Relation Between Kerma and Absorbed Dose in Photon Beams

Acta Radiologica: Oncology ISSN: 0349-652X (Print) (Online) Journal homepage: http://www.tandfonline.com/loi/ionc19 Relation Between Kerma and Absorbed Dose in Photon Beams B. Nilsson & A. Brahme To cite this article: B. Nilsson & A. Brahme (1983) Relation Between Kerma and Absorbed Dose in Photon Beams, Acta Radiologica: Oncology, 22:1, 77-85, DOI: 10.3109/02841868309134343 To link to this article: https://doi.org/10.3109/02841868309134343 Published online: 08 Jul 2009. Submit your article to this journal Article views: 4708 View related articles Citing articles: 12 View citing articles Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalinformation?journalcode=ionc19 Download by: [46.3.193.220] Date: 20 December 2017, At: 04:01

Actn Radiologica Oncology 22 (1983) Fuse. I FROM THE DEPARTMENT OF RADIATION PHYSICS, KAROLINSKA INSTITUTET, S-10401 STOCKHOLM, SWEDEN. RELATION BETWEEN KERMA AND ABSORBED DOSE IN PHOTON BEAMS B. NILSSON and A. BRAHME The determination of the relation between kerma and absorbed dose is one of the basic problems of dosimetry and it is treated in most textbooks on radiation physics and dosimetry (cf. WHYTE 1959, JOHNS & CUNNINGHAM 1969, GREENING 1981). Despite the importance of these concepts, there is no consistency in the treatments as pointed out already by ATTIX (1968, 1979). An error is also introduced when the depth integral of the kerma is equated with that of the absorbed dose (JOHNS& CUNNINGHAM, ANDERSON 1976) at least when it is assumed that the absorbed dose can be taken from experimental depth dose curves (CORMACK& JOHNS 1954, AN- DERSSON, ALMOND et COll. 1978). There are at least three reasons for having a good knowledge of the relation between different kerma concepts and absorbed dose. First, it is of importance for the fundamental understanding and the conceptual clarity of the different quantities that have the same unit: namely absorbed energy per unit mass. Secondly, it is of interest to know how these relations vary in different materials depending on the basic physical constants governing the electron transport in the materials, principally: the mass scattering power and the radiation and collision mass stopping powers. Thirdly, the ratio of kerma and absorbed dose is of practical importance in dosimetry with air kerma or exposure calibrated ionization chambers. In the present paper, the relation between kerma (K) and absorbed dose (D) is determined on the basis of the method of calculating the electron transport in a photon beam developed by NILSSON & BRAHME (1979, 1981) and on the partition of the kerma in a collision part (Kcol) and a radiative part (Krad) as introduced by ATTIX (1979). Fundamentals In a photon beam the kerma is defined as the initial kinetic energy of all charged particles, mainly electrons and positrons, liberated by photon interactions per unit mass in a medium (ICRU 1980). Since part of this kinetic energy may be converted back to energetic photons mainly through bremsstrahlung and annihilation in flight processes it is useful to analyse that part of the kerma which remains as kinetic energy of charged particles, namely the collision kerma, Kcol (ATTIX 1979). Therefore, if a photon beam is impinging on a semiinfinite half space of matter, the volume integral of the collison kerma over the half space must by definition be equal to the absorbed dose integral or the mean energy imparted, C (ICRU 1980), when that small part of the collision kerma that may escape through the entrance surface of the half space can be disregarded: ///, E, = J,D(r) e(r) d v // fl1(r) e(r) d v = (1) where e(r) is the density of the medium. A more relevant geometry for the use of uniform collimated photon beams in homogeneous media is to look at a cylindric volume aligned with the beam axis and located no less than one maximum electron Accepted for publication 14 October 1982. 77

