Sodium imaging at 7 Tesla:
Characterizing the relaxation behavior of quadrupolar nuclei in isotropic and anisotropic in-vivo environments using high-field MR MSc Thesis – Department of Physics, Leiden University Karina P. Soemarwoto First supervisor: P. de Bruin, PhD Second supervisor: Prof. Dr. A.G. Webb Second reader: Prof. Dr. Ir. T.H. Oosterkamp'
Table'of'Contents'
1.! Introduction,...,4! 2.! MRI,of,spin,½,particles,...,6! 2.1! Quantum,mechanical,description,...,6! 2.1.1! Overview!...!6! 2.1.2! Matrix!representation!...!7! 2.1.3! Zeeman!level!splitting!...!7! 2.1.4! Density!Matrices!and!Coherence!...!8! 2.1.5! The!Rotating!Frame,!Operators!and!RF!Pulses!...!10! 2.5.6! Relaxation!and!the!spectral!density!...!11! 3.! MRI,of,Spin,3/2,Particles,...,12! 3.1! Quadrupolar,interactions,...,12! 3.2! Density,Matrix,and,Coherence,for,spin,!",systems,...,13! 3.2.1! Irreducible!Spherical!Tensor!Operators!...!14! 3.3! Time,evolution,of,ISTOs,and,relaxation,behavior,in,Spin,3/2,systems,...,15! 3.3.1! Evolution!of!ISTOs!in!an!isotropic!environment:!<EFG>!=!0!...!16! 3.3.2! Evolution!of!ISTOs!in!the!presence!of!static!EFGs:!<EFG>!≠!0!...!17! 3.4! Pulse,sequences,using,ISTO,notation,...,18! 3.4.1! Multiple!quantum!filters!...!19! 4.! Spin,3/2,MRI,in,Biological,Tissue,...,21! 4.1! Structure,,Diffusion,and,the,sodium,spectrum,...,22! 4.2! The,cell,membrane,...,23! 4.3! B1,Inhomogeneity,...,24! 5.! Literature,Review,...,25! 5.1! Sodium,Environments,...,25! 5.1.1!Environmental!Anisotropy!...!25! 5.2! InQvivo,Imaging,Methods,...,27! 5.2.2! Chemical!Shift!Reagents!...!28! 5.3! Sodium,Quantification,...,29! 5.3.1! Methodology!...!29! 5.4! Sodium,in,Pathology,...,30! 5.4.1! Brain!...!30! 5.4.2! Kidney!...!32! 5.4.3! Cardiac!...!32! 5.4.4! Cartilage!...!33! 5.4.5! Tumors!...!34! 5.4.6! Muscle!...!34! 5.4.7! Other!...!36! 6.! Experimental,Examination,of,23Na,Environment,Influence,...,37! 6.1! The,influence,of,environmental,order,on,the,relative,flip,rate,in,sodium, MRI, 37! 6.2! The,effect,of,Sodium,concentration,on,T1,and,T2,relaxation,rates,...,39!6.4! Summary,...,Error!!Bookmark!not!defined.! 7.! Conclusion,...,43!
Appendix,...,45!
A2.! Tensor,evolution,under,a,hard,,90°,pulse,along,xQ,or,yQaxis.,...,45! References,...,46! !
!
1.'
Introduction'
! MRI!has!become!one!of!the!most!vital!tools!with!which!the!human!body!can!be! examined.!Much!of!the!development!in!MRI!has!focused!on!detecting!the! abundance!and!behavior!of!hydrogen!nuclei,!however,!as!the!trend!of!increasing! magnetic!field!strength!continues!the!appeal!of!studying!other!nuclei!grows.!A! prime!example!is!Sodium[23!(23Na)!which!provides!the!second!strongest!NMR! signal!of!all!nuclei!in!biological!tissue!due!to!its!relative!abundance!and!strong! magnetic!dipole!moment![1]!(the!strength!of!the!23Na!signal!in[vivo!is!still!6000! times!lower!than!it!is!for!proton!MRI![2]).!The!NMR!behavior!of!sodium!is! characterized!by!its!nuclear!spin!of!3/2.!Nuclei!with!a!nuclear!spin!of!3/2!or! higher!have!a!non[zero!quadrupole!moment!that!interacts!with!the!electric!field! gradient!to!which!the!nuclei!is!exposed.!This!interaction!results!in!a!lifting!of!the! degeneracy!of!the!transitions!between!the!energy!levels!of!the!nucleus!in!a! magnetic!field,!which!translates!to!detectable!changes!in!the!NMR!behavior!of! the!nuclei.!For!nuclei!exposed!to!strong,!static!quadrupolar!interactions,!for! instance,!this!lifting!of!the!degeneracy!results!in!the!single!sodium!resonance! splitting!in!to!three!separate!peaks!in!the!NMR!spectrum.!Understanding!these! changes!is!necessary!for!probing!the!structural!and!time[varying!environment!in! which!the!nuclei!are!located.!! ! Early!work!in!the!field!focused!on!a!theoretical!understanding!of!Sodium!NMR! spectra!in!biological!tissue.!A!mathematical!framework!using!irreducible!tensor! operators!was!developed!through!which!deciphering!the!contributions!of! quadrupolar!interactions!to!the!NMR!spectra!was!made!possible!and!pulse! sequences!(so[called!quantum!filters)!that!mapped!these!interactions!were! introduced.!Detailed!analysis!using!quantum!filtering!techniques!and!modeling! of!the!NMR!spectra!obtained!from!DNA![3]!and!bovine!cartilage![4],![5]!showed! that!quadrupolar!interactions!occurred!in!biological!tissue!and!that!they!arose!as! a!result!of!ordered!structures!in!biological!tissue.!Further!research!on! quadrupolar!interactions!in!the!tissue,!particularly!in[vivo!has!been!limited,! likely!due!to!a!combination!of!low!SNR!and!high[SAR!and!B1!sensitivity!of!the! quantum!filtering!pulse!sequences.! ! In[vivo!Sodium!MRI!has!primarily!focused!on!mapping!the!total!sodium! concentration!(TSC)!in!biological!tissue!as!an!indicator!of!disease.!The!typical! procedure!involves!placing!phantoms!with!a!known!sodium!concentration!and! similar!relaxation!time!to!the!tissue!under!study!near!the!region!of!interest.!The! sodium!concentration!of!the!tissue!is!subsequently!extrapolated!through!a! comparison!of!the!signal!intensity!from!the!tissue!to!that!of!the!phantoms.! Changes!in!TSC!have!been!linked!to!a!wide!range!of!conditions.!Elevation!in!TSC! has!been!observed!in!the!brain!of!some!multiple!sclerosis!patients![6]–[9]!and! appears!to!provide!a!method!for!distinction!between!differing!forms!of!the! disease.!Elevations!in!TSC!have!also!been!detected!in!the!brains!of!other! neurological!conditions!such!as!Huntington’s[10],!Alzheimer’s!disease!and! epilepsy![11].!Myocardial!infarctions!are!associated!with!a!breakdown!in!the! functioning!of!the!sodium!pump!causing!a!build[up!of!sodium!in!affected!tissue.! MRI!measurements!of!TSC!have!been!used!to!map!areas!affected!by!the! myocardial!infarction![12]–[14]!with!TSC!in!affected!tissue!elevated!by!50%.!! Sodium!plays!an!important!role!in!muscle!contraction!with!sodium!and! potassium!exchanged!rapidly!during!the!contraction!process.!Various!studies! have!found!that!exercise!results!in!an!elevation!in!the!measured!TSC!in!the! exercised!muscle![15],![16],!though!it!has!been!suggested!that!this!measured! increase!is!an!artifact!arising!due!to!changes!in!the!relaxation!behavior![15].! Acute!muscle!injuries!show!an!elevated!TSC!at!the!site!of!the!injury!and!this! elevation!persists!after!apparent!recovery!from!the!injury![17].!Patients!with! Duchene!muscular!dystrophy!have!been!shown!to!have!elevated!TSC!in!muscles! [18],![19].!Degeneration!of!cartilage,!as!seen!with!osteoarthritis,!has!been! associated!with!a!reduction!in!sodium!concentration!in!cartilage!tissue!in!the! knee![20]–[22].!The!loss!of!structure!seen!in!degeneration!of!cartilage!is!also! believed!to!be!responsible!to!changes!in!the!relaxation!behavior!of!sodium!in! osteoarthritic!cartilage![23].! ! The!clinical!potential!of!sodium!imaging!is!both!clear!and!wide[reaching.!The! non[zero!quadrupolar!nature!of!23Na!provides!a!potentially!powerful!tool!in! exploring!and!expanding!this!potential!further.!However,!it!also!complicated!the! NMR!behavior!of!sodium!nuclei!and!must!be!understood!in!order!to!accurately! compare!sodium!in!different!biological!conditions.!The!aim!of!this!thesis!is!two[ part:!(1)!to!examine!current!methods!of!sodium!imaging!and!(2)!quantification! in!order!to!determine!what!methods!are!most!suitable!for!different!tissue!types,! and!to!provide!an!overview!of!the!theoretical!underpinnings!of!23Na!NMR,!the! effect!of!the!electrical!environment!of!tissue!and!how!this!can!affect!the!NMR! response!of!23Na.!Ultimately!this!thesis!will!show!that!tissue!ordering!plays!an! important!role!in!the!strength!of!the!quadrupolar!interactions!witnessed!in!23Na! NMR!and!that!this!effect!introduces!both!a!tissue[ordering!and!orientation! dependent!variation!in!the!achieved!flip[angle.!The!role!of!tissue[order!in!the! strength!of!quadrupolar!interactions!may!lead!to!changes!in!the!relaxation! behavior!in!diseases!characterized!by!loss!of!tissue!order!and!could!explain!the! results!seen!in!osteoarthritic!cartilage![23].! ! The!structure!of!this!thesis!is!as!follows:!chapter!2!offers!an!outline!of!the! quantum!mechanical!description!of!spin!½!MRI.!The!quantum!framework!of!the! MRI!of!spin!½!particles!is!expanded!for!spin!3/2!in!Chapter!3.!The!quadrupolar! moment!is!introduced!as!an!additional!perturbation!to!the!system,!the! significantly!more!complex!relaxation!behavior!associated!with!it,!and!how!the! former!influences!the!latter.!A!number!of!theoretical!and!practical!considerations! for!performing!sodium!in[vivo!are!given!in!Chapter!4.!A!literature!review!of! previous!works!on!Sodium!magnetic!resonance!in!biological!tissue!is!given!in! Chapter!5.!An!examination!of!the!important!early!work!done!on!sodium!spectra! in!ex[vivo!samples!is!followed!by!an!overview!of!the!more!recent!work!which!has! seen!the!application!of!sodium!MRI!in[vivo!to!examine!a!wide!range!of! pathologies.!A!write!up!of!the!experiments!that!were!performed!to!examine!the! practical!nature!of!the!theoretical!study!provided!in!this!thesis!are!provided!in! Chapter!6.! !
