Biophysics BIOP3302 Module D - EMR 2008 (5.1) (5.2) is the initial incident intensity at the surface of the material.

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1 5 Ionising Radiation Attenuation of Ionising Radiation Ionising electromagnetic radiation incident on a material is attenuated by that material. The attenuation is caused by the scattering and interaction of each photon by the atoms, electrons and nuclei present in the material being irradiated. Given some initial intensity of incident radiation, the intensity of the unscattered radiation as a function of penetration depth x can be determined by considering that the fraction of photons attenuated at each depth is constant. In other words di dx = µ I (5.1) where µ is the linear coefficient of attenuation (units of per metre). Solving the differential equation in Equation 5.1 yields I µ x = I 0 e (5.) Radioactive Decay The differential equation given in Equation 5.1 to the left is a well known differential equation that also described radioactive decay. This equation has the solution of an exponential decay with position (attenuation) or time (radioactive decay). If the negative sign is removed then the solution is exponential growth with position or time. where I 0 is the initial incident intensity at the surface of the material. Figure 5.1 The intensity of radiation transmitted through a material having a linear attenuation coefficient of µ as a function of thickness. Sean Geoghegan Lectures 8 & 9

2 This function is shown in Figure 5.1. There are several factors which contribute to the linear attenuation coefficient. They are: 1) ω coherent scattering (Raleigh scattering), ) τ photoelectric effect, 3) σ Compton scattering, 4) κ pair production, and 5) π photodisintegration. The symbols given in the above list are the symbols used to indicate the various factors contributing to the linear attenuation coefficient where µ ω + τ + σ + κ + π =, (5.3) that is the various radiation transport mechanisms are assumed to be independent of each other. There are several ways of representing the attenuation coefficient for a material by scaling the linear attenuation coefficient by a relevant density. Typically three common densities are used, the mass density, the electron density and the atomic density. The mass attenuation coefficient is given by µ µ = ρ m cm /g (5.4) where ρ is the mass density. In this case the corresponding distance has units of g/cm. The electron attenuation coefficient is given by µ µ e = cm /electron (5.5) ρ e where ρ e is the electron density. In this case the corresponding distance has units of electrons/cm. The atomic attenuation coefficient is given by µ µ a = cm /atom (5.6) ρ where ρ a is the atomic density. In this case the corresponding distance has units of atoms/cm. Coherent Scattering (Raleigh Scattering) a In coherent scattering the photons are scattered or deflected with negligible loss in energy. The electron cloud of the atom or molecule acts coherently and little energy is deposited in the medium. This effect Red and Blue Skies The scattering of light by Raleigh Scattering (coherent scattering) is strongly dependent on wavelength with a dependence inversely proportional to wavelength to the fourth power, that is blue light is much more strongly scattered than red light. As such, during day time, the sky appears blue instead of the black of space unless we are looking towards the sun when the sky appears red when there is a large mass of air for the radiation to pass through (sunrise or sunset). Sean Geoghegan Lectures 8 & 9

