What can we learn with intravascular tracers?

1. What can we learn with intravascular tracers?

2. Good Modeling References Axel, L. Methods using Blood Pool Tracers in “Diffusion and Perfusion Magnetic Resonance Imaging”, D. Le Bihan (ed.), Raven, 1995. Thomas DL et al. Measuring diffusion and perfusion using MRI. Phys Med Biol (2000) R97–R138. (see sect. 3.2) (on website) Weisskoff RM, et al., Pitfalls in MR Measurement of Tissue Blood Flow with Intravenous Tracers: Which Mean Transit Time? MRM 29:553-559, 1993. Jacquez, J. “Compartmental Modeling in Biology and Medicine”, pages 193-203. U Michigan Press, 1984.

3. MTT CBV CBF Today’s Parametric ImagesWhat is the mapping from data to parameter?

4. Lets consider the data in time. See plots

5. Today’s deep thoughts: MTT = CBV / CBF MTT = proability-weighted average transit time

6. What do we mean by ‘blood flow’?Is that the same as CBF?What do we mean by ‘Perfusion’?

7. Lets examine the ‘Perfusion’ of this system. The is the U.S. Brain Trust. What’s the ‘model’? Map of NIH “Arterial” Inflow “Venous” Outflow

8. Q. What is the ‘perfusion’ of people within a single region (i.e., building)? “Arterial” Inflow “Venous” Outflow

9. Lets examine this single region in detail.

10. Each building (pixel) has an inflow and an outflow. But there are multiple paths through the building. p i x e l inflow outflow Analogies • A building (e.g., CC) is a ... pixel • Rate of people entering CC at inflow: F • Average time spent in CC building: MTT • Fraction of people passing through CC: V • (compared to other buildings)

11. How to understand the major parameters? • F is a measure of the (fractional) rate of flow supplying (i.e., ‘external’ to) a particular area. • V is a measure of (steady state) capacity of the given area. • MTT is a measure of the time spent inside a given area - perhaps due to internal ‘tortuosity’.

12. Method: Inject an “impulse” of runners into the system, then monitor their arrival(s) downstream. In Out

13. p i x e l inflow outflow Lets further idealize the picture In the ideal case, we would examine the inflow to, and the outflow from every region (i.e., pixel). Thus, we would expect the outflow signal to be equal to the inflow signal convolved with the impulse response:

14. What is the impulse response, h(t)? t t + Dt time The response to an impulse input is the distribution of all possible transit times through the system. (Think p.d.f.) h(t)dt is the fraction of “particles” that leave the system between t and t+Dt The Mean transit time is at the center of mass of the distribution, h(t). I.e., 1st moment.

15. Where to make our observations? Outflow from CC Inflow to CC In this idealization, we would need to image every inflow and outflow (i.e., impulse response) of every building (aka., pixel).

16. p i x e l inflow outflow But, consider our actualobservation points... • Rather than measure at inflow and outflow, we make observations of something equivalent to • signal at ~inflow (the arterial function) and, • signal from the entire pixel.

17. h(t) t t + Dt time R(t) = 1 - H(t) Q. How do our observations relate to the histogram of transit times, h(t)? The integral H(t), of the histogram is all the tracer that has LEFT the system. (Think c.d.f) The residue function, R(t), describes all tracer still remaining, at time t and NOT yet drained from the system. Our observations are related to R(t).

18. 100% 0% View at input View of ‘runners’ remaining within the pixel How to understand R(t)? In the case of an ideal input, the view from within the pixel would look like: Thus, R(t) is - in effect - the impulse response as viewed from within the pixel. Recall:

19. S S S S Ca S Practically, we image a convo-lution of the Residue function. Ct Ct Ct Ct Ct

20. S S S S Ca S What’s in a shape? What does the shape of R(t) mean? Ct Ct Short Transit time Dispersed (non-ideal) bolus. Ct Ct Ct Long Transit time

21. What do the Residue Functions that we get from deconvolution look like? See plots

22. What is MTT in terms of the residue function, R(t)? - 1. h(t) The Mean transit time is at the center of mass of the distribution, h(t). I.e., 1st/0th moments. Recall that the Residue function is related to the integral of the histogram.

23. What is MTT in terms of the residue function, R(t)? - 2. Substituting dR into the expression for MTT, Integrating by parts we see that, Recall that we measure the one entity which is the Scaled Residue Function, F*R(t), so we must divide accordingly. Where by convention Scale is the maximum point on the scaled residue curve.

24. Scale*R(t) What is MTT in terms of the residue function, R(t)? - 3. Is equivalent to area / height = 1/2 base. If we approximate the Residue function as a triangle, we can see that the MTT lies at mid-point of the base.

25. Why is the Output Equation Scaled by the FlowArriving at the Pixel? ‘Scale’ is the relative inflow, F, to the pixel because the fraction of tracer arriving at a given pixel is proportional to the fractional flow to that pixel.