2.'
MRI'of'spin'½'particles'
! 2.1' Quantum'mechanical'description' ! The!classical!mechanical!description!of!MRI!as!a!rotating!dipole!offers!an! intuitive!explanation!for!how!the!system!behaves!and!proves!accurate!for!much! of!the!work!that!is!done!on!spin[½!systems.!In!fact,!it!has!proven!so!useful!that! the!majority!of!researchers!rarely!concern!themselves!with!the!underlying! quantum!mechanics!and!tend!to!rely!solely!on!the!classical!description!for!their! developments!of!even!complex!pulse!sequences.!Yet,!MRI!is!one!of!the!major! success!stories!of!quantum!mechanics.!It!is!one!of!only!a!few!examples!where! quantum!mechanical!effects!can!be!directly!seen!on!the!macroscopic!scale.! Understanding!the!quantum!mechanics!of!MRI!is!furthermore!vital!to! understanding!in!detail!what!happens!in!the!MRI!of!particles!with!spin!greater! than!½.!In!this!section!I!lay!out!the!quantum!mechanics!of!spin!½!systems!and! show!that!it!indeed!yields!the!same!behavior!as!in!the!classical!description!in! Chapter!2.1.!The!quantum!description!of!spin!½!particles!will!be!expanded!to! spin!3/2!particles!in!Chapter!3.! ! 2.1.1' Overview' ! Quantum!mechanics!is!based!on!the!principle!that!a!particle!or!system!of! particles!can!be!described!by!the!time[dependent!Schrödinger!equation:! ! ! !"! !, ! = 1 !ℏ!!(!, !)! ! Where!!!is!the!Hamiltonian,!a!function!that!contains!all!of!the!contributions!to! the!total!energy!of!the!system,!typically!the!sum!of!the!kinetic!energy!T!and!the! potential!energy!V!and!!,!the!wave!equation,!a!function!that!describes!the! quantum!mechanical!state!of!the!system.!Unlike!in!classical!mechanics,!the! Hamiltonian!consists!of!operators,!which!in!themselves!are!functions!that!act!on! the!wave!equation.!A!quantum!state!is!a!certain!function!of!!!that!is!a!solution!to! the!Schrödinger!equation.!It!is!very!difficult!(impossible)!to!find!(exact)!solutions! to!the!Schrödinger!equation,!particularly!for!systems!that!have!many!different! interactions!contributing!to!the!Hamiltonian.!! ! One!of!the!early!“proofs”!of!quantum!mechanics!occurred!when!the!Schrödinger! equation!was!solved!for!a!hydrogen!atom.!It!showed!that!the!energy!spectrum! predicted!by!QM!exactly!matched!the!observed!spectral!lines!of!an!excited! Hydrogen!gas.!The!energy!levels!of!Hydrogen!were!described!by!a!set!of! quantum!numbers,!which!define!the!exact!form!of!the!wave!equation!of!the! solution!to!Schrödinger’s!equation!(see!Table!1).!The!spin!quantum!and!the!total! angular!momentum!number!play!the!most!important!role!in!determining!the! behavior!of!particles!in!a!magnetic!field!and!are!the!numbers!that!govern!the! quantum!behavior!that!gives!rise!to!the!behavior!used!in!MRI.!! !!
Quantum!number! Associated!physical!property! Possible!values!
!! Energy!level!of!system! ! = (0),1,2,3, … , ∞!!
!! Orbital!angular!momentum! ! = 0,1, … , !!!
!!! !! = −!, −! + 1, … , ! − 1, !!!
!!or!!! Spin!angular!momentum! Intrinsic!to!the!particle!
!!!or!!!! !! = −!, −! + 1, … , ! − 1, !!
!!
Total!angular!momentum! !
!!! !
Table,1.,The,different,quantum,numbers,that,describe,quantum,mechanical,states.,The,ground, state,,the,lowestQenergy,state,,can,be,given,by,either,! = !,or,! = !,depending,on,the,system.,
!
2.1.2' Matrix'representation' !
Much!of!quantum!mechanics!lends!itself!to!being!formulated!using!vectors!and! matrices.!An!examination!of!the!time[independent!Schrödinger!equation!reveals! why:!!! ! = !"(!),!in!this!notation!it!is!clear!that!! ! !can!be!seen!as!an! eigenvector!(more!commonly!referred!to!as!an!eigenstate)!of!the!Hamiltonian! with!an!associated!eigenvalue!E.!By!selecting!a!basis!that!consists!of!the!solutions! to!Schrödinger’s!equation,! !! ! , !! ! , … , !!(!) ,!the!Hamiltonian!matrix!is!a! diagonal!matrix!with!the!energy!of!each!state!on!the!diagonal.!This!basis!can!be! seen!as!a!basis!for!the!!!quantum!number,! ! = 1, ! = 2, … , ! = ! .!It!is!of!course! possible!to!extend!this!further!to!basis!of!!,!!,!and!!;!For!a!spin!½!particle!we!have!
two!possible!states,!spin!up!!! = +!!!and!spin!down!!! = −!!!giving!a!two!
dimensional!basis! !! = +!!, !! = −!! .!In!this!basis,!called!the!spinor!notation,! the!spin!up!state!is!given!by!the!vector! 1 0 !and!the!spin!down!state!by! 01 .! ! The!power!of!this!basis!notation!becomes!particularly!apparent!when!operators! are!written!in!the!form!of!matrices.!A!common!example!of!such!matrix!notation! is!the!Pauli!matrices.!The!!!!operator!gives!the!z[component!of!the!spin!vector,! and!applying!it!to!a!spin!state!gives:!!! !! = ℏ!! !! .!Applying!this!operator!to! the!spinor!gives!the!following!result:!!! 1 0 = ℏ ! 1 0 !and!!! 01 = − ℏ ! 0 1 .!This! behavior!is!identical!to!applying!the!matrix!ℏ! 1 0 0 −1 !to!the!spinor.!In!fact,!it! may!be!said!that!!! = ℏ! 1 0 0 −1 !or!more!commonly!!! = ℏ !!!!where!!! = 1 0 0 −1 !is!one!of!the!Pauli!matrices!while!the!other!two!give!the!!!!and!!!! operators:!!! = 0 1 1 0 !and!!! = 0 −!! 0 .!Notice!that!the!spin[up!and!spin[
down!states!are!only!eigenvectors!of!!!,!the!eigenvectors!of!!!!and!!!!are!linear!
combinations!of!the!two.! 2.1.3' Zeeman'level'splitting' !
I!previously!stated!that!it!is!not!possible!to!gain!an!exact!solution!to! Schrödinger’s!equation!for!Hamiltonians!that!contain!contributions!from!
multiple!interactions.!If,!however,!the!magnitude!of!these!additional! contributions!is!small!enough!they!can!be!considered!a!perturbation!to!the! system!that!leaves!the!quantum!states!of!the!system!with!the!additional! interactions!the!same!as!those!for!the!unperturbed!system.!In!other!words,!an! eigenstate!of!the!unperturbed!system!is!also!an!eigenstate!of!the!perturbed! system.!Importantly,!while!the!eigenstates!of!the!perturbed!system!are!the!same! as!the!same!as!for!the!unperturbed!system!the!energy!of!those!eigenstates!does! not!have!to!be!the!same.!Perturbations!often!transform!degenerate!systems!in!to! systems!where!each!eigenstate!has!a!uniquely!defined!energy!level,!this!is!called! lifting!of!degeneracy.!The!theoretical!treatment!of!these!types!of!situations!is! called!perturbation!theory.!! ! Perturbation!theory!was!used!by!Zeeman!to!describe!the!behavior!of!hydrogen! atoms!exposed!to!a!magnetic!field.!The!energy!associated!with!a!magnetic!dipole! in!an!external!magnetic!field!is!given!by!! = −! ⋅ !! = −!ℏ!!!!!with!the!latter! expression!assuming!the!!! = !!!.!This!can!be!treated!as!a!perturbation!as!long! as!the!external!magnetic!field!is!much!weaker!than!the!internal!fields!in!the! atom,!this!is!called!the!weak!Zeeman!effect.!The!result!of!this!perturbation!is!that! the!energy!of!a!particle!becomes!spin!dependent,!the!Zeeman!effect!lifts!the! degeneracy!of!particles!in!the!same!state!but!with!different!spin!orientations.! The!energy!difference!between!the!two!states!as!a!result!of!the!Zeeman!effect! corresponds!to!the!Larmor!frequency;!the!MR!signal!is!obtained!through!forcing! transitions!between!the!two!energy!levels.! ! 2.1.4' Density'Matrices'and'Coherence' ! This!section!has!so!far!examined!particles!that!are!in!a!well[defined!and!known! state.!However,!a!particle!is!not!normally!in!a!specific!state!as!it!instead!has!a! probability!of!being!in!a!certain!range!of!states.!Consider!the!spinor!notation! from!before,!a!particle’s!general!state!can!be!given!by! ! = !!! ! !where! !! !!is! the!probability!of!finding!the!particle!with!spin!up!and! !! !!for!spin!down;! !! !+ !! ! = 1.!! ! For!systems!that!consist!of!many!particles!it!becomes!useful!to!talk!about!density! matrices!rather!than!individual!spin!states.!The!matrices!are!constructed!from! the!spinor!we!defined!above:! ! ! = !!! ! !! ∗ ! !∗ = !!!!∗ !!!!∗ !!!!∗ ! !!!∗ .!The!mean! value!of!this!matrix!of!all!particles!in!the!ensemble!is!called!the!spin!density! matrix:! ! = 1 ! !! !! = ! !!! ! !!!! ∗ ! !!!∗ !!!!∗ !!!!∗ = !!!! !!" !" !!! ! The!diagonal!elements!are!called!populations!and!indicate!the!number!of! particles!that!are!in!the!up!and!down!state!while!the!off!diagonal!elements!are! called!coherences!(in!this!case!between!spin!up!and!spin!down).!The!meaning! behind!these!names!is!indicated!in!Figure!1,!where!!!! = !!,!!!! = !!,!
!!" = !!,!!!" = !!.!If!!! < !!!then!the!system!has!a!negative!net!
magnetization,!if!!! > !!!then!the!magnetization!points!up,!if!!! = !!!there!
is!no!net!magnetization.! !
