Radiat Environ Biophys (1992) 31:161-180

Radiation and Environmental Biophysics © Springer-Verlag

1992

LET, track structure and models* A review G. Kraft 1, M. Kriimer 1 and M. Scholz 1 1 Gesellschafl fiir Schwerionenforschung mbH, Planckstrasse 1, W-6100 Darmstadt, Federal Republic of Germany Received January 8, 1992 / Accepted in revised form February 27, 1992

Summary. Swift heavy ions when penetrating through matter strip off those electrons having a smaller orbital velocity than the ion velocity. The remaining electrons screen the nuclear charge yielding an effective charge. The effective charge of the ions interacts predominately with the target electrons causing excitation and ionizations of the target atoms. Using the Bethe Bloch formula for the energy loss combined with the Barkas formula for effective charge, the energy loss values as well as unrestricted and restricted linear transfer can be calculated within a few percent of accurancy. From the primary energy loss only a small fraction of 10% or less is transformed into excitation. The major part of the energy loss is used for the ionization of the target atoms and the emission of the corresponding electrons with a high kinetic energy. These electrons form the track around the trajectory of the primary ion in which two thirds of the primary energy is deposited by collisions of primary, secondary and later generations of electrons with the target molecules. In the electron diffusion process the energy is transported from the center of the track into the halo. The radial dose decreases with the square of the radial distance from the center. The diameter of the track is determined by the maximum range of the emitted electrons, i.e. by the maximum energy electrons. All ions having the same velocity i.e. the same specific energy produce electrons of the same energy and therefore tracks of the same diameters independent of the effective charge. But the dose inside the track increases with the square of the effective charge. Track structure models using this continuous dose distributions produce a better agreement with the experiment than models based on microdosimetry. The critical volume as used in microdosimetry is too large compared to the size of the DNA as critical structure inside the biological objects. Track structure models yield better results because the gross-structure of the track i.e. its lateral extension and the thin down toward the end of the track is included in these calculations. In a recent refinement the * Invited paper given on the fourth workshop on "Heavy Charged Particles in Biology and Medicine" GSI, Darmstadt, FRG, September 23--25, 1991

162 repair capacity of the cell has been included in a track structure model by using the complete shouldered x-ray survival curve as a template for the local damage produced by the particle tracks. This improved model yields presently the best agreement with the experiment.

I. Introduction

Heavy particles like protons or heavier ions are different in their biological efficiency compared to sparsely ionizing radiation like X or gamma rays and electrons. These differences have been found at all levels of biological organization and have been attributed to the different "quality" of radiation. From early cloud chamber photographs the dense tracks of heavy particles were known and compared to the more diffuse pattern of ionisations of X or gamma rays which are produced via Compton or photo-electrons and at higher energies by pair production. In order to differentiate between the types of radiation, radioprotection introduced as a quality factor the relative biological efficiency (RBE) which was attributed solely to the energy loss of the primary particle. The concept of quality factor is widely accepted in radiation research and is based on two assumptions : 1. The radiation effects caused by densely ionizing radiation are the same in their nature as those caused by sparsely ionizing radiation. But the energy deposition i.e. the dose necessary to produce these effects is different. 2. The quality of radiation can be characterized by the energy loss of the projectile or by the linear energy transfer (LET) to the biological object. The first assumption, the independence of the nature of the biological lesion from radiation quality was never a subject of a critical review because many of the test systems used in radiobiology as for instance the most frequently used survival test are not sensitive to these differences. In contrast, the second assumption the dependence of RBE from one single parameter like LET was always a subject of intense research and large controversies. From cloud chamber photographs as well as from tracks in nuclear emulsion it was obvious that the particle tracks have a lateral extension and an inhomogeneous dose distribution inside the track. In addition, many biological experiments confirmed that LET is not a good parameter for the description of the biological effects. Basically two different approaches have been used for a more detailed description of the changes in biological response to heavy charged particles: In microdosimetry, small volumes of tissue having a diameter of 1 gm are simulated with gasfilled counters and the distribution of the energy deposition in these volumes was regarded to be the important parameter for the biological response (Zaider and Rossi 1986). The microdosimetric approach clearly tries to take into account the inhomogeneous dose distribution inside the track as produced by the primary ionisation of the projectile and by the emerging electrons. In a second approach, the track structure calculation of Katz et al. (1971), the grainy pattern of the energy deposition by the electrons is not included at all. The dose is assumed to decrease continuously with the square of the radial distance from the center of the track. In a track structure model as developed by Katz et al. (1985) good approximation to the inactivation measure-

