Lecture 4 Excitation & Acquisition Lecture Notes by Assaf Tal. To Measure A Signal, Spins Must Be Excited

Transcription

1 Lecure 4 Eciaion & Acquisiion Lecure Noes b Assaf Tal To easure A Signal, Spins us e Ecied The Saic Nuclear agneic Field Is Too Weak To e Reliabl Deeced We ve previousl calculaed he bulk magneic momen of mm 3 of waer in a 3 T magneic field and found i o be abou ~ -8 J/T Imagine his voel is inside he human and mus be deeced in a coil wrapped around he sa, cm awa from i The magneic field of he voel is dipolar r 3 rr ˆ ˆ 4 r 3 If we consider a sphere of radius r around, he field will be maimal a he poles: aimal field where is value will be (puing r= m, = -8 J/T, =4-7 N/A ): 3 r 3 ~ T= pt r This is an incredibl small magneic field The eremel weak magneic fields produced b neural currens are - orders of magniude larger, and even hose require eremel specialied hardware based on super-cooled magneic field deecors known as SQUIDs These are used in magneoencephalograph (EG) machines, bu he are no sensiive enough o deec he sor of signals we re ineresed in I is possible o consruc sensiive enough deecors, bu even if we pu aside he engineering complei and cos of hese in addiion o he RI magne iself, we are sill faced wih furher problems: How can we separae he in field creaed b he nuclear spins from he much larger sources of magneic fields creaed b oher phenomena in he, eg neural currens? And how can we, b deecing he magneic fields ouside he, deduce he disribuion of spins wihin he? Tha is, can we image (r) b measuring (r) ouside he? These sor of problems are known as inverse problems and are ofen incredibl difficul o solve properl (he same issue plagues EG) Is here a beer wa o deec he RI signal? The answer is es, and i is linked o he pession of spins around he main field Pessing Spins Induce easurable Currens In The Receiver Coils The R in RI sands for resonance As we ve remarked previousl, an spin will pess abou a consan magneic field In paricular, if we place he spins in a saic, srong magneic field sa, he 3 Tesla field of a pical RI scanner i will pess For a hdrogen nucleus, his pession frequenc would be kh T 7 H mt Using a coil and he law of iproci we can measure he ime-depdenden flu induced b he spin This dnamic measuremen will creae a significanl larger signal han a saic one We are, however, faced wih a parado: a hermal equilibrium, he bulk magneic momen is parallel o, and hence he pession is degenerae remains saic (even hough he microscopic momens, m, will pess) For us o observe rue pession, mus make some non-ero angle wih :

2 he origin) The epression for he magneic field creaed b a loop of curren a is cener is well known from basic magneism: A hermal equilibrium, and are colinear, and no pession is observed R I nˆ R I Onl b creaing some nonero angle beween and ha is, onl b eciing - can we observe pession Creaing such an angle is called eciaion Puing ha aside for a momen, le s calculae he volage induced in a coil pu around he pessing spin via Farada s law and he law of iproci Here we ake a wo circular coils of radius R in he and planes, wih a poin magneic momen placed a he origin and performing some aion as a funcion of ime: where R is he ring s radius, I he curren, and ˆn a uni vecor normal o he plane of he ring We ake uni curren (I=) and so: d d ˆ R R cos R Number Time For =4-7 Vs/(Am), =7 H for pons a 3 Tesla, R = 5 cm (head coil), and = -8 J/T (previousl calculaed magneiaion of ml of waer), we ge 3 V, around he righ order of magniude for he volages deeced in magneic resonance The componens of he pessing spin have a sinusoidal ime dependence: cos sin LH aion abou Remember ha he law of iproci ells us ha he volage induced in he coil can be calculaed via v () dm r, where is he field creaed b a uni curren he loop a he posiion of he magneic momen (a This is a small bu deecable volage level wih oda s elecronics, and his is he basis of signal epion in modern RI The smaller he radius of he coils, R, he beer: alwas build coils ha are as small as possible! Furhermore, he signal is proporional o =, and increases wih (alhough an eac analsis of he SNR will awai a laer chaper) The Spins Can e Ecied Wih A Resonan RF Field A Small RF Field Can Have a Large Effec if i is Resonan The discussion of he previous secion has shown ha, in order o induce non-ero volage, we mus il he magneiaion vecor awa from equilibrium and have i pess Indeed, he basic R eperimen can be described as follows:

3 Thermal Equilibrium: A hermal equilibrium, he spins are aligned along and do no pess Eciaion: The spins are somehow ecied, ha is, iled o some angle wih respec o This usuall happens quickl and relaaion can be negleced Pession & Deecion: Once iled, he pess and give off a ime dependen magneic field The magneic field induces a volage in a nearb RF coil via Farada s law We can also furher manipulae he spins wih magneic fields during his period o bring ou paricular conras pes We usuall have a ime ~ T before decoherence "eas up" he observable pessing magneiaion Thermaliaion: Relaaion processes kick in The ransverse magneiaion decas wih a ime consan T while he longiudinal magneiaion builds up back up due o T relaaion If we wai for a ime 5T, he magneiaion will be back a is hermal equilibrium value Each such block (ecie-acquire-wai) is called a scan I is in fac no mandaor o wai for a ime 5T for he spins o reurn o hermal equilibrium; we'll see laer on ha waiing a shorer amoun of ime has boh benefis (shorer scan imes) and disadvanages (less signal per scan) For now, however, we'll assume ha is he case, so is equal o and poins along he -ais before he beginning of each scan We ve alread remarked ha RF << How can we hope o non-negligibl ecie he spins wih such a weak RF field? The answer is ha we use a resonan field ha oscillaes a he Larmor frequenc Namel, we are going o solve he loch equaions seing G=, and RF ("on resonance irradiaion") This means we will need o solve he loch equaions wih a ime dependen magneic field Alhough a numerical soluion is possible, we will emplo a frame ransformaion rick which will enable us o solve his problem wihou an Transforming o a Frame Which Roaes A The Same Frequenc As The RF Field akes i Appear Saic: The Roaing Frame In he laboraor frame, his amouns o solving he loch equaions wih a complicaed imedependen magneic field The loch equaions are easier o solve in a frame which aes around he -ais wih a frequenc given b RF We ackle his as follows: consider a saic (laboraor) frame wih ime independen, fied uni vecors ˆ, ˆ, ˆ, and a aing frame wih uni vecors ˆ', ˆ', ˆ' ' ' ' wih ˆ sin cos ˆ RF RF RF If he aing frame is aing wih an angular veloci abou an ais given b he uni vecor ˆn, hen each of he aes of he aing frame pess abou he vecor = ˆn This means Schemaic for a simple single eperimen ("scan") (no drawn o scale) Ecie he spins This usuall akes << T, T Le spins pess while acquiring a signal unil i decas due o T Wai for magneiaion o "build back up" due o T before eciing again Time

4 each obes a pession equaion idenical (formall) o he loch equaion: dˆ ' ˆ ' ω dˆ ' ˆ ' ω dˆ ' ˆ ' ω The magneiaion vecor can be epressed in eiher frame: ˆ ˆ ˆ ˆ' ˆ' ˆ' For eample, if () is,,, ˆ sin cos ˆ RF RF RF as given above, while he aing frame aes a he angular frequenc = RF abou he -ais ( = ˆ RF ) hen he componens of in he wo frames are: cos RF, sin RF,, Noe he componens change, bu he vecor is frame-independen since i is a geomerical quani Differeniaing () wih respec o ime, we obain d d d,, d, ˆ' ˆ' ˆ' dˆ' dˆ' dˆ',,, d d,, d, ˆ' ˆ' ˆ' ˆ' ˆ' ˆ' ω d ω,,, Equaing, we obain: d d ω This is pisel he loch equaion bu wih an ecive field ω The above equaion is rue for an aing frame However, in RI, when we speak of he aing frame, we will be referring o a frame which aes a a consan angular veloci RF abou he -ais according o he lef hand rule For he aing frame, ω ˆ ˆ RF When epressed in he aing frame, he componens of he ecive field ω are:,,, (in he aing frame: RF ) If we selec RF we are on resonance: he RF irradiaes he spins a he same frequenc as heir naural frequenc, In his case: On resonance: RF In he aing frame: RF An Analog From echanics Imagine he earh going around he sun in a circle: On he oher hand, he loch equaion sas

