A Statistical Description of Nuclear Reaction Models for Medical Radionuclides: the Paradigmatic Case of 47 Sc Production with Thick Vanadium Targets
AA Statistical Description of Nuclear Reaction Models for Medical Radionuclides:the Paradigmatic Case of Sc Production with Thick Vanadium Targets
M.P. Carante , F. Barbaro , L. Canton , A. Colombi , , A. Fontana INFN, Sezione di Pavia, Pavia, Italy, INFN, Sezione di Padova, Padova, Italy, Dipartimento di Fisica dell’Universit`a di Pavia, Pavia, Italy.
Theoretical and computational support is an essentialingredient to aid experimental investigations in findingefficient routes and conditions for the production ofemerging radionuclides for medical applications: thisis demonstrated for example by the studies on thescandium, tin and manganese radionuclides described inother contributions in this Annual Report [1–4]. Differentnuclear reaction codes have been developed and have nowbecome a reference to study the physics of nuclear reactionswith cyclotrons: however none of them can serve as optimaltool and their predictions are often different. Therefore, itis important to be able to assess their theoretical variabilityfor a given reaction or for a particular energy interval.In this note we focus our attention on the models thatare used in code TALYS [5] (version 1.9): this codeprovides a default configuration which has been used inmany calculations. On the contrary, some authors defined adifferent configuration of the code parameters (referred oftento as adjusted [6]) which is believed to provide more reliableresults in selected cases. In our approach we propose toinclude all the models provided by the code and to consider a”Best Theoretical Evaluation” (BTE) that could be used as areference value for the models, in analogy with the evaluated cross section that is recommended for example by IAEA forthe experimental data, or with the TENDL library.
METHODS
The nuclear reaction mechanisms relevant forradionuclide production at cyclotrons are dominatedby compound nucleus formation and by pre-equilibriumemission. TALYS is based on the Hauser-Fesbachmodel where the compound-nucleus formation implies anintermediate state of thermodynamic equilibrium beforeevaporation and de-excitation. The formation of thisstate depends on the level density at the excitation energycorresponding to the projectile incident energy. Regardingpre-equilibrium emission, the models used by TALYS canbe divided in two categories, the exciton model and themulti-step model, for which the code provides differentoptions. In general, TALYS has a built-in variety of fourmodels of pre-equilibrium (PE) and six models for nuclearlevel densities (LD), for a total of 24 different combinationsof models.It is important to assess the uncertainties involved in thecalculations which depend both on the reaction mechanism and on the availability of the relevant nuclear data. Toachieve this goal, we introduce some simple statisticalconcepts to deal with the theoretical variability provided bythe different models. Instead of plotting all the 24 curves, wecompare the different results by means of a statistical band(similarly to what was done in [7]). This band is constructedupon the interquartile range for all models, namely thedifference between the third ( Q ) and the first ( Q ) quartile.In this context we define for each energy the BTE crosssection by taking the average of the first and third quartile,and associate to it the uncertainty given by the half-width ofthe interquartile band: σ BTE = Q + Q , ∆σ BTE = Q − Q . This provides a reference value that takes into account thetheoretical models variability in a simple way. The sameprocedure is applied to evaluate the statistical band not onlyfor cross sections but also for yields, activities, isotopic, andradionuclidic purities, both for the desired nuclide and for itscontaminants. If applicable, the band can be plotted also asa function of time to study the time evolution of the relevantquantities.
Fig. 1. Theoretical results for the cross sections with the statisticalband description.
RESULTS
As application of these concepts we consider theproduction of the radionuclide Sc with proton beams a r X i v : . [ nu c l - t h ] J un mpinging on thick targets of natural Vanadium. Schas gained interest recently as an emerging theranosticthanks to the emission of β − particles that can delivercytotoxic doses to small-medium sized tumors, and γ -rayssuitable for SPECT imaging. The Vanadium target appearspromising due to the possibility to produce Sc with naturalmaterials since their costs are rather inexpensive if comparedwith alternative, enriched targets. On the other hand,isotopic and radionuclidic purity should be considered ofparamount importance in any study on production routes forradiopharmaceuticals application.We consider, in Fig. 1, the cross section for the nat V ( p , x ) Sc reaction. The available experimental data,including the ones recently measured within the INFNPASTA project, are compared with this new approach. Inthis case, we considered 3 PE models and 6 different LDmodels, for a total of 18 combinations. We excludedcalculations with the fourth PE model because the resultsturned out unphysical on various occasions. The dashedlines indicate the maximum and minimum values of allthe models and it is clear that the model variability is toohigh: depending on the selected models, the descriptioncan either over- or under-estimate by large the data. If weconsider the interquartile band, spanning quartiles Q1 andQ3, we get a more reasonable description with a narrowerband, where only 50% of the calculations are included.This description appears much more practical, since it keepsthe width reasonably small by trimming the calculations atthe edge of the set. It can be seen from the figure thatthe comparison theory-experiment is more meaningful inthis way. The solid black line is the BTE and representsthe center of the band. The half-thickness is the spreadgenerated by the variability of the models and we suggestit as a theoretical ”error” indicator.Next, we consider the production yield. As wellknown, this is calculated by an energy integration over theproduction cross section times the target density and dividedby the stopping power. The other factors in front of theintegral depend on the physical parameters of the beam andtarget and they are detailed in the quoted contribution by A.Colombi, et al. In Fig. 2 we plot the integral yield at theEnd of Bombardment for an irradiation of 1 h, consideringa sufficiently thick target so that the beam energy degradatesto zero, i.e. without leaving the target. We calculatedthe yields for all the 18 TALYS models and repeated thestatistical analysis, finding the upper and lower limits of theset of calculations (dashed lines). The narrower interquartileband (in blue) with the BTE given by the center of theband provides a good way to estimate the production yieldand includes by construction a simple way to assess thetheoretical indeterminations.The production yield of a target of given thickness canbe obtained from the difference of the integral yield (shownin Fig. 2) calculated at E in and E out , the energies of theincoming and outgoing beam with respect to the target.If we set E in = . E out = . nat V-target thickness of 1080.81 µ m, and the correspondingenergy interval is shown in Fig.2 with the green strip. Withthese irradiation conditions for the Sc production yieldwe obtain Y p = ±
46 KBq/ µ Ah with the theoreticaluncertainty derived from propagating the width of the bandshown in Fig.2. Similarly, we have evaluated the productionyield of Sc , the main Scandium contaminant, the only onewith an half-life longer than Sc. With the same irradiationcondition we get for Sc Y p = ± µ Ah, andthis confirms that the irradiation conditions are optimal forthe production with an high level of purity.
Fig. 2. Integral yield as a function of beam energy with the errorband.