Exponential Fitting Research Articles

Multiscale matrix pencils for separable reconstruction problems

The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the nonlinear parameters. The matrix pencil method, which reformulates the problem statement into a generalized eigenvalue problem for the nonlinear parameters and a structured linear system for the linear parameters, is generally considered as the more stable method to solve the problem computationally. In Section 2 the matrix pencil associated with the classical complex exponential fitting or sparse interpolation problem is summarized and the concepts of dilation and translation are introduced to obtain matrix pencils at different scales. Exponential analysis was earlier generalized to the use of several polynomial basis functions and some operator eigenfunctions. However, in most generalizations a computational scheme in terms of an eigenvalue problem is lacking. In the subsequent Sections 3–6 the matrix pencil formulation, including the dilation and translation paradigm, is generalized to more functions. Each of these periodic, polynomial or special function classes needs a tailored approach, where optimal use is made of the properties of the parameterized elementary or special function used in the sparse interpolation problem under consideration. With each generalization a structured linear matrix pencil is associated, immediately leading to a computational scheme for the nonlinear and linear parameters, respectively from a generalized eigenvalue problem and one or more structured linear systems. Finally, in Section 7 we illustrate the new methods.

A deep learning approach for accurate and robust T1 estimation from cardiac magnetic resonance MOLLI sequences

Abstract Funding Acknowledgements Type of funding sources: None. Background T1 parametric maps of the myocardium are a recent development in cardiac Magnetic Resonance Imaging (MRI). They can be used to characterize the composition of the heart and identify alterations that can be indicative of a pathology. A common method of T1 estimation involves pixel-wise non-linear curve fitting from Modified Look-Locker Inversion Recovery (MOLLI) series [1]. Whilst relatively accurate, this method presents several drawbacks. Firstly, it assumes magnetization recovery to be a purely exponential process dependent on T1, thus neglecting other parameters such as T2 and B1 that can interfere in the process. Secondly, it does not consider that each acquisition during an inversion recovery experiment inverts the magnetization vector by a given angle and thus interferes with the recovery leading to the measurement of a shorter T1* rather than the actual T1 [2]. Additionally, since it considers each pixel individually, neglecting spatial information, it is sensitive to image quality [3]. Finally, since non-linear curve fitting is an iterative algorithm, it is computationally expensive and time-consuming. Methods To address these issues a convolutional neural network, DeepBLESS [4], and two spatially aware variants of it, were trained to predict T1, T2, and B1 with a dataset of MOLLI series simulated from experimental data. The simulation was performed using Bloch simulations and parameters extracted from a dataset composed of 27 single-slice MOLLI series acquired with the 5(3s)3 protocol using a b-SSFP readout at a main magnetic field strength of 3 Tesla. Thus, a dataset could be generated without relying on exponential fitting for ground truth generation. Results The exponential fitting of T1 was shown to consistently underestimate true T1 values by 138 +- 90 ms. In comparison, the spatially naïve DeepBLESS approach showed a mean absolute error of 14 +- 9 ms and the spatially aware DeepBLESS approach (Median-DeepBLESS) showed a mean absolute error of 13 +- 9 ms. As can be seen in Figure 1, Median-DeepBLESS also showed considerably more resilience to noise than exponential fitting and the spatially naïve version of DeepBLESS. Additionally, the deep learning approaches showcased promising results with regards to the estimation of T2 and B1 from MOLLI sequences (See Figure 2). Conclusions It was shown that the Deep Learning approaches eliminated the underestimation bias of exponential fitting with the spatially aware network Median-DeepBLESS being more robust to noise than both exponential fitting and the standard DeepBLESS network for myocardial T1 estimation. Relatively accurate estimation of T2 and B1 from MOLLI sequences was also accomplished through the Deep Learning approach, which may lead to simultaneous T1, T2, and B1 estimation in common clinical practice in the future.

