# Model for [15O]H2O PET

The methods to measure perfusion with [15O]H2O (diffusible and inert tracer) are based on the principle of exchange of inert gas between blood and tissues (Kety and Schmidt, 1945; Kety, 1985), and on the Fick’s principle:

Venous administration of [15O]H2O can be replaced by inhalation of [15O]CO2, because it is instantly converted to [15O]H2O in the lungs by carbonic anhydrase. Being diffusible and inert, [15O]H2O is an optimal tracer for perfusion assessment, but the relatively even distribution of radioactivity shortly after injection makes it difficult to draw ROIs in the image. It takes ∼ 2 h for water (heavy water, D2O) to fully equilibrate in all body fluids, and arterial and venous concentrations reach the same level at about 40 min (Edelman, 1952).

Since (labelled) water diffuses nearly instantly from capillary blood to extra- and intracellular spaces, and back, the [15O]H2O concentrations in the tissue, CT and in the capillary and venous blood, CV, are in equilibrium (unless perfusion is so high that diffusion becomes partly limiting factor). Tissue-to-blood ratio, the partition coefficient (p) of water, is only dependent on the relative water contents of tissue and blood per volume, which are well known in the literature for normal tissues.

Venous [15O]H2O concentration in Eq (1) can be substituted by CT/p, giving equation

This is the formula of a compartmental model with one tissue compartment (i.e. two compartmental model), with K1=f and k2=f/p. It is used to describe mathematically the kinetics of [15O]H2O concentration in the tissue, CT(t), depending on the concentration in arterial blood, CA(t), perfusion or blood flow (f), and the partition coefficient of water, p. This ordinary differential equation (ODE) can be solved as usual.

When radioactivity in the volume of interest is measured with PET (CPET(t)), the radioactivity of vascular blood inside the measured volume should also be taken into account; VA is the arterial volume fraction in tissue:

Radioactivity concentration in the local venous blood is, by definition, the same as in the local tissue (or to be precise, CT/p), and venous blood volume thus does not need to be included in the formula. Since major part of blood volume in tissue is venous blood, the tissue concentration CT(t) in previous equation does not necessarily have to be scaled with (1-VA), and certainly not with (1-VB), which is customary in compartmental models for other radiopharmaceuticals, because both VB and the arterial fraction of it are unknown.

In practise, tissue volume may contain veins coming from other tissues or tissue vasculature where no or less substrate exchange takes place, leading to increase in apparent blood volume. Otherwise than that, non-nutritive (non-effective) blood flow is not affecting the perfusion estimate obtained using [15O]H2O PET (Lammertsma and Jones, 1983). For the quantification of total vascular volume [15O]CO PET would be more accurate, and could be a useful to combine with radiowater-PET when studying microvascular function.

Bolus infusion studies can be analyzed with kinetic model fitting or autoradiographic (ARG) method. Both methods are based on the same model, and both can be applied to dynamic PET scan data, but static PET data can only be analyzed with ARG method.

## Kinetic model vs. autoradiography (ARG) method

ARG method produces perfusion images that are of better quality (less noise) than images produced by kinetic (dynamic) model with low injected dose, enabling many repeated studies to the same subject. Kinetic technique can also generate reproducible perfusion measurements (de Langen et al., 2008).

ARG method can be applied to static PET images, which reduces the image reconstruction time and file size, and due to better count statistics, may provide better image quality. However, ARG method requires that the partition coefficient of water is known, and that it is uniform in all regions of interest. Dynamic method also avoids the tendency to underestimate the blood flow in the presence of flow heterogeneity within a ROI or pixel (Wells et al., 2003). Therefore, if precise quantitation is required or partition coefficient is of interest, dynamic PET imaging and kinetic modelling is recommended.

Quantitative perfusion estimation requires frequent arterial blood sampling. Blood sampling has been omitted in brain activation studies, where only relative perfusion changes are used, but these have been mostly replaced by MRI. Blood TAC is used instead of plasma TACs, because the permeability of red blood cell membranes is very high for water (Eichling et al., 1974). Venous blood cannot be used as model input because of the very high first-pass extraction of water; however, venous sampling could be used to scale input function derived by measuring the radioactivity of exhaled air (Koeppe et al., 1985). Image-derived, population-based, and model-based input functions are used, when possible, instead of arterial sampling.

## Kinetic model analysis

The ordinary differential equation for radiowater can be solved, and model parameters parameters fitted, using variable methods, including non-linear fitting and basis function approach. The differential equation for radiowater can also be integrated, assuming that all concentrations are zero at time zero, providing equation

, or, if arterial volume fraction is accounted for,

These equations already are in linear forms from which the parameters can be directly solved using standard linear algorithms, as suggested by Blomqvist (1984) for the first one. Alternatively, both sides of the first equation (where VA is omitted) can be divided by 0TCA to give equation for the Yokoi plot

Like with other multiple-time graphical analysis methods, including Patlak and Logan plots, line can be fitted to the linear phase of the plotted graph. The slope of the line represents -k2, y axis intercept represents K1, and x axis intercept is the p of radiowater (Yokoi et al., 1993).

