectiveness is strictly limited. The ultrasound frequencies most commonly used are 1 MHz and 3 MHz. The reasons for these being popular operating frequencies will become apparent in later sections of this chapter. In water and tissues of high water content the velocity of sound is close to 1500 m.s-1 thus the wavelength of 1 MHz ultrasound is (from equation 9.1) about 1.5 mm and that of 3 MHz ultrasound is about 0.5 mm. PRODUCTION OF THE WAVES The apparatus used to generate ultrasound waves consists of a high frequency oscillator, a power amplifier and a piezo-electric crystal which is mounted in a hand-held probe. A gating circuit is usually interposed between the oscillator and the power amplifier to provide pulsing of the ultrasound output. Figure 10.1 illustrates the arrangement. Figure 10.1 An ultrasound machine (schematic). THERAPEUTIC WAVES: ULTRASOUND 248 A power supply is needed to convert mains supplied 50 Hz AC into DC to power the Various crystals display subsections shown. piezo-electric properties but The heart of the circuit is the oscillator which produces high frequency sinusoidal AC two, quartz and barium (see chapter 5). This current is amplified and applied to a piezo-electric crystal, titanate (both ceramics) are causing it to vibrate (change in thickness) at the same frequency. The piezo-electric most useful for practical effect was discovered in 1880 by the brothers Pierre and Paul-Jaques Curie. They applications. found that certain crystals display the remarkable property of producing a small For example a pulsed potential difference between their faces when subjected to mechanical pressure. The ultrasound signal which is on reverse of this effect, namely that when a voltage is applied to a piezo-electric crystal it for 5 ms then off for 5 ms has changes in thickness, was discovered a short time afterwards. a duty cycle of 1:2 and a All piezo-electric crystals are found to exhibit a resonance effect - that is, they vibrate mark-space ratio of 1:1. A most efficiently at a certain (resonant) frequency. This natural frequency depends on signal which is on for 1 ms the dimensions, most importantly on the thickness, of the crystal. The resonant and off for 9 ms has a duty frequency of the oscillator (see chapter 5) is normally adjusted during manufacture to cycle of 1:10 and a mark- correspond to the crystal's resonant frequency. space ratio of 1:9. In continuous mode the gating circuit is not used and the piezo-electric crystal is supplied with high frequency AC continuously. In pulsed mode the AC is applied to the crystal in bursts. The burst frequency is normally 100 Hz; thus the time from the start of one burst to the start of the next is one-hundredth of a second or 10 milliseconds. The duty cycle is the ratio of 'on' time to total time ('on' plus 'off') for the output. In other words the duty cycle is the fraction of time for which ultrasound is being produced. Typical values of duty cycle for apparatus used in therapy are in the range 1:2 to 1:10. An alternative to specifying the duty cycle of pulsed ultrasound is to specify the mark- space ratio. The mark-space ratio is the ratio of 'on' time to 'off' time for the output. The rationale for the use of pulsed ultrasound will be discussed in a later section of this chapter.

THERAPEUTIC WAVES: ULTRASOUND 249 PATTERN OF THE ULTRASOUND FIELD The ultrasound generator produces a beam of ultrasonic waves by vibration of the metal end plate of the treatment head (the transducer) shown in figure 10.1. The plate is typically a few centimetres in diameter - perhaps twenty or thirty wavelengths. When the diameter of the transducer is many multiples of the wavelength, the sound The angle of divergence of a beam is cylindrical in shape and the beam divergence is low. At a frequency of 1 MHz, sound beam produced by a an ultrasound beam in water, produced by a typical size transducer (2.8 cm diameter) flat vibrating disk is given by: has a divergence of about 4o. This figure increases to about 40o at a frequency of 100 kHz and 90o at 65 kHz. Thus at frequencies of 65 kHz or less, there is no beam: sinθ = 0.61 v sound waves radiate in all directions. At MHz frequencies, the sound beam is pencil- f.r shaped and almost the same diameter as the transducer. where v is the wave velocity, f Although the beam has a relatively uniform, cylindrical shape, the relatively small size is the frequency and r is the of the treatment head of typical ultrasound machines results in a marked variation in transducer radius. ultrasound intensity across the width of the beam. To see how this occurs, consider a particular point in front of the transducer such as point A in figure 10.2. Every point on the transducer surface will act as a source of sound waves. The total wave amplitude, and hence total wave energy, at point A will depend on the contribution from all points on the transducer surface. Waves from some points will arrive in phase and reinforce each other; others will arrive out of phase and cancel. Figure 10.3 shows waves originating at only two points on the transducer surface: in this case the waves are out-of-phase and cancel. By adding (vectorially) the waves originating from all points on the Figure 10.2 transducer surface we can calculate the resulting intensity at any particular Interference of sound waves from a point. The calculations are made complex by the fact that the surface of the transducer does not remain planar, but flexes and undulates as it radiating source. oscillates. THERAPEUTIC WAVES: ULTRASOUND 250 The net result is that the ultrasound field is not uniform. Near the transducer a distinctive pattern of maxima and minima of intensity are produced. Beyond this region (called the near or interference field) the distant field is more homogenous and decreases smoothly in intensity with distance from the transducer. The effect is illustrated in figure 10.3 where the undulations of the transducer surface have been ignored for simplicity. The pattern of intensity was calculated for a frequency of 1 MHz and a transducer of diameter 2.8 cm (area 6.2 cm2): with larger diameters the pattern is qualitatively similar with the interference effects extending to greater distances. At an ultrasound frequency of 3 MHz the pattern would Figure 10.3 again be qualitatively similar but with the interference Intensity along the axis of a sound beam for effects extending over approximately three times the a transducer of diameter 2.8 cm, operated distance shown in figure 10.3. Figure 10.3 shows the variation in intensity of an ultrasound at 1 MHz frequency in water. beam at points along the central axis. In the near field, local 'hot-spots' or regions of maximum intensity are separated by 'cold-spots' or regions of minimum intensity. Off-axis, patterns of hot-spots and cold-spots are also observed. The location of their maxima and minima are, however, different. Averaged across the beam, the intensity is relatively constant, only decreasing slowly with distance. So at any particular distance, hot-spots and cold spots are produced in different locations across the beam, while the average energy is constant. Figure 10.4 shows another view of the energy distribution in an ultrasound beam. This time a two-dimensional view showing the high intensity regions off the central axis. The shaded areas indicate regions of high local ultrasound intensity. Note that regions of low intensity on the central axis have, alongside, regions of high intensity

THERAPEUTIC WAVES: ULTRASOUND 251 and vice-versa. Most of the ultrasound energy is confined within the area defined by the brown lines. There is a slight convergence of the beam in the near (interference) field and a small divergence in the far field. The complex interference pattern makes Figure 10.4 it essential that in therapeutic application Variation in intensity within the ultrasound of ultrasound the transducer be moved around over the area to be treated. If the sound-head (ultrasound transducer) were kept stationary, beam described in figure 10.3. localized 'hot spots' would be produced in tissue which could result in excessive local heating. By moving the sound-head in circular paths, production of local areas of high temperature rise is avoided. If the sound-head is moved in a circular path so as to produce a treated area of at least twice the diameter of the head, hot-spot production will be avoided. Beam Nonuniformity Ratio (BNR) A quantity of interest is the beam nonuniformity ratio or BNR. This is the ratio of the peak intensity to average intensity of the beam. Because there are always local regions of high intensity, the BNR is always greater than 1. In figure 10.4, the intensity pattern is that which would be produced by a piezo-electric If a hollow, doughnut-shaped crystal which was about the same size as the ultrasound treatment head. If the crystal crystal were used, the were appreciably smaller than the metal end-plate of the ultrasound treatment head, intensity pattern would again the metal end-plate would vibrate differently and the pattern shown in figure 10.4 be different to figure 10.4. In would be different. The high-intensity regions would be in different positions and, this case the highest peak more importantly, the peaks would be higher. So the BNR would be higher. (and the BNR) would be A low BNR is clearly an advantage but movement of the treatment head is of much lower. THERAPEUTIC WAVES: ULTRASOUND 252 more crucial importance in clinical practice. In the following sections we ignore beam nonuniformity and assume that the treatment head is moved to produce the effect of a uniform ultrasound beam. For a more accurate analysis we would have to consider the exact shape of the beam and its movement and also include the effect of tissue inhomogeneity in the calculations (for example the effect of blood vessels in fatty tissues). Much useful insight can, however, be gained with the simplifying assumptions used. TRANSFER OF ENERGY TO TISSUE: COUPLING Before examining the effect of ultrasound on tissue we must first consider the transfer When the two impedances of energy from the transducer to tissue. Previously we saw that reflection of waves at are the same, no reflection an interface depends on the difference in impedance of the two media. occurs while if the impedances differ greatly Table 10.1 (following) lists the acoustic impedances of air, water, steel and body most of the wave energy is tissues. Notice that the impedance of air and metal differ significantly from the rest reflected. with air showing by far the largest deviation. The acoustic impedance of air is only a tiny fraction of the impedance of body tissues. For this reason the ultrasound transducer must be in intimate contact with the skin for appreciable transfer of energy to the tissues. If the transducer is separated from the skin by even a tiny air gap most of the ultrasonic energy will be reflected back into the transducer from the air/tissue boundary. It is a useful exercise to calculate the amount of energy transmitted at an air/tissue interface. Using the values in table 10.1 and equation 9.5 ρ = ar = Z1 - Z2 .... (9.5) ai Z1 + Z2 we predict that the amplitude of the reflected wave will be 0.9997 times the amplitude of the incident wave for an air/tissue interface. Hence (0.9997)2 x 100 = 99.94 per cent of the incident energy is reflected! Clearly the amount of energy transmitted (0.06%) is negligible. Almost all of the wave energy is reflected back into the air.