5 Gravi This can be undersood b an observer in space he following wa: he Earh wans o go forward bu gravi pulls i inward, curving is pah ino a circle In ec, he Earh is coninuousl falling ino he sun, bu escaping doom hanks o is angenial veloci All his is all a consequence of Newon s second law, F=ma Ne, imagine how hings would look o an observer sanding on he sun and aing wih i Neglecing for he ime being he weaher on he surface, he Earh would appear saionar o such an observer: Gravi If ha observer would r o use Newon s law F=ma o undersand his world he would fail: according o F=F gravi =ma, earh should be falling owards he sun, bu i isn! The ruh is ha when ou ransform o a aing frame ou need o add a ficiious force Tha is, ou need o presuppose a force which doesn arise ou of an phsical source, called he cenripeal force, o eplain how i is possible for he earh o remain saionar: Cenripeal So, in mechanics when ou r o undersand hings in a aing frame ou need o do wo hings: Undersand how hings in he real frame would look in he aing frame (eg, he Earh would remain sill) Add ficiious forces (eg, he cenripeal force) A similar hing happens when ou go o a aing frame in magneic resonance, aing wih he same angular veloci as he RF field: Firs, he RF field appears saionar in he aing field which maches is aion frequenc (ie because RF ) Now we need o add he cor ficiious "force" - field, o be pise - given b fic To see, imaging a saic spin in he lab frame, wih no magneic field Now ransform o a frame aing wih an angular veloci abou he -ais In his frame, he spin would appear o ae wih an angular veloci fic, as if here was a ficiious field fic presen along he -ais The ulk agneiaion Pesses Around The Effecive Field In The Roaing Frame We've seen he magneiaion vecor obes he loch equaions in he aing frame, onl swapping he field for an ecive field, ω and epressing ha field in he aing frame basis (ie as i would appear o an observer aing wih he frame) This means pesses abou in he aing frame Saring from hermal equilibrium a ime =, poins along (aken o coincide wih he -ais) in boh he laboraor and he aing frames, which are also assumed o coincide for =: Gravi

6 ' ' Now we urn on he resonan RF field in he laboraor frame: ˆ sin cos ˆ RF RF RF This field aes in he -plane in he lab frame, and appears saionar in he aing frame Furhermore, if we assume our irradiaion is on resonance, RF=, he ecive field in he aing frame has no -componen: RF Lab Frame ˆ Lab Frame RF The magneiaion pesses abou he ais in he aing frame We can hus creae an angle we'd like beween i and he -ais, depending on how long we le i pess and how srong is Le's assume we have RF on for jus enough ime for he magneiaion o il o he plane - ha is, creae a 9 angle beween and Deducing he moion of in he lab frame is now merel a maer of ransforming back o he lab frame, which simpl aes a an angular veloci - relaive o he aing frame Tha is, in he lab frame performs a spiral as i descends and aes: ' ' The magneic field in he laboraor frame has a large -componen and a small, aing componen (no shown o scale) In he aing frame, assuming RF is on resonance (RF==) he ecive field is saic ' Ro Frame A ime = (hermal equilibrium), poins along he -ais in boh frames Ro Frame ' ' Seing The Radiofrequenc (RF) Pulse s (Area)=(Duraion)(Ampliude) Ses The Flip Angle We see he spins will perform a aion abou he -ais in he aing frame a a frequenc = Noe his is no he same as RF, (one is he ampliude of RF, he second is is oscillaing frequenc) Afer a ime, will have creaed an angle : Noe ha ampliude of RF duraion of RF This relaion is rue onl on resonance, when RF =, where has no -componen To ip he magneiaion ono he ais, we wai a ime 9 such ha: or Lab Frame Shown here is he rajecor of he magneiaion in he lab (lef) and aing (righ) frames The wo frames are conneced b a simple aion RF(), In he original laboraor (unaing) frame he spins eecue addiional moions, bu he imporan hing o realie is ha a spin which is in ' ' ' Ro Frame ' Ro Frame '