3 can be neglected at high energies, however it does reduce resolution for low energy photons. Photodisintegration In photodisintegration the photons interact directly with the nuclei. Because photodisintegration involves the incident photon knocking out a neutron from the original nucleus, the product may be radioactive. This effect is only important if the photon has a very high energy (> 10 MeV) with the exception of the interaction with beryllium-9 which has a threshold of 1.65 MeV for conversion to beryllium-8. Nuclear Fusion Nuclear fusion, in which the nuclei of atoms fuse together and release energy, is effectively the reverse of photodisintegration. 9 ν (5.7) 8 Be + h Be + n Because of the very high photon energies usually involved, photodisintegration can be neglected in the majority of cases. Photoelectric Effect The photoelectric effect, illustrated in Figure 5., is the excitation of an electron to such a high energy that it escapes that atom or molecule to which it was bound. The kinetic energy of the ejected electron depends on the workfunction of the atom or molecule as given by Equation 5.8 below: K max = hν φ (5.8) where φ is the workfunction (the energy required by the electron to leave the atom or molecule). If the electron is ejected from the inner core of the atom then a cascade effect can result, as shown in Figure 5.3. Not all the photon energy needs to be absorbed by the ejected electron and only a small fraction of the incident photon energy need be lost by the photon in this interaction. The contribution to the mass attenuation coefficient by the photoelectric process is proportional to the cube of the atomic number of the scattering atom and is inversely proportional to the cube of the energy of the incident photon (as summarised in Equations 5.9 and 5.10), ie. and τ 1 ( hν ) 3 m (5.9) τ z 3 m. (5.10) Solar Cells A variant of the photoelectric effect is used in solar cells in that the energy gained by the photoelectron is captured in the material and used to drive a current. This works at sufficiently low enough photon energies around the optical and ultraviolet parts of the electromagnetic spectrum. At higher photon energies the photoelectron has sufficient energy to escape the material and carry the energy away from the device meaning that no useful current is generated unless the photoelectron is captured by some other method. Even if this is achieved, large portion of the energy is lost as heat and the device is subject to radiation damage. Sean Geoghegan Lectures 8 & 9

4 Figure 5. Basic illustration of the photoelectric effect when a photon ejects an electron from matter. Figure 5.3 Examples of possible cascades resulting from the photoelectric ejection of an inner core electron. Sean Geoghegan Lectures 8 & 9

5 Figure 5.4 The angular distribution of intensity of the photoelectric electron following the photoelectric effect involving photons with energies specified in the diagram (reproduction of Figure 7-6 from Hendee, Medical Radiation Physics, 1970). Sean Geoghegan Lectures 8 & 9

6 Because of the photon energy dependence of the photoelectric effect cross-section, the photoelectric effect dominates at low energies. The angular distribution of the intensity of the ejected photoelectron depends on the initial energy of the photon, as shown in Figure 5.4. Exercise 41 What is the maximum kinetic energy of a photo electron ejected from the K shell of lead ( φ = 88 kev) by photoelectric absorption of a 100 kev photon. Compton Scattering At higher energies Compton scattering starts to dominate. The Compton effect is the scattering of an incident photon by free electrons. In order to treat Compton scattering accurately, the quantum nature of the photon needs to be taken into account, particularly the fact that each photon has momentum and can transfer this momentum to an electron. Both the classical and quantum mechanical models of the Compton effect are illustrated in Figure 5.5 showing the differences in the descriptions of the Compton effect. Note that the momentum transfer to the electron in the quantum mechanical model is correct. Radiobiology Compton scattering is the predominant mechanism of the transfer of energy from x-rays and gamma rays to biological materials and, as such, is important in radiobiology. By applying conservation of energy and momentum to the photon and electron before and after the interaction, Compton found that Figure 5.5 Illustration of the classical and quantum mechanical models of Compton scattering. Sean Geoghegan Lectures 8 & 9

7 h λ λ0 = 1 mc ( cosθ ), (5.11) ie the photon wavelength shift depends only on the scattering angle. The factor Compton wavelength h mc has units of wavelength, and is known as the c h = = mc λ nm. (5.1) In order to derive Equation 5.11 we need to write down the equations for the conservation of energy and the conservation of x and y components of momentum. The conservation of energy equation is hc hc = + KE λ λ0 (5.13) which, allowing for relativistic effects, becomes hc λ 0 hc = + γ λ ( 1) mc. (5.14) The conservation of momentum in the x direction is described by h λ 0 h = cosθ + γmv cosφ λ. (5.15) and the conservation of the y component of momentum is described by h = sinθ + γmvsinφ λ 0. (5.16) Eliminating v and φ from these equations (Equations 5.14 to 5.16) yields Equation Because Compton scattering is caused by the interaction between the photons and free electrons, the electronic linear attenuation coefficient is mainly dependent on the electronic density of the material through which the photons pass. There is a slight dependence on the photon energy, however materials with the same electronic density have the same linear attenuation coefficient as each other. There is no dependence on atomic number. The angular distribution of the intensity of the scattered photon and electron depends on the initial energy of the photon, as shown in Figure 5.6. Exercise 4 Show that, irrespective of the energy of the incident photon, the maximum energy of the photon is 55 kev for a photon scattered at 180º and 511 kev for a photon scattered at 90º. Sean Geoghegan Lectures 8 & 9

