Input function: Modelling A-V difference

Arterial blood sampling is regarded as the gold standard for obtaining the input function for quantitative PET data analysis. Based on Fick's principle, the arterial-venous (A-V) concentration difference is dependent on the blood flow, f, and the concentration change in the tissue that the vein drains:

$C_{A}(t) - C_{V}(t) = \frac{dC_{T}/dt}{f} \\ \qquad \Leftrightarrow \\ C_{V}(t) = C_{A}(t) - \frac{dC_{T}/dt}{f}$

As can be seen, if blood perfusion is high compared to the uptake/release rate into/from the tissue, concentration in the venous blood blood is similar to the arterial concentration. Blood flow can be increased by warming the hand to reduce the A-V difference at the sampling site ("arterialized" venous blood). Arterialized venous blood or venous blood is often sampled instead of arterial blood, but that leads to bias and higher variance in the results.

The radionuclides most often used used in PET studies have relatively short half-lives, limiting the length of PET studies to maximally few hours. PET radiopharmaceuticals are administered intravenously, leading to immediate appearance in the circulation and rapid concentration changes in blood and in the hand tissue. Thus the A-V concentration differences are can be significant, and the A-V difference is higher soon after administration. In pharmacokinetic (PK) studies the drugs are usually administered orally, leading to slower appearance in circulation and offering more time for the drug to reach equilibrium in the central compartment. With pharmacological doses the drug concentrations in blood can be measured for days; thus the A-V differences are relatively small, and stable, and PK studies indeed are usually based on venous plasma samples, although pharmacological effects are driven by the arterial concentration of the drug. Depending on the drug and venous sampling site, false results can be evident in PK studies, too (Brun et al., 1949; Chiou, 1989a, 1989b).

In PET studies, the most used approach is to conduct both arterial and venous or arterialized venous blood sampling, and then directly compare the curves, and/or analyse the tissue data with both input functions and compare the obtained final results. This way, for instance, was validated the use of arterialized venous plasma as input function if [¹⁸F]FDG studies (Phelps et al., 1979; van der Weerdt et al., 2002). For most PET radiopharmaceuticals, or for detailed compartmental model analyses, these comparisons will result in conclusion that venous or arterialized venous sampling is not adequate.

Obvious solution in PK and PET studies has been to model the A-V difference (McGuire et al., 1972; Rehling et al., 1989; Pitsiu et al., 2002; Syvänen et al., 2006). Basically, the concentration in the venous blood, C_V(t), can be calculated by convolution of the arterial blood curve, C_A(t), with transfer function h(t):

$C_{V}(t) = C_{A}(t) \: \otimes \: h(t) = \int_{0}^{t} h(t - \tau) C_{A}(\tau) d\tau$

Arterial concentration curve is estimated using deconvolution of the measured venous curve with the transfer function, if a reasonable approximation of the transfer function has been found. Estimation of the transfer function requires both arterial and venous sampling in a small subset of study subjects. A suitable transfer function (model) must be selected and validated. Model parameters need to be assessed by fitting venous curves, simulated with the model and arterial curve, to the measured venous data.

In theory, A-V difference modelling can be simple for a homogeneous tissue. However, venous samples are usually taken from the hand or arm, and the tissues that are drained by the sampled vein are very heterogeneous, including at least muscle, adipose tissue, connective tissue (including bone), and skin, and blood flow to skin can vary considerably (Detry et al., 1972; Heinonen et al., 2011). Therefore, A-V difference in the hand or arm cannot be modelled assuming a single compartment for a highly extracted radiotracer (Koeppe et al., 1985). In a nicotine study, Pitsiu et al (2002) found that the transfer function could be best characterized with biexponential function. In a [1-¹⁴C]glucose bolus infusion study three-exponential function was used (McGuire et al., 1972). Levitt (2004) added a "standard arm" model into physiologically based pharmacokinetic (PBPK) model that describes the drug kinetics in the whole body. The arm model contains tissues that are drained by the antecubital vein, and is applicable to any solute. The tissues in the model, and their proportion of the total blood flow through the arm, are the shunt (52.5%), muscle (5%), lipid (7.5%), skin (25%), and others (10%) (Levitt, 2004). The shunt compartment represents the arteriovenous anastomoses in the skin of the hand, and despite its name, the exchange in the shunts is not assumed negligible. The solute concentration in the antecubital vein is calculated as flow weighted mean of the venous concentrations from the tissue compartments. The models and model parameters of the tissue compartments were assumed to be the same as in the whole body PBPK model, except for the shunt compartment. Musther et al (2015) developed the model further, and got best results when the "others" was assigned to a true shunt using assuming arterial concentration, and the flow fractions were set to shunt-skin 52.5%, muscle 5%, adipose 7.5%, skin 25%, and shunt-arterial 10%.

