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) R97R138. (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 Todays Parametric Images What is the mapping from data to parameter? CBV CBF MTT Perfusion Imaging Lets consider the data in time. See plots Perfusion Imaging Todays deep thoughts: MTT = CBV / CBF

MTT = proability-weighted average transit time Perfusion Imaging What do we mean by blood flow? Is that the same as CBF? What do we mean by Perfusion? Perfusion Imaging Lets examine the Perfusion of this system. The is the U.S. Brain Trust. Whats the model? Map of NIH Arterial Inflow Perfusion Imaging

Venous Outflow Q. What is the perfusion of people within a single region (i.e., building)? Arterial Inflow Perfusion Imaging Venous Outflow Lets examine this single region in detail. Perfusion Imaging pixel Each building (pixel) has an inflow and an outflow. But there are multiple paths through the building.

inflow outflow Analogies A building (e.g., CC) is a ... pixel Rate of people entering CC at inflow: F MTT Fraction of people passing through CC: V Average time spent in CC building: (compared to other buildings) Perfusion Imaging 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 Method: Inject an impulse of runners into the system, then monitor their arrival(s) downstream. In Out Perfusion Imaging Lets further idealize the picture pixel inflow

outflow 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: S out Sin h Perfusion Imaging What is the impulse response, h(t)? 400 350 300 250 200 150 100 50

0 0 5 t t + t 10 15 20 25 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+t The Mean transit time is at the center of mass

of the distribution, h(t). I.e., 1st moment. t t h(t )dt 0 Perfusion Imaging Where to make our observations? Inflow to CC Outflow from 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... pixel

inflow outflow 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 Q. How do our observations relate to the histogram of transit times, h(t)? 400 350 h(t) 300 250 200

150 100 50 t t + t 0 0 5 10 15 20 25 time 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. 250 H (t ) h(t )dt 200 150 100 R(t) = 1 - H(t) 50 0 0 5

10 15 Our observations are related to R(t). Perfusion Imaging 20 25 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 convolution of the Residue function. S S Ct Ct S Ct

S Ca Ct S Ct t Ct (t ) Scale Ca (T ) R(t T )dT 0 Perfusion Imaging Whats in a shape? What does the shape of R(t) mean? S S

Ct Ct Short Transit time S Ct Dispersed (nonideal) bolus. S Ca Ct S Ct

Long Transit time t Ct (t ) Scale Ca (T ) R(t T )dT 0 Perfusion Imaging 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. h(t) The Mean transit time is at the center of mass

of the distribution, h(t). I.e., 1st/0th moments. t t h(t )dt 0 h(t )dt 0 t h(t )dt 0 Recall that the Residue function is related to the integral of the histogram. R (t ) 1 t

h( )d 0 dR h(t )dt Perfusion Imaging What is MTT in terms of the residue function, R(t)? - 2. Substituting dR into the expression for MTT, t t dR 0 Integrating by parts we see that, t Rt | 0

R ( t ) dt 0 R (t )dt 0 Recall that we measure the one entity which is the Scaled Residue Function, F*R(t), so we must divide accordingly. t Scale R (t )dt Scale 0

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. t Scale R(t )dt Max[ Scale R(t )] 0 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 midpoint of the base. Perfusion Imaging Why is the Output Equation

Scaled by the Flow Arriving at the Pixel? t Ct (t ) Scale Ca (T ) R(t T )dT 0 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 Q. What assumptions do we make in applying our simple input-output model? 1. Every pixel IQ! is supplied directly by g n i l

e d mo your input. Testthe 2. All dispersion of a bolus input is due Q! I g n i l mode r u o to multiple path-lengths inside a y t

Tes pixel IQ! draining vessels are g 3. Feeding and n i l e d o your m Testoutside the pixel ng IQ! i l e d o

ur m Test yo 4. No recirculation. Perfusion Imaging 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. FI Ideal FA ?

Actual 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. valid invalid 3a. Different tissue types may require different minimum pixels sizes 4. Recirculation must be removed before applying model. Perfusion Imaging What about recirculation? HW #1 Perfusion Imaging 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. C (t )dt CBV k C (t )dt t 0 0 Perfusion Imaging

a 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. 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 Exit Exit People dt 0 People dt

0 Exit People dt 0 Perfusion Imaging 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), C (t )dt 0 t is larger and CBV is overestimated. Perfusion Imaging Consequence of BBB Leakage

to Contrast Agent pixel inflow outflow 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. pixel inflow outflow

Freely diffusable tracers are the domain of PET Perfusion Imaging Hows it done? - Data Flow 1. Inject Gd-DTPA 2. Scan over time 3. Convert signal to concentration or 4. Find AIF 5. Fit First Pass 6. Calculate CBV, CBF, MTT 7. Post-process, tabulate stats

Perfusion Imaging CBV = CBF = CBFGMC BFWM =2 Sample Results CBV CBF MTT Take-off time

Recirc. time Normalized X2 Perfusion Imaging Perfusion Imaging Why a take-off threshold - 1. A generalized Gamma-Variate function has 4 (estimatable) parameters t0, K, , : G K (t t0 ) e (t t0 ) or G G ( K , t0 , , ) (1)

but equation (1) cannot be linearized for rapid computation. If we can find the take-off, t0 , graphically, then the model becomes: G K ( ) e ( ) (2) which, when log transformed to: ln G ln K ln( ) ( ) can be used to fit the (log-transformed) data via non-iterative multiple-linear regression. In the process, ln(K), , are estimated. Perfusion Imaging (3) Why a take-off threshold - 2. Thus, we identify the take-off, t0 , by extrapolating from near-threshold points back

to baseline. peak value 1st point above threshold take-off time, t0 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 ? C (t )dt CBV

C (t )dt K t 0 h a 0 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. 250 onset of recirculation

200 threshold % of peak 150 100 observed signal 1st pass recirculation 50 0 0 5 -50 10

15 20 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 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 . 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): t Ct (t ) K CBF Ca (T ) R(t T )dT h 0 (1)

We approximate equation (1) as follows: b Ax (2) Ct f AR (3) where: Ct (1) A1 A f 12 A13

C ( n ) t A14 Perfusion Imaging An1 R1 An Rn (4) Why an SVD threshold? - 2 According to SVD, we can represent the A matrix in terms of the diagonal matrix, , made up of singular

values, i : Ct f Q2 Q1T R (5) We then solve equation (2) by: f R Q1[diag (1 / j )]Q2T 1 0 0 1 1 0 2 Q1 0 0 0 T

Q2 1 n (6) (7) But very small singular values, i , 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