163 merits of many different objects as well as for the LET dependence of other biological endpoints has been achieved. In some other track structure calculations, the existence of a "track core" in the very center of the track is assumed in which one half of the energy loss of the primary particle should be localized (Maggee and Chatterjee 1979). In detailed Monte Carlo calculations in which the primary ionisation and 1 the cascade of liberated electrons is followed up individually, the r~ law of the radial dose distribution is confirmed but there is no evidence of an elevated energy deposition in the center of the track (Kr/imer and Kraft 1991). The same is true for microdosimetric measurements of the radial ionisation distribution (Metting et al. 1988). In addition, measurements of the angular and energy distribution of the emitted electrons confirmed the previous assumption of Bethe (1930) that the overwhelming part of the energy loss of the projectiles is transformed into kinetic energy of the emitted electrons (Schmidt et al. 1988). Therefore no energy is left over for the formation of a track core. In the following paragraphs, the basic formulas governing the energy loss of the primary particle and the resulting LET curves as function of the particle energies are discussed. The process of electron liberation in the ion atom collisions is given in greater detail. The radial dose profiles as calculated in Monte Carlo calculations and in the track structure models are compared with microdosimetric measurements. Finally the basic ideas of a new model are presented in which the radial dose distribution in a particle track is combined with the inactivation probability as given by the dose effect curves of x-ray experiments. This model yields presently the best agreement with the measured inactivation cross section.

II. Energy loss and LET The energy loss of heavy charged particles is mainly governed by the particle velocity. The velocity determines firstly the charge state and secondly the interaction mechanism between projectile and target atom.

Effective charge If the velocity of the projectile exceeds the orbital velocity of its own electrons, these electrons are stripped off in the first few collisions and only the stronger bound electrons of the inner shells stay with the projectile. When the projectiles are slowed down, more and more electrons are captured and the charge state of the projectile is reduced until it comes to rest and is neutralized. The dependence of the projectile charge from the velocity has been approximated by Barkas and Berger (1964) with the following expression Z~ff=Z[1 - - e x p ( -

125flZ-2/3)]

(1)

where Z is the atomic number of the projectile and /~ the velocity relative to that of light. In Fig. 1 the effective charge Z~ff is given as a function of the specific ion energy of the projectile. It should be noticed that the Zeff values

164 100u

.

.

.

... . .. . . .

.

I

.

.

.

I

. . . . . . . U

Zeff 10

1

O.I

I

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IO0

Spec. Energy [MeV/u] Fig. l. The effective charge of heavy charged particles according to Barkas et al. (1963) is given as function of the specific ion energy of the particles. Light ions are fully stripped at energies around 10 MeV/u and the effective charge equals the atomic number. For the very heavy ions like uranium, energies significantly greater than 100 MeV/u are necessary to remove the last electrons

of Barkas represent the mean value of a broad charge state distribution. This mean value as well as the width of this distribution differs between gas and solid absorbers when compared at the same velocity. Due to the high collision frequency in a solid, more electrons from the inner shells are ' p u m p e d ' by multiple collision into highly excited states from which ionisation is produced more easily (Kraft 1977).

Nuclear stopping At low projectile velocities corresponding to particle energies of 10 keV/u or lower, the nuclear stopping dominates the stopping process. The nuclear stopping represents the scattering of the projectile at the screened Coulomb potential of the target atom. The nuclear stopping process is characterized by a large energy and angular momentum transfer and has been treated by Schiott (1966) in detail. Nuclear stopping has a large biological efficiency because the target atoms are displaced directly and a collision cascade can be produced. In track segment experiments in the regime of nuclear stopping, cross sections larger than the geometrical cross section of the biological target have been reported (Schneider et al. 1990). However, the nuclear stopping process is only of importance at extremly low energies. At higher energies, nuclear stopping contributes a few percent to the total stopping cross sections. Therefore, it will not be discussed here in greater detail.

165

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Kr Ar Ne

0.1 C

0.0t

He i

0.1

i

,

.....

I

i

,

1

i

i i ,,,r

. . . . . . . .

I

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. . . . . . . .

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, ....

