A numerical method to optimise the spatial dose distribution in carbon ion radiotherapy planning.

The authors describe a numerical algorithm to optimise the entrance spectra of a composition of pristine carbon ion beams which delivers a pre-assumed dose-depth profile over a given depth range within the spread-out Bragg peak. The physical beam transport model is based on tabularised data generated using the SHIELD-HIT10A Monte-Carlo code. Depth-dose profile optimisation is achieved by minimising the deviation from the pre-assumed profile evaluated on a regular grid of points over a given depth range. This multi-dimensional minimisation problem is solved using the L-BFGS-B algorithm, with parallel processing support. Another multi-dimensional interpolation algorithm is used to calculate at given beam depths the cumulative energy-fluence spectra for primary and secondary ions in the optimised beam composition. Knowledge of such energy-fluence spectra for each ion is required by the mixed-field calculation of Katz's cellular Track Structure Theory (TST) that predicts the resulting depth-survival profile. The optimisation algorithm and the TST mixed-field calculation are essential tools in the development of a one-dimensional kernel of a carbon ion therapy planning system. All codes used in the work are generally accessible within the libamtrack open source platform.

The principles of Katz's cellular track structure radiobiological model.

The cellular track structure theory (TST), introduced by Katz in 1968, applies the concept of action cross section as the probability of targets in the radiation detector being activated to elicit the observed endpoint (e.g. cell killing). The ion beam radiation field is specified by the charge Z, speed ß (or energy), fluence and linear energy transfer (LET) of the ion, rather than by its total absorbed dose or dose-averaged LET. The detector is represented by radiosensitive elements of size a0 and radiosensitivity D0, its gamma-ray response being represented by c-hit or multi-target expressions rather than by the linear-quadratic formula. Key to TST is the Dδ(r) formula describing the radial distribution of delta-ray dose (RDD) around the ion path. This formula, when folded with the dose response of the detector and radially integrated, yields the 'point target' action cross section value, σPT. The averaged value of the cross section, σ, is obtained by radially integrating the a0-averaged RDD. In the 'track width' regime which may occur at the distal end of the ion's path, the value of σ may considerably exceed its geometrical value, [Formula: see text]. Several scaling principles are applied in TST, resulting in its simple analytic formulation. Multi-target detectors, such as cells, are represented in TST by m, D0, σ0 (the 'saturation value' of the cross section which replaces a0) and κ (a 'detector saturation index'), as the fourth model parameter. With increasing LET of the ion, the two-component formulation of TST allows for successive transition from shouldered survival curves at low LET values to exponential ones at radiobiological effectiveness (RBE) maximum, followed by 'thindown' at the end of the ion track. For a given cell line, having best-fitted the four model parameters (m, D0, σ0 and κ) to an available data set of measured survival curves, TST is able to quantitatively predict cell survival and RBE for this cell line after any other ion irradiation.

A TPS kernel for calculating survival vs. depth: distributions in a carbon radiotherapy beam, based on Katz's cellular Track Structure Theory.

An algorithm was developed of a treatment planning system (TPS) kernel for carbon radiotherapy in which Katz's Track Structure Theory of cellular survival (TST) is applied as its radiobiology component. The physical beam model is based on available tabularised data, prepared by Monte Carlo simulations of a set of pristine carbon beams of different input energies. An optimisation tool developed for this purpose is used to find the composition of pristine carbon beams of input energies and fluences which delivers a pre-selected depth-dose distribution profile over the spread-out Bragg peak (SOBP) region. Using an extrapolation algorithm, energy-fluence spectra of the primary carbon ions and of all their secondary fragments are obtained over regular steps of beam depths. To obtain survival vs. depth distributions, the TST calculation is applied to the energy-fluence spectra of the mixed field of primary ions and of their secondary products at the given beam depths. Katz's TST offers a unique analytical and quantitative prediction of cell survival in such mixed ion fields. By optimising the pristine beam composition to a published depth-dose profile over the SOBP region of a carbon beam and using TST model parameters representing the survival of CHO (Chinese Hamster Ovary) cells in vitro, it was possible to satisfactorily reproduce a published data set of CHO cell survival vs. depth measurements after carbon ion irradiation. The authors also show by a TST calculation that 'biological dose' is neither linear nor additive.

The application of amorphous track models to study cell survival in heavy ions beams.

In a study of amorphous track models, in the local effect model (LEM), the Kellerer algorithm was used, which folds radial dose distributions from different ion tracks. In representative set of 10 experimental cell survival curves of normal human skin fibroblast cells irradiated with carbon ions, the method that applies the Kellerer algorithm was found to be more accurate and 10(4) times faster than the usual Monte Carlo summation method based on a regular grid.