Time-dependent Volume Research Articles

Analysis of Processes in the Containment Using ATHLET-CD and COCOSYS Codes

A brief description of performed input deck modifications and results of stand-alone and coupled calculations of Dn 200 mm loss of coolant accident with simultaneous total station blackout accident scenario for Rivne Nuclear Power Plant Unit 1 (WWER‑440/V-213) with application of
 ATHLET-CD 3.1A and COCOSYS 2.4 codes are presented in the paper.
 ATHLET-CD stand-alone calculation was performed with constant containment pressure (a time dependent volume with constant pressure and temperature was used as a boundary volume for leakage). Further, mass and energy release and fission products from the primary system obtained during ATHLET‑CD stand-alone calculation were used to perform COCOSYS stand-alone calculation. In addition, coupled ATHLET-CD and COCOSYS calculation was performed. All the computer analyzes were performed until the lower head failure. ATHLET‑CD model was extended with core degradation module (ECORE), which allowed calculation of scenario until reactor pressure vessel failure.
 According to the results of comparative analysis, nearly the same behavior of the main parameters in the stand-alone and coupled calculation at an early phase of scenario was obtained. Some small differences occur due to distinction in behavior of water and steam mass flows released through the break and due to existence of heat transfer from the primary system structures to the containment compartments during coupled calculation of transient. As for middle and late phases of the accident, some differences between stand-alone and coupled calculation results for analyzed scenario are present. These differences are caused by different total fission products and aerosols release from the reactor coolant system to the containment compartments. The above information allows recommending application of coupled code/model versions for performing the computer severe accident analyses.

Effect of volumetric expansion on shock-induced ignition of H2–NO2/N2O4 mixtures

The competition between chemical energy release rate and volumetric expansion related to shock wave’s dynamics is of primary importance for a number of situations relevant to explosion safety. While studies have been performed on this topic over the years, they have been limited to mixtures with monotonous energy release profile. In the present study, the ignition of H2–NO2/N2O4 mixtures, which exhibit a single-step or a two-step energy release rate profile depending on the equivalence ratio, has been investigated under volumetric expansion conditions. The rate of expansion has been calculated using the Taylor–Sedov solution and accounted for using 0-D numerical simulations with time-dependent specific volume. The results were analyzed in terms of a Damkohler number defined as the ratio of the expansion to ignition times. For mixtures with non-monotonous energy release rate profiles, two critical Damkohler numbers can be identified for individual steps of energy release. It was also shown that the fluid element behind a shock propagating at the Chapman–Jouguet velocity is most likely to ignite. The thermo-chemical dynamics have been analyzed about the critical conditions using energy release rate per reaction, rate of production and sensitivity analyses.

Development of An Integrated Numerical Model for Simulating Wave Interaction with Permeable Submerged Breakwaters Using Extended Navier–Stokes Equations

An integrated two-dimensional vertical (2DV) model was developed to investigate wave interactions with permeable submerged breakwaters. The integrated model is capable of predicting the flow field in both surface water and porous media on the basis of the extended volume-averaged Reynolds-averaged Navier–Stokes equations (VARANS). The impact of porous medium was considered by the inclusion of the additional terms of drag and inertia forces into conventional Navier–Stokes equations. Finite volume method (FVM) in an arbitrary Lagrangian–Eulerian (ALE) formulation was adopted for discretization of the governing equations. Projection method was utilized to solve the unsteady incompressible extended Navier–Stokes equations. The time-dependent volume and surface porosities were calculated at each time step using the fraction of a grid open to water and the total porosity of porous medium. The numerical model was first verified against analytical solutions of small amplitude progressive Stokes wave and solitary wave propagation in the absence of a bottom-mounted barrier. Comparisons showed pleasing agreements between the numerical predictions and analytical solutions. The model was then further validated by comparing the numerical model results with the experimental measurements of wave propagation over a permeable submerged breakwater reported in the literature. Good agreements were obtained for the free surface elevations at various spatial and temporal scales, velocity fields around and inside the obstacle, as well as the velocity profiles.

Plasma-sheath transition in multi-fluid models with inertial terms under low pressure conditions: comparison with the classical and kinetic theory

We present high-order accuracy simulations of the plasma-sheath transition with an ion–electron multi-fluid model that considers the electron inertial terms and the ion pressure gradient. By means of a third-order accuracy time-dependent finite volume scheme, we solve the isothermal multi-fluid equations with realistic ion-to-electron mass ratios. We propose a numerical procedure and boundary conditions that retrieve a steady-state solution of a planar 1D floating sheath. The classical solution to this problem neglects the ion temperature since the model equations present a singularity for a finite ion temperature. The multi-fluid simulations provide a rigorous solution to the plasma-sheath transition without singularities. We compare our solution to the classical theory, finding perfect agreement when the ion-to-electron temperature tends to zero. We discuss the effect of the ion temperature, the electron inertia, and the elastic collisions with neutrals in low-pressure plasmas. Finally, relying on a kinetic approach, we derive analytical expressions for the electron macroscopic quantities inside a steady sheath that assumes a truncated Maxwellian electron velocity distribution function (EVDF) with a wall that collects the electron random flux. The derived analytical expressions are beyond the classical isothermal solution with Boltzmann electrons. The isothermal multi-fluid solution captures properly the first two moments of the EVDF inside the sheath. Our analytical expressions show the need for solving for higher-order moments to fully explain the electron physics inside the sheath, as opposite to the classical isothermal assumption. This work demonstrates that the multi-fluid simulations are able to capture the plasma-sheath transition under weakly collisional conditions with a solution that is consistent with the classical and the kinetic theory.

