Microdosimetric derivation of inactivation RBE values for heavy ions using data from track-segment survival experiments

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1 Microdosimetric derivation of inactivation RBE values for heavy ions using data from trac-segment survival experiments Gustavo A. Santa Cruz a a Comisión Nacional de Energía Atómica, Av. del Libertador 85 (49), Buenos Aires, Argentina. Abstract. We describe a methodology for deriving heavy ion RBE inactivation values using data obtained from trac-segment survival experiments, within the framewor of microdosimetry. We assume that the yield of lethal lesions can be expressed as a polynomial function of the specific energy accumulated within interaction regions belonging to the cell nucleus. The associated formalism intends to follow a natural extension of the Compound Dual Radiation Action (CDRA) theory, and sees to explain the production of higher-order damage entities (complex chromosome rearrangements), which are produced very efficiently by heavy ions. One of the goals of the present approach is to derive inactivation RBE values for heavy ion traectories in tissue that do not comply with the trac-segment approximation, e.g. trac lengths too short for assuming a constant LET. An important situation that can be described is the non-uniform production of heavy ions at microscopic scale, as in Boron Neutron Capture Therapy (BNCT). To explore the adequacy of the proposed approach, the coefficients of the polynomial function describing the yield of lesions are calculated from published V79 trac-segment survival data, using the simplest expressions for the survival coefficients alpha and beta that describe their general observed LET dependence. As a result, a sensitive site of about.8 microns is found to be a suitable microdosimetric volume. A cubic term in the specific energy appears to be appropriate for describing the moderate to high-let behavior of the survival initial slope. As an example of application, we derive V79 RBE values of the two most important neutron capture reactions in BNCT: the B and 4 N thermal neutron capture reactions. RBE inactivation values for V79 cells of.4 for boron reactions and.9 for nitrogen reactions are obtained at.% survival relative to X-rays, in good agreement with published data, showing small variations between different boron microdistributions. KEYWORDS: Microdosimetry, dual radiation action, heavy ion, BNCT.. Introduction. Formalism We assume the existence of a function ε ( z) that relates the specific energy deposited by one or more independent events in a sensitive site with the yield of lethal lesions in it. By integrating this function with the multi-event density function (MEDF) [], f ( zd,, ) of a particular radiation type, we obtain ε (, ) ε D = z f z D dz () where ε ( D) is the average number of lethal lesions produced in a site at the absorbed dose, D. It is important to note that there is no a priori reference to the sensitive site size or shape for which the ε z is described by a polynomial of degree Q in z: MEDF has to be calculated. Let us suppose that ε ( z) Q = A z = The principal lesions, double-strand breas (DSBs), are created with a yield proportional to z. They interact producing lethal damage entities (lethal DSBs, -brea aberrations, complex chromosome rearrangements, etc.). Their yields are proportional to the integer powers of z, with a maximum degree of complexity, given by Q. () Presenting author, santacr@cnea.gov.ar

2 We assume that ε ( ) = (the site starts to exist with at least one relevant energy transfer, i.e., after a principal lesion is formed). From () and () it follows that where ε ( D) M is the -order moment of f (, ) Since that f ( zd, ) Q = AM (3) = zd: M z f ( z, D) dz = (4) comes from a Poisson compound process, there exists a recursive relationship m between M and the moments of the single-event density function (SEDF) []: M m (5) + + = CM m+ n = where n is the expected number of events at dose D, and =! coefficient. Expression (5) can also be written as M = M C hmhq h h= (6) where q = m m ; l. The first moment, M l l = nm, is the dose, M = D. Let us suppose that the average yield of lethal lesions has a linear-quadratic (LQ) dependence on D, which is usually appropriate for the range of doses involved in radiobiology or radiotherapy: = + (7)!! C is the binomial ε D αd βd LQ For this particular dependence, expression (3) should admit only those contributions that contain terms proportional to M and M. From Eq. (6), the -order moment, expanded up to order in M is: () = + h h h h= M Mq M C qq Therefore, Q Q Q ε LQ( D) = AM = Aq D+ A C hqhq h D = = = h= and comparing with (7), the bracets in (9) are therefore identified with Q α = Aq () β (8) () = Q A C q q h h h = h= = () If we expand expressions () and () up to third order in, (assuming that Q > ) we obtain: α = A + A q + Aq 3 3 () β = A + 3Aq 3 Note that q = zd, the dose-mean specific energy. If the coefficient A 3 is zero, we obtain the familiar expressions of Dual Radiation Action (DRA) theories [-4], which include at most -brea chromosome aberrations.. Trac-Segment Approximation (TSA) In order to obtain the values of the coefficients A from trac-segment experiments, we have to mae some assumptions regarding the size of the site and the particle s traectory. We assume that the particle s traectory throughout the sensitive site is a straight-line segment of constant LET, and that the size of the site is small enough to ensure the validity of the TSA [5]. Therefore, the specific energy (9)

