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1. The nature of ionizing radiation
Ionizing radiation is the term used to describe highly energetic particles or waves which when they collide with atoms cause the target atoms to receive significant kinetic energy. This energy may cause inelastic collisions when the target atom absorbs a proportion of the energy and is placed into a higher energy state. Alternatively the incident energy may be divided between the source and the target, in which both are displaced with energies whose sum is the total incident energy. The specific mechanism which occurs is dependent on the nature of the incident radiation, the target and the incident energy level. Ionizing radiation is produced by one of several phenomena.
Cosmic radiation, which is mainly due to extra-terrestrial nuclear reactions. The nuclear processes which take place in the sun and the stars result in a small but measurable flux of high energy radiation which reaches and in some cases traverses the earth. This radiation is found (UNSC, 1982) to give rise to somewhat more than one tenth of our annual natural background radiation dose.
Nuclear decay of unstable elements on the earth. There are rocks in the earth's crust containing unstable elements which undergo nuclear decay. This process gives rise to a small amount of nuclear radiation, but as part of the decay, certain of the child products are also radioactive, and these in turn give rise to radiation. In particular, there are significant doses due to the decay of radon-222 and radon-220 through absorption through the lungs.
Radon is locally concentrated owing to the types of subsoil and building material used.
Artificial production of ionizing radiation from high energy sources. If an energetic electron collides with an atom, it may give up its energy inelastically and produce high energy photons, some of which are in the X ray region.
1.1. Sources of X rays
A medical X ray tube (FIG. 1) is built from a vacuum tube with a heated cathode which emits electrons. These are accelerated by a high electric potential (of up to around 300 kV) towards a target anode. The anode is built from a metal with a high atomic number to provide the best efficiency of conversion of the incident electron energy into photons. Nonetheless the typical efficiency is only about 0.7%. With typical tube currents of 10 to 500 mA, instantaneous input powers up to 100 kW may be used. As a result, there is a significant problem of anode heating. To reduce this problem, the anode is normally rotated at high speed (about 3600 rpm), and is normally made with a metal which has a high melting point. Frequently the anode target layer is relatively thin and is backed by copper to improve thermal conduction. In spite of these precautions, localized temperatures on the anode may reach around 2500°C. A typical tube is about 8-10 cm in diameter and 15-20 cm in length.
FIG. 1 X ray tube
FIG. 2 Simplified circuit for driving X ray tube
FIG. 2 shows a simplified schematic for an X ray power circuit. In practice, the tube is driven in modern sets from sophisticated supplies. In some forms of radiography which are outlined in Section 6 X ray pulses are used over a prolonged period to obtain images.
Frequently these require pulse durations of milliseconds. It is crucial in most of these cases that the high voltage power supply is stable to ensure that radiation of the required spectrum is produced.
1.1.1. X Ray Spectra
The spectral emission of an X ray tube is shown in FIG. 3.
FIG. 3 Form of X ray spectrum
FIG. 4 Filtered X ray spectrum
The radiation emitted by an X ray tube is primarily due to Bremsstrahlung--the effect of the deceleration of the high energy electron beam by the target material of the X ray tube. The incident electrons collide with the nuclei in the target, and some result in inelastic collisions in which a photon is emitted. The peak photon energy is controlled by the peak excitation potential used to drive the tube. For a thin target, the radiant photon energy is of a uniform distribution.
When a thick target is used, as is normally the case for diagnostic applications, the incident electrons may undergo a number of collisions in order to lose their energy. Collisions may therefore take place throughout the depth of the target anode. Through the thickness of the target, there is a progressive reduction of the mean incident energy. The result would then be a spectrum decreasing linearly from zero to the peak energy. However, the target material also absorbs a proportion of the generated X rays, preferentially at the low energy end of the spectrum. There are also strong spectral lines produced in the X ray spectrum as a result of the displacement of inner shell electrons by incident electrons. The form of the resultant spectrum is as shown in FIG. 3.
Protection against the low energy components of this spectrum may be enhanced by the use of filters. These are used when the low energy components would not penetrate the area of interest adequately, and therefore simply cause potential problems due to an unnecessary radiation dose. Low energy components may of course be used alone when they are adequate to pass through small volumes of tissue in applications such as mammography. The filtered spectrum is shown in FIG. 4.
1.2. Radioactive decay
The nuclei of large atoms tend to be unstable and susceptible to decay by one of several mechanisms.
1. Beta emission, in which an electron is emitted from the nucleus of the atom, approximately retaining the atomic weight of the atom, but incrementing the atomic number.
2. Alpha emission is due to the release of the nucleus of a helium atom from the decaying nucleus. Alpha particles are released with energies in the range 4-8 MeV, but readily lose their energy in collisions with other matter.
3. Neutrons are emitted when a nucleus reduces its mass, but does not change its atomic number. Neutrons are emitted either spontaneously or as a result of an unstable atom absorbing a colliding neutron and then splitting into two much smaller parts together with the release of further neutrons.
4. Gamma radiation is the emission of high energy photons from unstable atomic nuclei. This occurs frequently following either of the previously mentioned forms of decay, which often leave the resulting atomic nucleus in a metastable state.
Radioactive decay is a probabilistic process: at any time there is a constant probability that any atom will spontaneously decay by one of these processes. The decay is not immediate, since there is an energy well which must be traversed before it may take place: the probability of the decay event taking place is related to the depth of the well. Thus for a population of N atoms the radioactivity Q is given by
Q = -wV = dN/dt (1)
T--(In 2)/h (2)
The decay constant is related to the 'half-life' of the nuclide by: x
Radioactive decomposition of large nuclei takes place in a series of steps. For example, both uranium and radium are present in the earth's crust in significant quantities. They decay through a series of energy reducing steps until they ultimately become lead.
2. Physics of radiation absorption, types of collision
There are several characteristic modes of collision between ionizing radiation and matter: the mechanism depends on the form and energy of the radiation, and the matter on which it is incident. For our purposes, the following are the most significant.
1. The Photoelectric Effect, in which an incident photon gives up all its energy to a planetary electron. The electron is then emitted from the atom with the kinetic energy it received from the incident photon less the energy used to remove the electron from the atomic nucleus. Clearly this process can only occur when the incident photon energy is greater than the electron's binding energy. The probability of this interaction decreases as the photon energy increases.