!
Figure,1.,The,interpretation,behind,the,populations,and,coherences,of,the,density,matrix,[24].,
!
The!quantum!coherences!!!!and!!!!are!related!to!the!transverse!
magnetization,!where!a!non[zero!coherence!indicates!a!non[zero!transverse!
magnetization!(see!Figure!2).!!!!and!!!!are!complex!numbers!and!related!
through!conjugation,!!! = !!∗.! ! ! Figure,2.,Left),A,system,with,no,coherence.,Right),A,system,with,a,nonQzero,coherence.,The,phase,of, the,coherence,indicates,the,direction,in,which,it,points,,the,magnitude,of,the,coherence,is,the, magnitude,of,the,transverse,magnetization,[24]., Since!the!coherences!are!complex!numbers!they!can!be!written!in!polar!form:! !! = !!! !!!!,!the!magnitude!corresponds!to!the!size!of!the!transverse! magnetization,!and!!!!the!direction!of!the!transverse!magnetization.!The!x[ component!is!given!by!cos(!!),!the!y[component!by!sin !! .! ! An!useful!starting!point!in!understanding!systems!is!by!looking!at!their!behavior! in!thermal!equilibrium.!The!coherences!in!thermal!equilibrium!are!zero!–!there! is!no!transverse!magnetization.!The!difference!in!the!number!of!up!and!down! spins,!and!thus,!the!occupation!numbers!of!the!states,!is!determined!by!the! energy!differences!between!the!states.!This!is!given!by!the!Boltzmann! distribution.! !! = ! !!! !!! !!!!!!! ! !
In!the!case!of!a!spin!½!this!equation!can!be!simplified!by!using!the!so[called!
Boltzmann!factor:!! =!ℏ!! !
!!,!the!energy!difference!between!the!two!states.!The!
occupation!number!of!the!spin!up!and!spin!down!states!(in!equilibrium)!can!then! be!written!as!!!"! = !!(1 +!!!)!and!!!"! = !!(1 −!!!),!respectively.!This!yields!an! equilibrium!density!matrix!of!the!form:! ! !!" = 1 2 1 +1 2! 0 0 1 −1 2! = 1 21 + 1 2!!!! ! 2.1.5' The'Rotating'Frame,'Operators'and'RF'Pulses' ! Fundamental!to!MRI!is!the!evolution!of!the!coherence!of!the!density!matrix!over! time.!This!yields!the!NMR!signal!and!determines!the!transverse!relaxation!of!the! system.!Herein,!the!system!will!be!considered!from!the!rotating!reference!frame,! which!rotates!at!the!Larmor!frequency!relative!to!lab!frame.!The!relationship! between!the!lab!frame!and!the!rotating!frame!is!shown!in!Figure!3.!The!rotating! frame!is!a!rotation!about!the!z[axis!relative!to!the!lab!frame!by!a!time!dependent! angle!Φ = ! !!"#! +!!!"#.!The!quantum!mechanical!rotation!operators!for! rotations!about!the!x,!y!and!z!axis!by!an!angle!!!are!given!by:! ! !!(!) = !!!"!!! !!(!) = !!!"!!! !!(!) = !!!"!!! ! the!relationship!between!the!lab!frame!and!the!rotating!frame!is!then!!′! = !! Φ !! = !!!!!!!
! = cos Φ !! + sin(Φ)!!.!By!taking!!! = !!"#!the!x!and!y!
component!of!a!spin!that!is!precessing!at!the!Larmor!frequency!is!constant!in!the! rotating!frame!and!Φ!loses!its!time!dependence.!In!systems!of!multiple!spins!the! Larmor!frequency!can!differ!between!spins!due!to,!for!instance,!chemical!shift,!B0! inhomogeneity!or!quadrupolar!interactions.!In!such!systems!Φ!is!still!time! dependent!and!the!spin!will!rotate!in!the!rotating!frame.!The!rate!of!rotation!is! given!by!the!difference!between!the!Larmor!frequency!of!the!spin!and!the! frequency!of!rotation!of!the!reference!frame:!Ω! = !!− ! !"#.! ! Figure,3.,The,rotating,frame,(indicated,by,'),and,the,fixed,frame,,the,angle,!,is,given,by,! = !!!"#! + !!!"#.,The,rotating,reference,frame,rotates,at,the,same,frequency,as,!!"#,,as,a,result,!,loses, its,time,dependence,in,the,rotating,frame,(assuming,all,spins,rotate,at,the,same,frequency).,
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2.5.6' Relaxation'and'the'spectral'density'' !
Two!main!categories!of!relaxation!behavior!occur!in!NMR.!Firstly,!spin[lattice!
relaxation!is!the!process!by!which!the!spin!populations!!!!and!!!!return!to!
thermal!equilibrium.!Secondly,!spin[spin!relaxation!is!the!process!through!which!
the!coherences!!!!and!!!!return!to!zero.!These!relaxation!processes!are!
characterized!by!the!T1!and!T2!times,!respectively,!which!typically!vary!for!
different!tissues,!temperatures!and!magnetic!field!strengths!and!provide!a!useful! tool!for!generating!contrast!in!magnetic!resonance!imaging.!Several!physical! processes!contribute!to!the!relaxation!of!the!system:!dipole[dipole!coupling!(the! interaction!between!the!magnetic!dipoles!of!the!spins!in!the!system),!chemical! shift!anisotropy!caused!by!the!chemical!shift!not!being!isotropic!around!the! molecules!in!the!system!and!(as!will!later!be!discussed)!for!systems!with!spin! greater!than!½!the!quadrupolar!moment.! !! The!relaxation!behavior!of!spin!systems!is!governed!by!local!fluctuations!in!the! magnetic!field!to!which!the!spins!are!exposed,!typically!due!to!thermal!motion!of! the!spins.!The!relaxation!of!spins!is!characterized!by!the!autocorrelation!of!those! field!fluctuations!or,!more!frequently,!its!Fourier!transform,!the!spectral!density! !(!).!The!spectral!density!takes!the!form!of!! ! = !! !!!!! !!!where!!!!is!the! correlation!time,!a!term!which!describes!the!rate!at!which!the!local!field! fluctuations!take!place.!The!autocorrelation!function!can!be!linked!to!the!T1!and! T2!times!through!the!Solomon!equations,!which!yield!the!following!expressions:! 1 !! = 3 10 !!ℏ 4! !! !! ! ! !! + 4!(2!!) ! 1 !! = 3 20 !!ℏ 4! !! !! ! 3! 0 + 5! !! + 2!(2!!) ! ! with!r!the!distance!between!the!spins,!!!the!gyromagnetic!ratio!of!the!spins!and! !!!implicitly!included!in!!(!).!The!importance!of!the!spectral!density!in!the! relaxation!behavior!of!sodium!will!become!clear!when!looking!at!quadrupolar! interactions!in!ordered!sodium!structures.! ! ! !
3.'
MRI'of'Spin'3/2'Particles'
! Much!of!the!quantum!mechanical!description!that!applies!to!spin!½!particles!can! be!extended!to!spin!3/2!particles.! ! 3.1' Quadrupolar'interactions' ! A!significant!extra!interaction!has!to!be!considered!when!performing!NMR!on! particles!with!a!spin!greater!than!a!half.!This!interaction,!called!the!Quadrupolar! interaction,!comes!about!due!to!an!uneven!charge!distribution!in!the!nucleus! that!affects!only!particles!with!a!spin!greater!than!½.!The!quadrupolar! interaction!is!examined!by!means!of!a!multipole!expansion!of!the!charge! distribution!in!the!nucleus.!The!total!charge!distribution!is!seen!as!the!sum!of!a! set!of!charge!distributions!whose!shape!is!given!by!the!same!equations!that! govern!the!shape!of!electron!orbitals!(see!Figure!4).!The!0th!order!term!gives!the! total!charge!in!the!nucleus,!the!1st!order!term!is!the!electric!dipole!moment,! which!has!been!experimentally!measured!to!be!zero!in!all!nuclei,!the!2nd!order! term!is!the!quadrupolar!moment!and!is!non[zero.!The!order!of!the!terms!that! must!be!considered!for!the!nuclear!charge!distribution!is!determined!by!the!total! spin!of!the!nucleus.!All!terms!with!higher!order!than!! = 2!!are!zero.! ! ! Figure,4.,The,total,charge,distribution,in,the,nucleus,is,given,by,the,multipole,expansion.,The,total, charge,distribution,(left),is,the,sum,of,ℂ!(!),,ℂ!(!),,ℂ!(!),,…,,ℂ!(!),respectively.,The,highest, order,term,that,needs,to,be,considered,is,determined,by,the,nuclear,spin:,! = !",[24]., The!multipole!terms!interact!with!the!electric!field!of!the!environment.!The! interaction!of!the!nuclear!charge!distribution!with!the!electric!field!is!looked!at! using!a!similar!expansion!of!the!electric!potential!as!used!before!with!the!total! charge!distribution:! ! ! ! = ! ! ! + ! ! ! + ! ! ! + ⋯ + ! ! ! = ! ! ! + ∇! ! ! + ∇!! ! ! + ⋯ + ∇!! ! ! ! ! Only!interaction!between!the!same!ordered!multipole!and!electric!potential!terms!must!be!considered,!that!is!ℂ! ! !and!! ! ! ,!ℂ! ! !and!! ! ! etc.!The!
interaction!energy!is!then!given!by:! !
!!"!#(!) = ℂ! ! ! ! ! !"!
For!particles!with!spin!greater!than!½!this!means!there!is!an!additional!energy! term!that!needs!to!be!consider!when!there!is!a!non[zero!electric!field!gradient!in! the!sample!under!consideration.! ! If!the!quadrupolar!interaction!energy!is!much!smaller!than!the!Zeeman!effect,! the!frequency!shift!can!be!described!as!a!result!of!the!quadrupolar!interaction! using!the!first[order!quadrupolar!Hamiltonian:! ! ℋ!(!) = !!! 1 6(3!!!− !(! + 1)1)! ! with!!!(!)!the!first[order!quadrupolar!coupling!given!by:! ! !!(!) = 3!!!!! 2!(2! − 1)ℏ! !
where!!!! = !!!!!!!and!!!!!the!average!value!of!!!!!over!all!the!possible!orientation! (weighted!by!the!probability!of!that!orientation).!In!practice!the!first[order! quadrupolar!coupling!term!is!zero!for!isotropic!liquids!since!the!particle!is!free! to!orient!itself!in!any!direction!resulting!in!!!! = 0.!This!is!not!the!case!for! anisotropic!fluids!such!as!liquid!crystals!since!they!have!certain!preferred! orientation.!In!solids!!!(!)!gains!an!angular!dependency!due!to!the!fixed! orientation!of!the!molecules:! ! !!! Θ = 3!!! !(2! − 1)× 1 2(3!"#!!!− 1)! !
with!!!!the!angle!between!the!principle!axis!of!the!EFG!and!!!!and!!! =!