8 Figure 5.6 The angular distributions of intensities of the Compton scattered photon and electron for the incident photons having energies shown in the diagram (reproduction of Figure 7-9 from Hendee, Medical Radiation Physics, 1970). Sean Geoghegan Lectures 8 & 9

9 Figure 5.7 Illustration of pair and triplet production. Pair Production Pair production requires the photon to have a minimum energy of twice the rest mass energy of an electron/positron which, at MeV each, means that a minimum photon energy of 1.0 MeV is required. The excess energy of the photon is distributed between the produced electrons and positrons as well as a nearby object (typically an atom) which provides the momentum. This nearby object is required for pair production to take place in order for momentum to be conserved. Because an atom has a much larger mass than the combined masses of the positron and electron the following equation is a very accurate description of the energy distribution for pair production near an atom: h 1. 0 KE e KE ν MeV. (5.17) = e Occasionally pair production occurs near an electron and the distribution of the energy between the electron and the produced electron and positron is not described by Equation In this instance all three particles have a large fraction of the excess photon energy resulting in three high energy leptons. This case is referred to as triplet production. Because this event is relatively rare, the major influence on the probability of pair production occurring is the density of massive objects with which momentum can be shared and, as such, the attenuation coefficient for pair production is proportional to the square of the atomic number and proportional to the logarithm of photon energy, i.e. Bhabha Scattering When a positron and electron interact, there is not always a case of mutual annihilation and it is possible for the particles to scatter off each other by the mediation of a virtual photon. This scattering of an electron off a positron (or vice versa) is known as Bhabha Scattering. This is a quantum mechanical process described in quantum electrodynamics. Sean Geoghegan Lectures 8 & 9

10 κ z, and (5.18) ( ) κ ln hν. (5.19) Figure 5.8 The dominant scattering effect as a function of both atomic number and energy of the incident photon (reproduction of Figure 7-13 from Hendee, Medical Radiation Physics, 1970). 80 kev 300 kev 1.5 MeV Figure 5.9 Illustration of the effect of competing scattering effects from x-ray films taken at different photon energies (reproduction of Figure 7-11 from Hendee, Medical Radiation Physics, 1970). Sean Geoghegan Lectures 8 & 9

11 Exercise 43 A 5.0 MeV photon near a nucleus interacts by pair production. Residual energy is shared equally between the electron and positron. What are the kinetic energies of these particles? Exercise 44 Prove that, regardless of the energy of the incident photon, a photon scattered at an angel of 60º or greater during a Compton scattering event cannot undergo pair production. Production of X-Rays Up until this stage we have dealt with high energy photons eventually resulting in high energy electrons through the photo electric effect, Compton scattering and pair production. X-rays are generally produced by decelerating high energy electrons (Bremsstrahlung radiation) in a material. This is used in an x-ray tube by accelerating electrons into a high atomic mass target (eg. copper) as illustrated in Figure It is evident that high energy electrons produce high energy photons the energy of which is eventually absorbed through the photoelectric effect. The end result of all of this is that free electrons are created in the material together with ionised material which are highly reactive species. Figure 5.10 Sketch of an x-ray generating circuit. Sean Geoghegan Lectures 8 & 9