1000

Fig. 2. Comparison of energy loss values of various projectiles in carbon. Solid line: Heinrich et al. (1991). Dashed line: Northcliffe and Shilling (1970). Crosses: Hubert et al. (1990). Points: Experimental data H. Geissel et al. (1983)

Electronic stopping

At energies of a few MeV/u and higher; which are more relevant for radiobiology, the major part of the energy loss of the primary ions is due to the interaction with the target electrons which is called electronic stopping. In 1913, Bohr calculated the electronic stopping power in a classical approach (Bohr 1948) using two extreme situations: Direct hits ("knock o n " collisions) with small impact parameters between projectile and electrons which have a large energy transfer. The other extreme are large impact parameters, (" glancing collisions") which are connected to small energy transfers only. The original formula of Bohr is very simular to the Bethe Bloch formula (see below) except that the logarithmic term contains a Z1 contribution in contrast to the Bethe Bloch formula. It was one of the main motivations of Bethe (1930) to use quantum mechanics to abandon this artificial separation between large and small impact parameters. In first Born approximation, Bethe obtained for the stopping power: dE

4~e 4 Z~ N Z 2

dX-

me v2

2my 2

In ~

( + relativistic terms)

(2)

where e and me are the charge and the mass of the electrons respectively, Z1 and Z2 the atomic numbers of the projectiles and target atoms, N. Zz the density of the target electrons, v the velocity of the projectiles and I a mean ionization potential. For this ionisation potential Bloch (1933) calculated: I = I o Z2 with Io ~ 10eV. This formula of the electronic energy loss has been derived for hydrogen projectiles. For the application to heavier ions, the projectile atomic number

166 has to be replaced by the effective charge as given by the Barkas formula (1). The values of the Bethe Bloch formula are usually calculated numerically including some density and relativistic corrections and are given in tables by various authors for practical use. In Fig. 2 the most reliable tables of energy loss values are compared with a few existing measurements. In the range between 10 and 100 MeV, there is a good agreement between recent calculations of Huber et al. (1990) and Heinrich et al. (1991) with the experimental data of Geissel et al. (1983) but not with the older data of Northcliffe and Shilling (1970). At the higher energies, when all electrons are nearly stripped off from the projectile a good agreement exists between all calculations. But for this energy range systematic experiments do not exist which could confirm these calculations. Frequently the question was raised, whether the inner, more dense part of a track would act differently than the outer more diffuse part and whether the observed high-LET effects could be due to the inner part of the track only. A restriction of the linear energy transfer L E T to that fraction which forms the center part was proposed. Assuming an emission characteristic of the electrons mainly perpendicular to the ion trajectory, the inner part of the track will be formed mainly by the low energy electrons, the outer part by the high energy electrons. Therefore, instead of a spatial restriction to the track center, a cut off in the electron energy-spectrum is frequently used for the calculation of restricted L E T values [ICRU 16 (1970)]. Restricted L E T values have been published recently by Heinrich et al. 1991 : Except for the very heavy projectiles like uranium, the restricted L E T parallels the total energy loss curve for particle energies greater than 1 MeV/u. Therefore, replacing the unrestricted L E T by restricted L E T values represents a multiplication with a constant factor for all L E T values. Hence it is evident that deviations from a smooth dependence on L E T as for instances the o--LET hooks cannot be attributed simply to differences between unrestricted and restricted L E T values i.e. to a different energy dependence of the very inner part of the track compared to the outer part.

IIL Electron emission by heavy ion impact Electron spectra The primary energy distribution of the electrons together with their energy loss determine the radial dose distribution of the particle track and the track diameter. A typical spectrum of electrons emitted in heavy ion collisions is given in Fig. 3. It shows a broad continuum of electron emission overlayed by some line structures. These various structures originate from the following processes: The continuous 6-electron 1 emission is due to the interaction of the projectile with the bound electrons of the target atoms. The discrete lines originate from Auger processes i.e. auto ionisation processes in the target and in the projectile atoms (Stolterfoht 1987). The broad electron loss maximum - the so called 1 The term "6-electron" is applied for all electrons of the continuum in the electron spectrum. Frequently in radiobiology a difference is made between high and low energetic electron calling the high energetic electrons 6-electrons and the low energetic electrons just electrons. However, there is no clearly defined threshold energy and this difference is very artificial

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Fig. 3. Typical delta electron spectrum of a heavy ion atom encounter according to Stoltcrfoht et al. (1974). For detailed explanation see text