The application of A-V modelling to PET data (an undisclosed ¹¹C-labelled drug molecule) showed that the transfer function was best characterized with biexponential function (Syvänen et al., 2006).

$h(t)=a_1 e^{-b_1 t} + a_2 e^{-b_2 t}$

In case of this radiotracer, biexponential function could be constrained to have AUC=1, which means that the AUCs of the arterial and venous curves are the same, and the only difference is in the shape of the curve (dispersion).

$\begin{equation} \int_0^{\infty}{h(t)dt} = \frac{a_1}{b_1} + \frac{a_2}{b_2} = 1 \end{equation}$ , where a₁>0, a₂>0, b₁>0, and b₂>0.

Thus a₁=b₁(1-a₂/b₂), and there is one parameter less to fit (Syvänen et al., 2006). Another constraint that might prove useful could be to assume that the response function starts from one, that is a₂=1-a₁. Then a₁=b₁(1-b₂)/(b₁-b₂), and there would be only two parameters to fit, b₁ and b₂.

Syvänen et al. (2006) calculated the arterial curve numerically from venous curve, using the parameters of the biexponential function, with equation

$\begin{equation} C_A(t) = c_1 \frac{dC_V(t)}{dt} + c_2 C_V(t) + c_3 \int_0^{t}e^{-c_4(t-u)}C_V(u)du \\ \text{, where} \\ c_1 = \frac{1}{a_1 + a_2} \\ c_2 = \frac{a_1 b_1 + a_2 b_2}{(a_1 + a_2)^2} \\ c_3 = - \frac{a_1 a_1 (b_1^2 + b_2^2 - 2 b_1 b_2)}{(a_1 + a_2)^3} \\ c_4 = \frac{a_1 b_2 + a_2 b_1}{a_1 + a_2} \end{equation}$

Parameters of the transfer function (in the hand) were fitted together with parameters of one- or two-tissue compartmental model for the brain uptake curves from three regions that had markedly different uptake kinetics. Metabolite-corrected venous curve was used as initial input function, which was transformed into arterial curve to be used as input function for the brain model. The number of parameters to fit was too large for the optimization algorithm with two-tissue compartmental model, but one-tissue model performed reasonably well. Convolution procedure was used for the brain model, too, since both the arterial and brain curve could be easily simulated together as

$C_\text{brain}(t) = C_V(t) \otimes h^{-1}(t) \otimes g(t)$

, where g(t) is the transfer function of the brain model (mono- or biexponential function) and h^-1(t) is the inverse of the transform h(t) (Syvänen et al., 2006). Although this approach provided as good fits with venous input data as using measured arterial data, a possible problem with the approach of fitting the transfer function and brain model simultaneously is that the parameters of one model may balance the mismatch of the other model.

This approach was applied in analysis of [¹¹C]zolmitriptan brain data, except that A-V transfer function was set to

$\begin{equation} h(t) = \left\{ \begin{array}{l l} (1-E) \: \Delta T & \quad \small{\text{if } t \leq \Delta T } \\ \alpha \: E \: e^{-\beta (t-\Delta T} & \quad \small{\text{if } t > \Delta T } \end{array} \right. \end{equation}$

, where E is the extraction during passage from artery to the hand vein, ΔT is the transit time from artery to the venules, β is the washout rate from the hand tissue, and α is a constant to set the integral of h(t) to 1 (Bergström et al., 2006).

Similar transfer function was used to simulate venous curve from arterial data, in order to estimate the bias that would be caused by replacing arterial sampling with venous in Logan plot analysis of [¹⁸F]fluorocholine PET data (Blais & Lee, 2015).

Simulation of the difference between arterial and venous samples

The Fick's equation, shown above, can be used for simulating A-V difference, or venous blood curve, when arterial blood curve, C_A(t) is measured. The equation utilizes the change in tissue concentration, dC_A(t)/dt, in this case in hand, which is not measured, but can be simulated using compartmental models, for instance 3-tissue compartmental model.

Three-tissue compartmental model, compartments in series — Compartmental model with three tissue compartments in series.