78 9. NILSSON AND A. BRAHME range from the border of the photon beam. As a lateral charged particle equilibrium exists in this geometry, eq. (1) is still valid and may even be simplified one step as the photon fluence was assumed to be uniform and thus both the collision kerma and the absorbed dose will only depend on the depth along the beam: Interaction ymbol Name Primary and coherent scatter Incoherent scatter Bremsstrahlung where A is the cross sectional area of the cylindric volume. Thus, in this geometry the depth integral of the collision kerma is equal to that of the absorbed dose assuming again that the charged particle energy backscattered through the front surface can be disregarded. It is interesting that the relative importance of the backscattered electrons can be determined by measuring the relative build down at the back surface of a phantom. If spectral changes are disregarded the relative dose decrease at the back surface of a phantom, when a saturating backscattering block behind the phantom is removed, should be exactly equal to the extrapolated relative surface dose at the front surface of the phantom. In order to evaluate the integrals in eqs (1) and (2) it is important to be able to describe the depth dependence of energy fluence, kerma and absorbed dose in a correct and consistent way. By definition the energy fluence of primary photons, Yp, decreases at a rate given by the total linear attenuation coefficient, p: Yp(z) = Y&O) e-pz (3 a) This relation holds exactly for a monoenergetic photon beam. If the coherently scattered photons are included in the fluence of primaries a corresponding reduction of p has to be introduced. For beams with a finite width of the photon spectrum, p will necessarily vary with depth. This is so as the absorption of the different photon energies may be evaluated in terms of a depth dependent linear attenuation coefficient, p(z) defined by Yp(z) = 1 Yp(O, E) e-p(e)zde = Y P (0) e-p(z)'z (3 b) In the total photon fluence of an infinitely broad beam, where there is a lateral equilibrium both with - I -photon c -electroi - ----posit roi Annihilation in flight Annihilal ion 'at rest' Flourascent Total fluence Fig. 1. Illustration of the photon interactions contributing to the total fluence of photons: the primaries Yp, the most important build-up part (Ys+Yb+YJ and the essentially isotropic part Y. regard to scattered photons and charged particles, all generations of photons are included, mainly scattered photons, bremsstrahlung and annihilation quanta, but also fluorescent photons (Fig. 1). This implies that the depth dependence will be given by the mean linear energy absorption coefficient of the total energy fluence: YJZ) = ~ ~ ( e-'enz 0 ) (4) For finite beam cross sections, scattered photons will necessarily escape from the beam and the true mean attenuation coefficient, p, will be somewhere between pen and,u (Fig. 2). The shape of the beam cross section will therefore determine the exact value of p and the depth variation of the total energy fluence: Y,(z) = w,(o) e-pz (5) Depending on the shape of the primary photon spectrum the total fluence spectrum may change slowly with depth. This implies that p and pen may have a depth dependence in the above relations (cf. eq. 3 b). This dependence is generally quite weak, especially for bremsstrahlung spectra and at large depths where an equilibrium is reached in the absorption and production of scattered photons. However, for 6oCo it was found necessary to consider the slow change in p with depth.

KERMA AND ABSORBED DOSE IN PHOTON BEAMS 79 0.5 t I I I 1 Depth Att. dependence coeff. I, 0 1 2 3 z /cm Pen - P P Beam Fluence width Yt Yt Yp Infinite 1 Broad 1 Narrow Point monodirectional Fig. 2. Schematic illustration of the interrelation between different attenuation coefficients. Based on eqs (3) to (5) the depth dependence of kerma, collision kerma and absorbed dose may now be written: Here ptr, the linear energy transfer coefficient, expresses the fraction of the photon energy transferred to kinetic energy of charged particles. The pen value is a reduced p,, value taking into account the loss of kinetic energy of generated charged particles due to forward directed bremsstrahlung, annihilation in flight photons and to a less extent to essentially isotropic fluorescent photons. It is important that these photons are included in the total energy fluence of photons (Fig. l), as they are dosimetrically difficult to distinguish from the primary photons. Furthermore, if the depth dependence of the total energy fluence is determined by dosimetric measurements according to eqs (5) and (8) the mean attenuation coefficient determined will automatically include them. The correction factor p(z) (cf. LOEVINGER 1981) which takes the motion of the charged particles into account will be treated in considerable detail in the following, based on calculations of the charged particle transport using the diffusion equation and the basic equality given by eq. (2). (7) Method In order to calculate the integral of the collision kerma the effective attenuation coefficient for the photons has to be determined. This may be done by using build-up factors (ATTIX 1979). However, the build-up factors normally published do not include the contribution from bremsstrahlung and annihilation in flight photons and must consequently be corrected. An estimation of this correction for 6 MeV photons in aluminium gives a change in the effective attenuation coefficient of 2 per cent. For lower energies and lower atomic number materials the difference is smaller. An alternative way to determine the effective attenuation coefficient is by using experimental depth dose curves for broad beams corrected to infinite SSD. A detailed analysis of the experimental data shows that for 6oCo the change of the p value with depth must be considered. This was done by fitting the TAR values in water (Brit. Inst. Radiol. 1972) with three-exponential functions according to TAR(z) = k[e- mze-pi +(I -e-mz) e-'2'] (9) where k, m, pl and p2 are field size dependent factors determined to minimize the mean square deviations from experimental data. With this function the agreement with tabulated data was better than 0.5 per cent and most of the deviations were due to random errors in the tabulated TAR values. For a 4 cm x 4 cm field the factors obtained were 1.02305, 0.14918 cm-', 0.02668 cm-' and 0.05875 cm-', respectively. This will give a mean extrapolated effective attenuation coefficient of 0.029 cm- ' in the build-up region in agreement with experimentally obtained attenuation coefficients for the wall in ionization chambers (HOLT et coll. 1979, CUNNING- HAM & SONTAG 1980). For 21 MV roentgen radiation only a single p value of 0.0209 cm-' for the whole depth range was sufficient. Data for air and aluminium have been obtained by scaling data for water by the mass energy absorption coefficient. The effective attenuation coefficient can be used to calculate the depth integral of the absorbed dose for depths beyond the depth of transient equilibrium. The main problem is thus to determine p(z) or the shape of absorbed dose curve in the build-up region. In the present work the absorbed dose in the build-up region was obtained by using the method of

