Keywords: NMR; MRI; modified-bloch equation; Anisotropic anomalous diffusion; PFG

Transcription

1 Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b oe-bloch equaons Guong Ln* Calson chool o Ches an Boches Cla Unves Wocese A 6 UA Absac Anoalous uson ess wel n pole an bologcal sses. Pulse-el gaen PFG anoalous uson s coplcae especall n he ansoopc case whee le eseach has been epoe. An geneal PFG sgnal aenuaon epesson nclung he ne gaen pulse FGPW eec o ee geneal ansoopc aconal uson { < α β } base on he aconal evave has no been obane whee α an β ae e an space evave oes. I s essenal o eve a geneal PFG sgnal aenuaon epesson nclung he FGPW eec o PFG ansoopc anoalous uson eseach. In hs pape wo ecenl evelope oe-bloch equaons he acal eenal oe-bloch equaon an he aconal negal oe-bloch equaon wee eene o oban geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson. Vaous cases o PFG ansoopc anoalous uson wee nvesgae nclung couple an uncouple ansoopc anoalous uson. he connuous-e ano wal CRW sulaon was also cae ou o suppo he heoecal esuls. hee s goo ageeen beween he heo an he CRW sulaon. he obane sgnal aenuaon epessons an he hee-ensonal aconal oe-bloch equaons ae poan o analng PFG ansoopc anoalous uson n NR an RI. ewos: NR; RI; oe-bloch equaon; Ansoopc anoalous uson; PFG *Eal:gln@clau.eu. Inoucon Anoalous uson [3] has been nvesgae n an sses such as pole o bologcal sses [4] poous aeals [56] acal geoees [7] celle sses [8] an ohe sses [9]. he Anoalous uson has e evave oe α an space evave oe β. When he anoalous uson euces o noal uson. Unle noal uson anoalous uson has a non-gaussan popagao an s ean β-h powe o splaceen s no lneal popoonal o s uson e. Anoalous uson coul be oele b e-space aconal uson equaons base on acal evave see Appen A oel an aconal evave see Appen B oel [ ]. ue o he non-gaussan chaacesc eaues o anoalous uson s cul o nvesgae pulse el gaen PFG anoalous uson. he PFG echnque has been a poweul ool n sung noal uson [34] bu an PFG noal uson heoes a no be ecl applcable o he nvesgaon o anoalous uson whou ocaon. hee have been an eos evoe o sung PFG anoalous uson [ ] bu os o he ae elae o soopc anoalous uson. he heoecal eseach o PFG ansoopc anoalous uson s ve le [373839]. G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7

2 a b RF RF G δ G δ PGE PGE Fg. wo pcal PFG pulses: a PGE pulse sequences b PGE pulse sequence. he gaen pulse wh s an he uson ela s Δ. In PGE pulse sequence he agneaon s ansee o he Z econ b he secon ao equenc RF pulse o elnae he elaaon eec whch has he avanage ove he PGE pulse sequence. Ansoopc uson behavos coul es n an aeals [ ] such as seche o copesse poles bloc poles lqu csals uscle cells ban ssues an so on. In hese ansoopc sses he uson a belong o ansoopc anoalous uson. Copae o PFG ansoopc noal uson he PFG ansoopc anoalous uson s uch oe coplcae. Fs he PFG sgnal aenuaon n anoalous uson a be a seche eponenal uncon base aenuaon o he acal evave [-4] o a ag-lele uncon LF base aenuaon o he aconal evave [35-7]. When s no sall he LF uncon has E E E E whch s een o he ep ep ep ep n ansoopc noal uson. econ he chaacesc eaue o ansoopc noal uson s ha has hee een uson coecen coponens n aes o he pncpal as ae. Whle he ansoopc anoalous uson have hee ses o paaees nclung an whee an s he anoalous uson coecen along he as. In aon he anoalous uson along he hee pncpal aes a be coelae eee o as couple uson o non-coelae eee o as uncouple uson. Cuenl o e aconal uson an geneal aconal uson he heoecal esuls o PFG ansoopc anoalous uson base on he oe Bloch equaon eho n Re. [37] onl nclue he aenuaon o he s gaen pulse. In pcal pulse gaen spn echo PGE o pulse gaen sulae echo PGE epeens as shown n Fg. he PFG sgnal aenuaon aes place n all hee peos:. hus s poan o eve geneal PFG sgnal aenuaon epessons o he sung o PFG ansoopc anoalous uson. hs pape has hee goals. Fs o een he ecenl evelope acal eenal an aconal negal oe-bloch equaons [46] o anale he hee-ensonal ansoopc anoalous uson whch s poan o su hee-ensonal PFG anoalous uson. Vaous goups have popose een oe Bloch equaons o anoalous uson whch cobne he aconal uson wh G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7

