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Thermally activated delayed fluorescence enables organic semiconductors with charge transfer-type excitons to convert dark triplet states into bright singlets via reverse intersystem crossing. However, thus far, the contribution from the dielectric environment has received insufficient attention. Here we study the role of the dielectric environment in a range of thermally activated delayed fluorescence materials with varying changes in dipole moment upon optical excitation. In dipolar emitters, we observe how environmental reorganization after excitation triggers the full charge transfer exciton formation, minimizing the singlet-triplet energy gap, with the emergence of two (reactant-inactive) modes acting as a vibrational fingerprint of the charge transfer product. In contrast, the dielectric environment plays a smaller role in less dipolar materials. The analysis of energy-time trajectories and their free-energy functions reveals that the dielectric environment substantially reduces the activation energy for reverse intersystem crossing in dipolar thermally activated delayed fluorescence emitters, increasing the reverse intersystem crossing rate by three orders of magnitude versus the isolated molecule.
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Semiconductores , FluorescenciaRESUMEN
We investigate the charge transport physics of a previously unidentified class of electron-deficient conjugated polymers that do not contain any single bonds linking monomer units along the backbone but only double-bond linkages. Such polymers would be expected to behave as rigid rods, but little is known about their actual chain conformations and electronic structure. Here, we present a detailed study of the structural and charge transport properties of a family of four such polymers. By adopting a copolymer design, we achieve high electron mobilities up to 0.5 cm2 V-1 s-1 Field-induced electron spin resonance measurements of charge dynamics provide evidence for relatively slow hopping over, however, long distances. Our work provides important insights into the factors that limit charge transport in this unique class of polymers and allows us to identify molecular design strategies for achieving even higher levels of performance.
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INTRODUCTION: Interstitial, cartilage, and bone collagens have been proposed as biomarkers of joint deterioration in hemophilic arthropathy. The role of basement membrane (type IV and VIII) collagens as biomarkers of endothelial turnover in relation to acute joint bleeding is not understood. METHODS: Thirty-one adult patients with hemophilia were studied prospectively for 3 years with musculoskeletal ultrasound/power Doppler (MSKUS/PD) during pain-free intervals and painful events for joint bleed status, synovial vascular flow, and 10 plasma markers of collagen turnover. Joint health was determined using Hemophilia Joint Health Scores and Pettersson scores. In animal studies, bleeding was induced in factor VIII-/- mice by knee joint injury. Synovial vascular remodeling was assessed using MSKUS/PD and histology. Murine plasma samples were analyzed for type IV collagen turnover markers. RESULTS: Ninety-one patient visits were compiled. Twenty-five were due to acute painful episodes, with 16 confirmed hemarthroses. Type IV collagen turnover markers (PRO-C4 and C4M), and a type VIII collagen synthesis marker (PRO-C8), were transiently elevated during acute hemarthrosis. Hemarthrosis was accompanied by increased synovial microvascular flow (MSKUS/PD), and levels of type IV collagen markers correlated with PD signals in the joint. In factor VIII-deficient mice, plasma levels of type IV collagen turnover markers correlated negatively with synovial αSMA staining, indicating that reduced type IV collagen turnover was associated with thicker vessels. CONCLUSIONS: Our findings suggest that basement membrane turnover markers, closely linked to synovial vascular remodeling, may be systemic biomarkers of acute hemarthrosis. Vascular instability during neovascularization may be involved in the dynamics of hemarthrosis.
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Hemartrosis , Hemofilia A , Adulto , Animales , Membrana Basal , Biomarcadores , Hemofilia A/complicaciones , Humanos , Ratones , Remodelación VascularRESUMEN
Optoelectronic devices based on conjugated polymers often rely on multilayer device architectures, as it is difficult to design all the different functional requirements, in particular the need for efficient luminescence and fast carrier transport, into a single polymer. Here we study the photophysics of a recently discovered class of conjugated polymers with high charge carrier mobility and low degree of energetic disorder and investigate whether it is possible in this system to achieve by molecular design a high photoluminescence quantum yield without sacrificing carrier mobility. Tracing exciton dynamics over femtosecond to microsecond time scales, we show that nearly all nonradiative exciton recombination arises from interactions between chromophores on different chains. We evaluate the temperature dependence and role of electron-phonon coupling leading to fast internal conversion in systems with strong interchain coupling and the extent to which this can be turned off by varying side chain substitution. By sterically decreasing interchain interaction, we present an effective approach to increase the fluorescence quantum yield of low-energy gap polymers. We present a red-NIR-emitting amorphous polymer with the highest reported film luminescence quantum efficiency of 18% whose mobility concurrently exceeds that of amorphous-Si. This is a key result toward the development of single-layer optoelectronic devices that require both properties.