In the one-tissue compartment model for radiowater, both rate constants K1 and k2 contain the perfusion term f, and therefore perfusion can be measured using either one of the rate constants. Errors in PET data affect the reliability of parameters differently, and depending on the situation, K1- or k2-based perfusion estimates should be used; for example, k2 is not affected by attenuation and partial tissue effect; the quantification of myocardial perfusion is based on k2.

### Computation of blood flow images

Before calculation, make sure that both the blood and PET image are in the same calibration units (preferably kBq/mL or Bq/mL). If necessary, filter the dynamic PET image to reduce the noise level.

To compute the perfusion image (where K1 represents the perfusion), one of the CLI programs imgbfh2o (basis function approach) or imgflow (applying multilinear NNLS method to solve linearized 1TCM parameters) can be used. Optionally, k2 image can be saved and used to compute perfusion image.

The units in the resulting perfusion image are (ml blood)/(min × ml tissue) by default, but it can be changed to per 100 ml tissue with option -dl.

If perfusion is required in units (ml blood)/(min × 100 g tissue), the perfusion image need to be calculated with options -dl and -density=1.04 (1.04 g/mL is the density of the brain and myocardial tissue; density may be very different in other organs, especially in bone, adipose tissue, and tumours).

The results from parametric images should always be validated against results from regional average curves (Lodge et al., 2000). Noise in dynamic image may lead into biased results with distorted variance. Filtering of dynamic images may be needed to achieve the same quantitative results as in the regional analysis. To prevent artefacts and excessive loss of image resolution, the strength of filtering must not exceed the level that is required to achieve comparable results.

### Blood flow analysis from regional TACs

Perfusion model parameters can be estimated in CLI using nonlinear fitting with fit_h2o, or using basis function approach with bfmh2o. Both programs estimate the three radiowater model parameters, blood flow (perfusion), partition coefficient

, and arterial volume fraction. In addition, fit_h2o can optionally estimate and correct also the delay time.

Make sure that both the blood and tissue data are in the same calibration units (preferably kBq/mL or Bq/mL) and that the time unit is sec. You can view the files in text editor, or view and convert the units with tacunit.

Weights should be added to tissue data file using dftweigh. Weights can be calculated based on either SIF or the average tissue curves. If weights are extracted from the SIF, the command could be e.g.:

dftweigh uo268.dft uo268dy1.img.sif O-15


If SIF file is not available, the command could be:

dftweigh uo268.dft


Note that you may need to change the default lower and upper limits for the model parameters.

Many of the first quantitative PET studies using [15O]H2O, or actually [15O]CO2, were performed using steady-state technique (Frackowiak et al., 1980): A static PET scan is performed during continuous inhalation of [15O]CO2 (15O-labelled carbon dioxide), which is the same as continuous intravenous infusion of [15O]H2O. After about 10 min inhalation [15O]H2O concentrations in tissues has reached a dynamic equilibrium (“steady state”), in which the influx from arterial blood into tissue equals the efflux into venous blood and the radioactive decay. PET scan may have been started earlier to verify that the equilibrium has been achieved, but the PET data that is actually used in the analysis is collected from the equilibrium phase.

During the steady state, the radioactivity concentration in the tissue, CT, can be described by equation

, where CA is the radioactivity concentration in arterial blood, p is the partition coefficient of water between tissue and blood, f is the perfusion, and λ is the decay constant for 15O. Radioactivity concentrations are, by way of exception, not corrected for decay in steady-state formulas and computations.

By rearrangement, we get an equation for calculation of the tissue perfusion (Frackowiak et al., 1980; Jones et al., 1985; Ruotsalainen et al., 1997):

### Steady-state vs. kinetic method

Performing and analysis of steady-state study is very simple, enabling also computation of parametric perfusion images. Noise level in the data can be decreased simply by extending the PET scan length. However, steady-state technique leads to higher radiation dose.

Steady-state approach has the same disadvantages as the ARG method: partition coefficient of water must be known, and tissue heterogeneity causes underestimation in perfusion estimates.

## Whole-body model for [15O]H2O

Comprehensive whole-body pharmacokinetic models for [15O]H2O have been developed for dosimetry (Brihaye et al., 1995) and simulation (Narayana et al., 1997). For describing the [15O]H2O concentration in blood, simplified models with just few compartments have been developed (Bigler et al., 1981; Maguire et al., 2003). The study of Bigler et al. (1981) was mainly aimed at understanding the kinetics of [15O]O2 and its metabolite [15O]H2O in the brain; the model consists of a central compartment (blood water) and water in two reversible tissue compartments, one for slow and the other for rapidly exchanging water. The model is similar to the PK three-compartment model without the clearance term. The model contains four rate constants, and the model was be fitted to the D2O data by Edelman et al (1952).

Maguire et al. (2003) used a rectangular (boxcar) function to represent bolus infusion of the radiowater. The model was applied to arterial blood data from actual [15O]H2O PET studies. Central compartment is the site of tracer administration and the site of sampling; sampled blood is assumed to be delayed and dispersed. The authors tested if one of the tissue compartments could be removed, or assumed irreversible, and, the best model setting was found to be one with irreversible slow tissue compartment.

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Updated at: 2019-03-24
Created at: 2008-03-13
Written by: Vesa Oikonen