3 deposited along the particle s path is proportional to the length of the chord, l, and the -order specific energy moments are z = clm. µ (3) where c =.36 Gy/eV, L is the average LET, m is the site mass and µ is the -order moment of the site s isotropic chord-length density [5]. For a spherical site of diameter d and density g/cm 3 : 3 q = z z = ( cld ) (4) F + Therefore, expressions () can be written as α = B + B L+ B L ; α (5) β = d B + B L ; 3 3c 4c β (6) where B 3A ( cd ) ( ) = +. Eqs. (), (5) and (6) form the simplest set of expressions that could describe the general observed dependence of α and β with LET in a specific range, i.e., increasing of alpha values up to a maximum and decreasing for higher LET (which occurs if A 3 is negative), and a continuous decreasing of beta values for increasing LET (which is consistent with negative A 3 ). Once the coefficients A are determined, the general expressions () can be used for any other situation where the TSA is not suitable, e.g., short heavy ion tracs with highly variable stopping power along their traectory, such as boron neutron capture reactions. Moreover, it is nown in BNCT that the boron compounds used clinically appear to distribute non-uniformly at microscopic scale, and therefore dosimetric computations based on erma factors are not appropriate for calculating the boron dose to cells [6].. Results. Experimental Data In order to obtain the parameters for the LQ dependence, we use expression (5) to fit data provided by trac-segment heavy ion survival experiments, finding the B m coefficients from the initial slope α ( L). Figure (a) shows a fit of the alpha coefficient for V79 cells survival data compiled from several ion experiments [7]. Since the remaining unnown parameter is the site diameter, we fit the experimental β L values by adusting only the factor d in expression (6) (see Figure (b)). Figure. Fits of experimental alpha (a) and beta (b) values and 95% confidence bands. 3

4 Table summarizes the values for A and d. The diameter for the sensitive site obtained is in agreement with the TSA for heavy ions of moderate energy. Table. Values for the coefficients and site s diameter obtained from the experimental data. Parameter Value LL95% UL95% A (Gy - ) A (Gy - ) A 3 (Gy -3 ) -.87E-4-3.4E E-4 d (µm) The coefficient α reaches its maximum value when L max = B B 3. In this case, L max = 85.7 ev/µm.. Derivation of B and 4 N RBEs We have assumed that equations () describe the dependence of α and β for irradiated V79 cells with heavy ions of intermediate energy, and that the sensitive site is a small sphere of.8 microns diameter. If we now calculate the moments of the SEDF for boron reactions and use Eqs. (), we obtain the derived α and β coefficients and the corresponding RBEs as a function of erma in comparison with photon reference data. The boron or nitrogen SEDF moments can be calculated with the formalism developed by Santa Cruz et al. [7], which includes the case of non-uniform distributions. Values of the frequency-mean lineal energy, y, and calculated alpha and beta coefficients for uniform (U) boron and nitrogen F (ev/µm) microdistributions, for extrasite (E), and 3: (inside:outside) boron microdistributions, are given in Table. To obtain RBE values for the reactions considered, we will use as photon reference the survival data published by Gabel et al [8]. A LQ fit of these X-ray data yields the values α =.83 Gy and - β =.337 Gy. Table. Calculated frequency mean lineal energy, alpha and beta values for boron and nitrogen capture reactions. B (U) B (E) B (3:) 4 N (U) y F α Gy β ( Gy ) The RBE dependence on the heavy ion dose D, for LQ expressions is ξ 4 RBE ( D) = + ( ξρ D + χ D ) D ξ where ξ = α β for the reference radiation, subscript, ρ = α α is the ratio of initial slopes and χ = β β is the ratio of the beta values. Figure is a plot of RBE vs. survival for boron and nitrogen neutron capture reactions. (7) 4

