# Dispersion of input function

The measured blood curve after a radiotracer bolus is smeared out, because of inhomogeneous velocity fields in the vessels and in the catheter and detector assemblies. Also sticking of radiotracer to the tubing may add to this dispersion effect. While different distance and blood velocity in different arteries causes variable appearance time (delay) of radioactivity in the blood between sampling sites and the tissue being scanned, dispersion affects the shape of the blood curve, especially at the time of the blood curve peak after bolus administration. In manual blood sampling the volume of each sample is relatively large compared to the volume of blood in catheter, making dispersion effects in tubing negligible; with flow-through detectors long tube lengths can lead to considerable dispersion (Votaw & Shulman, 1998).

Dispersion does not affect the *AUC _{0-T}* of blood
TAC, after

*T*is large enough. Therefore, dispersion correction is necessary only when very fast kinetics are measured, for example perfusion measurement with radiowater (Kanno et al., 1987), and when precise estimates of

*K*and other individual parameters of compartmental models are fitted. Multiple-time graphical analysis methods (Logan and Patlak plots) are relatively insensitive to dispersion (Blais & Lee, 2015).

_{1}In theory, input function is the same for all organs and can be measured from arterial blood, but since dispersion effect differs for the measured and true input function to the region of interest, this is not strictly true (Figure 1), unless dispersion correction is applied. Venous blood sampling causes additional delay, dispersion and other biases.

Arterial time-activity curves following intravenous bolus administration have been observed to be of the form of the gamma variate function because of dispersion, or sums of gamma variate functions when recirculation is taken into account, too. The recirculation starts to affect the arterial curve very soon, because coronary circulation time is only few seconds. Arterial input curve is thus dispersed already before entering any tissue, and this part of dispersion, and recirculation effect, obviously must not be corrected (van den Hoff et al., 1993).

Iida et al (1986) suggested that a single
monoexponential function *d(t)* could be used to represent the effect of dispersion:

, where *τ* denotes the time constant of dispersion.
The measured arterial curve *C _{meas}(t)* and the true arterial curve

*C*are related as:

_{true}(t), where ⊗ denotes the operation of convolution. Essentially, this is a compartmental model, where the first compartment represents the concentration before dispersion, and the second compartment the concentration in the compartment that can be measured.

This model can be described with an ordinary differential equation (ODE):

, where *k = 1 / τ*. Equation can be integrated, assuming that at time zero all
concentrations are zero:

For discrete data the equation can be solved using definite integrals and Euler integration like a one-tissue compartment model, giving equation

, or, using dispersion time constant *τ* instead of *k*:

These equations can be used to *simulate* the effect of dispersion on input curves.
For example, the simulations in Figure 1 were done using those; data and script
can be downloaded from GitLab
repository.
Simulation of dispersion is also necessary as part of fitting procedure, when dispersion time
constant for blood sampling equipment is estimated.

In addition to these exponential models, dispersion has been modelled using laminar flow model (Hutchins et al., 1986) and turbulent fluid flow model (Wollenweber et al., 1997).

## Dispersion correction

Measured blood curve can be corrected for the dispersion using deconvolution methods. Dispersion correction should be done before time delay correction, or simultaneously (Meyer, 1989).

Dispersion corrected *AUC _{0-T}* of blood TAC can
be calculated from reorganized equation (4) when dispersion time constant

*τ*is known

With the help of the integral curve, dispersion corrected blood curve can be calculated applying 2nd order Adams-Moulton method, based on Euler integration:

While this works with noiseless simulated data, the measured data can be noisy, and dispersion correction increases the variation; dispersion correction is essentially deconvolution, which can blow up the noise level. Calculation over three adjacent sample point provides slight smoothing and less noisy dispersion corrected blood curve:

Generally, dispersion correction should not be accompanied by smoothing (Wollenweber et al., 1997). Munk et al (2004 and 2008) proposed a dispersion correction method with less noise problem.

If a function is fitted to the input curve, the dispersion correction can and should be included in the fitting: not only does it remove the problem with noise, but it improves the fit, too, since the functions are aimed for fitting input curves with little or no dispersion. Program fit_sinf can currently use up to two known dispersion time constants to fit function, representing dispersion corrected input curve, to the measured blood or plasma curve.

### Dispersion in the detector system

With Automatic blood sampling system (ABSS) blood needs to be drawn via long tubing to minimize background radiation from the subject to the detectors. Most ABSSs are not compatible with MR, requiring long distance from PET-MR, even when ABSS detector is well shielded. Certain ABSSs are MR-compatible (Breuer et al., 2010). While long tubing increases the dispersion error, O'Doherty et al (2015) have shown that dispersion correction can be successful even with 3 m arterial sampling tubing.

