1 INTRODUCTION

Recently, positron emission tomography (PET) has been more and more widely used for early diagnosis of malignant neoplasms at stages where morphological changes in tissues are not observed yet [1, 2]. Standard PET scanners identify events of two-photon positron annihilation— separation lines of annihilation photons (lines of response, LORs, Eγ = 511 keV). Positrons (е+) are produced in the organism by the β+ decay of the radiotracer (RT) injected in the patient’s blood. A normally used radiotracer is fluorodeoxiglucose, a radiopharmaceutical in which a number of fluorine atoms are substituted with F-18 isotopes. The F-18 nuclei undergo β+ decay (T1/2 = 110 min), emitting positrons with the maximum energy Emax = 633 keV and average initial energy 0.4Emax ≈ 250 keV.

A typical range of these positrons in water is about 1 mm, and their thermalization time is 5 to 10 ps. A thermalized positron annihilates with one of the environment electrons and emits two annihilation photons, each with the energy of 511 keV, or may form a positronium atom Ps, a bound state of е+ and е–. In living tissues and in water, positronium atoms are formed in about 40% of the cases. This process is a combination (sometimes called recombination) of the thermal е+ and one of the secondary (track) electrons resulting from ionization slowing down of a fast positron.

2 RESEARCH METHODS: DETAILS

To go further, the following should be taken into account. Depending on its total spin values (zero or ћ) corresponding to the para (p-Ps) or ortho (о-Ps) states respectively, the positronium atom annihilates into an even number of photons (as a rule, two photons with the energy 511 keV each) or an odd number of photons, usually three. The latter process is almost a thousand times slower. It is clear that the total energy of the annihilation photons is always 1.022 MeV.

However, in condensed matter there is a mechanism that reduces the o-Ps lifetime by a factor of tens. It is the so-called pick-off process, where the constituent positron of o-Ps annihilates not with a “fellow” electron but with an “alien” (molecular) one, whose spin is antiparallel to the е+ spin. In this case, “fast” two-photon annihilation becomes possible. As a result, only a small fraction (0.5%) of all positrons annihilate into three photons, while the major fraction of o-Ps undergoes 2γ annihilation.

2.1 PET Imaging

In a standard PET scanner, the “image” is reconstructed from LORs. Obviously, this 3D image reconstruction procedure turns out to be ill-defined and time-consuming, because positions of annihilation points on the LORs remain unknown. Moreover, this method visualizes the region of a higher radiopharmaceutical concentration than the standardized uptake value (SUV). And how much this higher concentration region correlates with the cell dysmetabolism region or the tumor localization region is a separate question to be specially examined.

2.2 Total Body PET and Ps Imaging

Now traditional PET scanners identify only few percent of all 2γ annihilations occurring in the patient’s organism. Annihilation photons diverge in opposite directions and are detected by coincidence annihilation detectors arranged in a circle (20 to 30 cm wide) around the patient [1, 2]. Recently, in Poland (Jagiellonian University, Cracow) and the United States (California), new annihilation tomographs, Total Body PET (TB-PET) scanners, have been developed, which are capable of detecting annihilation photons emitted into a much larger solid angle (larger scintillator arrangement region) [3, 4]. Thus, almost half of all two-photon annihilation events are detected. It is expected that an additional increase in the lifetime resolution from 200 to 100 ps will allow the PET sensitivity to be effectively increased by a factor of 100–200 [5, 6].

These apparatuses make it possible (i) to reconstruct coordinates of the point where annihilation occurred, (ii) determine the time when е+ annihilation occurred, and (iii) determine the ortho-positronium lifetime and formation probability. This can be done in two ways (see below), each with its particular advantages and disadvantages.

