Sub-millimeter precise photon interaction position determination in large monolithic scintillators via convolutional neural network algorithms

Physics in Medicine & Biology PAPER • OPEN ACCESS Sub-millime e p ecise pho on in e ac ion posi ion de e mina ion in la ge monoli hic scin illa o s ia con olu ional neu al ne wo k algo i hms To ci e his a icle: M Kawula e al 2021 Phys. Med. Biol. 66 135017 View he a icle online o upda es and enhancemen s. This con en was downloaded om IP add ess 193.136.214.165 on 20/08/2021 a 16:58 Phys. Med. Biol. 66 (2021)135017 h ps://doi.o g/10.1088/1361-6560/ac06e2 PAPER Sub-millime e p ecise pho on in e ac ion posi ion de e mina ion in la ge monoli hic scin illa o s ia con olu ional neu al ne wo k algo i hms M Kawula 1 , T M Binde 1,2 , S Lip andi 1 , R Viegas 1,3 , K Pa odi 1 and P G Thi ol 1,∗ 1 Depa men o Medical Physics, Ludwig-Maximilians-Uni e si ä München, Ga ching b. München, Ge many 2 KETEK GmbH, Munich, Ge many 3 Uni e si y o Coimb a, Po ugal ∗ Au ho o whom any co espondence should be add essed. E-mail: [email p o ec ed] and [email p o ec ed] Keywo ds: beam ange moni o ing, Comp on came a, had on he apy, monoli hic scin illa o , neu al ne wo ks, adia ion de ec ion, spa ial esolu ion Abs ac In his wo k, we p esen he de elopmen and applica ion o a con olu ional neu al ne wo k (CNN)- based algo i hm o p ecisely de e mine he in e ac ion posi ion o γ-quan a in la ge monoli hic scin illa o s. Those a e used as an abso be componen o a Comp on came a (CC)sys em unde de elopmen o ion beam ange e ifica ion ia p omp -gamma imaging. We examined wo scin illa ion c ys als: LaB 3 :Ce and CeB 3 . Each c ys al had dimensions o 50.8 mm ×50.8 mm × 30 mm and was coupled o a 64- old segmen ed mul i-anode pho omul iplie ube (PMT)wi h an 8 × 8 pixel a angemen . We de e mined he spa ial esolu ion o h ee pho on ene gies o 662, 1.17 and 1.33 MeV ob ained om 2D de ec o scans wi h igh ly collima ed 137 Cs and 60 Co pho on sou ces. Wi h he new algo i hm we achie ed a spa ial esolu ion o he CeB 3 c ys al below 1.11(8)mm and below 0.98(7)mm o he LaB 3:Ce de ec o o all in es iga ed ene gies be ween 662 keV and 1.33 MeV. We he eby imp o ed he pe o mance by mo e han a ac o o 2.5 compa ed o he p e iously used ca ego ical a e age pa e n algo i hm, which is a a ia ion o he well-es ablished k-nea es neighbo algo i hm. The ained CNN has a low memo y oo p in and enables he econs uc ion o up o 10 4 e en s pe second wi h only one GPU. Those imp o emen s a e c ucial on he way o u u e clinical in i o applicabili y o he CC o ion beam ange e ifica ion. 1. In oduc ion A he apeu ic had on beam used in pa icle he apy s ops inside he pa ien ʼs body, ende ing he in o ma ion on he dose deposi ion no as easily accessible as i is he case wi h pho on i adia ion. Among he nume ous ange e ifica ion echniques p esen ly unde s udy, he e a e hose ha ely on he de ec ion o an ionoacous ic signal o igina ing om he mal expansion ollowing localized hea ing in he dose deposi ion o a pulsed ion beam (Kellnbe ge e al 2016), seconda y cha ged pa icles (in case o ca bon ion beams)(Gwosch e al 2013)o delayed annihila ion γ- ays such as in posi on emission omog aphy (PET)(Zhu and El Fakh i 2013). Finally, he e a e me hods based on he de ec ion o p omp γ- ays such as mul i-sli came a (Smee s e al 2016), p omp -gamma spec oscopy (Ve bu g and Seco 2014), p omp -gamma iming (Golnik e al 2014),γ-PET, also called iple-γPET o whole gamma imaging (Lang e al 2014, Manzano e al 2015, Yoshida e al 2020),o Comp on came a (CC)based imaging sys ems (K imme e al 2015, Pol e al 2015, Llosá e al 2016, Aldawood e al 2017). In his pape he ocus is se on he la e app oach. The basic componen s o a CC sys em a e wo de ec o s: a sca e e and an abso be . Depending on he ene gy, he ini ial γ- ay unde goes Comp on sca e ing in he fi s de ec o and should hen be ully abso bed in he second one. The goal o a single-e en econs uc ion is o de e mine he so-called Comp on cone, which is a OPEN ACCESS RECEIVED 21 Feb ua y 2021 REVISED 11 May 2021 ACCEPTED FOR PUBLICATION 1 June 2021 PUBLISHED 2 July 2021 O iginal con en om his wo k may be used unde he e ms o he C ea i e Commons A ibu ion 4.0 licence. Any u he dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)and he i le o he wo k, jou nal ci a ion and DOI. © 2021 Ins i u e o Physics and Enginee ing in Medicine collec ion o all possible loca ions om which he de ec ed γ-quan um could ha e been emi ed. By sequen ially de ec ing mul iple γ- ays o igina ing om he same poin , i is possible o ge he sou ce loca ion a he in e sec ion o he econs uc ed Comp on cones. The use o monoli hic scin illa o s as an abso be componen is an al e na i e o pixela ed de ec o s, as being well-es ablished in PET scanne s, p o iding be e ene gy esolu ion, highe sensi i i y as well as cheape and easie manu ac u ing. All hese ad an ages, howe e , come a he cos o a mo e complex de i a ion o spa ial in o ma ion ha has o be ca ied ou wi h he help o designa ed algo i hms. Se e al algo i hms ha e been p oposed so a , no ably he k-nea es neighbo s (kNN)and ca ego ical a e aged pa e n (CAP)( an Dam e al 2011)algo i hms. The main p inciple o bo h a e lookup ables, he e called e e ence lib a ies, con aining he posi ion dependen de ec o esponse, i.e. he scin illa ion ligh ampli ude dis ibu ion ac oss he (segmen ed)pho osenso a ay. Du ing he econs uc ion p ocess, he ligh ampli ude dis ibu ion o an unknown e en is compa ed wi h all e e ence lib a y en ies. Then, acco ding o specific ules, he pho on in e ac ion posi ion is de e mined by he maximum o a 2D his og am, which con ains a selec able numbe o k mos simila e en s. Al hough bo h algo i hms achie e sa is ac o y spa ial esolu ion, hese me hods a e no applicable in eal- ime p ac ice due o hei long compu a ional imes. In his pape we p opose an algo i hm o de e mining he posi ion o γ- ay in e ac ions in a monoli hic scin illa ion c ys al, which is based on supe ised machine lea ning in ol ing con olu ional ne wo k ne wo ks (CNN). We show he de elopmen s eps s a ing om he ne wo k a chi ec u e op imiza ion, h ough he de e mina ion o he op imal da abase size and sugges an e ec i e aining schedule. Ou algo i hm is applicable o all ypes o de ec o s whe e a pho on in e ac ion posi ion can be associa ed wi h a cha ac e is ic ligh ampli ude dis ibu ion. The new me hod allows imp o ing he posi ioning accu acy by a ac o o abou 2.5 compa ed o he CAP algo i hm, has a low memo y oo p in and opens up p ospec s o eal- ime applica ion. 2. Ma e ials The CC p o o ype unde de elopmen in ou g oup o ion beam ange e ifica ion ia p omp -gamma imaging is a wo-s age de ec o sys em. The sca e e consis s o a s ack o six double-sided silicon s ip de ec o s, which p o ide di ec in o ma ion on he deposi ed ene gy as well as in e ac ion posi ions o bo h, he inciden γ- ay and he sca e ed elec on. The second componen , which is he ocus o his wo k, is a single hick monoli hic scin illa ion c ys al ead