Validation of precipitable water vapor estimates from an inexpensive infrared thermometer

Water vapor is a fundamental component of the Earth's atmosphere with a high spatial and temporal variability. This work studies to what extent low-cost infrared thermometers can infer precipitable water (variable commonly used to characterize atmospheric water vapor). In a calibration process, infrared thermometer readings recorded at Badajoz (Spain) during the 2015–2018 period are compared against precipitable water data measured with reference ground-based Global Navigation Satellite Systems (GNSS) in order to obtain conversion factors through regression analyses considering two exponential fits. After this calibration, using the equation of the best fit, thermometer readings for the year 2019 are transformed into precipitable water estimates. A validation analysis in which these estimates are compared with GNSS measurements yields rms differences of 19% and 17% when normal and seasonal calibration had been employed, respectively. These results are similar (or even better) to those obtained with satellite data. In addition, we explore if certain factors, such as solar elevation, precipitable water content, precipitable water measurements used as reference and equations to convert temperature readings into precipitable water estimates, can significantly affect the quality of the estimates. In view of our results, low-cost infrared thermometers could be used to create an extensive and dense network for a better characterization of the spatial and temporal variability of water vapor.

Using deep learning–derived image features in radiologic time series to make personalised predictions: proof of concept in colonic transit data

ObjectivesSiamese neural networks (SNN) were used to classify the presence of radiopaque beads as part of a colonic transit time study (CTS). The SNN output was then used as a feature in a time series model to predict progression through a CTS.MethodsThis retrospective study included all patients undergoing a CTS in a single institution from 2010 to 2020. Data were partitioned in an 80/20 Train/Test split. Deep learning models based on a SNN architecture were trained and tested to classify images according to the presence, absence, and number of radiopaque beads and to output the Euclidean distance between the feature representations of the input images. Time series models were used to predict the total duration of the study.ResultsIn total, 568 images of 229 patients (143, 62% female, mean age 57) patients were included. For the classification of the presence of beads, the best performing model (Siamese DenseNET trained with a contrastive loss with unfrozen weights) achieved an accuracy, precision, and recall of 0.988, 0.986, and 1. A Gaussian process regressor (GPR) trained on the outputs of the SNN outperformed both GPR using only the number of beads and basic statistical exponential curve fitting with MAE of 0.9 days compared to 2.3 and 6.3 days (p < 0.05) respectively.ConclusionsSNNs perform well at the identification of radiopaque beads in CTS. For time series prediction our methods were superior at identifying progression through the time series compared to statistical models, enabling more accurate personalised predictions.Clinical relevance statementOur radiologic time series model has potential clinical application in use cases where change assessment is critical (e.g. nodule surveillance, cancer treatment response, and screening programmes) by quantifying change and using it to make more personalised predictions.Key Points• Time series methods have improved but application to radiology lags behind computer vision. Colonic transit studies are a simple radiologic time series measuring function through serial radiographs.• We successfully employed a Siamese neural network (SNN) to compare between radiographs at different points in time and then used the output of SNN as a feature in a Gaussian process regression model to predict progression through the time series.• This novel use of features derived from a neural network on medical imaging data to predict progression has potential clinical application in more complex use cases where change assessment is critical such as in oncologic imaging, monitoring for treatment response, and screening programmes.

Exponentially fitted numerical method for solving singularly perturbed delay reaction-diffusion problem with nonlocal boundary condition

ObjectivesIn this article, a singularly perturbed delay reaction-diffusion problem with nonlocal boundary conditions is considered. The exponential fitting factor is introduced to treat the solutions inside the boundary layer which occur due to perturbation parameter. The considered problem has interior layer at s = 1 and strong boundary layers at s = 0 and s= 2. We proposed an exponentially fitted finite difference method to solve the considered problem. The nonlocal boundary condition is treated using Composite Simpson’s frac{1}{3} rule.ResultThe stability and uniform convergence analysis of the proposed approach are established. The error estimation of the developed method is shown to be second-order uniform convergent. Two test examples were carried out to validate the applicability of the developed numerical method. The numerical results reflect the theoretical estimations.

