Measuring Water Diffusion In Biological Systems Using Nuclear Magnetic Resonance Karl Helmer HST 583,

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 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 In Biological Tissue?

We Don’t. 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??? 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 1/2 = (2nDt) 1/2 D = Self-Diffusion Coefficient n = # 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 displacement of 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 a magnetic field added to B 0 that varies with position.  (z) =  (B 0 + 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 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  if no diffusion: signal = M 0 z

But, in reality, there is always diffusion so we find that: Apply -g(z) for a time  if diffusion: signal = M 0 e (-q 2 Dt) (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 = M 0 e (-q 2 Dt) ? One equation, but two unknowns (M 0, D) How do we get another equation?

q2tq2t ln(M) Slope = D Intercept = ln(M 0 ) Change the diffusion-sensitizing gradient to a different value and acquire more data. b = q 2 t = 0 b = q 2 t ≠ 0

Unrestricted Diffusion r r'

r Restricted Diffusion

The effect of barriers to the free diffusion of water molecules is to modify their probability distribution. P(z)  Diffusion coefficient decreases with increasing diffusion time

Determination of D? Slope = D 0  t dif Slope = ‘D’  t dif bead pack water a = 15.8  m bead pack, t dif = 50 ms,  = 1.5 ms, g(max) = 72.8 G/cm bulk water

Water Diffusion in an Ordered System – High q a = 15.8  m bead pack, t dif = 100 ms 2  /a q2q2

Short diffusion times: Long diffusion times:

 S/V  1/T t 1/2 [sec 1/2 ] ‘D’(t dif ) gives information on different length scales ] a = 15.8  m bead pack T = tortuosity S/V = surface-to-volume ratio ‘D’(t)

ln M(q,t)/M(0,t) q 2 [x10 -9 m -2 ] 42 ms 92 ms 192 ms 292 ms 492 ms DW-Weighted Tumor Data D(t)  Apparent Diffusion Coefficient (ADC) t dif =

ADC(t) for water in a RIF-1 Mouse Tumor D(t)  10 5 [cm 2 /s] (t) 1/2 [s 1/2 ] Necrosis!!

Control 1 x > 255 x10 -7 cm 2 /sec ADC Tumor Volume Day 1 Day 2 Day 3Day cm cm cm cm 3 Tumor Volume Day 5 Day 6 Histology 1.70 cm cm 3 ADC for water in a RIF-1 Mouse Tumor

Treatment, 100mg/kg 5-FU 1 x > 255 x10 -7 cm 2 /sec ADC Tumor Volume 0.76 cm cm cm cm cm cm 3 Day 1Day 2Day 3Day 4Day 5Day 6 ADC 1 x > 255 x10 -7 cm 2 /sec Day 7 Day 8Day 9Day 10 Day 11 Histology Tumor Volume1.13 cm cm cm cm cm 3

ROI Positions < 30 > 60 ADC (x10 -5 mm 2 /s) MCAO2 hr3 hr4 hr5 hr6 hr 7 hr8 hr9 hr10 hr11 hr12 hr ADC av 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.

ln M(q,t)/M(0,t) q 2 [x10 -9 m -2 ] 42 ms 92 ms 192 ms 292 ms 492 ms DW-Weighted Tumor Data What does non-monexponentiality tell us? t dif =

‘Fast’ and ‘Slow’ Diffusion? Slope = D slow  t dif Slope = ‘D fast ’  t dif 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 t dif sample structure sample tortuosity Clark et al. MRM 47, 623, 2002.

D ave (fast) D ave (slow) FA(fast) FA(slow) Clark et al. MRM 47, 623,  ‘slow’  ‘restricted’…

Do We Get More Information by Using the Entire Diffusion Attenuation Curve? ln M(q,t)/M(0,t) q 2 [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: I = true signal S = measured signal  = noise

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? q2tq2t ln(M) This or ?

Practical Issues in DWI q2tq2t ln(M) This or…? How to distribute the b-values?

Practical Issues in DWI q2tq2t 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 N high-b = N low-b  3.6 Note that depending on N and how you estimate the error, you can get different numbers for the optimum values, but Δb opt ~ 1(+)/D and N high-b ~ N low-b  4

What effect does the direction of the diffusion- sensitizing gradient have upon what we measure? x y In the 1- dimensional case (we measure D x or D y ): D y  D 0, the bulk value D x <(<) D 0 D / ADC is a scalar Diffusion Tensor Imaging

What effect does the direction of the diffusion-sensitizing gradient have upon what we measure? x y In the 3- dimensional case (we measure D x, D y and D z ): D y  D 0, the bulk value D x = D z <(<) D 0 D = (D x, D y, D z ) 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 Matter fiber tracts.

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 D av = (D xx + D yy + D zz )/3

Trace Imaging and b-value Strength

LeBihan et al., JMRI, 13, 534 (2001) Distribution of Gradient Sampling Directions Need at least 6 different sampling directions

Diffusion Tractography Follow Voxels With Largest Eigenvalues Being ‘Continuous’ Between Two Regions of Interest