THERAPEUTIC WAVES: ULTRASOUND 253 Only by having a coupling medium between the transducer and tissue can efficient It is also important that there transfer of energy be ensured. The coupling medium is spread on the surface of the be no tiny air bubbles within skin so that the ultrasound transducer contacts the skin via the coupling medium. No the coupling medium as air/tissue boundary is present. these too would produce Many different coupling media can be used. The desirable characteristics of the reflection and prevent efficient coupling medium are: transmission of the * It should be fluid, so as to completely fill the gap between skin and treatment ultrasound energy. Criterion five is least head and exclude air bubbles. important since a very thin * It should be viscous so that it stays on the skin rather than rapidly flowing and layer of couplant is used in practice so that energy spreading. absorption by virtue of the * It should not inhibit heat loss from the skin otherwise high temperatures may be penetration depth in the coupling medium is relatively produced in skin and subcutaneous tissue. insignificant. * It should have an impedance similar to that of steel and tissue, so as to minimize reflection. * It should absorb a negligible amount of the ultrasound energy: in other words, should have a high penetration depth. In practice the first two criteria listed above are the most important. The principal function of a coupling medium is to eliminate air gaps and provide contact between treatment head and tissue. Criterion three is also very important and water or water based gels are best in this regard. Criterion four is important but most liquids have similar values of acoustic impedance. Water meets all the above criteria with the exception of the second (viscosity). For this reason water is most often used either in a coupling cushion (a polythene or rubber bag filled with water) or in a bath - when the part to be treated can be immersed. Oils and liquid medicinal paraffin have appropriate viscosities and so can be used as THERAPEUTIC WAVES: ULTRASOUND 254 couplants, however they inhibit heat loss from the skin and produce greater superficial In this section we ignore the heating than water, water based gels or glycerol. effects of different coupling Glycerol is viscous and has similar acoustic properties to water. It makes a very good media and focus attention on coupling medium. what happens to the Thixotropic couplants are solids at room temperature which liquefy when ultrasound ultrasound energy which is is applied. They are ideally suited to treatment of a vertical surface as they will not run transmitted into tissue. down the skin. A number of thixotropic couplants are available. THERMAL EFFECTS OF ULTRASOUND At the intensity levels used for therapy the major effect of ultrasound waves on tissue is, as with all diathermic modalities, the production of heat. The amount of heat generated in a particular tissue depends on two factors: * the rate at which energy is absorbed by the tissue - which is determined by the penetration depth, δ. * the extent to which the waves are reflected back into the tissue on striking a tissue interface: determined by the difference in impedance between the two media. We considered in chapter 9 the rate of absorption of energy in different tissues. Values of the penetration depth are shown in table 9.1. The figures are not very accurate and vary by up to a factor of two between different tissue samples. They do, however, give a clear indication of the relative absorbing power of different tissues. The amount of reflection at a tissue interface is determined by the difference in impedance of the two materials. The acoustic impedance (Z) depends on the elasticity and density of a medium according to equation 10.1: .... (10.1)

THERAPEUTIC WAVES: ULTRASOUND 255 where ρ is the density and Y the modulus of elasticity (stiffness) of the medium. The velocity of sound in the medium, v, also depends on elasticity and density according to equation 10.2: .... (10.2) Combining these two equations we obtain a simple expression for the impedance in terms of velocity and density: Z = ρ.v .... (10.3) Table 10.1 lists the acoustic properties of air, water, various tissues and steel. As noted previously the acoustic impedance of air differs considerably from the remaining materials. The table also shows that there is little Material velocity density impedance difference in the acoustic impedance of muscle, (m.s-1) (kg.m-3) (kg.m.s-1) fatty tissue and water. For this reason we Air expect little reflection at a fat/muscle interface. Fatty Tissue 340 0.625 213 The reflection coefficient calculated using Muscle 1450 940 1.4 x 106 equation 9.5 is 0.10, thus the amount of energy Bone 1550 1100 1.7 x 106 reflected is 0.1 squared or 1%. Water 2800 1800 5.1 x 106 The impedance of bone is higher than that of Steel 1500 1000 1.5 x 106 muscle hence we expect significant reflection at 5850 8000 47.0 x 106 a muscle/bone interface. The reflection coefficient is 0.50 so we expect about 25% of the energy to be reflected. Heating Rate of Boneless Tissue Table 10.1 Acoustic properties of materials. Consider first an ultrasound beam travelling through fatty tissue and muscle with no bone present. Equation 9.5 tells us that reflection at the fat/muscle interface is THERAPEUTIC WAVES: ULTRASOUND 256 negligible and equation 9.12 indicates that refraction is minimal for incident angles up to about 50o. Thus we can consider the waves to be travelling in one direction in a straight line through the tissue. The wave intensity at a point is the energy per unit area per unit time; the area being taken perpendicular to the wave direction. Since energy varies with distance according to equation 9.2, the wave intensity I (in watts per square metre) is given by equation 10.4: I = Io e-x/δ .... (10.4) where x is the distance in the tissue and δ is the penetration depth. The rate of heating is equal to the rate of decrease of intensity with distance. It The lower the penetration depends on two factors, the wave intensity at a particular point and the rate of depth, the greater the rate of absorption of energy (specified, indirectly, by the penetration depth). heating. The rate of decrease of intensity with distance is obtained by differentiating equation 10.4 to give: .... (10.5) where Pv is the heat developed per unit volume per second. We can use equations 10.4 and 10.5 together with values for δ (from table 9.1) to calculate the wave intensity and heat development in different parts of a fatty tissue/muscle combination once we know the thickness of the fat and muscle layers. For example, suppose that we have a fat layer of uniform thickness (1 cm) on top of a thick muscle layer and that ultrasound of frequency 1 MHz is incident upon this tissue combination. The penetration depth in fatty tissue at this frequency is 7.2 cm (table 9.1) thus the wave intensity (equation 10.4) will be reduced by a factor of e-1/7.2 or 0.87 on traversing the fatty tissue - a decrease of only 13%. After travelling a distance of one centimetre in the muscle the intensity would be reduced by a factor of e-1/1.7 or

THERAPEUTIC WAVES: ULTRASOUND 257 0.56 so the intensity would be 56% of 87% or 49% of the original energy. In 2 cm of muscle the Figure 10.5 shows the overall reduction in wave intensity with distance in the tissue intensity would be reduced by and also the relative rate of heating of the tissue (equation 10.5). Calculated heating a factor of e-2/1.7 = e-1.2 = rates are scaled to a value of 100% at the muscle surface (because this is where 0.31 so the intensity would maximum heating occurs). be 31% of 87% or 27% of Even though we have made a number of simplifying assumptions (to be discussed the original value. shortly) the general implications of figure 10.5 are valid. It is clear that only modest heating is produced in the fatty tissue. Greatest heating is produced in the few centimetres of muscle tissue adjacent to the fat/muscle interface. Using our simplified model, even after penetrating 2 cm of muscle tissue the ultrasound is predicted to produce a higher rate of heating than at any point in the fatty tissue. This validates ultrasound as being classified as a diathermic modality. A greater rate of heating is produced at depth. [The depth efficiency of MHz frequency ultrasound is not, however, as great as implied by figure 10.5. Depth efficiency is best assessed by the rate of temperature increase, which is not the same as the rate of heating. A graph of temperature increase resulting from this heating pattern does not show as great a difference between fatty tissue and muscle. This is because temperature elevation in tissue depends on a number of factors other than rate of energy input. Temperature elevation will be considered later in this chapter]. Heating Rate of Tissues With Bone Figure 10.5 Wave intensity and relative rate of heating The same kind of calculation as made above can be performed for in fat and muscle tissue with ultrasound of the more complex tissue combination of fat/muscle/bone. In this case we can not ignore reflection. Equation 9.5 indicates that close frequency 1 MHz. to 25% of the wave energy incident upon the muscle/bone interface THERAPEUTIC WAVES: ULTRASOUND 258 is reflected. Figure 10.6 Wave intensity and relative rate of heating Figure 10.6 shows the relative rate of heating which is predicted for a in fat, muscle and bone with ultrasound of combination of 1 cm fatty tissue and 1 cm muscle overlying bone. The reflection has two effects: frequency 1 MHz. * a greater proportion of the total wave energy is absorbed in the fat and muscle tissue. * the reflected and incident waves will interfere and produce a standing wave pattern. The first of these effects is quite significant. Energy will be absorbed both as the wave travels through fat and muscle to the boundary with bone and as the reflected wave travels back through muscle then fatty tissue. Hence the total rate of heating of fat and muscle tissue at any depth is greater than without the bone (compare figures 10.6 and 10.5). The effect on fatty tissue is small. As might be expected, in muscle the effect is larger because the reflected wave, and thus the reflected wave energy, is larger. The second effect is of less practical importance. Certainly an interference pattern will be produced but consider the distance between nodes and antinodes (figure 9.11). The wavelength of the standing wave pattern is the same as that of the incident and reflected waves with nodes and also antinodes separated by one half of a wavelength. For ultrasound of frequency 1 MHz the wavelength is 1.5 mm so the antinodes will be separated by 0.75 mm. The antinodes represent points of maximum wave energy and hence maximum heat production. We have, then, that points of maximum heat production are only 0.75 mm apart. This is too close to be of practical significance, particularly if the treatment head is kept