!!" ! !the! quadrupolar!coupling!constant!in!units!of!hertz.!The!influence!of!quadrupolar! interactions!in!human!MRI!will!be!examined!further!in!section!5.1.6.! 3.2' Density'Matrix'and'Coherence'for'spin'! !'systems' ! The!density!matrices!that!describe!systems!of!spin!3 2!particles!are!4x4!matrices! that!contain!higher!order!coherences!than!those!seen!in!the!spin[½!system! discussed!earlier:! ! = !!! ! !!!!! !!!!! !!! ! !!!!! !!!! !!!! !!!!! !!!!! !!!! !!!! !!!!! !!! ! !!!!! !!!!! !!! ! ! ! !!! ! ,!!!! ! , !!! !!and!!!! !!again!represent!the!occupation!number!of!each! of!the!spin!states.!The!interpretations!of!the!12!coherences!are!given!in!Figure!5.! The!first!term!in!the!subscript!of!the!coherence!is!the!coherence!rank!and!is! given!by!the!difference!in!the!spin!number!!!!between!the!two!states!in!the!
coherence.!The!second!term!is!the!satellite!order!and!gives!the!difference!in!the! square!of!!!!between!the!states.! ! ! Figure,5.,Physical,model,of,the,quantum,coherences,in,a,spin,3/2,system., ! Evidently!three!different!orders!of!coherences!emerge!in!a!spin!3/2!system;!six! single!quantum!coherences!with!the!coherence!order!of!1,!four!double!quantum! coherences!with!coherence!order!of!2!and!two!triple!quantum!coherences!with! coherence!order!3.!First!order!quantum!coherences!are!the!only!quantum! coherences!that!can!be!directly!observed!using!NMR,!second!and!third!order! quantum!coherences!can!only!be!detected!indirectly.!Furthermore,!second!and! third!order!quantum!coherences!cannot!be!created!through!the!application!of!a! single!RF!pulse!but!rather!require!a!sequence!of!carefully[timed!pulses,!see! chapter!3.4.! ! Further!distinction!between!the!behavior!of!coherences!in!a!spin!3/2!systems! can!be!made;!The!frequency!of!the!!!!!!coherence!is!unaffected!by!any!
quadrupolar!effects,!whereas!the!!!!±!!coherence!shifts!by!±!!! !leading!to!a!
formation!of!a!central!peak!and!two!satellite!peaks!in!the!NMR!spectrum!of!a! spin!3/2!system!with!a!non[zero!quadrupolar!interaction.!The!center!peak!is!also! observed!as!having!a!higher!signal!amplitude!in!gradient!echo!imaging!than!the! two!satellite!peaks.!This!is!because!the!!!!!!coherence!has!a!larger!probability! amplitude!of!occurring!than!the!!!!±!!coherence,!resulting!in!a!3:4:3!signal! amplitude!ratio!between!the!!!!!!, !!!!, !!!!!!respectively.! ! 3.2.1' Irreducible'Spherical'Tensor'Operators' ! Describing!the!behavior!of!spin!3/2!systems!using!the!density!matrix!becomes! arduous,!particularly!when!dealing!with!complex!sequences!of!RF!pulses!as!well! as!in!the!presence!of!quadrupolar!interactions.!A!mathematical!formalism!using! so[called!irreducible!spherical!tensor!operators!(ISTOs)!has!been!produced! which!allows!for!tracking!the!changes!to!the!density!matrix!of!the!system! following!the!application!of!RF!pulses!using!(relatively)!simple!tensor!algebra! and!rotation!matrices![25],![26].!The!ISTOs!form!an!orthogonal!basis!in!which! the!density!matrix,!or!more!importantly!the!behavior!of!the!density!matrix!under! the!influence!of!the!Zeeman!and!both!the!static!and!fluctuating!quadrupolar!
Hamiltonian!can!be!solved.!ISTOs!are!used!extensively!in!the!formulation!of! quantum!filters,!the!workings!of!which!are!discussed!in!section!3.4.!ISTOs,! denoted!as!!!"!with!l!and!m!the!rank!and!order!of!the!tensor!respectively,!are! formulated!using!nuclear!spin!operators!and!their!commutation!relations;!see! Table!2.!The!matrix!form!of!the!ISTOs!for!spin!3/2!systems!can!be!found!in![27].! The!rank!and!order!of!the!ISTOs!are!closely!related!to!the!rank!and!order!of!the! various!coherences!in!the!density!matrix:!!!"!is!the!equilibrium!state!of!the! system!when!unaffected!by!quadrupolar!interactions,!!!±!!is!a!rank!2!double! quantum!coherence,!and!!!±!!is!a!rank!3!single!quantum!coherence.! ! !!!!!!!!!!!!!!!! = 1!! !!" = ! !(!!!!!! !!! )!! !!" = ! !"(!!!!! !! !!! !! !!)!! !!!!!!!!!!!!!!!"= !!!! !!±! = ∓!!!!,!± !!! !!±! =∓! !"! [!!!!!! !!! !!!,!!±]!! !!!!!!!!!!!!!!!±! =∓!!!±!! !!±! = !!!± !!! ! !±! = !![!!,!!±!]!!! ! ! !!±! =∓! !! !±!!! Table,2.,The,tensor,operators,for,a,I,=,3/2,system,written,in,terms,of,the,spin,operators,!!,and, !±= !!± !!!,,the,+,subscripted,after,the,commutator,indicates,that,it,is,the,antiQcommutator:, [!, !]!= !" + !",[28]., ! 3.3' Time'evolution'of'ISTOs'and'relaxation'behavior'in'Spin'3/2'systems' ! The!application!of!an!RF!pulse!will!transform!a!system!from!its!equilibrium!state! into!some!new!state.!The!type!of!RF!pulse!as!well!as!the!relaxation!behavior!of! the!system!in!its!new!state!determine!the!NMR!response!of!the!system.!In!spin[½! systems!the!classical!picture!of!an!ensemble!of!rotating!vectors!provides!a! powerful!and!intuitive!model!of!both!the!pulse!response!and!relaxation!behavior.! In!I!>!½!systems!the!behavior!of!the!system!is!significantly!more!complex!yet!the! commutation!relations!between!the!various!Hamiltonians!(purely!Zeeman,!static! quadrupolar!and!fluctuating!quadrupolar)!and!ISTOs!allow!for!understanding!of! the!dynamics!of!these!systems!under!various!conditions!and!pulse!sequences.!All! subsequent!calculations!are!done!in!the!rotating!reference!frame!so!that!!! = 0! and!the!Zeeman!Hamiltonian!can!be!ignored.! ! The!time!dependent!behavior!of!a!system!(under!a!time[invariant!Hamiltonian)! is!given!by!Heisenberg’s!equation!of!motion:! ! !"(!) !" = −![!!, !]! ! Two!types!of!pulse!are!typically!discussed!in!literature:!a!hard!RF!pulse!(a!short! pulse!that!ignores!any!relaxation!behavior!of!the!system!during!the!application! of!the!pulse),!and!a!soft!RF!pulse!(considered!to!be!a!longer!pulse!during!which! the!relaxation!behavior!cannot!be!ignored).!The!latter!case!is!significantly!more! complex!and!will!not!be!expanded!on!in!this!thesis.!A!(hard)!RF!pulse!is!the! application!!!!operator!to!the!system,!this!operator!can!be!written!in!terms!of! ISTOs!as!!! = !!(!!!!− !!!!) = !!! ! !(see!Table!2).!The!Hamiltonian!of!this! action!is!then!!! = !!!!!!(!) = !!!!!(!).!The!general!solution!to!the! Heisenberg’s!equation!of!motion!under!this!Hamiltonian!is!of!the!general!form:!
! ! ! = !!!!!!!!!!!!! ! The!full!list!of!the!evolution!of!the!ISTOs!during!the!application!of!an!RF!pulse! can!be!found!in!table!2!in![28].!Significantly,!the!application!of!a!hard!RF!pulse,! thus!ignoring!any!relaxation!behavior!during!the!RF!pulse,!cannot!change!the! rank!of!the!ISTOs,!only!the!order!is!affected.! ! 3.3.1' Evolution'of'ISTOs'in'an'isotropic'environment:'<EFG>'='0' ! A!system!where!the!electric!field!gradients!average!to!zero!in!the!time!scale!of! 1 ∆!!with!∆!!the!line!width!of!the!NMR!peak!is!said!to!be!an!isotropic!system.! The!Hamiltonian!of!a!system!exposed!only!to!EFG!fluctuations!that!average!to! zero!is!given!by:! !!" ! = !! (−1)!! !!!!"!!![!!!! ! − !!!! ] ! !!!! !
with!!!!the!quadrupolar!coupling!constant,!and!!!!!!the!EFG!tensor!
components!(see![29]!for!the!relevant!components!of!the!EFG!tensor).!The! differential!equation!which!governs!the!time!dependent!behavior!of!the!system! now!takes!on!the!form:! ! !"(!) !" = − !!!, !!!! , ! ! !!!! (!! !!! + !!!! !!! )! !
where!!!!and!!!!are!the!real!and!imaginary!components,!respectively,!of!the!
Fourier!Transform!of!the!autocorrelation!function!of!the!EFG![30].!The!most! important!characteristic!of!this!equation!is!that!it!allows!for!the!evolution!of! ISTOs!in!to!ISTOs!with!the!same!order!but!different!rank.!! ! In!the!isotropic!environment!the!0[coherence!order!tensors!T10!and!T30,! describing!the!longitudinal!magnetization!and!the!octupolar!spin!polarization,! respectively,!transform!as!follows:! ! !!"→ !!"!!!! ! + !!"!!"! ! ! !!" → !!"!!!! ! + !!"!!"! ! ! ! where!!!"! ! !are!the!bi[exponential!relaxation!functions!that!describe!the! relaxation!behavior!of!the!system[28].!This!relaxation,!marked!by!the!absence!of! a!net!quadrupolar!interaction,!is!called!quadrupolar!relaxation[31].!In!the!
extreme!narrowing!limit!!!!! ≪ 1!with!!!!the!correlation!time!of!the!EFG!
fluctuations!the!relaxation!functions!simplify!to!!!!! ! = !!!!!!and!!!"! ! =
!!!"! ! = 0!and!as!a!result!!!"!is!no!longer!formed![32].!The!!!"!tensor!is!
decoupled!from!both!!!"!and!!!",!it!cannot!transition!in!to!any!other!tensors,!and!
while!it!does!have!its!own!relaxation!rate,!this!rate!cannot!be!measured!since!the!
