# MPI with chemical microspheres

Labelled small microspheres that are trapped in tissue capillaries can be used to quantify nutritive blood flow to tissue (perfusion). Similarly, perfusion could be assessed with labelled radiopharmaceutical that is rapidly and irreversibly taken up by cells.

Concentration of the tracer in the blood (C_{B}) is an average of concentrations in
plasma (C_{P}) and red blood cells (C_{RBC}), weighted by
haematocrit (HCT):

The arterial plasma TAC, not the whole blood TAC, is the correct input function in this model since the radiopharmaceutical in blood cells is assumed to be trapped. Since we want to assess blood flow instead of plasma flow, we write the differential equations for tissue and blood cell concentrations as

, where (1-HCT)×C_{P} is the plasma radioactivity per volume of blood.
After integration and rearrangement,

, which clearly shows that in this chemical microsphere model the TACs of muscle tissues and RBCs have the same shape, the integral of plasma TAC as a function of time, but with their own scaling factors.

Solving Eq 4 for C_{RBC}(t) and substituting it into
Eq 1 gives equation

which can be solved for (1-V_{B})×C_{P}(t), which is then substituted in
Eq 2, giving

This is similar to the differential equation of the commonly used
reversible one-tissue compartment model
with k_{2}=k_{C}×HCT.

The regional TACs from myocardial PET studies are distorted by Spillover and partial volume effects. In theory, this can be accounted for by modelling the left ventricular (LV) muscle and cavity TACs as:

*α*is the tissue fraction inside myocardial muscle ROI and

*V*is the blood volume fraction, and

_{B}*β*is the recovery coefficient of LV cavity ROI. The

*V*is assumed to represent the spillover and partial volume effects from the blood in LV cavity; the contribution of vascular blood volume in muscle is assumed to be negligible; majority of that would be venous blood, with curve shape determined by the concentration in red blood cells and thus similar to the muscle tissue in this model (see Eq 4).

_{B}The β is dependent on the image resolution, and could be measured with
[^{15}O]CO PET.
In myocardial radiowater PET studies the model can be first
fitted to the data from a large ROI covering most of the left ventricular myocardium
(possibly excluding septum and areas with marked spillover from the liver), enabling calculation of
C_{B}(t), which is then used for analysis of smaller myocardial regions
(Iida et al., 1988,
1991, and
1992).
The resolution of modern PET may allow using the blood curve derived from a tiny ROI placed in
the LV cavity directly, and therefore we assume that β=1, and

While the α can be estimated in myocardial PET studies with radiowater that is a diffusible tracer with known equilibrium tissue distribution volume, that is not possible with a trapped radiopharmaceutical. Instead we must draw muscle ROI avoiding carefully the tissues outside of the heart, and use the simple geometrical model

C_{T}(t) and dC_{T}(t)/dt are solved from this equation and its derivative,
respectively, and placed in Eq 6, providing the
ODE and its integral

This multilinear equation has three parameters,
V_{B}, K_{1}, and k_{2}=k_{C}×HCT, and all concentrations are
measured with PET.
The parameters can be fitted with non-linear optimisation methods, or solved using
non-negative least squares (NNLS) optimisation method,
like has been done for example to fit muscle data from radiowater PET studies
(Burchert et al., 1997).
The NNLS method is computationally very fast, allowing pixel-by-pixel calculation and production
of parametric K_{1} and V_{B} image.

The last of the parameters, k_{2}=k_{C}×HCT, is related only to
the kinetics of tracer in blood and plasma, and therefore common to all tissue regions.
Therefore it is possible to either perform the fits of all tissue regions simultaneously,
with a single k_{2} parameter, or to first fit a large (less noisy) myocardial region, and
fit the (noisier) TACs of smaller myocardial regions with fixed k_{2}.
If k_{2} is known, the remaining two parameters can be solved using NNLS from multilinear
equation

With k_{2}, the plasma component of measured, discrete blood data curve could be computed
from ODE that could, for example, be solved using implicit
second-order Adams-Moulton corrector, resulting in
equation

, where Δt is the sample time difference.
Program b2ptrap can make the plasma curve with this method
using known k_{2}, or it can fit the k_{2} based on blood data alone with assumption
that radioactivity in plasma is zero after a certain time.
Program fitmtrap, which can be used to fit the model in
Eq 10 to regional data, can also make the plasma curve with this method.