THERAPEUTIC WAVES: ULTRASOUND 259 moving. We have already seen why the treatment head can not be kept stationary: In summary, in figure 10.6 we movement is necessary to smooth out the effects of variations in ultrasound intensity have included the effect of with depth shown in figure 10.4. This same movement will produce variations in reflection which occurs at the tissue thickness well in excess of 0.75 mm. The net result will be an averaging of any muscle/bone interface. This standing wave pattern as the treatment head is moved: so much so that no evidence results in a less rapid drop- for standing waves would be detected. In addition factors such as the pulsatile nature off in heat production with of blood flow through tissue and muscle contraction will result in variations in the depth in the fat and muscle thickness of tissue layers: the standing wave pattern will hence shift back and forth, tissue when compared with further smoothing the pattern of heat production. figure 10.5. The effect of The most significant feature of figure 10.6 is the high rate of heating of the bone standing waves has been surface. Most of the wave energy transmitted into the bone is absorbed in the first few ignored for the reasons millimetres. This is predicted from the value of penetration depth given in table 9.1. described. The result is substantial heating. As can be seen, the heating rate is predicted to be There have been reports of about three times greater at the bone surface than anywhere in the muscle tissue. deep tissue burns being Heat development is confined to the first few millimetres of bone but is quite inflicted by unqualified substantial. In practice, heat production at the bone surface is often the factor which practitioners who neglected limits the intensity which can be used in therapeutic application of 1 MHz ultrasound. to move the ultrasound head Too great an intensity or too prolonged a treatment can result in periosteal pain and during treatment. significant tissue damage (a periosteal burn). The risk of periosteal burns is reduced by movement of the ultrasound transducer (treatment head). Movement distributes the ultrasound energy over a larger area of the bone surface, thus reducing the average energy in a specific location. The pattern of heat production shown in figures 10.5 and 10.6 indicate the value of 1 MHz ultrasound for heating of deeply located tissue. Figure 10.6 also highlights the risk when the soft tissue layers are thin and underlying bone is exposed to the ultrasound beam. If a frequency of 3 MHz is used rather than 1 MHz, values of penetration depth are smaller (table 9.1). The ultrasound intensity decreases more rapidly so heat production is greater in the superficial tissues. A less pronounced 'deep heating' effect results but there is less energy remaining at depth to heat underlying bone. THERAPEUTIC WAVES: ULTRASOUND 260 Figure 10.7 shows the wave intensity and relative rate of heating calculated for ultrasound of frequency 3 MHz in a tissue combination with the same dimensions as assumed in figure 10.6. Note that with the assumptions made, the peak heating rate at the bone surface does not exceed that at the muscle surface. Comparison of figures 10.6 and 10.7 bears out the qualitative observation made earlier: if a maximum depth efficiency of heating is required then 1 MHz ultrasound is the modality of choice. For less deeply located structures, 3 MHz ultrasound may be preferred to avoid excessive heating of the bone. Figures 10.6 and 10.7 indicate the great usefulness of ultrasound for heating of joints, particularly those located under thick tissue layers. Heat developed at the bone surface will be transferred to heat the adjoining tissue. Experimental work in which the temperature elevation of the hip joint was measured directly confirms that ultrasonic therapy is very useful in this regard. Let us now briefly summarize the approximations made Figure 10.7 in calculating the results shown in figures 10.6 and 10.7: Wave intensity and relative rate of heating * We have assumed that the ultrasound beam is in fat, muscle and bone with ultrasound of uniform: that is, we have neglected the original beam frequency 3 MHz. shape. This is a reasonable approximation to make if the treatment head is moving and the transfer of heat between and within tissues is taken into account. * We have neglected reflection and refraction at the fat/muscle interface. A reasonable approximation as only about 1% of the energy is reflected and the refraction effect is very small (equation

THERAPEUTIC WAVES: ULTRASOUND 261 9.12 and table 10.1). Reflection at the bone surface was taken into account - The rate of temperature refraction in bone is unimportant as the penetration depth is so small. increase depends not just on * We have assumed the tissue layers to be homogeneous. A valid assumption the rate of energy absorption, when we are considering heating of the tissue as a whole. We return to this point but also on heat capacity of shortly in considering the mechanical effects of ultrasound. the tissue and the rate of heat * We have neglected heat losses to the bloodstream and heat transfer to adjacent transfer to adjacent tissue. tissues. These effects are considered next. Water has a much higher heat capacity than most other Temperature Distribution in Tissue substances. This is illustrated by the fact that a The results shown in Figures 10.5 to 10.7 correctly describe the rate of heating due to cup of lukewarm tea has a the absorption of wave energy. They do not, however, describe the resulting rate of higher heat content than a temperature increase. The rate of temperature increase depends on the heat capacity (large) red-hot nail. of the tissue. The heat capacity is measured by how much heat must be provided to increase the temperature by one degree (see chapter 7). A tissue with a high heat capacity will require more heat to increase the temperature by 1o than a tissue with a low heat capacity. Muscle tissue (principally because of its high water content) has a higher heat capacity than fatty tissue or bone. Consequently, for the same amount of heat energy, the temperature increase in muscle will be less than that of fatty tissue or bone. At the commencement of treatment, the relative rate of heating will be more or less as indicated in the figures. Graphs of rate of temperature increase would, however, show an overall lesser rate of temperature increase in muscle than fat or bone, because of the heat capacity effect. A second factor affecting the rate of temperature increase in tissue is heat loss to the surrounding tissue and, more importantly, the blood vessels. Muscle has a much higher vascularity and therefore volume rate of blood flow, than fatty tissue or bone. We would thus expect convective cooling of muscle (i.e. heat carried away by the bloodstream) to reduce the rate of temperature increase still further. A third factor is that any temperature increase in muscle would also be expected to trigger reflex dilation, whereby arterioles dilate to increase the blood flow in response THERAPEUTIC WAVES: ULTRASOUND 262 to an increase in temperature. This would further decrease the rate of temperature Equation 7.6 tells us that if increase. the specific heat capacity is The main points relating temperature increase to heating rate are as follows: low the rate of temperature * the low specific heat capacity of fatty tissue and poor thermal conductivity will elevation will be greater for a fixed rate of heat energy result in a greater temperature rise than indicated by the graphs. In addition the supplied to the tissue. thermal conductivity of fatty tissue is low and its vascularity is not as good as muscle; consequently heat can not be removed as rapidly. This adds to the temperature elevation of fatty tissue as compared to muscle. * efficient heat transfer through muscle tissue and to blood vessels will result in more uniform heating of muscle and less temperature rise than might otherwise be expected. At the same time heat transfer to the adjacent fatty tissue will reduce the temperature elevation of muscle near the fat/muscle interface. * bone is a relatively good conductor of heat. The heat will be rapidly distributed in the bone and also transferred to the periosteum. The higher thermal conductivity partially compensates for the rapid absorption of energy near the bone surface and reduces the selective heating. It is still possible, however, to produce a maximum temperature elevation in the periosteum when the intervening tissue layers are not very thick. This gives rise to the periosteal pain mentioned previously. Despite these limitations, some of which also apply to other diathermic modalities, ultrasound is an effective deep-heating modality. The principal factor limiting the temperature elevation which can be produced at depth is heating of the periosteum. Mechanical Effects The predominant physiological effects of ultrasound therapy are due to a rise in temperature of the treated tissues. Certain effects are, however, produced which are a direct result of the mechanical vibration of tissue.

THERAPEUTIC WAVES: ULTRASOUND 263 As an ultrasound wave propagates the particles in the medium experience rapidly alternating compressions and rarefactions. The pressure varies with distance as shown in figure 10.8. Regions of high pressure (dark shading) are separated by one wavelength - about 1.5 mm for 1 MHz ultrasound. For a wave intensity of 2 to 4 watts per square centimetre (near the upper limit for therapeutic application) the pressure amplitude of the waves is about 2 to 4 atmospheres (20 to 40 newtons per square centimetre). This means that the pressure extremes - a Figure 10.8 difference of 4 to 8 atmospheres - are separated in tissue by only Pressure variation in an ultrasound wave. one half of a wavelength (0.75 mm). Any tissue component or substructure with dimensions of about one wavelength will be subject to substantial mechanical stresses, alternating at a frequency of 1 MHz. Smaller structures such as tissue cells will experience lesser, though still substantial stresses and will be vibrated back and forth by the pressure changes. Figures 10.4 and 10.8 are two important views of the ultrasound intensity variation in a Therapeutic ultrasound beam. Figure 10.4 is at a gross, large-scale level and shows the variation in intensity produces large stresses in within a beam. Regions of high intensity (shown by the deepest blue colouration) are biological tissues, acting over separated by distances measured in centimetres and the positions are stationary in distances of a fraction of a the beam. Viewed on a smaller scale, figure 10.8 shows a sound wave within the millimetre. The stresses are beam. At any point within the beam shown in figure 10.4, the pressure varies from greatest in the regions shown maximum to minimum over a distance of half of one wavelength and regions of high in figure 10.4. pressure move through the medium (tissue, air or water) at a high velocity. The sound velocity, v, is about 1500 m.s-2 in water and soft tissue. Listed below are some examples of the effects of ultrasound where mechanical stresses are thought to play a significant role. It should be emphasized that in all instances heating contributes to the observed results: in most cases it is difficult to ascertain the relative contribution of thermal and mechanical effects. * Experiments have shown that the extensibility of connective tissue can be increased by exposure to ultrasound. Since connective tissue fibres are key THERAPEUTIC WAVES: ULTRASOUND 264 constituents of tendons, scar tissue, joint capsules and muscle the results are of For example, the diffusion of major significance for therapy. Part of the effect is attributed to the increase in sodium ions across a temperature on exposure to ultrasound: the separation of fibres and loosening of membrane will deplete the the structure as a result of the mechanical stresses would also be expected to concentration close to one contribute. side of the membrane and * The rate of diffusion of ions across cell membranes is found to increase on increase it on the other side: exposure to ultrasound. An effect is observed over and above that due to heating this will result in a decrease alone. A possible explanation is that a stirring effect is produced which in the rate of diffusion. If the increases the concentration gradient of ions and other materials. In any diffusion fluids are agitated, mixing will process there will be a narrow region on either side of the membrane where the proceed more rapidly and the ion concentrations are not the same as in the bulk of the fluid. Another possible concentration gradient will be contributing factor involves the fluidity of the cell membrane. If we picture the cell more efficiently maintained. membrane as a thixotropic barrier we predict that the fluidity, and hence the permeability, of the cell membrane will increase in response to the mechanical agitation of the ultrasound waves. * Ultrasound is useful in relieving pain and muscle spasm. While any form of heating is useful in this regard, it appears that ultrasound can have an effect other than via direct heating. The mechanism of this action has not been conclusively established but it is interesting to note that an optimum effect appears to be produced using pulsed ultrasound beams. The pulse frequency normally available is 100 Hz - the same frequency used to produce analgesia by electrical stimulation. PULSED ULTRASOUND Most ultrasound apparatus makes provision for either pulsed or continuous output. In pulsed mode the ultrasound is produced in bursts, normally with a frequency of 50 or 100 Hz. If the duty cycle ('on' time to 'on + off' time) is 1:5 then the apparatus is 'on' for only one fifth of the time: consequently the rate of transfer of energy is one fifth of that obtained using the continuous mode at the same intensity. If the dose required (continuous mode) necessitates treatment for 20 minutes then to obtain a similar thermal effect using pulsed ultrasound we would have to extend the treatment time or

