Measuring Water Diffusion In Biological Systems Using
Nuclear Magnetic Resonance
Karl HelmerHST 583, 2006
http://www.medicineau.net.au/clinical/Radiology/Radiolog1768.html
Why Would We Want to Measure the Self - Diffusion Coefficient of Water
In Biological Tissue?
We Don’t.
Why Would We Want to Measure the Self - Diffusion Coefficient of Water
In Biological Tissue?
We Don’t.
What we are really interested in is howwhat we measure for a diffusion-weighted signal
reflects the structure of the sample.
Why Would We Want to Measure the Self - Diffusion Coefficient of Water
In Biological Tissue?
We Don’t.
What we are really interested in is howwhat we measure for a diffusion-weighted signal
reflects the structure of the sample.
So, what are we measuring???
Why Would We Want to Measure the Self - Diffusion Coefficient of Water?
How Can the Diffusion Coefficient Reflect Sample Structure?
Self-diffusion in bulk samples is a well-understood random process -
Displacement (z) has a Gaussian probability distribution
<z2>1/2 = (2nDt)1/2
D = Self-Diffusion Coefficientn = # of dimensions
z
H.C. Berg, 1993
proba-bility(t)
How Can We Measure the Diffusion Coefficient of Water
Using NMR?
We Can’t.
How Can We Measure the Diffusion Coefficient of Water
Using NMR?
Instead we measure the displacementof the ensemble of spins in our sample
and infer the diffusion coefficient.
We Can’t.
How Can We Measure the Diffusion Coefficient of Water
Using NMR?
How can we measures the (mean) displacement of water molecules using NMR?
g(z) is amagnetic field added to B0 that varies with position.
(z) = (B0 + g(z)z)
How can we measures the (mean) displacement of water molecules using NMR?
Applying g(z) for a time results in a phase shift
that depends upon location
in z
z
z = 0
Tagging the initial positionusing phase
of M
Now, after waiting a time ∆ we apply an equal gradient, but with the opposite sign
Apply -g(z) for a time
if no diffusion:signal = M0
z
But, in reality, there is always diffusion sowe find that:
Apply -g(z) for a time
if diffusion:signal = M0e(-q2Dt)
(t = ∆ - /3)q = q(g)
z
Pulse Sequences
DW Spin Echo/2
= gradient duration = separation of gradient leading edges
But what do we do with:signal = M = M0e(-q2Dt)?
One equation, but two unknowns (M0, D)
How do we get another equation?
q2t
ln(M)
Slope = DIntercept = ln(M0)
Change the diffusion-sensitizing gradient to a different value and acquire more data.
b = q2 t = 0
b = q2 t ≠ 0
Unrestricted Diffusion
r
r'
r
r'
Restricted Diffusion
The effect of barriers to the free diffusion of water molecules is to modify their
probability distribution.
P(z)
Diffusioncoefficient decreaseswith increasingdiffusion time
Determination of D?
-7
-6
-5
-4
-3
-2
-1
0
0.0 0.5 1.0 1.5
q2 x 107 [1/cm2]
ln(M
/M0)
Slope = D0tdif
Slope = ‘D’tdif
bead pack water
a = 15.8 m bead pack, tdif = 50 ms, = 1.5 ms, g(max) = 72.8 G/cm
bulk water
-7
-6
-5
-4
-3
-2
-1
0
0 1 2 3 4 5 6
k2 x 107 [1/cm2]
ln(M
/M 0)
Water Diffusion in an Ordered System – High q
a = 15.8 m bead pack, tdif = 100 ms
2/a
q2
Short diffusion times:
Long diffusion times:
0
40
80
120
160
0 0.2 0.4 0.6 0.8 1
D(t
) x
10-7 [
cm2 /sec
] S/V
t
1/T
t1/2 [sec 1/2]
‘D’(tdif) gives information on different length scales
]
a = 15.8 m bead pack
T = tortuosityS/V = surface-to-volume ratio
‘D’(t
)
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0ln
M(q
,t)/M
(0,t)
150100500
q2 [x10
-9 m
-2]
42 ms
92 ms
192 ms292 ms492 ms
DW-Weighted Tumor Data
D(t) Apparent Diffusion Coefficient (ADC)
tdif =
ADC(t) for water in a RIF-1 Mouse Tumor
D(t)
10
5 [c
m2 /s
]
(t)1/2 [s1/2]
0.10
0.240.60 0.75
0.10
2.55
Necrosis!!
Control
1 x 10-7
> 255 x10-7
cm2 /s
ec
ADC
ADC
Tumor Volume
Day 1 Day 2 Day 3 Day 4
1.42 cm31.26 cm30.97 cm30.68 cm3
Tumor Volume
Day 5 Day 6 Histology
1.70 cm3 2.04 cm3
ADC for water in a RIF-1 Mouse Tumor
ADC for water in a RIF-1 Mouse Tumor
Treatment, 100mg/kg 5-FU
1 x 10-7
> 255 x10-7
cm2 /s
ecADC
Tumor Volume 0.76 cm30.71 cm30.86 cm30.95 cm30.70 cm30.60 cm3
Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
ADC1 x 10-7
> 255 x10-7
cm2 /s
ec
Day 7 Day 8 Day 9 Day 10 Day 11 Histology
Tumor Volume 1.13 cm3 1.36 cm3 1.60 cm3 1.79 cm3 2.08 cm3
ROI Positions < 30 > 60
ADC (x10-5 mm2/s)
MCAO 2 hr 3 hr 4 hr 5 hr 6 hr
7 hr 8 hr 9 hr 10 hr 11 hr 12 hr
ADCav Maps vs Post-Occlusion Time Rat Brain – 30 min Occlusion
Temporal ADC Changes in the Caudoputamen: 30-minute Transient Occlusion (n = 4)
30
35
40
45
50
55
60
65
70
75
80
Rep 1 2 3 4 5 6 7 8 9 10 11 12
Time (hours post reperfusion)
ADC
(x10
-5 m
m2 /s
)
Ipsilateral
Contralateral
ADCav Maps vs Post-Occlusion Time Rat Brain – 30 min Occlusion
Issues with Interpreting DW Data
In biological tissue, there are alwaysrestrictions. How then can we interpret the diffusion attenuation curve?