The result of the electron loss is to ionize the atom, and if one of the inner shell electrons is removed, to leave the atom in an excited state. The atom leaves the excited state when an electron descends from an outer orbit to replace the vacancy, and a photon may be emitted, again having X ray energy. The photoelectric effect is the predominant means of absorption of ionizing radiation when the incident photon energy is low.
2. The Compton Effect occurs when an incident electron collides with a free electron. The free electron receives part of the energy of the incident photon, and a photon of longer wavelength is scattered. This effect is responsible for the production of lower energy photons which are detected in nuclear medicine systems (see Section 6). Their reduced energy means that they may subsequently be recognized by the detection system and largely eliminated from the resulting image.
3. Pair Production occurs when a highly energetic photon interacts with an atomic nucleus.
Its energy is converted into the mass and kinetic energy of a pair of positive and negative electrons. This process may only occur once the incident energy exceeds the mass equivalent of two electrons (Le. 2mc2 = 1.02 MeV).
4. Neutron Collisions result in a wide range of recoil phenomena: in the simplest case, the target atomic nucleus receives some kinetic energy from the incident neutron, which is itself deflected with a reduced energy. Other forms of collision take place, including the capture of incident neutrons leaving the atom in an excited state from which it must relax by further emission of energy.
For a fuller description of these effects, the reader is referred to specialist texts, such as Greening (1981).
3. Radiation Measurement and Dosimetry
3.1. Dosimetric Units
We must firstly consider what is meant by radiation dose. Ionizing radiation incident on matter interacts with it, possibly by one of the means outlined above. In doing so, it releases at least part of its energy to the matter.
As a simple measure, we may look at the rate of arrival of radiation incident on a sphere of cross section da. The fluence is
= dN/da
where dN is the number of incident photons or particles, and the fluence rate is
41 = dcD/dt.
The unit of this measurement is therefore m"s-'. If the energy carried by the particles is now considered, the energy fluence rote may be derived comparatively in units of Wm-*. The spectral intensity of the incident radiation is dependent on a number of factors: it is often important to be able to assess the spectral distribution of the incident radiation.
The unit of decay activity of radionuclides was the Curie, which became standardized at 3.7~10'~s-'. This is approximately the disintegration rate of a gram of radium. As the SI unit of rate is s-I, this unit has now been superseded by the Bequerel (Bq) with unit s-I when applied to radioactive decay.
The unit of absorbed dose, being the amount of energy absorbed by unit mass of material, was originally the Rad, or 100 erg g-1 J kg-') of absorbed energy. This has now also been superseded by the SI unit the Gray (Gy), which is defined as 1 J kg-I. Another unit of interest relates to exposure to ionizing photon radiation. This measure quantifies the ionization of air as a result of incident energy. The Roentgen (R) is defined as 2.58 x 10^-4 C kg-I. The term 'Dose Equivalent' is used to denote a weighted measure of radiation dose: the weighting factor is derived from the stopping power in water for that type and energy of radiation. This measure is normally expressed in the unit Sievert (Sv) which has the same dimensions as the Gray, but is given a special name to denote its different basis.
The 'Effective Dose Equivalent' is the measure used to denote dose equivalent when it has been adjusted to take account of the differing susceptibilities of different corporal organs to radiation. The Effective Dose Equivalent is defined as: The weighting factors employed here vary between 0.25 for the gonads to 0.03 for bone surface, and 0.3 for the bulk of body tissue.
3.2 Outline of Major Dosimetric Methods
A wide range of methods exists for the measurement of radiation dose. They include fundamental methods which rely on calorimetric measurement and measurement of ionization: these are required as standards for the assessment of other techniques. Scintillation counters, which also help to characterize the received radiation, are described in Section 6.3. However, in practical terms, two main methods outlined below are used in monitoring individual exposures to radiation. In addition, as a basic protection, it is frequently wise to have available radiation counters when dealing with radioactive materials as they provide real time readings of the level of radiation present.
3.2.1. Thermoluminescence
Many crystalline materials when irradiated store electron energy in traps. These arc energy wells from which the electron must be excited in order for it to return via the conduction band to a rest potential. The return of the electron to the rest state from the conduction band is accompanied by the release of a photon which may be detected by a photomultiplier. The trapping and thermoluminescent release processes are shown in FIG. 5. If the trap state is sufficiently deep, the probability of the electron escaping spontaneously may be sufficiently low for the material to retain the electron in the excited state for a long period: it can be released by heating the material and observing the total light output. Various materials are used, but they should ideally have similar atomic number components to that of tissue if the radiation absorption characteristics are to have similar energy dependencies. The materials are used either in a powder form in capsules or alternatively embedded in a plastic matrix.
FIG. 5 Trapping and thermoluminescence
3.2.2. Film Badge
The photographic film badge is a familiar and rough and readily portable transducer for the measurement of radiation dose. The film is blackened by incident radiation, although unfortunately its energy response does not closely match that of tissue. The badge holder therefore contains various metal filters which provide a degree of discrimination between different types of and energies of incident radiation. The badges worn by radiation workers are typically swapped and read out on a monthly basis to provide a continuing record of their exposure to radiation.
4. Outline of the Application of Radiation in Medicine Radiology, Radiotherapy
Ionizing radiation is used in medicine in two main applications. As the radiation is in some cases very energetic it is able to pass through body tissue with limited absorption. The differential absorption of radiation in different types of tissue makes it possible to obtain images of the internal structures of the body by looking at the remaining radiation if a beam of X or gamma radiation is shone through a region of the body. Absorbed doses (see section 3.1) from diagnostic investigations are typically around 0.1 mSv.
Additionally, radioactive substances may be injected into the body as 'labels' in biochemical materials which are designed to localize themselves to particular organs or parts of organs.
The radiation emitted from the decay of these materials may be examined externally to derive an image of the organ's condition. As ionizing radiation presents a significant risk of causing biological damage to tissue, if large doses of radiation may be administered to specific areas of body tissue it is possible to destroy cancerous tissue selectively and without the risks entailed with surgery. The doses involved in radiotherapy are much higher, being typically localized doses of tens of Gy delivered in smaller doses of a few Gy at intervals of several days.
5. Physics of NMR
Nuclear Magnetic Resonance is a physical effect which has become increasingly used in medical imaging since the 1970s. This section provides a simple outline of the physics of the NMR process. An overview of the instrumentation which is used to obtain images from this process is presented in Section 4.
