RedeemingtheClinicalPromiseof
DiffusionMRI
inSupportoftheNeurosurgicalWorkflow
LucFlorackTMCV,July262017,Hawaii,US
EconomicCostofBrainDisordersinEurope2010:€798billion…
EuropeanJournalofNeurology2012,19:155–162
(N)MRIscanner
“everythingmustbemadeassimpleaspossible,butnotonebitsimpler”
aMributedtoAlbertEinstein
operaPonalmodel:brain=constrainedwater
operaPonalmodel:brain=constrainedwater
hypothesis
Pssuemicrostructureimpartsnon-randombarrierstowaterdiffusion
©C.Beaulieu,NMRBiomed.2002,vol.15,nr.7-8,DOI:10.1002/nbm.782
operaPonalmodel:brain=constrainedwater
Riemannianparadigm
extrinsicdiffusiononEuclideanspace≈intrinsicgeometryofaRiemannianspace
operaPonalmodel:brain=constrainedwater
Riemannianparadigm
extrinsicdiffusiononEuclideanspace≈intrinsicgeometryofaRiemannianspace
operaPonalmodel:brain=constrainedwater
Riemannianparadigm
extrinsicdiffusiononEuclideanspace≈intrinsicgeometryofaRiemannianspace
localgaugefigureRiemanngeometry
gaugefigure=unitsphere=indicatrix=Riemannianmetric=innerproduct
localgaugefigureRiemanngeometry
localgaugefigureRiemanngeometry
localgaugefigureRiemanngeometry
localgaugefigureRiemanngeometry
length2=6
localgaugefigureRiemanngeometry
length2=9
localgaugefigureRiemanngeometry
geodesictractography
‘short’geodesic
‘long’geodesic
Riemannianlength:Euclideanlength
5.0:6.0<1
7.5:6.0>1
geodesictractography
DiffusionTensorImagingversus
localgaugefigure
DiffusionTensorImagingphysicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
E" = �1
2�~Bz
E# = +1
2�~Bz
�E = �~Bz = ~!Larmor
nuclearspinquantization→ Zeemansplitting
E = �µ ·B
N"N#
= exp
�E
KT
�⇡ 1 +
�~Bz
KT
Boltzmannstatistics→magnetization(typicalclinical3TMRIscanner)
N" �N#N" +N#
⇡ �~Bz
2KT⇡ 10�5
Mz =N" �N#N" +N#
Mmax
⇡ (N" +N#)�2~2Bz
4KT
`lowsensitivitymodality’
`bigxsmall=measurable’
Bloch-Torrey/Stejskal-Tanner/Basser-Majello-LeBihan:GaussiansignalaMenuaPoninq-space
Bloch-Torreyequation→DTI
@M?@t
= �i�(M?Bk �MkB?)�M?T2
+r ·DrM?
S(x, q, ⌧) = S0(x) exp(�⌧q ·D(x)q)
D(x) = �1
⌧
r2q ln
S(x, q, ⌧)
S0(x)
Bloch-Torreyequation→DTIfurtherreading
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
geodesictractographymathematicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
geodesictractographymathematicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
geodesictractographymathematicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
geodesictractographymathematicspipeline
DTI→ localgaugefigure
DTIsignalmodel:S(x, q, ⌧) = S0(x)e
�⌧q
TD(x)q
Riemannmetric: G(⇠, ⇠) |x
= ⇠
TG(x)⇠
Fusteretal.: G(x).
= D
adj(x)
Lengletetal./O’Donnelletal.: G(x).