ou by a mul i-anode PMT ha abso bs he sca e ed pho on. A cha ac e is ic ligh dis ibu ion pa e n egis e ed by he PMT co ela es wi h he pho on in e ac ion posi ion inside he c ys al. Two monoli hic scin illa ion c ys als wi h dimensions o 50.8 mm ×50.8 mm ×30 mm we e examined as po en ial candida es o he abso be componen : LaB 3 :Ce and CeB 3 . The c ys als we e w apped in a eflec i e ma e ial which, despi e a mo e inhomogeneous spa ial esponse o he de ec o compa ed o an abso bing coa ing, has a significan ly be e ene gy and ime esolu ion and hus a be e o e all pe o mance (Aldawood e al 2015). Bo h c ys als show hyg oscopic p ope ies and we e he e o e p o ec ed om humidi y wi hin an aluminum housing. Bo h scin illa ion ma e ials exhibi sho decay imes o 16–19 ns, high ligh yield in he ange o 6.3–6.8 ×10 4 ph MeV −1 and an excellen ela i e ene gy esolu ion o 3.5%–4.1% a 662 keV. While LaB 3 :Ce ca ies an in insic adioac i i y o abou 2 Bq cm −3 , CeB 3 , being ee o his disad an age, shows a highe signal- o noise a io (Sain -Gobain 2016, Scionix 2018). We used a 64- old segmen ed mul i-anode PMT wi h an 8 ×8 pixel a angemen om Hamama su (models H8500C and H12700 Hamama su 2015), which has been shown o be supe io when compa ed o a highe segmen ed 256-channel PMT (Viegas 2018). Each pixel has a size o 5.8 mm ×5.8 mm, he ou e dimension o he PMT a ay is 49 mm ×49 mm, he e o e sligh ly smalle han he size o he scin illa ion c ys als. The pho osenso and abso be we e coupled wi h an op ical g ease whose e ac i e index is simila o ha o he abso be ma e ial and he pho omul iplie en ance window made o bo osilica e glass. The signals o he 64 PMT channels a e sen o adap e boa ds ia ou 16-pin coaxial ibbon cables and ans e ed u he in o ou Cons an F ac ion Disc imina o modules ia LEMO cables. The MCFDs ampli y he ene gy signals and c ea e ampli ude-independen logical signals ha ac as indi idual ga es o he cha ge digi ize modules. Bo h ypes o signals a e hen ed in o cha ge- o-digi al con e e modules ha in eg a e he incoming signals o e he du a ion o he indi idual ga es, p o iding ene gy in o ma ion o each channel. In addi ion o he 64 signals, a signal om he PMT sum-dynode is ed o a sepa a e MCFD module, whose common logic ou pu is sen u he o a TRIVA 5 VME igge uni (Ho man 2009)as igge o he da a acquisi ion sys em, as well as o a Quad Coincidence Logic Uni , ha spli s his mas e ga e in o iden ical copies which a e sen o each MQDC module, opening an acquisi ion ime window o ensu e synch onized da a acquisi ion ac oss all modules. The acqui ed da a is sen o he con ol PC ia a VME on end CPU (RIO-3 ope a ed unde he eal- ime ope a ional sys em LynxOS). A block o he da a acquisi ion sys em is p esen ed in figu e 1. 2 Phys. Med. Biol. 66 (2021)135017 M Kawula e al 3. Me hods 3.1. Da a acquisi ion The aining o a CNN equi es an ex ensi e collec ion o labeled da a (he e e e ed o as e en s), which a e in ou case 2D ligh ampli ude dis ibu ions egis e ed by a PMT and associa ed o hei espec i e i adia ion posi ion coo dina es. We adap ed exis ing e e ence lib a ies, which se ed p e iously as lookup ables o posi ion econs uc ion using he kNN and CAP algo i hms. The da a acquisi ion o he ligh ampli ude dis ibu ions was pe o med by i adia ing he c ys al’s on su ace wi h igh ly collima ed γ- ay sou ces: 137 Cs emi ing 662 keV γ- ays and 60 Co emi ing wo coinciden pho ons o ene gies o 1.17 and 1.33 MeV. Ou collima o was a 10 cm long DENSIMENT od wi h a cen al 1 mm diame e channel, a