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Efficient conjugated polymer optoelectronic devices benefit from concomitantly high luminescence and high charge carrier mobility. This is difficult to achieve, as interchain interactions, which are needed to ensure efficient charge transport, tend also to reduce radiative recombination and lead to solid-state quenching effects. Many studies detail strategies for reducing these interactions to increase luminescence, or modifying chain packing motifs to improve percolation charge transport; however achieving these properties together has proved elusive. Here, we show that properly designed amorphous donor-alt-acceptor conjugated polymers can circumvent this problem; combining a tuneable energy gap, fast radiative recombination rates and luminescence quantum efficiencies >15% with high carrier mobilities exceeding 2.4 cm2/Vs. We use photoluminescence from exciton states pinned to close-crossing points to study the interplay between mobility and luminescence. These materials show promise towards realising advanced optoelectronic devices based on conjugated polymers, including electrically-driven polymer lasers.
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The original version of this Article contained an error in the spelling of the author Dan Credgington, which was incorrectly given as Dan Credington. This has now been corrected in both the PDF and HTML versions of the Article.
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OBJECTIVES: Musculoskeletal ultrasound (US) is used increasingly to examine hemophilic arthropathy. However, quantitative algorithms to document findings are lacking. We developed and sought to validate a protocol quantifying hemophilic joint abnormalities. METHODS: Thirty-one patients with hemophilia were examined serially for 2 years with musculoskeletal US (≈600 joint examinations and ≈6000 images). Based on the spectrum of pathologies, a quantitative algorithm, named Joint Tissue Activity and Damage Examination (JADE), was developed for soft tissue and osteochondral measurements, including power Doppler, using nominal group techniques. To study intra- and inter-rater reliability, 8 musculoskeletal US-experienced hemophilia providers performed anatomic landmark recognition and tissue measurements on 86 images with arthropathic changes, with repetition 1 month later. Twenty-three musculoskeletal US-inexperienced providers performed similar assessments. Inter-operator reliability was established by 6 musculoskeletal US-experienced hemophilia providers, each acquiring images and JADE assessments of 3 hemophilic arthropathic joints. A radiologist and musculoskeletal sonographer functioned as adjudicators. The statistical analysis was performed with the intraclass correlation coefficient (ICC), Fleiss κ, and Cohen κ where appropriate. RESULTS: The musculoskeletal US-experienced providers showed excellent intra-and inter-rater reliability for tissue measurements (ICCs, 0.94-0.96). Agreement was good to excellent for landmark recognition (Fleiss κ, 0.87-0.94). Inter-operator reliability was excellent for measurements and landmark recognition (ICC, 0.90; Fleiss κ, 1.0). Agreement with adjudicators was mostly good to excellent. Musculoskeletal US-inexperienced providers showed excellent inter-rater reliability for measurements (ICC, 0.96) and moderate agreement for landmark recognition (Fleiss κ, 0.58). CONCLUSIONS: The JADE protocol appears feasible for quantifying hemophilic intra-articular abnormalities. Musculoskeletal US-trained hemophilia providers showed high intra-rater, inter-rater, and inter-operator reliability, supporting JADE as a protocol for clinical management and research.