80 B. NILSSON AND A. BRAHME 9.0 8.0 7.0-6.0-5.0-4.0-3.0-1.0, o 0 5 10 15 20 Fig. 3. Approximation of the incoherent electron angle distributions according to the Klein-Nishina equation with three Gaussian distributions. Photon energy: 5 MeV. Mean electron ener- NILSSON & BRAHME (1979, 1981) to calculate the contribution to the dose distribution for each photon energy and group of secondary electrons according to: D(z) K(0) (7 N SC& = absorbed dose due to secondary electrons at depth z, = kerma at phantom surface, =the cross section for production of secondary electrons belonging to the group under consideration (cf. Fig. 3), = the density of atoms in the medium, =mean collision mass stopping power for the secondary electrons, 0 /degree gies of the groups 1, 2 and 3 are 4.51 MeV, 2.33 MeV and 2.33 MeV, respectively. = mean mass energy transfer coefficient in water, = mean attenuation coefficient of the beam, = mean extrapolated range of the secondary electrons, = coefficients describing the transmission of a parallel beam of electrons as taken from TABATA & IT0 (1974), = initial mean square scattering angle of the electrons emitted in a photon interaction, the factor (1 +B$/2) modifies the transmission functions to take the angular distribution of emitted electrons into account, = the ratio of fluence to planar fluence for the secondary electrons. fr is calculated from the equation

KERMA AND ABSORBED DOSE tn PHOTON BEAMS 81 0.2 - where g(8) = l/)cos 81 for l8-x/21~ 8, and g(8) = Usin8, elsewhere (BRAHME 1975) and $(z) =the mean square scattering angle at depth z. The cut off angle for cos 8 was taken from 8; = Td, where To = the linear scattering power, A = mean free path of the electrons. In order to take the energy and angular distributions of the electrons into account the following assumptions were made. For incoherent scattering the differential cross section for the recoil of a free electron at angle 8 with respect to the direction of the photon (dddq) was approximated by a sum of three Gaussian distributions (Fig. 3). Thus, it was possible to divide the recoil electrons in one high energy small scattering angle group and a low energy large angle group (cf. BRAHME 1982). For the pair production interactions the electron energy varies only slightly with the emission angle as seen from the angular dependence of the mean positron energy (Fig. 4). For low photon energies the mean energy was calculated from the differential for 21 MV roentgen radiation the spectrum was approximated by three energy groups. Beside the absence of a true d-particle equilibrium eq. (10) should ideally give the absolute value of the absorbed dose. However, by using very few energy groups the calculations include some uncertainty, especially through the energy dependence of the S,,, also without normalization. After normalization for each energy the contribution from the different electron energies were weighted and added together and the absorbed dose in the build-up region was obtained. The absorbed dose distribution for larger depths was then added and the total absorbed dose distribution could be obtained. An alternative method to using a fluence build-up factorf, and the planar fluence transmission eq. (10) is to use the energy deposition algorithm suggested by TABATA & ITO, eq. (10) then becomes (12) where I(z)dz is taken from the energy deposition at depth z, according to TABATA & ITO, but corrected for the mean initial angular spread of the emitted electrons. This method was used for comparison in a few cases. Results and Discussion In Fig. 5 the absorbed dose distribution in the build-up region for 6oCo y-rays is plotted for three different materials. A noticeable difference was 6-838101