anged poin ing pe pendicula o he abso be su ace. The c ys al was i adia ed on a 2D g id o 102 ×102 e e ence posi ions wi h a 0.5 mm s ep size ( ia a scanning sys em wi h emo ely con olled and mo o ized (x,y) ansla ion s ages). Fo he LaB 3 :Ce c ys al 800 pho opeak e en s o igina ing om he 137 Cs i adia ion and 600 e en s om 60 Co i adia ion (sepa a ely o bo h cobal ene gies)we e collec ed a each posi ion. The gene a ion o he a o emen ioned 137 Cs lib a y (sou ce ac i i y 72 MBq)o 600 ×102 ×102 en ies ook abou 7 d o con inuous measu emen ime. The ac i i y o he 60 Co sou ce was significan ly lowe (27 MBq), hus he acquisi ion o 600 ×102 ×102 pho opeak e en s equi ed abou 18 d unin e up ed da a acquisi ion. In o de o sepa a e sca e ed e en s om pho opeak e en s, an ene gy ga e was se o selec he pho opeak ene gy. Due o he absence o in e nal adioac i i y o he CeB 3 and he e o e he highe signal- o-noise a io, we we e able o educe (compa ed o he LaB 3 :Ce c ys al i adia ion) he numbe o e e ence lib a y en ies o 600 and 480 o he i adia ion wi h 137 Cs and 60 Co sou ces, espec i ely. A ske ch o he expe imen al se up is p esen ed in figu e 2(a). 3.2. Da a p ep ocessing Each e en en y belonging o a e e ence lib a y consis s o a wo-dimensional, 8 ×8 ea u e ec o , ep esen ing he ligh ampli ude dis ibu ion in each channel o he PMT, and a a ge ec o indica ing he (x,y) coo dina es o he i adia ion posi ion. The a ge ec o s we e cons uc ed using he one-ho , also known as Figu e 1. Block diag am o he abso be de ec o signal ead ou and da a acquisi ion elec onics. 3 Phys. Med. Biol. 66 (2021)135017 M Kawula e al dummy, encoding scheme (Alkha usi 2012). All inpu ec o s we e no malized such ha he sum o e all PMT channels is equal o one. An exempla y ligh ampli ude dis ibu ion, a e aged o e 400 e en s egis e ed a he same i adia ion posi ion, oge he wi h he co esponding a ge ec o s is p esen ed in figu e 2(b). App oxima ely 20%–30% o he e e ence e en s we e se aside o se e la e as a es se being a p oxy o he new unlabeled da a. The emaining 70%–80% we e used as aining and alida ion da a o op imize he model pa ame e s. 3.3. Wo kflow We ca ego ized he ask o finding he in e ac ion posi ion o a γ- ay, based on he PMT esponse, as an image, i.e. pa e n, ecogni ion p oblem. CNN building blocks we e used odesign a new econs uc ion algo i hm, while he la ge da ase s o labeled da a enabled us o ca y ou supe ised lea ning. CNN algo i hms a e known o hei la ge numbe o unco ela ed hype pa ame e s ha mus be se p io o he aining. An a emp o op imizing hem all simul aneously would be equi alen o finding he absolu e minimum o a mul idimensional unc ion, which due o compu a ional cons ain s is no possible o be ca ied ou in p ac ice. Fo ha eason, he p ocess o sea ching hei op imum combina ion was di ided in o se e al s ages wi h he ollowing p io i ies: (1)model specific hype pa ame e s (numbe o con olu ional laye s and ke nels),(2)da abase size (3)op imize hype pa ame e s (lea ning a e and numbe o epochs o aining). 