If both arterial plasma and blood curves would available, a very simple linear equation can be formed with only two parameters to be fitted:

The reversible one-tissue model with parameters V_{B}, K_{1}, and k_{2}
is included in all PET modelling software, and the parameters V_{B} and K_{1} from
those tools can directly be interpreted as the parameters of the chemical microsphere model.

## K_{1} and perfusion

The unidirectional transport rate of the radioligand from the blood to the tissue (K_{1})
and blood flow (F), according to the Fick's principle and
Renkin-Crone capillary model, are related by first-pass extraction fraction *E*,

, where *E* depends on perfusion, and the product of capillary permeability
*P* and capillary surface area *S*, *PS*.
Only highly diffusive (*E ≅ 1*) tracers can be used to reliably measure regional
perfusion. If *E* is substantially reduced when perfusion is high, a non-linear function
is needed to convert K_{1} to perfusion.
*E* cannot be determined from PET data, but comparison of K_{1} estimates to
perfusion values measured with gold-standard methods can provide the non-linear function that can
be used to obtain perfusion from K_{1}.

## Simulations

### Late-time tissue-to-blood ratio (TBR)

An intravenously administered radiotracer that traps into cells is rapidly cleared from
the plasma. As plasma curve approaches zero, its area-under-curve (integral) stabilizes to
a certain level. Likewise, the concentrations in blood cells and tissue stabilize
to a level determined by the integral (area-under-curve) of the plasma concentration and
the rate constants k_{2}=k_{c}×HCT and K_{1}
(see Eq 4).
At this phase, since C_{P}=0, the concentration in blood is

and the late-time regional tissue-to-blood ratio, based on Eq 7 and
C_{T}/C_{RBC} from Eq 4, is

We can see that the late-time TBR correlates with K_{1}, but is affected by V_{B}
(which represents the partial volume and spillover effects in myocardial model) and
k_{2}=k_{c}×HCT (the rate of tracer uptake in blood cells).
The relationship of K_{1} and late-time TBR is shown in Fig 3.
When K_{1} < k_{C}×HCT, the late-time concentration in myocardial regions
will be lower than concentration in blood (LV cavity), which complicates the drawing of
regions-of-interest to areas with perfusion defect.

Haematocrit has a linear effect on late-time TBR (Eq 13). The tissue uptake is related to plasma flow, and its proportion of blood flow, (1-HCT)×F, is higher when haematocrit is lower, which explains the higher late-time TBR with lower haematocrit in the right panel of Fig 3.

The late-time TBR correlates linearly with K_{1} when other model parameters are
assumed constant. V_{B} however has strong effect on the regional tissue concentrations.
When drawing the regions on the PET image it is difficult to control the contribution of blood in
LV cavity to the muscle regions; usage of tissue-to-blood ratio (TBR) or late-time SUV in
quantification of myocardial perfusion cannot therefore be recommended, especially as
the compartment model solutions are readily available, providing K_{1} that is not affected
by V_{B}, k_{C}, or haematocrit. However, the late SUV or TBR image may be useful
in visual analyses.

### Kinetic data simulation

In this simulation, a sum of exponentials
function (Feng et al., 1993a and
1993b) is used to describe
the input function of the model, the arterial plasma curve (C_{P}(t)), shown in
Fig 4.
In the present model the intravenously administered radiotracer is assumed to become rapidly trapped
in cells, including blood cells, and therefore the function is set to approach zero in few minutes.
The tracer concentration in blood cells is simulated based on Eq 3, and
the arterial blood curve is calculated as the sum of tracer concentrations in blood cells and plasma
(Fig 4).