External dispersion of the input function can be determined by measuring the rising/falling edges of the detector system's response to an input step function (Iida et al., 1986; Senda et al., 1988; Weinberg et al., 1988; Votaw & Shulman, 1998; Boellaard et al., 2001; Convert et al., 2007). Delay and dispersion between original and dispersed curves can be fitted for example using fit_disp. Alternatively, a separate β-microprobe placed close to the arterial catheter could be used to measure the shape of the blood curve without most of the dispersion effects of the main detector system and blood tubing (Seki et al., 1998). Microprobe system could also work without the pump, relying only on the impedance of the system to control the blood flow (Wollenweber et al., 1996).

### Internal dispersion

Dispersion and time delay in the arterial blood curves are different at the site of arterial blood sampling and at the region-of-interest measured with PET; generally, blood arrives earlier and with less dispersion into organs with high perfusion than into radial artery which is typical blood sampling site; delay time and dispersion in less perfused peripheral tissues are higher. Thus, dispersion correction of sampled blood curve may require either removal or adding dispersion effect to the input data. In the case of image-derived blood curve, dispersion and delay time is always higher in arteries of the organ of interest than in the image region from where the blood curve is extracted; thus the dispersion correction involves adding dispersion to the blood curve, which does not add noise.

The internal between the radial artery and the brain is 4-6 sec (Iida et al., 1986). The liver is especially difficult case with dual input (Keiding, 2012).

In heart PET studies, delay and dispersion (both internal and detector system) can be corrected by fitting the compartmental model (van den Hoff et al., 1993) to the curve of the LV cavity (Iida et al., 1989) or ascending aorta (Harms et al., 2014).

The methods for estimating delay and dispersion simultaneously
have been developed with [^{15}O]H_{2}O data,
where these effects can cause especially high biases, but applied to other radiotracers, too.
[^{15}O]H_{2}O data is analyzed using single-tissue compartment model, which is
applicable for delay and dispersion modelling with other radiopharmaceuticals when only
the initial part of the curves are used.
More precise estimates of delay and dispersion can be obtained with radiotracers that stay in
vasculature, such as [^{15}O]CO
(Kuwabara et al., 1993).

Meyer (1989) gave the formula for simulating the tissue curve in the optimization procedure (notations changed):

, where *C _{A}^{m}(t)* is the measured arterial blood curve containing
both delay,

*Δt*, and dispersion,

*τ*, errors. This was further investigated by Lüdemann et al (2006). The linearized version of this, proposed by van den Hoff et al (1993), with the same notations is

Neither of these formulas include the vascular volume fraction,
*V _{A}*, which in case of radiowater represents only arterial volume fraction, and
can quite safely be neglected in brain studies where even the total blood volume is <5%.
The single-tissue compartmental model equation that includes the vascular volume fraction, with
corrected input function,

*C*, is

_{A}(t)Comparison of this to the previous equations used for fitting delay and dispersion shows that dispersion time constant and vascular volume fraction cannot be estimated independently. This can be utilized in time delay correction: dispersion does not affect the estimated delay time between input function and tissue curve, because vascular volume fraction accounts for the dispersion (Mourik et al., 2008).

The time delay from left ventricular cavity of the heart to a peripheral artery is linearly related to the dispersion time constant between these sites (Iida et al., 1989, Fig 7); the resulting linear equation

could potentially be used to estimate and correct for dispersion based on time delay, when arterial input function is derived from the PET image.

### Dispersion correction in Turku PET Centre

In Turku PET Centre, when the dispersion correction is necessary, it is usually done automatically by the blood data processing software: first, the external dispersion is corrected, and secondly the internal dispersion is corrected, if an estimate of internal dispersion time constant is available.

In Turku PET Centre, the external dispersion time constant has been estimated to be 2.5 sec for the assemblies that are currently in use. If tubing is changed, the dispersion time constant should be measured again.

Internal dispersion time constant of 5 sec has been used for the brain studies.

The corrections are done using disp4dft. Alternatively, the two dispersion time constants can be given to program fit_sinf to fit measured curve with function that represents dispersion corrected input curve.

The impact of dispersion can be simulated by adding dispersion to input curves. This can be
done using simdisp. Alternatively, same result can be
obtained with convexpf that convolves TAC with
exponential functions; set *a _{1}=b_{1}=1/τ* and

*a*.

_{2}=b_{2}=a_{3}=b_{3}=0## See also:

- Time-delay correction
- Blood sampling
- Preprocessing arterial input data
- Input function
- Fitting PET input curves
- Input for simulations
- Circulatory system

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Tags: Dispersion, Input function, Blood, ABSS, Delay correction

Updated at: 2023-06-16

Written by: Vesa Oikonen