7 he plane in he aing frame, mus also be in he -plane in he laboraor frame (alhough where in he plane is a differen sor!) Number Time We ve remarked ha,ma ~ T for an RI scanner For pons, one would need 9 ~5 mso ecie he 3 spins ono he -plane For C, Signal 9 Is Acquired ~ ms Via Farada s Law Relaaion Can e Negleced During Eciaion Since os Pulses Are Shorer Than T, T Our calculaions in he previous secion have shown ha eciaion mosl happens on he imescale of milliseconds in RI, which is much shorer han T, T Hence, o an ecellen approimaion, relaaion ecs can be negleced for mos pulses and mos issue pes in he We will make some remarks abou he ecs of relaaion laer on bu, in general, will neglec i unless specificall saed oherwise The Phase of he Pulse Deermines The Phase of he Ecied agneiaion We have so far modeled RF in he lab frame as: cos c RF, lab sin c Since we have full conrol over he and componen we have no problem modulaing boh () and adding a ime-dependen phase () o he RF field: cos RF, sin Le's keep () and () fied Then he consan phase ()= is called he phase of he pulse, and is equal o he angle he RF field makes wih he -ais deermines where he RF pulse will poin in he ransverse plane The phase of he magneiaion is defined as he angle made b he ransverse componen of he magneiaion vecor (ie is projecion on he plane) wih he -ais ecause he magneiaion ges ipped a righ angles o he RF field following he lef hand rule, he relaion beween he pulse's and magneiaion's phase is given b: m The sandard noaion for a consan RF pulse hen assumes he form, where is is flip angle and is (consan) phase The following convenions are also used: RF : 9 : 8 : 7 : Some eamples are shown below (magneiaion is assumed o sar ou from, and is he blue vecor; he RF is he red vecor): RF, lab cos c sin c In he aing frame, his will look like his : R, where To prove his, use lab R c is a RH aion mari abou he -ais b an c angle c (he aing frame aes wih a lef handed aion and angular frequenc c; in i, i appears he RF field aes a he same angular frequenc bu in he opposie diion) There is a bi of algebra and rigonomer involved bu he proof is sraighforward

8 (9) pulse (45) pulse On he oher hand, he pulse's duraion, 9 magneiaion in plane (phase: 9) (9)- pulse 9 magneiaion in plane (phase: ) Eciaion Flip Angles < 9 Are Used To inimie The Duraion, Decrease Power Deposiion A The Cos Of SNR An eciaion pulse need no ip he spins b 9, and can creae an angle beween and he main field The disadvanage of his is is reduced signal: in our simple model we've seen ha he volage, d, R is proporional o he ime derivaive of (reoriening he coil would inroduce he ime derivaive of, and would no change our conclusions) The magniude of will be proporional o he flip angle Hence, he signal iself will also be proporional o sin() and decrease wih he flip angle : 45 magneiaion in plane (phase: 9) (45) pulse 45 magneiaion in plane (phase: 8), is proporional o he flip angle and decreases linearl (assuming we keep fied) Anoher advanage of shor pulses is ha he have reduced specific absorpion rae (SAR) Some of he RF energ is absorbed in he paien s issue and causes undesired heaing The amoun of SAR is proporional o he square of and he pulse s duraion: consan pulse, duraion T SAR We observe SAR reduces linearl wih he flip angle The amoun of SAR is limied b mos modern scanners' hardware based on our undersanding of he ec of SAR on biological issues odern RF coils deposi power on par wih modern cell-phones and are generall considered safe as long as guidelines are observed Off-Resonan Eciaion And The Concep of Selecive Eciaion The -Field Can Var as a Funcion of Posiion, Which Leads o Non-ero Offses In The Roaing Frame So far our approach has been o make disappear b moving o a aing frame a a frequenc, in which he ficiious field negaes compleel: lab signal, sin We will see laer on his decrease is acuall miigaed in mos sequences where pulses are applied rapidl (on he order of, or faser han T) and don afford he magneiaion enough ime o reurn o hermal equilibrium before he ne eciaion However, when varies as a funcion of posiion, r, i is impossible o make he -componen of he field disappear a ever poin: lab r r Some sources of variaion could include:

9 Imperfecions in he main magne field Suscepibili arifacs in he sample, in which he eernal field induces microscopic magneic momens which hemselves disor he main field (in all diions, bu predominanl in he diion on ) 3 Some paiens migh have meal implans which disor he magneic field again, in man diions, bu heir ec is mos pronounced along 4 Ofen we inenionall creae hese inhomogeneiies, as is he case wih gradien coils, in which we creae a linear dependence of he -field on posiion: G r For eample, when a gradien is urned on, lab G r G r, and, when he RF is urned on: G r If we have some form of spaial inhomogenei due o hardware imperfecions/suscepibili arifacs, we could wrie i as lab r, and in he aing frame is -componen will be lab r Number Time A gradien will creae a range of frequencies given b G over a spaial region of wih Across a mm piel, his will be G 4 H Suscepibili arifacs a 3 Tesla will creae spaial variaions across he head on he order of hundreds of H, mosl in regions where air-issue inerfaces eis such as he prefronal core, close o he oral cavi or ears, and so forh In he case he -componen is no compleel eroed ou, we mus anale and undersand he case for which The quani will be referred o as he offse of he spins Since we will be looking a a specific poin in space we can assume is jus a consan A Qualiaive Soluion: All Pulses Are Selecive Wih A Finie andwih Given ~ / I is fairl simple o divide our analsis ino wo ereme cases: in one, <<, and we can neglec i, obaining: We hus over he previous case in which we ecie he spins as usual, as if he were on resonance On he oher ereme,, and he RF eciaion will have no ec, resuling in no eciaion We can guess and erapolae beween hese wo eremes, saing ha here is a cuoff o he ec of when ~ In oher words, a range of offses ~ will be ecied This is known as he bandwih of he pulse: he range of offses (or frequencies) i will ecie W This can also be undersood graphicall, b ploing he pession cone of he spins abou he ecive field, saring ou from hermal equilibrium (ie along he -ais):

10 = poins in he -plane sa, along he -ais in his eample - and he spin pesses in a circle in he plane << sars iling up in he plane, causing he aion cone o sar folding = In his dividing case, he - and - componens of are equal The pession cone jus ouches he plane >> now becomes close o he -ais The pession cone becomes ver narrow: even if we wai for a long amoun of ime he spins will no sra far from he -ais A Consan Pulse's Eciaion Profile Le us eplore wha happens when our pulse is no hard and has a finie duraion We ll look a a 9 pulse, alhough our conclusions will appl o an pulse flip angle For a 9 flip angle, he duraion mus be 9 4 However, as previousl discussed, his onl ensures a 9 flip angle for spins when = As we increase, he magniude and diion of and is diion var, and he corresponding final posiion of he magneiaion - assumed o sar ou from hermal equilibrium along he -ais - varies In fac, even if is applied along, spins no a he cener do no even remain in he plane anmore The following diagram shows he ecive field and he pise rajecor raced b he magneiaion vecor during he pulse's duraion, 9, for he four cases oulined previousl: = >> A Hard Pulse Is One Wih High Peak Power And An Infinie andwih When we are ineresed in flipping all of he spins ono he -plane regardless of heir offse we mus creae a bandwih larger han he range of offses in our sample Since W, his means we need o have a ver high >>range of offses in our sample The duraion of an (on-resonance) 9-pulse, 9 4 We see ha hard pulses are shor and have a high peak power Such pulses are called hard pulses in R jargon The ideal hard pulse is one for which, duraion, such ha duraion equals he desired flip angle = << Insead of hese picorial diagrams, one can plo he componens of as a funcion of he offse, This is known as he pulse's response or pulse profile Such a response is ploed below for = kh ( 35 T), 9=5 ms ( applied along he -ais as in he above diagrams, ie has phase):

11 pulse we had =, = Le us see wha happens if we var () in a sinc-like manner: (kh) ime (ms) (kh) ime (ms) In he above, The magneiaion was assumed o have an arbirar magniude of uni, and he verical ais sreches from The dashed red lines signif he poins a which kh, which define he bandwih of he pulse I is quie clear ha he concep of bandwih has some arbirariness o i since he profile of and are no sharp The profile of acuall looks somewha wider han, which is a resul of he magneiaion vecor s consan magniude, and he relaion beween and (kh) For eample, if =9, hen 8 and 436 So, even if is almos unperurbed, migh sill appear o be quie siable, leading o is wider profile Also noe he eensive wiggles in ouside he slice, indicaing ha some eciaion occurs even for >> We will deal wih his in a momen b inroducing shaped pulses Shaping The Pulse Affecs The Pulse's Profile odern RF ransmiers have he capabili of shaping he RF pulse, RF i ; ha is, conrolling is - and - componens Such pulses are called shaped pulses For he consan The new pulse mainains he same peak and same area (and hence flip angle) as he angular pulse, bu is necessaril longer (since he negaive lobes of he sinc derac from he area) The frequenc response of his pulse can be calculaed b solving he loch equaions numericall, ielding: (kh) The response is shown using he same scaling and plo range as he angular pulse for a "fair" comparison The dashed red lines represen he same bandwih (= ) calculaed for he angular pulse The ensuing response is significanl beer-behaved, wih less wiggles and sharper ransiion lines We won' go ino he heor of shaped pulses in his course, bu we will remark wihou proof ha for ip angles up unil abou 9 he profile of resembles he Fourier ransform of RF ()

12 The Slice s Cener Can e Shifed Sinusoidall odulaing The Pulse s Phase The pulses discussed so far ecie a bandwih abou a cenral frequenc There is a ver simple wa o shif he cener of he ecied slice, b modulaing he pulse s phase wih a linear erm ahemaicall, his means: c shif cos e ff sin c To undersand wh his shifs he pulse, imagine shif being given in he aing frame, where i aes around he -ais according o he lef hand rule wih angular frequenc c performing a second aing frame ransformaion, ino a frame which aes wih c relaive o he original ( firs ) aing frame, we fi and add an addiional ficiious field: nd aing shif frame c becomes i i cos i cos i e i c c sin c c sin c The comple noaion makes he ec seem much simpler (ie onl mulipling he comple RF i c waveform b ) e Slice Selecion Ofen, We Are Ineresed In Eciing A Single Slice We now come o he firs form of spaial selecivi in RI: selecive eciaion, in which onl a par of he sample is ecied We will confine ourselves o he simple scenario of one-dimensional eciaion, meaning selecivel eciing a range of posiions along a fied ais: In his nd aing frame he -componen has a fied offse All of our previous argumens can be repeaed, bu now he cener of he profile would no occur a, bu raher a a frequenc defined b, or: c c Curren RI hardware enables one o conrol he phase and ampliude of he RF pulse as a funcion of ime, making shifing he profile s cener eas (we can generae an pracical frequenc c ) I also places almos no demands on he hardware 3 I s ineresing o noe ha, using comple noaion, our consan original field Wihou such selecive eciaion, an RF pulse would ecie he enire sample Alhough his can be and is someimes done, a slice-selecive approach also has is own meris Appling A Pulse In The Presence Of A Gradien Will Ecie A Slice In he join presence of a gradien and an RF irradiaion, he ecive field in he aing frame is: 3 This rick for shifing he pulse's profile and he relevan analsis oulined, also holds for non-consan pulses r, G r

13 We will assume for simplici our gradien is consan and urned on along he -diion, so G G ˆ, and so: G RF, RF, G The gradien creaes a linearl increasing offse along : Offse Pos G G Finall, jus as he profile of a pulse can be shifed in frequenc space as a funcion of he offse, hus he slice's cener can also be shifed b simpl modulaing he RF pulse's shape: The gradien assigns frequencies o posiions via G, and hence an pulse ha ecies a range of frequencies W will, in he presence of a gradien, ecie a range of posiions given b: W G G This is, in fac, he slice hickness The ecied slice will be perpendicular o he diion of he gradien vecor G For eample appling he same pulse wih a gradien in a differen diion - sa, a 45 o he -ais - will ecie a slice ha is iself iled b 45, since now our gradien will creae a linear correspondence beween frequenc and he +=cons planes: Acquisiion The Time Evoluion Of The agneiaion In The Roaing Frame In The Absence Of RF Irradiaion In he absence of RF irradiaion, he loch equaions consis onl of a -field, made up of is posiion-dependen offse due o gradiens, inhomogeneiies and so forh: r r We would like o solve he loch equaions: macro, macro, T macro, macro, T macro, macro, T Subsiuing he field above, we obain: r r T T T

