Positron emission tomography (PET) is a unique nuclear medicine test using positron emitters such as ¹⁸F and ¹¹C. In PET tests, various kinds of functional aspects of human bodies can be evaluated by using compounds labeled by these positron emitters. Recently, combined scanners of PET and anatomical imaging modalities such as CT and MRI have been developed and functional information with anatomical location can be easily obtained, increasing the usefulness of PET tests.
PET tests are now essential imaging tools to diagnose various kinds of disease with functional abnormalities. In the field of oncology, ¹⁸F-fluorodeoxy glucose PET tests are routinely used in clinical practice under health insurance. In the field of neurology, PET tests are actively used to investigate cerebral function by labeled neurotransmitters and so on. Currently, brain PET tests to detect beta-amyloid are applied to the diagnosis of dementia. In the field of cardiology, cardiac perfusion and myocardial metabolism are quantitatively measured by using PET and obtained results have successfully revealed the pathogenesis of intractable cardiac diseases. Future technical advances will enhance the usefulness of PET tests more and more.

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¹⁸F-FDG is a most popular radiopharmaceutical for tumor diagnosis in the world. In addition, ¹¹C-methionine, ¹⁸F-FLT and ¹¹C-choline have been used to compensate for drawbacks of ¹⁸F-FDG. Now, novel radiopharmaceuticals are required to estimate or predict therapeutic efficacy because we have many strategies to treat tumors. Radiotherapy which damage DNA by producing free radicals is commonly used to treat various types of tumors. Hypoxia is closely associated with resistance to chemo- and/ or radiotherapy and is a common feature of solid tumors. Recently, understanding of tumor hypoxia in oncology has led to development of radiopharmaceuticals for hypoxia imaging. This review provides an overview of PET radiopharmaceuticals for hypoxia imaging and ¹⁸F-FBPA which is used for boron neutron capture therapy.

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Positron emission tomography (PET) plays important roles in cancer diagnosis, neuroimaging and molecular imaging research; but potential points remain for which big improvements could be made, including spatial resolution, sensitivity and manufacturing costs. Higher spatial resolution is essential to enable earlier diagnosis, and improved sensitivity results in reduced radiation exposure and shortened measurement time. Therefore, research on next generation PET technologies remains a hot topic worldwide. In this paper, innovation and future technologies for the next generation PET scanners, such as time-of-flight measurement and simultaneous PET/MRI measurement, are described. Among them, depth-of-interaction (DOI) measurement in the radiation sensor will be a key technology to get any significant improvement in sensitivity while maintaining high spatial resolution. DOI measurement also has a potential to expand PET application fields because it allows for more flexible detector arrangement. As an example, the world’s first, open-type PET geometry “OpenPET”, which is expected to lead to PET imaging during treatment, is under development. The DOI detector itself continues to evolve with the help of recently developed semiconductor photodetectors, often referred to as silicon photomultipliers.

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Positron emission tomography (PET) is widely used for image diagnostics; making judgments for early diagnostics, differential diagnostics, staging and treatment effect. As for undertaking the large clinical trial and the multicenter study using several diagnostics, the quantitative standardization of PET images is important. We should maintain the safety and the accuracy of daily clinical images. Moreover, we have to develop a safety treatment manual for instruments, apparatus and radiopharmaceuticals in order to produce PET studies of the highest diagnostic accuracy. In addition, daily quality assurance (QA) and quality control (QC) are very important in order to achieve efficiency and safety of PET studies. The importance of the QA and the QC have been recognized from the view of clinical incident protection points. PET will become more advanced in the future, and therefore the QA and the QC for PET images will continue to important in our work. In the view-risk management, we should reaffirm the importance of both QA and QC. Furthermore, we underline the importance of the constant management system and organization in order to gain the quality enhancement of PET imaging.

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Quantitative assessment of 18F-FDG PET can predict treatment responses or outcomes. Here, I briefly describe some world trends in standardizing PET images for image-based assessments of treatment responses, followed by present and future strategies for defining the optimal acquisition conditions for quantitative PET imaging. Finally, information is provided about new technical approaches to improving the quantitation of semi-quantitative indexes such as point spread function, time-of-flight and respiratory gating.

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Because radiotherapy is local treatment, it is very important to define target volume and critical organs based on accurate lesion area. The PET using an index such as the SUV is quantifiable noninvasively with information of the molecular biology for individual case/lesion. In particular, PET with 18F-fluorodeoxyglucose (FDG-PET) has been used for the diagnosis and treatment evaluation of various tumors. The radiation therapy based on PET enables the treatment planning that reflected metabolic activity of the lesion.
The PET produce an error by various factors, therefore, we must handle the PET image in consideration of this error when apply PET to radiotherapy.

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In this issue, symbolic methods for solving differential equations were firstly introduced. Of the symbolic methods, Laplace transform method was also introduced together with some examples, in which this method was applied to solving the differential equations derived from a two-compartment kinetic model and an equivalent circuit model for membrane potential. Second, series expansion methods for solving differential equations were introduced together with some examples, in which these methods were used to solve Bessel’s and Legendre’s differential equations.
In the next issue, simultaneous differential equations and various methods for solving these differential equations will be introduced together with some examples in medical physics.

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