3 pecesson an elaaon n sgncanl een annes [93746]. he oe-bloch equaons popose n Re. [46] wee eploe hee base on he ollowng conseaons: he pove he sgnal aenuaon o he whole gaen pulse sequence an he esuls can be euce o hose obane b he sho gaen pulse GP appoaon [46]. Aonall when he aconal uson becoes ve slow o evenuall hee s no aconal uson he oe Bloch equaons n Re. [46] euce o he aonal lnea Laou pecesson equaon. oeove he nuecal evaluaon o he PFG sgnal aenuaon base on hs oe-bloch equaon s no cul. he secon goal o hs pape s o oban he geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b hese oe-bloch equaons. h o help us unesan he PFG anoalous uson an o es he esuls o he phenoenologcal oe-bloch equaon he connuous-e ano wal CRW sulaon [47] s peoe. Rano wal sulaon s a sochasc up pocess whch s a poweul nuecal eho use o oel noal an anoalous uson n phscs ches an an ohe scplnes [47]. he ano wal can convenenl sulae he PFG sgnal aenuaon whch can be use o es aheacall acable sses an help us unesan coplcae sses ha a be aheacall nacable n PFG noal uson an PFG anoalous uson. he CRW eho use n hs pape has been use n eeences [33] an [36] o sulae PFG sgnal aenuaon n ee an esce anoalous usons. hs CRW sulaon eho s base on wo een oels. he connuous wang e an up lengh ae base on he CRW oel suggese n eeence [47] whose applcaons ae use o oel anoalous uson n phscs an econocs. Whle he phase change an he PFG sgnal aenuaon ae sulae base on he algoh o he lace oel evelope n eeences [4849] whch has been use o oel PFG uson n pole sses [5]. Boh he ansoopc usons wh coelae o non-coelae econ usons wll be sulae. he sulaon esuls agee peecl wh he esuls o he aconal negal oe Bloch equaon. All he heoecal esuls obane hee can be euce o he epoe ansoopc noal uson when α = an β = [34] an soopc anoalous uson esuls when he hee paaees an ae he sae n een as econs. Aonall een nuecal appoaches can be use o calculae he heoecal PFG sgnal aenuaon. In pacula o he aconal evave he PFG sgnal aenuaon can be nuecall evaluae ehe b he Aoan ecoposon eho [ ] o a ec negaon eho see Appen C ha s popose n hs pape. Alhough he nuecal values obane o hese wo ehos ae encal he calculaon spee o he ec negaon eho s ascall ase han ha o he Aoan ecoposon eho. he esuls hee help us o unesan PFG ansoopc anoalous uson an pove a valuable oals o PFG ansoopc anoalous uson.. heo In PFG ansoopc uson he eecve gaen g n he pncpal as ae can be obane b g Rg whee g s he gaen veco n he obsevaonal eeence ae an R s he oaon a ha oaes a veco o enso o he obsevaonal eeence ae o he pncpal as ae. When he uson enso can be ansoe b G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 3

4 ~ R R whee an ae he uson ensos n he obsevaonal eeence ae an he pncpal as ae especvel. In PFG epeens he accuulang phase o a spn s [463] g g 3 whee s he goagnec ao an ae he e-epenen poson vecos n he obsevaonal eeence ae an he pncpal as ae especvel. Aveagng ove all possble leas o sgnal aenuaon [46] A P ep 4 whee A s he sgnal aenuaon an ae he sgnal aplues a he begnnng o he s gaen pulse an a he en o he las gaen pulse especvel an P s he pobabl sbuon uncon o he accuulang phase. Fo splc wll be se as hen A. wll be s use o enoe he sgnal aenuaon n he es o hs pape. As P ep alwas ue n geneal he sgnal nens aece b uson s aenuae whch can be onoe n PFG epeens egaless o whehe he uson s noal o anoalous. Fo Eqs. 3 an 4 he PFG sgnal aenuaon epesson can be anale ecl nse he pncpal as ae. In ansoopc uson he usons n he hee pncpal aes ae couple o uncouple whch can be seen o he connuous-e ano wal [47]. In CRW sulaon o couple uson he change o each ano up belongng o one o he hee aes s popoonal o cean aos. Fo a sple case he couple uson a have an pobabl o upng n a gven econ ao equals 3. I s sll possble ha he couple uson has een on each as bu he oal up e o each as s een. Fo an uncouple uson he uson along each as econ s nepenen o he uson along ohe aes econs. Fo such an uncouple uson he paaee se o hee econs coul be een. uch an uncouple uson can be eae as a cobnaon o hee one-ensonal usons.. Couple ansoopc uson We s su he sple couple ansoopc anoalous uson ha has he sae bu a have een on hee pncpal aes... Faconal eenal oe-equaon base on he acal evave In Re. [46] he one-ensonal aconal eenal oe Bloch equaon base on he acal evave oel s G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 4