26. Test your modeling IQ! Test your modeling IQ! Test your modeling IQ! Test your modeling IQ! Q. What assumptions do we make in applying our simple input-output model? 2. All dispersion of a bolus input is due to multiple path-lengths inside a ‘pixel’ 1. Every pixel is supplied directly by the input. 3. Feeding and draining vessels are ‘outside’ the pixel 4. No recirculation.

27. FI FA ? Ideal Actual What implications are there to our assumptions? 1. An impulse input at the artery would arrive at the ‘pixel’ as an impulse. 2. Measured CBF is an upper bound. So, MTT = CBV/CBF may be biased. 3. Model is only valid for regions on the order of the size of the capillary bed. I.e., with its own supplying arteriole and draining venule. 3a. Different tissue types may require different minimum pixels sizes 4. Recirculation must be removed before applying model. valid invalid

28. What about recirculation? HW #1

29. What is Volume Fraction, V? CBV is a measure of relative blood carrying capacity of a region. We measure it as the ratio of all the tracer that passes through a voxel over time to all the tracer that passes through a point in the vasuclature over all time.

30. Why measure CBV? 1. Vasodilation (increased CBV ) may occur distal to narrowed carotid arteries. 2. Decreased CBV/CBF may reflect slowed cerebral circulation. 3. CBV necessary to measure CMRO2

31. An analogy to understand CBV as relative capacity. • Consider a multiplex movie theatre • But, all theatres in the multiplex play the same movie. • People spread themselves across all theatres at constant concentration of people per seats. • The fraction of patrons that enter a given theatre over all time is a measure of the relative size of that theatre.

32. V: Total # people to enter is proportional to capacity Exit Exit Exit

33. CBV - Assumptions • All people entering leave after ‘residing’ (i.e., no staying for a second show). • Implication: Leakage of Blood Brain Barrier violates the model.

34. With Leakage Ideal Consequence of BBB Leakage to Contrast Agent If contrast agent does NOT stay wholly intravascular (as in case of damage to BBB), and CBV is overestimated.

35. p i x e l inflow outflow Consequence of BBB Leakage to Contrast Agent If CBV is overestimated, then MTT = CBV/CBF is also overestimated. This makes sense: leakage makes the effective mean path-length longer

36. p i x e l inflow outflow A Contrast Agent that leaks across the BBB is also called a “freely diffusable tracer”.Freely diffusable tracers are the domain of PET…

37. Gd-DTPA or CBV = CBF = CBFGMCBFWM = 2 How’s it done? - Data Flow 1. Inject 2. Scan over time 3. Convert signal to concentration 4. Find AIF 5. Fit First Pass 6. Calculate CBV, CBF, MTT 7. Post-process, tabulate stats

38. CBV CBF MTT Normalized X 2 Take-off time Recirc. time Sample Results

40. Why a take-off threshold - 1. but equation (1) cannot be linearized for rapid computation. If we can find the take-off, t0 , ‘graphically’, then the model becomes: A generalized Gamma-Variate function has 4 (estimatable) parameters t0, K, b, a: which, when log transformed to: can be used to fit the (log-transformed) data via non-iterative multiple-linear regression. In the process, ln(K), b, a are estimated.

41. peak value 1st point above threshold take-off time, t0 Why a take-off threshold - 2. Thus, we identify the take-off, t0 , by extrapolating from near-threshold points back to baseline. The threshold - defined as percent of peak - determines the points to be used in extrapolation. Only pre-peak points are used in finding take-off.

42. Why a Recirculation Threshold ? Because volume fraction (relCBV) is based on the total amount of tracer, that drains from an ‘open’ system, we must find a way to identify and integrate the first-pass response, independent of recirculation effects. onset of recirculation threshold % of peak observed signal 1st pass recirculation ignore signal A common approach is to set a threshold relative to peak and ignore all later data that dips below that threshold.

43. Thresh ; CBV bias ; Fit Quality . CBV- Effect of Recirculation Threshold Thresh = 50 % CBV = 0.37 X2= 0.008 Thresh = 30 % CBV = 0.42 X2= 0.010 Thresh = 20 % CBV = 0.49 X2= 0.180

44. Why an SVD threshold? - 1 Singular Value Decomposition is used to solve an approximation to expression (1) which relates the convolution of the arterial input function Ca(t) and the Residue function, R(t), to the tissue concentration, Ct(t): We approximate equation (1) as follows: where:

45. Why an SVD threshold? - 2 According to SVD, we can represent the A matrix in terms of the diagonal matrix, S, made up of singular values, si : We then solve equation (2) by: But very small singular values, si , that may result from roundoff error will wreak havoc with the solution. Therefore , we zero all singular values less than a specified (threshold) percentage of the maximum singular value.