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Calibration of the NPTC Range Verifier

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Calibration of the NPTC Range Verifier Calibration of the NPTC range verifier Bernard Gottschalk∗ Harvard Cyclotron Laboratory 44 Oxford St., Cambridge, MA 01238, USA January 16, 2001 Abstract The Range Verifier (RV) is a multi-layer Faraday Cup with relatively thick brass plates. The stopping peak of a monoenergetic proton beam is coarsely binned by the RV so that the main signal appears in only three or four channels. We search for the best way to evaluate this spectrum. Fitting the data with various functions—for instance a Gaussian—works reasonably well, but a straightforward two-step computation of the mean x0 is better and much faster. We study the integral and differential lin- earity of this method using runs at many beam energies, each with several different degrader combinations. We then compare RV measurements with interleaved measurements of the range d80 in a Scanditronix (SCX) water tank. The relation between d80 and x0 requires a small quadratic term because range in brass is not exactly proportional to range in water. The calibration function is x x 2 d =0.0618 + 40.3957 0 +0.9117 0 80 35 35 and agrees with theory. The water tank measurements also yield informa- tion about the difference between d80 and d90 and about the accuracy of the Range at Nozzle Entrance (RNE) set by the NPTC treatment control program. ∗This work was performed for Ion Beam Applications s.a. (IBA). 1 Contents 1 Introduction 3 1.1 The Range Verifier . 3 1.2ErrorinRVData........................... 4 1.3 The Mean Proton Range: r0, d80 and d90 ............. 5 1.4 Equivalent Thicknesses . 5 2 Analysis of the RV Spectrum 7 2.1 Converting to Charge, Merging, Correcting for Beam Steering . 7 2.2 Computing x0 ............................. 7 2.2.1 Fitting the Spectrum . 7 2.2.2 Obtaining x0 Directly.................... 8 3 Results 8 3.1Resolution............................... 9 3.2 RNE-based Runs: Integral and Differential Linearity . 9 3.3 SCX-based Runs: x0 v. Range in Water . 10 3.4 RV Calibration Curve . 10 3.5 d80, d90 andRNE........................... 11 4 Summary and Acknowledgement 11 2 1 Introduction Each proton nozzle of the Northeast Proton Therapy Center (NPTC) includes a Range Verifier (RV) built into one set of jaws of the the variable collimator. This report analyzes recent data taken with the X jaws open and the Y jaws closed (that is, intercepting the full beam) and with nominally monochromatic beams (no depth modulation). We mainly address three questions: • What is the best way to evaluate RV range spectra? • What are the integral linearity, differential linearity and resolution? • How well do RV ranges agree with ranges from Bragg peaks? The remainder of this Introduction, however, is devoted to a brief description of the RV and three related issues: the measurement error in RV data; the definition of mean range and the quantities r0, d80 and d90 and finally, the concept of ‘water equivalent’ and of equivalent thicknesses in general. 1.1 The Range Verifier The RV consists of a Multi-Layer Faraday Cup (MLFC) in each of the two opposing downstream (Y ) jaws of the 4-jaw adjustable collimator, which are ≈ 40 cm downstream of the second scatterer and just upstream of the second ionization chamber set. A MLFC is a stack of metal collecting plates (here 2 mm brass) separated from each other and from ground by thin insulating sheets (here 0.025 mm polyethylene, two thicknesses per gap). Each collector goes to a current integrator. The MLFC has two favorable properties that may not be immediately obvious. First, the ejection of electrons from insulators into collectors and vice versa has no net effect because each such electron remains ‘bound’ to the positive ion it left behind and therefore does not contribute to the measured current. Second, protons that stop in insulators attract a mirror charge into the nearest collector, and are therefore counted just as well as protons that stop in collectors. Therefore the MLFC counts only the incident charge and counts all of it, accurately measuring differential fluence v. depth. The distribution of charge in a MLFC after exposure to monoenergetic pro- tons consists (Fig.1) of a continuum and a sharp peak. The continuum comes from the charged decay products (mostly protons) of the inelastic nuclear inter- actions experienced by some of the incident protons. The peak comes from those protons that stop entirely by electromagnetic interactions. The total charge col- lected equals the beam current entering the device, integrated over time. The continuum can be exactly accounted for by appropriate