5 G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 5 g 5 whee s he spn agneaon n he obsevaonal eeence ae s he poson an s he spn-spn elaaon e. he one-ensonal oe-bloch equaon 5 can be eene o hee ensons. he hee-ensonal aconal eenal oe Bloch equaon base on he acal evave o soopc uson wll be g 6 whee. Whle o ansoopc uson he oe Bloch equaon n he pncpal as ae can be wen as g 7 whee s he spn agneaon n he pncpal as ae g s he eecve gaen n he pncpal as ae s he coponen o aconal uson coecen enso n he pncpal as ae an he uns o ae s. When an he ansoopc uson euces o an soopc uson. Fo a sse wh hoogeneous spn ens he agneaon n he pncpal as ae can be escbe as ep ep [3746] whee g s he wavenube n he pncpal as ae. ubsung ep ep no Eq. 7 we ge. 8 he soluon o he Eq. 8 s : ep ep ep ep b A. 9 whee R R ~ ~ b a an s ene as

6 G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 6 R R R. b Fo splc s sae all heoecal calculaons an he CRW sulaons n hs pape ae peoe n he pncple as ae. In eal applcaon g an can be obane b Eq. an Eq. b especvel... Faconal negal oe-bloch equaon base on he aconal evave he one-ensonal oe-bloch equaon o PFG aconal uson base on he aconal evave s an negal pe equaon [46]! g a whee an s ene n Appen B. In Eq. a he elaaon eec epens on he local agneaon nens a each nsan o e. Whle a be possble ha he elaaon oes no epen on he local agneaon nens an s nepenen o he uson nuce aenuaon hen he oe-bloch equaon coul be wen as! ep g. b Whehe o no he elaaon eec epens on he local agneaon nens n anoalous uson s sll no clea whch eques uue epeenal su o cla. In he es o hs pape onl he elaaon eec escbe b Eq. a wll be consee. Fo Eq. a he hee-ensonal aconal negal oe Bloch equaon can be wen as! g. whee. Fo ansoopc uson he negal oe-bloch equaon base on he aconal evave n he pncpal as ae wll be

7 G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 7! g. 3 Fo space-aconal uson whose Eq. 3 euces o g. 4 he soluon o Eq. 4 can be assue as ep ep whch can be subsue no Eq. 4 o gve ep ep. 5 Fo PGE o PGE epeens Eq. 5 can be negae o gve g ep ep. 6 Fo a sse wh hoogeneous spn ens he agneaon o a geneal aconal uson n he pncpal as ae can be escbe as ep. ubsung ep no Eq. 3 we ge!. 7 Neglecng he elaaon we have α *. 8 he sla pe o aconal equaon o Eq. 8 has been solve b he Aoan ecoposon eho [ 5-55]. Accong o hese eeences he soluon o Eq. 8 s [465-55] n n 9a whee! 9b an

8 G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 8 n n n n a J 9c whee a. When [5746] an ae use because n pcal PGE o PGE epeens o a sall aenuaon we have : b E E whee b s ene b Eq. a. hus he esuls o he oe Bloch equaon an ha o IA eho appoaon ae close o each ohe a a sall level o sgnal aenuaon...3 Coplcae couple case I s sll possble ha he usons n hee ensons have een bu ae couple n a cean anne. Fo nsance a coelae ano wal aes place n hee econs. he possbl o a gven up havng a econ along one o hee pncpal as s popoonal o s oal e sang n ha aes n he whole uson pocess. he oal uson e ' whee s he oal e on as. Fo he acal evave oel b seng c o c 3 whch s conssen wh he sple case n secon.. c he aenuaon a be obane base on Eq. 9 as c ep ep. Fo he aconal evave oel b seng c o c 3 we have c c. hus base on Eq. 9 he sgnal aenuaon a be gven as

9 n a n whee b! an J n a c c n n. c. Uncouple ansoopc uson wh een on aes Fo he uncouple uson he usons n aes ae nepenen o each ohe an all he hee anoalous uson paaees on he hee aes n he pncpal ae a be een. In such a suaon he sgnal aenuaon can be obane b 3 whee s he aenuaon epesson o one-ensonal anoalous uson whch can be euce o Eqs. 9 an 9. Fo he acal evave oel o Eq. 9 we have ep 4 whle o he aconal evave oel o Eq. 9 we ge n n n n!. 5 As ep ep ep ep he seche eponenal aenuaon o such uncouple ansoopc uson wll obe he sae equaon as Eq. 9. Whle o he uson escbe base on he aconal evave oel as E E E E he sgnal aenuaon base on Eqs. 3 an 5 s een o ha base on Eqs. 9a-c. 3. CRW sulaon he connuous e ano wal has been eploe o sulae PFG soopc aconal uson [3336]. he sulaon wll be oe o ve he heoecal esuls o ansoopc aconal uson. In he sulaon he ano wal o a spn n he eal space consss o a sequence o nepenen ano wang es... 3 an a sequence o ano ups n 3 whee. wo oels CRW oel [47] an Lace oel [4849] wee use n he n G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 9