! The!1st!order!coherence!tensors!! !±!!and!!!±!!evolve!similarly!as!the!0[order! coherences!but!with!different!relaxation!functions.! ! !!±! → !!±!!!!±! ! + !!±!!!"±! ! ! !!±! → !!±!!!!±! ! + !!±!!!"±! ! ! ! In!the!extreme!narrowing!limit,!the!relaxation!functions!become!mono[
exponential!and!are!given!by!!!!±! ! = !!!!!!and!!!"±! ! = !!"±! ! = 0,!the!
formation!of!the!third[rank!tensor!is!suppressed!and!the!first!rank!tensor!relaxes!
with!a!single!relaxation!time!T2.!The!!!±!,!!!±!,!!!±!!and!!!±!!coherences!are!all!
fully!decoupled!from!all!other!tensors!and!are!characterized!by!their!own! relaxation!times.!These!tensors!are,!however,!not!directly!observable!and!as!a! result!these!relaxations!cannot!be!measured.! ! 3.3.2' Evolution'of'ISTOs'in'the'presence'of'static'EFGs:'<EFG>'≠'0' ! Local!structures!can!result!in!the!presence!of!non[zero!averaged!EFG!and! consequently!non[zero!averaged!quadrupolar!interactions.!The!Hamiltonian!that! describe!the!non[zero[averaged!energy!of!the!system!is:! ! !!" = !!!!"! !
with!!!!the!residual!quadrupolar!frequency!shift![33].!Since!!!"!and!!!"(!)!
commute!the!time!dependent!behavior!of!the!density!matrix!is!described!by!the! sum!of!the!effects!of!the!two!Hamiltonians! ! !"(!) !" = − !!! 6 !!", ! − !!!, !!! ! , ! ! !!!! (!! !!! + !!!! !!! )! ! The!zero[order!coherences,!associated!with!the!T1!relaxation!rate!are!unaffected! by!the!additional!term!in!the!differential!equation.!While!the!extreme!narrowing! limit!no!longer!holds!for!systems!with!residual!quadrupolar!interactions!(the! correlation!times!of!the!EFGs!are!reduced)!the!longitudinal!relaxation!can!still!be! treated!as!mono[exponential!for!weak!residual!quadrupolar!interactions[34].! For!non[zero!order!coherences!the!additional!quadrupolar!term!results!in!many! of!the!tensors!no!longer!being!decoupled.! ! The!first[order!coherences!now!readily!transform!into!a!superposition!of!all! other!first[order!coherences!and!consequently!formation!of!second!rank!tensors! becomes!possible:! ! !!±! → !!!!!±!!!!! ! ∓ !!±!!!"! ! + !!±!!!"! ! ! !!±! → ∓!!±!!!"! ! + !!±!!!!! (!) ∓ !!±!!!"! ! ! !!±! → !!! !!±!!!"! ! ∓ !!±!!!"! ! + !!±!!!!! ! ! !
The!biexponential!relaxation!functions!are!now!significantly!more!complicated! functions!that!depend!on!several!different!decay!parameters!(see![35]).!The! decay!parameters!relevant!to!the!NMR!spectrum!depend!on!the!pulse!sequences! used!in!the!acquisition!of!the!spectrum!and!will!be!covered!in!more!detail!in! chapter!3.4.!Importantly,!the!two!satellite!transitions!(i.e.!from!!! =!!!→!!!!and! !! = !!→ ! !!will!decay!at!a!faster!rate!compared!to!the!central!transition[36]!and! is!typically!more!likely!to!exhibit!biexponential!decay!than!the!central!transition.! ! In!short,!under!the!application!of!a!hard!(non[selective)!RF!pulse,!the!rank!of!the! ISTOs!doesn’t!change!but!the!order!can!change!with!the!result!typically!a! superposition!of!several!ISTOs.!In!the!relaxation!process!the!rank!of!the! coherence!can!change.!In!the!absence!of!a!non[zero[averaged!EFG!second!rank! coherences!cannot!be!formed.!In!the!presence!of!non[vanishing!quadrupolar! interactions!the!decay!behavior!becomes!more!complex!and!second[rank! coherences!become!accessible.!Several!different!pulse!sequences!will!be!covered! in!chapter!3.4!and!their!effect!and!implementation!in!studies!will!be!covered!in! chapter!5.! ! 3.4' Pulse'sequences'using'ISTO'notation' ! The!framework!set!out!in!the!previous!chapters,!can!now!be!used!to!examine!the! behavior!of!! = 3 2!spin!systems!under!a!series!of!pulses!and!relaxations.!The! inversion!recovery!sequence!is!a!sequence!consisting!of!(at!least)!two!pulses,!a! 180°!pulse!followed!after!an!inversion!time!(TI!time)!by!a!90°!pulse.!Variation!of! the!TI!time!allows!for!signal!suppression!based!on!T1!times,!additionally,! acquisition!of!data!over!a!range!of!TI!times!allows!for!the!determination!of!the!T1! time!in!a!particular!environment!and!is!used!in!section!7.3.1!to!determine!the!T1! times!in!various!agar!and!water!phantoms.!In!thermal!equilibrium!the!density! matrix!of!a!system!of!3/2!spins!is!given!by!the!!!"!tensor,!the!application!of!a! 180°!pulse!results!in!an!inversion!of!this!state!to!−!!"!(see!Appendix!1).!The! system!will!begin!to!undergo!relaxation!immediately!after!the!application!of!the! RF[pulse;!after!time!t1!the!state!of!the!system!is:! ! ! !! = 5 !!" 1 − 2!!!! !! − 2!!"!!"! (!!) ! ! After!a!certain!inversion!time!the!inversion!recovery!sequence!gives!a!90°!pulse! along!the!y[axis!of!the!rotating!frame,!immediately!after!the!application!of!the! pulse!the!state!of!the!system!is:! ! ! !! = 5 !!! ! 1 − 2!!!! !" + 3 2 !!" ! !!"! !" −! 5 2 !!!!!"! !" ! ! The!system!consists!of!three!tensors,!two!1st!order!coherences!and!one!third! order!coherence.!Signal!can,!however,!only!(directly)!be!observed!from!1st!order! coherence!and!thus!the!observable!signal!a!time!t2!after!the!application!of!the!90°! pulse!is!proportional!to:! !
! !", !! ∝ 1 − !!!! !" !!!! !! + 3 2 !!"! !" !!"! !! ! ! where!the!second!term!comes!from!the!relaxation!of!the!!!" ! !tensor!to!the! !!! ! !tensor!after!the!application!of!the!90°!pulse.!The!observed!signal!contains! contributions!from!four!different!relaxation!functions!(see!Appendix!A2).!In! most!systems!the!different!longitudinal!relaxation!rates!are!so!similar!they!are! difficult!to!detect!and!the!longitudinal!relaxation!looks!mono[exponential.! ! 3.4.1' Multiple'quantum'filters' ! In!systems!with!non[zero!residual!quadrupolar!interaction,!i.e.!one!where!the! EFGs!do!not!average!to!zero!in!a!time[scale!of!the!inverse!Larmor!frequency,!the! degeneracy!of!the!system!is!lifted!(see!the!type[a!energy!levels!diagram!in!Figure! 7).!In!cases!where!the!residual!quadrupolar!interaction!is!large!there!will!be!an! observable!split!in!the!NMR!signal!corresponding!to!the!different!∆!!of!the!three! coherences.!In!most!biological!tissue!EFGs!are!small!and!rotational!movement! and!diffusion!of!molecules!result!in!a!residual!quadrupolar!interaction!that!is!so! small!the!resultant!frequency!shift!is!smaller!than!the!FWHM!of!the!central! sodium!peak.!The!resultant!NMR!spectrum!contains!just!one!central!peak!despite! the!degeneracy!of!the!system!being!lifted!(type[c!spectrum!in!Figure!7).!In!order! to!detect!residual!quadrupolar!interactions!in!cases!where!the!frequency!split!is! not!observable!so[called!multiple!quantum!filters,!first!developed!by!Jaccard!et! al[32],!can!be!used.! ! ! Figure,6.,Double,and,triple,quantum,filtering,pulse,sequence,(with,optional,additional,refocusing, pulse).,The,difference,between,the,two,pulse,sequences,is,mainly,due,to,the,different,phase,cycling, used,for,the,DQF,and,TQF.,The,TQF,is,only,able,to,yield,signal,from,the,!!!,tensor,whereas,the,DQF, sequence,can,yield,signal,from,both,secondQorder,(in,cases,where,there,are,nonQzeroQaveraged, EFGs),and,third,order,coherences,depending,on,flipQangle,selection,(figure,adapted,from,[27])., The!magnetization!is!moved!from!equilibrium!by!an!initial!90°!pulse!creating!a! rank[1!1st!order!coherence!which!then!relax!in!to!various!different[ranked! tensors!depending!on!both!the!phase!of!the!applied!RF!pulse!and!the!electric! environment!to!which!the!sodium!ions!are!exposed.!By!using!different!phases!on!
the!RF!pulses!different!orders!of!the!quantum!coherences!can!be!excited.!The!
triple!quantum!filter!yields!only!signal!from!the!!!±!!tensor,!the!presence!of!this!
tensor!indicating!that!the!extreme!narrowing!limit!cannot!be!applied.!The!double!
quantum!filter!contains!signal!from!both!the!!!±!!and!!!±!!tensor!with!the!
presence!of!the!!!±!!tensor!arising!due!to!the!non[zero[averaged!residual!
quadrupolar!interaction.!Signal!from!the!!!±!!tensor!can!be!suppressed!by!using!
a!flip[angle!of!54.7°!for!the!!!!and!!!!pulses![37]!leaving!just!the!signal!from!the!
residual!quadrupolar!interactions.! ! Pulse! !!! !! !!! !!! Acquisition! DQF! !!! −(!!− !!)! −(!!− !!)! 0°! 2!!+ 3!!! TQF! !!! !!+ !!! !!+ !!! !!+ !!! 2!!+ 4!!+ !4!!! Table,3.,Phase,cycling,for,the,phases,of,the,four,pulses,in,a,DQF,and,TQF,experiment.,For,the,DQF:,
!!,and,!!,cycle,through,0°,,90°,,180°,and,270°,,and,!!,cycles,through,0°,,60°,,120°,,180°,,240°,,300°.,
For,the,TQF:,!!,=,0,,!!,cycles,through,0°,,90°,,180°,and,270°,,!!,and,!!,cycle,through,0°,,60°,,120°,,
4.'