THERAPEUTIC WAVES: ULTRASOUND 265 increase the intensity to compensate. An increase in the treatment time alone will not Temperature elevation compensate adequately. Suppose the duty cycle is 1:5 then a 20 minute (continuous) depends not just on the dose treatment could be increased to 100 minutes (pulsed). Although the total energy (the but also on the dose rate. dose) supplied to the tissue is the same in both cases, spreading the treatment over The referances here are to 100 minutes will considerably reduce the temperature elevation produced. two old but reliable papers: Increasing the intensity by a factor of five will result in the same rate of energy transfer Dyson M et al. Clinical to tissue (dose rate) but the much higher peak intensities could result in tissue Science 35, 273-285 (1968) damage through gaseous cavitation - the rapid formation and collapse of tiny gas and Dyson M & Suckling J. bubbles in the tissue fluid. The cavitation effect will be described more fully in chapter Physiotherapy 64, 105-108 12 along with other potentially harmful effects. (1978). Proponents of the use of pulsed ultrasound argue that heat production is rarely the sole objective of therapy and that in some applications it may even be undesirable. By use of pulsed ultrasound, at low to moderate intensities, mechanical effects are produced while heat production is kept to a minimum. Of course the same (low) rate of heat production could be achieved using the continuous mode at one fifth of the peak intensity. We would, however, expect some differences in the mechanical effects produced: continuous mild mechanical agitation does not necessarily produce the same effect as brief vigorous mechanical agitation. The idea is that mechanical effects do not depend linearly on intensity: that there is a threshold intensity level below which the mechanical effects are negligible. Pulsed ultrasound would ensure that intensities above threshold are achieved while keeping heat production to a minimum. One study which indicates the possibility of therapeutically significant mechanical effects was carried out by Dyson et al. (1968). These authors examined the rate of tissue repair using continuous output treatment compared with pulsed mode treatment using different duty cycles. The frequency used was 3 MHz and the output in pulsed mode was adjusted to keep the average power the same in each experiment. Tissue growth rate was increased using a duty cycle of 1:5 but retarded when a duty cycle of 1:80 was used. It seems that modest duty cycles may promote repair activity but that (for the same average power) too small a duty cycle involves peak power levels which are damaging to tissue. The results of this and other relevant studies THERAPEUTIC WAVES: ULTRASOUND 266 are summarized by Dyson and Suckling (1978). There is no clinical evidence To date, insufficient research has been done to quantitatively determine the relative that pulsed has greater contributions of heating and mechanical effects in different therapeutic applications of therapeutic benefit than ultrasound. When such information is available the therapist will be in a better continuous ultrasound. position to select between pulsed and continuous mode and, when pulsed mode is Biophysical evidence shows indicated, to chose the appropriate duty cycle. that there are different effects, At this stage it is possible, at least, to say that if ultrasound is chosen (rather than but whether this translates another diathermic modality) on the basis of the predicted pattern of heat production, into clinical practice is then a continuous output is indicated. Where mechanical effects are a desirable part unknown at present. of the therapy, continuous or pulsed mode might be selected. EXERCISES 1 Consider the schematic diagram of an ultrasound generator shown in figure 10.1. (a) Briefly explain the function of each subsection. (b) What is the function of the piezoelectric crystal shown in the probe? Explain what happens when an AC signal is applied to opposite faces of the crystal. 2 (a) Explain why the ultrasound beam produced by apparatus used in therapy is non-uniform. (b) What is meant by the terms 'near field' and 'far field' as applied to therapeutic ultrasound? 3 Figure 10.3 shows the distribution of ultrasound energy with distance along the central axis of an ultrasound transducer. In this case the interference pattern is generated by a 2.8 cm diameter, 1 MHz frequency, ultrasound source in water.

THERAPEUTIC WAVES: ULTRASOUND 267 Describe how the interference pattern would change if: (a) the transducer diameter was made smaller (b) the operating frequency was increased (to 3 MHz, say) (c) fatty tissue rather than water was used. What is the effect of the different wave velocity in fatty tissue? 4 Use equation 9.5 and the impedance values given in table 10.1 to calculate the reflection coefficient of a metal/air boundary and an air/fatty tissue boundary. (a) What percentage of the incident ultrasound energy would be transmitted at these boundaries? (b) What are the implications of your results for ultrasound therapy? 5 (a) What is a coupling medium and why is one needed in ultrasound therapy? (b) List the characteristics of a 'good' coupling medium and describe the relative importance of each. (c) State the relative advantages and disadvantages of water, oils, glycerol and thixotropic fluids as coupling media. 6 Ultrasound of frequency 1 MHz travels through water and strikes fatty tissue. Use the data in table 10.1 to calculate the reflection coefficient of the water/fat boundary. What percentage of the incident wave energy is transmitted? 7 Use equation 9.5 and the figures in table 10.1 to calculate the reflection coefficient of ultrasound at the following boundaries: (a) fatty tissue/muscle (b) muscle/bone What are the practical implications of these figures? THERAPEUTIC WAVES: ULTRASOUND 268 8 (a) Refer to figure 10.5 and explain how the graph of wave intensity vs distance can be used to obtain the graph of relative rate of heating versus distance. (b) What is the relationship between penetration depth (table 9.1) and relative rate of heating (figure 10.5)? 9 For ultrasound of frequency 2 MHz (table 9.1) calculate the fraction of energy remaining after travelling through: (a) 2 cm fat (b) 2 cm muscle (c) 2 cm bone 10 Ultrasound of frequencies 1, 2 and 3 MHz travels through 3 cm of fatty tissue and strikes a fat/muscle interface. Assuming no reflection at the interface, what fraction of the original energy reaches the muscle? Use the data in table 9.1 and equation 9.2. 11 Consider 3 MHz ultrasound travelling through a tissue combination of 2 cm fatty tissue over muscle. (a) Use equation 10.4 and the data in table 9.1 to calculate the wave energy at depths of 1, 2, 3 .... 10 cm in the tissue combination. You may assume no reflection at the fat/muscle interface. (b) Use equation 10.5 to calculate the relative rate of heating from the intensity figures obtained in part (a). (c) Plot a graph of wave intensity versus depth and relative rate of heating versus depth for comparison with figure 10.5. (d) What similarities and differences are evident in comparing your graphs with figure 10.5? 12 Compare figures 10.5 and 10.6 and explain why the wave intensity appears to diminish less rapidly with distance in fat and muscle in figure 10.6.

THERAPEUTIC WAVES: ULTRASOUND 269 13 Consider figure 10.6, which shows the wave intensity at different depths in a muscle/fat/bone tissue combination. The graph is obtained assuming that 25% of the incident energy is reflected at the muscle/bone interface. (a) How would the graph of intensity versus depth differ if reflection at the muscle/bone interface was negligible? (b) Reflection of waves at an interface results in the production of a standing wave pattern. Why is this not shown in figure 10.6? (c) Under what conditions will the production of standing waves be of practical significance for 1 MHz ultrasound in complex tissue combinations? 14 Compare figures 10.6 and 10.7 and summarize the advantages and disadvantages of 1 MHz ultrasound compared with 3 MHz ultrasound for different thicknesses of tissue over bone. 15 The results shown in figures 10.5 to 10.7 take no account of heat losses to the air, the bloodstream and between adjacent tissues. Redraw figure 10.7 to show, qualitatively, the relative rate of heating when heat loss and heat transfer are taken into account. 16 Figures 10.5 to 10.7 show the relative rate of heating of different tissue exposed to ultrasound. (a) What additional factor must be taken into account to predict the initial rate of temperature increase in each tissue? (b) Draw a diagram, based upon figure 10.7 to show (qualitatively) the initial rate of temperature increase in each tissue. You may assume that the specific heat capacity of muscle is twice that of fatty tissue and bone. 17 Explain why graphs such as those shown in figures 10.5 to 10.7 can be used to accurately predict the initial rate of temperature increase in tissue but not the final temperature increase in therapy. What additional factors must be taken into account to predict the final temperature elevation? THERAPEUTIC WAVES: ULTRASOUND 270 18 (a) Briefly explain why ultrasound would be expected to have mechanical effects on tissue in addition to thermal effects. (b) List and briefly describe the effects of ultrasound where mechanical vibration is thought to play a direct role. 19 Compare the following two methods of treating a patient with ultrasound: (i) Exposure at a level of 2 W.cm-2 for 10 minutes in continuous mode (ii) Exposure at a level of 2 W.cm-2 peak for 20 minutes using a duty cycle of 1:2 Would you expect: (a) the same mechanical effect? (b) the same total heat production? (c) the same temperature elevation? How do the dose and dose rate compare in each case? 20 Compare the following two methods of treating a patient with ultrasound: (i) exposure at a level of 1 W.cm-2 for 15 minutes in continuous mode. (ii) exposure at a level of 2 W.cm-2 peak for 15 minutes in pulsed mode (duty cycle 1:2). Would you expect: (a) the same mechanical effects? (b) the same total heat production? (c) the same temperature elevation? How do the dose and dose rate compare in each case?

