Diﬀusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann

RedeemingtheClinicalPromiseof

DiffusionMRI

inSupportoftheNeurosurgicalWorkflow

LucFlorackTMCV,July262017,Hawaii,US

EconomicCostofBrainDisordersinEurope2010:€798billion…

EuropeanJournalofNeurology2012,19:155–162

(N)MRIscanner

“everythingmustbemadeassimpleaspossible,butnotonebitsimpler”

aMributedtoAlbertEinstein

operaPonalmodel:brain=constrainedwater


hypothesis

Pssuemicrostructureimpartsnon-randombarrierstowaterdiffusion

©C.Beaulieu,NMRBiomed.2002,vol.15,nr.7-8,DOI:10.1002/nbm.782


Riemannianparadigm

extrinsicdiffusiononEuclideanspace≈intrinsicgeometryofaRiemannianspace


Riemannianparadigm



Riemannianparadigm


localgaugefigureRiemanngeometry

gaugefigure=unitsphere=indicatrix=Riemannianmetric=innerproduct





length2=6


length2=9


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:





physicsinanutshell:





physicsinanutshell:





physicsinanutshell:





physicsinanutshell:





physicsinanutshell:





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:




geodesictractographymathematicspipeline

physicsinanutshell:





physicsinanutshell:





physicsinanutshell:





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



x = 0


i +X

jk

�ijk(x) x

jx

k = 0




x = 0


i +X

jk

�ijk(x) x

jx

k = 0


Euclideangeodesic



x = 0


i +X

jk

�ijk(x) x

jx

k = 0


Riemanniangeodesic



x = 0


i +X

jk

�ijk(x) x

jx

k = 0


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


DTI ⊂ HARDI

⬍ ⬍

Riemanngeometry

⊂ Finslergeometry


DTI ⊂ HARDI

⬍ ⬍

Riemanngeometry

⊂ Finslergeometry


DTI ⊂ HARDI

⬍ ⬍

Riemanngeometry

⊂ Finslergeometry…


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}


familyofosculatingindicatrices⬍

single(convex)indicatrix

gaugefigure=unitsphere=indicatrix=Finslermetric≠innerproduct


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


StephanMeestersetal.,electronicposter3476,ISMRM2017,Hawaii

preoperative postoperative

© Stephan Meesters


preoperative postoperative

© Stephan Meesters


©DaichiKominami


Bernhard Riemann Sophus Lie Elie Cartan Albert Einstein Paul Finsler

conclusion

• FinslergeometryisagenericandpotentiallypowerfulframeworkfordiffusionMRIbeyondclassicalDTI

• thisframeworkallowsustoexploitarichbodyofknowledgegainedovermorethanacenturybygreatscientists

…

Documents

Diﬀusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann