4D maximum a posteriori reconstruction in dynamic SPECT using a compartmental model-based prior

A 4D ordered-subsets maximum a posteriori (OSMAP) algorithm for dynamic SPECT is described which uses a temporal prior that constrains each voxel's behaviour in time to conform to a compartmental model. No a priori limitations on kinetic parameters are applied; rather, the parameter estimates evolve as the algorithm iterates to a solution. The estimated parameters and time-activity curves are used within the reconstruction algorithm to model changes in the activity distribution as the camera rotates, avoiding artefacts due to inconsistencies of data between projection views. This potentially allows for fewer, longer-duration scans to be used and may have implications for noise reduction. The algorithm was evaluated qualitatively using dynamic 99mTc-teboroxime SPECT scans in two patients, and quantitatively using a series of simulated phantom experiments. The OSMAP algorithm resulted in images with better myocardial uniformity and definition, gave time-activity curves with reduced noise variations, and provided wash-in parameter estimates with better accuracy and lower statistical uncertainty than those obtained from conventional ordered-subsets expectation-maximization (OSEM) processing followed by compartmental modelling. The new algorithm effectively removed the bias in k21 estimates due to inconsistent projections for sampling schedules as slow as 60 s per timeframe, but no improvement in wash-out parameter estimates was observed in this work. The proposed dynamic OSMAP algorithm provides a flexible framework which may benefit a variety of dynamic tomographic imaging applications.     

Investigation of dynamic SPECT measurements of the arterial input function in human subjects using simulation, phantom and human studies

Computer simulations, a phantom study and a human study were performed to determine whether a slowly rotating single-photon computed emission tomography (SPECT) system could provide accurate arterial input functions for quantification of myocardial perfusion imaging using kinetic models. The errors induced by data inconsistency associated with imaging with slow camera rotation during tracer injection were evaluated with an approach called SPECT/P (dynamic SPECT from positron emission tomography (PET)) and SPECT/D (dynamic SPECT from database of SPECT phantom projections). SPECT/P simulated SPECT-like dynamic projections using reprojections of reconstructed dynamic (94)Tc-methoxyisobutylisonitrile ((94)Tc-MIBI) PET images acquired in three human subjects (1 min infusion). This approach was used to evaluate the accuracy of estimating myocardial wash-in rate parameters K(1) for rotation speeds providing 180° of projection data every 27 or 54 s. Blood input and myocardium tissue time-activity curves (TACs) were estimated using spatiotemporal splines. These were fit to a one-compartment perfusion model to obtain wash-in rate parameters K(1). For the second method (SPECT/D), an anthropomorphic cardiac torso phantom was used to create real SPECT dynamic projection data of a tracer distribution derived from (94)Tc-MIBI PET scans in the blood pool, myocardium, liver and background. This method introduced attenuation, collimation and scatter into the modeling of dynamic SPECT projections. Both approaches were used to evaluate the accuracy of estimating myocardial wash-in parameters for rotation speeds providing 180° of projection data every 27 and 54 s. Dynamic cardiac SPECT was also performed in a human subject at rest using a hybrid SPECT/CT scanner. Dynamic measurements of (99m)Tc-tetrofosmin in the myocardium were obtained using an infusion time of 2 min. Blood input, myocardium tissue and liver TACs were estimated using the same spatiotemporal splines. The spatiotemporal maximum-likelihood expectation-maximization (4D ML-EM) reconstructions gave more accurate reconstructions than did standard frame-by-frame static 3D ML-EM reconstructions. The SPECT/P results showed that 4D ML-EM reconstruction gave higher and more accurate estimates of K(1) than did 3D ML-EM, yielding anywhere from a 44% underestimation to 24% overestimation for the three patients. The SPECT/D results showed that 4D ML-EM reconstruction gave an overestimation of 28% and 3D ML-EM gave an underestimation of 1% for K(1). For the patient study the 4D ML-EM reconstruction provided continuous images as a function of time of the concentration in both ventricular cavities and myocardium during the 2 min infusion. It is demonstrated that a 2 min infusion with a two-headed SPECT system rotating 180° every 54 s can produce measurements of blood pool and myocardial TACs, though the SPECT simulation studies showed that one must sample at least every 30 s to capture a 1 min infusion input function.     