Detection of Ningnanmycin Using Fluorescence Spectroscopy Combined with BP Neural Network.

Ningnanmycin is a new antibiotic pesticide with good bactericidal and antiviral efficacy, which is widely used in the control of fruit and vegetable diseases, and the excessive pesticide residues pose a serious threat to the environment and human health. In this study, we used fluorescence spectrometer to scan the three-dimensional spectrum of ningnanmycin samples. We used a BP neural network to complete the regression analysis of content prediction based on the fluorescence spectra. After that, the prediction performance of the BP neural network was compared with the exponential fitting method. The results of the BP neural network modeling based on the obtained samples showed that the mean square error of the prediction results of the test set is less than 10-4, the R-square is greater than 0.99, the average recovery is 99.11%, and the model performance of the BP neural network is better than exponential fitting. Studies have shown that fluorescence spectroscopy combined with BP neural network can effectively predict the concentration of ningnanmycin.

Thermochromic Pr3+ doped CaMoO4 phosphor with diverse thermal responses for temperature sensing

CaMoO4:Pr3+ thermochromic phosphors with diverse thermal responses for temperature sensing were prepared by the traditional solid-phase reaction method. The typical CaMoO4:Pr3+ had scheelite structure belonging to tetragonal crystal system and space group of I41/a (88). Pr3+ ions can be easily substituted for Ca2+ ions of host CaMoO4 because of similar ionic radius. CaMoO4: 1.5% Pr3+ have the block structure with mean size of 6.84 μm. The Eg (∼3.93 eV) value of pure CaMoO4 is bigger than that (∼3.65 eV) of CaMoO4: 1.5%Pr3+, attributing to the existence of intermediate defect energy levels. Appropriate Pr3+ doping concentration is 1.5%, and the concentration quenching phenomenon can be explained by the concrete electric multipole type of d-d interaction. The emission peak at ∼605 nm from 1D2→3H4 transition have a good thermal stability of 99.452%@423 K, while the wide band centered at ∼490 nm from 3T1,2 → 1A1 transition in the MoO42− complex and 3P0→3H4 transition in Pr3+ have a poor thermal stability of 27.572%@423 K. Calculated activation energy is 0.239 eV. Temperature-dependent FIR for optical thermometry was constructed due to their diverse thermal responses. CaMoO4: Pr3+ phosphor had good relative sensitivities of 2.216%, 0.969% and 0.932% based on FIR of I605 nm/I490 nm with Boltzmann distribution, modified Boltzmann distribution and exponential equation fitting. Thermochromic behavior and thermal quenching mechanism are investigated. The obtained relative sensitivity is better than that of most phosphors, implying that CaMoO4: Pr3+ has a potential for application in optical thermometry.

Fractional exponential fitting backward differential formulas for solving differential equations of fractional order

Fractional exponential fitting backward differential formulas for solving differential equations of fractional order

An optimization method for surface urban heat island footprint extraction based on anisotropy assumption

The footprint (FP) is the main indicator used to quantitatively analyse the surface urban heat island (SUHI) effect. Currently, SUHI FP extraction methods ignore the interference caused by the city shape and are mostly based on the isotropic expansion assumption, resulting in an exaggerated range. Therefore, a new approach for an optimization schema that leverages the assumptions of angle segmentation and anisotropy is proposed herein. This method divides an urban area and its buffer rings into 16 sectors with equal central angles and selects appropriate land surface temperature (LST) curves for an exponential decay fit. The final FP was determined using the SUHI extension distance and the reference rural LST. To verify the effectiveness of this method, SUHI FPs of 27 cities in the central region of the Yangtze River Delta (YRD) were extracted during the summer of 2019, and the results were compared with those derived from traditional methods. The results showed that the FPs calculated using this method were more detailed because they minimized the influence of urban shape, topography, and the surrounding landscape. Moreover, this method played a vital role in revealing the effect of the land use allocation outside urban area on the SUHI effect. This state-of-the-art method is useful for unveiling SUHI spatial distribution patterns and for providing scientific support for land planning for sustainable urban development.