ELECTROMAGNETIC WAVES FOR THERAPY 271 11 Electromagnetic Waves for Therapy Ultraviolet and infrared radiation have low We saw, in chapter 9, that Maxwell's equations predict that whenever charges are penetration depth but are accelerated electromagnetic waves are produced. In this chapter we consider the useful for therapy in electromagnetic waves used in therapy: how they are produced and why they are applications other than useful to physiotherapists. diathermy. Three main kinds of electromagnetic wave are used in therapy: microwaves, infrared When discussing ultraviolet and ultraviolet radiation. Of these, only microwaves are able to penetrate tissue radiation it is a common significantly and so be classed as diathermic. convention to talk in terms of The different therapeutic applications of these radiations arise from their differing wavelength rather than effect on tissue. These effects, in turn, are determined by the wavelength (or frequency. frequency) of the waves. Before considering the effect on tissue we examine the way in which each kind of electromagnetic wave is produced: this gives a first insight into their physical and physiological effects. PRODUCTION OF WAVES AROUND OPTICAL FREQUENCIES In what follows we consider the way in which infrared and ultraviolet radiation are produced for therapy. Both kinds of radiation are normally produced by similar apparatus: more fundamental are the similarities in the molecular processes involved. Production Of Ultraviolet Radiation Electromagnetic waves with frequencies from 0.75 x 1015 Hz to 3.00 x 1015 Hz are classified as ultraviolet radiation (see figure 9.5). Their frequencies are above those of visible light and below those of X-rays. Ultraviolet radiation has wavelengths between 400 nm and 100 nm. The wavelengths used in therapy are restricted to the high end of this range: 190 nm to 400 nm, as wavelengths less than 190 nm are strongly absorbed in air. By international convention the ultraviolet spectrum is divided into three regions. ELECTROMAGNETIC WAVES FOR THERAPY 272 These are: When deciding whether to * UV-A: wavelengths between 400 nm and 315 nm use UV for therapy, the * UV-B: wavelengths between 315 nm and 280 nm clinician must, as with all * UV-C: wavelengths between 280 nm and 100 nm. interventions, weigh the benefits against the risks. UV-C radiation is used to sterilize things when you don't want to boil them. This is Indeed, we have evidence for because UV-C, at sufficiently high intensities, destroys bacteria. It does this by gaseous conduction at damaging the bacterial DNA. UV-C exposure will also damage human cells in the normal temperature and same way and can produce malignancies (cancer). UV-C and, in fact, UV-B and -A pressure with every lightning have an extremely low penetration depth, so most of the absorption of UV is by the flash in a thunderstorm. skin. The low penetration depth of UV is the reason that UV exposure (in particular, exposure to UV-C) is associated with skin cancer. The usual means of producing ultraviolet light is by the passage of an electric current through an ionized gas or vapour. Gases at normal temperature and pressure are very poor conductors. They can, however, be made to conduct at high temperature or low pressure in the presence of a sufficiently strong electric field. Ultraviolet radiation for therapeutic application is usually produced by current flow through mercury vapour. Mercury under reduced pressure is contained in a sealed envelope of quartz or special glass with an electrode inserted in each end. The device is similar to the strip-lights (fluorescent lights) commonly found in the kitchen at home and the office or tutorial room. The difference is that UV lights operate at lower pressures than household or business lights. This means that more energy is required to initiate conduction and charges are accelerated over greater distances so that when they collide, the energy release is larger and, as a result of the higher energies, UV rather than visible light is produced. The arrangement used with a mercury vapour lamp is shown in figure 11.1. Figure 11.1 Schematic diagram of a mercury vapour lamp.

ELECTROMAGNETIC WAVES FOR THERAPY 273 The reduced pressure in the lamp ensures that mercury vapour is present, but in Ionization occurs continuous- order for current to flow the vapour must be ionized. This means that electrons must ly in all materials: it is brought be separated from the parent atoms. Cosmic rays and gamma-rays are high about by various radiations frequency and high energy and can 'kick' electrons from their orbitals, so producing always present at low levels; positive ions and free electrons. Under normal circumstances the electron returns to cosmic rays and natural its parent atom, because of the attraction between positive and negative charges. radioactivity are examples. However, in a sufficiently strong electric field (as in the lamp) the excited electron can It is generally necessary to accelerate and collide with other atoms. If the electric field is strong enough, the help initiate the avalanche, or electron can gain enough energy to cause further ionization and produce an discharge, by pulsing the 'avalanche' effect: one electron is accelerated and collides, producing more metal lamp with a high voltage. ions and free electrons which in turn accelerate, collide and cause further ionization. Once the discharge is started The glow of a mercury-vapour lamp is a consequence of the avalanche of ionization. the current must be regulated After participating briefly in the avalanche, electrons reattach to ions, dropping into a to limit and control the output particular orbital and releasing energy in the form of electromagnetic waves. of light from the lamp. In the discharge process ions are being continually formed and are continually recombining with electrons. As ions and electrons recombine energy is released in the form of electromagnetic radiation which has frequencies characteristic of the parent atom. The range of frequencies put out by the lamp is modified by the pressure within the lamp and further modified in passing through the glass envelope which contains the vapour. Figure 11.2 compares the range of wavelengths over which various lamps put out appreciable energy. The spectral range for an ordinary incandescent (tungsten filament) lamp, a fluorescent tube (strip-light) and for sunlight is also included for comparison. Also indicated in the figure are approximate proportions of ultraviolet, visible and infrared radiation expressed as a percentage of the total energy output. The proportions vary with the pressure of mercury vapour in the lamp or tube and with the thickness and composition of the lamp envelope. Percentages are not shown for fluorescent tubes ('strip lights') or incandescent lamps (normal globes) as the ELECTROMAGNETIC WAVES FOR THERAPY 274 proportions vary depending on the construction of the device, the power rating and whether filters are used to block-out certain wavelengths. Figure 11.2 Spectral range of various lamps and sunlight Mercury vapour lamps operating at lower pressure put out more radiation in the high frequency region of the spectrum (towards the far ultraviolet region). Even so, all ultraviolet lamps produce a considerable amount of energy in the infrared and visible region. Both the visible and far ultraviolet radiation can be removed by the use of filters. If water cooling of the lamp is used - as with the Kromayer lamp - the water serves the dual role of keeping the lamp cool and absorbing the infrared radiation.

ELECTROMAGNETIC WAVES FOR THERAPY 275 Within the ultraviolet region of the spectrum there are significant differences in the Quartz or special composition output of mercury vapour lamps and tubes: glass must be used for the * Low pressure mercury vapour lamps, otherwise known as cold quartz lamps envelope of all ultraviolet light sources as normal glass when the envelope material is quartz, emit most of their ultraviolet radiation in the absorbs strongly over almost UV-C region, at a wavelength of 253.7 nm. The operating temperature of the the whole of the ultraviolet lamp envelope rarely exceeds about 60oC. spectrum. * High pressure mercury vapour lamps, known as hot quartz lamps when the The presence of a vapour, envelope material is quartz, put out a proportion of their ultraviolet energy at a even at atmospheric wavelength of 366.0 nm (in the UV-A region). There is also significant output at pressure, enables a specific wavelengths in the UV-B and UV-C regions. The amount of energy in discharge to be sustained each region depends on the construction of the lamp. The normal operating with a gap up to around 5 cm temperature of these lamps is several hundred degrees Celsius: if they are to be when about 80 volts potential used close to, or in contact with the patient they must be cooled by a water jacket difference is applied. (Kromayer lamps) or an air blower. * Fluorescent ultraviolet tubes are usually low pressure mercury lamps in the form of a long tube. The tube is coated on the inside with fluorescent substances (phosphors). The purpose of the phosphor is to absorb the original ultraviolet radiation and re-emit it at longer wavelengths. Different phosphors have different wavelengths for re-emission of radiation. The commonly used ultraviolet tubes put out most of their energy in the UV-A region. Special tubes are available which produce a maximum output in the UV-B region. A negligible amount of UV-C radiation is emitted from any of these light sources. In the past carbon arcs were used extensively for the production of ultraviolet radiation. Two carbon rods are brought into contact with each other and a current is passed through them. With a small point of contact the high current density heats and vapourises the carbon. The rods are then separated and the presence of carbon vapour enables a current to flow in the form of an arc discharge between the ends of the rod. The spectrum produced by carbon arcs has a range close to that of sunlight (figure 11.2): the proportions of ultraviolet, visible and infrared radiation are also similar. ELECTROMAGNETIC WAVES FOR THERAPY 276 Carbon arcs are rarely used today in physiotherapy departments: they have been Any object will be emitting largely superseded by mercury vapour lamps which are cleaner and easier to operate. and absorbing infrared radiation on an ongoing Production Of Infrared Radiation basis. Whether emission outweighs absorption Infrared radiation - sometimes referred to as radiant heat - is produced (and depends on the temperature absorbed) by all materials at temperatures above absolute zero. of the object relative to its Absorption of infrared radiation results in changes in molecular and atomic motion of surroundings. a material; the continuous agitation and changes in the motion of molecules, and within molecules also results in the emission of infrared radiation. For example, chemical bonds in molecules can absorb energy and 'stretch', changing the bond length and thus the energy of the bonding electrons. When the bond reverts to its original size, infrared radiation is produced at a frequency characteristic of the bond. Any molecule may, as a result of absorption of radiation or collision, change its state of rotation or vibration, or both simultaneously. On changing to a rotation or vibration state of lower energy, infrared radiation is produced. A particular kind of molecule has very many possible states of rotation and vibration and therefore many options for going from one state to some other. At a given temperature a body will emit a continuous spectrum of radiation - the maximum intensity occurring at a particular frequency but with significant intensities extending over a wide range of frequencies. The frequency of maximum production of radiation is directly proportional to the absolute temperature of the source. Since wavelength and frequency are inversely related (by equation 9.1, v = f.λ), it follows that the wavelength of maximum production of radiation is inversely proportional to the absolute temperature of the source. (This is called Wien's Law). As the source of radiation becomes progressively warmer, the wavelength of maximum emission becomes progressively shorter: thus an iron bar turns from black to 'red-hot' to 'white hot' to 'blue hot' as its temperature increases. In the black to red- hot temperature range both near infrared (770 to about 4000 nm) and far infrared