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Hemofilia A/complicaciones , Artropatías/complicaciones , Artropatías/diagnóstico por imagen , Ultrasonografía/métodos , Adulto , Femenino , Humanos , Articulaciones/diagnóstico por imagen , Masculino , Persona de Mediana Edad , Reproducibilidad de los ResultadosRESUMEN
CdSe/CdTe core-crown type-II nanoplatelet heterostructures are two-dimensional semiconductors that have attracted interest for use in light-emitting technologies due to their ease of fabrication, outstanding emission yields, and tunable properties. Despite this, the exciton dynamics of these complex materials, and in particular how they are influenced by phonons, is not yet well understood. Here, we use a combination of femtosecond vibrational spectroscopy, temperature-resolved photoluminescence (PL), and temperature-dependent structural measurements to investigate CdSe/CdTe nanoplatelets with a thickness of four monolayers. We show that charge-transfer (CT) excitons across the CdSe/CdTe interface are formed on two distinct time scales: initially from an ultrafast (â¼70 fs) electron transfer and then on longer time scales (â¼5 ps) from the diffusion of domain excitons to the interface. We find that the CT excitons are influenced by an interfacial phonon mode at â¼120 cm-1, which localizes them to the interface. Using low-temperature PL spectroscopy we reveal that this same phonon mode is the dominant mechanism in broadening the CT PL. On cooling to 4 K, the total PL quantum yield reaches close to unity, with an â¼85% contribution from CT emission and the remainder from an emissive sub-band-gap state. At room temperature, incomplete diffusion of domain excitons to the interface and scattering between CT excitons and phonons limit the PL quantum yield to â¼50%. Our results provide a detailed picture of the nature of exciton-phonon interactions at the interfaces of 2D heterostructures and explain both the broad shape of the CT PL spectrum and the origin of PL quantum yield losses. Furthermore, they suggest that to maximize the PL quantum yield both improved engineering of the interfacial crystal structure and diffusion of domain excitons to the interface, e.g., by altering the relative core/crown size, are required.
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Organometal halide perovskites (OHP) are promising materials for low-cost, high-efficiency light-emitting diodes. In films with a distribution of two-dimensional OHP nanosheets and small three-dimensional nanocrystals, an energy funnel can be realized that concentrates the excitations in highly efficient radiative recombination centers. However, this energy funnel is likely to contain inefficient pathways as the size distribution of nanocrystals, the phase separation between the OHP and the organic phase. Here, we demonstrate that the OHP crystallite distribution and phase separation can be precisely controlled by adding a molecule that suppresses crystallization of the organic phase. We use these improved material properties to achieve OHP light-emitting diodes with an external quantum efficiency of 15.5%. Our results demonstrate that through the addition of judiciously selected molecular additives, sufficient carrier confinement with first-order recombination characteristics, and efficient suppression of non-radiative recombination can be achieved while retaining efficient charge transport characteristics.
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Electrically injected charge carriers in organic light-emitting devices (OLEDs) undergo recombination events to form singlet and triplet states in a 1:3 ratio, representing a fundamental hurdle for achieving high quantum efficiency. Dopants based on thermally activated delayed fluorescence (TADF) have emerged as promising candidates for addressing the spin statistics issue in OLEDs. In these materials, reverse singlet-triplet intersystem crossing (rISC) becomes efficient, thereby activating luminescence pathways for weakly emissive triplet states. However, despite a growing consensus that torsional vibrations facilitate spin-orbit-coupling- (SOC-) driven ISC in these molecules, there is a shortage of experimental evidence. We use transient electron spin resonance and theory to show unambiguously that SOC interactions drive spin conversion and that ISC is a dynamic process gated by conformational fluctuations for benchmark carbazolyl-dicyanobenzene TADF emitters.
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OBJECTIVES: Point-of-care musculoskeletal ultrasound (US) is increasingly used by hemophilia providers to guide management; however, pathologic tissue differentiation with US is uncertain. We sought to determine the extent to which point-of-care musculoskeletal US can identify and discriminate pathologic soft tissue changes in hemophilic arthropathy. METHODS: Thirty-six adult patients with hemophilia A/B were prospectively enrolled. Point-of-care musculoskeletal US examinations were performed on arthropathic joints (16 knees, 10 ankles, and 10 elbows) using standard views by a musculoskeletal US-trained and certified hematologist, who recorded abnormal intra-articular soft tissue accumulation. Within 3 days, magnetic resonance imaging was performed using conventional and multiecho ultrashort echo time sequences. Soft tissue identification (synovial proliferation with or without hemosiderin, fat, and/or blood products) was performed by a musculoskeletal radiologist. Findings obtained with both imaging modalities were compared and correlated in a blinded fashion. RESULTS: There was perfect agreement between the modalities on the presence of abnormal soft tissue (34 of 36 cases). However, musculoskeletal US was unable to discriminate between coagulated blood, synovium, intrasynovial or extrasynovial fat tissue, or hemosiderin deposits because of wide variations in echogenicity. CONCLUSIONS: Musculoskeletal US is valuable for point-of-care imaging to determine the presence of soft tissue accumulation in discrete areas. However, because of limitations of musculoskeletal US in discriminating the nature of pathologic soft tissues and detecting hemosiderin, magnetic resonance imaging will be required if such discrimination is clinically important.