Factor analysis with a priori knowledge--application in dynamic cardiac SPECT

Two factor analysis of dynamic structures (FADS) methods for the extraction of time-activity curves (TACs) from cardiac dynamic SPECT data sequences were investigated. One method was based on a least squares (LS) approach which was subject to positivity constraints. The other method was the well known apex-seeking (AS) method. A post-processing step utilizing a priori information was employed to correct for the non-uniqueness of the FADS solution. These methods were used to extract 99mTc-teboroxime TACs from computer simulations and from experimental canine and patient studies. In computer simulations, the LS and AS methods, which are completely different algorithms, yielded very similar and accurate results after application of the correction for non-uniqueness. FADS-obtained blood curves correlated well with curves derived from region of interest (ROI) measurements in the experimental studies. The results indicate that the factor analysis techniques can be used for semi-automatic estimation of activity curves derived from cardiac dynamic SPECT images, and that they can be used for separation of physiologically different regions in dynamic cardiac SPECT studies.     

Blind estimation of compartmental model parameters

Computation of physiologically relevant kinetic parameters from dynamic PET or SPECT imaging requires knowledge of the blood input function. This work is concerned with developing methods to accurately estimate these kinetic parameters blindly; that is, without use of a directly measured blood input function. Instead, only measurements of the output functions--the tissue time-activity curves--are used. The blind estimation method employed here minimizes a set of cross-relation equations, from which the blood term has been factored out, to determine compartmental model parameters. The method was tested with simulated data appropriate for dynamic SPECT cardiac perfusion imaging with 99mTc-teboroxime and for dynamic PET cerebral blood flow imaging with 15O water. The simulations did not model the tomographic process. Noise levels typical of the respective modalities were employed. From three to eight different regions were simulated, each with different time-activity curves. The time-activity curve (24 or 70 time points) for each region was simulated with a compartment model. The simulation used a biexponential blood input function and washin rates between 0.2 and 1.3 min(-1) and washout rates between 0.2 and 1.0 min(-1). The system of equations was solved numerically and included constraints to bound the range of possible solutions. From the cardiac simulations, washin was determined to within a scale factor of the true washin parameters with less than 6% bias and 12% variability. 99mTc-teboroxime washout results had less than 5% bias, but variability ranged from 14% to 43%. The cerebral blood flow washin parameters were determined with less than 5% bias and 4% variability. The washout parameters were determined with less than 4% bias, but had 15-30% variability. Since washin is often the parameter of most use in clinical studies, the blind estimation approach may eliminate the current necessity of measuring the input function when performing certain dynamic studies.     

Non-iterative methods incorporating a priori source distribution and data information for suppression of image noise and artefacts in 3D SPECT

Non-iterative methods have been developed for image reconstruction in 3D SPECT with uniform attenuation and distance-dependent spatial resolution. It was observed that these methods can, in general, be susceptible to data noise and other errors, yielding conspicuous image artefacts. In this work, we developed and evaluated a regularized inverse-filtering approach for effective suppression of noise and artefacts in 3D SPECT images without significantly compromising image resolution. The proposed approach allows the incorporation of a priori random image field and data information and can thus robustly control the degree of suppression of noise and artefacts in 3D SPECT images. Using computer simulations, we evaluated and compared quantitatively images reconstructed from data sets of various noise levels by the use of the proposed methods and the existing non-iterative methods. These numerical results clearly demonstrated that the proposed regularized inverse-filtering approach can effectively suppress image noise and artefacts that plague the existing non-iterative methods, thus yielding quantitatively more accurate 3D SPECT images. The proposed regularized inverse-filtering approach can also be generalized to other imaging modalities.     

Joint penalized-likelihood reconstruction of time-activity curves and regions-of-interest from projection data in brain PET

This paper presents a novel statistical approach for joint estimation of regions-of-interest (ROIs) and the corresponding time-activity curves (TACs) from dynamic positron emission tomography (PET) brain projection data. It is based on optimizing the joint objective function that consists of a data log-likelihood term and two penalty terms reflecting the available a priori information about the human brain anatomy. The developed local optimization strategy iteratively updates both the ROI and TAC parameters and is guaranteed to monotonically increase the objective function. The quantitative evaluation of the algorithm is performed with numerically and Monte Carlo-simulated dynamic PET brain data of the 11C-Raclopride and 18F-FDG tracers. The results demonstrate that the method outperforms the existing sequential ROI quantification approaches in terms of accuracy, and can noticeably reduce the errors in TACs arising due to the finite spatial resolution and ROI delineation.     