12 Electron Scattering High energy electrons mainly interact with only two parts of matter, ie. with the nuclei or with the electrons in a material. Such interactions can either be elastic or inelastic which thus results in a total of four phenomena that we need to consider. However, because we are mainly interested in the deposition of energy into the medium, only the inelastic interactions need be considered in detail. The important effect of elastic interactions is that it leads to beam broadening and we shall treat this effect qualitatively. Elastic Scattering of Electrons by Electrons The incident electron is deflected with little change in energy, the probability of which is proportional to the atomic number. This is analogous to Raleigh scattering in the interaction of photons with matter. prob z (5.0) Elastic Scattering of Electrons by Nuclei Møller Scattering The scattering of an electron off another electron is called Møller Scattering. This is a quantum mechanical process described by Quantum Field Theory. The incident electron is deflected with little change in energy, similar to Raleigh scattering. The probability of this occurring is proportional to the atomic number squared and inversely proportional to the kinetic energy of the incident electron squared. z prob (5.1) E k Inelastic Scattering of Electrons by Electrons Collisional Process The incident electron transfers some of its energy to the electrons in the medium. An electron receiving energy can be ejected from the atom with kinetic energy given by. E E = (5.) k E b where E is the energy lost by the incident electron and E b is the binding energy of the target electron to the atom. The probability of this type of interaction increases with atomic number of the absorber and decreases with incident kinetic energy. The actual form of this will be discussed later in the discussion on stopping powers. Inelastic Scattering of Electrons by Nuclei Radiative Process The incident electron is deflected with a reduced velocity which implies that deceleration has taken place. Because the electron has an associated electric field, the deceleration causes radiation to be emitted. Sean Geoghegan Lectures 8 & 9

13 Figure 5.11 Illustration of how brehmsstrahlung is generated as a high energy electron passes close to a nucleus of an atom. The electron loses kinetic energy to the emitted radiation due to the deceleration of the electron by the nucleus of the atom. Figure 5.1 Angular dependence of intensity of bremsstrahlung for incident electrons with kinetic energies shown (reproduction of Figure 3-3 from Hendee, Medical Radiation Physics, 1970). Sean Geoghegan Lectures 8 & 9

14 Figure 5.13 Bremsstrahlung spectrum for electrons with a kinetic energy of 0 kev incident on a molybdenum target (reproduction of Figure 3-4 from Hendee, Medical Radiation Physics, 1970). The radiated energy is known as bremsstrahlung (braking radiation). A bremsstrahlung photon may possess any energy up to the kinetic energy of the incident electron. The probability of this interaction taking place is proportional to the square of the atomic number of the absorber, ie. prob z. (5.3) The ratio of the probabilities of the radiative process and the collisional process is approximately given by Equation 5.4 below ( loss) ( collisional loss) ( z + 1 ) prob radiative E. prob 800 where the energy of the incident electron is given in MeV. Electron Stopping Powers (5.4) In a similar manner to the definition of the attenuation coefficients used in the scattering of photons, an energy absorption coefficient is used to describe the interaction of high energy electrons with matter. The electron stopping power is defined to represent the energy transferred from the electron to the medium by collisional and radiative processes. Shielding Electrons High energy electrons that interact with high atomic number materials produce higher energy photons via bremsstrahlung than when low atomic number materials are used. This leads to the somewhat counterintuitive result that shielding against high energy electrons with lead is dangerous and plastics or woods are safer materials to use. This is because the high energy photons produced when using lead penetrate the lead more easily than the low energy photons produced in plastic or wood can penetrate the plastic or wood radiation shields. Sean Geoghegan Lectures 8 & 9

15 Figure 5.14 The energy loss along a photon radiation track with a magnified view of one of the resulting electron radiation tracks which includes the delta ray side branches to the secondary electron track. The quantification of this concept is though defining the stopping power to be the rate of energy loss per unit length, ie. S E x =. (5.5) This concept, together with that for attenuation, is illustrated in Figure In the case of electrons, a more useful representation is the mass stopping power which is defined to be the stopping power divided by the density of the medium. Separating the mass stopping power into the two energy loss processes, we have S ρ tot S = ρ Collision Stopping Power col S + ρ rad. (5.6) The collision stopping power depends on energy and atomic number in a complicated manner. The equation describing this dependence is A Power is a Force It is unfortunate that the term stopping power has been chosen to label the ability of a material to remove energy from energetic particles. The misnomer is understandable given that power is technically the rate of change of energy with time, whereas stopping power is the rate of change of energy with distance. Nevertheless, the true units of stopping power, being J/m in SI, are equivalent to those of force, N in SI. S ρ col πr mc N = β A A z ( τ ) τ + ( I mc ) + F ( τ ) δ ln (5.7) where Sean Geoghegan Lectures 8 & 9