High-Precision State of Charge Estimation of Urban-Road-Condition Lithium-Ion Batteries Based on Optimized High-Order Term Compensation-Adaptive Extended Kalman Filtering

It is significant to improve the accuracy of estimating the state of charge (SOC) of lithium-ion batteries under different working conditions on urban roads. In this study, an improved second-order polarized equivalent circuit (SO-PEC) modeling method is proposed. Accuracy test using segmented parallel exponential fitting parameter identification method. Online parameter identification using recursive least squares with variable forgetting factors(VFFRLS). An optimized higher-order term compensation-adaptive extended Kalman filter (HTC-AEKF) is proposed in the process of estimating SOC. The algorithm incorporates a noise-adaptive algorithm that introduces noise covariance into the recursive process in real-time to reduce the impact of process noise and observation noise on the accuracy of SOC estimation. Multiple iterations are performed for some of the processes in the extended Kalman filter(EKF) to compensate for the accuracy impact of missing higher-order terms in the linearization process. Model validation results show over 98% accuracy. The results after comparing with the EKF algorithm show a 4.1% improvement in SOC estimation accuracy under Hybrid Pulse Power Characterization(HPPC) working conditions. 2.7% improvement in accuracy in Dynamic Stress Test(DST) working conditions. 2.1% improvement in Beijing Bus Dynamic Stress Test(BBDST) working conditions. The superiority of the algorithm is demonstrated.

Time-resolved investigation of nanometric cell membrane patches with a mid-infrared laser microscope

The proton pump Bacteriorhodopsin (BR) undergoes repeated photocycles including reversible conformational changes upon visible light illumination. Exploiting the sensitivity of infrared (IR) spectra to the conformation, we have determined the reaction kinetic parameters of the conductive intermediate M for the wild-type protein and for its slow mutant D96N during its photocycle. Time-resolved IR micro-spectroscopy using an in-house developed confocal laser microscope operating in the mid-IR is employed to record absorption changes of 10−4 at wavelengths λ1 = 6.08 μm and λ2 = 6.35 μm, assigned to backbone and retinal structural modifications, respectively. Protein samples were embedded in dried lipid bilayers deposited on ultraflat gold supports to enhance the surface field. The signals were analyzed according to a simplified photocycle model with only two dominant states: the dark-adapted state BR* and the intermediate M. We obtained the excitation and relaxation times of the intermediate M from exponential fits to the absorption change time traces. Our results constitute a first step towards future plasmonic-assisted nanoscale time-resolved mid-IR spectrometers for the characterization of bioelectronic and light-harvesting nanodevices based on BR.

Universal Evolution of Fickian Non-Gaussian Diffusion in Two- and Three-Dimensional Glass-Forming Liquids.

Recent works show that glass-forming liquids display Fickian non-Gaussian Diffusion, with non-Gaussian displacement distributions persisting even at very long times, when linearity in the mean square displacement (Fickianity) has already been attained. Such non-Gaussian deviations temporarily exhibit distinctive exponential tails, with a decay length λ growing in time as a power-law. We herein carefully examine data from four different glass-forming systems with isotropic interactions, both in two and three dimensions, namely, three numerical models of molecular liquids and one experimentally investigated colloidal suspension. Drawing on the identification of a proper time range for reliable exponential fits, we find that a scaling law λ(t)∝tα, with α≃1/3, holds for all considered systems, independently from dimensionality. We further show that, for each system, data at different temperatures/concentration can be collapsed onto a master-curve, identifying a characteristic time for the disappearance of exponential tails and the recovery of Gaussianity. We find that such characteristic time is always related through a power-law to the onset time of Fickianity. The present findings suggest that FnGD in glass-formers may be characterized by a "universal" evolution of the distribution tails, independent from system dimensionality, at least for liquids with isotropic potential.