In essence we will find that images using NMR are effectively maps of the concentration of hydrogen atoms. The images obtained are of high resolution. The display is derived from details of a subject's morphology based on factors different from those examined when conventional radiological studies are made. The examination technique has fewer apparent inherent dangers than does the use of ionizing radiation, but has the serious drawback of the high capital cost of the equipment used to obtain images.
5.1. Precessional Motion
Probably the easiest point to start an understanding of NMR is by looking at the motion of a spinning particle in a field. Consider a child's spinning top. If it is placed spinning so that one end of its axis is pivoted, then the mass of the top acts with the earth's gravitational field and the reaction of the pivot to form a couple which tends to rotate the spinning angular momentum vector downwards (FIG. 6). Since however angular momentum is conserved, a couple is produced which causes the top to make a precessional motion about its pivot.
Expressing this in mathematical notation, and using the symbols from the diagram (note that bold type refers to vector quantities), a torque is caused by gravity acting on the mass of the top:
z=rxmg (4)
FIG. 6 Forces acting on a gyroscope
Here the symbol x is the vector cross product. This torque acts on the gyroscope whose angular momentum is L to modify it, so that:
Z=d%t (5)
In a short time t the angular momentum of the gyroscope is modified by a small amount AL acting perpendicularly to L. The precessional angular velocity of the gyroscope, which is the rate at which its axis rotates about the z co-ordinate, may now be derived:
OP = Since we are looking at a small change in AL << L, the small angle A$ is
AL =At Lsin0 Lsin0 A$=-=
and the precessional velocity from equation 6 above is
A+ = 0 =-=At Lsin8 (7)
Substituting for z from equation 4, we obtain an expression for the magnitude of the angular velocity of the precessional motion: This tells us that the precessional angular velocity is proportional to the force due to the field (mg) and inversely proportional to the body's angular momentum.
The NMR phenomenon is analogous. A spinning charge (in the simplest case a proton, the nucleus of a hydrogen atom) if placed in a magnetic field precesses about the field. The spin vector representing angular momentum may be either directed with or against the magnetic field: the two directions possible with hydrogen represent two different energy states. Evaluation of the concentration of hydrogen is undertaken by stimulating a proportion of the nuclei into the higher energy state with a radio frequency electromagnetic pulse and then examining the energy released as they decay into the lower state. The following paragraphs provide a mathematical statement of the effect so that it may be quantified.
Firstly, a rotating charge has a magnetic moment : m=yI (10)
in which m is the magnetic moment, and I the angular momentum, and l/gamma the gyromagnetic ratio. In classical physics, y is e/2m where e is the charge and m the mass of the particle.
If the rotating charge is placed in a magnetic field of strength B, the field causes a torque which makes the particle's magnetic moment and, as a result, also its momentum vector, precess about the direction of the field. The rate of change of the particle's momentum then is given by Now substitute in equation 10 to yield In the steady state, the precession continues indefinitely with an angular velocity given by 63=-yB (13)
This expression has the same form as that of the expression which we derived for a spinning top. In this case the precessional velocity is proportional to the strength of the applied magnetic field.
We now may briefly extend our view to include a quantum mechanical description of the motion. In this view, energy states and angular momentum are discrete rather than a continuum of values. In the case of a hydrogen nucleus, the permitted values of the spin quantum numbers are +-&, representing spin vectors with and against the magnetic field. The respective energy states are E = + y AlBl (14)
AE = yAlBl (15)
where A is 27ch and h is Plank's constant. The separation of the levels is These expressions describe the precession of the momentum vector in terms of a fixed system of 'Laboratory Co-ordinates'. We could instead describe the equations in terms of some other set of co-ordinates. It will turn out to be easier to understand the origin of the later expressions and visualize the processes if we transform equation 12, which is known as the Larmor Equation, into a rotating co-ordinate system.
As a first step, consider a vector A which is fixed in a co-ordinate system which is rotating with angular frequency 0,. This is shown pictorial form in FIG. 7. In time 6t, its end point is displaced by an amount 6A, so that in terms of the fixed co-ordinate system
6A = (oGt)Asine = (o,xA)Gt lim (6A/6t) =dA/dt =( o,x A) (16)
and the velocity of A in the fixed system is (17)
&-to If now A is not fixed in the rotating system, but is itself moving at a velocity DAIDt, its velocity in the laboratory co-ordinate system is dA/dt = DA/Dt +(o,xA) (18)
FIG. 7 Rotating co-ordinate system
Note that the newly introduced notation of the form DADt refers to a separate differentiation operation. Using the form of expression shown in equation 18, we may rewrite the precessional motion as dm/dt = Dm/Dt + (O xm) (19)
and now substituting this result into equation 12 we obtain
Dm/Dt = y m XB -(a xm)
= ym xB+m xw
= ym x( B+y)
This expression demonstrates that in a rotating co-ordinate system, the body is subjected to an apparent magnetic field given by (B,,,=B+wly), and that the apparent rotational velocity is decreased by the velocity of rotation of the co-ordinate system. We may now remove terms which become constant in the rotating reference frame.
5.2. Resonant Motion
We now apply a circularly polarized magnetic field B, in a plane normal to the steady field B, and view this from within the rotating co-ordinate system. Note that we may decompose a circularly polarized field into two counter-rotating sinusoidal fields of the same frequency. If the additional field B, rotates at the same frequency as the new co-ordinate system, the spinning particle experiences an apparent field in the sense of B, which is denoted Bapp. It would be seen to precess (by an observer in the rotating system) about the resultant of B, and B,,, namely BreS. These fields are shown schematically in FIG. 8.
FIG. 8 Summation of fields in the rotating co-ordinate system.
B,, reduces to B, when B, = Bapp. The magnetic moment m then rotates around B,, becoming parallel and antiparallel to B,. In this condition, the precession frequency
o P = -YB, =oL (23)
has the same frequency as the natural oscillation of precession of the particle's magnetic moment (the Larmor frequency). This is a forced resonance condition, in which the frequency of resonance is proportional to the applied field B,.