3.4. A chi ec u e design The fi s s age o he hype pa ame e op imiza ion, pe o med o de e mine he numbe o con olu ional laye s and 3 ×3 ke nels building he so-called con olu ional block, was ca ied ou h ough he andom sea ch me hod, meaning ha se e al highly dissimila ne wo k a chi ec u es we e in es iga ed o de e mine a smalle sub-space o he second s age o op imiza ion, he g id sea ch. A e e e y con olu ion ec ified linea uni (ReLU)ac i a ion (Gonzalez and Woods 2018)and ba ch no maliza ion (Io e and Szegedy 2015)we e applied. To p e en he image om sh inking, each laye was ze o-padded be o e pe o ming con olu ion ope a ion. To gene a e p edic ions, sepa a ely o he xand ycoo dina es, he ne wo k was spli in o wo ou pu blocks, each made up o wo ully connec ed laye s wi h so max (Gonzalez and Woods 2018)as he final ac i a ion unc ion. The ou pu had he o m o wo 102-dimensional p obabili y ec o s, o igina ing om he 102 e e ence lib a y i adia ion posi ions along each axis. The coo dina e wi h he maximum p obabili y alue was conside ed as he econs uc ed in e ac ion posi ion. 3.5. T aining We ca ied ou he aining using a ca ego ical c oss-en opy loss unc ion, Adam op imize and in oduced a da a spli in o mini-ba ches o size 400. E en hough he Adam op imize p o ides a sepa a e lea ning a e (l ) o each ne wo k pa ame e , which is adap ed as lea ning un olds, we ound i beneficial o di ide he aining in o se e al phases, g adually dec easing he ini ial lea ning a e o he op imize as ollows: l ={10 −5 ,5×10 −6 , 10 −6 } o app oxima ely 100, 50 and 50 epochs, espec i ely. We p epa ed sepa a e ne wo ks, o each ene gy o he inciden γ- ay and each c ys al. Howe e , o exploi common ea u es o he ligh dis ibu ions, da a om all lib a ies we e included a he beginning o he aining. Figu e 2. (a)Ske ch o he se up o he 2D de ec o scan using collima edγ- ay sou ces. The scin illa ion c ys al ②is a ached o a mo o ized ansla ion s age ①while he collima o ④ oge he wi h i s su ounding lead shielding block ③is kep s a iona y. (b) Schema ic ligh in ensi y dis ibu ion map wi h indica ed in e ac ion poin (in e sec ion o he ed dashed a ows)and co esponding one-ho -encoded a ge ec o s. 4 Phys. Med. Biol. 66 (2021)135017 M Kawula e al The final pa ame e uning was ca ied ou wi h he e en s o he specific ene gy and o he dedica ed c ys al only. 3.6. Da abase op imiza ion In heo y, he neu al ne wo k pe o mance should imp o e wi h inc easing da abase size, bu e en ually a sa u a ion poin is eached and he model does no longe benefi om new aining examples. Since e e ence lib a y acquisi ion is a leng hy p ocess (7 and 18 d o collec 137 Cs and 60 Co da a, espec i ely), i is ad an ageous o de e mine he minimum da abase size equi ed o achie e he desi ed model accu acy. The e o e, he aining was ca ied ou o 13 di e en numbe s o pho opeak e en s pe i adia ion posi ion: = n 10, 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600 , epp ain {} ∣ while addi ional 50 e en s pe posi ion we e used as alida ion se . Model weigh s and biases we e ini ialized wi h he iden ical se o andom numbe s. Six di e en alues o he numbe o pho opeak e en s pe i adia ion posi ion n epp| es we e examined, in o de o de e mine he minimum numbe o es e en s ha p oduce a smoo h his og am, sui able o ull wid h a hal de e mina ion: = n 1, 5, 10, 15, 20, 25 . epp es {} ∣ 3.7. De e mina ion o he algo i hm spa ial esolu ion In e ac ion posi ions o e en s om he es se we e p edic ed by he neu al ne wo k and compa ed o he g ound- u h. The di e ences be ween hese wo we e used o fill a 2D e o his og am c ea ing a na ow peak simila in shape o a 2D Gaussian o Lo en zian. The spa ial esolu ion o he algo i hm was calcula ed as he a e age FWHM o he his og am p ojec ions along bo h axes. An exempla y e o his og am and i s p ojec ion in xdi ec ion a e depic ed in figu e 3. In o de o quan i y he s a is ical unce ain y o he spa ial esolu ion de e mina ion, he es da ase was di ided in o fi e subg oups