10 sulaon. he CRW oel escbe n eeence [47] s use o pouce he wang e an up lengh o each ano up whch s popose o oel een aconal uson sses n phscs an econocs [47]. he wang e sbuon n he sulaon ollows a ag-lele uncon whch can be appoae as a seche eponenal uncon a sall value an a powe law a lage value [47]. he wang e s gven b [47] sn log U cos V 6 an whee s a scale consan an U V ae wo nepenen ano nubes. Whle he up lengh sbuon behaves as a Lev α-sable uncon whch s a genealaon o a Gaussan uncon [47]. he up lengh s gven b [47] logu cos cos sn 7 cos whee V an s a scale consan. Reeence [47] shows ha he pobabl ens uncon base on he CRW oel sases he aconal uson equaon n he usve l. In he sulaon o he couple ansoopc uson hee ae equal possbles n he econs an each up wll ae one o hose econs he coplcae ansoopc anoalous uson n secon..3 s no sulae hee. Whle o he uncouple o non-coelae ansoopc uson hee nepenen ano wals n econs ae place sulaneousl. Aonall he algoh use n he lace oel sulaon o eeences [4849] was oe o eco he e an splaceen change o spn phase n he h wal s [ ] o he CRW ano wal. he ne n g 8 whee g s he gaen sengh a he e an s he e ne o o n. In he sulaon a scee e wh equal e neval was use o eco he connuousl evolvng wh l phase whch s calculae b g g l l l. he e g epesens he paal phase evoluon o he l l l l l l h up G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7

11 gnal aenuaon sep ha he ecoe e belongs. he anne o phase econg oes no aec he phase evoluon pocess. hus ehe a scee econg o a connuous e econg can be use. he oal noale sgnal aenuaon can be ecl obane o he sulaon va [495] N wals cos. 9 N wals A oal o a leas wals wee peoe o each sulaon. Onl he subuson base on he aconal evave oel was sulae because he oel n eeence [47] s popose speccall o subuson base on he aconal evave. 4. Resuls an scusson.8 s Faconal evave Facal evave s Fg. Copason o he PFG sgnal aenuaon o he oe-bloch equaons base on he acal evave Eq. 9 wh hese base on he aconal evave Eq. 9. he paaees use n hese heoecal pecons ae.75 =.46-9 β s α an g =.5. Relaaon eec s neglece. he ecenl evelope oe-bloch equaon eho [46] was eene o gve he geneal PFG sgnal aenuaon epessons nclung he FGPW eec o ee ansoopc anoalous uson. he oe-bloch equaons ae base on wo een oels he acal evave oel an he aconal evave oel. he aenuaon base on he acal evave oel an ha base on he aconal evave oel have slaes an eences. he acal eenal oe-bloch equaon els G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7

12 a seche eponenal uncon EF aenuaon whle he aconal negal oe-bloch equaon els a ag-lele uncon LF base sgnal aenuaon. he sla an eence beween EF an LF aenuaons ae shown n Fg.. A sall level o sgnal aenuaon he LF can be appoae as a EF whle a lage sgnal aenuaon he LF aenuaes oe slowl han he EF a subuson. Onl he subuson s shown n Fg. he supeuson s oe coplcae an heeoe no he ocus o hs pape. he esuls o space-aconal uson agee wh hose obane b he een oe Bloch equaons popose b ohe goups [37-39]. Fo geneal aconal uson he esuls base on he aconal negal oe-bloch equaon agee wh he pevousl epoe esul base on a een oe-bloch equaon n Re. [37] a he s gaen pulse he epoe esul n Re. [37] onl gves he aenuaon n he s gaen pulse. he esuls can be euce o he epoe ansoopc noal uson esul when an. hese esuls can also be euce o he epoe soopc anoalous uson esul when an. Boh he esuls can be euce o he one-ensonal PFG anoalous uson esuls base on he oe-bloch equaon n Re. [46]. he esuls o he oe-bloch equaon eho also agee wh he eecve phase sh uson eho an he obsevng he sgnal nens a he ogn eho see Appen. he esuls o he oe-bloch equaon ae n goo ageeen wh he CRW sulaons. Fg. 3 shows he copason beween he heoecal pecng base on Eq. 9 wh he CRW sulaon o couple ansoopc uson wh sae evave paaees bu een uson consan coponens whee. 75 ae use o all hee aes an : : : 4 :. 5 wh =.7 - s.75. Fg. 4 shows he copason o he heoecal pecon base on Eq. 9 wh CRW sulaon o couple ansoopc uson wh he sae bu een an uson consan coponens whee. 75 ' '. 5 '. 9 = s.75 an = s.75. G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7