The ordinary differential equations for the three-tissue compartmental model (where compartments are in series) are:

$\frac{dC_{1}(t)}{dt} = K_{1} C_{A}(t) - (k_{2} + k_{3}) C_{1}(t) + k_{4} C_{2}(t) \\ \frac{dC_{2}(t)}{dt} = k_{3} C_{1}(t) - (k_{4} + k_{5}) C_{2}(t) + k_{6} C_{3}(t) \\ \frac{dC_{3}(t)}{dt} = k_{5} C_{2}(t) - k_{6} C_{3}(t)$

and the differential equation for the total tissue concentration is the sum of individual tissue compartments:

$\frac{dC_{T}(t)}{dt} = \frac{dC_{1}(t)}{dt} + \frac{dC_{2}(t)}{dt} + \frac{dC_{3}(t)}{dt} \ = K_{1} C_{A}(t) - k_{2} C_{1}(t)$

Thus, venous blood concentration, C_V(t), can be calculated as

$C_{V}(t) = C_{A}(t) - \frac{K_{1} C_{A}(t) - k_{2} C_{1}(t)}{f}$

Program sim_av can be used for this purpose. The hand contains several tissue types, and a comprehensive simulation would require simulation for each tissue type, and averaging the curves weighted by their relative perfusion.

Rate constant K₁ is dependent on blood flow (f) and permeability (P×S), and k₂ is defined in terms of K₁ and distribution volume of the radiotracer in the first tissue compartment (V₁):

$K_{1} = f \: \left(1 - e^{-\frac{PS}{f}} \right) \\ k_{2} = \frac{K_{1}}{V_1}$

When K₁ is strongly limited by the P×S of the capillary endothelium, as is the case in central nervous system because of the blood-brain barrier, it can be assumed that the first tissue compartment (C₁) represents the interstitial and intracellular spaces, and the second tissue compartment (C₂) represents either a metabolic or receptor-bound compartment inside the tissue, and the third tissue compartment is usually not needed.

Perfusion limited uptake

If P×S >> f (blood flow limited uptake), then e^-PS/f=0 and K₁=f. Accordingly, k₂=f/V₁, and C_V(t)=C₁(t)/V₁. Assuming one-tissue compartmental model (k₃=k₄=k₅=k₆=0), this would equal the model for radiowater, where all tissue compartments are lumped into one because of the rapid diffusion of water in the tissues.

The liver is an example of an organ where the capillary endothelium has large openings and is very permeable to all radiotracers. Therefore, blood flow is the first limiting factor for uptake, and the first tissue compartment (C₁) represents extracellular volume (including vascular and interstitial space). The second tissue compartment (C₂) represents the intracellular space in hepatocytes, and the third tissue compartment (C₃, if there is any) could represent a metabolic compartment inside hepatocytes. Thus, K₁=f, and k₃ and k₄ represent the transport rates between extracellular and intracellular compartments. Similar model may apply to tissues, such as resting skeletal muscle, where the capillary endothelium is much tighter than in liver, but perfusion is very low.

References

Bergström M, Yates R, Wall A, Kågedal M, Syvänen S, Långström B. Blood-brain barrier penetration of zolmitriptan--modelling of positron emission tomography data. J Pharmacokinet Pharmacodyn. 2006; 33(1): 75-91. doi: 10.1007/s10928-005-9001-1.

Chiou WL. The Phenomenon and rationale of marked dependence of drug concentration on blood sampling site - Implications in pharmacokinetics, pharmacodynamics, toxicology and therapeutics (Part I) Clin Pharmacokinet. 1989a; 17(3): 175-199. doi: 10.2165/00003088-198917030-00004.

Chiou WL. The Phenomenon and rationale of marked dependence of drug concentration on blood sampling site - Implications in pharmacokinetics, pharmacodynamics, toxicology and therapeutics (Part II) Clin Pharmacokinet. 1989b; 17(4): 275-290. doi: 10.2165/00003088-198917040-00005.

Levitt DG. Physiologically based pharmacokinetic modeling of arterial - antecubital vein concentration difference. BMC Clin Pharmacol. 2004; 4:2. doi: 10.1186/1472-6904-4-2.

McGuire EA, Helderman JH, Tobin JD, Andres R, Berman M. Effects of arterial versus venous sampling on analysis of glucose kinetics in man. J Appl Physiol. 1976; 41(4): 565-573. doi: 10.1152/jappl.1976.41.4.565.

Syvänen S, Blomquist G, Appel L, Hammarlund-Udenaes M, Långström B, Bergström M. Predicting brain concentrations of drug using positron emission tomography and venous input: modeling of arterial-venous concentration differences. Eur J Clin Pharmacol. 2006; 62: 839-848. doi: 10.1007/s00228-006-0179-y.

Tags: Input function, Blood, Venous blood, Extraction, Biexponential

Updated at: 2019-02-13
Created at: 2018-11-11
Written by: Vesa Oikonen

Input function: Modelling A-V difference

Simulation of the difference between arterial and venous samples

Perfusion limited uptake

See also:

References