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Identifying and localising tumor hypoxia in cancer patients is a challenge in oncology imaging. There are many ongoing clinical trials using conventional positron emission tomography (PET) scans and PET agents as cellular markers for detection of tumour hypoxia depending on the concept of the basic physics of 2gamma annihilation. However, Kacperski and Spyrou (2004) proposed, for the first time, to use 3gamma annihilation as a new PET molecular imaging modality, where the positronium and its annihilation, could serve as an oxygen-sensitive marker. The effective yield of 3gamma annihilation depends on the rates of formation and quenching. The formation of positronium is not only sensitive to the physics parameters, but also behaves as an active chemical particle. A hypoxic cell is a microenviromnent which has an inadequate amount of oxygen. Oxygen is known to be a strong positronium quencher where 2gamma annihilation replaces the 3gamma process. It is thus possible for hypoxic cells to be characterised by higher 3gamma rates than those cells which are well oxygenated. The measurement of 3gamma annihilation to differentiate between oxygenated and non-oxygenated biological samples in order to extract useful information in PET for oncology is a challenging project. It opens up very interesting applications in nuclear medicine imaging as the 3gamma yield has not been measured before in biological tissue and in particular hypoxic tumour cells. The project is multidisciplinary involving physics, biology and chemistry. There are many factors which affect the dissolved oxygen in mineral water, defibrinated horse blood and serum samples. These factors and the challenges to prepare in-vitro hypoxic samples have been experimentally measured using polarography with different chemical reactions e.g. carbon dioxide, nitrogen and AnaeroGen(TM). Results have shown that polarography was more suitable than colorimetery in the measurement of dissolved oxygen in blood due to the high absorption characteristic of blood. AnaeroGen(TM) is the method of choice, unlike carbon dioxide or nitrogen treatment, for preparing hypoxic samples due to the good agreement of the behaviour of the oxygen reduction as in the oxyhaemoglobin dissociation curve which is caused by a decrease in pH, an increase in partial pressure of carbon dioxide and an increase in temperature. The 3gamma yield was measured in normoxic and hypoxic environments using the triple coincidence measurement of three high-energy resolution detectors (HPGe). The AnaeroGen(TM) was used to generate a hypoxic environment. The percentage of the coincidence events qualified as 3gamma was 26.5% higher in the hypoxic environment. This reseach work is a step towards the application of the novel modality of 3gamma PET which in conjunction with conventional 2gamma PET could serve as a non-invasive oxygen sensitive marker. The combination of 3gamma and 2gamma coincidences in nuclear medicine imaging systems may contribute important information for the development and validation of appropriate hypoxia markers. The relative 3gamma/2gamma yield was measured for a positron emitter 22Na with the new generation of scintillator LaC13:Ce and LaBr3:Ce detectors, which had been characterised together with NaI(Tl) and HPGe detectors. The experimental focus was on measuring the relative 3gamma/2gamma yield in different samples by applying the peak-to-peak and the peak-to-valley methods. The value of the ratio 3gamma/2gamma depends on the specimen and is of the order of 10-2. The relative 3gamma/2gamma yields obtained for the peak to peak method in the silica sample were, for example, (3.41+/-0.18)x10 -2, (2.98+/-0.13)x10-2, (4.01+/-0.16)x10 -2 and (2.12+/-0.14)x10-2 for LaBr3:Ce, LaCl3:Ce, NaI(Tl) and HPGe detectors, respectively. The results show that the lanthanum based crystals, LaBr3:Ce and LaCl3:Ce, have the potential to replace NaI(Tl) and HPGe due to both good energy resolution and good detection efficiency and can be the scintillator of choice for determining the yield of 3gamma. The peak-to-valley method was applied to measure the relative yield of 3gamma/2gamma positron annihilation using 18F in 11 haematological samples of different oxygenation levels. The relative 3gamma/2gamma yield was found to vary as much as 11% between the components investigated.