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Measuring Water Diffusion In Biological Systems

Using Nuclear Magnetic Resonance Karl

Helmer HST 583, 2006

http//www.medicineau.net.au/clinical/Radiology/Ra

diolog1768.html

Why Would We Want to Measure the Self -

Diffusion Coefficient of Water In Biological

Tissue?

Why Would We Want to Measure the Self -

Diffusion Coefficient of Water In Biological

Tissue?

We Dont.

Why Would We Want to Measure the Self -

Diffusion Coefficient of Water In Biological

Tissue?

We Dont.

What we are really interested in is how what 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?

We Dont.

What we are really interested in is how what we

measure for a diffusion-weighted signal reflects

the structure of the sample.

So, what are we measuring???

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

ltz2gt1/2 (2nDt)1/2 D Self-Diffusion

Coefficient n of dimensions

proba- bility(t)

z

H.C. Berg, 1993

How Can We Measure the Diffusion Coefficient of

Water Using NMR?

How Can We Measure the Diffusion Coefficient of

Water Using NMR?

We Cant.

How Can We Measure the Diffusion Coefficient of

Water Using NMR?

We Cant.

Instead we measure the displacement of the

ensemble of spins in our sample and infer the

diffusion coefficient.

How can we measures the (mean) displacement of

water molecules using NMR?

g(z) is a magnetic 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?

z 0

z

Applying g(z) for a time ? results in a phase

shift that depends upon location in z

Tagging the initial position using phase of M

Now, after waiting a time ? we apply an equal

gradient, but with the opposite sign

Apply -g(z) for a time ?

z

if no diffusion signal M0

But, in reality, there is always diffusion so we

find that

Apply -g(z) for a time ?

z

if diffusion signal M0e(-q2Dt) (t ? - ?/3) q

q(g)

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?

Change the diffusion-sensitizing gradient to a

different value and acquire more data.

q2t

b q2 t 0

Slope D Intercept ln(M0)

ln(M)

b q2 t ? 0

Unrestricted Diffusion

r'

r

Restricted Diffusion

r

r'

The effect of barriers to the free diffusion of

water molecules is to modify their probability

distribution.

- Diffusion
- coefficient
- decreases
- with increasing
- diffusion time

P(z)

Determination of D?

Slope D?tdif

bead pack water

bulk water

Slope D0?tdif

a 15.8 ?m bead pack, tdif 50 ms, ? 1.5 ms,

g(max) 72.8 G/cm

Water Diffusion in an Ordered System High q

2?/a

q2

a 15.8 ?m bead pack, tdif 100 ms

Short diffusion times

Long diffusion times

D(tdif) gives information on different

length scales

T tortuosity S/V surface-to-volume ratio

D(t)

t1/2 sec 1/2

a 15.8 ?m bead pack

DW-Weighted Tumor Data

0.0

-0.5

tdif

-1.0

-1.5

ln M(q,t)/M(0,t)

-2.0

-2.5

-3.0

-3.5

D(t) ? Apparent Diffusion Coefficient (ADC)

ADC(t) for water in a RIF-1 Mouse Tumor

Necrosis!!

0.10

0.24

D(t) ?105 cm2/s

0.60

0.75

0.10

(t)1/2 s1/2

2.55

ADC for water in a RIF-1 Mouse Tumor

ADC for water in a RIF-1 Mouse Tumor

ADCav Maps vs Post-Occlusion Time Rat Brain 30

min Occlusion

ADC (x10-5 mm2/s)

ROI Positions

lt 30

gt 60

ADCav Maps vs Post-Occlusion Time Rat Brain 30

min Occlusion

Issues with Interpreting DW Data

In biological tissue, there are

always restrictions. 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.

Fast Exchange

Slow Exchange

But real systems are rarely either/or.

DW-Weighted Tumor Data

0.0

-0.5

tdif

-1.0

-1.5

ln M(q,t)/M(0,t)

-2.0

-2.5

-3.0

-3.5

What does non-monexponentiality tell us?

Fast and Slow Diffusion?

Slope Dfast?tdif

Slope Dslow?tdif

bulk water

Does Fast and Slow Mean Extracellular and

Intracellular?

- No, because
- 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. MRM 47, 623, 2002.

Dave(fast)

Dave(slow)

FA(slow)

FA(fast)

Clark et al. MRM 47, 623, 2002.

?slow ? restricted

Do We Get More Information by Using the Entire

Diffusion Attenuation Curve?

0.0

-0.5

-1.0

-1.5

ln M(q,t)/M(0,t)

-2.0

-2.5

-3.0

-3.5

Practical Issues in DWI

How do I choose my lowest b-value?

- 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.

Practical Issues in DWI

How do I choose my highest b-value? 1. Greatest

SNR in calculated ADC

I true signal S measured signal ? noise

Practical Issues in DWI

Practical Issues in DWI

How do I choose my highest b-value? 2. Greatest

sensitivity to ?ADC

Practical Issues in DWI

How to distribute the b-values?

q2t

This or ?

ln(M)

Practical Issues in DWI

How to distribute the b-values?

q2t

This or?

ln(M)

Practical Issues in DWI

How to distribute the b-values?

q2t

This?

ln(M)

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 different numbers for the

optimum values, but ?bopt 1()/D and

Nhigh-b Nlow-b ? 4

Diffusion Tensor Imaging

What effect does the direction of the

diffusion-sensitizing gradient have upon what we

measure?

In the 1- dimensional case (we measure Dx or

Dy) Dy ? D0, the bulk value Dx lt(lt) D0 D /

ADC is a scalar

y

x

What effect does the direction of the

diffusion-sensitizing gradient have upon what we

measure?

In the 3- dimensional case (we measure Dx, Dy and

Dz) Dy ? D0, the bulk value Dx Dz lt(lt)

D0 D (Dx, Dy, Dz)

z

y

x

Diffusion Tensor Imaging

Why not stick with vectors? Because is not

z

x

y

The ADC is greatest along White Matter fiber

tracts.

Taylor et al., Biol Physhiatry, 55, 201 (2004)

1. There is nothing special about using

tensors to characterize anisotropic diffusion.

Rotate to principal frame 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.edu8000/pages/papers/ma

ier/radiology2001.pdf

Distribution of Gradient Sampling Directions

Need at least 6 different sampling directions

LeBihan et al., JMRI, 13, 534 (2001)

Diffusion Tractography

Follow Voxels With Largest Eigenvalues Being

Continuous Between Two Regions of Interest

http//splweb.bwh.harvard.edu8000/pages/papers/ma

rtha/DTI_Tech354.pdf