5.3. Relaxation Processes
Forcing energy is delivered as a pulse of electromagnetic radiation with energy in the resonant frequency region. Once forced into a resonance condition, the energy acquired by magnetic dipoles requires a time to allow it to be given up to the surrounding material. The resonant effect is then observed by examining the release of that energy to the surrounding material as the nuclear spin returns to alignment along the B, axis. Firstly we see from the diagram that the magnetization M rotates in the resonant condition about the forcing function B,. For a field strength of around lo-' T, the precessional rate is in the order of lo6 rad s-l. This means that it is necessary to administer pulses in the order of 1 ps duration.
We have so far described the resonance phenomenon from the viewpoint of a single spinning particle. We now describe the system in terms of the net magnetization M which is the sum Cm, over all nuclei in a unit volume.
The first form of decay process to observe is the spin-lattice relaxation time TI. This is the process in which the stimulated nuclei (normally in our cases protons) release their excess energy to the lattice so that the system returns to a thermodynamic balance. The relaxation process of the magnetization M is described by
-dM--(Mo -M) dt TI
and M, the equilibrium magnetization. This relaxation time is about 2 seconds for water, but values are typically in the range between lo" and lo4 s. The relaxation processes use a number of different physical mechanisms by which energy is transferred to the lattice from the resonating nuclei: see Lerski (1985) for a description of various physical models.
In addition to this effect, the spins of neighboring nuclei may interact. A precessing nucleus produces a local field disturbance = le T in its nearest neighbor in water causing a de-phasing of protons in--lo-" s owing to their frequency differences. The spin-spin interaction time is commonly denoted T2.
These relaxation processes effectively limit the rate at which an image may be acquired using NMR and its spectral resolution. T, means that having stimulated one region, the signal from that area must decay before another area may be stimulated in order to determine its proton population.
6. Ultrasound
Sound is the perception of pressure fluctuations travelling through a medium; its waves are transmitted as a series of compressions and rarefactions. There are a number of ways in which this pressure fluctuation can be transmitted which give rise to three classes of wave which are outlined below.
Ultrasound is defined as sound above the range of hearing of the human ear. This is usually taken to be 20 kHz although the appreciation of sound above 16 kHz is exceptional. FIG. 9 gives an indication of the classification of sound and some natural and manmade phenomena and uses.
FIG. 9 The sonic spectrum
6.1. Longitudinal or Pressure Waves
In a Longitudinal wave the particles of the transmission medium move with respect to their rest position. The particle movement causes a series of compressions and rarefactions. The wave front travels in the same direction as the particle motion. The particle movement and subsequent compressions cause corresponding changes in the local density and optical refractive index of the material of the medium.
6.2. Shear or Transverse Waves
In shear waves, the wave front moves at right angles to the particle motion. Shear waves are often produced when a longitudinal wave meets a boundary at an oblique angle.
6.3. Surface, Rayleigh or Lamb Waves
Rayleigh or Lamb waves occur at the surface of materials and only penetrate a few wavelengths deep. These waves occur only in solids. Some semiconductor filters have been developed which rely on the properties of surface waves travelling in crystalline materials.
For medical applications we need only consider longitudinal waves as both Imaging and Doppler techniques rely on the propagation of longitudinal waves. Shear waves can propagate in fluids: however, they are not intentionally produced.
7. Physics of Ultrasound
7.1. Velocity of the Propagating Wave
The velocity (c) of a longitudinal wave travelling through a fluid medium is given by the ratio of its bulk modulus to its density.
where K = bulk modulus
p = density
7.2. Characteristic Acoustic Impedance
The relationship between particle pressure and the particle velocity is analogous to Ohm's law. Pressure and velocity correspond to voltage and current respectively. The acoustic impedance is therefore a quantity analogous to impedance in electrical circuits. It is related to particle pressure and velocity by the following equation: p=zv (26)
where p= particle pressure
v= particle velocity
Z= acoustic impedance
Acoustic impedance can be expressed as a complex quantity in the manner of electrical impedance. However for most practical medical applications it can be considered in a simple form. The characteristic acoustic impedance of a material is the product of the density and the speed of sound in the medium:
z=pc (27)
where p = density in kg mF3.
Hence, materials with high densities have high acoustic impedances. For instance steel has a higher acoustic impedance than perspex. The following table shows materials with similar and dissimilar acoustic impedances.
Similar Z Iron--Steel Water--Oil Fat--Muscle Dissimilar Z Water -Air Steel--Fat
The dimensions of the acoustic impedance are kg m-2 s-'. Most materials found in the human body or used in transducers have acoustic impedances of the order of 10^6 kg m-2 s-I; therefore, the commonly expressed unit of acoustic impedance is the Rayle. One Rayle is 1x106 kg m-* s-I .
The acoustic impedance of a number of materials is presented in FIG. 10.
FIG. 10 Table of acoustic impedance values (see Wells 1977, Duck 1990)
3.7.3. Acoustic Intensity
Consider a particle vibrating with Simple Harmonic Motion (SHM) in a lossless medium. The total energy of the particle (e_total) is the sum of its potential and kinetic energies. If the medium is lossless the total energy is constant. The total energy of the particle when at zero displacement from its resting position is given by its kinetic energy:
e_total =(1/2) mv) ^2 [28]
where v= velocity when at zero displacement
m = particle mass
The total mass of particles contained within unit volume is given by the density of the medium (p). Therefore the total energy of the particles in unit volume is given by
(29)
The intensity (0 of a wave can be defined as the energy passing through unit area in unit time.
The wave velocity is the rate at which this particle energy passes through the medium.
Therefore in unit time a unit area will travel a distance of c meters, defining a volume c. As the total energy per unit volume is E_Tuv, the energy passing through unit area in unit time will be given by:
The intensity can also be expressed in terms of pressure.
[31]
This equation's dimensions are:
I = m s-I x kg m-3 x m2 s-~ = kg s-~.
The units of intensity are watts per square meter, which is equivalent to kg s-~.
7.4. Reflection
If a longitudinal wave travelling through a medium meets an interface with a different medium, reflection or transmission of the wave will occur. The laws of geometric reflection can be applied as long as the wavelength of the ultrasound is small compared to the dimensions of the interface. If this is so the reflection is said to be 'specula'. However, if this condition does not apply then scattering occurs. This will be considered in section 7.7.
Consider a wave travelling through a medium and impinging upon an interface at an angle theta_i (FIG. 11), a portion of the wave will be reflected at an angle 8, equal to the angle of incidence. Some of the wave is transmitted at an angle 8, given by Snell's law.
