Fourier deconvolution reveals the role of the Lorentz function as the convolution kernel of narrow photon beams.

The two-dimensional lateral dose profiles D(x, y) of narrow photon beams, typically used for beamlet-based IMRT, stereotactic radiosurgery and tomotherapy, can be regarded as resulting from the convolution of a two-dimensional rectangular function R(x, y), which represents the photon fluence profile within the field borders, with a rotation-symmetric convolution kernel K(r). This kernel accounts not only for the lateral transport of secondary electrons and small-angle scattered photons in the absorber, but also for the 'geometrical spread' of each pencil beam due to the phase-space distribution of the photon source. The present investigation of the convolution kernel was based on an experimental study of the associated line-spread function K(x). Systematic cross-plane scans of rectangular and quadratic fields of variable side lengths were made by utilizing the linear current versus dose rate relationship and small energy dependence of the unshielded Si diode PTW 60012 as well as its narrow spatial resolution function. By application of the Fourier convolution theorem, it was observed that the values of the Fourier transform of K(x) could be closely fitted by an exponential function exp(-2pilambdanu(x)) of the spatial frequency nu(x). Thereby, the line-spread function K(x) was identified as the Lorentz function K(x) = (lambda/pi)[1/(x(2) + lambda(2))], a single-parameter, bell-shaped but non-Gaussian function with a narrow core, wide curve tail, full half-width 2lambda and convenient convolution properties. The variation of the 'kernel width parameter' lambda with the photon energy, field size and thickness of a water-equivalent absorber was systematically studied. The convolution of a rectangular fluence profile with K(x) in the local space results in a simple equation accurately reproducing the measured lateral dose profiles. The underlying 2D convolution kernel (point-spread function) was identified as K(r) = (lambda/2pi)[1/(r(2) + lambda(2))](3/2), fitting experimental results as well. These results are discussed in terms of their use for narrow-beam treatment planning.

Dosimetric characteristics of an unshielded p-type Si diode: linearity, photon energy dependence and spatial resolution.

The unshielded Si diode PTW 60012, used for accurate measurements of the transversal dose profiles of narrow photon beams, has been investigated with regard to its linearity, photon energy dependence and spatial resolution. The diode shows a slight supralinearity, i.e., increase of the response with pulse dose, by 3% over the pulse dose range 0.1 to 0.8 mGy. In p-type silicon, supralinearity results from the increased chance for radiation-induced electrons to escape recombination when the pulse dose increases. Over the energy range from 6 to 15 MV, the response decreases by about 4%. This small variation of the response results from partial compensation between the influences of the secondary electron energy on the mass stopping power ratio silicon/water and on electron backscattering from the silicon chip. The lateral response function of the examined diode has a full half width of 1.3 mm. Dose profiles of 5 mm half-width can still be recorded with negligible error.

The effect of a carbon-fiber couch on the depth-dose curves and transmission properties for megavoltage photon beams.

PURPOSE: To investigate the attenuation of a carbon-fiber tabletop and a combiboard, alongside with the depth-dose profile in a solid-water phantom. MATERIAL AND METHODS: Depth-dose measurements were performed with a Roos chamber for 6- and 10-MV beams for a typical field size (15 cm x 15 cm, SSD [source-surface distance] 100 cm). A rigid-stem ionization chamber was used to measure transmission factors. RESULTS: Transmission factors varied between 93.6% and 97.3% for the 6-MV beam, and 95.1% and 97.7% for the 10-MV photon beam. The lowest transmission factors were observed for the oblique gantry angle of 150 degrees with the table-combiboard combination. The surface dose normalized to a depth of 5 cm increased from 59.4% (without table, 0 degrees gantry), to 108.6% (tabletop present, 180 degrees gantry), and further to 120% (table-combiboard combination) for 6-MV photon beam. For 10 MV, the increase was from 39.6% (without table), to 88.9% (with table), and to 105.6% (table-combiboard combination). For the 150 degrees angle (tablecombiboard combination), the dose increased from 59.4% to 120% (6 MV) and from 39% to 108.1% (10 MV). CONCLUSION: Transmission factors for tabletops and accessories directly interfering with the treatment beam should be measured and implemented into the treatment-planning process. The increased surface dose to the skin should be considered.

Spatial resolution of 2D ionization chamber arrays for IMRT dose verification: single-detector size and sampling step width.