2.3 3γ Annihilation of Ortho-Positronium

An increase in the detection efficiency of annihilation photons and a large solid angle allows one to use the PET imaging technique that involves a rather rare three-photon annihilation channel (0.5% of all annihilation events) of ortho-positronium atoms for determining the o-Ps place of annihillation, lifetime, and formation probability. In this case, both the spatial coordinates of the places where annihilation photons hit the scintillators and the corresponding times of these events are determined [7]. This technique is attractive because it is possible (i) to still use the radiopharmaceutical without the “start” γ quantum accompanying production of the positron (fluorodeoxiglucose) and (ii) to more accurately identify the o-Ps annihilation points (instead of obtaining just the lines in which these points lie).

The three-photon event is initially detected by the TB-PET detectors in terms of the Xi, Yi, Zi, and Ti (i = 1, 2, 3) coordinates referenced to the scanner (hit positions and hit times). These three points at which annihilation photons are detected and the point where o-Ps annihilation occurred lie in one plane (total momentum of annihilation photons is zero). Therefore, one can consider the given annihilation event in the “annihilation” plane or, in other words, go over from the Xi, Yi, Zi, and Ti coordinates to the two-dimensional coordinates xi, yi, and ti (i = 1, 2, 3) of the same annihilation event (Fig. 1).

Fig. 1.
figure 1

(Color online) Example of o-Ps three-photon annihilation (considered in the annihilation plane).

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Solving a set of equations that describes propagation of photons from the o-Ps annihilation (or formation) point to the photon detection points in the scintillators we find two-dimensional coordinates of the o-Ps annihilation point (x, y, t) from the equations

$$\begin{gathered} {{({{x}_{1}} - x)}^{2}} + {{({{y}_{1}}-y)}^{2}} = {{c}^{2}}{{({{t}_{1}} - t)}^{2}}, \hfill \\ {{({{x}_{2}} - x)}^{2}} + {{({{y}_{2}}-y)}^{2}} = {{c}^{2}}{{({{t}_{2}} - t)}^{2}}, \hfill \\ {{({{x}_{3}} - x)}^{2}} + {{({{y}_{3}} - y)}^{2}}~ = {{c}^{2}}{{({{t}_{3}} - t)}^{2}}. \hfill \\ \end{gathered} $$

Note that in [8] a somewhat different procedure was proposed for reconstructing coordinates of the three-photon o-Ps annihilation points. It is based on measuring in each annihilation event the energies E1, E2, and E3 of the three annihilation photons and the coordinates Xi, Yi, Zi, and Ti (i = 1, 2, 3) of their detection points in the detectors. Then, writing the condition for equality of the total momentum of these photons to zero (component by component, three equations), we can reconstruct three coordinates of the point where o-Ps annihilation initially occurred. In this procedure it is not necessary to know the times of arrival of the γ quanta at the detectors (t1, t2, t3) but their energies must be measured.

2.4 Detection of  β+γ–2γ Events. Ortho-Positronium Pick-off Annihilation

If isotopes emitting γ quanta at the time of е+ production (start photons) are used as е+ emitters, it is possible to determine е+ lifetimes (including o-Ps lifetimes), places of е+ formation (or annihilation), and Ps formation probability. An advantage of this method is that a much larger number of annihilation events are used. The Ps formation probability in biological tissues is quite high, ~40%. Furthermore, as was already said, if the o-Ps atom is in a medium, annihilation via the three-photon channel is low probable, and the pick-off process dominates. However, radiotracers that do not emit start photons (e.g., fluorodeoxyglucose) cannot be used in this case, and the so-called β+γ isotopes, e.g., Sc-44, must be utilized.

A complicating factor in this β+γ–2γ approach is the necessity of the energy discrimination of the detected photons. More energetic start photons (for Sc-44, these are γ quanta with the energy of 1.16 MeV) should be separated from less energetic photons (511 keV). To this end, a special time-over-threshold (TOT) method was developed [9]. It is based on the fact that more energetic photons have anode signals elongated in time.

To find the o-Ps annihilation point, the time-of-flight (TOF) method should be used. This can be done using modern TB-PET scanners.

