Diffusion MRI in the Neurosciences
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Transcript of Diffusion MRI in the Neurosciences
Mitg
lied
der H
elm
holtz-G
em
ein
schaft
Diffusion MRI in the Neurosciences
N. Jon Shah
Institute of Neuroscience and Medicine – 4
Research Centre Juelich
52425 Juelich
GERMANY
April 20, 2023 Slide 3Institute of Neuroscience and Medicine
Outline Introduction
Qualitative issues of diffusion. Spins diffusing under a background gradient.
Bipolar gradient-echo sequence. Stejkal-Tanner spin-echo sequence.
Approaches to the problem Microscopic approach. Quantum mechanical treatment.
The diffusion propagator. Solution for the Stejskal-Tanner experiment.
Macroscopic approach. The Bloch-Torrey equation. Solution for the bipolar gradient-echo. Solution for the Stejkal Tanner equation.
Diffusion in biological tissue. Hindered diffusion. The ADC approach. Signal attenuation.
Anisotropic systems. The DTI approach. Invariant parameters. Fractional anisotropy (FA), Mean Diffusivity
(MD), etc. Fibre tracking.
April 20, 2023 Slide 4Institute of Neuroscience and Medicine
• 1827 – Brown describes what is now known as “Brownian Motion”.
History …
Definition: diffusion is the thermal motion of all particles at temperatures above the absolute zero in a liquid or gas. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles.
• 1905 – Einstein describes the statistical mechanics of diffusion.
Dtrr 2)0()0(2
Mean Square Displacement at the time t (averaged over the particle ensemble). Can be measure with MRI.
Diffusion Coefficient. Estimated from MRI signal attenuation.
x
y
)(tr
)0(r
April 20, 2023 Slide 5Institute of Neuroscience and Medicine
Classical representation of NMR diffusion measurements
Transverse Magnetization
Gradient Echo
The constant magnetic field is superimposed by an inhomogeneous field
90°
RF
Gradient gδ δ
Δ -g
Assuming no field inhomogeneities more than the caused by the gradient
gzB 0 )()( 0 gzBz
April 20, 2023 Slide 6Institute of Neuroscience and Medicine
RF
Hahn spin-echo
Gradient
Transverse Magnetization
90° 180°
gδ δ
Δ
τ
2τ
April 20, 2023 Slide 7Institute of Neuroscience and Medicine
Finally
033)(
000
0),,()()(
rrdderrPrpM
tM
V
rrgi where:
}{Tr 20 yIbM
AssumingV
rp1
)( 0
, a gradient in the z-direction and using the gaussian propagator,
DgE eSTS
2)(0)(
So one can estimate D by measuring changes in the signal while varying the time parameters or the gradient strength.
the amplitude of the signal results
x
y2r
1r
Drr 22
12
22)(6
1
0)(rg
E eSTS
April 20, 2023 Slide 8Institute of Neuroscience and Medicine
),(ˆˆˆ
),(),(
1
0
20 trmDe
T
mm
T
ememBtrm
t
trm Tz
zyyxx �
A macroscopic approach:The Bloch-Torrey equation.
H.C. Torrey showed in 1956 how the Bloch equations would change for an ensemble of spins diffusing in the presence of a magnetic field gradient.
The Bloch-Torrey equation for the evolution of the distribution of magnetization m in the presence of a magnetic field Bz is,
Diffusion term
Larmor precession 0B
Transversal relaxation
Longitudinal relaxation
zz
yz
xz e
z
Be
y
Be
x
Btg ˆˆˆ)(
rtgBtrBz )(),( 0
April 20, 2023 Slide 9Institute of Neuroscience and Medicine
In the absence of diffusion, m+ is exponentially damped with relaxation time T2.
One gets the next differential equation
�
Drgit
T)(
20),(),( Tttietrtrm
Proposing the next solution (going to the rotating frame)
Solution for an infinite system.
Defining m+ as the circular component of the transverse spatial magnetization distribution
),(),(),( trimtrmtrm yx
mDT
mBmi
t
m Tz
�
2
)()0,( 0 rprm
(Initial condition)
April 20, 2023 Slide 10Institute of Neuroscience and Medicine
Assuming D = constant, tt
dttgdttgDdt
tAd00
2 ')'(')'()(ln
t ttdttgdttgdtD
A
tA0
''
0
''
0
2
0
')'(')'(''exp)(
Relative signal amplitude.
Gradient
t
dttgrietAtr 0
')'()(),(
The general solution is
April 20, 2023 Slide 11Institute of Neuroscience and Medicine
Solution for a Gradient Echo bipolar pulse.90°
Bipolar pulse
RF
τ
)2''(
''')'(
''
0 tg
gtdttg
t
322
0 3
2exp
)(Dg
A
tA
2τg
-g
Solution for the Stejkal-Tanner pulse sequence.
90°
RF
180°
Gradient2τ
gδ δ
τ
)exp(3
exp)( 222
0
bDDgA
tA
April 20, 2023 Slide 12Institute of Neuroscience and Medicine
Diffusion in biological tissue.Apparent Diffusion Coefficient (ADC) approach.
• ADC is a phenomenological parameter that incorporates integrative information on the tissue microstructure.• It is insensitive to microstructure details.• It is a very important biomarker for identifying pathologies.
Bundles of axons in the corpus callosum in the human brain.
Then one needs to measure S0, the non-diffusion-weighted value (no gradients), and the signal for at least a non-zero b-value.
The usual range for the b-values is (0-1000) s/mm2. For grater b-values, the attenuation is not longer monoexponential.
For a given gradient direction, the signal attenuation at the echo time can be written as
ADCbeSbS 0)(
3)( 2 gb
April 20, 2023 Slide 13Institute of Neuroscience and Medicine
Typical diffusion signal attenuations.