16 v = c β, (5.8) mc = γ τ, (5.9) ( + 1) mc ( τ ) = ( 1 β )( 1+ τ 8 ( τ + 1) ln ) F, (5.30) I is the mean excitation energy and δ is the density effect correction term. Radiative Stopping Power The radiative stopping power dependencies on atomic number and energy are simpler than that for the collisional stopping power. The equation describing the radiative stopping power dependencies on atomic number and energy is S ρ rad = 4N A z ( z + 1) A r E z (5.31) Effects of Ionising Radiation on Humans Absorbed Dose The absorbed dose, D, is measured in Gy (J/kg) and is the measure of energy deposited in the tissue of the human. The absorbed dose determine the acute effects of ionising radiation on the target. Because different high energy ions, electrons, and photons have different biological effects, the absorbed dose needs to be converted to a biological equivalent dose to determine the probability of long term effects. Exercise 45 What is the temperature rise is water exposed to 1 Gy of ionising radiation? Biological Equivalent Dose The biological equivalent dose, H, is measured in sieverts (Sv) where the number of sieverts is related to the absorbed dose by H = wd Sv (5.3) where w is the weighting factor and D is the absorbed dose in Gy. The weighting factor is unitless, that is the unit of biological equivalent dose is J/kg (the same as the Gy). Another term for the weighting factor Random Effects The biological effects of radiation are essentially based on random effects: the probability of dose being deposited in a cell in an organ, the probability of cancer or cell death resulting and the probability of organ failure. These random events combine to create two types of effects: Stochastic and Deterministic. Stochastic effects include cancer and, for a given dose, have a certain probability (not unity) of occurring. Deterministic effects occur after a certain dose threshold is reached (with a probability of unity) and either are early (that is acute within days to a few months) or late (up to years later). Sean Geoghegan Lectures 8 & 9

17 is the relative biological effect (RBE). A list of weighting factors is given in Table 5.1. Effective Dose For radiation inhomogeneously distributed through the body, the biological equivalent dose is multiplied by the tissue weighting factor to obtain the effective dose, E. This is described by where E = w H Sv (5.33) T w T is the tissue weighting factor and H the equivalent dose in Sv. A list of tissue weighting factors is given in Table 5.. Low Doses The long term survival of a human whose whole body has been exposed to absorbed doses less than 1 Gy is almost guaranteed, that is, no critical organs or biological functions have been sufficiently impaired to cause failure of a life preserving function. However, short term effects (also known as deterministic effects) such as nausea, vomiting, lethargy, confusion and anxiety, among others, may result from an acute exposure to this level of ionising radiation. Deterministic Effects Skin reddening, hair loss, organ failure, cataracts, infertility, skin necrosis, paralysation and death are deterministic effects (guaranteed to occur) once an individual or an organ is exposed to a dose exceeding some threshold (e.g. 45 Gy for the spinal cord when given over about 6 weeks). Because of the underlying random nature of dose deposition and because each person has a slightly different radiosensitivity there is a narrow range of doses about which the probability of a deterministic effect increases from zero to unity. Radiation Weighting Factor Photons (all energies) 1 Electrons (all energies) 1 Neutrons (< 10 kev) (10 kev to 100 kev) (100 kev to MeV) ( MeV to 0 MeV) (> 0 MeV) Protons (> MeV) ~5 ~10 ~0 ~10 ~5 ~5 Alpha particles (all energies) ~0 Fission fragments, ions (all energies) ~0 Table 5.1 Weighting factors for different ionising radiation types and energies. Sean Geoghegan Lectures 8 & 9