FIG. 11 Snell's law
sine;--c, sine, c2
--where c1 and c2 are the velocities of the wave in media 1 and 2 respectively. The subscripts i, t, r refer to the incident, transmitted and reflected waves respectively.
For a particular interface, as the angle of incidence increases, the angle of transmission also increases until the point of total internal reflection is reached. Total internal reflection occurs when the angle of the transmitted wave is equal to n/2. Therefore from equation (32) the incident angle for total reflection to occur is given by:
€Ii = sin'5 as sin%=l (33)
c2 if c2 'CI
7.4.1. Pressure Relationship
The particle pressure at an interface must be continuous. Therefore the sum of the particle pressure on one side is equal to that on the other or
Pi + Pr = Pr (34)
Consider a wave with particle velocity vj impinging upon an interface at an angle 8;. The velocity either side of the interface is also continuous and therefore.
vi cos ei--v, cos e, = v, cos e, (35)
As the particle velocity is a vector, the reflected velocity is negative (in the opposite direction) with respect to the incident wave.
Recalling equation (26) equations (34) and (35) can now be combined pJpj is known as the pressure reflectivity and pip, is known as the pressure transmittivity.
Equation (36) can be solved to yield: p,--z2 cos ei--z, COS e, pi z2 cosei + z, case, -_ and (37)
These equations are often shortened by assuming the incidence to be normal so all the cosine terms are 1. Therefore equations (37) and (38) reduce to: There will therefore be no reflection at an interface between two materials if their acoustic impedances are equal.
Consider an ultrasound wave travelling from medium 1 to medium 2 with acoustic impedances Z, and Z2 respectively. If Z, > Z2 the reflected wave will be n radians out of phase with the incident wave. However, if Z, < Z, the reflected wave will be in phase with the incident wave.
7.4.2. Intensity Relationship
The preceding equations define the transmission of a pressure wave across a boundary. By following the derivation for obtaining pressure expressions we may arrive at equations which define the intensity of waves at a boundary. Recall equation (31) which may be substituted into equations (37) and (38) to describe the wave intensity.
where Ir/l; is known as the intensity reflectivity and I,/Ij is known as the intensity transmittivity. These equations are often simplified by assuming normal incidence so equating all the cosine terms to 1. Therefore equation (40) becomes: 2 ;=(-) and ?=(z2+zl)2 42122 Hence, the degree of transmission or reflection of the pressure or the intensity of an acoustic wave incident on a boundary between two materials is related to their acoustic impedances.
Recalling the table of material pairs with similar and dissimilar acoustic impedances, clearly there will be minimal transmission and almost total reflection between the dissimilar materials. Conversely negligible reflection and almost total transmission occurs between similar materials.
Reflections from soft tissue
Kidney I Muscle = 0.03; Soft Tissue I Bone = 0.65; Tissue Air Coupling = 0.999
7.4.3. Transmission Through Thin Layers
The preceding analysis determined equations relating the intensity of a wave incident on an interface to the acoustic impedance of the two materials. The transmission of ultrasound through a thin layer is given by the following equation. It is a special case and will be considered as it has important implications for transducer design and practical application of ultrasound in medicine (Hill 1986). 2 (42)
z~ (z, +z3)2cos'2x~+(~2 h2 +?) sin22n~ h2 T=
Where T is the transmission and t2 is the thickness of the thin layer with impedance 2, between media Z, and 5.
There are three situations when this equation can be simplified.
1. If Z, >> Z, and Z3 >> Z2 then the right hand side of the denominator will be large and
therefore the transmission of ultrasound through the thin layer will be negligible. This situation occurs when there is a layer of air trapped between an ultrasound transducer and a patient.
If cos227[:2=1 i.e. when t2 =nh2 where n=l,2,3,4,5,6 ,... then t 2.
A2 In this instance the thickness of the thin layer is chosen such that transmission through it is independent of its acoustic properties. This is known as a half wave matching layer.
If sin2 2x2 = 1 i.e. when t2 = (2n--1)L where n = 1,2,3,4,5,6 ... then t h A2 4 3.
If the impedance of the second material can be chosen such that it is equal to Z, = Jm then the transmission through the layer can be total. This situation is known as a quarter wave matching layer.
Both quarter and half wave matching layers are used in ultrasonics (section 9.3); however, the properties of these layers depend on the wavelength in the second medium and therefore as the wavelength changes with frequency they are frequency specific.
7.5. Attenuation
So far we have referred to the conducting medium for ultrasonic propagation as lossless. However, in all practical situations the intensity of a wave diminishes with its passage. The reduction in the intensity or pressure of a wave passing through a medium in the x direction is referred to as the attenuation of the medium. The reduction in the wave can be attributed to a number of effects: namely reflection, wave mode conversion (longitudinal to shear), beam spreading, scattering and absorption. Attenuation varies with frequency as both scattering and absorption are frequency dependent.
The attenuation of a medium is expressed in terms of dB cm-1 at a particular frequency.
Attenuation can be determined for the pressure or intensity of a wave. The intensity attenuation coefficient is given by and the pressure attenuation coefficient by In each case x is the displacement between the points 1 and 2 where intensity and pressure I,,P, and 12,P2 were measured.
7.6. Absorption
An ultrasonic wave travelling through a medium is absorbed when wave energy is dissipated as heat. Absorption occurs when the pressure and density changes within the medium caused by the travelling wave become out of phase. When this happens wave energy is lost to the medium. The fluctuations become out of phase with the density changes as the stress with the medium causes the flow of energy to other forms. In section 3.7.3 we derived an expression for the intensity of a wave travelling through a lossless medium by considering the energy of a particle to be composed entirely of potential and kinetic energy.
In a real medium, the total wave energy is shared between a number of forms which include molecular vibration and structural energy. During the compression cycle of the longitudinal wave, mechanical potential energy is transferred to other forms. During the rarefaction of the medium the energy transfer reverses and the energy is returned to the wave. The energy transfer is referred to as a relaxation process.
