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.
MTT CBV CBF Today’s Parametric ImagesWhat is the mapping from data to parameter?
Lets consider the data in time. See plots
Today’s deep thoughts: MTT = CBV / CBF MTT = proability-weighted average transit time
What do we mean by ‘blood flow’?Is that the same as CBF?What do we mean by ‘Perfusion’?
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
Q. What is the ‘perfusion’ of people within a single region (i.e., building)? “Arterial” Inflow “Venous” Outflow
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)
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’.
Method: Inject an “impulse” of runners into the system, then monitor their arrival(s) downstream. In Out
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:
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.
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).
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.
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).
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:
S S S S Ca S Practically, we image a convo-lution of the Residue function. Ct Ct Ct Ct Ct
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
What do the Residue Functions that we get from deconvolution look like? See plots
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.
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.
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.
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.
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.
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
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.
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
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.
V: Total # people to enter is proportional to capacity Exit Exit Exit
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.
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.
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
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…
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