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Positronium is abundantly produced within the molecular voids of a patient's body during positron emission tomography (PET). Its properties dynamically respond to the submolecular architecture of the tissue and the partial pressure of oxygen. Current PET systems record only two annihilation photons and cannot provide information about the positronium lifetime. This study presents the in vivo images of positronium lifetime in a human, for a patient with a glioblastoma brain tumor, by using the dedicated Jagiellonian PET system enabling simultaneous detection of annihilation photons and prompt gamma emitted by a radionuclide. The prompt gamma provides information on the time of positronium formation. The photons from positronium annihilation are used to reconstruct the place and time of its decay. In the presented case study, the determined positron and positronium lifetimes in glioblastoma cells are shorter than those in salivary glands and those in healthy brain tissues, indicating that positronium imaging could be used to diagnose disease in vivo.

Positron emission tomography (PET) is a widely utilized medical imaging modality that uses positron-emitting radiotracers to visualize biochemical processes in a living body. The spatiotemporal distribution of a radiotracer is estimated by detecting the coincidence photon pairs generated through positron annihilations. In human tissue, about 40% of the positrons form positroniums prior to the annihilation. The lifetime of these positroniums is influenced by the microenvironment in the tissue and could provide valuable information for better understanding of disease progression and treatment response. Currently, there are few methods available for reconstructing high-resolution lifetime images in practical applications. This paper presents an efficient statistical image reconstruction method for positronium lifetime imaging (PLI). We also analyze the random triple-coincidence events in PLI and propose a correction method for random events, which is essential for real applications. Both simulation and experimental studies demonstrate that the proposed method can produce lifetime images with high numerical accuracy, low variance, and resolution comparable to that of the activity images generated by a PET scanner with currently available time-of-flight resolution.

Purpose: The aim of this work is to investigate the feasibility of the Jagiellonian Positron Emission Tomography (J-PET) scanner for intra-treatment proton beam range monitoring., Methods: The Monte Carlo simulation studies with GATE and PET image reconstruction with CASToR were performed in order to compare six J-PET scanner geometries. We simulated proton irradiation of a PMMA phantom with a Single Pencil Beam (SPB) and Spread-Out Bragg Peak (SOBP) of various ranges. The sensitivity and precision of each scanner were calculated, and considering the setup's cost-effectiveness, we indicated potentially optimal geometries for the J-PET scanner prototype dedicated to the proton beam range assessment., Results: The investigations indicate that the double-layer cylindrical and triple-layer double-head configurations are the most promising for clinical application. We found that the scanner sensitivity is of the order of 10 -5 coincidences per primary proton, while the precision of the range assessment for both SPB and SOBP irradiation plans was found below 1 mm. Among the scanners with the same number of detector modules, the best results are found for the triple-layer dual-head geometry. The results indicate that the double-layer cylindrical and triple-layer double-head configurations are the most promising for the clinical application, CONCLUSIONS:: We performed simulation studies demonstrating that the feasibility of the J-PET detector for PET-based proton beam therapy range monitoring is possible with reasonable sensitivity and precision enabling its pre-clinical tests in the clinical proton therapy environment. Considering the sensitivity, precision and cost-effectiveness, the double-layer cylindrical and triple-layer dual-head J-PET geometry configurations seem promising for future clinical application., Competing Interests: Declaration of competing interest The Authors declare no conflict of interest., (Copyright © 2024 Associazione Italiana di Fisica Medica e Sanitaria. Published by Elsevier Ltd. All rights reserved.)

Discrete symmetries play an important role in particle physics with violation of CP connected to the matter-antimatter imbalance in the Universe. We report the most precise test of P, T and CP invariance in decays of ortho-positronium, performed with methodology involving polarization of photons from these decays. Positronium, the simplest bound state of an electron and positron, is of recent interest with discrepancies reported between measured hyperfine energy structure and theory at the level of 10 -4 signaling a need for better understanding of the positronium system at this level. We test discrete symmetries using photon polarizations determined via Compton scattering in the dedicated J-PET tomograph on an event-by-event basis and without the need to control the spin of the positronium with an external magnetic field, in contrast to previous experiments. Our result is consistent with QED expectations at the level of 0.0007 and one standard deviation., (© 2024. The Author(s).)

Background: Alongside the benefits of Total-Body imaging modalities, such as higher sensitivity, single-bed position, low dose imaging, etc., their final construction cost prevents worldwide utilization. The main aim of this study is to present a simulation-based comparison of the sensitivities of existing and currently developed tomographs to introduce a cost-efficient solution for constructing a Total-Body PET scanner based on plastic scintillators., Methods: For the case of this study, eight tomographs based on the uEXPLORER configuration with different scintillator materials (BGO, LYSO), axial field-of-view (97.4 cm and 194.8 cm), and detector configurations (full and sparse) were simulated. In addition, 8 J-PET scanners with different configurations, such as various axial field-of-view (200 cm and 250 cm), different cross sections of plastic scintillator, and multiple numbers of plastic scintillator layers (2, 3, and 4), based on J-PET technology have been simulated by GATE software. Furthermore, Siemens' Biograph Vision has been simulated to compare the results with standard PET scans. Two types of simulations have been performed. The first one with a centrally located source with a diameter of 1 mm and a length of 250 cm, and the second one with the same source inside a water-filled cylindrical phantom with a diameter of 20 cm and a length of 183 cm., Results: With regards to sensitivity, among all the proposed scanners, the ones constructed with BGO crystals give the best performance ([Formula: see text] 350 cps/kBq at the center). The utilization of sparse geometry or LYSO crystals significantly lowers the achievable sensitivity of such systems. The J-PET design gives a similar sensitivity to the sparse LYSO crystal-based detectors while having full detector coverage over the body. Moreover, it provides uniform sensitivity over the body with additional gain on its sides and provides the possibility for high-quality brain imaging., Conclusion: Taking into account not only the sensitivity but also the price of Total-Body PET tomographs, which till now was one of the main obstacles in their widespread clinical availability, the J-PET tomography system based on plastic scintillators could be a cost-efficient alternative for Total-Body PET scanners., (© 2023. Springer Nature Switzerland AG.)

In positron emission tomography (PET) studies, convolutional neural networks (CNNs) may be applied directly to the reconstructed distribution of radioactive tracers injected into the patient's body, as a pattern recognition tool. Nonetheless, unprocessed PET coincidence data exist in tabular format. This paper develops the transformation of tabular data into n-dimensional matrices, as a preparation stage for classification based on CNNs. This method explicitly introduces a nonlinear transformation at the feature engineering stage and then uses principal component analysis to create the images. We apply the proposed methodology to the classification of simulated PET coincidence events originating from NEMA IEC and anthropomorphic XCAT phantom. Comparative studies of neural network architectures, including multilayer perceptron and convolutional networks, were conducted. The developed method increased the initial number of features from 6 to 209 and gave the best precision results (79.8) for all tested neural network architectures; it also showed the smallest decrease when changing the test data to another phantom.

Objective: The Jagiellonian PET (J-PET) technology, based on plastic scintillators, has been proposed as a cost effective tool for detecting range deviations during proton therapy. This study investigates the feasibility of using J-PET for range monitoring by means of a detailed Monte Carlo simulation study of 95 patients who underwent proton therapy at the Cyclotron Centre Bronowice (CCB) in Krakow, Poland. 
Approach: Discrepancies between prescribed and delivered treatments were artificially introduced in the simulations by means of shifts in patient positioning and in the Hounsfield unit to the relative proton stopping power calibration curve. A dual-layer, cylindrical J-PET geometry was simulated in an in-room monitoring scenario and a triple-layer, dual-head geometry in an in-beam protocol. The distribution of range shifts in reconstructed PET activity was visualised in the beam's eye view. Linear prediction models were constructed from all patients in the cohort, using the mean shift in reconstructed PET activity as a predictor of the mean proton range deviation. 
Main results: Maps of deviations in the range of reconstructed PET distributions showed agreement with those of deviations in dose range in most patients. The linear prediction model showed a good fit, with coefficient of determination r^2 = 0.84 (in-room) and 0.75 (in-beam). Residual standard error was below 1 mm: 0.33 mm (in-room) and 0.23 mm (in-beam). 
Significance: The precision of the proposed prediction models shows the sensitivity of the proposed J-PET scanners to shifts in proton range for a wide range of clinical treatment plans. Furthermore, it motivates the use of such models as a tool for predicting proton range deviations and opens up new prospects for investigations into the use of intra-treatment PET images for predicting clinical metrics that aid in the assessment of the quality of delivered treatment.&#xD., (Creative Commons Attribution license.)

Background: The Jagiellonian Positron Emission Tomograph is the 3-layer prototype of the first scanner based on plastic scintillators, consisting of 192 half-metre-long strips with readouts at both ends. Compared to crystal-based detectors, plastic scintillators are several times cheaper and could be considered as a more economical alternative to crystal scintillators in future PETs. JPET is also a first multi-photon PET prototype. For the development of multi-photon detection, with photon characterized by the continuous energy spectrum, it is important to estimate the efficiency of J-PET as a function of energy deposition. The aim of this work is to determine the registration efficiency of the J-PET tomograph as a function of energy deposition by incident photons and the intrinsic efficiency of the J-PET scanner in detecting photons of different incident energies. In this study, 3-hit events are investigated, where 2-hits are caused by 511 keV photons emitted in [Formula: see text] annihilations, while the third hit is caused by one of the scattered photons. The scattered photon is used to accurately measure the scattering angle and thus the energy deposition. Two hits by a primary and a scattered photon are sufficient to calculate the scattering angle of a photon, while the third hit ensures the precise labeling of the 511 keV photons., Results: By comparing experimental and simulated energy distribution spectra, the registration efficiency of the J-PET scanner was determined in the energy deposition range of 70-270 keV, where it varies between 20 and 100[Formula: see text]. In addition, the intrinsic efficiency of the J-PET was also determined as a function of the energy of the incident photons., Conclusion: A method for determining registration efficiency as a function of energy deposition and intrinsic efficiency as a function of incident photon energy of the J-PET scanner was demonstrated. This study is crucial for evaluating the performance of the scanner based on plastic scintillators and its applications as a standard and multi-photon PET systems. The method may be also used in the calibration of Compton-cameras developed for the ion-beam therapy monitoring and simultaneous multi-radionuclide imaging in nuclear medicine., (© 2023. The Author(s).)

Objective: This paper reports on the implementation and shows examples of the use of the ProTheRaMon framework for simulating the delivery of proton therapy treatment plans and range monitoring using positron emission tomography (PET). ProTheRaMon offers complete processing of proton therapy treatment plans, patient CT geometries, and intra-treatment PET imaging, taking into account therapy and imaging coordinate systems and activity decay during the PET imaging protocol specific to a given proton therapy facility. We present the ProTheRaMon framework and illustrate its potential use case and data processing steps for a patient treated at the Cyclotron Centre Bronowice (CCB) proton therapy center in Krakow, Poland., Approach: The ProTheRaMon framework is based on GATE Monte Carlo software, the CASToR reconstruction package and in-house developed Python and bash scripts. The framework consists of five separated simulation and data processing steps, that can be further optimized according to the user's needs and specific settings of a given proton therapy facility and PET scanner design., Main Results: ProTheRaMon is presented using example data from a patient treated at CCB and the J-PET scanner to demonstrate the application of the framework for proton therapy range monitoring. The output of each simulation and data processing stage is described and visualized., Significance: We demonstrate that the ProTheRaMon simulation platform is a high-performance tool, capable of running on a computational cluster and suitable for multi-parameter studies, with databases consisting of large number of patients, as well as different PET scanner geometries and settings for range monitoring in a clinical environment. Due to its modular structure, the ProTheRaMon framework can be adjusted for different proton therapy centers and/or different PET detector geometries. It is available to the community via github., (Creative Commons Attribution license.)

In vivo assessment of cancer and precise location of altered tissues at initial stages of molecular disorders are important diagnostic challenges. Positronium is copiously formed in the free molecular spaces in the patient’s body during positron emission tomography (PET). The positronium properties vary according to the size of inter- and intramolecular voids and the concentration of molecules in them such as, e.g., molecular oxygen, O 2 ; therefore, positronium imaging may provide information about disease progression during the initial stages of molecular alterations. Current PET systems do not allow acquisition of positronium images. This study presents a new method that enables positronium imaging by simultaneous registration of annihilation photons and deexcitation photons from pharmaceuticals labeled with radionuclides. The first positronium imaging of a phantom built from cardiac myxoma and adipose tissue is demonstrated. It is anticipated that positronium imaging will substantially enhance the specificity of PET diagnostics.

We perform a parametric study of the newly developed time-of-flight (TOF) image reconstruction algorithm, proposed for the real-time imaging in total-body Jagiellonian PET (J-PET) scanners. The asymmetric 3D filtering kernel is applied at each most likely position of electron-positron annihilation, estimated from the emissions of back-to-back γ-photons. The optimisation of its parameters is studied using Monte Carlo simulations of a 1-mm spherical source, NEMA IEC and XCAT phantoms inside the ideal J-PET scanner. The combination of high-pass filters which included the TOF filtered back-projection (FBP), resulted in spatial resolution, 1.5 times higher in the axial direction than for the conventional 3D FBP. For realistic 10-minute scans of NEMA IEC and XCAT, which require a trade-off between the noise and spatial resolution, the need for Gaussian TOF kernel components, coupled with median post-filtering, is demonstrated. The best sets of 3D filter parameters were obtained by the Nelder-Mead minimisation of the mean squared error between the resulting and reference images. The approach allows training the reconstruction algorithm for custom scans, using the IEC phantom, when the temporal resolution is below 50 ps. The image quality parameters, estimated for the best outcomes, were systematically better than for the non-TOF FBP., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2021. Published by Elsevier B.V.)

Charged lepton system symmetry under combined charge, parity, and time-reversal transformation (CPT) remains scarcely tested. Despite stringent quantum-electrodynamic limits, discrepancies in predictions for the electron-positron bound state (positronium atom) motivate further investigation, including fundamental symmetry tests. While CPT noninvariance effects could be manifested in non-vanishing angular correlations between final-state photons and spin of annihilating positronium, measurements were previously limited by knowledge of the latter. Here, we demonstrate tomographic reconstruction techniques applied to three-photon annihilations of ortho-positronium atoms to estimate their spin polarisation without magnetic field or polarised positronium source. We use a plastic-scintillator-based positron-emission-tomography scanner to record ortho-positronium (o-Ps) annihilations with single-event estimation of o-Ps spin and determine the complete spectrum of an angular correlation operator sensitive to CPT-violating effects. We find no violation at the precision level of 10 -4 , with an over threefold improvement on the previous measurement., (© 2021. The Author(s).)

The purpose of the presented research is estimation of the performance characteristics of the economic total-body Jagiellonian-PET system (TB-J-PET) constructed from plastic scintillators. The characteristics are estimated according to the NEMA NU-2-2018 standards utilizing the GATE package. The simulated detector consists of 24 modules, each built out of 32 plastic scintillator strips (each with cross section of 6 mm times 30 mm and length of 140 or 200 cm) arranged in two layers in regular 24-sided polygon circumscribing a circle with the diameter of 78.6 cm. For the TB-J-PET with an axial field-of-view (AFOV) of 200 cm, a spatial resolutions (SRs) of 3.7 mm (transversal) and 4.9 mm (axial) are achieved. The noise equivalent count rate (NECR) peak of 630 kcps is expected at 30 kBq cc -1 . Activity concentration and the sensitivity at the center amounts to 38 cps kBq -1 . The scatter fraction (SF) is estimated to 36.2 %. The values of SF and SR are comparable to those obtained for the state-of-the-art clinical PET scanners and the first total-body tomographs: uExplorer and PennPET. With respect to the standard PET systems with AFOV in the range from 16 to 26 cm, the TB-J-PET is characterized by an increase in NECR approximately by factor of 4 and by the increase of the whole-body sensitivity by factor of 12.6 to 38. The time-of-flight resolution for the TB-J-PET is expected to be at the level of CRT = 240 ps full width at half maximum. For the TB-J-PET with an AFOV of 140 cm, an image quality of the reconstructed images of a NEMA IEC phantom was presented with a contrast recovery coefficient and a background variability parameters. The increase of the whole-body sensitivity and NECR estimated for the TB-J-PET with respect to current commercial PET systems makes the TB-J-PET a promising cost-effective solution for the broad clinical applications of total-body PET scanners. TB-J-PET may constitute an economic alternative for the crystal TB-PET scanners, since plastic scintillators are much cheaper than BGO or LYSO crystals and axial arrangement of the strips significantly reduces the costs of readout electronics and SiPMs., (Creative Commons Attribution license.)

In this paper we introduce a semi-analytic algorithm for 3-dimensional image reconstruction for positron emission tomography (PET). The method consists of the back-projection of the acquired data into the most likely image voxel according to time-of-flight (TOF) information, followed by the filtering step in the image space using an iterative optimization algorithm with a total variation (TV) regularization. TV regularization in image space is more computationally efficient than usual iterative optimization methods for PET reconstruction with full system matrix that use TV regularization. The efficiency comes from the one-time TOF back-projection step that might also be described as a reformatting of the acquired data. An important aspect of our work concerns the evaluation of the filter operator of the linear transform mapping an original radioactive tracer distribution into the TOF back-projected image. We obtain concise, closed-form analytical formula for the filter operator. The proposed method is validated with the Monte Carlo simulations of the NEMA IEC phantom using a one-layer, 50 cm-long cylindrical device called Jagiellonian PET scanner. The results show a better image quality compared with the reference TOF maximum likelihood expectation maximization algorithm., (Copyright © 2020 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.)

Purpose: The time-over-threshold (TOT) technique is being used widely due to itsimplications in developing the multi-channel readouts, mainly when fast signal processing is required. Using the TOT technique, as a measure of energy loss instead of charge integration methods, significantly reduces the signal readout costs by combining the time and energy information. Therefore, this approach can potentially be utilized in J-PET tomograph which is built from plastic scintillators characterized by fast light signals. The drawback in adopting this technique lies in the non-linear correlation between input energy loss and TOT of the signal. The main motivation behind this work is to develop the relationship between TOT and energy loss and validate it by the J-PET tomograph setup., Methods: The experiment was performed using a 22 Na beta emitter source placed in the center of the J-PET tomograph. This isotope produces photons of two different energies: 511 keV photons from the positron annihilation (direct annihilation or through the formation of a para-positronium atom or pick-off process of ortho-positronium atoms) and a 1275 keV prompt photon. This allows the study of the correlation between TOT values and energy loss for energy ranges up to 1000 keV. Since the photon interacts predominantly via Compton scattering inside the plastic scintillator, there is no direct information of the energy deposition. However, using the J-PET geometry, one can measure the scattering angle of the interacting photon. Since the 22 Na source emits photons of two different energies, it is necessary to know unambiguously the energy of incident photons and their corresponding scattering angles in order to estimate energy deposition. In summary, this work presents a dedicated algorithm developed to tag photons of different energies and studying their scattering angles to calculate the energy deposition by the interacting photons., Results: A new method was elaborated to measure the energy loss by photons interacting with plastic scintillators used in the J-PET tomograph. We find the relationship between the energy loss and TOT is non-linear and can be described by the functions TOT = A0 + A1 * ln(E dep + A2) + A3 * (ln(E dep + A2)) 2 and TOT = A0 - A1 * A2[Formula: see text]. In addition, we also introduced a theoretical model to calculate the TOT as a function of energy loss in plastic scintillators., Conclusions: A relationship between TOT and energy loss by photons interacting inside the plastic scintillators used in J-PET scanner is established for a deposited energy range of 100-1000 keV.

A detection system of the conventional PET tomograph is set-up to record data from [Formula: see text] annihilation into two photons with energy of 511 keV, and it gives information on the density distribution of a radiopharmaceutical in the body of the object. In this paper we explore the possibility of performing the three gamma photons imaging based on ortho-positronium annihilation, as well as the possibility of positronium mean lifetime imaging with the J-PET tomograph constructed from plastic scintillators. For this purposes simulations of the ortho-positronium formation and its annihilation into three photons were performed taking into account distributions of photons' momenta as predicted by the theory of quantum electrodynamics and the response of the J-PET tomograph. In order to test the proposed ortho-positronium lifetime image reconstruction method, we concentrate on the decay of the ortho-positronium into three photons and applications of radiopharmaceuticals labeled with isotopes emitting a prompt gamma. The proposed method of imaging is based on the determination of hit-times and hit-positions of registered photons which enables the reconstruction of the time and position of the annihilation point as well as the lifetime of the ortho-positronium on an event-by-event basis. We have simulated the production of the positronium in point-like sources and in a cylindrical phantom composed of a set of different materials in which the ortho-positronium lifetime varied from 2.0 ns to 3.0 ns, as expected for ortho-positronium created in the human body. The presented reconstruction method for total-body J-PET like detector allows to achieve a mean lifetime resolution of  ∼40 ps. Recent positron annihilation lifetime spectroscopy measurements of cancerous and healthy uterine tissues show that this sensitivity may allow to study the morphological changes in cell structures.

A novel approach to tomographic data processing has been developed and evaluated using the Jagiellonian positron emission tomography scanner as an example. We propose a system in which there is no need for powerful, local to the scanner processing facility, capable to reconstruct images on the fly. Instead, we introduce a field programmable gate array system-on-chip platform connected directly to data streams coming from the scanner, which can perform event building, filtering, coincidence search, and region-of-response reconstruction by the programmable logic and visualization by the integrated processors. The platform significantly reduces data volume converting raw data to a list-mode representation, while generating visualization on the fly.

A novel whole-body positron emission tomography (PET) system based on plastic scintillators is developed by the J-PET Collaboration. It consists of plastic scintillator strips arranged axially in the form of a cylinder, allowing the cost-effective construction of the total-body PET system. In order to determine the properties of the scanner prototype and optimize its geometry, advanced computer simulations were performed using the GATE (Geant4 application for tomographic emission) software. The spatial resolution, sensitivity, scatter fraction and noise equivalent count rate were estimated according to the National Electrical Manufacturers Association norm, as a function of the length of the tomograph, the number of detection layers, the diameter of the tomographic chamber and for various types of applied readout. For the single-layer geometry with a diameter of 85 cm, a strip length of 100 cm, a cross-section of 4 mm  ×  20 mm and silicon photomultipliers with an additional layer of wavelength shifter as the readout, the spatial resolution (full width at half maximum) in the centre of the scanner is equal to 3 mm (radial, tangential) and 6 mm (axial). For the analogous double-layer geometry with the same readout, diameter and scintillator length, with a strip cross-section of 7 mm  ×  20 mm, a noise equivalent count rate peak of 300 kcps was reached at 40 kBq cc -1 activity concentration, the scatter fraction is estimated to be about 35% and the sensitivity at the centre amounts to 14.9 cps kBq -1 . Sensitivity profiles were also determined.

J-PET is a detector optimized for registration of photons from the electron-positron annihilation via plastic scintillators where photons interact predominantly via Compton scattering. Registration of both primary and scattered photons enables to determinate the linear polarization of the primary photon on the event by event basis with a certain probability. Here we present quantitative results on the feasibility of such polarization measurements of photons from the decay of positronium with the J-PET and explore the physical limitations for the resolution of the polarization determination of 511 keV photons via Compton scattering. For scattering angles of about 82 ∘ (where the best contrast for polarization measurement is theoretically predicted) we find that the single event resolution for the determination of the polarization is about 40 ∘ (predominantly due to properties of the Compton effect). However, for samples larger than ten thousand events the J-PET is capable of determining relative average polarization of these photons with the precision of about few degrees. The obtained results open new perspectives for studies of various physics phenomena such as quantum entanglement and tests of discrete symmetries in decays of positronium and extend the energy range of polarization measurements by five orders of magnitude beyond the optical wavelength regime.

We have estimated the Ginzburg number G governing crossover from the asymptotic to the classical critical behaviour near the normal-incommensurate phase transition in [N(CH(3))(4)](2)ZnCl(4) (TMAZC) crystals and the dependence of G on the hydrostatic pressure (0.1-330 MPa), following from the experimental data for the optical birefringence and the quantitative analysis of temperature derivatives of the birefringence in the framework of the approximation of weak Gaussian fluctuations. The Ginzburg number found experimentally for TMAZC is G ∼ 8 × 10(-3) at the atmospheric pressure and a considerable part of it is assumed to refer to structural defects. It is shown that the G value for TMAZC decreases with increasing hydrostatic pressure and, based on analysis of the literature birefringence data for Cs(2)CdBr(4) and Cs(2)HgBr(4) crystals, this behaviour might be expected to be generally typical for all the A(2)BX(4) family. The results obtained are discussed using a phenomenological theory of second-order structural phase transitions. In particular, they provide a basis for estimating the limits of the asymptotic critical region in TMAZC and demonstrate that the major part of the incommensurate phase should belong to the crossover region.