Biology-based Model:Intracellular and extracellular compartments
Biexponential Model with a distribution of cell sizes and shapes.
))1((
)1(
21110
2111
bDbD efefSS
DfDfD
Fast Exchange
Slow Exchange
But real systems are rarely either/or.
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0ln
M(q
,t)/M
(0,t)
150100500
q2 [x10
-9 m
-2]
42 ms
92 ms
192 ms292 ms492 ms
DW-Weighted Tumor Data
What does non-monexponentiality tell us?
tdif =
‘Fast’ and ‘Slow’ Diffusion?
-7
-6
-5
-4
-3
-2
-1
0
0.0 0.5 1.0 1.5
q2 x 107 [1/cm2]
ln(M
/M0)
Slope = Dslowtdif
Slope = ‘Dfast’tdif
bulk water
Does ‘Fast’ and ‘Slow’ Mean ‘Extracellular’ and ‘Intracellular’?
No, because:
1)The same shape of curve can be found in the diffusion attenuation curve of single compartment systems (e.g., beads).
2) It gives almost exactly the opposite values for extra- and intracellular volume fractions (20/80 instead of 80/20 for IC/EC).
Exchange?
What does ‘fast’ and ‘slow’ measure?
Answer: It depends on…•range of b-values•TE•tdif
•sample structure•sample tortuosity
Clark et al. MRM47, 623, 2002.
Dave(fast) Dave(slow)
FA(fast) FA(slow)
Clark et al. MRM47, 623, 2002. ‘slow’ ‘restricted’…
Do We Get More Information by Usingthe Entire Diffusion Attenuation Curve?
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
ln M
(q,t)
/M(0
,t)
150100500
q2 [x10
-9 m
-2]
Practical Issues in DWI
1)Diffusion gradients act like primer-crusher pairs. Therefore, slice profile of g = 0 image will be different from g 0 image.
2) Diffusion gradients also suppress flowing spins.
Therefore, the use of a g = 0 image is discouraged.
How do I choose my lowest b-value?
Practical Issues in DWI
How do I choose my highest b-value?
1. Greatest SNR in calculated ADC:
2/12
0
ii
Dbi
ISeII i I = true signal
S = measured signal
= noise
tqbb
SSD 201 ,lnln
)1()(1 220
2
221
202
2 bDDD e
Ibb
Practical Issues in DWI
0)(
)1(0
2/12 IbDD
D SNRbDFIebDDSNR
Practical Issues in DWI
How do I choose my highest b-value?
2. Greatest sensitivity to %ADC:
0.1|max bDDI
Practical Issues in DWI
How to distribute the b-values?q2t
ln(M)
This or ?
Practical Issues in DWI
q2t
ln(M)
This or…?
How to distribute the b-values?
Practical Issues in DWI
q2t
ln(M)
This?
How to distribute the b-values?
Multiple measurements of 2 b-values are better than multiple different b-values.
If the number of measurements can be large,then Nhigh-b = Nlow-b 3.6
Note that depending on N and how you estimate the error, you can get differentnumbers for the optimum values, but
Δbopt ~ 1(+)/D and Nhigh-b ~ Nlow-b 4
What effect does the direction of the diffusion-sensitizing gradient have upon what we measure?
x
yIn the 1- dimensional case(we measure Dx or Dy):
Dy D0, the bulk value
Dx <(<) D0
D / ADC is a scalar
Diffusion Tensor Imaging
What effect does the direction ofthe diffusion-sensitizing gradienthave upon what we measure?
x
y
In the 3- dimensional case(we measure Dx, Dy and Dz):
Dy D0, the bulk value
Dx = Dz <(<) D0
D = (Dx, Dy, Dz)
z
Why not stick with vectors?
Because is not
x
y
z
Diffusion Tensor Imaging
Taylor et al.,Biol Physhiatry, 55, 201 (2004)
The ADC is greatest along White Matterfiber tracts.
1. There is nothing special about using tensors to characterize anisotropic diffusion.
Rotate to principalframe to get eigen-values.
Rotational Invariants for 3D Tensors.
Eigenvalues = D1, D2, D3 or 1, 2, 3
Dav = (Dxx + Dyy + Dzz)/3
Trace Imaging and b-value Strength
http://splweb.bwh.harvard.edu:8000/pages/papers/maier/radiology2001.pdf
LeBihan et al.,JMRI, 13, 534 (2001)
Distribution of Gradient Sampling Directions
Need at least6 different samplingdirections
Diffusion Tractography
Follow Voxels With Largest EigenvaluesBeing ‘Continuous’Between Two Regions of Interest
http://splweb.bwh.harvard.edu:8000/pages/papers/martha/DTI_Tech354.pdf