14 The firs hing o noe is ha he ransverse () and longiudinal () componens of () are decoupled: he ransverse magneiaion pesses in he -plane and decas wih a ime consan T, while he longiudinal componen builds up owards hermal equilibrium wih a ime consan T, and is unaffeced b he -componen of he field We can solve for independen of he gradiens, field inhomogeneiies, ec he soluion having been alread inroduced in lecure 4: T / T / e e The equaions for and are mied ogeher, or coupled in mahemaical erms We will herefore rea he ransverse magneiaion ( and ) as one eni: i We now mulipl he second equaion b i and add up he firs wo equaions: d i d i r, i T The LHS is merel he ime derivaive of Similar simplificaions can be made for he RHS, and we obain: d This equaion is of he form wih i r, T d a air, T If a were consan, is soluion would be simple: a e Once a is ime dependen we mus break down he ime inerval ino small seps such ha a() is consan in each ime sep Then he soluion in each inerval is a e The full soluion is obained b concaenaing he shor-ime soluions: a a 3 a 4 3 a3 e e e e e e e aa aaa a3aaa This can be coninued b inducion, wih he sum urning ino an inegral as : a ' ' e Reurning back o our case and subsiuing he epression for a()= r,, we obain: ', ' / T,, r r r e Usuall, = will correspond o he poin in ime righ afer he eciaion pulse, so (r,) will be he ransverse magneiaion righ afer eciaion Noe ha we haven' reall said if we're in he aing or lab frame, and our derivaion - and final epression - are valid for boh Since lab r, r,, we can also deduce ha lab r e r, i, which makes sense, because he lab and aing frames differ b a consan aion abou he - ais b an angular frequenc The Acquired Signal In RI Is Proporional To The Transverse agneiaion We now sae a ver fundamenal relaion in RI: he acquired signal is proporional o he inegral of he magneiaion, weighed b he eiver profile:

15 r r s dv, where is he spaial field dependence of he eiver coil when we drive a uni curren hrough i In he remainder of his secion we derive his equaion, bu ou can skip his wihou loss of coninui Proof: Armed wih our epression for he ime course of he magneiaion, we urn o deriving a usable epression for he acquired signal (volage) in our eiver coil We've seen in Lecure 3 ha, for an eiver coil, d r, v r dv d d d dv Here and are boh measured in he laboraor frame Firs, he ime derivaive of he -componen of he magneiaion, which changes wih a ime consan T (on he order of H) is much smaller han he - and - componens, which pess wih a frequenc of H Therefore, we can neglec he -componen o an ecellen approimaion: d d v dv Since we have an epression for in he lab frame, we can differeniae i Since varies much faser han T and G()r, we can neglec boh erms and obain 4 : so d lab r, lab i r, v Re i r r, dv The ineresing signal is modulaed b a rapid i (~ H) phase erm, e, which is "hidden" inside I is beneficial o ge rid of i for boh convenience, as well as o lessen he burden on he analog-o-digial converer (which needs o deal wih slower varing signals) This is called demodulaion To do his, v is spli ino he idenical copies, wih one muliplied b cos( ) and he oher b sin( ), and each is hen passed hrough a low-pass filer (LPF) v sin( ) cos( ) LPF LPF ADC ADC Real channel Imaginar channel We define i, i and noe: so i i i Re Using and wriing cos sin e e e e i i i i i Therefore: lab d v Re dv 4 This assumpion needs o be modified when suding solids wih ver shor Ts (on he order of microseconds) However, such shor Ts are rarel observed in RI since he lead o signals well below he noise levels

16 lab lab i r, r, r, e i r, r, r, e we obain, righ before he LPF: i i Re im v i e e dv i i Re re v e e dv Since he LPF removes he fas changing componen, we obain, afer he LPFs: i Re im v i e dv sin dv i Re re v e dv cos dv We hen form he comple signal in he compuer: Since s v i v re im lab i lab e dv, and since he aing and lab frame magneiaion vecors are relaed via lab i r, r, e, we can simplif: r r s dv, We've omied he consan of proporionali since he acual measured signal's magniude will depend anwa on he elecronics, amplifiers and so on