measuring water diffusion in biological systems using nuclear magnetic resonance karl helmer hst...

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