The relaxation process takes a finite amount of time, known as the relaxation time (the inverse of which is known as the relaxation frequency). If the wave is at low frequency then the energy transfer can be completed. However, as the frequency increases, the energy transfer becomes out of phase with the wave, energy is lost and absorption occurs. The absorption increases with frequency reaching a maximum at the relaxation frequency. At frequencies above the relaxation frequency the absorption decreases as there is insufficient time for the initial energy transfer to take place.
FIG. 12a shows the variation of absorption with frequency for a single relaxation process.
If one considers two relaxation processes with different relaxation frequencies, one would find that, generally, the higher frequency process would cause greater absorption. This situation is depicted in FIG. 12b.
FIG. 12 Relaxation process
In biological materials there is a large number of different relaxation processes, each of which has a characteristic differing relaxation frequency. Therefore, the absorption characteristic of tissue increases approximately linearly with frequency and is attributable to the summation of absorption from a large number of relaxation processes.
7.7. Scattering
If a wave with wavelength h impinges upon a boundary whose dimensions are large compared to the wavelength, then specular reflection will occur. However, if the obstacle is smaller than the wavelength or of comparable size the laws of geometric reflection will not apply. In this instance, the wave is said to scattered using one of two different processes, Rayleigh and Stochastic.
1. The Rayleigh region is when the dimensions of the scattering object are very much less than the wavelength of the incident ultrasound. In the Rayleigh region incident ultrasound is scattered equally in all directions. The relationship determining the degree of scattering is the same as that derived for light. See, for example, Longhurst (1967). 4 Scattering oc ($) oc f4
2. If the dimensions of the scatterer are similar to the wavelength of the incident ultrasound then the scattering is stochastic. In this region there is a square law relationship between the degree of scattering and frequency.
The ratio of the incident ultrasonic intensity to the power scattered at a particular angle is known as the scattering cross section. If SI is the power of the scattered ultrasound and I, is the intensity of the incident ultrasound then a, the scattering cross section, is given by a=SI 11
In Doppler blood flow detection and in medical imaging the majority of the detected signal originates from scattered ultrasound. Therefore the variation of scattering with angle is of importance. The ratio of the intensity of the ultrasound scattered at a particular angle to the intensity of the incident ultrasound is the differential scattering cross section (the scattering cross section at a particular angle). Of most importance in medical imaging and Doppler blood flow studies is the scattering cross section at 180°, which corresponds to ultrasound transmitted directly back to the source as this determines the signal detected by the system.
7.8. Attenuation in Biological Tissues
The attenuation in biological materials has been measured both in vivo and in vitro. Tests are conducted at a given temperature, pressure and frequency. The standard values determined may find some clinical importance: for example, attenuation in tumor tissue is different from attenuation in breast tissue. However, attenuation by tissue is not at present used routinely in clinical situations. The attenuation of various tissues is represented in FIG. 13. These values are important when designing any ultrasound system as they determine the strength of the echoes received from a certain depth in either ultrasonic imaging or Doppler studies.
FIG. 13 Table of attenuation values (Duck 1990)
8. The Doppler Effect
8.1. Introduction
The Doppler effect was first derived in 1845 by the German physicist C.J. Doppler (1803-1853). He noted that there was a change in the detected frequency when a source of sound moved relative to an observer.
The Doppler effect will have been noticed by readers as the world we live in is full of examples of the slight change in the sound detected from a moving object. For example, when an ambulance with a siren or a motor bike passes, the note we hear is affected by the velocity of the source.
The sounds we hear are characterized by their frequencies. When a sound is emitted from a moving source the apparent frequency a stationary observer detects is affected. The apparent frequency will increase if the velocity of an emitter is positive, towards the detector, conversely the frequency will be lowered if the velocity is negative (the sign of the frequency shift is therefore dependent on the sign of the velocity). This is why the effect is most noticeable when the source passes us, as the velocity becomes negative and the Doppler shift suddenly changes from being positive to negative. The magnitude of the Doppler effect depends on the magnitude of the velocity.
The Doppler effect has been used for many years for military and commercial Radar allowing the velocity and the position of an airplane to be determined. In medicine, Doppler techniques have been substantially developed for blood flow studies enabling determination of blood flow velocity, detection of turbulence associated with pathological disturbances and the detection of fetal heart beats.
8.2. Derivation Of Doppler Equations
8.2.1. Stationary Detector Moving Source
FIG. 14 is a diagrammatic representation of the effect of the moving source. If the velocity is away from the detector then the apparent wavelength is increased. Conversely movement towards the detector shortens the apparent wavelength and increases the frequency.
Think of an object emitting sound moving directly away from an observer and at constant velocity. Then the apparent wavelength detected by the observer will be elongated by the distance that the source moves while that wave is being emitted.
FIG. 14 Moving source
the velocity of sound in the medium is c ms-'
the velocity of the source is v ms-'
the frequency emitted from the source is f Hz the wavelength of the emitted wave is h meters
the apparent wavelength of the detected wave is ha meters
The apparent wavelength is the distance travelled by the wave front in time At divided by the number of oscillations in time At.
This is the Doppler equation for a moving source, the sign of the denominator is positive for movement away from the detector and negative for movement towards the detector.
8.2.2. Special case for v<<c
If the velocity of the source v is small compared with the velocity of the wave in the medium, then equation (52) can be simplified by using a series expansion.
The series expansion of:
In Doppler analysis of blood flow v << c as the velocity of sound is approximately 1500 ms-' in most soft tissues and the blood flow velocity is in the range 0 to 5 ms-I. The Doppler shift is therefore directly proportional to the source frequency, the source velocity and inversely proportional to the speed of sound in the medium of interest.
8.2.3. Moving Detector and Stationary Source
Consider a stationary source emitting sound waves and a detector moving with velocity v in a straight line towards the source as depicted in FIG. 15. The wavelength in this situation stays constant and the apparent velocity increases causing the detector to cross a greater number of wave crests leading to a consequently increased frequency. Hence the frequency increases by the extra number of waves received in time At.
This is the same result as for a moving source.
FIG. 15 Moving detector
FIG. 16 Velocity vector
FIG. 17 Doppler flow measurement
When considering Doppler shifts caused by moving interfaces, the vector component of the velocity has to be considered. Therefore, only the component of the scattering object's velocity acting along the axis of the ultrasound wave is considered (see FIG. 16a). If a scatterer is moving at a angle $ to the ultrasound beam then the component of the velocity acting along the axis is given by vcos$ . Normally, the ultrasound probe consists of a transmitting transducer and a separate receiving transducer. These two transducers necessarily make different angles to the velocity vector of the scatterer (see FIG. 16b).
8.3. Doppler Blood Flow Studies
In Doppler blood flow studies ultrasound is scattered from a moving object back to a transmitting transducer as depicted in FIG. 17.
Ultrasound is transmitted from a stationary source and scattered back from a moving scatterer.
Therefore there are two Doppler shifts: The first shift occurs as the ultrasound strikes the moving scatterer, the situation is that of a stationary source and a moving detector.
The second frequency shift occurs as the ultrasound is scattered back, this situation is that of a stationary detector and a moving source.
This can perhaps be appreciated if you consider the radar detection of an airplane’s velocity.
If a radar signal is transmitted from the ground, the frequency detected by the plane's radar operator will be shifted due to the plane's velocity (the first Doppler shift). The radar signal reflected back to the ground will neither be that originally transmitted nor that detected on the plane but rather a further shifted signal (the second Doppler shift). Consider the situation where an object moves towards the ultrasound probe. The first Doppler shift will be given by equation (64).
FIG. 18 Doppler scattering angles
9. Generation and Detection of Ultrasound
Ultrasound is defined as acoustic vibration above the range of human hearing. As the frequency lies between 20 kHz and 20 MHz there is a variety of methods of generation and detection. At low frequencies ultrasound may be generated by specialized loud speakers and sirens. At higher frequencies however, these mechanisms become difficult, and piezo electric and magnetostrictive transducers are used.
In medical application with frequencies between 1 and 12 MHz piezo electric transducers are used for both generation and detection of ultrasound.
FIG. 19 The piezo electric effect
9.1. Piezo Electric Materials
When a piezo electric material is stressed a potential difference is produced. Conversely when a potential is applied a stress is produced within the material. These two phenomena are exploited respectively in the detection and generation of ultrasound. FIG. 19 demonstrates how a piezo electric material may be used to generate and detect ultrasound.
Piezo electric properties are due to the chemical morphology of the material. In a piezo electric material the constituent molecules are electrically polarized i.e. one end of the molecule is electrically negative and the other end positive. The polarization is due to the unequal sharing of electrons within the molecule. For example when fluorine combines with other elements to form a molecule the fluorine has a strong attraction for electrons and therefore is electro negative with respect to the rest of the molecule. Electrically polarized molecules are referred to as dipoles In naturally occurring materials the electrical dipoles are normally randomly arranged and the material therefore exhibits weak piezo electric properties.
9.1.1. Electro Mechanical Properties of Piezo Electric Elements
The characteristics of piezo electric materials are classified by a number of coefficients. The electromechanical coupling factor (k2) is the ratio of the mechanical energy converted to electrical energy to the total mechanical energy input to the material.
Conversely the factor may be defined as the ratio of electrical energy converted to mechanical energy to the total electrical energy input to the material.
...energy converted ktotal energy (73)
The ratio of the electrical potential developed by a piezo electric element in response to an applied mechanical stress is denoted by the coefficient g known as the piezo electric voltage constant.
potential produced = applied stress (74)
When a piezo electric element is subjected to a force a charge is developed, the coefficient d is defined as the ratio of the charge produced to the applied force and is referred to as the piezo electric charge constant. Alternatively d may be defined as the resultant deformation due to an applied potential.
charge produced applied force d= (75)
The value of g should be high for a piezo electric element used in a receiving mode. Generally it is desirable for transducer materials to have high values of k2, d and g.
A transducer element exhibits piezo electric behavior in three dimensions. The coefficients k2, d and g can be expressed in terms of the relationship between quantities applied in the same or different planes. For instance k2 is expressed for thickness mode vibrations as k2,0r k2,, where the subscripts refer to the 3rd plane of the element or the thickness. Alternatively k2,, refers to the coupling of energy between the third and first plane of the transducer element.
Table 1 Electro mechanical properties of piezo electric materials
The constants d, g and k2 vary with temperature. A material's piezo electric properties disappear at the Curie Point, and are not constant with time. In addition a piezo electric element may become depolarized by high stresses and high alternating fields.
9.1.2. Types of Piezo Electric Materials
There are naturally occurring piezo electric materials such as quartz which display piezo electric properties when cut along a particular plane to obtain domain alignment. Quartz has been used for many years as a piezo electric material in transducers. Some biological materials, such as bone, are weakly piezo electric. However, the majority of piezo electric materials used in ultrasonic transducers are manmade. There are three types of manmade piezo electric material: crystalline, ceramic and polymer.
Crystalline materials such lithium niobate are grown in solution and are, therefore, difficult to produce and shape by machining. Ceramic materials such as lead zirconium titanate (PZT) and Polymer materials such as polyvinyl diflouride (PVDF) are heated close to a material dependent temperature known as the Curie Point or Temperature. When heated above this temperature the piezo electric domains disappear, reappearing when the material temperature falls. If the material temperature is close to the Curie Point the domains can move and therefore a potential may be applied to attain domain alignment as the material is cooled.
Polymer materials are stretched to increase their piezo electric properties.
Ceramic materials have high piezo electric coefficients when compared to polymer materials.
However, the acoustic impedance of ceramic materials (approximately 30 Rayle) is an order of magnitude greater than that of soft body tissues (fat approximately 1.5 Rayle). Ceramic materials produce high mechanical and electrical responses when used as either generators or detectors of ultrasound. However they suffer from poor transmission of ultrasound through the tissue-transducer interface. Polymer materials have an acoustic impedance well matched to the acoustic impedance of body tissues but have low piezo electric coupling factors (k2). To produce a material with good piezo electric properties and low acoustic impedance some further transducer materials have been developed which are called copolymers. These consist of powdered piezo electric ceramic within a polymer material.
9.1.3. Transducer Properties
In medical applications transducer elements are formed from discs of the piezo electric material (FIG. 20). When a piezo electric element is excited it can vibrate in three dimensions. The required vibration for an ultrasonic transducer in a medical application is in thickness mode, that is vibrating along the x axis in FIG. 20. The disc is supported around its rim and polarized along the x axis.
When a transducer is excited ultrasound is emitted from both faces of the disc. Part of the energy arriving at the front face is reflected back into the transducer element. This reflected signal is itself partially reflected by the rear face and returned in the original direction of the wave. Some of the energy arriving at the front face will therefore have travelled a distance equal to twice the disc thickness. If this extra trip distance is equal to one wavelength, constructive interference takes place.
