What can we learn with intravascular tracers? Perfusion Imaging

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. Perfusion Imaging

Today’s Parametric Images What is the mapping from data to parameter? E Morris Monday, July 10, 2000 Today’s Parametric Images What is the mapping from data to parameter? CBV CBF Here is a preview of results. Our primary goal is to generate CBV, CBF and MTT maps from dynamic data. We will explain how to generate these images shortly. First lets see what model is required to make our output images and what assumptions we must ‘buy’ into. MTT Perfusion Imaging

Lets consider the data in time. E Morris Monday, July 10, 2000 Lets consider the data in time. See plots Perfusion Imaging

Today’s deep thoughts: E Morris Monday, July 10, 2000 Today’s deep thoughts: MTT = CBV / CBF MTT = proability-weighted average transit time Lets consider the theory behind perfusion in simple terms. I will introduce a few analogies which I hope will be helpful in demonstrating what it is that we are actually measuring/calculating. Perfusion Imaging

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

E Morris Monday, July 10, 2000 Lets examine the ‘Perfusion’ of this system. The is the U.S. Brain Trust. What’s the ‘model’? We use the above image of a familiar ‘map’ for illustrative purposes. How would we go about measuring local perfusion in this complicated system? Here we think of perfusion as a measure of usage of individual buildings on campus. Perhaps Congress wants to check up on NIH?? How well supplied are they with scientists? How long does the average scientist spend in the building. Map of NIH “Arterial” Inflow “Venous” Outflow Perfusion Imaging

E Morris Monday, July 10, 2000 Q. What is the ‘perfusion’ of people within a single region (i.e., building)? Lets focus in on a single building which for our purposes we consider the smallest region - or component - of our image. That is, a building can be thought of as a pixel. Lets look more closely at a single bulding... “Arterial” Inflow “Venous” Outflow Perfusion Imaging

Lets examine this single region in detail. E Morris Monday, July 10, 2000 Lets examine this single region in detail. The Clinical Center will serve as our region or pixel of interest. Now since we have equated the CC building with a pixel in our image it means that we are only going to Image the building as a whole - but in doing so, what can we tell about its internals? Perfusion Imaging

E Morris Monday, July 10, 2000 Each building (pixel) has an inflow and an outflow. But there are multiple paths through the building. p i x e l Lets assume that every building has a single input and output point. (In the case of CC, both of these points are in the front of the building.) Once a scientist enters the building, he/she can take one of many different paths to get to the exit. For a given flow rate - which is fixed within the pixel - the population of different length paths will give rise to a distribution of different residence times. 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) Perfusion Imaging

How to understand the major parameters? E Morris Monday, July 10, 2000 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’. Perfusion Imaging

E Morris Monday, July 10, 2000 Method: Inject an “impulse” of runners into the system, then monitor their arrival(s) downstream. In Out Perfusion Imaging

Lets further idealize the picture E Morris Monday, July 10, 2000 Lets further idealize the picture outflow inflow p i x e l 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: Perfusion Imaging

What is the impulse response, h(t)? E Morris Monday, July 10, 2000 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. Perfusion Imaging

Where to make our observations? E Morris Monday, July 10, 2000 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). Perfusion Imaging

But, consider our actual observation points... E Morris Monday, July 10, 2000 But, consider our actual observation points... outflow inflow p i x e l 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. Perfusion Imaging

Our observations are related to R(t). E Morris Monday, July 10, 2000 Q. How do our observations relate to the histogram of transit times, h(t)? t t + Dt time 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. R(t) = 1 - H(t) Our observations are related to R(t). Perfusion Imaging

E Morris Monday, July 10, 2000 How to understand R(t)? In the case of an ideal input, the view from within the pixel would look like: 100% 0% View at input View of ‘runners’ remaining within the pixel Thus, R(t) is - in effect - the impulse response as viewed from within the pixel. Recall: Perfusion Imaging

