Perfusion Imaging

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

MTT

CBV

CBF

Today’s Parametric ImagesWhat is the mapping from

data to parameter?

Perfusion Imaging

Lets consider the data in time.

See plots

Perfusion Imaging

Today’s deep thoughts:MTT = CBV /

CBFMTT = 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

“Arterial” Inflow“Arterial” Inflow “Venous” Outflow“Venous” Outflow

Lets examine the ‘Perfusion’ of this system. The is the U.S. Brain

Trust. What’s the ‘model’?

Map of NIH

Perfusion Imaging

“Arterial” Inflow“Arterial” Inflow “Venous” Outflow“Venous” Outflow

Q. What is the ‘perfusion’ of people within a single region

(i.e., building)?

Perfusion Imaging

Lets examine this single region in detail.

Perfusion Imaging

Each building (pixel) has an inflow and an outflow. But there are

multiple paths through the building.

inflow outflow

Analogies

p i x

e l

• 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?

• 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

In

Out

Method: Inject an “impulse” of runners into the system, then monitor

their arrival(s) downstream.

Perfusion Imaging

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:

outflowinflow

p i x e l

hSS inout

Perfusion Imaging

What is the impulse response, h(t)?

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.

0

50

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250

300

350

400

0 5 10 15 20 25t t + t time

0

)( dtthtt

Perfusion Imaging

Where to make our observations?

In this idealization, we would need to image every inflow and outflow (i.e., impulse response)

of every building (aka., pixel).

Inflow to CCOutflow from CC

Perfusion Imaging

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.

outflowinflow

p i x e l

Perfusion Imaging

Q. How do our observations relate to the

histogram of transit times, h(t)?

0

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0 5 10 15 20 25t t + t 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)

dtthtH )()(

0

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0 5 10 15 20 25

Our observations are related to R(t).

Perfusion Imaging

How to understand R(t)?

Thus, R(t) is - in effect - the impulse response as viewed from within the pixel. Recall:

In the case of an ideal input, the view from within the pixel would look like:

View at input

View of ‘runners’ remaining within the pixel

100%

0%

Perfusion Imaging

Practically, we image a convo-lution of the Residue function.

S

S

SS

S

t

at dTTtRTCScaletC0

)()()(

CaCt

Ct

Ct

CtCt

Perfusion Imaging

What’s in a shape? What does the shape of R(t) mean?

S

S

SS

S

t

at dTTtRTCScaletC0

)()()(

CaCt

Ct

Ct

CtCt

Long Transit time

Short Transit time

Dispersed (non-ideal) bolus.

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.

0

00

)(

)()(

dttht

dtthdtthtt

The Mean transit time is at the center of mass of the distribution, h(t). I.e., 1st/0th moments.

h(t)

Recall that the Residue function is related to the integral of the histogram.

dtthdR

dhtRt

)(

)(1)(0

Perfusion Imaging

What is MTT in terms of the residue function, R(t)? - 2.

0

00

)(

)(|

dttR

dttRRtt

Substituting dR into the expression for MTT,

Integrating by parts we see that,

0

dRtt

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

ScaledttRScalet

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.

Is equivalent to area / height = 1/2 base.

)]([)(0

tRScaleMaxdttRScalet

If we approximate the Residue function as a triangle, we can see that the MTT lies at mid-point of the base.

Scale*R(t)

Perfusion Imaging

Why is the Output Equation Scaled by the Flow Arriving at

the Pixel?

t

at dTTtRTCScaletC0

)()()(

‘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 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.

Test your modeling IQ!

Test your modeling IQ!

Test your modeling IQ!

Test your modeling IQ!

Perfusion Imaging

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.

What implications are there to our assumptions?

FI

FA ?

Ideal Actual

invalid

valid

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.

0

0

)(

)(

dttCk

dttCCBV

a

t

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.

• 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

dtPeople

0

dtPeople

0

0

dtPeople

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.

Perfusion Imaging

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.

Ideal

With Leakage

larger is )(0

dttCt

Perfusion Imaging

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

outflowinflow

p i x e l

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…

outflowinflow

p i x e l

Perfusion Imaging

How’s it done? - Data Flow1. 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

CBFGMC

BFWM

= 2

CBV = CBF =

or

Perfusion Imaging