Joint estimation of dynamic PET images and temporal basis functions using fully 4D ML-EM

A fully 4D joint-estimation approach to reconstruction of temporal sequences of 3D positron emission tomography (PET) images is proposed. The method estimates both a set of temporal basis functions and the corresponding coefficient for each basis function at each spatial location within the image. The joint estimation is performed through a fully 4D version of the maximum likelihood expectation maximization (ML-EM) algorithm in conjunction with two different models of the mean of the Poisson measured data. The first model regards the coefficients of the temporal basis functions as the unknown parameters to be estimated and the second model regards the temporal basis functions themselves as the unknown parameters. The fully 4D methodology is compared to the conventional frame-by-frame independent reconstruction approach (3D ML-EM) for varying levels of both spatial and temporal post-reconstruction smoothing. It is found that using a set of temporally extensive basis functions (estimated from the data by 4D ML-EM) significantly reduces the spatial noise when compared to the independent method for a given level of image resolution. In addition to spatial image quality advantages, for smaller regions of interest (where statistical quality is often limited) the reconstructed time-activity curves show a lower level of bias and a lower level of noise compared to the independent reconstruction approach. Finally, the method is demonstrated on clinical 4D PET data.     

Spatio-temporal diffusion of dynamic PET images

Positron emission tomography (PET) images are corrupted by noise. This is especially true in dynamic PET imaging where short frames are required to capture the peak of activity concentration after the radiotracer injection. High noise results in a possible bias in quantification, as the compartmental models used to estimate the kinetic parameters are sensitive to noise. This paper describes a new post-reconstruction filter to increase the signal-to-noise ratio in dynamic PET imaging. It consists in a spatio-temporal robust diffusion of the 4D image based on the time activity curve (TAC) in each voxel. It reduces the noise in homogeneous areas while preserving the distinct kinetics in regions of interest corresponding to different underlying physiological processes. Neither anatomical priors nor the kinetic model are required. We propose an automatic selection of the scale parameter involved in the diffusion process based on a robust statistical analysis of the distances between TACs. The method is evaluated using Monte Carlo simulations of brain activity distributions. We demonstrate the usefulness of the method and its superior performance over two other post-reconstruction spatial and temporal filters. Our simulations suggest that the proposed method can be used to significantly increase the signal-to-noise ratio in dynamic PET imaging.     

The effect of acquisition interval and spatial resolution on dynamic cardiac imaging with a stationary SPECT camera

The current SPECT scanning paradigm that acquires images by slow rotation of multiple detectors in body-contoured orbits around the patient is not suited to the rapid collection of tomographically complete data. During rapid image acquisition, mechanical and patient safety constraints limit the detector orbit to circular paths at increased distances from the patient, resulting in decreased spatial resolution. We consider a novel dynamic rotating slant-hole (DyRoSH) SPECT camera that can collect full tomographic data every 2 s, employing three stationary detectors mounted with slant-hole collimators that rotate at 30 rpm. Because the detectors are stationary, they can be placed much closer to the patient than is possible with conventional SPECT systems. We propose that the decoupling of the detector position from the mechanics of rapid image acquisition offers an additional degree of freedom which can be used to improve accuracy in measured kinetic parameter estimates. With simulations and list-mode reconstructions, we consider the effects of different acquisition intervals on dynamic cardiac imaging, comparing a conventional three detector SPECT system with the proposed DyRoSH SPECT system. Kinetic parameters of a two-compartment model of myocardial perfusion for technetium-99m-teboroxime were estimated. When compared to a conventional SPECT scanner for the same acquisition periods, the proposed DyRoSH system shows equivalent or reduced bias or standard deviation values for the kinetic parameter estimates. The DyRoSH camera with a 2 s acquisition period does not show any improvement compared to a DyRoSH camera with a 10 s acquisition period.     