13 <' > Fg. 3a <' > <' > = ' +' Value Eo '.46e-9 9.8e-3-9 Chsq R 5.84e NA NA uson e s G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 3

14 gnal aenuaon Fg. 3b.8.6 ' : ' : ' s CRW_ Bloch_ CRW_ Bloch_ CRW_3 Bloch_3 CRW_ Bloch_ s Fg. 3 Copason o he PFG sgnal aenuaon pece o he oe-bloch equaon wh he CRW sulaon esuls o anoalous uson wh he sae vaables bu een.75 an : : 4:. 5 : a vesus o he sulaon he : ng gves =.46-9 β s α b ne gaen pulse wh eec wh equalng s s 3 s an s g equalng.5. G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 4

15 gnal aenuaon.8.6 ' : ' : ' ' =.5 ' =.5 ' =.9.4. CRW Bloch s Fg. 4 Copason o he PFG sgnal aenuaon pece o he oe-bloch equaon wh he CRW sulaon esuls o ansoopc anoalous uson wh sae bu een an paaees ae. 75 ' '. 5 '. 9 = s.75 =.. he -9.9 s.75 = s an g equalng.5. hee s a clea eence beween he sgnal aenuaon o couple an uncouple aconal uson base on he aconal evave oel. Fg. 5 shows he copason o CRW sulaon o uncouple uson wh he heoecal pecons base on Eqs. 3 an 5 o uncouple uson an Eq. 9 o couple uson. Fo Fg. 5 he pecon o he couple aenuaon Eq. 9 s salle han ha o he CRW sulaon a lage aenuaon whle he sgnal aenuaon o he Eqs. 3 an 5 o he uncouple ansoopc uson agees well wh he CRW sulaon acoss he whole cuve. he eence beween he couple an uncouple anoalous uson shoul ase o ha E E whch s sgncanl een o he EF aenuaon. Fg. 5 ncaes ha n eal applcaons a nee o ell couple o uncouple usons. G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 5

16 gnal aenuaon Couple uson.vs. uncouple uson.8 =.5 =.7 ' ' ' = = = ' ' '.6.4. CRW_uncouple Bloch_Couple Bloch_uncouple s Fg. 5 Copason he PFG sgnal aenuaon beween uncouple an couple ansoopc anoalous uson o CRW an heoecal pecons. he heoecal pecons o couple uson ae calculae b Eq. 9 whle he heoecal pecon o uncouple uson s calculae b Eqs. 3 an 5. he paaees use ae '. 5 '. 7 '. 9 ' ' ' = s.5 = s.7 = s.9 = s an g equalng.5. I s woh nong ha ~ R R a onl be use une he conon. I hs conon oes no hol he uns o ae s whch ae een aong he hee pncpal aes. heeoe one a nee o anse g o g Rg an calculae he PFG sgnal aenuaon n he pncpal as ae ecl. hs s een o ansoopc noal uson. In eal uson sses he paaees an ae elae wh he wang e sbuon an up lengh sbuon whch ae aece b he aeal opholog an nac pope. he paaees an coul be epeenall eene n PFG epeens. A sall level o sgnal aenuaon he sgnal aenuaon epesson Eqs. 9 an s equvalen o Eq. 9 nael E b : ep b : 3 G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 6

17 when b s sall. Base on Eq. 3 a sall sgnal aenuaon we have : ln ln A ln A c c ln g c ln 3 ln when an when when an g an g ae e g s e 3 s e g whee c 3 s a consan an. When he lnea ng s use o epeenal aa g base on Eq. 3 he sgnal aenuaon shoul be sall. Howeve he sall sgnal aenuaon equeen a be slghl elae b peong a polnoal ng o he cuve whch can gve he an values o he coecen o he s oe e. Fo Re. [33] he eene values base on sulaon o epeenal aa ae whn a naow ange whle he values show a lage eence beween he ngs base on he EF an LF aenuaons.6 an.7 o LF whle.8 o EF whch s ue o he ac ha EF aenuaon s ase han LF aenuaon [8]. Because he values coul be eene whn a naow ange shoul be eene o he sall sgnal aenuaon epeenal aa s whch a no onl be ease bu coul also pove he accuac o analss. hen he obane values can be use o eene he values o he lage sgnal aenuaon epeenal aa. Aonall values can be eene b ohe ehos o epeenal aa. Fo eaple n eeence [36] s oun ha usng. vs. o ng he esce uson aa can el a value whn a naow ange. Geneall n eal applcaons he paaees an shoul no be abal selece. he oe-bloch equaon s ve sla o he aconal eacon-uson equaon [56]. hese wo es g an a be vewe as eacon es. In a spn sse he gaen el g an he spn-spn elaaon aec he phase evoluon o spn oens bu hese wo eacons alwas ae place n he NR saple whou ega o how as o slow he anslaonal uson s. In ac he phase evoluon aece b he gaen el can be vewe as a phase uson pocess whch can be eplane b he eecve phase sh uson equaon eho [8]. he oe- Bloch equaon s a acoscopc appoach ha nclues boh he anslaonal uson an he phase evoluon o a spn sse. he nuecal evaluaon o he PFG sgnal aenuaon can be peoe wh he copue asssance. he calculaon o he seche eponenal uncon base PFG sgnal aenuaon s ease han ha o ag-lele uncon base PFG sgnal aenuaon. Beses he Aoan ecoposon eho [5-55] he ec negaon eho see Appen C can be use o nuecall evaluae he ag-lele uncon base PFG sgnal aenuaon. he esuls o he ec negaon eho agee wh hose obane b he Aoan ecoposon eho as shown n Fg. 6a. he ec negaon eho s G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 7