and o each o hem a sepa a e e o his og am wi h i s FWHM was calcula ed. F om he esul ing fi e da a poin s an a e age FWHM and i s s anda d de ia ion we e de e mined. The sys ema ic unce ain y has many sou ces. Fi s ly, bo h hyg oscopic abso be c ys als a e encapsula ed in an aluminum housing and he exac posi ion o he c ys al edges has o be de e mined indi ec ly by he so-called ‘edge scan’, i.e. 1D scans in xand ydi ec ions, espec i ely, ollowed by sigmoid fi s o he esul ing in ensi y p ofiles, wi h a p ecision o abou ±0.3 mm (Lip andi 2018). Secondly, he beam is no infini esimally na ow, bu collima ed by a 10cm long ube wi h a 1 mm diame e opening. Assuming o simplici y, ha all pho on in e ac ions ake place a he c ys al on su ace, which is 0.2–0.3 mm away om he ea end o he collima o , we ha e expe imen ally de e mined he beam spo size o ha e app oxima ely 1.2 mm diame e (Binde 2017) wi h a Gaussian in ensi y p ofile. Figu e 3. (a)Two-dimensional e o his og am. (b)An exempla y p ojec ion o he e o his og am on o he xaxis. The o ange cu e p esen s he linea in e pola ion be ween adjacen bins. The ed ho izon al line indica e he FWHM le el. 5 Phys. Med. Biol. 66 (2021)135017 M Kawula e al 4. Resul s 4.1. A chi ec u e design Using a andom sea ch me hod we de e mined he uppe limi o he numbe o con olu ional laye s and 3×3 ke nels o n con =6 and n ke n =4, espec i ely. Mo e complex a chi ec u es did no lead o u he imp o emen s. The second s age o op imiza ion was pe o med employing he g id sea ch echnique o all combina ions o he ollowing pa ame e alues: n con ={2, 3, 4, 5, 6}and n ke n ={2, 3, 4}. Figu e 4p esen he co esponding ou comes o he op imiza ion e e ing o he LaB 3 :Ce c ys al and he h ee in es iga ed pho on ene gies, illus a ed by showing he esul ing spa ial esolu ion as exp essed by he FWHM alue o he e o his og am as in oduced in sec ion 3.7. The e is no clea c i e ion o an unambiguous selec ion o he bes se o pa ame e s. None heless, he a chi ec u e comp ising fi e con olu ional laye s o h ee ke nels wi h ReLU ac i a ion and ba ch no maliza ion almos always p o ided he bes esul s, was easy o ain, and he e o e was selec ed o se e as he final model. Figu e 5shows he ske ch o he CNN a chi ec u e. The same aining was ca ied ou o he CeB 3 scin illa o . A summa y o he algo i hm pe o mance, exp essed by he achie ed spa ial esolu ion ia he FWHM o he e o his og am, using he bes final CNN a chi ec u e o bo h c ys als as a unc ion o he ini ial γ- ay ene gies is p esen ed in figu e 6. 4.2. C ys al pe o mance Figu e 7p esen s he spa ial esolu ion achie ed wi h he CNN algo i hm as a unc ion o he dis ance om he c ys al cen al axis (0, 0) o h ee geome ical egions: cen al (|x|,|y|8.5 mm), in e media e (8.5 mm <|x|, |y|17 mm)and ou e (17 mm <|x|,|y|25.5 mm). I can be obse ed ha he spa ial esolu ion imp o es owa ds he c ys al’s cen al axis. This can be a ibu ed o he less equen sca e ing o scin illa ion ligh i compa ed o he c ys al edges, dis o ing he ligh dis ibu ion on he PMT g id and enhancing abso p ion. The beha io o bo h 60 Co cu es is e y simila , Figu e 4. Pe o mance o he CNN algo i hm o h ee di e en ene gies o inciden γ- ays, as a unc ion o he numbe o laye s and ke nels ( wo-blue, h ee-o ange, ou -g een) o he LaB 3 :Ce c ys al. The spa ial esolu ion wascalcula ed as FWHM o hee o his og am. Figu e 5. Ske ch o he bes pe o ming CNN a chi ec u e ( o mo e de ails see ex ). 