Estimation of myocardial glucose utilisation with PET using the left ventricular time-activity curve as a non-invasive input function

The validation study is described of a new modelling method that has been developed, using tracer kinetic modelling with positron emission tomography (PET) to achieve non-invasive measurement of myocardial metabolic rate of glucose (MMRGlc). Eight data sets obtained from dynamic cardiac PET 2-[¹⁸F]fluoro-2-deoxy-D-glucose (FDG) studies on human subjects are employed, and the estimation of MMRGlc using both the new and traditional methods is compared. The results from all eight human FDG studies are consistent with those from previous computer simulations. With the new method, the estimated mean of K (a parameter directly proportional to MMRGlc) increases by about 8%, and that of k₄ (the rate constant of FDG dephosphorylation) decreases by about 48%. The approach should be more suitable for use in dynamic cardiac PET studies when non-invasive means are used to obtain the plasma time-activity curve from left-ventricle PET images.     

A study of the image discrepancies due to object time-dependence in transmission and emission tomography

In conventional computed tomography (CT) imaging of a point object, projection filtering causes the back-projected contributions to image positions away from the point to sum to zero. If the point object intensity is time-dependent, and all the projections are not acquired simultaneously, this cancellation cannot be complete and artefacts result. Loss of spatial invariance makes a general linear-systems approach to the problem impossible. We have studied the properties of such artefacts by the computer simulation of decaying exponential time-dependence in three different spatial distributions and four transmission and emission CT geometries. Spatially complex time-dependent objects typically produce artefacts that can be treated as an additional broad-spectrum noise source with a power comparable to that of other CT noises. Artefacts from broad ranges of similar time-dependence can add coherently to cause patches of artefact, particularly in geometries with a strong correlation between projection acquisition time and projection angle. As expected, artefacts are reduced for all geometries as scan duration is reduced. In our model, with a most rapid decay constant of 1.2 min-1, negligible artefacts were observed for a six second scan duration.     

Closed-form kinetic parameter estimation solution to the truncated data problem

In a dedicated cardiac single photon emission computed tomography (SPECT) system, the detectors are focused on the heart and the background is truncated in the projections. Reconstruction using truncated data results in biased images, leading to inaccurate kinetic parameter estimates. This paper has developed a closed-form kinetic parameter estimation solution to the dynamic emission imaging problem. This solution is insensitive to the bias in the reconstructed images that is caused by the projection data truncation. This paper introduces two new ideas: (1) it includes background bias as an additional parameter to estimate, and (2) it presents a closed-form solution for compartment models. The method is based on the following two assumptions: (i) the amount of the bias is directly proportional to the truncated activities in the projection data, and (ii) the background concentration is directly proportional to the concentration in the myocardium. In other words, the method assumes that the image slice contains only the heart and the background, without other organs, that the heart is not truncated, and that the background radioactivity is directly proportional to the radioactivity in the blood pool. As long as the background activity can be modeled, the proposed method is applicable regardless of the number of compartments in the model. For simplicity, the proposed method is presented and verified using a single compartment model with computer simulations using both noiseless and noisy projections.     

Analytical propagation of errors in dynamic SPECT: estimators, degrading factors, bias and noise.

Dynamic SPECT is a relatively new technique that may potentially benefit many imaging applications. Though similar to dynamic PET, the accuracy and precision of dynamic SPECT parameter estimates are degraded by factors that differ from those encountered in PET. In this work we formulate a methodology for analytically studying the propagation of errors from dynamic projection data to kinetic parameter estimates. This methodology is used to study the relationships between reconstruction estimators, image degrading factors, bias and statistical noise for the application of dynamic cardiac imaging with 99mTc-teboroxime. Dynamic data were simulated for a torso phantom, and the effects of attenuation, detector response and scatter were successively included to produce several data sets. The data were reconstructed to obtain both weighted and unweighted least squares solutions, and the kinetic rate parameters for a two-compartment model were estimated. The expected values and standard deviations describing the statistical distribution of parameters that would be estimated from noisy data were calculated analytically. The results of this analysis present several interesting implications for dynamic SPECT. Statistically weighted estimators performed only marginally better than unweighted ones, implying that more computationally efficient unweighted estimators may be appropriate. This also suggests that it may be beneficial to focus future research efforts upon regularization methods with beneficial bias-variance trade-offs. Other aspects of the study describe the fundamental limits of the bias variance trade-off regarding physical degrading factors and their compensation. The results characterize the effects of attenuation, detector response and scatter, and they are intended to guide future research into dynamic SPECT reconstruction and compensation methods.     