EtecfronEnergy[keV] cusp peak - is caused by electrons which are liberated from the projectile and travel with the velocity of the projectile. The overwhelming part of all emitted electrons is found in the continuum. Therefore the dose distribution inside the track depends mainly on the energy distribution of these electrons. In the following the discrete lines of Auger and cusp electrons will not be treated further. The different contributions to the electron continua depend strongly on the impact parameter i.e. the distance of closest approach between the projectile carrying the charge e / e ft and the target electrons. For small impact parameters the energy transfer from the projectile to the target electrons is large compared to the binding energy of the target electrons and the collision process can be treated in the Binary Encounter Approximation (BEA) (Bonsen and Vriens 1970). In this model, in a binary encounter the quasifree target electrons are scattered at projetile ions like classical billard balls. The electron emission energy Ee and the emission angle 0 and the particle energy are correlated in the following way: E e = m~4me Ep cos 2 0

(3)

where me and mp are the electron and projectile mass, respectivly. E~ is the projectile energy. According to this formula, each emission angle corresponds to only one electron energy. However, the target electrons are bound before the scattering

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Fig. 4. Angular plot of the double differential cross section for electron emission. In this plot the center of the circles coincides with the reaction center. The direction and the distances from this point represent the angle and cross section of the electron emission. In the extreme case of very heavy ion impact ( U - Ar) the angular distribution shows a strong forward directed emission characteristic. The maximum at 60° is due to multiple scattering in the target atom (Schmidt et al. 1989) process and have an initial m o m e n t u m distribution which depends on the quantum numbers of the initial state. This m o m e n t u m distribution has to be added to the m o m e n t u m transferred in the collision which yields a continuous distribution from low to the m a x i m u m possible energy for all emission angles but peaked around the binary energies. I f the impact parameter between projectile and target electrons is large, the energy transferred to the electron is small and the electrons can not be regarded to be quasifree which was a basis of the binary encounter approximation. In this case, the collision has to be treated as a three body interaction including the target nucleus. The electrons are consequently emitted in all angles between 0 and J80 ° and their angular distribution should be more or less isotropical. In a heavy ion collision, however, polarisation effects of the targets are introduced by the high charge of the projectile which again increases the emission probabilities into forward angles for all energies. In addition to the forward peaked emission characteristic for the heavy ions, a small m a x i m u m of emission has been found at intermediate angles (Fig. 4) which was interpreted to be due to a multiple scattering process in the target a t o m (Schmidt-B6cking et al. 1992; Kelbch et al. 1991). Another special effect of the very heavy ion projectiles is the multiple ionisation in which more than one electron is liberated from the same target a t o m and again emitted preferably to forward direction (Schmidt-B6cking et al. i 992).

Other models of the electron distribution In m a n y radiobiological model calculations (Butts and Katz 1967; Paretzke 1988; Kiefer and Straaten 1986)) the collision process is regarded as a two

169 body interaction only between the projectile and unbound electrons in the target. A straightforward calculation based on classical Coulomb scattering yields the following expression for the electron distribution dn

CZ~ff dE f12

E2

with

C

2ge 4 N Z 2

(4)

me C2

where dn is the number of electrons produced in an energy interval between E and E + d E . Combined with the kinematics of the Coulomb scattering this formula yields a strong emission maximum perpendicular to the projectile trajectory. However even with such a simplified emission characteristic reasonable agreement of the dose profiles between calculation and experiment can be obtained because the electrons suffer many collisions during the slowing down process. This homogenizes widely the angular distribution after a few collisions and differences in the initial angular distribution become less important.

IV. Condensed phase effects - the track core

There is a difference in the charge state of projectile and its electronic configuration, when the projectiles are passing through a solid instead of a gas absorber. This was mentioned before and has been studied in power experiments by H. Geissel et al. (1982, 1983). But there are also differences for the target atoms in the reaction mechanism when gas and condensed matter are compared:

Wake effect

In a solid target collective excitations are produced by heavy ions independent from the degree of ionisation of the target atoms. By the passage of the high charge of a swift heavy ion, the electron shells of neutral target atoms become polarized. In the case of a gas target, the target atoms are insulated and their interaction is negligible. In a solid target, atoms of similar polarization are close together and interact forming a common potential (Gemell 1979). Because the polarization is caused by the projectile ion, this potential travels like a wake behind the projectile (Fig. 5). One consequence of this plasmon wake is the enhanced emission of cusp electrons which are called "convoy electrons" in the case of a solid target. Such an enhancement of the forward emission

Fig. 5. Wake potential behind a fast ion having an effective charge Zofe and velocity v. The potential is due to the polarization of the target atoms by the effective charge of the projectile (Gemmell 1979)

170

l

e-

e-

I. ~÷ ~÷ I+ i÷ /e-+~+~e -

"

eff •

Recombination

le. ~

Coulombgxptosion

"e-

Ionization

Fig. 6. Schematic view of the Coulomb explosion mechanism. When the electrons are emitted from the target atoms, positively charged ions remain in the center of the track. These ions are pushed from their regular position if the Coulomb forces are greater than the binding energy. Due to the axial symmetry, the Coulomb explosion produces a zone of reduced density and broken chemical bindings in the center of the track of electrons may contribute to the Coulomb explosion mechanism because the primary electrons together with their secondaries will produce a higher degree of ionisation in the center of the track.

Coulomb explosion For higher energy loss values many electrons are liberated from target atoms in close proximity. Therefore a high local charge of positive ions is created in the center of the track (Fig. 6). These atoms are pulled out of their regular positions in the molecule by the " C o u l o m b explosion" when the Coulomb energy is greater than the binding energy of the molecule. The dimensions of the volume in which the original chemical structure is destroyed by Coulomb explosion depend strongly on the target material. Threshold values of the energy loss between 10 and several hundred keV/pm have been found for the production of a track in different nuclear track detectors. Detailed experiments using neutron scattering (Albrecht et al. 1985) confirmed the expected strong correlation between the dimensions of Coulomb exploded zone and the electronic energy loss. Meanwhile it was also possible to study the area of local disorder as generated by Coulomb explosion directly in a raster tunnel electron microscope (Fig. 7). These direct observations are in perfect agreement with the neutron scattering data. Rough estimates on the kinematics of the Coulomb explosion could be made for metallic glasses (Klaumfinzer et al. 1986). While the primary effect of electron emission occurs within 10 - i s s or even faster, the Coulomb explosion and the following recombination of the electrons in the now disordered molecular lattice of the original atoms take place within 10 -14 s. This time is much shorter than that for the diffusion controlled chemical changes. In consequence, the majority of atoms are displaced in the center of the particle track before chemical and biochemical reactions start. The Coulomb explosion and the nuclear collision process are the only processes of a direct atomic displacement which are confirmed experimentally. But their energy and L E T dependence is very different from the frequently postulated " c o r e " effects in the core-penumbra model.

171

Fig. 7. Scanning tunneling microscopy image of damage produced by a 15 MeV/u Au ion in highly oriented pyrolytic graphite. The microscope has a resolution of atomic dimensions and each maximum indicates the position of one C-atom. A damaged area of the extension of a few atoms only is visible in the center of the image (Courtesy of R. Neumann, GSI)

V. Radial dose distribution in the track halo According to various authors (Bethe 1930; Groeneveld et al. 1980; Paretzke 1988) f r o m the energy loss of the projectile a fraction of approximately 5 - 1 5 % is spent for the excitation of the neutral atoms, and 15-25% to overcome the binding energy of the liberated electrons. But m o r e than two thirds of the energy loss are transformed into kinetic energy of the liberated electrons. Therefore the m a j o r fraction o f the energy deposition is found in the radial dose distribution o f the track halo. The track halo is frequently called " p e n u m b r a " . But the term p e n u m b r a implies tacitly the existence of a " d a r k " center part for which no evidence has been found. There are two different theoretical approaches to calculate the radial dose distribution: The Monte Carlo method simulates the elementary collision processes for a large n u m b e r of electrons. The track structure model calculates radial dose distribution on the basis of empirical range energy relations but does not use any detailed knowledge of the elementary collision processes. These models have to be c o m p a r e d with experiments where the energy deposition events are measured as a function o f the radial distance of the projectile trajectory in diluted gas target.