6 Phys. Med. Biol. 66 (2021)135017 M Kawula e al because o hei small ene gy di e ence o only 0.16 MeV. The ne wo k pe o mance o 137 Cs is in e io o he highe ene gies o 60 Co due o he lowe pho on ene gies. In ac , less ene ge ic γ- ays end o in e ac u he away om he PMT’s en ance window, inc easing he likelihood o sca e ing and a enua ion o he scin illa ion ligh on i s pa h o he pho osenso . 4.3. Da abase op imiza ion We de e mined he op imum numbe o e en s pe i adia ion posi ion equi ed o an e ficien aining o be n epp| ain =400 (350 o he aining se and 50 o alida ion). Using smalle da abases esul ed in o e fi ing o led o subop imal pe o mance. Abo e ha poin he spa ial esolu ion (FWHM)did no imp o e u he . Fo he op imiza ion o he es se , e o his og ams co esponding o n epp| es 15 su e ed om oo low s a is ics, so he use o a linea in e pola ion o de e mine he FWHM o hese peaks could no be jus ified. I is necessa y o use a alida ion da ase wi h n epp| es 20. To ge he unce ain y o he spa ial esolu ion de e mina ion we used fi e se s wi h n epp| es =20 e en s. Table 1summa izes he compa ison be ween he CAP and CNN algo i hms o he LaB 3 :Ce scin illa ion c ys al conce ning he compu a ional ime, memo y oo p in and he bes spa ial esolu ion achie ed wi h bo h echniques. Apa om he supe io posi ioning accu acy, he CNN algo i hm is up o 10 4 imes as e and has a much lowe memo y oo p in compa ed o he CAP algo i hm. This can be unde s ood by analyzing he wo king p inciples o he CAP algo i hm, which assumes compa isons o he unknown e en wi h all en ies om he ex ensi e e e ence lib a y being ime and memo y consuming. In con a y, he CNN equi es a single o wa d pass h ough a ai ly simple ne wo k. Figu e 6. Spa ial esolu ion achie ed wi h he CNN algo i hm (fi e con olu ional laye s, h ee ke nels)as a unc ion o he ini ial γ- ay ene gy, o he wo scin illa o c ys als LaB 3 :Ce and CeB 3 . Figu e 7. Spa ial esolu ion achie ed wi h he CNN algo i hm o he LaB 3 :Ce and CeB 3 monoli hic scin illa o s as a unc ion o he dis ance o he i adia ion posi ion om he c ys al’s cen al axis: ou e , in e media e and cen al. The esul s o h ee in es iga ed ene gies o he inciden γ- ays a e p esen ed. 7 Phys. Med. Biol. 66 (2021)135017 M Kawula e al 5. Discussion The de e mina ion o he pho on in e ac ion posi ion inside a monoli hic scin illa ion c ys al has been pe o med so a using algo i hms like kNN o i s op imized e sion CAP. Howe e , hose su e ed om long compu a ional imes and high memo y equi emen s. This mo i a ed us o de elop an al e na i e econs uc ion p ocedu e ha wo ks as e and does no sac ifice he algo i hm posi ioning pe o mance. E en hough he me hod p esen ed he e was de eloped o wo la ge 50.8 mm ×50.8 mm ×30 mm monoli hic LaB 3 :Ce and CeB 3 scin illa ion c ys als, coupled o 64- old segmen ed pho osenso s wi h an 8 ×8 pixel a angemen , i can be applied o all kinds o de ec o s whe e a pho on in e ac ion posi ion is associa ed wi h a cha ac e is ic ligh ampli ude dis ibu ion. The newly designed CNN consis s o fi e con olu ional laye s, each comp ising h ee 3 ×3 ke nels wi h ReLU ac i a ion, ba ch no maliza ion and ze o padding applied. To gene a e p edic ions, sepa a ely o he x and ycoo dina es, he ne wo k was spli in o wo ou pu blocks, each made up o wo ully connec ed laye s wi h so max as he final ac i a ion unc ion. The coo dina es wi h he highes p obabili y a e conside ed as he econs uc ed posi ion. Using mo e complica ed a chi ec u es inc eased he aining complexi y and did no lead o an imp o emen in he pe o mance. Employing his me hod, we achie