Variance in parametric images: direct estimation from parametric projections

Recent work has shown that it is possible to apply linear kinetic models to dynamic projection data in PET in order to calculate parameter projections. These can subsequently be back-projected to form parametric images--maps of parameters of physiological interest. Critical to the application of these maps, to test for significant changes between normal and pathophysiology, is an assessment of the statistical uncertainty. In this context, parametric images also include simple integral images from, e.g., [O-15]-water used to calculate statistical parametric maps (SPMs). This paper revisits the concept of parameter projections and presents a more general formulation of the parameter projection derivation as well as a method to estimate parameter variance in projection space, showing which analysis methods (models) can be used. Using simulated pharmacokinetic image data we show that a method based on an analysis in projection space inherently calculates the mathematically rigorous pixel variance. This results in an estimation which is as accurate as either estimating variance in image space during model fitting, or estimation by comparison across sets of parametric images--as might be done between individuals in a group pharmacokinetic PET study. The method based on projections has, however, a higher computational efficiency, and is also shown to be more precise, as reflected in smooth variance distribution images when compared to the other methods.     

A multipinhole small animal SPECT system with submillimeter spatial resolution

Single photon emission computed tomography (SPECT) is an important technology for molecular imaging studies of small animals. In this arena, there is an increasing demand for high performance imaging systems that offer improved spatial resolution and detection efficiency. We have designed a multipinhole small animal imaging system based on position sensitive avalanche photodiode (PSAPD) detectors with the goal of submillimeter spatial resolution and high detection efficiency, which will allow us to minimize the radiation dose to the animal and to shorten the time needed for the imaging study. Our design will use 8 x 24 mm2 PSAPD detector modules coupled to thallium-doped cesium iodide [CsI(Tl)] scintillators, which can achieve an intrinsic spatial resolution of 0.5 mm at 140 keV. These detectors will be arranged in rings of 24 modules each; the animal is positioned in the center of the 9 stationary detector rings which capture projection data from the animal with a cylindrical tungsten multipinhole collimator. The animal is supported on a bed which can be rocked about the central axis to increase angular sampling of the object. In contrast to conventional SPECT pinhole systems, in our design each pinhole views only a portion of the object. However, the ensemble of projection data from all of the multipinhole detectors provide angular sampling that is sufficient to reconstruct tomographic data from the object. The performance of this multipinhole PSAPD imaging system was simulated using a ray tracing program that models the appropriate point spread functions and then was compared against the performance of a dual-headed pinhole SPECT system. The detection efficiency of both systems was simulated and projection data of a hot rod phantom were generated and reconstructed to assess spatial resolution. Appropriate Poisson noise was added to the data to simulate an acquisition time of 15 min and an activity of 18.5 MBq distributed in the phantom. Both sets of data were reconstructed with an ML-EM reconstruction algorithm. In addition, the imaging performance of both systems was evaluated with a uniformity phantom and a realistic digital mouse phantom. Simulations show that our proposed system produces a spatial resolution of 0.8 mm and an average detection efficiency of 630 cps/MBq. In contrast, simulations of the dual-headed pinhole SPECT system produce a spatial resolution of 1.1 mm and an average detection efficiency of 53 cps/MBq. These results suggest that our novel design will achieve high spatial resolution and will improve the detection efficiency by more than an order of magnitude compared to a dual-headed pinhole SPECT system. We expect that this system can perform SPECT with submillimeter spatial resolution, high throughput, and low radiation dose suitable for in vivo imaging of small animals.     

Use of 3D reconstruction to correct for patient motion in SPECT

Patient motion occurring during data acquisition in single photon emission computed tomography (SPET) can cause serious reconstruction artefacts. We have developed a new approach to correct for head motion in brain SPECT. Prior to motion, projections are assigned to conventional projections. When head motion occurs, it is measured by a motion monitoring system, and subsequent projection data are mapped to 'virtual' projections. The appropriate position of each virtual projection is determined by applying the converse of the patient's accumulated motion to the actual camera projection. Conventional and virtual projections, taken together, form a consistent set that can be reconstructed using a three-dimensional (3D) algorithm. The technique has been tested on a range of simulated rotational movements, both within and out of the transaxial plane. For all simulated movements, the motion corrected images exhibited better agreement with a motion free reconstruction than did the uncorrected images. This technique may help to overcome one of the major remaining limitations on image quality and quantitative accuracy in SPECT.     