172

Monte Carlo calculations In the Monte Carlo calculations the ionization and energy deposition events are recorded as a function of the radial distance from the center of the track. For the electron transport calculations the following input data are necessary (Paretzke 1988) - cross section double differential in energy and angle for the primary electron emission - elastic cross sections for scattering - inelastic cross sections for excitation and ionization of target molecules by electrons The double differential cross sections of electron production give the number of electrons, their energy and emission angle. For these cross sections frequently the electron distribution of formula (4) is used although it does not agree with the experimental findings. The cross section for the elastic scattering describes the scattering process in which essentially no energy is transferred from the electron to the target atom but the direction of the momentum is changed. But the elastic scattering determines the spatial evolution of the electron diffusion. Finally, the inelastic scattering produces excitation and ionisafion in the course of the slowing down process of the electrons. It should be noted that no interaction, especially no slowing down occurs between two subsequent collisions. The energy loss of the electrons is calculated discontinuously as a result of many individual collisions 2. In a recent Monte Carlo calculation (Krfimer and Kraft 1991) using the binary encounter approximation as the starting point of the electrons good agreement with all measured radial dose distributions has been achieved (Fig. 8).

Track structure calculations. A completely different approach to calculate the radial dose distribution has been followed by Katz and coworkers (1971, 1985). Starting from the classical two body interaction and using an approximation for the energy dependence of the electron range, the radial dose has been calculated for cylindrical shells around the partical track. In this calculation it is assumed that all electrons are emitted perpendicular to the particle trajectory and that they are not deflected during their penetration. In addition, a linear relationship between electron range and energy is assumed which is equivalent a constant energy loss independent of the electron energy. A straight forward calculation of the dose distribution based on these assumptions yields for the radial dose (Butts and Katz 1967):

D(r)= 2~fi~ r

with

C

mec2

(5) 1

According to this formula the dose decreases with r~ up to a maximum value 2 Frequently the electron stopping is calculated using the continuous slowing down approximation CSDA. The use of CSDA is justified if a large number of collision processes are necessary to stop the electrons. For the low energy electrons of a few hundred electron volt CSDA does not reflect the physical reality

173

2.57 MeV/u O -> H20 I111

107 10 6

105

lO,t (/)

o 103

D

102 Fig. 8. Radial dose profile of 2.6 MeV/u O ions in water vapor. Points: experiment by Varma et al. (1977). Histogram: Monte Carlo calculation (M. Kr/imer 1991)

10 ~

10o

- , ...... a , , , , , , J

10-8

10-7

....

.a

,,,l~

10~ 10-5 Q [cm]

lO-,t

of the radial distance rmax which is given by the maximum lateral penetration of the electrons. 3 As explained before, the radial dose distribution derived in this way is frequently in good agreement with the experimental values and Monte Carlo calculations.

Other models o f radial dose distribution A frequently used approach of radial dose distribution is the " c o r e and penumb r a " model developed by Magee and Chatterjee (1979). Following the model of " k n o c k o n " and "glancing collisions" it is assumed that the glancing collisions produce with a higher probability target exitation, whereas knock on collisions are mostly responsible for the liberation of the &-electrons. Exitation is assumed to form a core of high energy deposition in the center of the particle track while the &-electrons transport their energy away from the " c o r e " forming 3 For the range energy relation a linear dependence is assumed r=k'Ee with k=10 gg/cm2 keV. However, it would be naive to assume that the radial dose has a sharp cut off at a range r=rmax. The electron emission process as well as the radial diffusion produce broad energy distributions. Therefore, a maximum radial range of the electrons i.e. a track diameter is not sharply defined

174 a "penumbra" around it. An equipartition of the energy deposition between the core and a penumbra is postulated and for the radial dose distribution 1 outside the core again a 7z law is assumed. For increasing particle energies both diameters of core and penumbra are assumed to increase linear with the energy. The core-penumbra model does not reflect the physical reality: More than 80% of the total energy loss is used to liberate electrons from the target atom and to emit these &electrons with high energies. Less than 10% is left over for excitation. Therefore an equipartition of the energy between core and penumbra does not hold. In addition, excitation does not result in chemical damage except for a few supercited states. Excitation is therefore not relevant for the biological action. Another model of track structure using relativistic collision dynamics has been developed by Kiefer and Straaten (1986). It is based on the assumption that the angular distribution of the liberated electrons is not given by classical but relativistic collision dynamics. Secondly the energy distribution of the electrons follows the usual dn ~_~L law. Finally, the electrons do not undergo multiple collisions but travel in straight lines. As a consequence of the relativistic treatment of the problem, the electrons are preferentially emitted into forward directions which is in reasonable agreement with the experimental findings. But due to the omission of the multiple scattering process in the target material the high energy electrons are not scattered in the direction perpendicular to the particle trajectory. Therefore the maximum penetration of the electrons perpendicular to the particle track is reduced and the track diameters are found mostly to be too small compared with experiments. Microdosimetric measurements of the radial dose distribution