Chiral-Controlled Cyclic Chemiluminescence Reactions for the Analysis of Enantiomer Amino Acids

The similarity and complexity of chiral amino acids (AAs) in complex samples remain a significant challenge in their analysis. In this work, the chiral metal-organic framework (MOF)-controlled cyclic chemiluminescence (CCL) reaction is developed and utilized in the analysis of enantiomer AAs. The chiral MOF of d-Co0.75Zn0.25-MOF-74 is designed and prepared by modifying the Co0.75Zn0.25-MOF-74 with d-tartaric acid. The developed chiral bimetallic MOF can not only offer the chiral recognize sites but also act as the catalyst in the cyclic luminol-H2O2 reaction. Moreover, a distinguishable CCL signal can be obtained on enantiomer AAs via the luminol-H2O2 reaction with the control of d-Co0.75Zn0.25-MOF-74. The amplified difference of enantiomer AAs can be quantified by the decay coefficient (k-values) which are calculated from the exponential decay fitting of their obtained CCL signals. According to simulation results, the selective recognition of 19 pairs of AAs is controlled by the pore size of the MOF-74 and their hydrogen-bond interaction with d-tartaric acid on the chiral MOF. Furthermore, the k-values can also be used to estimate the change of chiral AAs in complex samples. Consequently, this chiral MOF-controlled CCL reaction is applied to differentiate enantiomer AAs involved in the quality monitoring of dairy products and auxiliary diagnosis, which provides a new approach for chiral studies and their potential applications.

Exponential fitting of spread of excitation response measurements in cochlear implants

BackgroundHearing performance in cochlear implant (CI) users is variable. An objective measure which can allow a prediction of this performance is desirable. Spread of neural excitation (SoE) curves are an objective measure that can be obtained using the fitting software of cochlear implants and might be able to be used as a predictor. A novel method to interpret SoE curves is presented. New MethodSpread of excitation measurements for three recording sites were fitted using two exponential functions. An asymmetric width measure was developed, defined as the distance in mm to the point, where 25% or 50% of peak normalized amplitude was reached, for each half of the SoE separately. Also, a novel population of subjects with MED-EL CIs is used. Furthermore, speech perception (speech reception threshold, SRT) was evaluated using a matrix sentence test in a multi-source noise field. ResultsSoE width was narrowest for the basal recording site and widest for the apical recording site. Fitted SoE exponential functions were most asymmetric for the apical recording site. A significant positive correlation between sentence test SRT and SoE width at the apical recording site was found. Comparison with Existing MethodsThe use of an asymmetric width measure correlated strongly and positively with speech perception for apical recording sites, unlike the symmetric width measure used in previous studies. Presumably, longer electrodes allow stimulation of a more apical part of the cochlear. At the apical part of the cochlea, dendrites from a large region of the basilar membrane map to a narrow portion on the spiral ganglion, which might explain the observed asymmetry. ConclusionsFor subjects implanted with long electrode arrays, an asymmetric width measure correlates positively with apical SoE distance. However, due to lack of a sufficient amount of data, the results are currently less conclusive and need to be consolidated in a larger cohort of subjects.

18F]FE-PE2I DAT correlates with Parkinson’s disease duration, stage, and rigidity/bradykinesia scores: a PET radioligand validation study