Performance of the dynamic single photon emission computed tomography (dSPECT) method for decreasing or increasing activity changes

Radionuclide imaging is now widely used whenever functional information is required. We present a new approach to dynamic SPECT imaging (dSPECT method) that uses a single slow rotation of a conventional camera and allows us to reconstruct a series of 3D images corresponding to the radiotracer distribution in the body at various times. Using simulations of various camera configurations and acquisition protocols, we have shown that this method is able to reconstruct washout half-lives with an accuracy greater than 90% when used with triple-head SPECT cameras. Accuracy decreases when using fewer camera heads, but dual-head geometries still give an accuracy greater than 80% for short and 90% for long half-lives and about 50-75% for single-head systems. Dynamic phantom experiments have yielded similar results. Presence of attenuation and background activity does not affect the accuracy of the dSPECT reconstructions. In all situations investigated satisfactory dynamic images were produced. A preliminary normal volunteer study measuring renal function was performed. The reconstructed dynamic images may be presented as a three-dimensional movie showing movement of the tracer through the kidneys and the measurement of the regional renal function can be performed. The time-activity curves determined from this dSPECT data are very similar to those obtained from dynamic planar scans.     

Effects of attenuation in single slow rotation dynamic SPECT

Dynamic imaging using SPECT has been a topic of research interest for many years. Several proposed approaches have considered the reconstruction of dynamic images from SPECT data acquired with a conventional single slow rotation of the camera, which results in an extremely underdetermined reconstruction problem. Accurate attenuation correction (AC) is particularly important in this context, in order to distinguish the actual dynamic behavior of the tracer within a region from the effects of attenuation on the projection data as the camera rotates around the patient. In this paper, we demonstrate that the standard approach to AC used in conventional SPECT imaging is not sufficient to account for the effects of attenuation in dynamic imaging of this type. As a result, artifacts may be created in the reconstructed images. Using realistic dynamic 3D phantom simulations, as well as real-life dynamic renal SPECT data, we assess the severity of these artifacts and investigate a method to eliminate them. The proposed method is shown to substantially improve the accuracy of the reconstructed image.     

Fully 4D motion-compensated reconstruction of cardiac SPECT images

In this paper, we investigate the benefits of a spatiotemporal approach for reconstruction of image sequences. In the proposed approach, we introduce a temporal prior in the form of motion compensation to account for the statistical correlations among the frames in a sequence, and reconstruct all the frames collectively as a single function of space and time. The reconstruction algorithm is derived based on the maximum a posteriori estimate, for which the one-step late expectation-maximization algorithm is used. We demonstrated the method in our experiments using simulated single photon emission computed tomography (SPECT) cardiac perfusion images. The four-dimensional (4D) gated mathematical cardiac-torso phantom was used for simulation of gated SPECT perfusion imaging with Tc-99m-sestamibi. In addition to bias-variance analysis and time activity curves, we also used a channelized Hotelling observer to evaluate the detectability of perfusion defects in the reconstructed images. Our experimental results demonstrated that the incorporation of temporal regularization into image reconstruction could significantly improve the accuracy of cardiac images without causing any significant cross-frame blurring that may arise from the cardiac motion. This could lead to not only improved detection of perfusion defects, but also improved reconstruction of the heart wall which is important for functional assessment of the myocardium.     

Error-corrected estimation of regional kinetic parameter histograms directly from pet projections

Positron emission tomography (PET) is a unique method to investigate physiology in the living body. Kinetic models with kinetic rate constants describe the dynamic radioactive tracer uptake in living tissue. If the variation of the kinetic parameter values within a specific tissue region could be determined accurately, it would give valuable quantitative information about the tissue heterogeneity. In this study we developed a unique method to estimate the variation from the regional kinetic parameter histograms. To determine the kinetic parameter values, we chose non-penalized maximum likelihood (ML) estimation due to the specific statistical error properties of the ML estimates. The parameter values were estimated directly from the time series of PET projections. The choice of the estimation method enabled us to utilize the ML theory in error correction. We developed a Monte Carlo approach to determine the regional error distributions. The true variation of the kinetic parameters could then be revealed by correcting the regional ML estimate histograms with the estimated error distributions. The method was tested with simulated data. In simulations both the average and the deviation of the kinetic parameters were determined from the error-corrected histograms with good numerical accuracy for the selected region of interest.