Tumor control probability model for alpha-particle-emitting radionuclides.

Alpha-particle emitters are currently being evaluated for the treatment of metastatic disease. The dosimetry of alpha-particle emitters is a challenge, however, because the stochastic patterns of energy deposition within cellular targets must be taken into account. We propose a model for the tumor control probability of alpha-particle emitters which takes into account these stochastic effects. An expression for cell survival, which is a function of the microdosimetric single-event specific-energy distribution, is multiplied by the number of cells within the tumor cluster. Poisson statistics is used to model the probability of zero surviving cells within the cluster. Based on this analysis, a number of observations have been made: (1) The dose required to eradicate a tumor is nearly a linear function of the cell survival parameter z(0). (2) Cells with smaller nuclei will require more dose to achieve the same level of tumor control probability, relative to cells with larger nuclei, for an identical source-target configuration and cell sensitivity. (3) As the targeting of alpha-particle emitters becomes more specific, the dose required to achieve a given level of tumor control decreases. (4) Additional secondary effects include cell shape and the initial alpha-particle energy.

Values of "S," , and <(z1)2> for dosimetry using alpha-particle emitters.

In a recent paper [J. Nucl. Med. 38, 1923-1929 (1997)], the authors presented a dosimetry system which combines the computational ease of the MIRD schema with additional information provided by microdosimetry for use with alpha-particle emitters. In addition to the absorbed dose (average specific energy) to the targets (cell nuclei), this system gives the spread (standard deviation) in values of this specific energy received by individual targets. It also gives the fraction of targets receiving zero (or any number of) hits. In this paper, input quantities are presented for alpha-particle energies and cell and nuclear sizes appropriate for the radionuclides being investigated. The quantities include S values for the usual determination of the absorbed dose along with the microdosimetric quantities, and <(z1)2>, the average and average square, respectively, of the single-hit specific energy. Using analytical procedures described previously [Med. Phys. 19, 1385-1393 (1992)], the single-hit distributions of specific energy are determined for the given alpha-particle energies, source locations, and target sizes. From these distributions, the values for the input quantities are calculated. Sources considered are (1) those located inside and on the surface of the target cell and an unbounded source in the medium external to the cell; (2) those distributed uniformly on either side of a plane boundary or on the surface of the plane with a spherical target at various distances from the plane; and (3) those located either inside or on the surface of a spherical boundary centered externally to the target. Examples show how the input quantities are used to provide the spread in specific-energy values and the probability of any number of hits for nuclei of cells exposed to these sources. Thus a complete micro-dosimetric analysis involving the calculation of multi-hit specific energy distributions is not necessary to provide this information. Such information may be useful in interpreting the biological response due to alpha-particle emitters.

The use of microdosimetric moments in evaluating cell survival for therapeutic alpha-particle emitters.

In evaluating the efficacy of alpha-particle emitters, a cell survival curve is often determined for a particular source-target configuration. Investigators often wish to use this information about survival for a different source-target configuration which might be more appropriate for a therapeutic application. Since the population cell survival parameter, D0, is a function of the source-target configuration, it is important to determine the individual cell survival parameter, z0, which is more fundamental. Unlike D0, z0 does not depend upon the microdosimetric variations in the specific energy distribution resulting from changes in the source-target configuration. Instead it is determined by the cell sensitivity and the radiation quality. However, the calculation of z0 from the data on survival involves computing the microdosimetric specific energy distributions of the radiation. This paper describes an approximate but sufficiently accurate method for determining z0 from D0 if the first and second moments of the single-hit specific energy distributions are known or can be estimated. Examples of applications are given. This may alleviate the need for multihit microdosimetric calculations.

Dosimetric framework for therapeutic alpha-particle emitters.