18 Organ Tissue Weighting Factor Gonads 0.0 Bone marrow (red) 0.1 Colon 0.1 Lung 0.1 Stomach 0.1 Bladder 0.05 Breast 0.05 Liver 0.05 Oesophagus 0.05 Thyroid 0.05 Skin 0.01 Bone surface 0.01 Remainder 0.05 Table 5. Tissue weighting factors. The long term effects (also known as stochastic effects) of exposure to ionising radiation is an increase in the rate of carcinogenesis, the spontaneous generation of a cancerous cell in the body of exposed person, (both fatal and non-fatal) and severe hereditary effects. These typically manifest decades after the exposure event and the total probability has an approximate value of 5.6% per Sv per lifetime for workers and 7.3% per Sv per lifetime for the population as a whole (including children). Radiation Syndromes At absorbed doses greater than 1 Gy, deterministic effects start to become evident. It appears that the deterministic effects are proportionate to the absorbed dose rather than the effective dose. For whole body absorbed doses from 1 to 6 Gy the bone marrow syndrome occurs, for doses from 6 to 10 Gy the gastrointestinal syndrome predominates and for doses greater than 10 Gy the central nervous system syndrome determines the outcome of the exposure. Bone Marrow Syndrome At absorbed acute doses greater than 1 Gy the haematopoietic stem cells (the precursors of red blood cells, white blood cells and platelets) in the bone marrow start to be destroyed. If enough are destroyed, eg. for Stochastic Effects At low doses, where deterministic effects are not expected to occur, the effects of ionising radiation on biological systems can take years to become apparent. These time scales are longer than late deterministic effects (such as paralysis due to necrosis of the spinal cord over a year or two following exposure), and are in the form of cancer and severe hereditary defects. These effects occur with a probability approximately proportional to the total dose received by the organism. This approximation of proportionality is conservative and used for protection. Sean Geoghegan Lectures 8 & 9

19 a whole body exposure to a large enough dose, then there are not enough replacement red blood cells to support life and the casualty, without medical aid, dies within 30 days. Even though no white blood cells are produced, the life time of the white blood cells is approximately 5 years which means that opportunistic infections are not usually the cause of death. Anaemia is usually the cause of death. For doses from1 to Gy survival is probable but for does from to 6 Gy survival is increasingly unlikely. The dose required to kill 50% of humans within 30 days is somewhere between 3 and 5 Gy, typically quoted to be 4.5 Gy. Gastrointestinal Syndrome For absorbed doses greater than 6 Gy, sufficient damage is caused to the crypt cells in the jejunum (which is naturally replaced every three days) that after 3 days whole sections of the jejunum are destroyed. This results in severe infections that, without medical aid, result in a rapid death of the casualty. Death occurs within 10 days. Central Nervous System Syndrome For absorbed doses greater than 10 Gy the central nervous system is severely damaged and the casualty falls into a coma very quickly (after having experienced great pain, nausea, vomiting, distress, anxiety and confusion). Without medical intervention death occurs within 36 hours and even with the most advanced medical intervention death is assured, although death can be delayed for weeks or even months with the patient in a comatose state. Cancer Treatment The doses used to treat cancerous tumours using radiotherapy are typically of the order to 70 Gy delivered over 35 days (7 weeks) at Gy per fraction (a single treatment on a particular day). This dose is delivered to the tumour volume and some surrounding local tissue, ie. the dose is targeted to the tumour. In an ideal practical delivery a small volume of healthy tissue receives at most 95% of the treatment dose delivered to the tumour volume, the vast majority receiving less than 10% of the treatment dose. In most cases less than ideal treatments are often given (mainly due to limitations of technology and the nature of dose prescription methods) and a very small volume of healthy tissue may receive up to 130% (in extreme cases) of the prescribed dose. As can be seen, these large doses of radiation far exceed the doses (if given to the whole body) that would kill a person. Exercise 46 With regard to the answer to Exercise 45, are the biological effects of ionising radiation due to thermal or athermal mechanisms? Briefly describe the mechanisms by which ionising radiation induces change in biological systems. Sean Geoghegan Lectures 8 & 9