BackgroundCorrelations between dopamine transporter (DAT) availability and Parkinson’s disease (PD) motor symptoms vary depending on the imaging modality, choice of regions of interest and clinical measures. We aimed to validate the PET radioligand [18F]FE-PE2I as a clinical biomarker in PD, hypothesizing negative correlations between DAT availability in specified nigrostriatal regions with symptom duration, disease stage and motor symptom scores.MethodsWe included 41 PD patients (age 45–79 years; H&Y stage < 3) and 37 healthy control subjects in a cross-sectional study with dynamic [18F]FE-PE2I PET. Binding potential (BPND) was estimated in the caudate nucleus, putamen, ventral striatum, sensorimotor striatum, and substantia nigra using the cerebellum as reference region.ResultsWe found negative correlations (p < 0.02) between symptom duration and BPND in the putamen and sensorimotor striatum (rs = − .42; rs = − .51), and between H&Y stage and BPND in caudate nucleus, putamen, sensorimotor striatum, and substantia nigra (rs between − .40 and − .54). The first correlations were better described with exponential fitting. MDS-UPDRS-III in ‘OFF’ state correlated negatively (p < 0.04) with BPND in the sensorimotor striatum (rs = − .47), and excluding tremor score also in the putamen (rs = − .45).ConclusionResults are in agreement with earlier findings in in vivo and post-mortem studies and validate [18F]FE-PE2I as a functional PD biomarker for PD severity.Trial registration: EudraCT 2011-0020050, Registered April 26 2011; EudraCT 2017-003327-29, Registered October 08 2017; EudraCT 2017-001585-19, Registered August 2 2017. https://eudract.ema.europa.eu/.

VLF and Ionospheric D‐Region Perturbations Associated With WWLLN‐Detected Lightning in the South Pacific Region

AbstractThe subionospheric early very low frequency (VLF) perturbations observed on NWC (19.8 kHz) navigational transmitter signal monitored at a low‐latitude station, Suva (18.1°S, 178.5°E), Fiji, during campaign periods of November 2011, 2012, and 2014 and December 2014, are presented. Early VLF events are associated with D‐region conductivity changes mainly produced by lightning‐generated transient luminous events (TLEs). Early VLF events occurred both during daytime and nighttime, with a considerably higher occurrence at nighttime. VLF perturbations caused by lightning strokes located up to 100 km off the transmitter‐receiver great circle path (TRGCP) are attributed to narrow‐angle scattering, while lightning strokes 100–500 km off the TRGCP are considered to cause early VLF events by wide‐angle scattering. Using the World Wide Lightning Location Network data, for the first time, we have studied the relationship between the energy of lightning strokes and the level of VLF perturbations. Greater is the energy of lightning, greater would be the strength of the VLF perturbation. However, the low‐energy lightning stroke can also produce a comparable level of perturbation to that of strong lightning. The modeling results of scattered amplitude (M) and echo phase (ϕE) of the unusually long recovery early/fast VLF event showed a better exponential fit (r ∼ 0.9) than the logarithmic fit. Long‐wavelength propagation capability (LWPC) code modeling of nighttime early VLF events considering causative TLE width of 50 km column indicated a decrease in the D‐region reference height (Hʹ) by up to 30 km and an increase in the sharpness factor (β) by 0.25 km−1.

A hybrid mixed finite element method for convection-diffusion-reaction equation with local exponential fitting technique

A new hybrid mixed finite element method is proposed for solving the convection-diffusion-reaction equation with local exponential fitting technique. In this method, the exponential fitting technique in [7] is used to discretize the convection-diffusion-reaction equation by introducing a new variable at element level; then the hybrid mixed discontinuous Galerkin finite element method is used to approximate the discretized problem. The convergence of the proposed method is analyzed, and the corresponding a priori error estimate is derived. The numerical results are presented to confirm our theoretical analysis.

Novel apparatus for monitoring simultaneously color and °Brix of selected fruit juices through RGB image analysis