Spin'3/2'MRI'in'Biological'Tissue'
! The!NMR!spectrum!of!sodium!ions!is!determined!by!the!electrical!environment! in!which!they!are!located!as!well!as!the!mobility!of!the!ions.!Four!different! environments!are!typically!discussed!for!spin!3/2!particles!in!biological!tissue! (see!Figure!7);!type[a!is!characterized!by!strong!quadrupolar!splitting!and! exhibits!3!peaks!with!relative!peak!amplitudes!of!3:4:3.!This!spectrum! corresponds!to!a!stationary!electric!field!relative!to!the!sodium!ions!and!can!be! obtained!from!crystalline!forms!of!sodium.!The!peak!split!can!be!extremely!large! (upwards!of!1!MHz)[38]!depending!on!crystalline!structure,!B0!field!strength!and! the!orientation!of!the!principle!axis!of!the!crystal!relative!to!the!B0!field.!!The! type[b!spectrum!still!displays!peak!splitting!but!the!relative!amplitude!of!the! peaks!is!no!longer!fixed!at!3:4:3,!broadening!of!the!central!peak!is!also!present.! The!type[b!spectrum!corresponds!to!a!fluctuating!EFG!interaction!of!the! quadrupolar!molecule!on!a!time!scale!longer!than!!!!!!with!! !!the!Larmor! frequency.!The!type[c!spectrum!exhibits!no!peak!splitting!but!displays!a!broad! central!peak.!This!spectrum!is!obtained!when!the!EFG!fluctuations!occur!on!the! same!time!scale!as!!!!!.!The!type[d!spectrum!shows!a!single!narrow!peak!and!is! obtained!in!the!absence!of!an!electric!field!gradient!or!when!the!EFG!interactions! occur!on!a!much!shorter!time!scale!than!!!!!!and!corresponds!to!the!extreme! narrowing!limit!discussed!in!chapter!3.!Type[b!and!c!spectra!are!the!most! commonly!obtained!spectra!in!tissue,!type[d!can!be!obtained!from!NaCl!solutions! in!water[39].!In!this!chapter!I!will!expand!on!how!the!biological! microenvironment!affects!the!sodium!signal!and!affects!in[vivo!sodium!MRI.! ! ! ! Figure,7.,Sodium,spectrum,in,four,different,electrical,environments.,Type,a),A,constant,,strong, electric,field,interaction,,peak,split,is,seen.,The,relative,amplitude,of,the,three,peaks,is,3:4:3.,Type, b),Very,slow,EFG,fluctuations,compared,to,the,Larmor,frequency,,some,peak,splitting,is,still,present, in,the,spectrum.,Type,c),Fluctuations,in,the,EFG,of,the,order,of,the,inverse,Larmor,frequency,,no, speak,splitting,is,present,,the,single,peak,is,broader,than,the,type,d,spectrum,peal.,Type,d),No,(net),electric,field,interactions.,Fluctuations,in,the,EFG,occur,on,a,much,shorter,time,scale,than,the, inverse,Larmor,frequency.[39], ! 4.1' Structure,'Diffusion'and'the'sodium'spectrum' ! Fluctuations!in!the!EFG!clearly!play!a!fundamental!role!in!the!NMR!behavior!of! Sodium.!Understanding!the!relationship!between!the!fluctuations!in!the!EFG!and! the!obtained!sodium!spectrum!provides!a!very!useful!tool!for!probing!the! biological!microenvironment!in!which!the!sodium!is!located.!In!chapter!3!the! correlation!time!of!EFGs!was!introduced!and!used!to!identify!the!regimes!in! which!second[rank!ISTO!were!decoupled!from!first!and!third!rank!tensors.!This! model,!with!a!singular!variable!(the!correlation!time),!is!the!Debye!model!where! the!spectral!densities!are!given!by[39]:! ! !! !!! = 24 5 ℎ!!!"# !" ! !! 1 + !!!!! ! ! !
with!!!!the!Larmor!frequency,!!!!the!frequency!of!the!quadrupolar!interactions!
and!!!!the!correlation!time!of!the!quadrupolar!fluctuations.!A!simulated! spectrum!using!the!Debye!model!with!!!!=!94.5!MHz,!!!!=!25!MHz!and!!!!=!20!ns! and!104!ns!is!given!in!Figure!8.!The!model!shows!a!clear!transition!from!a!type[c! spectrum!to!a!type[b!spectrum!as!the!correlation!time!of!the!EFG!fluctuations! increase.!The!model!fails!however,!to!match!experimental!data!obtained!from! various!biological!samples!(see!the!literature!review).! ! ! ! ! Figure,8.,Simulated,NMR,spectrum,with,the,EFG,fluctuations,modeled,using,the,Debye,model.,The, model,was,simulated,with,a,correlation,time,of,20,ns,on,the,left,and,10,us,on,the,right.,The, spectrum,shifts,from,typeQc,to,typeQb,as,the,correlation,time,in,increases., ! The!Berendsen!model![3]!looks!at!the!interaction!between!sodium!ions!and! water!molecules!in!the!so[called!hydration!shell!of!large!macromolecules!such!as!
proteins.!The!hydration!shell!is!formed!by!the!polar!interaction!of!water! molecules!with!the!charged!surface!of!protein!molecules!and!produces!local! electric!field!gradients!described!by!the!Hamiltonian:! ! !!"# !!" =!!!! 3!"#!! − 1 ! ! with!!!"!the!location!of!the!sodium!ion,!r!the!distance!to!the!water!molecule,!!! the!angle!between!the!vector!connecting!the!water!molecule!and!the!B0!magnetic! field!(see!Figure!9)!and!!!!the!proton!magnetic!dipole!moment.!The!EFG! produced!by!this!field!is!given!by!!!! =! !!!"(3!!"#!! − 1)!which!yields!the! angular[dependent!expression!for!!!!given!in!chapter!3.!The!size!of!the! perturbation!to!the!system!caused!by!the!quadrupolar!coupling!is!thus! dependent!on!the!angle!formed!by!the!local!director!of!the!molecule!and!the! magnetic!field,!with!maximum!splitting!occurring!when!the!director!is!aligned! with!the!magnetic!field.!!! ! ! ! Figure,9.,A,model,of,the,hydration,shell,used,by,Berendsen,to,describe,the,effect,of,the,Sodium, quadrupole,on,the,NMR,spectrum,in,macromolecules,[3]., ! The!magnitude!of!the!splitting!is!determined!by!the!length!of!the!correlation!time! (which!in!turn!is!related!to!the!mean!level!of!order!to!which!the!sodium!ions!in! the!sample!are!exposed![40])!relative!to!the!Larmor!frequency!of!sodium!and! gives!rise!to!the!four!spectra!shown!in!Figure!7![41].!! ! 4.2' The'cell'membrane' ! There!is!a!large!difference!in!the!intra[!and!extra[cellular!ion!concentration! which!causes!a!potential!difference!between!the!two!compartments.!This! potential!difference!exists!across!the!cell!membrane!and!is!called!the!membrane! potential.!In!human!cells!the!cell!membrane!potential!is!typically!of!the!order!of! 40!to!90!mV!in!healthy!tissue!and!is!affected!by!pathology![42].!The!presence!of! such!a!membrane!potential!may!give!rise!to!a!residual!quadrupolar!interaction.! Studies!examining!red!blood!cells!(RBCs)!have!shown!that!there!is!a!non[ vanishing!quadrupolar!interaction!(characterized!by!the!presence!of!a!T21!
tensor)!present!in!RBCs!but!do!not!attribute!the!cause![43].!A!different!study! examined!various!RBCs!in!various!conditions,!intact,!unsealed!ghosts!and!sealed! ghosts.!The!study!found!that!there!was!a!significant!change!in!the!level!of! residual!quadrupolar!coupling!between!the!intact!RBCs!and!the!sealed!ghosts,! despite!there!being!little!difference!in!the!ion!distribution!in!the!cells!(and! consequently!unlikely!to!be!a!significant!change!in!membrane!potential).Thus,!it! is!unlikely!that!the!membrane!potential!is!a!cause!for!(significant)!residual! quadrupolar!coupling,!rather!the!authors!speculate!that!it!is!the!high!level!of! tissue!ordering!in!the!cell!membrane!itself!that!causes!the!residual!quadrupolar! coupling![40].! ! 4.3' B1'Inhomogeneity' ! Signal!intensity!obtained!from!a!sample!always!displays!a!dependence!on!the!flip! angle!applied!to!the!spins!in!that!sample.!Measurements!of!the!sodium! concentration,!by!extension,!also!depend!on!the!local!B1!fields.!Ideally!RF!coils! produce!a!perfectly!homogenous!B1+!field!in!the!sample!resulting!in!an!equal!flip! angle!over!the!entire!sample.!In!reality,!the!B1+!field!distribution!is!strongly! determined!by!many!factors!including!tissue!conductivity,!the!relevant!Larmor! frequency!(and!consequently!the!strength!of!the!B0!field),!the!design!of!the!RF! coil!etc.!RF!pulse!transmission!is!typically!performed!using!a!birdcage[like!coil! design!which!offers!a!very!homogenous!B1+!field.!At!high!frequencies!this!may!no! longer!hold!true!becoming!problematic!in!proton!imaging!in!UHF!MRI!(B0!>!3T,!f0! >!127!MHz).!While!the!sodium!frequency!at!7T!is!much!lower!(79.1!MHz)!than! the!proton!frequency,!multi[pulse!sequences!used!in!sodium!imaging!can!have!a! very!strong!flip!angle!dependence!and!are!thus!highly!sensitive!to!B1+! inhomogeneities.!The!Triple[Quantum[Filter!experiment,!for!instance,!has!a! sin5(α)!relationship![32].!In!multi[quantum!filtering!experiments!a!flip!angle!of!
54.7°!eliminates!the!signal!from!the!!!"!tensor,!variations!in!the!B1+!will!mean!
that!this!tensor!is!only!cancelled!in!certain!regimes.!Close!proximity!to!the!coil! itself!can!also!cause!a!local!increase!in!B1+!intensity!compared!to!that!seen!in!the! center!of!the!coil.!Sodium!quantification!methods!whereby!reference!phantoms! with!a!known!sodium!concentration!are!placed!!next!to!the!body!(and!as!a!result! closer!to!the!coil!than!the!region!of!interest),!are!vulnerable!to!this!type!of!B1+! variation![44].!Mapping!the!B1+!distribution!during!the!imaging!sequences!allows! for!corrections!to!the!data!caused!by!spatial!variations!in!the!obtained!flip!angle! [45],![46].!! ! ! !