UNLABELLED: Alpha-particle emitters are currently being considered for the treatment of metastatic disease. However, the dosimetry of alpha-particle emitters is a challenge because the dimensions of subcellular targets (e.g., the cell nucleus) are of the same order of magnitude as the range of the particle. Hence, a single dose value is often not representative of the dose delivered to a cell population. Here, we propose a dosimetry system that combines the calculational ease of the Medical Internal Radiation Dosimetry (MIRD) system with the additional information provided by microdosimetry. METHODS: Beginning with the microdosimetric, single-event specific-energy spectrum, we derived expressions for the first and second moments. Using the MIRD S-factor along with these moments, we determined not only the mean absorbed dose but also the s.d. and the fraction of cells receiving zero (or any number of) hits. RESULTS: Using the formalism developed here, we have generated tables for a simple example to demonstrate the use of this method. CONCLUSION: We have developed a formalism for the rapid determination of not only the mean absorbed dose but also the s.d. and fraction of cells receiving zero hits. These parameters are potentially useful in analyzing the biological outcome for cells exposed to alpha-particle irradiation.

Relationships between cell survival and specific energy spectra for therapeutic alpha-particle emitters.

Cell survival studies are a means of quantifying the biological effects of radiation. However, for alpha-particle sources, the dose-response relationship is complicated by the dominance of microdosimetric effects. In this work, we relate observed cell survival to the microdosimetric energy deposition spectra. The chord length distributions through spherical cell nuclei for sources distributed inside of, on the surface of and outside of the critical target are used as approximate analytical representations of the single-event specific energy spectra. Mathematical relationships are derived which relate cell survival to the Laplace transform of the single-event specific energy spectrum. The result is an analytical relationship between D0 (the observed slope of the cell survival curve) and Z0 (the specific energy required to reduce the survival probability of a single cell to 1/e). These studies indicate that for small energy deposition events, Z0 is approximately equal to D0. However, as the maximum energy deposited by a single event is increased, there are marked deviations between Z0 and D0. These differences between Z0 and D0 are also related to the shape of the single-event spectrum. This technique provides a powerful tool for relating observed cell survival to microdosimetric quantities for therapeutic alpha-particle emitters.

Analysis of survival of C-18 cells after irradiation in suspension with chelated and ionic bismuth-212 using microdosimetry.

A previous analysis of non-stochastic dose (Jostes et al., Radiat. Res. 127, 211-219, 1991; Schwartz et al., Health Phys. 62, 458-461, 1992) based on data obtained during irradiations of C-18 cells in suspension by alpha particles emitted from two forms (chelated and ionic) of 212Bi was made using survival curves. No appreciable difference in slope (1/D0) was found between the two forms. Such non-stochastic analyses do not account for the large differences in specific energies deposited in the individual cell nuclei. This microdosimetric (stochastic) analysis aims to determine the survival sensitivity (1/z0) of the individual C-18 cells using the distribution of specific energies deposited in the individual cell nuclei. The resulting sensitivity is greater for the alpha particles emitted from the chelated 212Bi than from the ionic 212Bi. An attempt to account for this greater sensitivity in terms of greater LET of alpha particles passing through the cell nuclei from the chelated 212Bi is unsuccessful. Instead the greater sensitivity disappears if the microdosimetric analysis uses average values for the radii of the cell and of its nucleus rather than the values (from the peak in the cell size distribution) used by the non-stochastic dose analysis.

Analytic microdosimetry for radioimmunotherapeutic alpha emitters.

Analytic microdosimetry using Fourier transform techniques has been applied to internal alpha emitters. These techniques need revision and simplification for use with short-lived radionuclides such as those which may be useful for radioimmunotherapy. Analytic methods may have advantages over Monte Carlo methods in some cases (e.g., where time is important). Applications to eight different source geometries show the usefulness of these techniques. Comparisons of some of the results of Monte Carlo calculations prove its accuracy. For a uniform source of 5.867-MeV alphas spread throughout the volume outside a cell surface, the two methods agree well. Results are within 1% both for the average specific energy and for the number of hits. Analytic microdosimetry provides an alternate method to use for the critical evaluation of models that seek to predict the relation between alpha energy deposition and cell survival data. Similarly, it may be helpful to point the way toward the rational interpretation of general biological results for antibodies labeled with alpha emitters.

Obtaining S values for rectangular-solid tumors inside rectangular-solid host organs.