18 gnal aenuaon uch ase han he Aoan ecoposon eho. Aonall he ec negaon eho can be use o calculae ag-lele pe uncons E c an E. he ag-lele uncon calculae b Eq C.3 agee wh hose obane b he eho n Re. [57] an he Pae appoaon [58] whch s shown n Fg. 6b. he FORRAN coe o he calculaon base on Eq. C.3 an Pae appoaon can be obane o Re. [59]. hs ec negaon calculaon eho s sple an as. Especall oes no cause ovelow n copuaon. Fg. 6a Aoan ecoposon Eq. 9 ec Inegaon Eq. C s G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 8

19 E - Fg. 6b.9 Nuecal Evaluaon o LF =.5.8 LF_b ec negaon LF_Re. [57] LF b Pae Appoaon Re. [58] Fg. 6 he nuecal evaluaon b he ec negaon eho I: a he goo ageeen beween he ec negaon eho an he Aoan ecoposon eho n he nuecal evaluaon o PFG sgnal aenuaon he paaees use ae '. 5 '. 7 '. 9 ' ' ' = s.5 = s.7 = s.9 = s an g equalng.5 b he goo ageeen o he ec negaon eho n calculaon o he ag-lele uncon E beween he ec negaon eho wh hese b eho n Re. [57] an Pae appoaon eho n Re. [58]. G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 9

20 G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 he PFG ansoopc anoalous uson esuls pove new oalss o PFG anoalous uson sues n NR an RI. In pacula can poenall be a convenen ool n uson enso agng I [6 ]. In eal applcaon sses he uson an elaaon echans ae coplcae whch eques uch eo o avance he heoecal an epeenal sues n hs el. Appen A. enon o acal evave o eeence [-]. l P P P. A. Appen B. enon o aconal evave [35-7]. he Capuo aconal evave * s ene as [5-7]. : *. B. he space aconal evave s ene as [35-7] β β cos B. whee β B.3 an β. B.4 Appen C. ec negaon eho o nuecal evaluaon o ag-lele pe PFG sgnal aenuaon. he Aoan ecoposon eho poves an analcal appoae soluon o Eq. 7. Howeve a ec negaon eho can be popose as an alenae wa o he nuecal calculaon o PFG sgnal aenuaon. he algoh o he ec negaon s sple: s ve he e no ; secon calculae he. a a C. sep b sep o = o n whee s he sgnal nens a e an s use n he calculaon. In hs calculaon each s calculae onl once whle n he Aoan ecoposon eho n n whch s a supeposon o an es. hus he

21 G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 calculaon spee o he ec negaon eho can be a ew oes o agnue ase han he Aoan ecoposon eho. he esuls o he ec negaon eho agee wh hose obane b he Aoan ecoposon eho as shown n Fg. 6a. oeove une GP appoaon [84] g c g a s a consan; o Eqs. 7 an 9 we can ge E c an c E whch ples c ce c E. C. When c = E c E heeoe he ec negaon eho can also be use o calculae ag-lele uncon an s evave b he ollowng scee o o equaons:. a E a E C.3 E E E. C.4 he ag-lele uncon calculae b Eq. C.3 agee wh hose obane b he eho n Re. [57] an he Pae appoaon [58] whch s shown n Fg. 6b. he FORRAN coe o he calculaon base on Eq. C.3 an Pae appoaon can be obane o Re. [58]. hs ec negaon calculaon eho s sple an as. Especall has no ovelow poble n copuaon. Appen. Geneal PFG soluons b ohe ehos: I. EPE eho: Fo he acal evave oel he FGPW eec can be obane b he eecve phase sh uson equaon eho. he ansoopc eecve phase sh uson equaon can be wen as [46] ' ' P P. whee s he eecve phase uson coecen. he soluon o Eq.. can be obane slal o ha o he one enson uson n e. [46] as c P 3 4 ep 4.. he sgnal aenuaon can be obane as [846] GP P ep ep.3