The microdosimetric approach originates from an early interpretation of chromosomal exchange figures where an interaction length up to one micrometer was assumed for two separately produced primary lesions. Therefore, volumes having this extensions have been regarded to be of the critical size and the total energy deposition e in such a microvolume regardless of its internal pattern was postulated to be the critical parameter for the cellular response. In analogy to the macroscopically defined parameters of dose and LET, the specific energy z =--m as corresponding to the dose and the lineal energy transfer y = l as analogon to LET have been defined for these microvolumes (Kellerer and Rossi 1978) (m -~ mass of the microvolume, l= mean cord length). Because it is not possible to determine the energy deposition events experimentally in such small volumes of tissue or water, the equivalent volumes have been simulated in a diluted gas with a wall-less ionisation chamber which measures the individual ionisation events. Details of the experimental set up and the data analysis of such as microdosimetry experiment have been given recently by Metting et al. (1988). In these experiments, the specific energy is usually determined as a function of the radial distance from the primary particle trajectory. A typical result of such a measurement is shown in Fig. 9 for 600 MeV/u

175

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o.,r °.3F/

/

/ ooo.II

/~

Z"

i

i

1 "'~1_k

t

/

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0.01 0.1 1 10 -Specific Energy, Gy Fig. 9. Measured distribution of the specific energy z = ~/m as function of the radial distance from the center of the track. The vertical line in the spectra indicate the mean value ~ (according to Metting et al. 1988)

Fe-beam. For radial distances larger than 5 I.tm the distribution of the z-values and especially the mean value of z does not change significantly with the radial distance from the center. This indicates that the measured events are produced by an always similar mixture of stopping electrons causing large energy deposition events and traversing electrons having smaller energy depositions. For small radial distances, the distribution of the specific energy is shifted to greater values of energy deposition. Finally, the primary beam penetrates the ionisation chamber and a very sharp peak of events is found at specific energies hundred times higher than the values of the stopping electrons. Because individual electrons cannot produce a greater energy deposition than in the maximum of their stopping power, these high z-values are due to the simultaneous action of many electrons. The individual events are not resolved in time because they are produced simultaneously by the passage of the same primary projectile. However this high energy deposition events can be dispersed in different ways over the total microvolume of 1 gm in diameter which is huge compared to the dimensions of DNA molecules. One cubic micrometer would contain the DNA of thousand bacteria cells, which demonstrates that the dimensions of microdosimetry are to great. In the experiment, the radial dependence of the mean value of the specific 1

energy does not follow a r~ law, but levels off at high of individual energy deposition events. In order to obtain the radial dose distribution, the z values have to be multiplied with their probability of occurence. Then the radial dose 1 distribution follows the r~ law very precisely. However it should be kept in mind that the small doses in the track halo are made out of single events baring a high energy deposition each but having a very low probability to occur. Only a few microdosimetric measurements of the radial dose distributions have been published up to now. But these data are the experimental basis

176

for all model calculations including the Monte Carlo calculation of the radial dose distribution.

VI. Comparison between the experiment and the model calculation

Experimentalfindings The biological effects of particles have been investigated for different endpoints and objects over a wide range of LET values (for review see Kraft 1987). The action cross sections for the incidence of heavy ion damage on the different levels of biological organisation exhibit a very similar LET dependence for effects which originate from DNA double strand breaks. As typical examples the inactivation cross section of the bacteria, yeast and mammalian cells are given in Fig. 10. For LET values below 10 keV/pm the cross section increase proportional to the LET indicating that the inactivation depends only on the

........

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101

102

103 LET (keV/p.m)

t0 L'

Fig. 10. Inactivation cross sections of bacteria (bottom), yeast (top) as function of the linear energy transfer (LET)

(middle)

and mammalian cells

177 energy deposition and is nearly independent from track structure. A separation between curves of different atomic number is not measurable. For LET values between 10 and 100 kev/gm the cross sections increase more than proportional to the LET (The overproportional increase is found for the induction of double strand breaks and other biological events related to double strand breaks. This is in contrast to single strand breaks where the cross sections always increase proportional or less than proportional to LET in this range). This overproportional increase of the inactivation cross sections corresponds to RBE values greater than one and indicates that the increased ionisation density produces lethal lesions with a higher efficiency. For LET values greater than 100 keV/l.tm the common curves separate into individual curves for each ion in order of increasing atomic numbers and the cross sections decrease for further increasing energy deposition forming individual o--LET hooks for each atomic number. These hooks of the cross section occur always at the end of the particle tracks when the energy deposition is high but the diameter of the track decreases. In this "thin down region", the local ionisation density and the geometry of the track dominates over the LET dependence.