ed an accu acy o he posi ion de e mina ion eaching an op imum alue o 0.9(±0.2)mm o CeB 3 a 60 Co pho on ene gies o 1.17 and 1.33 MeV and unde he same condi ions o LaB 3 :Ce a esolu ion o 0.96 (±0.02)mm. This accu acy is supe io by mo e han a ac o o 2.5 compa ed o he bes alue o 2.9(0.1)mm achie ed by he CAP algo i hm o he LaB 3 :Ce scin illa o a 1.3 MeV (Lip andi 2018). The lack o in e nal adioac i i y in CeB 3 explains i s be e pe o mance i compa ed o LaB 3 :Ce. We obse ed a end o sligh ly be e pe o mance o inc easing pho on ene gies which can be explained, fi s ly, by he la ge pene a ion powe a highe γ- ay ene gies, causing mo e ene ge ic pho ons o in e ac close o he PMT en ance window and secondly, by mo e seconda y scin illa ion pho ons gene a ed by highe ene ge ic p ima y pho ons. To he bes o ou knowledge, he CNN algo i hm as p esen ed he e achie es he bes epo ed spa ial esolu ion o 2D posi ion de e mina ion inside la ge monoli hic scin illa o s. On he compu a ional side he new me hod is compa ible wi h bo h CPUs and GPUs and allows o de e mine he in e ac ion posi ion o abou 10 4 e en s pe second using a single GPU node (N idia Quad o P1000), which, when conside ing mul i-mode ins umen a ion, will be su ficien o a po en ial eal- ime applica ion in p omp -gamma imaging. This is a significan imp o emen o e he CAP algo i hm, ha equi ed a compu e clus e and was able o econs uc only abou one e en pe second pe single CPU node. Exploi ing la ge da ase s o labeled da a, ha we e o iginally acqui ed o se e as lookup ables o he kNN and CAP algo i hms, we ca ied ou supe ised aining. A each o 102 ×102 i adia ion posi ions 460–800 pho opeak e en s we e collec ed and each o hem was assigned wo labels, co esponding o he xand y coo dina es. Conside ing his ask as a eg ession p oblem, in con as o he classifica ion pa adigm, equi ed a long aining and did no lead o sa is ying esul s. An ex ensi e da abase is c ucial o p o ide a ep esen a i e aining se o ain models ha gene alize well and a e esis an o o e fi ing. Ye , he da a acquisi ion is a ime consuming p ocess ( akes up o a couple o weeks)and i possible should be kep a minimum. Fo his eason we se a lowe h eshold o 400 and 100 (5×20)e en s pe i adia ion posi ion o aining and es ing, espec i ely. This acili a es a as aining wi hou o e fi ing and smoo h e o his og ams necessa y o FWHM de e mina ion. The accu acy o he ne wo k is bes in he cen al a ea o he c ys al and de e io a es sligh ly owa ds he ou side egions. This eflec s he la ge p obabili y o sca e ing and a enua ion o he scin illa ion ligh a he scin illa o edges compa ed o he mo e cen al in e ac ions. The e o his og ams de i ed om he compa ison o calcula ed and g ound- u h in e ac ion posi ions exhibi a non-Gaussian shape (see figu e 3)and hus hei FWHM is no linea ly ela ed o he s anda d de ia ion σ(as in he case o no mal dis ibu ion whe e s =FWHM 2 2 ln 2 ). Howe e , he FWHM o he e o his og am s ill se es as a alid measu e o he ne wo k pe o mance, enabling a quan i a i e compa ison Table 1. Quan i a i e compa ison be ween he CAP and CNN algo i hms o he LaB 3 :Ce scin illa ion c ys al. The aining ime, econs uc ion speed pe one GPU (Quad o P1000), memo y equi emen s and he bes spa ial esolu ion achie ed wi h bo h echniques a e gi en. T aining Recons uc ion Memo y Spa ial esolu ion o Algo i hm ime (h)speed (e en s s −1 ) oo p in (GB)LaB 3 :Ce a 1.33 MeV (mm) CAP —1 10 2.9(1) CNN 10 10 4 <1 0.96(2) 8 Phys. Med. Biol. 66 (2021)135017 M Kawula e al