82 B. NILSSON AND A. BRAHME - D (11 K (01 1.0 0.5 0 Fig. 5. Absorbed dose distribution in water (-), air (---) and aluminium (---) in the build-up region for 6oCo y-rays calcualted according to eq. (10). The kerma andcollision kerma distributions D (21 - K 101 1.0 z/gcm2 in water are also included. They cannot, however, be distinguished in this scale. Data are normalized to kerma at surface. 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Fig. 6. Absorbed dose distribution in water in the build-up region for o C @' y-rays calculated by different methods: the present electron diffusion model eq. (10) (-), the energy deposition algorithm model eq. (12) (TABATA & ITO 1974, ---), the exponential function model eq. (13) (GREENING 1981,..) and the Monte Carlo method using the electron energy deposition distributions according to Berger (cf. ALLISY 1967) (J).

KERMA AND ABSORBED DOSE IN PHOTON BEAMS 83 - D IZl K 101 1.0 0.5 0 0 0.1 0.2 0.3 0.4 z /gem= Fig. 7. Comparison between calculated absorbed dose in the build-up region, eq. (10). with experimental data for mco y-rays. The experimental data are measured for small fields and include collimator scattered electrons which are not included in the calculated curve. Compton electrons from the air are, however, included. RICHARDSON et coll. (I954 A). SMITH & SUTHERLAND (1976 0). LEUNG et coll. (1976 w). Present investigation (4. - D fzl K (0) 1.0 0.5 0 0 2.o 4.O 6.0 8.0 z/gcm2 Fig. 8. Absorbed dose distribution in water and air in the build-up region for 21 MV roentgen radiation calculated according to eq. (10). The kerma and collision kerma distributions in water are also included. Data are normalized to kerma at surface. found between air, water and aluminium, principally depending on the differences in scattering and stopping powers of these materials. Aluminium has higher mass scattering power than water and air which implies that the dose rises faster. The slightly deeper penetration in air compared with water is due to the smaller mass stopping power of air which results in a somewhat larger range of the electrons as the mass scattering power of these materials is almost identical. In Fig. 6, eq. (10) is compared with data obtained by using the energy deposition algorithm given by eq. (12). This algorithm seems to give a faster rise in absorbed dose. In the figure is also plotted data obtained by integrating the energy deposition distribution for a plane parallel 6oCo compton electron source in air calculated by Berger (cf. ALLISY 1967). The later data give a lower dose in the first two mm. Another way to express the depth absorbed dose distribution is by two-exponential functions (JOHNS et coll. 1949, BRAHME & SVENSSON 1979, GREEN- ING). The relation between absorbed dose and kerma then becomes: WZ) - Pe (e-pz-e-~e~ NO) P-Pe ) (13) where pe is an effective linear attenuation coefficient for the electrons. pe was determined by linear exponential regression to 8.55 cm-'. This gives a quite good agreement (cf. Fig. 6) with the calculated depth dose distribution. For comparison some experimental data are included in Fig. 7 (RICHARDSON et COIL 1954, SMITH & SUTHERLAND 1976, LEUNG et coll. 1976) where contamination of air electrons (NILSSON& BRAHME 1979) and backscattered electrons and photons have been added to the results obtained by eq. (10). A linear absorption of the air electrons in the phantom has been assumed. The contribution from backscattered electrons was taken into account by using moments method calculations of SPENCER (1959) giving the absorbed dose at the surface. The calculated curve seems to rise too fast during the first mm. The differences are, however, small and the agreement is good considering the approximations made. In Fig. 8 the kerma (K) and collision kerma (I&,) for water and 21 MV roentgen radiation are plotted together with the absorbed dose (D) normalized to 1.00 for the kerma at the surface. The ratios of absorbed dose to collision kerma and kerma are

84 B. NILSSON AND A. BRAHME Table Absorbed dose and kerma ratios Photon Material Field size DIK,,, DIK km energy (m2) 1.25 MeV* (60co) Water Air lox 10 4x4 lox 10 4x4 Aluminium lox 10 4x4 6 MeV** Aluminium Broad beam 21 MV Water 20x20 Air 20x20 * Data in parentheses from LOEVINGER (1981). ** Data in parentheses from ATTIX (1979). tabulated in the Table. The DIK,,, values for the 4 cm x 4 cm field agree well with those calculated by LOEVINGER using the mean distance, X, that the secondary electrons travel before they deposit their energy as suggested by ROESCH (1958, 1968). LOE- VINGER used a narrow beam attenuation coefficient in his calculations, which holds for the primary photons and is rather close to the effective attenuation for a small beam. For the 10 cm x 10 cm field the values are somewhat lower depending on the larger effective attenuation coefficient. Using the present model, R can easily be calculated from the relationin[p(z)l,/,l(z). It is interesting to note that because /3(z) stays constant independent of depth beyond the depth of transient equilibrium, X will decrease slowly with depth due to the slow increase with depth of p(z). The absorbed dose and the kerma practically coincide in the transient equilibrium region because for 6oCo (1-g) is 0.996 and /3 is 1.005 (REICH& TRIER 1981, GREENING). When calibrating an air equivalent ionization chamber free in air, a correction for the photon attenuation in the wall has to be made. An interesting quantity for dosimetry is therefore the ratio between the absorbed dose at maximum build-up and the collision kerma at the surface as the latter quantity can easily be obtained from an exposure or air kerma calibration and the former quantity is equal to 1.0051 1.0059 1.0051 1.0059 (1.007) 1.0043 I,0049 (1.006) 1.017 (1.018) 1.023 1.022 1.001 1 1.0019 1.001 1 1.0019 0.998 0.997 0.966 (0.967) 1.001 1.000 0.997 0.992 0.997 0.991 0.997 0.991 0.948 0.910 0.908 the absorbed dose to the air in the cavity. The ratio between the absorbed dose at maximum build-up and the collision kerma at the surface is given in the Table under the heading k,,,. These values can be used to correct for wall attenuation when using plane parallel chambers in a phantom. When using a cylindric ionization chamber free in air this ratio is not strictly applicable because the geometry is somewhat different. However, the attenuation factor for the 4 cm x 4 cm field is 0.991 for air close to what is obtained experimentally for cylindric chambers (JOHANSSON et coll. 1978, HOLT et coll.) and by Monte Carlo methods (BOND et coll. 1978). This indicates that the effect of the cylindric shape of the thimble chamber is of small importance for the value of the attenuation factors. The data can also be used to investigate the material dependence of the attenuation factor. According to the present results the material dependence is very weak (Table) with a difference of less than 0.1 per cent, as also found experimentally by JOHANS- SON et COll. For 21 MV roentgen radiation very similar results are obtained for water and air. Surprisingly enough, also at this high energy the kerma still practically coincides with the absorbed dose for depths beyond transient equilibrium as (1-g) /3 is still very close to unity. In order to compare with the results obtained by

KERMA AND ABSORBED DOSE IN PHOTON BEAMS 85 ATTIX (1979) calculations were made for 6 MeV photons in aluminium. Practically the same results were obtained with both methods (Table). SUMMARY The relation between kerma and absorbed dose has been calculated using a diffusion equation approximation for the electron transport in photon beams and by equating the depth integral of the collision kerma with that of the absorbed dose. The results show that the absorbed dose in the transient equilibrium region practically coincides with the total kerma in water both for *Co and 2 1 MV roentgen rays. The ratio between the absorbed dose at maximum build-up and the collision kerma at the surface (katt) was calculated. The results show that the dependence of k,, on the atomic number is very small. For @'Co kart was determined for water and air to 0.992 and 0.991, respectively. REFERENCES ALLISY A.: Contribution a la mesure de I'exposition produite par les photons emis par le @'Co. 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