AbstractThis work introduces a novel apparatus for monitoring both color and °Brix of selected fruit juices. These parameters are essential for quality and processing control at concentration and dilution operations. The proposed apparatus performs the red, green, and blue (RGB) image analysis of the fruit juice as it flows through a transparent cell equipped with a webcam and a light source. The viewing cell is inside a box for controlling temperature and for ensuring homogeneous lighting. This apparatus not only measures the RGB color space of the fruit juice online but also allows relating it to the °Brix. The relationship between RGB and °Brix was obtained by measuring the RGB color space of various fruit juice samples, of which the °Brix were previously determined using a digital refractometer. RGB image analysis showed an excellent relationship with °Brix (R2 &gt; 0.99, using an exponential fit), allowing for monitoring reliably this parameter during continuous fruit juice concentration and dilution experiments. These results show the potential of this novel apparatus to substitute conventional instruments such as colorimeters and refractometers for online process monitoring. Furthermore, it has the advantage of not requiring sampling, which reduces measurement errors and saves time.Practical applicationsThis work introduces a novel apparatus based on RGB image analysis for monitoring simultaneously the color and the °Brix of fruit juices. These parameters are important to control the production and quality of fruit juices, especially at concentration and dilution operations. The apparatus proposed here is a viewing cell where the RGB color space is quantified using the software RGBview® as the fruit juice flows through it. The °Brix are determined from the RGB color space using a suitable fit obtained from a previous calibration with juice samples with known °Brix. This is an alternative to conventional analytical instruments such as colorimeters and refractometers but with the advantage of performing an online measurement. This configuration allows for the reduction of analytical errors associated with sampling and saves time. Moreover, this apparatus has the potential to monitor other physicochemical properties related to the color of fruit juices.

Low Prevalence of Late Myocardial Injury on Cardiac MRI Following COVID-19 Infection.

The prevalence of abnormal cardiac magnetic resonance imaging (MRI) findings indicative of myocardial injury in patients who recovered from coronavirus disease 2019 (COVID-19) is currently unclear, with a high variability in the reported prevalence. To assess the prevalence of myocardial injury after a COVID-19 infection. Prospective, bicentric study. Seventy consecutive patients who recovered from COVID-19 and were previously hospitalized. Mean age was 57 years and 39% of the patients were female. Ten healthy controls and a comparator group of 75 nonischemic cardiomyopathy (NICM) patients were employed. 1.5-T, steady-state free precession (SSFP) gradient-echo sequence, modified Look-Locker inversion recovery sequence with balanced SSFP readout, T2-prepared spiral readout sequence and a T1-weighted inversion recovery fast gradient-echo sequence was acquired ~4-5 months after recovery from COVID-19. The SSFP sequence was utilized for the calculation of left and right ventricular volumes and ejection fractions (LVEF and RVEF) following manual endocardial contouring. T1 and T2 mapping was performed by pixel-wise exponential fitting, and T1 and T2 values were computed by manual contouring of the left ventricular endocardial and epicardial walls. Late gadolinium enhancement (LGE) images were graded qualitatively as LGE present or absent. T-tests and the χ2 or Fisher's exact tests were used to compare continuous and categorical variables respectively between the COVID-19 and NICM groups. Inter-rater agreement was evaluated by the intraclass correlation coefficient for continuous variables and Cohen's kappa test for LGE. Reduced RVEF occurred in 10%, LGE and elevated native T1 in 9%, reduced LVEF in 4%, and elevated T2 in 3% of COVID-19 patients, respectively. Patients with NICM had lower mean LVEF (41.6% ± 6% vs. 60% ± 7%), RVEF (46% ± 5% vs. 61% ± 9%), and a significantly higher prevalence of LGE (27% vs. 9%) when compared to those post-COVID-19. Abnormal cardiac MRI findings may show a low prevalence in patients who recovered from COVID-19 and were previously hospitalized. 2 TECHNICAL EFFICACY: Stage 2.

Using a Spreadsheet Program to Fit a Gaussian Function

Due to their accessibility and relatively quick learning curve, spreadsheet programs (such as Microsoft Excel and Google Sheets) are a common choice for students when faced with data analysis work. As is well known by most physicists, however, these spreadsheet programs cannot always perform all the tasks that may be necessary for analyzing data. One such area where these programs tend to lag is in nonlinear curve fitting. Most of these programs contain a limited basic set of fit functions, such as linear, polynomial, and exponential fits, to name a few. Outside of these basic functions, one either needs to get creative or turn to a more complex data analysis platform that may have a steep learning curve or cost a considerable amount of money, making it inaccessible to students. In this paper, I will present a method to fit a specific nonlinear curve, a Gaussian function, to a set of collected data using Microsoft Excel.