2.5 Ps Imaging

Methods that allow determining both the Ps lifetime and Ps formation probability in a patient’s organism are called Ps imaging [10]. Both of the quantities to be measured depend, first, on the biological liquids/media in which Ps is formed and, second, on the presence of biologically active molecules, e.g., dissolved oxygen, in the close vicinity of Ps and the presence of radiolytic products of radical nature, such as OH radicals and solvated electrons, in the е+ track.

For example, if the Ps atom and the О2 molecule are in a close vicinity, the Ps oxidation reaction (Ps + О2 => е+ + \({\text{O}}_{2}^{ - }\), i.e., transition of an electron from Ps to О2 may occur, because the oxygen molecule is a strong oxidizer). At the same time, O2 may initiate the ortho–para conversion reaction, p-Ps <=> o-Ps, since the О2 molecule is also a strong “converter”, because it has a magnetic moment. Both these processes appreciably decrease the o-Ps lifetime.

On the other hand, it is well known that the O2 concentration is noticeably lower in malignant neoplasms that in healthy tissues [11]. Thus, Ps imaging (based on the o-Ps lifetime measurement) allows gaining additional important information on the О2 concentration in the tissues under study and, therefore, on their structure and metabolism. Thus, one can discriminate healthy tissues from malignant ones in vivo. By now, this possibility has already been confirmed experimentally [12, 13].

Thus, the prospect of improving the spatial resolution of PET and the additional possibilities provided by Ps imaging outweigh the difficulties to be faced on this way.

3 INTERACTION OF Ps WITH О2 IN VARIOUS LIQUIDS: EXPERIMENT

In 2019, we measured positron annihilation lifetime (PAL) spectra in organic liquids (isooctane, cyclohexane, isopropanol) for determining correlation between the o-Ps lifetime and the local concentration of dissolved oxygen [14]. It turned out that the shape of the PAL spectra substantially changed depending on whether the liquid under investigation was bubbled with an inert gas (diluted O2 is completely removed), the air, or pure oxygen (the O2 concentration comes to its maximum). The measurements allowed the reaction rate constants to be found for Ps oxidation and Ps spin ortho–para conversion caused by interaction of Ps with О2 molecules.

To remove dissolved O2 from the liquid phase, we dismissed the use of the standard freeze-thaw method to avoid possible damage of the е+ source under freezing, thawing, and vacuum degassing of the liquid under study. Oxygen was removed by bubbling argon through the liquids. To more accurately determine the Ps–О2 reaction rate constants, we also used bubbling with the air and pure oxygen. The gases were additionally purified by pre-blowing them through dry alkali KОН and then through 0.3-nm Metrohm nanosieves. The gas was supplied to an air-tight vial with the liquid under study using medical needles tightly glued into the vial cap. One to three gas bubbles per second were blown in through a needle.

The positron source was a 10-mm titanium washer 0.2 mm thick with its central part soldered with 10-µm titanium foils between which there was titanium tetrachloride powder that contained the Ti-44 isotope with the activity of 7.5 MBq. The positron source was fixed in the liquid in the vial by a thin stainless-steel clip tightly fastened to the vial cap.

4 DISCUSSION OF RESULTS

Figure 2 shows, as an example, transformation of cyclohexane PAL spectra at room temperature as the oxygen concentration changes. It is seen that the o-Ps lifetime noticeably decreases under bubbling with oxygen (when the O2 concentration is at its maximum). The resulting rate λo of o-Ps annihilation is related to the oxygen concentration \({{c}_{{{{{\text{O}}}_{{\text{2}}}}}}}\) as follows:

$${{\lambda }_{{\text{o}}}}({{c}_{{{{{\text{O}}}_{{\text{2}}}}}}}) = {{\lambda }_{{{\text{po}}}}} + {{k}_{{{\text{Ps}}{\kern 1pt} {\text{ + }}{\kern 1pt} {{{\text{O}}}_{2}}}}}\,{{c}_{{{{{\text{O}}}_{{\text{2}}}}}}}.$$
(1)
Fig. 2.
figure 2

(Color online) Effect of dissolved O2 on the shape of the PAL spectra in cyclohexane.