A method is described for obtaining S values between a tumor and its host organ for use with the MIRD formalism. It applies the point-source specific absorbed fractions for an infinite water medium, tabulated by Berger, to a rectangular solid of arbitrary dimensions which contains a rectangular tumor of arbitrary dimensions. Contributions from pairs of source and target volume elements are summed for the S values between the tumor and itself, between the remaining healthy host organ and itself, and between the tumor and the remaining healthy host organ, with the reciprocity theorem assumed for the last. This method labeled MTUMOR, is interfaced with the widely used MIRDOSE program which incorporates the MIRD formalism. An example is calculated.

Correlation of microdosimetric measurements with relative biological effectiveness from clinical experience for two neutron therapy beams.

Microdosimetric measurements were made for the neutron therapy beams at the University of Chicago and at the Cleveland Clinic with the same geometry and phantom material using the same tissue-equivalent spherical proportional counter and standard techniques. The energy deposition spectra (dose distributions in lineal energy) are compared for these beams and for their scattered components (direct beam blocked). The model of dual radiation action (DRA) of Kellerer and Rossi is employed to interpret these data in terms of biological effectiveness over this limited range of radiation qualities. The site-diameter parameter of the DRA theory is determined for the Cleveland beam by setting the biological effectiveness (relative to 60Co gamma radiation) equal to the relative biological effectiveness value deduced from radiobiology experiments and clinical experience. The resulting value of this site-diameter parameter is then used to predict the biological effectiveness of the Chicago beam. The prediction agrees with the value deduced from radiobiology and clinical experience. The biological effectiveness of the scattered components of both beams is also estimated using the model.

Neutron quality parameters versus energy below 4 MeV from microdosimetric calculations.

Charged-particle production by neutrons and the resulting energy-deposition spectra in micron-sized spheres of tissue of varying diameters were calculated from thermal energies to 4 MeV. These data were used to obtain dose-average values of several quality-indicating parameters as functions of neutron energy and of tissue sphere diameter. The contrast among the parameters is shown and discussed. Applications are made to two neutron spectra, one a fission spectrum in air and the other a moderated spectrum at the center of an irradiated cube of water.

Neutron dose equivalent next to the target shield of a neutron therapy facility using an LET counter.

The use of a spherical tissue-equivalent proportional counter for measurements of the lineal energy (y) and derivations of the linear energy transfer (LET) for fast neutrons has the advantage of giving distributions of dose and dose equivalent as functions of either LET or y. A measurement next to the target shielding of the neutron therapy facility at the University of Chicago Hospitals and Clinics (UCHC) is described, and the data processing is outlined. The distributions are presented and compared to those from measurements in the neutron beam. The average quality factors are presented.

Comparison of the microdosimetric event-size method and the twin-chamber method of separating dose into neutron and gamma components.

Microdosimetric measurements of event-size spectra, made with a proportional counter, are being used increasingly for separation of dose components in mixed n-gamma fields. Measurements in fields produced by 8.3 MeV deuteron bombardment of thick beryllium and deuterium targets were made in air and at 6 and 12 cm depth in water with a spherical tissue-equivalent (TE) proportional counter and with a pair of calibrated ion chambers (TE-TE and Mg-Ar). The dose results obtained with the two methods agree well for the neutron components, but the gamma components do not demonstrate consistent agreement. An important source of error in the microdosimetric method is the matching of the spectra measured at different gain settings to cover the large range of event sizes. The effect of this and other sources of error is analysed.

Calculations of charged-particle recoils, slowing-down spectra, LET and event-size distributions for fast neutrons and comparisons with measurements.

A rapid system has been developed for computing charged-particle distributions generated in tissue by any neutron spectra less than 4 MeV. Oxygen and carbon recoils are derived form R-matrix theory, and hydrogen recoils are obtained from cross-section evaluation. Application to two quite different fission-neutron spectra demonstrates the flexibility of this method for providing spectral details of the different types of charged-particle recoils. Comparisons are made between calculations and measurements of event-size distributions for a sphere of tissue 1 micrometer in diameter irradiated by these two neutron spectra. LET distributions have been calculated from computed charged-particle recoils and also derived from measurements using the conventional approximation that all charged particles traverse the chamber. The limitations of the approximation for these neutron spectra are discussed.