Monte Carlo models of the nuclear interaction [1]. Note especially that Fig.1 does not look like a Bragg peak and is not a Bragg peak. It is a differential stopping curve, rather than a differential energy deposition curve. For that reason the measured mean proton range corresponds to the peak value rather than the distal 80% value, as will be discussed more fully below. For reasons of economy the present RV is a bit more complicated than a simple stack of collectors. Each of the two jaws has 29 brass plates, each roughly equivalent to 1.1 cm of water. To improve the overall range resolution, one of the jaws has an additional half-thickness plate which is passive (not connected to an integrator). Precise range calculations must also take into account the 2 mm aluminum cover plates, the insulating sheets between the brass plates, and the fact that the brass plates may not have exactly the nominal thickness. If we use the rule (Sec. 1.4) that the effective energy in a finely divided stopping 3 stack is 0.52 × the incident energy, we find that the RV is ≈ 0.4% polyethylene by stopping power. When taking data with monochromatic beams, some upstream scatterer should be used to broaden the beam at the RV so as to reduce the fraction grazing the edge of the plates. In the runs presented below we used at least 1 mm of lead (lollipop #4).1 The transverse proton distribution at the RV can be inferred from the ion chambers downstream of it. The vertical distribution is sharp because the Y jaws are closed, whereas the horizontal distribution reflects the width of the ribbon of beam coming through the small gap between the Y jaws (Fig. 9). Near the maximum energy the distribution at the RV has a FWHM of ≈7 cm for a 1 mm lead scatterer, which appears to be adequate to get a clean spectrum in the RV. 1.2 Error in RV Data Because raw RV numbers are usually referred to as ‘counts’ and because of the superficial resemblance between an RV spectrum and a pulse height or Monte Carlo spectrum, there is an erroneous tendency to think of the error of each RV datum as having something to do with the square root of the number. Actually the error is far smaller. Each RV channel represents a charge measurement, not a statistical sample. The error comes from the resolution of the measurement, ∆Q = 10 pC, and is constant, independent of the ‘counts’. This is illustrated by Fig. 2, where a portion of the spectrum is shown for seven runs taken back- to-back. Because the count interval varied a bit (no beam monitor), the spectra have been normalized to the same total charge. Even the funny-looking peak channel (≈ 260 ‘counts’) has an rms spread√ of only 0.6%. If this were a statistical sample the relative error would be 1/ 260 or about 6%! The expected order of magnitude of the error is easily found. A recycling integrator with its counter is equivalent to an analog-to-digital converter having charge resolution ∆Q. The measured charge Q0 always equals ∆Q times an integer e.g. 1.000 nC for 100 counts. Only the counter, not the residual charge, is reset at the beginning of a measurement. The true charge Q that came in during the measurement interval may be nearly ∆Q less than Q0 (if the residual charge was nearly ∆Q) or it may be nearly ∆Q more (if the residual charge was nearly 0). Therefore the measurement error Q − Q0 is uniformly distributed between −∆Q and +∆Q if we assume all values of residual charge are equally likely. The rms error is therefore√ the rms deviation of a square distribution of half-width ∆Q, namely ∆Q/ 3 ≈ 0.6∆Q. Of course it is frequently the case that not all values of residual charge are equally likely. The runs may be taken back-to-back as here, leading to correlations. The measured rms deviation for all active channels in Fig. 2 varies considerably but averages only 0.4∆Q. However, without belaboring the point further we may conclude that the error in any point is at most a fraction of a count. Therefore a hundred counts or so in the peak channel is generally enough for a good measurement. That corresponds to a nanoAmp for a few seconds if the RV intercepts most of the beam. Under those circumstance the overall error in the range determination probably depends more on the algorithm used to find the range than on the number of counts, as will be seen below. 1A ‘lollipop’, so called because of its shape, is one of six lead or three Lexan degraders that may be inserted into the beam just upstream of the range modulator. 4 1.3 The Mean Proton Range: r0, d80 and d90 The distal 80% point d80 and the distal 90% point d90 of the Bragg peak are both relevant to proton radiation therapy, leading to potential confusion and annoyance.
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