Cluster Size Research Articles

Multiscale approach to the theory of nonisothermal homogeneous nucleation.

Molecular dynamics (MD) of the Lennard-Jones cluster in the environment of supersaturated vapor at fixed temperature and density is used for the investigation of nonisothermal nucleation. The results allow one to single out different processes occurring at different time scales, the Ornstein-Uhlenbeck fluctuations at the short time scale and a combination of slow diffusion and drift of the fluctuation packet that represents a cluster, at the long time scale. The multiscale approach is developed, in which a separate treatment of different time scales makes it possible to consider strongly correlated cluster size and temperature. This reduces the nonisothermal cluster evolution to a one-dimensional problem. The fluctuation packet drift velocity and diffusivity are calculated based on the cluster microscopic thermophysical parameters determined in this work from MD data for isothermal clusters. The proposed approach is consistent with the results of our MD simulation.

Testing for marginal covariate effect when the subgroup size induced by the covariate is informative.

In many cluster-correlated data analyses, informative cluster size poses a challenge that can potentially introduce bias in statistical analyses. Different methodologies have been introduced in statistical literature to address this bias. In this study, we consider a complex form of informativeness where the number of observations corresponding to latent levels of a unit-level continuous covariate within a cluster is associated with the response variable. This type of informativeness has not been explored in prior research. We present a novel test statistic designed to evaluate the effect of the continuous covariate while accounting for the presence of informativeness. The covariate induces a continuum of latent subgroups within the clusters, and our test statistic is formulated by aggregating values from an established statistic that accounts for informative subgroup sizes when comparing group-specific marginal distributions. Through carefully designed simulations, we compare our test with four traditional methods commonly employed in the analysis of cluster-correlated data. Only our test maintains the size across all data-generating scenarios with informativeness. We illustrate the proposed method to test for marginal associations in periodontal data with this distinctive form of informativeness.

Rapid survey for assessing effective cataract surgical coverage (eCSC) and effective refractive error coverage (eREC)-Novel indicators of universal eye health.

As per the recent World Health Organization estimates, approximately 2.2 billion people have near and distance vision impairment (VI) globally, and out of this almost 50% is avoidable. The Rapid Assessment of Avoidable Visual Impairment survey was a cross-sectional study conducted in September 2021, using cluster random sampling in 42 clusters with a cluster size of 140, giving a total of 6000 participants. Two teams comprising of trained optometrists and social workers conducted the ocular examination which included unaided, pinhole, and aided visual acuity assessments followed by examination of the anterior segment and lens. Distance visual acuity was measured using simplified tumbling "E" charts of different sizes for VA of 6/12, 6/18, and 6/60. The lens assessment was done in an un-dilated pupil with torch light by the optometrist. Overall, 6520 individuals aged 6 years and above were enumerated, of whom 5440 (83.4%) were examined. The response rate for examination was better among females (93.1%) than males (73.9%), and it decreased from 93.8% in the age group 6-15 years to 77.1% in the 45+ age group. The prevalence of blindness and VI were 0.18% (95% CI: 0.06-0.29) and 4.19% (95%CI: 3.65-4.72), respectively. The major causes of VI in all age groups were uncorrected refractive error (65.4%), cataract (23.7%), cataract surgical complications (2.6%), corneal opacity (0.4%), and other posterior segment diseases (7.5%). The effective cataract surgical coverage (eCSC) was 61.8%, effective refractive error coverage (eREC) for distance vision was 59.8%, and eREC for near vision was 47.0%. The RAAVI methodology is suitable to measure effective coverage in the general population, both for baseline measurement and periodic monitoring. The 2030 targets for the surveyed district are 90% eCSC and 100% eREC. Such exercises need to be conducted in each district of the country to determine the baseline and target values of effective coverage.

Isolation and Structure Elucidation of JBIR-157, a Skeletally Novel Aromatic Polyketide Produced by the Heterologous Expression of a Cryptic Gene Cluster.

Heterologous expression of natural compound biosynthetic gene clusters (BGCs) is a robust approach for not only revealing the biosynthetic mechanisms leading to the compounds, but also for discovering new products from uncharacterized BGCs. We established a heterologous expression technique applicable to huge biosynthetic gene clusters for generating large molecular secondary metabolites such as type-I polyketides. As an example, we targeted concanamycin BGC from Streptomyces neyagawaensis IFO13477 (the cluster size of 99 kbp), and obtained a bacterial artificial chromosome (BAC) clone with an insert size of 211 kbp that contains the entire concanamycin BGC. Interestingly, heterologous expression for this BAC clone resulted in two additional aromatic polyketides, ent-gephyromycin, and a new compound designated as JBIR-157, together with the expected concanamycin. Bioinformatic and biochemical analyses revealed that a cryptic biosynthetic gene cluster in this BAC clone was responsible for the production of these type-II polyketide synthases (PKS) compounds. Here, we describe the production, isolation, and structure elucidation of JBIR-157, determined primarily by a series of NMR spectral analyses.

Travelling waves for a fast reaction limit of a discrete coagulation-fragmentation model with diffusion and proliferation.

We study traveling wave solutions for a reaction-diffusion model, introduced in the article Calvez et al. (Regime switching on the propagation speed of travelling waves of some size-structured myxobacteriapopulation models, 2023), describing the spread of the social bacterium Myxococcus xanthus. This model describes the spatial dynamics of two different cluster sizes: isolated bacteria and paired bacteria. Two isolated bacteria can coagulate to form a cluster of two bacteria and conversely, a pair of bacteria can fragment into two isolated bacteria. Coagulation and fragmentation are assumed to occur at a certain rate denoted by k. In this article we study theoretically the limit of fast coagulation fragmentation corresponding mathematically to the limit when the value of the parameter k tends to . For this regime, we demonstrate the existence and uniqueness of a transition between pulled and pushed fronts for a certain critical ratio between the diffusion coefficient of isolated bacteria and the diffusion coefficient of paired bacteria. When the ratio is below , the critical front speed is constant and corresponds to the linear speed. Conversely, when the ratio is above the critical threshold, the critical spreading speed becomes strictly greater than the linear speed.

Case-area targeted interventions during a large-scale cholera epidemic: A prospective cohort study in Northeast Nigeria.

Cholera outbreaks are on the rise globally, with conflict-affected settings particularly at risk. Case-area targeted interventions (CATIs), a strategy whereby teams provide a package of interventions to case and neighboring households within a predefined "ring," are increasingly employed in cholera responses. However, evidence on their ability to attenuate incidence is limited. We conducted a prospective observational cohort study in 3 conflict-affected states in Nigeria in 2021. Enumerators within rapid response teams observed CATI implementation during a cholera outbreak and collected data on household demographics; existing water, sanitation, and hygiene (WASH) infrastructure; and CATI interventions. Descriptive statistics showed that CATIs were delivered to 46,864 case and neighbor households, with 80.0% of cases and 33.5% of neighbors receiving all intended supplies and activities, in a context with operational challenges of population density, supply stock outs, and security constraints. We then applied prospective Poisson space-time scan statistics (STSS) across 3 models for each state: (1) an unadjusted model with case and population data; (2) an environmentally adjusted model adjusting for distance to cholera treatment centers and existing WASH infrastructure (improved water source, improved latrine, and handwashing station); and (3) a fully adjusted model adjusting for environmental and CATI variables (supply of Aquatabs and soap, hygiene promotion, bedding and latrine disinfection activities, ring coverage, and response timeliness). We ran the STSS each day of our study period to evaluate the space-time dynamics of the cholera outbreaks. Compared to the unadjusted model, significant cholera clustering was attenuated in the environmentally adjusted model (from 572 to 18 clusters) but there was still risk of cholera transmission. Two states still yielded significant clusters (range 8-10 total clusters, relative risk of 2.2-5.5, 16.6-19.9 day duration, including 11.1-56.8 cholera cases). Cholera clustering was completely attenuated in the fully adjusted model, with no significant anomalous clusters across time and space. Associated measures including quantity, relative risk, significance, likelihood of recurrence, size, and duration of clusters reinforced the results. Key limitations include selection bias, remote data monitoring, and the lack of a control group. CATIs were associated with significant reductions in cholera clustering in Northeast Nigeria despite operational challenges. Our results provide a strong justification for rapid implementation and scale-up CATIs in cholera-response, particularly in conflict settings where WASH access is often limited.

Coarse grained MD simulation of bulk and interfacial behavior of mixture of CTAB/SDS surfactants.

This study involves simulating the process of inhibiting corrosion through the formation of micelles by surfactants and their deposition on iron (Fe) surfaces. The primary focus is on examining CTAB/SDS mixtures in aqueous solutions with different concentrations. Micelle properties, including size, shape, aggregation number, cluster size, and surfactant diffusion, were calculated and validated with experimental data. The coarse-grained Fe surface was modeled and validated against experimental water contact-angle data. Subsequently, the deposition of CTAB/SDS mixtures on the Fe surface and air-water interface was studied systematically. We found that the relative ratio of CTAB/SDS in the solution directly influences surfactant deposition behavior, which might impact the corrosion inhibition efficiency. All the MD simulations were performed using the GROMACS software with MARTINI2 force field and Martini polar water. The molecules are packed using PACKMOL software. Both NVT and NPT simulations are caried out at temperature and pressure of 303 K and 1 bar respectively, with a nonbonded interaction cut-off (rcut) of 1.1 nm. The LJ potential was shifted from 0.9 nm to rcut, while the electrostatic potential was shifted from 0.0 nm to rcut. For electrostatics, reaction-field coulomb type is used, relative dielectric constant (epsilon-r) and the reaction field dielectric constant (epsilon-rf) are equal to 2.5 and infinity respectively. The dielectric constant below rcut is epsilon-r, and beyond the cut-off is epsilon-rf. Coulomb-modifier used as potential-shift which leads to shift in the coulomb potential by a constant such that it is zero at the rcut. This makes the potential of the integral of the force . The neighbor list was updated every 10 steps, employing a neighbor list cut-off equal to rcut. Using a polar water model, we used a constant time step of 0.02 ps throughout the simulation. The used epsilon-r = 2.5, is recommended for polar water.

MODIFICATION OF PINDIGA BENTONITE CLAY WITH DIDODECYLDIMETHYLAMMONIUM BROMIDES SURFACTANT FOR FOUNDRY USE

Bentonite clay from Pindiga, Gombe State, Nigeria, was surface modified with didodecyldimethylammonium bromides (DDAB) by the Base Exchange (cation exchange capacity) method for its possible use in foundries. The surfaces of the unmodified and modified Pindiga bentonite clay were characterised by means of spectroscopic, diffraction, and scanning methods (X-Ray fluorescence spectrometry (XRF)Fourier-transformed infrared spectroscopy (FTIR), X-ray diffraction (XRD), and scanning electron microscopy (SEM).XRF showed that unmodified sample was rich of montmorillonite (MMt) with ca +2 as exchangeable cation, FTIR analysis revealed that unmodified water absorption causes stretching and bending vibrations for hydroxyl groups, with the most intense bands in the low-frequency region, while DDAB organo-bentonites showed slight changes in wave bands, with a strong methyl band and a weak methyl band indicative of long-chain linear aliphatic structure. The basal spacing of the unmodified bentonite sample was 1.52 nm, and after modification with DDAB, it improved to 2.1 nm as determined from the X-ray diffraction (XRD) analysis. Results of the SEM images revealed rougher surfaces with uniformly dispersed small particles of grain-like structures characteristic of the unmodified, while theorgano-modified bentoniterevealed clusters of different sizes and shapes, larger Fe 3+ ions replacing smaller Al 3+ ions at octahedral sites, discontinuous layers enveloping large grains, polygonal growth foams with octahedral and tetrahedral shapes, mixed layers of calcium bentonite, and particle sizes less than 200nm. The study concluded that modified bentonite with didodecyldimethylammonium bromides gives good microstructure, increases rigidity, and improve thermal stability, to the organoclays. Therefore, it is recommended that DDAB bentonite be used as binder for foundries.

Cluster partition-based two-layer expansion planning of grid–resource–storage for distribution networks

In order to realize the optimal planning of grid–resource–storage for distribution networks (DNs) with high penetrated distributed photovoltaics (PVs), a cluster partition-based two-layer expansion planning for DNs is proposed. First, a comprehensive cluster partition index-based cluster partition method is proposed, which involves the indexes such as electrical distance, power balance of the cluster, and cluster size. Second, a cluster partition-based two-layer expansion planning model is proposed. In the upper layer, a line planning model for clusters is established to carry out the planning of cluster connection lines. In the lower layer, a robust source-storage planning model is established with the uncertainty of PVs and loads, and then, the optimal location and capacity of PVs and energy storages (ESs) can be obtained. In addition, the uncertainty regulation parameter is utilized to control the range of uncertainty sets, which can reduce the conservatism of the optimization. Finally, the proposed method is carried out in a real DN in China, which can effectively improve the economy of DN planning.

Probing Isolated Water Molecules in Aqueous Acetonitrile Solutions Using Oxygen K-Edge X-ray Absorption Spectroscopy.

Oxygen K-edge X-ray absorption spectroscopy (XAS) of an aqueous acetonitrile solution exhibited a sharp peak at approximately 537 eV, which was similar to that of water vapor and was not observed in liquid water. The inner-shell spectra of isolated water molecules and water clusters of different sizes surrounded by acetonitrile molecules were obtained by extracting these water structures from the liquid structures of aqueous acetonitrile solutions, as calculated using molecular dynamics simulations. The sharp peak profiles of the O K-edge XAS spectra were derived not from water clusters but from isolated water molecules surrounded by acetonitrile molecules. The present study proposes that isolated water molecules are easily formed in aqueous acetonitrile solutions and that the electronic structures of the isolated water molecules can be analyzed using O K-edge XAS spectra, which separates the contributions of small water clusters.

Ionization Detail Parameters for DNA Damage Evaluation in Charged Particle Radiotherapy: Simulation Study Based on Cell Survival Database

Details of excitation and ionization acts hide a description of the biological effects of charged particle traversal through living tissue. Nanodosimetry enables the introduction of novel quantities that characterize and quantify the particle track structure while also serving as a foundation for assessing biological effects based on this quantification. This presents an opportunity to enhance the planning of charged particle radiotherapy by taking into account the ionization detail. This work uses Monte Carlo simulations with Geant4-DNA code for a wide variety of charged particles and their radiation qualities to analyze the distribution of ionization cluster sizes within nanometer-scale volumes, similar to DNA diameter. By correlating these results with biological parameters extracted from the PIDE database for the V79 cell line, a novel parameter R2 based on ionization details is proposed for the evaluation of radiation quality in terms of biological consequences, i.e., radiobiological cross section for inactivation. By incorporating the probability p of sub-lethal damage caused by a single ionization, we address limitations associated with the usually proposed nanodosimetric parameter Fk for characterizing the biological effects of radiation. We show that the new parameter R2 correlates well with radiobiological data and can be used to predict biological outcomes.

Anomalous criticality coexists with giant cluster in the uniform forest model

We show by extensive simulations that the whole supercritical phase of the three-dimensional uniform forest model simultaneously exhibits an infinite tree and a rich variety of critical phenomena. Besides typical scalings like algebraically decaying correlation, power-law distribution of cluster sizes, and divergent correlation length, a number of anomalous behaviors emerge. The fractal dimensions for off-giant trees take different values when being measured by linear system size or gyration radius. The giant-tree size displays two-length scaling fluctuations, instead of following the central-limit theorem.

Improvements of magnetic nanoparticle assays for SARS-CoV-2 detection using a mimic virus approach

Immunoassays with magnetic nanoparticles (MNPs) as markers are a promising approach for the fast and sensitive virus detection. Upon binding of antibody-functionalized MNP on virus proteins, the hydrodynamic diameter increases and a change in the Brownian relaxation time can be measured. In this study, we detect the whole SARS-CoV-2 by mimicking it with streptavidin-coated polystyrene beads with biotinylated spike proteins. Changes of the MNP dynamics are measured by alternating current susceptometry and magnetic particle spectroscopy. Due to the multiple binding sites of MNP and virus, crosslinking enlarges the change of the hydrodynamic diameter. In order to improve the sensitivity and the limit of detection of the assay, the ratio of the virus to the MNP amount RMV/MNP is investigated in detail. High RMV/MNP ratios lead to a saturation of the MNPs with viruses, so that the cluster size and therefore the sensitivity decrease again. Additionally, it is found that the smallest virus concentrations can be detected for small MNP concentrations. It is also shown that the RMV/MNP range that can be used for an unambiguous detection of viruses depends on the virus/MNP concentration; it shifts with increasing MNP concentration to smaller RMV/MNP values. For very small virus concentrations, an increase of the Brownian relaxation time is detected implying a decrease of the hydrodynamic diameter. Furthermore, the optimal antibody concentration for MNP functionalization was determined. It is also found that a washing process with a centrifuge improves the sensitivity by reliably removing unbound antibodies and eliminating small MNPs with improper functionalization.

Quasi-static filling of a disordered nanoporous medium with a non-wetting liquid as a process of self-organized criticality

The paper shows that the quasi-static process of filling a nanoporous medium with a non-wetting liquid is a process of self-organized criticality. In the system, there are distributed stepwise noncorrelated avalanches, separated both in space and time. In addition, it is shown that in the vicinity of the critical point (the threshold pressure of the filling) and above it the “avalanche” clusters of all possible sizes arise. The revealed mechanism allows to pass further to consideration of anomalously slow relaxation as the SOC process and to understanding of uniform internal mechanisms underlying the observed phenomena in such systems.

Abstract PO5-18-04: Effect of Digoxin on clusters of circulating tumor cells in patients with metastatic breast cancer (DICCT)

Abstract Background: The presence of circulating tumor cell (CTC) clusters is associated with tumor progression and with a bad prognosis in various cancer types1,2. In breast cancer, a pre-clinical model demonstrated that Na/K-ATPase inhibitors such as digoxin could dissolve CTC clusters and reduce metastasis3. DICCT is a single-arm, therapeutic exploratory phase I study aimed to examine whether digoxin can disrupt CTC clusters in metastatic breast cancer patients (NCT03928210). Material and methods: Metastatic breast cancer patients in whom clusters of CTCs have been detected were eligible. Patients with concomitant heart disease were excluded. After progressive disease in any line of therapy, a window of opportunity before initiation of subsequent systemic therapy was used to administer digoxin treatment for seven (7) days. For CTC detection, blood samples were processed with the FDA-cleared Parsortix system (ANGLE plc, UK). Captured CTCs were stained for EPCAM, HER2, EGFR (positive CTC selection) and CD45 (exclusion marker). To gain insight into the cluster-dissolution ability of digoxin, the primary endpoint was size of CTC clusters (i.e. number of cells within each cluster) under digoxin therapy. The cumulative effect of digoxin treatment on the primary endpoint was assessed by a linear regression model. Results: Nine (9) patients with CTC clusters detected at baseline were enrolled. A target level of digoxin above 0.7 ng/ml could be confirmed at day 7 for all patients. For quantitative and qualitative analysis of CTC clusters, 5.5 to 27 ml of blood were analyzed at baseline, at day 0 pre-treatment, and post 2 hours, 3 days and 7 days of digoxin treatment. The average cluster size ranged between 2 and 11.8 cells (median 2.9) considering all samples from all patients. Digoxin treatment reduced cluster size in 8 patients (88.9%) to different extents, and the proportion of CTC cluster over single CTCs was also reduced in 6 patients (66.7%). The effect of digoxin was evident in both homotypic (cancer cells only) and heterotypic (aggregates of CTCs and white blood cells) clusters. The treatment was well tolerated, and no adverse events related to the study treatment occurred. All patients completed the treatment according to the protocol. Conclusion: DICCT provides the first-in-human proof-of-principle that digoxin induces a partial dissolution of CTC clusters in patients with metastatic breast cancer at drug levels that are safe and well tolerated. Conceptualization of follow-up trials including Na/K-ATPase inhibitors and patient outcome endpoints are underway.

Theoretical prediction of low-energy photoelectron spectra of AlnNi- clusters (n = 1-13).

Mixed-metal clusters have long been studied because of their peculiar properties and how they change with cluster size, composition, and charge state and their potential roles in catalysis. The characterization of these clusters is therefore a very important issue. One of the main experimental tools for characterizing their electronic structure is photoelectron spectroscopy. Theoretical computation completes the task by fully determining the structural properties and matching the theoretical predictions to the measured spectra. We present density functional theory computations of the structural, magnetic, and electronic properties of negatively charged mixed AlnNi- clusters with up to 13 Al atoms. The lowest energy structures of the anionic clusters with up to 7 atoms are also found to be low-energy isomers of the neutral counterparts found in the literature. The 13-atom cluster is found to be a quartet and a perfect icosahedron. The predicted photoelectron spectra are also presented and can be used to interpret future experimental data. We also presented indicators that can be used to determine the potential of these systems for single-atom catalysis. These indicators point to smaller clusters to be more reactive as the gap between the Fermi energy and the center of the d-band increases with cluster size and that Ni occupies an internal site for n = 11-13. We speculate that reactivity can be enhanced if one adds an additional Ni atom. The DFT calculations were performed using the Becke exchange and Perdew-Wang/91 correlation functionals (BPW91), a DFT-optimized all-electron basis set for the aluminum atom, and the Stuttgart small core pseudopotential for the Ni atom. All of the computations used the Gaussian 03 software.

Scalable semidefinite programming approach to variational embedding for quantum many-body problems

In quantum embedding theories, a quantum many-body system is divided into localized clusters of sites which are treated with an accurate ‘high-level’ theory and glued together self-consistently by a less accurate ‘low-level’ theory at the global scale. The recently introduced variational embedding approach for quantum many-body problems combines the insights of semidefinite relaxation and quantum embedding theory to provide a lower bound on the ground-state energy that improves as the cluster size is increased. The variational embedding method is formulated as a semidefinite program (SDP), which can suffer from poor computational scaling when treated with black-box solvers. We exploit the interpretation of this SDP as an embedding method to develop an algorithm which alternates parallelizable local updates of the high-level quantities with updates that enforce the low-level global constraints. Moreover, we show how translation invariance in lattice systems can be exploited to reduce the complexity of projecting a key matrix to the positive semidefinite cone.

The Next Generation Virgo Cluster Survey (NGVS). XXVII. The Size and Structure of Globular Cluster Systems and Their Connection to Dark Matter Halos

We study the size and structure of globular cluster (GC) systems of 118 early-type galaxies from the NGVS, MATLAS, and ACSVCS surveys. Fitting Sérsic profiles, we investigate the relationship between effective radii of GC systems (R e,gc) and galaxy properties. GC systems are 2–4 times more extended than host galaxies across the entire stellar mass range of our sample (108.3 M ⊙ < M * < 1011.6 M ⊙). The relationship between R e,gc and galaxy stellar mass exhibits a characteristic “knee” at a stellar mass of M p ≃ 1010.8, similar to the galaxy R e –stellar mass relationship. We present a new characterization of the traditional blue and red GC color subpopulations, describing them with respect to host galaxy (g′−i′) color (Δgi): GCs with similar colors to their hosts have a “red” Δgi, and those significantly bluer GCs have a “blue” Δgi. The GC populations with red Δgi, even in dwarf galaxies, are twice as extended as the stars, suggesting that formation or survival mechanisms favor the outer regions. We find a tight correlation between R e,gc and the total number of GCs, with intrinsic scatter ≲0.1 dex spanning two and three orders of magnitude in size and number, respectively. This holds for both red and blue subpopulations, albeit with different slopes. Assuming that N GC,Total correlates with M 200, we find that the red GC systems have effective radii of roughly 1%–5% R 200, while the blue GC systems in massive galaxies can have sizes as large as ∼10% R 200. Environmental dependence on R e,gc is also found, with lower-density environments exhibiting more extended GC systems at fixed mass.

Novel Transmission Electron Microscopic Method to Quantify Protein Clustering on the Cell Surface.

The cell surface distribution patterns (clustering) of membrane proteins have been widely investigated in cell biology. Here we describe a novel transmission electron microscopic (TEM) protocol designed to improve the quality of information obtained about the protein distribution patterns detected. This novel method makes it possible to study the clustering of all transmembrane proteins on one half of the cytoplasmic membrane of a whole cell. To achieve better imaging, we combine various methods, including critical-point drying, fixation of gold beads with a carbon layer, and a newly developed chemical thinning method. In addition, in our image-processing algorithm, we implemented pair correlation and pair cross-correlation functions, providing more details and better quantitative accuracy in characterizing the size and numbers of possible protein clusters. © 2024 The Authors. Current Protocols published by Wiley Periodicals LLC. Basic Protocol 1: Sample preparation and transmission electron micrography Alternate Protocol: Direct cell labeling for transmission electron micrography Basic Protocol 2: Analysis of TEM images to detect immunogold-labeled proteins.

Sample size and power calculation for testing treatment effect heterogeneity in cluster randomized crossover designs.

The cluster randomized crossover design has been proposed to improve efficiency over the traditional parallel-arm cluster randomized design. While statistical methods have been developed for designing cluster randomized crossover trials, they have exclusively focused on testing the overall average treatment effect, with little attention to differential treatment effects across subpopulations. Recently, interest has grown in understanding whether treatment effects may vary across pre-specified patient subpopulations, such as those defined by demographic or clinical characteristics. In this article, we consider the two-treatment two-period cluster randomized crossover design under either a cross-sectional or closed-cohort sampling scheme, where it is of interest to detect the heterogeneity of treatment effect via an interaction test. Assuming a patterned correlation structure for both the covariate and the outcome, we derive new sample size formulas for testing the heterogeneity of treatment effect with continuous outcomes based on linear mixed models. Our formulas also address unequal cluster sizes and therefore allow us to analytically assess the impact of unequal cluster sizes on the power of the interaction test in cluster randomized crossover designs. We conduct simulations to confirm the accuracy of the proposed methods, and illustrate their application in two real cluster randomized crossover trials.