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Hemofilia A/complicaciones , Artropatías/complicaciones , Artropatías/diagnóstico por imagen , Articulaciones/diagnóstico por imagen , Ultrasonografía/métodos , Adulto , Anciano , Femenino , Humanos , Masculino , Persona de Mediana Edad , Sistema Musculoesquelético/diagnóstico por imagen , Sistemas de Atención de Punto , Estudios Prospectivos , Índice de Severidad de la Enfermedad , Adulto JovenRESUMEN
We investigate the origin of the broadband visible emission in layered hybrid lead-halide perovskites and its connection with structural and photophysical properties. We study ⟨001⟩ oriented thin films of hexylammonium (HA) lead iodide, (C6H16N)2PbI4, and dodecylammonium (DA) lead iodide, (C12H28N)2PbI4, by combining first-principles simulations with time-resolved photoluminescence, steady-state absorption and X-ray diffraction measurements on cooling from 300 to 4 K. Ultrafast transient absorption and photoluminescence measurements are used to track the formation and recombination of emissive states. In addition to the excitonic photoluminescence near the absorption edge, we find a red-shifted, broadband (full-width at half-maximum of about 0.4 eV), emission band below 200 K, similar to emission from ⟨110⟩ oriented bromide 2D perovskites at room temperature. The lifetime of this sub-band-gap emission exceeds that of the excitonic transition by orders of magnitude. We use X-ray diffraction measurements to study the changes in crystal lattice with temperature. We report changes in the octahedral tilt and lattice spacing in both materials, together with a phase change around 200 K in DA2PbI4. DFT simulations of the HA2PbI4 crystal structure indicate that the low-energy emission is due to interstitial iodide and related Frenkel defects. Our results demonstrate that white-light emission is not limited to ⟨110⟩ oriented bromide 2D perovskites but a general property of this class of system, and highlight the importance of defect control for the formation of low-energy emissive sites, which can provide a pathway to design tailored white-light emitters.
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Self-assembled hybrid perovskite quantum wells have attracted attention due to their tunable emission properties, ease of fabrication, and device integration. However, the dynamics of excitons in these materials, especially how they couple to phonons, remains an open question. Here, we investigate two widely used materials, namely, butylammonium lead iodide (CH3(CH2)3NH3)2PbI4 and hexylammonium lead iodide (CH3(CH2)5NH3)2PbI4, both of which exhibit broad photoluminescence tails at room temperature. We performed femtosecond vibrational spectroscopy to obtain a real-time picture of the exciton-phonon interaction and directly identified the vibrational modes that couple to excitons. We show that the choice of the organic cation controls which vibrational modes the exciton couples to. In butylammonium lead iodide, excitons dominantly couple to a 100 cm-1 phonon mode, whereas in hexylammonium lead iodide, excitons interact with phonons with frequencies of 88 and 137 cm-1. Using the determined optical phonon energies, we analyzed photoluminescence broadening mechanisms. At low temperatures (<100 K), the broadening is due to acoustic phonon scattering, whereas at high temperatures, LO phonon-exciton coupling is the dominant mechanism. Our results help explain the broad photoluminescence line shape observed in hybrid perovskite quantum wells and provide insights into the mechanism of exciton-phonon coupling in these materials.
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OBJECTIVE: Hemophilic arthropathy is associated with pronounced vascular joint remodeling. Also, compared to the general population, PWH have a higher prevalence of hypertension not explained by usual risk factors. As vascular remodeling in various vascular beds is a hallmark of hypertension, we hypothesized that vascular joint remodeling is associated with elevated blood pressures and hypertension. METHODS: Elbows, knees, and ankles of 28 adult PWH were evaluated for vascular abnormalities with MSKUS/PD, as well as for radiographic and clinical status and pain. Logistic and linear regression models were fitted to examine associations between hypertension, blood pressure, and PD score. RESULTS: The extent of vascular abnormalities was associated with hypertension and blood pressures. Hypertensive patients had a higher PD score compared to nonhypertensive patients, and the risk of hypertension increased steeply with PD score. SBP was also strongly associated with PD score, while DBP was only weakly associated. CONCLUSIONS: Vascular remodeling in hemophilic joints is associated with hypertension and elevated blood pressures. As hypertension is a grave risk factor for intracranial hemorrhage, a prominent cause of mortality in hemophilia patients, future studies are needed to address the causal pathways between vascular joint remodeling and blood pressure.