Myocardial muscle tissue curves (C_{T}) were then simulated using
Eq 2 with K_{1} values ranging from 0.2 to 3.0 mL Blood/(min ×
mL tissue). These, and previously simulated blood curve, were used to calculate regional muscle
curves (C_{PET}) with different values for V_{B} (Fig 5).

The simulated TACs reach a steady level ∼5 min after tracer administration, but the exact
time depends on the input function and model parameters.
As expected, the simulated, noiseless, regional tissue data could be well fitted with reversible
one-tissue compartment model using blood curve as input function, and it provided K_{1} and
V_{B} estimates that were close to the ones used to simulate the data (results not shown).
Extending the PET scan long after the steady phase is achieved would not help the parameter
estimation; in real-world PET studies it may be detrimental, if radioactive metabolites of the
tracer are released from cells at later times, invalidating the model.

### PET image simulation

As a rudimentary mock-up of a myocardial dynamic PET image was made by filling segments
of a ring with myocardial muscle TACs simulated with different K_{1} values, and centre
with blood TAC, representing LV cavity (Fig 6 and Fig 7).
The input curves simulated previously were used.

Dynamic image was then constructed with these TACs. In this simulation, no noise was added. Partial volume and spillover effects were simulated by strong smoothing with Gaussian filter; Fig 7 shows the sum images from time range 0-1 min and 5-10 min.

The blood curve from then extracted from middle of the area representing cavity, and model
in Eq 9 was applied pixel-by-pixel to the dynamic image, producing parametric
V_{B} and K_{1} images (Fig 8).
The spillover leads to apparent presence of muscle tissue far inside the cavity, in such extent
that K_{1} can be estimated there, but not in the middle of the cavity where V_{B}
approaches 1, and K_{1} can no longer be estimated (represented by the black hole in the
middle of the image where K_{1} is set to zero).
Partial volume effect leads to underestimation of K_{1} in the region where the muscle
actually is located, but less so at the border of muscle and cavity, because the spillover
between cavity and muscle is corrected in the model (with the model parameter V_{B}).

### The effect of spillover on LV cavity

In the present model and previous simulations the assumption is that true arterial blood curve
can be extracted from a small ROI drawn in the middle of the LV cavity.
In practise, the blood curve may be contaminated by myocardial muscle curve because of spillover
and partial volume effects.
In this simulation, the cavity curves were simulated with β values 100%, 95%, 90%, and 85%,
meaning contamination of 0%, 5%, 10%, and 15% from myocardium, respectively.
Since the level of tissue curve is dependent on perfusion, the simulation is
performed with three different K_{1} values, 0.5, 1.0, and 2.0.
The simulated data were then fitted using fitmtrap
according to Eq 9. The model fitted the simulated curves perfectly (not shown).
The simulated cavity curves and myocardial curve, and the bias induced to the estimated model
parameters, are shown in Fig 10.
The simulations suggest that tissue-contaminated blood curve leads to overestimation of all model
parameters. The bias in both K_{1} and V_{B} is similar to the contamination-% of
the cavity curve. While the higher perfusion leads to higher overestimation of cavity curve,
this does not influence the bias in estimated K_{1} values, which seems to be overestimated
by about the same percentage in all perfusion levels.
Most of the bias seems to be absorbed by the parameter k_{2}.

Overall, the results of this simulation suggest that the possible contamination of LV cavity
curve by myocardial muscle causes overestimation of K_{1}, the bias remains at acceptable
level. On the other hand, the nuisance parameter k_{2} is severely biased, suggesting that
k_{2} should not be constrained if contamination of LV cavity curve is suspected.

Script and data used to make these simulations are available in
https://gitlab.utu.fi/vesoik/simulations/.

## See also:

Tags: Perfusion

Updated at: 2023-03-18

Created at: 2023-02-27

Written by: Vesa Oikonen