ELECTROMAGNETIC WAVES FOR THERAPY 277 (4000 to 15 000 nm) radiation is produced in appreciable amounts. For an ordinary household A suitable device for producing such radiation consists of a coil of wire through which light bulb the tungsten a current is passed. If the coil is wound on an insulator such as a ceramic rod, both filament is at about 3000 K the wire and the ceramic will emit radiation. The ceramic, being at a lower and the wavelength of temperature will produce more far infrared radiation. The common household electric maximum emission is about heater is usually of this kind. A way of producing most radiation in the far infrared 960 nm - that is, in the near region of the spectrum is to encase the heating element inside a ceramic rod or infrared. For skin at about mount it behind a plate so that the major source of radiation is the rod or plate. If a 300 K it would be 9600 nm, in reflector is used, the reflector will absorb some radiation and re-emit it at higher the far infrared. wavelengths thus adding to the far infrared component. These devices are often used for therapy. Use is also made of incandescent infrared lamps which produce a significantly greater proportion of near infrared radiation. Incandescent infrared lamps (similar to household lamps - consisting of a tungsten filament mounted in a glass envelope) have maximum emission at a wavelength around 1000 nm: some visible and ultraviolet light is produced but the ultraviolet is absorbed by the glass envelope and not transmitted. Use may be made of specially shaped lamps with internal reflectors. The reflectors may be shaped to give a floodlight beam - suitable for treating large areas - or a spotlight beam for treatment of localized areas. Some lamps have a clear glass lens while others have a red lens: there is little difference in the therapeutic effects of each. EFFECTS OF ULTRAVIOLET AND INFRARED RADIATION ON TISSUE Infrared and ultraviolet radiation share the common feature that their effects are produced in the surface layers of the skin. This was mentioned briefly in chapter 9(link to p227). Figure 11.3 summarizes the penetrating properties of these radiations. The penetration depth of waves of these frequencies clearly distinguishes them from waves used for diathermy. Considering first the infrared radiation (wavelengths of 770 nm and above) the figure indicates that shorter infrared waves (770 to 1200 nm) penetrate to the deeper parts of the dermis while the longer wavelengths are absorbed in the superficial epidermis. ELECTROMAGNETIC WAVES FOR THERAPY 278 From a penetration depth of a few millimetres at 1200 nm there is a decrease to about 0.1 mm at 3000 nm. Wavelengths above 3000 nm are absorbed by moisture on the surface of the skin. The trend does not continue indefinitely and we find that in the far infrared region from 10 000 to 40000 nm, the penetration depth increases to several centimetres. In effect, the tissues become much more transparent. Figure 11.3 Penetration of radiation into skin in the infrared to ultraviolet region of the electromagnetic spectrum Over the whole of the near infrared spectrum and up to about 20000 nm in the far infrared, reflection is minimal. Close to 95 per cent of energy incident perpendicular to the skin is absorbed - only about 5 per cent is reflected. To a reasonable approximation then, we can consider infrared radiation to be wholly absorbed by tissue. The region of the ultraviolet spectrum of interest in therapy extends from about 180 nm to 390 nm. From figure 11.3 we can see that most of this radiation is absorbed in the epidermis. In the region from 220 nm to 300 nm about 5 to 8 per cent of incident radiation is reflected. The reflectance increases to about 20 per cent at 390 nm. Within the range there are regions of very low reflectance corresponding to specific absorption by particular molecules in the skin - for example,

ELECTROMAGNETIC WAVES FOR THERAPY 279 nucleic acids absorb strongly at frequencies between 250 and 260 nm and at 280 nm. The physiological effects of infrared radiation differ from Heating by Infrared Radiation those of other forms of heating (e.g. shortwave From the foregoing discussion it is clear that the major effect of infrared radiation is diathermy) only in the thermal: to increase the temperature of cutaneous tissue. The penetration depth is location of heat production. very small but some heat will be transferred to the subcutaneous tissues via the The effects of infrared capillaries. radiation are not damaging The main effects of treatment are: unless the temperature * An increase in metabolic rate in the superficial tissues. This is the direct effect of elevation is too high. temperature on the rate of chemical reactions generally. As a result there will be an increased demand for oxygen and an increased output of waste products. * Dilatation of capillaries and arterioles due directly to the heating and also as a reflex reaction to the presence of increased concentrations of metabolites. The flow of blood to the superficial tissues is thus increased producing a reddening of the skin (erythema) and an increased supply of oxygen and nutrients. The erythema produced by infrared therapy, unlike that resulting from ultraviolet treatment, appears quite rapidly and begins to fade soon after treatment ceases. * Sensory sedation. Mild heating has a 'sedatory' effect on sensory nerves and is thus useful for the relief of pain. * Muscle spasm relief. This results from both the effect of heat on nerve fibres and the direct effect of heat which is transferred to muscle from the superficial tissues. Effects of Ultraviolet Radiation The effects of ultraviolet radiation are mainly non-thermal and due to cellular damage and protective responses. While damage might seem an undesirable consequence, there are therapeutic benefits of treatment. Five principal effects of therapeutic significance are found to result from treatment with ultraviolet radiation: ELECTROMAGNETIC WAVES FOR THERAPY 280 * An increased blood supply to the skin results from dilation of the capillaries and arterioles. Dilation does not result from heating of the tissue but as a reflex response to destruction of cells. Cells are destroyed as a result of chemical changes caused by the absorption of radiation, and reddening of the skin (erythema) results. The effects are similar to the changes observed in inflammation. Two groups of waves produce this reaction, one with wavelengths in the UV-C region around 250 nm and one with wavelength close to 300 nm (UV-B). * Production of vitamin D. Ultraviolet radiation in the range 250 to 300 nm initiates Nowadays, vitamin tablets a sequence of chemical reactions by which vitamins of the D group are provide a means of achieving synthesized. The effect has been used in the past for the treatment of rickets and results more quickly and tetany, but is not used any longer. economically when treating * Pigmentation. The amino-acid tyrosine is converted, via a sequence of reactions, vitamin D deficiency. to the pigment melanin. The accumulation of melanin in the epidermis is triggered by the same wavelengths of ultraviolet radiation responsible for erythema production - in addition UV-A wavelengths around 340 nm in low doses can produce tanning without erythema. * Sterilization. Shorter wavelength ultraviolet radiation (UV-C, around 250 nm) is In laboratories and effective in destroying bacteria. In therapy this effect finds application in the pharmaceutical preparation treatment of indolent ulcers: ultraviolet treatment is found to promote and areas, contamination by accelerate the healing process. It is not clear to what extent the sterilization bacteria must be avoided, so contributes as compared to erythema production. The increased blood supply lamps producing UV-C are evidenced by erythema will increase the number of white blood cells and used to irradiate the areas. antibodies in the area, hence reinforcing the body's defence mechanism. * Desquamation occurs some time after exposure to ultraviolet rays - it is a casting-off of dead cells from the surface of the body. The amount of peeling varies with the strength of the dose: it ranges from virtually imperceptible through powdery peeling to free peeling of epidermal layers. This can be of value in the treatment of skin diseases such as acne and psoriasis.