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Hemofilia A/complicaciones , Hipertensión/patología , Remodelación Vascular , Adulto , Humanos , Artropatías , Articulaciones/irrigación sanguínea , Articulaciones/diagnóstico por imagen , Persona de Mediana EdadRESUMEN
Organic light-emitting diodes (OLEDs) promise highly efficient lighting and display technologies. We introduce a new class of linear donor-bridge-acceptor light-emitting molecules, which enable solution-processed OLEDs with near-100% internal quantum efficiency at high brightness. Key to this performance is their rapid and efficient utilization of triplet states. Using time-resolved spectroscopy, we establish that luminescence via triplets occurs within 350 nanoseconds at ambient temperature, after reverse intersystem crossing to singlets. We find that molecular geometries exist at which the singlet-triplet energy gap (exchange energy) is close to zero, so that rapid interconversion is possible. Calculations indicate that exchange energy is tuned by relative rotation of the donor and acceptor moieties about the bridge. Unlike other systems with low exchange energy, substantial oscillator strength is sustained at the singlet-triplet degeneracy point.
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Fundamental understanding of the charge transport physics of hybrid lead halide perovskite semiconductors is important for advancing their use in high-performance optoelectronics. We use field-effect transistors (FETs) to probe the charge transport mechanism in thin films of methylammonium lead iodide (MAPbI3). We show that through optimization of thin-film microstructure and source-drain contact modifications, it is possible to significantly minimize instability and hysteresis in FET characteristics and demonstrate an electron field-effect mobility (µFET) of 0.5 cm2/Vs at room temperature. Temperature-dependent transport studies revealed a negative coefficient of mobility with three different temperature regimes. On the basis of electrical and spectroscopic studies, we attribute the three different regimes to transport limited by ion migration due to point defects associated with grain boundaries, polarization disorder of the MA+ cations, and thermal vibrations of the lead halide inorganic cages.
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OBJECTIVE: Critically ill patients are at high risk of developing pressure ulcers (PU), with the sacrum and heels being highly susceptible to pressure injuries. The objective of our study was to evaluate the clinical effectiveness of a new multi-layer, self-adhesive soft silicone foam heel dressing to prevent PU development in trauma and critically ill patients in the intensive care unit (ICU). METHOD: A cohort of critically ill patients were enrolled at the Royal Melbourne Hospital. Each patient had the multi-layer soft silicone foam dressing applied to each heel on admission to the emergency department. The dressings were retained with a tubular bandage for the duration of the patients' stay in the ICU. The skin under the dressings was examined daily and the dressings were replaced every three days. The comparator for our cohort study was the control group from the recently completed Border Trial. RESULTS: Of the 191 patients in the initial cohort, excluding deaths, loss to follow-up and transfers to another ward, 150 patients were included in the final analysis. There was no difference in key demographic or physiological variables between the cohorts, apart from a longer ICU length of stay for our current cohort. No PUs developed in any of our intervention cohort patients compared with 14 patients in the control cohort (n=152; p<0.001) who developed a total of 19 heel PUs. CONCLUSION: We conclude, based on our results, that the multi-layer soft silicone foam dressing under investigation was clinically effective in reducing ICU-acquired heel PUs. The findings also support previous research on the clinical effectiveness of multi-layer soft silicone foam dressings for PU prevention in the ICU.