5.'
Literature'Review'
! A!wide!body!of!literature!looking!at!the!23Na!in!biological!tissue!exists.!Much!of! the!early!work!focused!on!understanding!the!NMR!behavior!of!sodium!by! studying!various!samples!of,!typically!non[human!tissue,!ex[vivo.!More!recently! the!focus!has!shifted!from!NMR!to!MRI!of!sodium,!driven!by!the!availability!of! high[field!MRI!systems!and!the!introduction!of!pulse!sequences!suited!to!the!extremely!short!T2!times!typically!seen!in!23Na!in!tissue.!
! 5.1' Sodium'Environments' ! Springer![38]!examines!the!behavior!of!several!biologically!important!S!=!3/2! ions!in!4!different!tissue!environments:!the!vascular,!interstitial,!cytoplasmic,! and!intraorganellar!compartments.!However,!due!to!difficulties!in!isolating!these! compartments,!the!study!limits!itself!to!studying!the!so[called!intracellular! compartment!(the!combined!cytoplasm!and!intraorganellar!compartments)!and! refers!to!the!combined!vascular!and!interstitial!compartments!as!the! extracellular!compartment.!The!paper!highlights!that!the!sodium!concentration! for!the!various!extracellular!compartments!is!roughly!the!same.!No!commentary! is!provided!on!any!differing!relaxation!behavior!between!the!different! extracellular!compartments!however.!Estimates!for!the!relative!volume!fractions! of!compartments!are!provided!in![1],!with!the!intracellular!compartment! occupying!an!average!volume!fraction!of!0.8!and!the!extracellular!compartment! an!average!of!0.2.!The!work!by!Springer!reviews!works!by!others!that!indicate! several!important!parameters:!The!sodium!concentration!is!reported!to!be!142! mM/L,!147!mM/L!and!10!mM/L!in!the!plasma!aqueous,!interstitial!space!and! intracellular!compartment!respectively.!! ! 5.1.1'Environmental'Anisotropy' ! Several!different!quadrupolar!behaviors!are!described!in!Chapter!4,!the!relevant! interaction!depends!entirely!on!the!electrical!environment!that!the!sodium!ions! are!located!in.!Typically!it!is!assumed!that!the!in[vivo!sodium!environments’! electric!field!gradient!fluctuations!(EFGF)!are!sufficiently!fast!as!to!average!the! EFG!to!zero!over!any!meaningful!timeframe[38].!This!is!certainly!true!for!sodium! in!highly!mobile!environments!such!as!the!blood!and!the!extracellular!space.!The! presence!of!lipids!and!proteins!provide!a!more!fixed!electric!environment,! however,!and!it!had!been!speculated!that!this!may!impact!the!23Na!signal!in!a! noticeable!manner![39].!Woessner!et!al![47]!show!the!impact!of!the!residual! quadrupolar!effect!in!samples!of!Xantham,!carefully!prepared!as!to!ensure! maximum!ordering,!show!strong!residual!quadrupolar!coupling.!A!type!b! spectrum!is!observed!due!to!the!formation!of!several!domains!of!locally!ordered! Xantham!fibres!(see!Figure!10).!
! Figure,10.,23Na,spectrum,of,Xantham,gum,obtained,at,4.7T.,The,spectrum,represents,a,typeQb, spectrum,due,to,the,formation,of,locally,ordered,domains,in,the,Xantham,samples,,the,quadrupolar, coupling,constant,is,reported,to,be,~4,KHz,[47]., The!effect!of!a!highly!ordered!environment!in!biological!matter!is!studied!by! Eliav!et!al[48],!motivated!by!the!23Na!spectra!obtained!from!bovine!nasal! cartilage!obtained!by!double!and!triple!quantum!filtering.!Experimentally!the! study!found!that!a!non[zero!T21!tensor!was!formed!after!the!first!pulse!in!the! DQF!in!the!bovine!nasal!cartilage!and!attribute!this!to!a!non[vanishing! quadrupolar!interaction.!Further!modeling!of!the!23Na!spectrum!of!bovine!nasal! cartilage!by!Eliav!and!Navon![4]!shows!that!the!obtained!spectrum!is!most! closely!fitted!by!a!model!that!considers!an!effect!of!the!angle!of!fibers!relative!to! the!B0!magnetic!field.!This!model!is!based!on!a!fiber!model!where!diffusion!is! able!to!take!place!along!the!length!of!the!fiber!but!is!suppressed!in!the!radial! direction,!initially!proposed!by!Halle[49].!! ! The!appearance!of!this!non[vanishing!quadrupolar!moment!due!to!the! heterogeneity!of!tissue!significantly!affects!the!success!of!multi[quantum[filtered! sequences!to!discriminate!between!intra[!and!extracellular!fluid!since!these! sequences!work!by!only!revealing!signal!from!sodium!with!a!net!quadrupolar! moment,!which!is!assumed!to!only!be!found!in!intracellular!sodium[50].!Work!by! Shinar!et!al!found!that!careful!selection!of!flip!angles!used!in!DQF!sequences!
allows!for!the!suppression!of!the!T32!signal!with!just!the!T21!signal!(hypothesized!
to!be!due!to!orientation!effects)!remaining[43].!They!use!this!method!to!examine! the!effect!of!the!plasma!membrane!of!red!blood!cells!on!the!sodium!quadrupole.!
DQF!performed!with!a!flip[angle!of!54.7°!(yielding!only!signal!from!the!T21!
tensor)!shows!an!NMR!signal,!after!destruction!of!the!cytoskeleton!of!the!red! blood!cells!this!signal!disappears!indicating!that!the!plasma!membrane!is!
responsible!for!the!formation!of!the!T21!tensor.!The!reported!∆!!!(at!either!8.46!
or!9.4!Tesla,!the!paper!is!unclear)!as!a!result!of!the!plasma!membrane!in!red! blood!cells!is!22.3!rads[1.!! ! The!same!method!is!also!used!to!by!Reddy!et!al!to!show!the!presence!of!residual! quadrupolar!coupling!in!the!brain!and!gastrocnemius![31]!and!knee!cartilage! [51]!and!by!Duvvuri!et!al!in!human!breast!tissue![52].!The!presence!of!residual! quadrupolar!interactions!in!the!gastrocnemius!is!surprising!given!the!alignment!
of!the!muscle!fibers!with!the!B0!magnetic!field!but!could!be!due!to!the!inclusion!
of!the!tendon!attachments!in!the!voxels.!An!explanation!by!the!authors!is!not! provided.!The!spectra!reported!by!Reddy!et!al!and!Duvvuri!et!al![31],![51],![52]! have!been!obtained!using!surface!coils,!yet!DQF!relies!on!flip[angle!selection!to!
discriminate!between!the!T21!and!T31!tensors.!Both!authors!provide!reference!
work!by!Reddy!et!al![37]!yet!this!work!only!looks!at!the!influence!of!B1!
inhomogeneity!on!the!T31!tensor,!not!on!the!T21!tensor.!A!question!then!remains!
on!what!the!influence!of!this!variable!flip!angle!is!on!the!spectra,!particularly! since!no!voxel!size!is!reported!in!any!of!these!papers.!DQF!spectra!obtained!from! isotropic!sodium!samples,!such!as!saline!solution,!that!should!show!no!signal! from!the!T21!tensor!could!provide!some!insight!but!are!not!provided.! ! A!study!by!Stobbe!and!Beaulieu![53]!looked!at!the!impact!of!residual! quadrupolar!interactions!on!sodium!MRI!in!the!brain!using!the!change!in!the! effective!flip!angle!of!sodium!as!a!result!of!the!presence!of!quadrupolar! interactions.!They!suggest!that!there!is!indeed!a!marginal!net!quadrupolar! interaction!in!the!white!matter!and!that!the!magnitude!of!this!interaction!is! dependent!on!the!direction!of!the!white!matter!tracks!relative!to!the!B0! interaction,!as!is!expected!for!quadrupolar!interactions.!Madelin!&!Regatta![1]! suggest!that!similar!effects!may!be!seen!in!other!ordered!structures!such!as! muscles!and!cartilage!though!provide!no!further!evidence!in!support.!A!good! understanding!of!which!tissues!are!affected!by!residual!quadrupolar!interactions! is!vital!in!the!accurate!quantification!of!sodium!since!quadrupolar!interactions! cause!the!bi[exponential!decay!behavior!that!is!used!for!modeling!sodium! concentration.!! ! 5.2' In]vivo'Imaging'Methods'' ! Much!of!the!scientific!literature!examining!sodium!in[vivo!using!MRI!has!looked! at!mapping!the!total!sodium!concentration!in!specific!regions!in!the!body.!The!bi[ exponential!nature!of!sodium!in!biological!tissue!poses!a!significant!challenge!to! this.!The!so[called!fast!component!of!the!sodium!spectrum,!representing!a! significant!fraction!of!the!total!sodium!signal,!decays!within!a!few!milliseconds.! In!order!to!accurately!map!the!sodium!environment!in[vivo!specialized!pulse! sequences!are!needed.!Most!commonly!these!are!adaptations!of!pulse!sequences! used!in!proton!imaging!with!very!short!echo!times.!Alternative!methods!that! allow!for!the!discrimination!of!tissues!with!differing!T2!times!are!also!used!but! present!their!own!challenges.! ! A!theoretical!examination!of!the!sodium!signal!behavior!under!various!pulse! sequences!showed!that!3D!imaging!using!a!non[selective!RF!pulse!allowed!for!a! reduction!in!the!minimum!achievable!echo!time![54].!Methods!of!efficiently! sampling!k[space!in!3!dimensions!using!non[Cartesian!sampling!have!been! developed!and!are!widely!used!in!sodium!imaging,!these!include!twisted! projection!imaging![55]–[57],!radial!projection![58]–[60]!and!3D!Cones![61].! These!pulse!sequences!allow!for!an!echo[time!of!less!than!1ms,!somewhat!lower! than!the!fast[component!of!the!T2!times!of!sodium!in!tissue!(typically!reported!to! bea!round!2!ms).!Drawbacks!to!using!these!methods!is!that!they!are!relatively!
