Can arterial spin labelling techniques quantify cerebral blood flow (CBF)?

Transcription

1 Can arterial spin labelling techniques quantify cerebral blood flow (CBF)? Christian Kerskens Bruker User Meeting 12. October 2016 Neuroimaging & theoretical neuroscience Trinity College Institute of Neuroscience

2 What is CBF? flow? For incompressible fluids, volume flow can be defined as dv = V/t= Av = constant dt If a concentration volu c is a conserved quantity then c t + V ( dv dt c ) = 0. which is for F=constant a transport equation c t = F c V The transport incorporates equation the describes transport only; the general solution is c(v-ft).

3 What is CBF? For CBF measurements, Kety et al introduce equation based on 1. Fick law dc i dt = CBF [ C a C i ] Motivation diffusion Consider a collection of particles performing a random walk in one dimension. At a given time step, half of the particles would move left and half would move right. From this observation 1. Fick law can be derived. For the blood flow opposing direction are consider as inand outflow; or arterial in- and venous output.

4 CASL and PASL Pictures are taken from the Recommended Implementation of Arterial Spin-Labeled Perfusion MRI for Clinical Applications: A Consensus of the ISMRM Perfusion Study Group and the European Consortium for ASL in Dementia

5 CBF measurements in ASL based on Recommended Implementation of Arterial Spin-Labeled Perfusion MRI for Clinical Applications: A Consensus of the ISMRM Perfusion Study Group and the European Consortium for ASL in Dementia (most cited paper in MRM!), CBF can be measured under the following assumptions: 1. The entire labeled bolus is delivered to the target tissue. This is the case when PLD > ATT for PCASL, or (TI-TI1) > ATT for QUIPSS II PASL. 2. There is no outflow of labeled blood water. Because the tissue water pool is much larger than the blood water pool, and water exchange between blood and tissue is rapid, this is generally a valid assumption. 3. The relaxation of the labeled spins are governed by blood T1. While this assumption is not likely to be strictly true, the errors introduced by this assumption, which are related to the difference in T1 between blood and tissue, are typically relatively small.

6 Rational behind assumption assumption 1: needed because Fick law doesn't describe any transport from labelling slice to imaging slice. How the transport is covered is not clear. assumption 2: not clear, presumable authors believe to cover inflow only in their derivation. Fick law as used clearly defines in- and outflow. assumption 3: T1 values of blood and tissue are close for most field strength.

7 CBF measurements in ASL: no outflow? what does that mean for the Fick law? dc i dt = CBF C a because C i = C v PASL: CASL: CBF = CBF = TI 3000 S exp( ) T lim!1 b 1 = 3000 T b 1!1 TI 1 SI PD SI PD PLD 3000 S exp( ) T lim b 1 T b 1!1 T b 1 SI PD (1 exp( T b 1 S TI 1, )) = 3000 SI PD S which translate for both cases into dc i dt = CBF 3000 C i

8 Testing linearity The model is linear. Any solution must be linear, too. Simple test: divide one bolus/slice into two boluses/slices, then for CASL: is linear. bolus into tw = S(,PLD)= S( 1,PLD)+ S( 2,PLD+ 1 ) and for PASL: TI 1 = TI TI2 1 S(TI 1,TI)= S(TI 1 1,TI)+ S(TI 2 1,TI TI 1 1) is nonlinear. TI 1 = TI 1 1 +exp( TI1 1 T b 1 )TI 2 1

9 How is CBF related to ΔS For CASL: CBF ¼ 6000 l ðsi control SI label Þe 2 a T 1;blood SI PD ð1-e - t T PLD 1; blood T 1;bloodÞ ½ml=100 g=minš For PASL: CBF= S[Par] ΔS CBF ¼ 6000 l ðsi T control SI label Þe 1; blood ½ml=100 g=minš 2 a TI 1 SI PD TI S[Par] includes all parameters which are flow independent

10 Where does it go wrong: error sources in derivation Buxton et al use a convolution with a residue function: AM(t) = 2A40bf\o*c(tr)r(t - t )m(t - t )dt r(t) is an outflow function with Ca=0 and solution from r(f) = exp[-ft/a] ion pulse, the m dc i dt = - CBF [ C i ] If k approaches 0 then f approaches 0, too. -=-+- 1 l f TI TI A

11 Where does it go wrong: limitation of Fick law Simplification only possible for linear concentration gradient. This is never the case for a dynamic measurement where bolus durations are shorter than the MTT Fick law doesn t consider transport This is a BIG problem. The distance between the labelling plane and the tissue is to far. 1. Fick law is a first order equation, therefore it can not contain 3 locations for the simplification which are Ca, Ct and Cv dc i dt = F V [C a C v ],

12 extending Fick law Let us go back to the original Fick law flux= j = dq dt = adding a flow and change F into P flux= F @V combining with the

13 Solution of Fokker-planck c c = F t V + P 2 c V c 2 T 1 incorporates the three factors tha Initial conditions y c = c 0 (V,0)fort = 0,, it is necessary to determ c(v, t) = exp( t/t 1) 4πPt c 0 (V, 0) exp ( (V V Ft) 2 ) dv 4Pt la determines the distribution of any contrast agent (T )intermsof Boundary conditions c(v, t) = c 0 (t) for V = 0, c(v, t) = 0 for t = 0,V >0, c(v, t) = C 0 t 0 exp( (t τ)/t 1 ) 4πP(t τ) ( ) V (t τ) exp (V F(t τ))2 dτ. 4P(t τ)

14 Moments of Fokker-Planck equation For CASL, we can write the results as a convolution cðv; tþ ¼ Z t 0 c 0 ðtþ ffl{zffl} AIF ðt tþ exp T 1 fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} mðt tþ! V qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð V Ft ð t ÞÞ2 dt 4pPðt tþ 3 4Pðt tþ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} rðv;t tþ The residue function r(v,t) works as a point spread ð Þ function. It helps to understand the parameters! MTT ¼ R1 0 R1 t rðv; tþ dt 0 rðv; tþ dt ¼ V F CTT ¼ 1 2 R1 0 ðt t 1 Þ 2 rðv; tþ dt R1 0 t rðv; tþ dt ¼ P F 2 first moment second moment

15 Application: Bolus length Bolus length variation of 1.5, 2.0 and 3.0 s in three subjects 2.0 (b) Mean MTT, CTT and ATT variation for varying bolus length C C B Time (secs) s 2.0 s 3.0 s 0.0 Mean MTT Mean CTT Mean ATT

16 Application: fmri Figure 5 (A) Change in MTT, (B) CTT, and (C) rcbv lw during neuronal activation. A statistically significant decrease in both transit times and a statistically significant increase in rcbv lw was measured in the activated S1FL when compared with the same S1FL region in the control experiment (two-tailed paired t-test: P = for MTT, P = for CTT and, P = for rcbv lw ). The error bars represent one standard deviation from the mean.

17 Summary ASL can t quantify CBF as defined by Fick law The derivation of recommended equations to quantify CBF holds errors 20 years-old approach can solve many problems. still ignored by the community