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Vendajes , Enfermería de Cuidados Críticos/métodos , Úlcera del Pie/enfermería , Talón/lesiones , Úlcera por Presión/prevención & control , Siliconas/uso terapéutico , Adulto , Anciano , Estudios de Cohortes , Femenino , Humanos , Masculino , Persona de Mediana Edad , Resultado del Tratamiento , Cicatrización de HeridasRESUMEN
Hemophilic arthropathy is a debilitating condition that can develop as a consequence of frequent joint bleeding despite adequate clotting factor replacement. The mechanisms leading to repeated spontaneous bleeding are unknown. We investigated synovial, vascular, stromal, and cartilage changes in response to a single induced hemarthrosis in the FVIII-deficient mouse. We found soft-tissue hyperproliferation with marked induction of neoangiogenesis and evolving abnormal vascular architecture. While soft-tissue changes were rapidly reversible, abnormal vascularity persisted for months and, surprisingly, was also seen in uninjured joints. Vascular changes in FVIII-deficient mice involved pronounced remodeling with expression of α-Smooth Muscle Actin (SMA), Endoglin (CD105), and vascular endothelial growth factor, as well as alterations of joint perfusion as determined by in vivo imaging. Vascular architecture changes and pronounced expression of α-SMA appeared unique to hemophilia, as these were not found in joint tissue obtained from mouse models of rheumatoid arthritis and osteoarthritis and from patients with the same conditions. Evidence that vascular changes in hemophilia were significantly associated with bleeding and joint deterioration was obtained prospectively by dynamic in vivo imaging with musculoskeletal ultrasound and power Doppler of 156 joints (elbows, knees, and ankles) in a cohort of 26 patients with hemophilia at baseline and during painful episodes. These observations support the hypothesis that vascular remodeling contributes significantly to bleed propagation and development of hemophilic arthropathy. Based on these findings, the development of molecular targets for angiogenesis inhibition may be considered in this disease.
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Factor VIII/genética , Hemartrosis/patología , Hemofilia A/patología , Neovascularización Patológica/patología , Remodelación Vascular , Actinas/genética , Actinas/metabolismo , Animales , Tobillo/irrigación sanguínea , Tobillo/patología , Modelos Animales de Enfermedad , Articulación del Codo/irrigación sanguínea , Articulación del Codo/metabolismo , Articulación del Codo/patología , Endoglina , Factor VIII/metabolismo , Expresión Génica , Hemartrosis/genética , Hemartrosis/metabolismo , Hemofilia A/genética , Hemofilia A/metabolismo , Humanos , Péptidos y Proteínas de Señalización Intracelular/genética , Péptidos y Proteínas de Señalización Intracelular/metabolismo , Articulación de la Rodilla/irrigación sanguínea , Articulación de la Rodilla/metabolismo , Articulación de la Rodilla/patología , Ratones , Ratones Endogámicos BALB C , Neovascularización Patológica/genética , Neovascularización Patológica/metabolismo , Factor A de Crecimiento Endotelial Vascular/genética , Factor A de Crecimiento Endotelial Vascular/metabolismoRESUMEN
BACKGROUND CONTEXT: Nonunion of the posterior arch of the atlas is an uncommon but normal developmental variant. It is usually asymptomatic in the patient but may be associated with greater incidence of fracture because of increased stress on the anterior arch. PURPOSE: We sought to determine whether anterior arch hypertrophy is present in cases of congenital nonunion of the posterior arch of the atlas. STUDY DESIGN/SETTING: A retrospective analysis of 1 year (February 2005-January 2006) of computed tomography cervical spine studies requested by the University of California San Diego Medical Center Trauma Department was undertaken. PATIENT SAMPLE: All patients matching the search criteria (see Study design) were included. OUTCOME MEASURES: Area density product, defined as the midline cross-sectional area of the anterior arch on sagittal reformat multiplied by the average areal radiodensity in Hounsfield units (HU) as measured by two raters, was calculated for cases and controls. METHODS: Cases of posterior arch nonunion were identified and matched to controls. The significance of differences in area density product between cases and controls were established by the Student t test. Interrater correlation was calculated. RESULTS: Posterior arch nonunion was identified in 26 individuals (3.1% of 839 studies reviewed). Compared with age- and sex-matched controls, a 21% increase in area density product of the midline anterior arch was observed in posterior arch nonunion cases (773 HU-cm2 in cases vs. 637 HU-cm2 in controls; p<.001). This increase was attributable to a 21% increase in cross-sectional area (1.05 cm2 in cases vs. 0.87 cm2 in controls; p<.002). In contrast, there was no significant difference with regard to increased average radiodensity. CONCLUSIONS: It has long been subjectively recognized but not objectively quantified, until the present study, that the anterior arch of the atlas is hypertrophied in cases of posterior arch nonunion. Anterior arch hypertrophy may represent an adaptive response to chronically elevated mechanical stress and loss of hoop strength in cases of posterior nonunion.