Fitting the blood-to-plasma or plasma-to-blood ratios

In the analysis of PET data, conversion of blood to plasma data is often needed. For the conversion, we need to know the dynamic relation between blood and plasma concentrations, and therefore the concentration in blood and plasma must be measured from blood samples taken during the PET scan. The plasma must be separated from blood, and the radioactivity and mass of both plasma and blood samples (or the precipitant left in the tube after most of plasma is removed) must be measured. Radioactivity measurement is hampered by the fast decay of radioactivity, especially with C-11 labelled radiopharmaceuticals. Fitting of a mathematical function to the measured ratios reduces variation and enables interpolation and extrapolation of the ratio curve, for the conversion of blood TAC to plasma.

Individually fitted ratios can be applied in the conversion of blood TAC to plasma using taccalc, but often a population mean curve is determined and implemented in programs b2plasma and p2blood.

The plasma-to-blood and blood-to-plasma ratio curves can be very different for different radiotracers, requiring development of a model of its own for each radiotracer, or empirically selecting a suitable function.

Erythrocytes (red blood cells, RBCs) are the main cellular component of blood, and the concentration difference between plasma and blood is dependent on partitioning of radiopharmaceutical and its labelled metabolites between water spaces in plasma and RBCs, and the rate of transport across RBC membrane. Binding to plasma proteins has major impact on the distribution, as well as binding to RBC proteins (mainly haemoglobin). RBC-to-plasma ratio (r_RBC/P) as function of time determines the blood-to-plasma ratio, but concentrations can be directly measured only from plasma and blood samples, not from RBCs. RBC-to-plasma ratio can be converted to plasma-to-blood ratio with equation

${r_{P/B}} = \frac{1}{(1 - HCT) + HCT \times r_{RBC/P}}$

, where HCT is the haematocrit. Plasma-to-blood ratio is preferred to blood-to-plasma ratio, because divider is not zero if either plasma or RBCs contain any radioactivity. Most radiotracers are administered intravenously into the plasma space, and in that case RBC-to-plasma ratio at zero time is zero, and plasma-to-blood ratio is only dependent on the haematocrit:

${r_{P/B}} = \frac{1}{1 - HCT}$

This is also the maximum value that the plasma-to-blood curve can get at any time point. In the opposite case, if practically all radioactivity is found in RBCs ([¹⁵O]CO, [¹⁵O]O₂), r_RBC/P is very high, and then r_P/B approaches zero. HCT cannot be much higher than 0.5, and thus r_P/B must be between 0 and ∼2.

After intravenous administration, radiotracer immediately starts to form equilibrium with the constituents of the blood, and at least partial equilibrium is achieved before bolus reaches the blood sampling site. While radiolabelled metabolites usually appear in the blood at sampling site only after one full circulation, requiring a delay time term in functions and models for fraction of metabolites in plasma, the delay time term is not needed in functions for RBC-to-plasma ratio.

Models

Time-dependent plasma-to-blood ratio can be described using compartmental models. For instance, Lee et al (2008) proposed a simple model for [¹⁸F]FDG rat studies. Asselin et al (2002) developed a unified compartment model for 6-[¹⁸F]fluoro-L-meta-tyrosine and its metabolites in blood and partitioning between plasma and blood cells.

Measurement of rapid concentration changes in blood and plasma soon after radiotracer administration is difficult and prone to sampling and timing errors. Unfortunately the modelling of radiotracer kinetics between plasma and blood cells has reached little attention, and in most studies only simplistic empiric functions, at best, have been applied to the data.

Functions

The r_RBC/P curve could be described with variable functions. Simulated r_RBC/P curves based on surge function with recirculation

${r_{RBC/P}(t)} = p_1 t e^{-p_2 t} + \frac{p_3 p_1}{(p_{2})^{2}} (1 - e^{-p_2 t} (1 + p_2 t))$

are shown inFigure 1 (left side), and plasma-to-blood ratios calculated from the same curves using Eq 1 are shown on the right side of the figure.

Example RBC-to-plasma ratios, represented with surge function plus recirculation — **Figure 1.** On the left, functions representing typical RBC-to-plasma ratios. On the right, the same functions converted to plasma-to-blood ratios, using *HCT*=0.45. Simulations are stored in GitLab.
The initial phase of radiotracer uptake into RBC (decrease in plasma-to-blood ratio) can be very rapid (see the purple curve). This can pass unnoticed in PET studies because of limited blood samples and continued plasma-to-RBC transport in blood sample tubes.

Example plasma-to-blood ratios, represented with surge function plus recirculation — **Figure 1.** On the left, functions representing typical RBC-to-plasma ratios. On the right, the same functions converted to plasma-to-blood ratios, using *HCT*=0.45. Simulations are stored in GitLab.
The initial phase of radiotracer uptake into RBC (decrease in plasma-to-blood ratio) can be very rapid (see the purple curve). This can pass unnoticed in PET studies because of limited blood samples and continued plasma-to-RBC transport in blood sample tubes.

Program fit_pbr can fit plasma-to-blood ratio curve with this function and HCT.

Exponential or two-exponential function with constant term has sometimes been used for fitting plasma-to-blood or blood-to-plasma ratio curves (Lee et al., 2008; Wu et al., 2016; Mu et al., 2020; Galovic et al., 2021). The two-exponential function for plasma-to-blood ratio

${r_{P/B}(t)} = p_1 e^{-p_2 t} + p_3 (1 - e^{-p_4 t})$

(Mu et al., 2020) can fit similar data as the surge function with recirculation (Eq 1), but parameter limits need to be set carefully to prevent non-physiological fit results, such as initially negative concentration in blood cells, or plasma-to-blood ratio exceeding its maximum 1/(1-HCT).

Similar r_RBC/P curves can also be fitted using rational functions. Program fit_pbr can fit this function and HCT to plasma-to-blood data:

${r_{RBC/P}(t)} = \frac{p_1 t + p_3 t^2 + p_5 t^3}{1 + p_2 t + p_4 t^2 + p_6 t^3}$

Usually, at least the terms p₅ and p₆ can be set to 0. Examples of simple r_RBC/P curves that can be fitted with rational function are shown in Figure 2 (left side), and plasma-to-blood ratios calculated from the same curves are shown on the right side of the figure.

Example RBC-to-plasma ratios, represented with rational function — **Figure 2.** On the left, rational functions representing typical RBC-to-plasma ratios. On the right, the same functions converted to plasma-to-blood ratios, using *HCT*=0.45.

Example plasma-to-blood ratios, calculated from rational functions for RBC-to-plasma ratios — **Figure 2.** On the left, rational functions representing typical RBC-to-plasma ratios. On the right, the same functions converted to plasma-to-blood ratios, using *HCT*=0.45.

More complicated data can be fitted with this function by freeing the parameter p₅. However, rational functions can easily result into non-physiological results and the fits cannot be used to extrapolate the ratio.

Complex RBC-to-plasma ratio curves can also be described with "model 2" function by Feng et al (1993), which is a combination of the surge function and exponentials:

$r_{RBC/P}(t) = \left(A_1 t - A_2 - A_3 \right) e^{\lambda_1 t} + A_2 e^{\lambda_2 t} + A_3 e^{\lambda_3 t}$

and, like before, with haematocrit this function can be used to fit plasma-to-blood ratio curves (examples in Fig 3). Program fit_pbr can fit this function with option -FM2.

Example RBC-to-plasma ratios, represented with Feng M2 function — **Figure 3.** On the left, Feng M2 functions representing RBC-to-plasma ratios. On the right, the same functions converted to plasma-to-blood ratios, using *HCT*=0.50.

Example plasma-to-blood ratios, calculated from Feng M2 functions for RBC-to-plasma ratios — **Figure 3.** On the left, Feng M2 functions representing RBC-to-plasma ratios. On the right, the same functions converted to plasma-to-blood ratios, using *HCT*=0.50.

Due to its many parameters and usually few samples, some of the parameters may need to be constrained to population means. Constraining parameter A₁ to zero leaves function

$r_{RBC/P}(t) = A_2 \left( e^{\lambda_2 t} - e^{\lambda_1 t} \right) + A_3 \left( e^{\lambda_3 t} - e^{\lambda_1 t} \right)$

, which too may be useful for some radiotracers.

Edison et al (2009) used extended Hill (sigmoidal) function for fitting [¹¹C]PIB plasma-to-blood ratios (r_P/Bs), and Bloomfield et al (2016) applied it to [¹¹C]PBR28:

$r_{P/B}(t)=1 - \frac{p_1 + p_2 \times t}{(\frac{p_3}{t})^{p_4} + 1}$

, where p₁ < 0, and t is normalised time (sample time divided by study length, thus t < 1). Function value approaches 1 at very small t, which means that concentration in blood cells at administration time is assumed to be the same as in plasma.

Somewhat similar function was used to fit blood-to-plasma ratio curves by Tarkia et al (2012):

$r_{B/P}(t)=\frac{ {p_1} \times {{t}^{p_2}} }{ {{t}^{p_2}} + p_3 } + p_4$

With this function at t=0 the blood-to-plasma ratio is p₄, which thus can be fixed to 1-HCT, if HCT is measured, and we assume that the concentration in blood cells is initially zero. For RBC-to-plasma ratio the function can be written as

$r_{RBC/P}(t)=\frac{ {p_1} \times {{t}^{p_2}} }{ {{t}^{p_2}} + p_3 }$

from which the plasma-to-blood ratio can be calculated using Eq 1 and haematocrit. This function can be fitted to plasma-to-blood ratio data with program fit_pbr.

Example plasma-to-blood ratios, represented with Hill function — **Figure 4.** Examples of plasma-to-blood ratio curves that can be fitted with Hill function. These may be useful when a radioactive metabolite is taken up by red blood cells while the parent tracer is not.

Software

Plasma-to-blood ratio data may be fitted using fit_pbr. Applications fit_bpr and fit_sigm may be useful in fitting functions to blood-to-plasma or RBC-to-plasma ratio data.

Function parameters are saved into specific fit file format, which are ASCII text files.

Program fit2dat can be used to calculate the fitted ratio curve from function parameter file. This can then be used to for converting blood curve to plasma or vice versa using program taccalc.

Literature

Asselin M-C, Wahl LM, Cunningham VJ, Amano S, Nahmias C. In vivo metabolism and partitioning of 6-[¹⁸F]fluoro-L-meta-tyrosine in whole blood: a unified compartment model. Phys Med Biol. 2002; 47: 1961-1977. doi: 10.1088/0031-9155/47/11/309.

Lee J-S, Su K-H, Lin J-C, Chuang Y-T, Chueh H-S, Liu R-S, Wang S-J, Chen J-C. A novel blood-cell-two-compartment model for transferring a whole blood time activity curve to plasma in rodents. Comput Methods Programs Biomed. 2008; 92(3): 299-304. doi: 10.1016/j.cmpb.2008.02.006.

Tags: Input function, Blood, Plasma, RBC, Fitting

Updated at: 2023-05-09
Created at: 2016-04-06
Written by: Vesa Oikonen

Fitting the blood-to-plasma or plasma-to-blood ratios

Models

Functions

Software

See also:

Literature