FIG. 20 Mounting crystals
The frequency which results from that wavelength is the natural or resonant frequency of the disc. The disc may resonate at odd multiples of this frequency giving rise to harmonics. There is therefore a number of resonant frequencies of a transducer disc. However, the strength of the resonance decreases with increasing frequency. The transducer element exhibits resonant behavior both when transmitting and receiving ultrasound.
The quality of resonance of any system can be described in terms of the width of its half energy amplitude (-3 dB) (see FIG. 21). This factor is termed the Quality or Q of the resonance and is defined as:
A high Q system is by definition of narrow bandwidth.
The resonance of a transducer element is restricted by losses in the material. Crystalline and ceramic materials have low internal losses and therefore resonate strongly. In contrast polymer materials have high losses and a weak resonance and, therefore, exhibit a broad frequency response with low Q.
FIG. 21 Q of resonance
9.2. Transducer Characteristics
Transducer characteristics can be damped (to reduce their Q) by backing the disc with an energy absorbing material with an acoustic impedance similar to that of the transducer element. In practice the Q of a transducer can be reduced most easily by backing the element with the same material as the active element but in an un-polarized form, or with tungsten loaded epoxy resin. The energy transmitted through the rear surface of the transducer is then dissipated in the backing. High Q transducers are constructed with air backing which deliberately provides poor coupling.
9.2.1. Transducer Fields
The acoustic field produced by a transducer depends on the designed frequency, the active element diameter and shape. The calculation of a transducer's acoustic field is achieved by modeling the surface of the piezo electric crystal as a series of infinitely small piston generators. The field pattern is then calculated by determining the interference between the waves generated by each piston. This method of calculating the transducer field is referred to as Huygen's Method.
The acoustic field produced by a transducer is complicated. Two forms of approximation are used to provide simplified working models. Close to the transducer, the Fresnel approximation provides the better estimates of the field, whereas at greater distances, the Fraunhofer approximation is employed. The intensity calculated using the Fresnel approximation in the vicinity of a transducer of diameter D is an oscillatory pattern over a region of approximately the transducer's dimensions. At depths of D2/4h the oscillatory pattern gives way to a region of uniformly divergent intensity. At a distance of xneX given by
xnear= D2/ h (77)
the half power width of the beam equals the transducer diameter. Whilst the Fresnel model is valid at all depths, the analysis required to obtain it is more complex, so at greater depths the Fraunhofer approximation is appropriate. In this case the width of the beam is defined by xUD (where x is the distance from the transducer face), which effectively limits the lateral resolution of the transducer. An analytic treatment of these models is provided by Macovski (1983).
FIG. 22 Field shapes
FIG. 22 shows the characteristic field shape produced by a disc transducer when transmitting continuously. The field can be considered as consisting of two regions, the near field and the far field. FIG. 22 shows these two regions. The limit of the near field can be defined as the distance from the transducer face when the 3 dB beam width is equal to the diameter of the transducer. The pulsed field distribution is represented schematically in FIG. 22. It differs from the continuous field as the pressure distribution is approximately uniform in the near field.
In medical applications transducers are designed to perform in the near field region which is up to the point where the field rapidly diverges. Therefore, the maximum depth of scan is effectively controlled by the degradation of the lateral resolution which is related to the disc diameter and operating frequency.
9.3. Basic Transducer Design
The basic design of an ultrasonic transducer is shown in FIG. 23.
The choice of piezo electric material was described in section 9.1.
The electrodes which connect to the transducer are thin layers of either gold or aluminum.
The layers are deposited by evaporating metallic wire in a vacuum. The piezo electric material is masked to ensure that the desired regions only are coated with the metal. The thickness of the layer is minimized to reduce interference with the emitted ultrasound.
The choice of backing material is dependent on the required use. For Doppler applications a mismatched backing allows the disc to resonate at its natural frequency: this maximizes the output power at this specific frequency. In this instance almost no ultrasound is transmitted through the backing so the majority of energy produced is transmitted usefully through the front face of the transducer. Doppler signals have a narrow bandwidth compared to the response of the transducers at high frequencies and so the transducer design can utilize a high Q resonant response.
FIG. 23 Transducer design
For imaging purposes the backing material is chosen to match the impedance of the transducer material. To avoid an excessive reflected pulse being returned by the rear face of the transducer and thus prolonging the emitted wave, the backing should provide high attenuation. The backing is loaded with tungsten powder.
4. Thickness mode vibrations at the desired frequency are obtained by choosing the element thickness to be half the wavelength at the resonant frequency.
5. For medical imaging purposes the object is required to be within the near field of the transducer since the field spreads after this point and therefore reduces the lateral resolution. Therefore having chosen the operating frequency and the required depth of field the disc diameter can be determined from equation (77).
6. The proportion of acoustic power transmitted from the active element to the tissue depends on the acoustic impedance match of the patient to the transducer element. Recalling the result for the transmission of sound through a thin layer in section 4.3, total transmission can be achieved if the impedance of a layer one quarter of a wavelength thick is chosen so that its acoustic impedance is equal to:
where Z,, Z, and Z, are the acoustic impedances of the transducer element, matching layer and patient respectively.
Therefore the acoustic impedance of a transducer element can be matched to that of the body and total transmission achieved. However, thickness of the layer will only equate to one quarter of the wavelength at a particular frequency. Therefore, although useful, matching layers tend to reinforce the narrow bandwidth characteristics of ceramic transducers.
7. The electrical impedance of the transducer element can be matched further to reduce the Q of the resonance and therefore increase the transducer bandwidth. This is achieved by using a shunt inductor to match the impedance of the transducer disc at its resonant frequency. The capacitance of the transducer disc can be determined from its thickness, the dielectric constant of the piezo electric material and the electrode area. The inductor required to match this at resonance is given by the following equation.
1 L= (f 24* c
where f is the resonant frequency of the transducer and C is the capacitance of the transducer.
The impedance of the transducer may be matched to that of the receiving amplifier by a transformer.
8. The transducer housing protects the relatively fragile transducer element and provides electrical screening. The housing may also be shaped so that reflections from the back surface are not directed back to the piezo electric element.