Valid range for the ADC approach.
Background noise.
Diffusion is restricted, i.e. the initial slope is smaller than in the free-diffusion case.
Free water diffusion.
ADCbS
bS
0
)(ln The ADC is given by the slope of the signal attenuation
in a logarithmic scale.
ADC = 2.3*10-3 mm2/s for free water at 24°C.
April 20, 2023 Slide 14Institute of Neuroscience and Medicine
600
Diffusion weighted imaging (DWI) in the human brain.
April 20, 2023 Slide 15Institute of Neuroscience and Medicine
DWI signal acquisition. Gradient along z-direction case.
April 20, 2023 Slide 18Institute of Neuroscience and Medicine
An application of DWI. Early detection of stroke.
http://www.radiologyassistant.nl/en/483910a4b6f14
DWIs permit earlier detection of stroke than other methods such as T2WI (T2 weighted image).
April 20, 2023 Slide 19Institute of Neuroscience and Medicine
Diffusion Anisotropy.Diffusion Tensor Imaging (DTI) approach.There are some cases in which diffusion depends on the spatial direction along which the gradient is applied.
April 20, 2023 Slide 20Institute of Neuroscience and Medicine
The presence of elongated cells, such as axons and dendrites, affects the trajectory of diffusion molecules, making diffusion along the main directions less obstructed than in a direction perpendicular to the main axes.
According to Einstein’s relation, the mean square displacement in a given direction “i”, and a diffusion time td, will be:
dii tDr 22
21r
23r
22r
April 20, 2023 Slide 21Institute of Neuroscience and Medicine
ji
ijijDb
eSS ,
0)(b
T t t
jiij dtdttgdttgb0 0 0
2 ')'(')'(
Formally, anisotropic diffusion is characterized by a diffusion tensor Dij and the signal attenuation can be written as:
Signal attenuation. b-matrix (matrix of b-values).
00
00
00
zzyzxz
yzyyxy
xzxyxx
DDD
DDD
DDD
D�
12
3
1
23
General coordinate frame.
Principal axes frame.
Since D is a second-order symmetric positive definite tensor, one needs to know the ADC for at least 6 non-collinear directions.
April 20, 2023 Slide 22Institute of Neuroscience and Medicine
• For N = 6, the solution is exact.
The number of gradient directions available for clinical applications ranges from 6 up to 256.
ji
ijkij
k DbS
S,0
ln
From the experiment to the diffusion tensor.One has to measure a non-weighted signal value S0 (b = 0 s/mm2) and N diffusion-weighted signals Sk (k = 1..N ≥ 6). Then, one has to solve the next system of equations
• For N ≥ 6, the system is overdetermined: tensor estimation is more robust
6 directions 12 directions 30 directions
April 20, 2023 Slide 27Institute of Neuroscience and Medicine
Tensor invariants.
These parameters are proposed to account for physical information regardless the choice of the frame of reference.
)(3
1321 MD
zzyzxz
yzyyxy
xzxyxx
DDD
DDD
DDD
D�
Diagonalization
eigenvalues.
eigenvectors.
i
iv
Mean Diffusivity
23
22
21
23
22
21 )()()(
2
3
MDMDMD
FAFractional Anisotropy
0 ≥ FA ≥ 1 Isotropic system Diffusion allowed in only one direction.
April 20, 2023 Slide 28Institute of Neuroscience and Medicine
Colour-encoded FA
3v
Eigenvector associated to the main eigenvalue, modulated by FA
MD FA Colour-encoded FA
• red = left-right.• green = anterior-posterior.• blue = top-bottom.
April 20, 2023 Slide 31Institute of Neuroscience and Medicine
• Assuming that the largest principal axis of the diffusion tensor aligns with the predominant fibre orientation.
• Construct a vector field
In-plane
Through-plane
April 20, 2023 Slide 33Institute of Neuroscience and Medicine
• Connect voxels on the basis of discrete vector fields (local principal eigenvector orientation)
• FA threshold (0.25-0.35). Helps to exclude gray matter and to segment white matter tracts that are separated by gray matter.
Discrete vector field
April 20, 2023 Slide 35Institute of Neuroscience and Medicine
Fibre tracking by seeding a region of region of interest in the corpus callosum with increasing fibre densities.
http://www.bioimagesuite.org/doc/node24.html#SECTION06310000000000000000
Choose to seed the tracking using a single point, a region of interest or a volume.
Corpus callosum tracking
April 20, 2023 Slide 36Institute of Neuroscience and Medicine
Pretty Pictures…
Isotropic resolution diffusion tensor imaging with whole brain acquisition in a clinically acceptable time
–D.K. Jones, S.C.R. Williams, D. Gasston, M.A. Horsfield, A. Simmons, R. Howard–Human Brain Mapping 15, 216-230 (2002)
April 20, 2023 Slide 38Institute of Neuroscience and Medicine
Limits of DTI.The problem of multiple fibres populations.
An inherent problem of DTI is the impossibility to recognize crossing fibres.
April 20, 2023 Slide 39Institute of Neuroscience and Medicine
Suggested notes
• H. C. Torrey, “Bloch Equations with Diffusion Terms”, Physical Review, vol. 104, 563 (1956).• A. Abragam, “Principles of Nuclear Magnetism”. Oxford Univ. Press. London and New York (1961).• J. Kärger, H. Pfeifer and W. Heink, ”Principles and Application of Self-Diffusion Measurements by Nuclear Magnetic Resonance”. Advances in Magnetic Resonance, vol. 12 (1988).• D. A. Yablonskiy and A. L. Sukstanskii, “Theoretical models of the diffusion weighted MR signal”. NMR in Biomedicine, vol. 23, 661 (2010).