5 Figure. RBE values vs. survival level for boron and nitrogen capture reactions. The values determined here for a survival level of.% are in agreement with the RBEs obtained by Gabel et al. [8] i.e.,.3 for boric acid (which is assumed that distributes uniformly throughout the cells) and.9 for nitrogen reactions. Very slight differences are observed between the different microdistributions within the calculated errors (Figure ). If we plot the boron initial slope as a function of the relative concentration a (see Figure 3) we find that its value is relatively constant for extrasite, uniform and slightly intrasite distributions, and as the microdistribution becomes more internal, the slope gradually decreases. This can be explained by considering the alpha dependence on the frequency-mean lineal energy (Figure 3, right axis), which correlates with the experimental data (Figure ), although as a function of LET instead of lineal energy. Figure 3. Alpha coefficient and frequency-mean lineal energy, as a function of the relative boron concentration. 5

6 4. Discussion and conclusions Although no attempt has been made to identify the sensitive site with any particular cell structure, it is assumed that the sites actually represent volumes that belong to the cell nucleus and can be present in it with some multiplicity. These volumes may correspond to spatially limited interaction regions where damaged entities combine to from a lethal lesion. A negative value for the term of order 3 could be associated with a combinatorial competition per unit dose between higher and lower order damage entities. The approximation of constant LET in trac-segment experiments with heavy ions of moderate velocity is in general a satisfactory assumption [5], and permits calculating the coefficients A in a very straightforward way. Once they are obtained, the more general relations with specific energy (Eqs. ()) can be used to derive the inactivation parameters corresponding to a different situation, e.g., the inactivation by boron and nitrogen neutron capture reactions. Since we restricted to third order the dependence of the yield of lesions on the specific energy, it is not advisable to predict inactivation parameters beyond 3 ev/µm, where the inactivation cross-section usually shows saturation. In order to do so, higher order terms should be included, and correspondingly higher LET inactivation data. The use of beta values to derive the sensitive site volume is not accurate, since the data have associated very large uncertainties. However, it can be shown that the resulting single-hit inactivation parameter alpha is rather insensitive to variations in the site size (less than 4% for site diameters between.4 µm and. µm), although the quadratic coefficient beta shows important differences. The size of the microdosimetric volume seems to be not crucial for boron reactions, but is critical for nitrogen reactions (and in general, for LETs below 4 ev/µm, e.g., the LET of proton recoils from the neutron elastic scattering reaction with hydrogen). This methodology, together with the present approach can be of importance to predict pure RBEs, in the sense that they are derived from data obtained in more controlled experiments, free from possible interfering factors such as synergistic interactions with other radiation components, or non uniform microdistributions of boron compounds. REFERENCES [] ROSSI, H. H. AND ZAIDER, M., Microdosimetry and its Applications, p., Springer, Berlin Heidelberg, New Yor, 996. [] KELLERER, A. M. AND ROSSI, H., The Theory of Dual Radiation Action. Curr. Top. Radiat. Res. Q. 8, (97). [3] KELLERER, A. M. AND ROSSI, H. H.. A Generalized Formulation of Dual Radiation Action. Radiat. Res. 75, (978). [4] ROSSI, H. H. AND ZAIDER, M., Compound Dual Radiation Action. I. General Aspects. Radiat. Res. 3, (99). [5] ICRU REPORT 36, MICRODOSIMETRY, p.4, ICRU, Bethesda, Maryland, 983. [6] SANTA CRUZ, G. A. AND ZAMENHOF, R. G., The Microdosimetry of the B Reaction in Boron Neutron Capture Therapy: A New Generalized Theory, Radiat. Res. 6, 7 7 (4). [7] BELLONI, F. et al., Inactivation Cross Sections for Mammalian Cells Exposed to Charged Particles: a Phenomenological Approach, Radiat. Prot. Dosim. 99 (-4), 99- (). [8] GABEL, D., FOSTER, S. AND FAIRCHILD, R. G., The Monte Carlo Simulation of the Biological Effect of the B(n,α)7Li Reaction in Cells and Tissue and its Implication for Boron Neutron Capture Therapy, Radiat. Res. (), 4-5 (987). 6