20 Radiobiological Models In order to describe the deterministic effects of ionising radiation of target cells, a model of the radiobiology of cells has been proposed. It has been suggested that the effects of ionising radiation are purely stochastic (even the deterministic effects), however, that there is a dose range over which the evident random nature of ionising radiation makes a transition to a region in which the probability of a group of cells being completely killed is so close to unity as to be taken to be unity (ie. deterministic). When this occurs the deterministic effects become evident, whereas the random effects (such as cancer induction and hereditary effects) will only become evident with time. A simple model of radiobiology is the single hit model. In this model a cell is said to dies if hit once. This can be described mathematically using dn dd = αn (5.34) where α is the rate at which cells are killed per unit dose (units of per Gy). The solution to Equation 5.34 is N αd = N 0 e (5.35) where N 0 is the initial number of cells. Dividing equation 5.35 by N 0 calculates the survival, ie. S αd = e (5.36) Plotting Equation 5.36 on a semi-log plot ( ln S as a function of D ) yields a straight line, the slope of which is α as shown in Figure Mammalian cells are characterised by a shoulder region at low doses, that is higher doses are required before an initial kill can be made. This is assumed to be due to the requirement for multiple hits to kill each cell as well as due to the repair Sean Geoghegan Lectures 8 & 9

21 Figure 5.15 Illustration of the survival as a function of dose for the single hit model. Figure 5.16 Illustration of the survival fraction for two different model cell types as a function of dose for the linear-quadratic model showing the shoulder typically found in mammalian cells: (i) fast proliferating cells with α = Gy -1 and β = Gy - ie. α β = 10 and (ii) slowly proliferating cells with α = 0.15 Gy -1 and = α β. β Gy - ie. = 3 Sean Geoghegan Lectures 8 & 9

22 mechanisms within the cells. An equation describing this behaviour is αd βd S = e (5.37) where the quadratic term β has been introduced to model these effects. This model is called the linear-quadratic model and is currently used to describe the behaviour of healthy tissue and cancer tumours in radiotherapy. A survival curve for the linear-quadratic model is shown in Figure 5.16 in which the typical mammalian survival shoulder is shown. The alpha-beta ratio has units of dose and is tissue type dependent. Fast reacting tissues have large alpha-beta ratios, typically in the range from 10 to 0 Gy, whereas late responding tissues have alpha-beta ratios in the range from to 5 Gy. When determining the effect of ionising radiation on tissues, the alpha-beta ratio determines which tissues are affected and at which time the effect manifest. Early effects occur within hours to days of exposure, depending on the replication rate of the cells. Late effects can occur up to several years following exposure to ionising radiation, but typically occur within several months of exposure. In treatment of cancers, the tumour is an early responder whereas sensitive organs (eg. the spinal cord) are often late responders (skin, bone marrow and intestines being some of the exceptions). Tumour cells typically have a larger value for α than normal tissues and the survival curves of tumour cells and normal tissues are maximally separated at a dose of about Gy (clinically the range 1.8 Gy to. Gy is relevant), meaning that significantly more tumour cells than normal cells are killed if a dose of around Gy is given. This leads to the use of fractionation in many radiotherapy treatments for cancer. Shown in Figure 5.16 are the survival curves for two model cell lines which are maximally separated at around 6.6 Gy, however a fraction dose of 6.6 Gy is not often given because of the large detriment to normal tissue. An exception is in brachytherapy where the dose fall off with distance from the brachytherapy sources means that normal tissues receive much less dose than the tumour. Both late and early effects are deterministic having a threshold dose above which the effect (eg. erythema, tissue necrosis, organ failure) will occur. Stochastic effects are randomly distributed within a population of exposed individuals and occur randomly within several decades of exposure. Sean Geoghegan Lectures 8 & 9