What can we learn with intravascular tracers?

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What can we learn with intravascular tracers?
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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 29553-559, 1993.
• Jacquez, J. Compartmental Modeling in Biology
  and Medicine, pages 193-203. U Michigan Press,
  1984.

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Todays Parametric ImagesWhat is the mapping
from data to parameter?
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Lets consider the data in time.
See plots
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Todays deep thoughts
MTT CBV / CBF
MTT proability-weighted average transit time
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What do we mean by blood flow?Is that the
same as CBF?What do we mean by Perfusion?
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Lets examine the Perfusion of this system. The
is the U.S. Brain Trust. Whats the model?
Map of NIH
Arterial Inflow
Venous Outflow
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Q. What is the perfusion of people within a
single region (i.e., building)?
Arterial Inflow
Venous Outflow
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Lets examine this single region in detail.
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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)

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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.

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Method Inject an impulse of runners into the
system, then monitor their arrival(s) downstream.
In
Out
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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
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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 tDt The Mean transit time is at
the center of mass of the distribution, h(t).
I.e., 1st moment.
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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).
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But, consider our actual observation 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.

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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).
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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

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Practically, we image a convo-lution of the
Residue function.
Ct
Ct
Ct
Ct
Ct
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Whats in a shape? What does the shape of R(t)
mean?
Ct
Ct
Short Transit time
Ct
Dispersed (non-ideal) bolus.
Ct
Ct
Long Transit time
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What do the Residue Functions that we get from
deconvolution look like?
See plots
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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.
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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, FR(t), so we must
divide accordingly.
Where by convention Scale is the maximum point on
the scaled residue curve.
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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.
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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.
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Q. What assumptions do we make in applying our
simple input-output model?

• 1. Every pixel is supplied directly by the input.

2. All dispersion of a bolus input is due to
multiple path-lengths inside a pixel
3. Feeding and draining vessels are outside
the pixel
4. No recirculation.
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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
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What about recirculation?
HW 1
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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.

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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
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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.

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V Total people to enter is proportional to
capacity
Exit
Exit
Exit
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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.

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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.

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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

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A Contrast Agent that leaks across the BBB is
also called a freely diffusable
tracer.Freely diffusable tracers are
the domain of PET
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Hows 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
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Sample Results
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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.
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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.
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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.
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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
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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
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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.