The spatial resolution of 2D detector arrays equipped with ionization chambers or diodes, used for the dose verification of IMRT treatment plans, is limited by the size of the single detector and the centre-to-centre distance between the detectors. Optimization criteria with regard to these parameters have been developed by combining concepts of dosimetry and pattern analysis. The 2D-ARRAY Type 10024 (PTW-Freiburg, Germany), single-chamber cross section 5 x 5 mm(2), centre-to-centre distance between chambers in each row and column 10 mm, served as an example. Additional frames of given dose distributions can be taken by shifting the whole array parallel or perpendicular to the MLC leaves by, e.g., 5 mm. The size of the single detector is characterized by its lateral response function, a trapezoid with 5 mm top width and 9 mm base width. Therefore, values measured with the 2D array are regarded as sample values from the convolution product of the accelerator generated dose distribution and this lateral response function. Consequently, the dose verification, e.g., by means of the gamma index, is performed by comparing the measured values of the 2D array with the values of the convolution product of the treatment planning system (TPS) calculated dose distribution and the single-detector lateral response function. Sufficiently small misalignments of the measured dose distributions in comparison with the calculated ones can be detected since the lateral response function is symmetric with respect to the centre of the chamber, and the change of dose gradients due to the convolution is sufficiently small. The sampling step width of the 2D array should provide a set of sample values representative of the sampled distribution, which is achieved if the highest spatial frequency contained in this function does not exceed the 'Nyquist frequency', one half of the sampling frequency. Since the convolution products of IMRT-typical dose distributions and the single-detector lateral response function have no or very small frequency contributions beyond 0.1 mm(-1), the mathematical approach introduced by Nyquist and Shannon shows that the sampling frequency of 0.2 mm(-1) is appropriate. Overall it is shown that the spatial resolution of the 2D-ARRAY Type 10024 is appropriate for the dose verification of IMRT plans. The insights obtained are also applied in the discussion of other available two-dimensional detector arrays.

Two-dimensional ionization chamber arrays for IMRT plan verification.

In this paper we describe a concept for dosimetric treatment plan verification using two-dimensional ionization chamber arrays. Two different versions of the 2D-ARRAY (PTW-Freiburg, Germany) will be presented, a matrix of 16 x 16 chambers (chamber cross section 8 mm x 8 mm; the distance between chamber centers, 16 mm) and a matrix of 27 x 27 chambers (chamber cross section 5 mm x 5 mm; the distance between chamber centers is 10 mm). The two-dimensional response function of a single chamber is experimentally determined by scanning it with a slit beam. For dosimetric plan verification, the expected two-dimensional distribution of the array signals is calculated via convolution of the planned dose distribution, obtained from the treatment planning system, with the two-dimensional response function of a single chamber. By comparing the measured two-dimensional distribution of the array signals with the expected one, a distribution of deviations is obtained that can be subjected to verification criteria, such as the gamma index criterion. As an example, this verification method is discussed for one sequence of an IMRT plan. The error detection capability is demonstrated in a case study. Both versions of two-dimensional ionization chamber arrays, together with the developed treatment plan verification strategy, have been found to provide a suitable and easy-to-handle quality assurance instrument for IMRT.

The dose-area product, a new parameter for the dosimetry of narrow photon beams.

In the dosimetry of narrow photon fields with side lengths of the order of 1 cm, the traditional parametrisation via the absolute dose on the beam axis and the relative lateral dose distribution has to deal with the difficulty to find sufficiently small detectors and to adjust them accurately on the narrow-beam axis. This can be avoided by reconsidering the parametrisation, using as normalization factor the surface integral of the dose in the plane perpendicular to the beam axis, abbreviated as the "dose-area product" (DAP). We investigated and confirmed the ability of a large-area parallel-plate ionisation chamber, with a sensitive volume shaped as a flat cylinder of 81.6 mm diameter and 2 mm thickness, to perform the integration over the full lateral dose profile of narrow photon beams with side lengths up to 5 cm. The lateral adjustment of this large-area detector relative to a narrow photon beam is not critical. The large-area ionisation chamber was calibrated in terms of the DAP by reference to a 0.3 cm3 ionisation chamber. A field-size dependent "modified output factor" was defined as the ratio of the DAP measured at 5 cm phantom depth for 100 cm SSD, and the monitor reading. A prominent phenomenon of narrow photon fields is the field-size and source-distance independence of the relative axial profile of the DAP as function of the thickness of a pre-absorber or of the depth in a phantom. For narrow-beam treatment planning in IMRT, the DAP is combined with the energy- and field size-dependent relative lateral dose distribution which is represented, for example, by a Gaussian convolution kernel. Another useful feature of the DAP is the possibility of its direct control during patient irradiation by means of an on-line monitor with spatial resolution, arranged in the accessory holder.

[Resolution and sensitivity of a two-dimensional ionisation chamber array (PTW type 10024)]. / Uber das Auflösungsvermögen und die Empfindlichkeit eines zweidimensionalen Ionisationskammer-Arrays (PTW: Typ 10024).

The two-dimensional verification of intensity-modulated radiation plans is one of the major requirements for the safe application of this technique. The present study examines the resolution and sensitivity of a two-dimensional ionisation-chamber array (PTW2D-Array, type 10024), which can be used for plan verification instead of films. According to the Shannon-Nyquist theorem, the resolution of the 2D-Array is sufficient for dose distributions with a minimal field size of 2 cm x 2 cm. The minimal field size can be reduced to 1 cm x 1 cm by shifting the array 5 mm in the direction of the MLC movement and by repeating the measurements. The high sensitivity against a monitor decalibration for a single field of a sequence is demonstrated on the basis of an individual case. The minimal threshold for MLC misalignment detected by a chamber of the array is less than 1 mm. Therefore, the resolution capabilities of the 2D-Array are sufficient for most intensity-modulated radiation therapy (IMRT)fields.