22 G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 whch can be euce o he one-ensonal anoalous uson esul n Res. [83346] an he ansoopc noal uson esul when n Re. [457]. Whle o he aconal evave oel he phase uson equaon base on he aconal evave oel popose b he eecve phase sh uson equaon eho [8] s P P α *.4 whch s ha o solve. B peong Foue anso on boh ses o Eq..4 an seng q q q q P he ollowng PFG sgnal aenuaon equaon can be obane α *.5 whch s he sae as Eq. 8. II. Obsevng he sgnal nens a he ogn eho: he hee-ensonal aconal uson equaon can be wen as α *..6 ubsung ep no Eq..6 els α *..7 A he ogn whee an g he gaen el has no eec on he agneaon phase an we have ep hus we ge α *..8 Eq..8 epouces Eqs. 8 an.5. Onl he aconal evave oel wll be consee hee. he acal evave oel can be eve slal. Acnowlegeens he lngusc help o hoas Cawoo Chsan Fallen o wng cene n Cla Unves an Ao Ln s acnowlege. Reeences [] R. ele J. lae he ano wal s gue o anoalous uson: a aconal nacs appoach Phs. Rep [] J. lae I.. oolov Fs sep n ano wal. Fo ools o Applcaons Oo Unves Pess New Yo. [3] A.I. achev G.. Zaslavs Faconal nec equaons: soluons an applcaons Chaos

23 [4] N. Faulln R. ch heo o el-gaen NR usoe o pole segen splaceens n he ube-epaon oel PHY. REV. E [5] Y. L G. Fahe an R. ch Phs. Rev. E [6] J. Zhong G. Ln W.-Y. Wen A.A. Jones. elan B.. Feean anslaon an oaon o peneans n ulapeeable nanocopose ebane o pol-bsluooehl-45- luoo-3-oole-coealuooehlene an Fue lca acoolecules [7]. Poo A. Bune. Havln H.E Roan ucual an nacal popees o he pecolaon bacbone n wo an hee ensons Phs. Rev. E [8] R. Angelco A. Cegle U. Olsson G. Palao L. Abosone Anoalous suacan uson n a lvng pole sse Phs. Rev. E [9] V. Gupa.. Nvah A.V. ccoc an H.. avs Che. Phs. Le []. Hahn J. äge an V. ula Phs. Rev. Le [] W. Chen e space abc unelng anoalous uson Chaos olons Fac [] W. Chen H. un X. Zhang. ooša Anoalous uson oelng b acal an aconal evaves Copu. ah. Appl [3] H. un.. eeschae Y. Zhang J. Zhu W. Chen A acal Rchas equaon o capue he non-bolann scalng o wae anspo n unsauae ea Avances n Wae Resouces [4] A.. Balann B. E. Elaaa Honacs o acal connuu low Phs. Rev. E [5] F. ana Y. Lucho an G. Pagnn he unaenal soluon o he space-e-aconal uson equaon Fac. Calc. Appl. Anal [6] R. Goenlo F. ana Faconal uson Pocesses: Pobabl sbuons an Connuous e Rano Wal pnge Lecue Noes n Phscs No 6 Beln 3 pp [7] R. Balescu V-Langevn equaons connuous e ano wals an aconal uson Chaos olons Fac [8] G. Ln An eecve phase sh uson equaon eho o analss o PFG noal an aconal usons Jounal o agnec Resonance [9] R.L. agn O. Abullah. Baleanu X.J. Zhou Anoalous uson epesse hough aconal oe eenal opeaos n he Bloch oe equaon J. agn. Reson [] E.L. Hahn pn echoes Phs. Rev []. W. ccall. C. ouglass E. W. Aneson an B. Bunsenges Phs. Che [] E. O. esal an J. E. anne J. Che. Phs ; o: [3] P. Callaghan anslaonal nacs an agnec Resonance: Pncples o Pulse Gaen pn Echo NR Oo Unves Pess. [4] W.. Pce Pulse-el gaen nuclea agnec esonance as a ool o sung anslaonal uson: Pa. Basc heo Conceps agn. Reson [5] J. äge H. Pee G. Voa e coelaon ung anoalous uson n acal sses an sgnal aenuaon n NR el-gaen specoscop Phs. Rev. A [6] R. ch NR: oogaph usoe Relaoe pnge-velag Heelbeg 997. G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 3