demanding!on!the!gradient!system!and!that!the!reconstruction!of!the!data!is! more!complicated!due!to!the!non[Cartesian!sampling!of!k[space.!Due!to!the!low! SNR!of!sodium!imaging!relatively!large!voxels!are!typically!used!in!data! acquisition.!In!anatomical!regions!these!large!voxels!can!introduce!significant! partial[volume!effects,!particularly!when!the!surrounding!environment!has!a! particularly!large!sodium!concentration,!as!is!typically!seen!in!joints!where!care! has!to!be!taken!that!signal!from!the!synovial!fluid!is!not!included!when! examining!cartilage.!Several!studies!have!used!inversion!recovery!combined! with!non[carthesian!sampling!of!k[space!to!suppress!the!signal!coming!from! liquid!in[vivo![62]–[64].! ! Single[Point!Ramped!Imaging!with!T1!Enhancement!(SPRITE)!imaging! modalities,!an!extension!of!the!SPI!technique,!were!initially!developed!for!the! examination!of!unusual!materials!such!as!gases[65]!and!concrete[66][67]!that!
exhibit!very!short!T1!and/or!T2/T2*!relaxation!times!and!have!since!been!applied!
by!several!authors!for!the!study!of!sodium!concentration!in!tissue.!Three!SPRITE! techniques!that!traverse!k[space!in!different!manners!were!compared!by! Romanzetti!et!al[68]!and!show!an!excellent!linear!dependence!of!image!signal!on! sodium!concentration!with!SPRITE!using!a!conical!k[space!trajectory!had!both! the!shortest!acquisition!time!and!highest!SNR!of!the!SPRITE!techniques! discussed!in!this!paper.!So!far!there!has!been!little!application!of!these!method!in! human!in[vivo!sodium!MRI!with!the!only!images!provided!in!the!paper!by! Romanzetti.!Veliyulin!et!al[69].!Steps!for!reducing!scan!duration!and!SAR! levels[70]!may!provide!easier!implementation!of!this!imaging!method!in!human! studies.!! , Mapping!of!regions!affected!by!quadrupolar!interactions!is!important!not!only!in! aiding!the!understanding!of!the!in[vivo!sodium!environment!but!also!is!vital!in! ensuring!accuracy!in!measurements!of!in[vivo!sodium!concentration.!Stobbe!and! Beaulieu![53]!map!the!quadrupolar!interaction!based!on!the!assumption!that! sodium!ions!experiencing!net!quadrupolar!interactions!will!feel!an!increased! effective!flip!angle,!as!described!theoretically!by!Pandey!et!al![71].!They!run!a!T1! weighted!gradient!echo!sequence,!one!with!a!60°!tip!angle!and!one!with!a!120°! tip!angle!and!use!the!standard!signal!equation!to!extract!a!map!of!the!relative!flip! rate.!Multiple!quantum!filtering!has!shown!promise!as!a!method!of!acquiring! intricate!knowledge!of!the!biological!sodium!environment!(as!previously! discussed!in!this!thesis).!Implementation!of!multiple!quantum!filtering!in!in[vivo! sodium!imaging!has,!however,!been!very!limited,!likely!due!to!SAR!limitations! associated!with!these!multi[pulse!sequences.!Examples!of!double!and!triple! quantum!filtered!do!exist!in!literature![72]–[74]!showing!that!it!is!possible!to! apply!this!technique!in!human!studies.! ! 5.2.2' Chemical'Shift'Reagents' ! Several!studies!compare!the!intra!and!extracellular!sodium!concentrations!using! so[called!chemical!shift!reagents!(CSRs)!that!shift!the!resonance!frequency!of! sodium.!By!choosing!a!CSR!that!is!unable!to!penetrate!the!cell!wall,!only!the! extracellular!sodium!has!its!frequency!shifted.!The!(potential)!toxicity!of!these!
materials!have!limited!the!application!of!CSRs!to!ex[vivo!studies!where!they! provide!a!simple!method!for!differentiating!between!the!intra!and!extra!cellular! sodium!concentration.!Liebling!and!Gupta![75]!use!the!membrane[impenetrable! dysprosium!tripolyphosphate!to!shift!the!frequency!of!sodium!in!the! extracellular!space!and!leaving!the!intracellular!sodium!frequency!untouched.! Cowen!et!al![76]!place!red!blood!cells!in!a!saline!solution!with!a!known!sodium! concentration!and!5mMol/l!Dy(PPP)2!to!determine!the!ratio!of!intra[!and!extra[! cellular!sodium!by!comparing!the!height!of!the!shifted!and!non[shifted!sodium! peaks.!An!intra[cellular!sodium!concentration!of!10.7!mmol/L!is!reported,! corresponding!to!a!sodium!membrane!potential!of!+65mV.! ! Exchange!between!sodium!environments!is!a!continually!occurring!process!due! to!sodium!leaking!in!to!the!cell!and!being!transported!out!of!the!cell!by!the! sodium!potassium!pump.!The!rate!at!which!this!exchange!takes!place!could!have! an!impact!on!the!sodium!spectrum!as!is!discussed!in!section!5.1.2.!Several!papers! have!used!CSRs!to!examine!the!rate!of!sodium!exchange!by!linking!the!rate!of! exchange!to!the!spectrum!shape!using!the!framework!described!in!section!5.1.2.! Elgavish!et!al![77]!use!this!method!to!model!the!rate!of!sodium!exchange!in! brush!border!membranes!found!in!the!kidney,!arriving!at!an!exchange!rate!of!! 580!s[1,!much!larger!than!the!reported!frequency!shift!induced!by!the!CSR,!as! such!the!sodium!in!these!kidney!membranes!should!be!treated!as!being!in!the!so[ called!fast!exchange!spectrum!as!explained!in!section!5.1.2., ! 5.3' Sodium'Quantification' ! A!primary!incentive!driving!the!development!of!sodium!MRI!is!the!need!map!the! sodium!concentration!in!various!tissue.!This!incentive!differs!from!that!typically! pursued!in!proton!imaging!where!the!primary!focus!is!on!mapping!the!internal! structure!of!the!body.!This!is!an!important!distinction.!When!performing!proton! imaging!the!signal!intensity!obtained!from!a!ROI!mainly!affects!the!clarity!with! which!anatomical!region!can!be!discriminated!by!influencing!the!tissue!contrast! and!SNR.!In!sodium!imaging!a!third!dimension!is!added,!the!signal!obtained!must! be!directly!relatable!to!a!specific!sodium!concentration.!A!common!standard!has! been!developed!where!the!signal!is!compared!to!known!concentrations!of! sodium!inside!phantoms!placed!in!the!RF!coil!together!with!the!patient.!In!this! section!we!will!examine!the!methodology!used!and!variables!that!need!to!be! accounted!for.! ! 5.3.1' Methodology' ! The!approach!of!comparing!signal!intensities!of!phantoms!with!known!sodium! concentration!to!interpolate!the!total!sodium!concentration!in!a!sample!is! introduced!by!Thulborne!and!Ackerman![44].!!Accurate!quantification!of!sodium! in!tissue!that!exhibits!bi[exponential!decay!requires!ultra[short!TE!imaging! sequences!to!ensure!the!signal!from!the!fast!component!T2!relaxation!is!not! decayed!significantly!during!acquisition[78].!Furthermore,!the!relaxation! behavior!of!the!reference!phantoms!should!mimic!that!seen!in!the!tissue!under! examination!to!ensure!accuracy!of!the!sodium!quantification,!using!a!reference!
phantom!that!shows!no!significant!fast[component!in!the!T2!to!map!sodium!in! tissue!that!does!will!lead!to!an!underestimation!of!the!sodium!concentration!in! tissue!due!to!more!rapid!signal!decay!in!tissue!compared!to!the!reference! phantoms.!Extra!care!should!be!taken!in!the!comparison!between!healthy!and! diseased!tissue,!Thulborn!et!al!observed!a!significant!change!in!the!volume! fraction!occupied!by!the!intra[!and!extra[!cellular!sodium!environment!in!stroke! victims![79],!such!changes!in!the!signal!ratio!can!lead!to!changes!in!the!T2! relaxation!of!the!tissue!sodium!signal!and!may!lead!to!inaccuracies!in!mapping!of! the!sodium!concentration!to!sodium!signal!intensity.!Placement!of!the!phantoms! in!the!coil!should!be!done!in!locations!that!have!similar!|B1+|!to!the!tissue!under! examination!to!ensure!similar!flip[angles!are!achieved!in!the!tissue!and!the! phantoms.!! ! 5.4' Sodium'in'Pathology' ! Changes!in!total!sodium!concentration!or!the!ratio!of!intra!and!extra[cellular! sodium!have!been!identified!in!a!wide!range!of!conditions.!In!the!following! section!I!will!provide!an!overview!of!some!of!the!different!pathologies!examined! and!the!imaging!sequences!used!in!those!studies.! ! 5.4.1' Brain' ! The!distribution!and!concentration!of!sodium!in!the!human!brain!has!been! extensively!studies!at!a!range!of!field!strengths!and!in!many!different! pathologies.!Measurements!of!a!healthy!human!brain!at!9.4!T!show!an! intracellular!and!extracellular!sodium!concentration!of!12!and!145!mMol/L! respectively[80].!Minor!differences!in!sodium!concentration!are!seen!between! grey!and!white!matter,!with!reported!values!in!literature!of!36.6!µmol/g!and! 27.6!µmol/g!respectively![57].!T1!relaxation!times!in!tissue!and!CSF!are!reported! to!be!25!ms!and!55!ms,!respectively![1].! ! Sodium!MRI!in!the!brain!has!been!used!extensively!to!study!the!effect!of!ischemic! stroke!and!shows!promise!as!a!clinical!tool!for!improving!the!accuracy!with! which!salvageable!tissue!can!be!identified.!A!model!for!determining!which!tissue! may!be!viable!and/or!at!risk!uses!perfusion!weighted!imaging!to!show!regions!of! reduced!perfusion!and!diffusion!weighted!imaging!to!show!dead!tissue.!In! regions!showing!both!a!reduction!in!PWI!and!DWI!signal!tissue!is!lost,!in!regions! showing!a!reduced!PWI!but!not!DWI!(a!so[called!PWI[DWI!mismatch)!tissue!is!at! risk!of!damage!but!not!yet!dead!and!may!be!eligible!for!emergency!intervention.! DWI!may,!however,!not!accurately!describe!regions!of!tissue!death!and!it!had! been!suggested!sodium!imaging!may!provide!a!more!accurate!mapping!of! regions!of!tissue!loss![81].! ! TSC!in!regions!affected!by!ischemic!stroke!increases!with!time!after!stroke,!with! the!rate!of!increase!levelling!off!after!24+!hours!indicating!that!the!increase!in! sodium!signal!is!associated!with!progressive!tissue!death![79],![82],![83].!Data! were!acquired!at!3T!using!a!twisted!projection!imaging!(TPI)!sequence!