Practically, we image a convo-lution of the Residue function. E Morris Monday, July 10, 2000 Practically, we image a convo-lution of the Residue function. S Ct Ct Ct Ca Ct Ct Perfusion Imaging

What’s in a shape? What does the shape of R(t) mean? E Morris Monday, July 10, 2000 What’s in a shape? What does the shape of R(t) mean? S Ct Ct Short Transit time Dispersed (non-ideal) bolus. Ct Ca Ct Ct Long Transit time Perfusion Imaging

E Morris Monday, July 10, 2000 What do the Residue Functions that we get from deconvolution look like? See plots Perfusion Imaging

What is MTT in terms of the residue function, R(t)? - 1. E Morris Monday, July 10, 2000 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. Perfusion Imaging

What is MTT in terms of the residue function, R(t)? - 2. E Morris Monday, July 10, 2000 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. Perfusion Imaging

What is MTT in terms of the residue function, R(t)? - 3. E Morris Monday, July 10, 2000 What is MTT in terms of the residue function, R(t)? - 3. Is equivalent to area / height = 1/2 base. Scale*R(t) If we approximate the Residue function as a triangle, we can see that the MTT lies at mid-point of the base. Perfusion Imaging

Why is the Output Equation Scaled by the Flow Arriving at the Pixel? E Morris Monday, July 10, 2000 Why is the Output Equation Scaled by the Flow Arriving 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. Perfusion Imaging

E Morris Monday, July 10, 2000 Q. What assumptions do we make in applying our simple input-output model? Test your modeling IQ! 1. Every pixel is supplied directly by the input. Test your modeling IQ! 2. All dispersion of a bolus input is due to multiple path-lengths inside a ‘pixel’ Test your modeling IQ! 3. Feeding and draining vessels are ‘outside’ the pixel 4. No recirculation. Test your modeling IQ! Perfusion Imaging

What implications are there to our assumptions? E Morris Monday, July 10, 2000 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. FI FA ? Ideal Actual valid invalid Perfusion Imaging

What about recirculation? HW #1 Perfusion Imaging

What is Volume Fraction, V? E Morris Monday, July 10, 2000 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. Perfusion Imaging

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 Perfusion Imaging

An analogy to understand CBV as relative capacity. E Morris Monday, July 10, 2000 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. Perfusion Imaging

V: Total # people to enter is proportional to capacity E Morris Monday, July 10, 2000 V: Total # people to enter is proportional to capacity Exit Exit Exit Perfusion Imaging

E Morris Monday, July 10, 2000 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. Perfusion Imaging

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

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

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

How’s it done? - Data Flow E Morris How’s it done? - Data Flow Monday, July 10, 2000 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 Gd-DTPA or CBV = CBF = CBFGM CBFWM = 2 Perfusion Imaging

Sample Results CBV CBF MTT Take-off time Recirc. time Normalized X 2 E Morris Monday, July 10, 2000 Sample Results CBV CBF MTT Take-off time Recirc. time Normalized X 2 Perfusion Imaging

Perfusion Imaging

Why a take-off threshold - 1. E Morris Monday, July 10, 2000 Why a take-off threshold - 1. A generalized Gamma-Variate function has 4 (estimatable) parameters t0, K, b, a : but equation (1) cannot be linearized for rapid computation. If we can find the take-off, t0 , ‘graphically’, then the model becomes: 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. Perfusion Imaging

Why a take-off threshold - 2. E Morris Monday, July 10, 2000 Why a take-off threshold - 2. Thus, we identify the take-off, t0 , by extrapolating from near-threshold points back to baseline. peak value threshold take-off time, t0 1st point above 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. Perfusion Imaging

Why a Recirculation Threshold ? E Morris Monday, July 10, 2000 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. Perfusion Imaging

CBV- Effect of Recirculation Threshold E Morris Monday, July 10, 2000 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 Thresh ; CBV bias ; Fit Quality . Perfusion Imaging

E Morris Monday, July 10, 2000 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: Perfusion Imaging

E Morris Monday, July 10, 2000 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. Perfusion Imaging