24 [7]. Zavaa N. ülan R. ch.f. Nonnenache Popagao epesenaon o anoalous uson: he oenaonal sucue aco oals n NR Phs. Rev. E [8] R. A. aon an.j. Pace Pecons o Pulse-Fel-Gaen NR Epeens o uson n Facal paces Poceengs: aheacal Phscal an Engneeng cences [9].. Benne.. chana R.. Benne.B. Rowe H. Lu J.. He Chaaceaon o connuousl sbue cocal wae uson aes wh a seche-eponenal oel agn. Reson. e [3].. Benne J.. He.. chana Wae uson heeogene ne n he huan ban s nsensve o he oenaon o apple agnec el gaens agn. Reson. e [3] C. Ingo R.L. agn L. Colon-Pee W. ple.h. aec On ano wals an enop n uson-weghe agnec esonance agng sues o neual ssue agnec Resonance n ecne [3] Gnbeg F Cobanu L Fahe E hah NJ uson uoss agng an log-noal sbuon uncon agng enhance he vsualsaon o lesons n anal soe oels. NR Boe [33] G. Ln Insananeous sgnal aenuaon eho o analss o PFG aconal usons J. agn. Reson [34] G. Ln Analng sgnal aenuaon n PFG anoalous uson va oe gaussan appoaon base on acal evave Phsca. A [35] G. Ln Analng sgnal aenuaon n PFG anoalous uson va a non-gaussan phase sbuon appoaon appoach b aconal evaves J. Che. Phs [36] G. Ln. Zheng X. Lao gnal aenuaon o PFG esce anoalous usons n plae sphee an clne J. agn. Reson [37] A. Hanga. eeńsa Ansoop n hgh-esoluon uson-weghe RI an anoalous uson Jounal o agnec Resonance [38] J.J. GaEla R.L. agn.. eeschae lva Capuan aco Palobo Anan ua an Ale. Leow econal behavo o anoalous uson epesse hough a ulensonal aconalaon o he Bloch-oe equaon. IEEE Jounal on Eegng an elece opcs n Ccus an ses. 3 3: [39].. eeschae R.L. agn A.Q. Ye Ansoopc aconal uson enso agng J. Vb Conol o: [4] U. F. ocs C. N. oé H. R. Wen eue an Ansoop: Peee Oenaons n Polcsals an he Eec on aeals Cabge Unves Pess. [4]. Le Bhan olecula uson ssue conacs an cosucue NR Boe [4] E. W. Hsu N. R. Aen. J. Blacban A u o uson Isoop n ngle Neuons b Usng NR coscop agn. Reson. e [43] E. Öaslan.H. aec Geneale uson enso agng an analcal elaonshps beween uson enso agng an hgh angula esoluon uson agng agn. Reson. e G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 4

25 [44] an ӧge: oels o Polec an Ansoopc Lqus Lec. Noes Phs [45].I. oo J.. PopeR.. Wella gal Pocessng o uson-enso Iages o Avascula ssues n: ecal Iage Pocessng: echnques An Applcaons Bologcal an ecal Phscs Boecal Engneeng pnge New Yo pp [46] G. Ln Faconal eenal an aconal negal oe-bloch equaons o PFG anoalous uson an he geneal soluons axv:7.76. [47] G. Geano. Pol E. calas R. L. chllng Phs. Rev. E [48].. Cceone P. A. Wagne.. Ege anslaonal uson on Heeogeneous Laces: A oel o nacs n Glass Fong aeals. J. Phs. Che. B [49] G. Ln J. Zhang H. Cao A. A. Jones J. Phs. Che. B [5] G. Ln. Aucon. Goo A. Canel W. Wen A.A. Jones acoolecules [5] G. Aoan olvng Fone Pobles o Phscs: he ecoposon eho luwe Acaec oech994. [5] G. Aoan R. Rach Inveson o nonlnea sochasc opeaos J. ah. Anal. Appl [53] G. Aoan On he soluon o algebac equaons b he ecoposon eho. J. ah. Anal. an Appl [54] R. C. al an R. Nga oluon o aconal nego-eenal equaons b Aoan ecoposon eho In. J. o Appl. ah. an ech [55] J.-. uan R. Rach. Baleanu A.-. Wawa A evew o he Aoan ecoposon eho an s applcaons o aconal eenal equaons Coun. Fac. Calc [56]. ben-avaha. Havln uson an eacons n acals an soee sses Cabge Unves Pess. [57] R. Goenlo J. Loucho Y. Lucho Copuaon o he ag-lele uncon Eαβ an s evave Fac. Calc. Appl. Anal [58] C. Zeng Y. Chen Global Pae appoaons o he geneale ag-lele uncon an s nvese Fac. Calc. Appl. Anal [59] G. Ln ag-lele uncon calculae b ec negaon hps:ghub.cogln7ag-lele-uncon-calculae-b-ec-inegaon. [6]. Le Bhan J.-F. angn C. Poupon C. A. Cla. Pappaa N. olo H. Chaba uson enso Iagng: Conceps an Applcaons. J. AGN. REON. IAG G. Ln Geneal PFG sgnal aenuaon epessons o ansoopc anoalous uson b 7 5