Model calculations Surprisingly, there are no detailed calculations based on microdosimetry which would describe the dependence of the biological action cross section over an extended range of particles and energies. This is not only true for the complex biological reactions like mutation or inactivation, this holds also for the simpler effects of the induction of DNA strand breaks. The few attempts of microdosimetry to calculate the cross sections (ICRU 40, 1986) are far from the success of the track structure models (Katz et al. 1985). The track structure calculations are based on the radial dose distribution as given in formula (4). The action cross section for a point target which is inactivated by a one hit process follows Poisson statistics and can be written as the inactivation probability as compared to x-rays integrated over a cylindrical surface (Butts and Katz 1967). S = 2 rc Sr d r [1 - exp DD37j(r)]

(6)

where D37 is the inactivation dose for x-rays and D(r) the dose deposited by the particle track. Inactivation cross sections obtained by numerical integration of (6) show the basic features of the measured action cross section: One common curve for low LET values and a hook structure for the high LET values in which the o- curves separate for each atomic number. Lateron, the track structure model has been refined and two different modes of inactivation, 7-kill and ion kill have been introduced in order to accommodate for the supra-linear increase at the lower LET values. With this assumption it is possible to describe the general feature of the action cross sections over a large range of energies and particles and for many biological objects. In order to achieve a quantitative agreement of the experimental data with the track structure model, free parameters have to be fitted and the size of the critical target as well as the maximum range of the electrons have to be adapted.

178 ii, I

''""'1

' ''m"l

" ''"'"1

' ''m'l

~

''~'"'1

102

/,4#

~'E 10' C O

k,.

S~S£C. cerevis~e(xlO)

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Fig. 11. Inactivation cross section calculated according to a track structure model (M. Scholz, G. Kraft 1991) in which the local dose produces an effect as given by the measured shouldered x-ray survival curves. In this calculation no parameters have been fitted

?

10-~ m~

10o

101

102 103 LET (keV/tJm)

10 '~

10 S

Recently, a parameter free track structure model for the calculation of heavy ion inactivation cross sections has been developed (Scholz and Kraft 1991). The basic input data are the following: - The radial dose distribution D (r) is assumed to follow the usual 1/r 2 law - The whole cell nucleus represents the critical target for the inactivation process; the sensitivity is assumed to be uniformly distributed over the cell nucleus and the size of the critical target is taken from measurements - The dose effect curve is assumed to be a shouldered survival curve with exponential tail for high doses as measured in x-ray experiments. The most important improvement compared to the Katz model is the method to calculate the inactivation probability. In the new model, the probability density to produce a lethal event by a local dose D(r) is taken from the dose effect curves measured in x-ray experiments. In order to calculate the total inactivation, the probability density for the different local doses is calculated and then integrated over the whole intersecting region of the particle track and the cell nucleus. In contrast, in the Katz model the total energy in the critical target is integrated first and then the inactivation probability is determined according to the x-ray curve. Since the x-ray survival curve is usually nonlinear, the result of the calculation depends critically on the order of the two steps, i.e. integration and determination of the effect and yields different results when first the total

179 d o s e o f the critical t a r g e t is i n t e g r a t e d a n d c o m p a r e d to the x - r a y d o s e effect curve ( K a t z ) o r w h e n the d i f f e r e n t i n a c t i v a t i o n p r o b a b i l i t i e s are c a l c u l a t e d for the different local doses a n d t h e n i n t e g r a t e d . W i t h the new t h e o r y it is p o s s i b l e to d e s c r i b e the L E T d e p e n d e n c e o f h e a v y i o n i n a c t i v a t i o n cross sections c o m pletely b y m e a n s o f m e a s u r a b l e q u a n t i t i e s as s h o w n in Fig. 11,

Acknowledgement. We appreciate many discussions with all the members of the biophysics group at GSI and the atomic physics groups of IKF Frankfurt. Especially the discussion with H. Schmidt-B6cking and R.E. Olson have been very important. We thank Heike Strehl for the preparation of the manuscript.

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