= D
inv(x)
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
Euclideangeodesic
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
Riemanniangeodesic
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
Riemann-DTIparadigm:
• neuralfiberbundlescorrespondtorelativelyshortgeodesicsinaRiemannian‘brainspace’
• theRiemannianstructurecanbeinferredfromDTI
geodesiccompleteness=
redundantconnecPons
ellipsoidalgaugefigure=
poorangularresoluPon
Riemann-DTIparadigmpros&cons
pro:pixels→geodesiccongruences
con:destructiveinterferenceoforientationpreferences
specificmodel(DTI): S(x, q, ⌧) = S0(x) exp(�D(x, q, ⌧))
D(x, q, ⌧) = ⌧q
TD(x) q
with6d.o.f.’sperpointsample
⬍6d.o.f.’soflocalgaugefigure
S(x, q, ⌧) =1X
k=0
S
i1...ik(x, ⌧)�i1...ik(q)genericmodel(HARDI):
D(x, q, ⌧) =1X
k=0
D
i1...ik(x, ⌧) i1...ik(q)
or ∞ d.o.f.’sperpointsample
beyondtheRiemann-DTIparadigmaparadigmshift
beyondtheRiemann-DTIparadigmaparadigmshift
DTI ⊂ HARDI
⬍ ⬍
Riemanngeometry
⊂ Finslergeometry
beyondtheRiemann-DTIparadigmaparadigmshift
DTI ⊂ HARDI
⬍ ⬍
Riemanngeometry
⊂ Finslergeometry
beyondtheRiemann-DTIparadigmaparadigmshift
DTI ⊂ HARDI
⬍ ⬍
Riemanngeometry
⊂ Finslergeometry…
beyondtheRiemann-DTIparadigmaparadigmshift
DTI ⊂ HARDI
⬍ ⬍
Riemanngeometry
⊂ Finslergeometry
Finslergeometryheuristics
w
x
wosculatingindicatrices
basemanifold: x 2 R3 (x,w) 2 R3 ⇥ S2→ (spherebundle)
x
basemanifold: x 2 R3 → (slittangentbundle)(x, ⇠) 2 R3 ⇥ TR3\{0}
Finslergeometryheuristics
familyofosculatingindicatrices⬍
single(convex)indicatrix
gaugefigure=unitsphere=indicatrix=Finslermetric≠innerproduct
Finslergeometryheuristics
in: Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data, Springer 2014
connections
Cartan tensor
HV-splittingFinslergeometry
axiomatics
distance: d(x1, x2) = inf
⇢Z
�F (�(t), �(t)) dt
���� � 2 C
1([t1, t2],R3) , �(t1) = x1 , �(t2) = x2
�
Finslermetric:F
2(x, ⇠) = gij(x, ⇠)⇠i⇠
j , gij(x, ⇠) =1
2@⇠i@⇠jF
2(x, ⇠)
Cijk(x, ⇠) =1
2@⇠kgij(x, ⇠) =
1
4@⇠i⇠j⇠kF
2(x, ⇠)Cartantensor:
Finslergeometryaxiomatics
Riemannianlimit:F
2(x, ⇠) = gij(x)⇠i⇠
j , gij(x) =1
2@⇠i@⇠jF
2(x, ⇠)
Cijk(x, ⇠) = 0Riemannianlimit: (Deicke’sTheorem)
Deicke’sTheorem:
SpaceisRiemannianifftheCartantensorvanishes.
Finslergeometryaxiomatics
HARDI~Finsleriangeometry(generalizednorm):
F (x,�y) = |�|F (x, y)
F
⇤(x,�q) = |�|F ⇤(x, q)
DTI~Riemanniangeometry(innerproductnorm):
F (x, y) =qy
TDinv(x)y
F
⇤(x, q) =pq
TD(x)q
Finsler-DTIparadigmgeodesictractography
Finsler-DTIparadigmgeodesictractography
Finslermetric:lengthsGv(v, v) = kvk2F
Chern-Rund(orother)connection:paralleltransportrx
x = 0
formalChristoffelsymbols:“pseudo-forces”x
i +X
jk
�ijk(x, x) x
jx
k = 0
• neuralfiberbundlescorrespondtorelativelyshortgeodesicsinthe3-dimensional‘horizontalpart’ofa5-dimensional`brainspace’furnishedwithaFinslerianstructure
• thisFinslerianstructurecanbeinferredfromdiffusionMRImeasurements
• theFinsleriandualmetriccanbeinterpretedasa‘5-dimensional(3x3)DTI’tensor
• FinslergeometryencompassesRiemanniangeometryasaspecialcase
• theFinslermetricadmits∞d.o.f.’sperspatialpointasopposedto6d.o.f.’sfortheRiemannianlimit
• theFinsler-DTIparadigmadmitsversatiledimensionalityreductionintrade-offwithacquisitiontime
*TomDelaHaije,PhDthesis,May162017,Eindhoven
Finsler-DTIparadigmoperationalization*
FinslerfuncPon:
F (x,�y) = |�|F (x, y)
F
⇤(x,�q) = |�|F ⇤(x, q)
Finsler-DTIparadigmsummary
FinslerfuncPon:
F (x,�y) = |�|F (x, y)
F
⇤(x,�q) = |�|F ⇤(x, q)
Finsler-DTIparadigmsummary
FLAIR DTI
© Neda Sepasian, Joep Kamps, Andrea Fuster
applications&outlook
applications&outlook
StephanMeestersetal.,electronicposter3476,ISMRM2017,Hawaii
preoperative postoperative
© Stephan Meesters
applications&outlook
preoperative postoperative
© Stephan Meesters
applications&outlook
©DaichiKominami
applications&outlook
Bernhard Riemann Sophus Lie Elie Cartan Albert Einstein Paul Finsler
conclusion
• FinslergeometryisagenericandpotentiallypowerfulframeworkfordiffusionMRIbeyondclassicalDTI
• thisframeworkallowsustoexploitarichbodyofknowledgegainedovermorethanacenturybygreatscientists
…