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Here λpo is the pick-off annihilation rate, and \({{k}_{{{\text{Ps}}{\kern 1pt} {\text{ + }}{\kern 1pt} {{{\text{O}}}_{2}}}}}\) is the total rate constant of the oxidation and spin conversion reactions. Decomposing the PAL spectra into three exponents decreasing with time (with the lifetime resolution of the spectrometer taken into account), we determined \({{\lambda }_{{{\text{po}}}}}({{c}_{{{{{\text{O}}}_{{\text{2}}}}}}})\), Fig. 3. Further, based on the slope of these dependences on the O2 concentration and using Eq. (1), we found the constants \({{k}_{{{\text{Ps}}{\kern 1pt} {\text{ + }}{\kern 1pt} {{{\text{O}}}_{2}}}}}\) and the rates λpo (Table 1).

Fig. 3.
figure 3

(Color online) Dependence of o-Ps annihilation rates on the dissolved oxygen concentration in various liquids.

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Table 1.   Ostwald coefficients \({{c_{{{{{\text{O}}}_{2}}}}^{{(L)}}} \mathord{\left/ {\vphantom {{c_{{{{{\text{O}}}_{2}}}}^{{(L)}}} {c_{{{{{\text{O}}}_{2}}}}^{{(G)}}}}} \right. \kern-0em} {c_{{{{{\text{O}}}_{2}}}}^{{(G)}}}}\) and О2 concentrations in liquids bubbled with oxygen and air at room temperature (20°С) and pressure of 1 atm. Values of \({{k}_{{{\text{Ps + }}{{{\text{O}}}_{2}}}}}\) and λpo are obtained by fitting PAL spectra with equation (1) [14, 15]
Full size table

For 2,3-butanediol, PAL spectra were also measured and o-Ps pick-off annihilation rates were obtained under bubbling with Ar (0.32(1), 1/ns), air (0.32(1), 1/ns), and oxygen (0.33(1), 1/ns). As we do not know solubility of О2 in butanediol, we cannot plot these data here.

It is impossible to separate the rate constants of the oxidation and conversion reactions on the basis of the concentration dependence of the annihilation rate \({{\lambda }_{{\text{o}}}}({{c}_{{{{{\text{O}}}_{{\text{2}}}}}}})\) alone. Therefore, it is impossible to state which of the processes dominates. This requires a more scrupulous analysis of the entire shape of the annihilation spectra rather than only their long-time tail (see Fig. 2) [14].

Concentrations of О2 dissolved in dioxane were found using the Henry constant \({{{\text{H}}}_{{{{{\text{O}}}_{2}}}}}\) = 16.32 kPa m3/mol [15]. If the data [16] are used, the oxygen concentration turns out to be about 1.5 times lower, but these data were omitted in the later issue of the Solubility Data Series (2014).

The rate constant \({{k}_{{{\text{Ps + }}{{{\text{O}}}_{{\text{2}}}}}}}\) as a function of the inverse medium viscosity is shown in Fig. 4. This dependence and its consequences are discussed in more detail in [14].

Fig. 4.
figure 4

(Color online) Linear dependence of the rate constants of the reaction between the ortho-positronium and dissolved oxygen in various liquids on inverse viscosity (see Table 1).

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5 CONCLUSIONS

The data obtained suggest that dissolved oxygen noticeably affects the behavior of the Ps atom in dissolved media. This arises from high solubility of molecular oxygen in those liquids. In pure water, the oxygen effect is the lowest because of low oxygen solubility therein.

Probably, the above-mentioned relationship between the Ps lifetime and the concentration of dis-solved oxygen in the media under investigation (or, in other words, the degree of oxygenation of tissues) will appreciably influence development of positron tomography in the near future. The relevant researches are well under way in Japan [17], Poland, and the United States. It can be believed that in the near future PET scanners will be given additional capabilities for localizing malignant tumors on the basis of precision spatially-resolved measurements of the positronium lifetime and formation probability.