ELECTROMAGNETIC WAVES FOR THERAPY 281 The degree of erythema production is used to characterize the dose in ultra-violet The counter-irritation effect of therapy using UV-B fluorescent tubes or mercury vapour lamps. The reaction is a fourth degree erythema has graded into four levels: been used in the past as a * A first-degree erythema is a slight reddening of the skin which takes from six to quick and effective method of relieving pain from joints and eight hours to develop. The erythema has faded in about twenty four hours other deep structures in leaving the skin apparently unchanged. A minimum erythema dose (MED) is also degenerative arthritis and a slight reddening which takes from six to eight hours to develop but in this case rheumatic conditions. the erythema is still just visible at twenty four hours. * A second-degree erythema is a more marked reddening of the skin (resembling mild sunburn). There is a slight soreness. The reaction fades in about two days and is followed by pigmentation. After one or two weeks desquamation (peeling, usually powdery) occurs. * A third-degree erythema resembles severe sunburn. The skin may begin to show the effects as soon as two hours after treatment. The reaction is severe and the skin becomes hot, sore and oedematous. Effects subside gradually over several days and the skin often peels off in sheets or flakes. * A fourth-degree erythema is similar to a third-degree reaction but exudation and oedema are so marked that blisters form. Production of a third or fourth degree erythema in a small localized area results in a counter-irritation effect. Dose characterization in this way is appropriate for sources which produce an appreciable proportion of UV-B radiation. When using UV-A fluorescent tubes, dosage can not be assessed in this way as erythema production is minimal except at extremely high dose levels. In practice this is not a problem as the principal use of UV-A is in conjunction with a photosensitizing drug, 8-methoxy-psoralen, for the treatment of psoriasis. For psoralen - UV-A, therapy a special procedure is used for dose characterization. The procedure is described in chapter 12. ELECTROMAGNETIC WAVES FOR THERAPY 282 PRODUCTION OF MICROWAVES Electromagnetic waves travel more slowly in biological Having considered the low penetration electromagnetic waves - infrared, visible and tissues than air. The higher ultraviolet - we now turn to lower frequency waves used in therapy; microwaves. the dielectric constant and Microwaves occupy the region of the electromagnetic spectrum between radio waves conductivity, the lower the and infrared radiation: their wavelengths are in the range from about a centimetre to a wave velocity. meter - corresponding to frequencies in the range 300 MHz to 30 000 MHz. Three main As we will see, a frequency of frequencies are used for physiotherapy, 2450 MHz (wavelength 12 cm), 915 MHz 2450 MHz is not the best (wavelength 33 cm) and 433.9 MHz (wavelength 69 cm). Note that the wavelengths choice for therapeutic quoted are in air. In biological tissues the wavelength is significantly lower because applications and for some the wave velocity is lower. years the use of lower Radio waves can be produced by first generating a very high frequency AC signal in an frequencies has been ordinary electronic circuit and then applying this signal to a suitable antenna. The high advocated. frequency alternating current in the antenna results in radio frequency waves being produced and radiated. The limit to the frequencies that can be produced by standard electronic circuits is determined by the time it takes for an electron to travel through a transistor. If the transit-time, the time taken, becomes comparable to the time of oscillation or period of the wave we wish to produce, then the transistor can no longer function at this frequency. Microwave frequencies are extremely high, by electronic standards, and are at the limit of those which can be produced by transistors. Although vacuum tubes (valves) are an older design and are generally more inefficient than transistors, two vacuum tube devices which can operate at microwave frequencies were developed many years ago: these are the magnetron and the klystron. The magnetron valve, first described by Hull in 1921, was developed for radar use during the second world war. It is more useful for high power applications than the klystron. After the war, apparatus operating at a frequency of 2450 MHz (the standard radar frequency) was made available to physiotherapists. Microwave apparatus (figure 11.4) consists of a device (a magnetron or klystron), powered by an electronic circuit. The high frequency alternating current which is produced is fed to an antenna. The current flowing in the antenna results in the

ELECTROMAGNETIC WAVES FOR THERAPY 283 production of electromagnetic waves (chapter 9) which are beamed by the reflector. Figure 11.4 Schematic diagram: microwave apparatus The frequency of the microwaves is equal to the frequency of the AC produced by the magnetron. This is determined by the physical construction of the magnetron and is fixed during manufacture. A number of differently shaped antennas and reflectors may be used for directing the beam. Each gives a different beam shape though none gives a perfectly uniform beam. To obtain a collimated uniform beam (like a searchlight) would require a parabolic reflector with a point source of radiation as shown in figure 11.5(a). If a point source of radiation is placed at the focus of the parabola the beam emerges with a uniform cylindrical shape as shown. In the case of microwaves used by physiotherapists, the most common frequency is 2450 MHz and the wavelength in air is 12 cm. The source of radiation is normally a half-wave antenna; a rod shaped conductor about 6 cm long. Placed in a small parabolic reflector the antenna would produce a highly non-uniform beam (figure 11.5b). To produce a reasonably uniform beam the antenna would need to be placed in a reflector very much larger than its 6 cm length. A reflector with a focal length of a ELECTROMAGNETIC WAVES FOR THERAPY 284 metre or more and a diameter of several metres would be needed - producing a beam which is metres in diameter. Figure 11.5 (a) a uniform beam from a parabolic reflector and point source, (b) a non-uniform beam from a parabolic reflector and extended source For therapeutic application, a microwave beam only 10 to 20 cm in diameter is desirable, in order to localize the microwave energy. Reflectors 10 or 20 cm in diameter with antennas about 6 cm in length cannot produce a uniform beam but can be designed to produce a diverging beam. The beams obtained from reflectors presently used in therapy diverge considerably - the wave intensity decreasing rapidly with distance from the reflector. The reflectors must be designed this way: if a less divergent beam is produced part of the beam will be divergent, part will be parallel and part focussed at some point in front of the reflector as in figure 11.5(b). This has the obvious risk of producing a local hot-spot in the patient's tissue and causing tissue damage. Microwave applicators are available to produce a number of beam patterns. The pattern is not obvious from inspection of the shape of the reflector but the manufacturers do supply this information.

ELECTROMAGNETIC WAVES FOR THERAPY 285 EFFECT OF MICROWAVES ON TISSUE The penetration depth of microwaves (table 9.1) indicates that the waves are useful for diathermy. The three factors determining the depth efficiency of waves generally are (chapter 9) the penetration depth (δ) of the waves in a particular tissue and the extent of reflection and refraction at tissue interfaces. Considering first penetration depth, we make the observation that tissues with high values of dielectric constant (ε) and conductivity (σ) absorb electromagnetic radiation more rapidly than tissues with low values of ε and σ. The reasons were given in chapter 9 previously. Values of ε and σ are significantly different at microwave frequencies to those appropriate to shortwave diathermy at 27 MHz (table 6.2). Table 11.1 lists the values applicable at microwave frequencies. Notice that fatty tissue and bone marrow have quite similar values of ε and σ - this explains why the penetration depth of microwaves (table 9.1) is almost the same in both tissues. The relatively high values of ε and σ for muscle result in a greater rate of microwave absorption and hence a lower value for the penetration depth in this tissue. The extent of reflection at an Table 11.1 interface is calculated from Dielectric constant and conductivity of tissue equation 9.5: it is determined by the mismatch in impedance of the at microwave frequencies. tissues. Since we are talking about electromagnetic waves the imped- ELECTROMAGNETIC WAVES FOR THERAPY 286 ance of interest is the electrical impedance - determined by the dielectric constant (ε) Clearly if we wish to calculate and conductivity (σ) of the tissue. Reflection of microwaves at the fat/muscle and the pattern of heating in tissue we must take account muscle/bone interfaces will be pronounced due to the difference in electrical of both the penetration depth properties (changed by a factor of 10) on either side of the boundary concerned. in each tissue and the The amount of refraction at an interface is calculated from equation 9.12: it is amount of reflection and determined by the mismatch in wave velocity of the tissues. The wave velocity in turn refraction at each tissue is determined by the dielectric constant and conductivity of the tissue. interface. Each factor will have a significant effect on Because of the large difference in the electrical properties (ε and σ) of air, fatty tissue, the heating pattern. See the chapter by Schwan muscle and bone, refraction effects will be significant unless the microwave beam in: Licht, S H, Therapeutic strikes each boundary at a right angle (zero angle of incidence). Heat and Cold, (2nd Edition), New Haven (1968). The Fraction of Total Energy Absorbed A knowledge of the dielectric constant and conductivity of each tissue enables us to calculate the relative rate of heating of each tissue. This information alone does not allow us to predict the actual amount of heat produced since much of the microwave energy is reflected at the air/skin interface. The significant difference in the electrical properties of air (for which ε ≈ 1 and σ ≈ 0) and soft tissue will result in a considerable amount of the energy incident upon the skin being reflected. The total percentage of microwave energy absorbed deeper in the body tissues and hence converted into heat also depends on the thickness of the skin/fatty tissue layer. This is because a proportion of the wave energy reflected from the fat/muscle interface will penetrate the skin and be re-radiated into the air. Some decades ago, H. P. Schwan (see Licht (1968)) calculated the percentage of total energy reflected at different frequencies and various thicknesses of skin and fat. His results show that: * At frequencies less than 1000 MHz, 60 to 70% of the energy is reflected this almost independently of skin and fat thickness.

ELECTROMAGNETIC WAVES FOR THERAPY 287 * Between 1000 and 3000 MHz reflection depends critically and in a complex way Another practical implication on tissue thickness. Between 0 and 80% of the energy is reflected. of the large amount of reflection is the need to avoid * Above 3000 MHz around 60% of the energy is reflected - again almost unintentional exposure of independently of tissue thickness. body parts (including those of the therapist). One major implication of the above results is that at a frequency of 2450 MHz the effective dosage is virtually impossible to determine in a clinical situation, due to the practical difficulty in establishing skin and fat thickness which may vary considerably in the treated area. Clearly a frequency above or below the range 1000 to 3000 MHz is to be preferred on these grounds. As we will see in what follows, a lower frequency is preferable. The Distribution of Absorbed Energy We examine now the absorption of the proportion of microwave energy which is not reflected by the skin or re-radiated. Consider a 2 cm fatty tissue layer adjoining muscle tissue. For simplicity we begin by making two assumptions: * that no bone is present. We will take bone into account in subsequent examples. * that refraction can be ignored. In other words the angle of incidence is assumed to be zero. Refraction effects will be described separately. The relative rate of heating can be calculated from the dielectric constant and conductivity of each tissue: the two factors which determine the amount of reflection and the penetration depth. The method of calculation is described by Schwan (see Licht (1968)). Figure 11.6 shows the pattern of heat production for microwaves at the relatively high frequency of 8500 MHz (wavelength 3.5 cm in air). A standing-wave pattern (see chapter 9) is produced in the fatty tissue: this is because of reflection at the fat/muscle interface. ELECTROMAGNETIC WAVES FOR THERAPY 288 Figure 11.6 Heating pattern predicted for microwaves of frequency 8500 MHz in a specimen of 2 cm fatty tissue over muscle. The standing-wave pattern in the fatty tissue is not ideal since reflection is not 100% When unequal size waves and the wave is progressively absorbed in its travel. The actual pattern is a interfere, the standing-wave combination of an exponential decrease (determined by the penetration depth, δ) and effect produces peaks and interference of unequal size waves (figure 9.12). troughs in the heating-rate At this frequency, most heat is produced in the fatty tissue close to the skin and in the pattern, but there are no true superficial region of the muscle. A reasonable heating rate is obtained at the muscle nodes (points where the surface but the effect extends to only a fraction of a centimetre into the muscle tissue. intensity is zero). The total amount of heat produced in each tissue is indicated by the area under the curves in figure 11.6. It is evident that there is greater overall heat production in the fatty tissue. This problem is typical of higher microwave frequencies. The peaks in the heating pattern in the fatty tissue are separated by one half of a wavelength (see chapter 9 - this is close to 1 cm in figure 11.6) so the wavelength of the microwaves in fatty tissue is about 2 cm.

ELECTROMAGNETIC WAVES FOR THERAPY 289 At a frequency of 2450 MHz, the frequency most commonly used in therapy, the relative rate of heating is as shown in figure 11.7. Figure 11.7 Heating pattern predicted for microwaves of frequency 2450 MHz in a specimen of 2 cm fatty tissue over muscle. Again a standing wave pattern is found in the fatty tissue where most heat is produced. At this lower frequency, the wavelength is greater (since equation 9.1 holds: v = f.λ) so only a single peak is seen in the heating pattern in fatty tissue. Heat production in the muscle tissue is improved over the 8500 MHz results but is still limited to the first centimetre or so. Evidently the lower frequency is preferable from a 'deep heating' point of view - but we saw earlier that frequencies in the range 1000-3000 MHz result in uncertain dosage. What of frequencies below 2450 MHz? Figure 11.8 shows the relative rate of heating predicted for a microwave frequency of 915 MHz in a tissue specimen with the same dimensions as used previously. Figure 11.8 Heating pattern predicted for microwaves of frequency 915 MHz in a specimen of 2 cm fatty tissue over muscle. ELECTROMAGNETIC WAVES FOR THERAPY 290 At 915 MHz, a standing wave pattern is still produced in the fatty tissue but the The wavelength in fatty tissue wavelength is so large that no peaks are evident. at 915 Mhz is about 18 cm so Figure 11.9 shows the relative rate of heating predicted for a microwave frequency of a peak and a trough would be 434 MHz in the same tissue specimen. separated by 4.5 cm (one quarter of a wavelength). Figure 11.9 Heating pattern predicted for microwaves of frequency 434 MHz in a specimen consisting of 2 cm fatty tissue over muscle. The depth efficiency of lower frequency microwaves is apparent from figures 11.8 and 11.9. Both frequencies give maximum heating in the muscle with much the same decrease in heating rate with distance into the tissue. The lowest frequency (434 MHz) produces least heating of fatty tissue; the difference being most noticeable near the tissue surface. Both frequencies give a heating pattern which is suitable for diathermy and dosage is reasonably predictable. The heating of the fatty tissue surface with 915 MHz microwaves can be compensated for by using a contact applicator with surface cooling. The microwave director (applicator) is designed to be used in direct contact with the patient. Cooling air is blown through the applicator and on to the patients' skin during treatment in order to minimize the temperature elevation of superficial tissues.

ELECTROMAGNETIC WAVES FOR THERAPY 291 When we consider the three-layer system of fat/muscle/bone we predict reflection at both the fat/muscle interface and at the muscle/bone interface. In consequence a complex heating pattern is produced in both the fat and muscle tissue. Figures 11.10, 11.11 and 11.12 show the patterns predicted for frequencies of 2450 MHz, 915 MHz and 434 MHz respectively. Tissue dimensions are the same as those chosen to illustrate the heating pattern for ultrasound (figures 10.6 and 10.7). Figure 11.10 Heating pattern predicted for a microwave frequency of 2450 MHz in a tissue combination of 2 cm fat, 2 cm muscle and 2 cm bone. Figure 11.11 Heating pattern predicted for a microwave frequency of 915 MHz in a tissue combination of 2 cm fat, 2 cm muscle and 2 cm bone. ELECTROMAGNETIC WAVES FOR THERAPY 292 For each frequency, heat production in bone is Figure 11.12 minimal. Both 915 and 434 MHz microwaves Heating pattern predicted for a microwave produce maximum heating in the muscle layer: frequency of 434 MHz in a tissue combination the lower frequency having greater depth of 2 cm fat, 2 cm muscle and 2 cm bone. efficiency. Not too much significance can be attributed to the actual positions of maxima and minima of heat production as these vary with the tissue dimensions and electrical properties assumed. The general implicat- ions of the figures are, however, clear: frequencies below 1000 MHz are needed if tissues located beneath a few centimetres of fat are to be effectively heated. For treating structures located closer to the skin surface - for example a knee or elbow joint which is not covered by a thick layer of fat - a frequency of 2450 MHz is adequate, though the dose will be somewhat unpredictable. More deeply located structures - for example, the hip joint - are not heated appreciably at this frequency. TISSUE GEOMETRY AND REFRACTION EFFECTS The heating patterns shown in figures 11.6 to 11.12 were calculated ignoring refraction effects. This is appropriate for a uniform microwave beam incident upon a plane surface with tissues of constant thickness beneath. When microwaves are incident upon a curved surface then, even if the beam is uniform, refraction will occur. This is illustrated in figure 11.13, where reflected waves are omitted for clarity. The amount of refraction depends on the curvature of the tissue surfaces an the electrical characteristics of the tissues. When the curvature of the tissues is pronounced, as for example with an arm or leg, the amount of refraction is considerable. The smaller is the radius of the limb, the greater is the refraction effect.

ELECTROMAGNETIC WAVES FOR THERAPY 293 Tissues of high dielectric constant and conductivity have a low electrical Figure 11.13 impedance and consequently a low wave velocity. Thus the velocity decreases as Refraction of a microwave beam the wave progresses from air to fatty tissue and then to muscle. This means that waves will be refracted to produce convergence of the beam. A focussing effect is at tissue interfaces. produced in the tissues. What effect will the beam convergence have on the heating pattern? As the beam Ho HS. Health Physics, 31, travels through fatty tissue and muscle it is progressively absorbed. The wave 97-108 (1976). energy is converted into heat energy and the wave intensity (energy per unit area) Lehmann, J F, Therapeutic decreases. Convergence of the beam will result in the energy remaining any Heat and Cold, (3rd Edition), particular depth being concentrated in a smaller area. This tends to increase the Williams and Wilkins (1982). beam intensity. Thus with curved tissue surfaces as shown in figure 9.12 the beam intensity does not diminish as rapidly as would occur with plane surface. Consequently the depth efficiency for heat production is greater. H. S. Ho (1976) has calculated the relative rate of heating for cylindric models with dimensions approximating to adult human arms and legs. His results are qualitatively similar to those shown in figures 11.10 to 11.12 but the relative rate of heating of muscle is significantly higher. Nonetheless the conclusions to be drawn from Ho's work are those described earlier. For patient treatment, better heating patterns are produced with frequencies lower than the 2450 MHz currently used. Ho's results indicate an optimum frequency of around 750 MHz for efficient and relatively uniform heating of muscle tissue. For a description of depth efficiency calculations using different frequencies and other geometric shapes resembling parts of the human body see A. W. Guy in Lehmann (1982), chapter 6. In summary we may conclude that 2450 MHz microwaves have low depth efficiency. This frequency is best suited to heat production in fatty tissue and the superficial region of muscle. 915 MHz and 434 MHz microwaves produce greater depth heating of muscle and less heating of fatty tissue. The optimum frequency for selective and uniform heating of muscle tissue being around 750 MHz. ELECTROMAGNETIC WAVES FOR THERAPY 294 Microwaves are intrinsically unsuited to heating of bone (see figures 11.10 to 11.12) The difference between because of its electrical characteristics: for this reason joints can only be heated when heating rate and rate of the overlying tissue layers are very thin. For heating of deeply located joints, temperature increase, and ultrasound or shortwave diathermy would be more effective. the relationship between As a final point it should be stressed that the graphs shown in figures 11.6 to 11.12 these quantities, was show where heat is produced but not the temperature increase in each tissue. The discussed in chapter 7. temperature increase depends on such factors as the specific heat capacity of the The divergence of a laser tissue and heat transfer within and between tissues and to the bloodstream (see beam is so small that a chapter 7). beam pointed at the moon could illuminate a target less LASERS than a metre across. The acronym 'laser' stands for 'light amplification by stimulated emission of radiation'. Lasers are electromagnetic wave amplifiers which can produce beams of electro- magnetic waves with two special properties: * the beam has very little divergence. It has a pencil-like shape. * the beam is coherent. That is, all the waves in the beam are of exactly the same frequency and wavelength and are synchronized with each other. The pencil-like beam of the laser means that the wave energy is always concentrated on the same area: the intensity (which is the energy per unit area) does not decreased appreciably with distance due to beam-spreading. Production of a laser beam Visible light can be produced by excitation of atoms. For example if crystals of a copper salt, such as copper sulphate, are heated in a flame, the flame turns blue. When strontium salts are heated, the flame turns violet. Sodium salts produce a yellow colouration. This is because electrons in the copper, strontium or sodium atoms are kicked from their 'ground state' orbitals by the heat energy of the flame and