Derivation of a local volume-averaged model and a stable numerical algorithm for multi-dimensional simulations of conversion batteries

In this article, we derive a general form of local volume-averaging theory and apply it to a model of zinc-air conversion batteries. Volume-averaging techniques are frequently used for the macroscopic description of micro-porous electrodes. We extend the existing method by including reactions between different phases and time-dependent volume fractions of the solid phases as these are continuously dissolved and reconstructed during operation of conversion batteries. We find that the constraint of incompressibility for multi-component fluids causes numerical instabilities in simulations of zinc-air battery cells. Therefore, we develop a stable sequential semi-implicit algorithm which converges against the fully implicit solution. Our method reduces the coupling of the variables by splitting the system of equations and introducing an additional iteration step.

On volume subregion complexity in Vaidya spacetime

We study holographic subregion volume complexity for a line segment in the AdS3 Vaidya geometry. On the field theory side, this gravity background corresponds to a sudden quench which leads to the thermalization of the strongly-coupled dual conformal field theory. We find the time-dependent extremal volume surface by numerically solving a partial differential equation with boundary condition given by the Hubeny-Rangamani- Takayanagi surface, and we use this solution to compute holographic subregion complexity as a function of time. Approximate analytical expressions valid at early and at late times are derived.

Population pharmacokinetics of inotuzumab ozogamicin in relapsed/refractory acute lymphoblastic leukemia and non-Hodgkin lymphoma

This population pharmacokinetics analysis evaluated the target-mediated drug disposition of inotuzumab ozogamicin (InO) through an empirical time-dependent clearance (CLt) term and identified potential covariates that may be important predictors of variability in InO distribution and elimination. This analysis was conducted by pooling data from 2 studies of single-agent InO in patients with relapsed or refractory (R/R) B cell acute lymphoblastic leukemia (ALL), 3 studies of single-agent InO, 5 studies of InO plus rituximab (R-InO), and 1 study of R-InO plus chemotherapy in patients with R/R B-cell non-Hodgkin lymphoma (NHL). Pharmacokinetic data included 8361 InO concentration–time observations that were modeled using nonlinear mixed-effects analysis. Covariate relations were identified using generalized additive modeling on base model parameters and then tested in a stepwise manner via covariate modeling. InO concentration was described with a 2-compartment model with linear and time-dependent clearance components. Based on the final model, baseline body surface area was a covariate of the linear and time-dependent clearance components and volume of distribution in the central compartment; baseline percentage of blasts in the peripheral blood was a covariate of the decay coefficient of the time-dependent clearance term (CLt); and concomitant rituximab treatment was a covariate of the linear clearance component (CL1). The magnitude of change of each pharmacokinetic parameter due to these covariates was not considered clinically relevant. Therefore, no dose adjustment of InO for the treatment of patients with R/R B-cell ALL or NHL is needed on the basis of selected covariates.

In Vitro-In Vivo Correlation for the Degradation of Tetra-PEG Hydrogel Microspheres with Tunable β-Eliminative Crosslink Cleavage Rates

The degradation of Tetra-PEG hydrogels containing β-eliminative crosslinks has been studied in order to provide an in vitro-in vivo correlation for the use of these hydrogels in our chemically controlled drug delivery system. We measured time-dependent gel mass loss and ultrasound volume changes of 13 subcutaneously implanted Tetra-PEG hydrogel microspheres having degradation times ranging from ~3 to 250 days. Applying a previously developed model of Tetra-PEG hydrogel degradation, the mass changes correlate well with the in vitro rates of crosslink cleavage and hydrogel degelation. These results allow prediction of in vivo biodegradation properties of these hydrogels based on readily obtained in vitro rates, despite having degradation times that span 2 orders of magnitude. These results support the optimization of drug-releasing hydrogels and their development into long-acting therapeutics. The use of ultrasound volume measurements further provides a noninvasive technique for monitoring hydrogel degradation in the subcutaneous space.

A Bi-Objective Timetable Optimization Model for Urban Rail Transit Based on the Time-Dependent Passenger Volume

In urban rail transit systems, energy conservation is a challenging problem due to the rising environmental and social issues. The existing literature on this topic usually ignores time-variant characteristics of passenger demand at each station. Based on the real-world time-dependent smart-card automated fare collection data, this paper develops a bi-objective timetable optimization model to minimize the total passenger waiting time and the pure energy consumption. In the model formulation, the total passenger waiting time is subjected to the train capacity in the oversaturated condition, and the pure energy consumption is represented by the difference between the traction energy consumption and the regenerative energy within a given period. Numerical examples based on the real-world data from Beijing Yizhuang metro line are conducted. The results indicate that the developed model can improve passenger service and reduce energy consumption efficiently in comparisons with the timetable used currently.

Beam Frame Representation for Ultrawideband Radiation From Volume Source Distributions: Frequency-Domain and Time-Domain Formulations

We present two novel beam-summation schemes for radiation from time-harmonic or time-dependent volume source distributions, where the field is expanded using a discrete phase-space set of beam-wave propagators. The generic term “beams” is used here for both the frequency-domain and the time-domain formulations where the propagators are isodiffracting Gaussian beam or isodiffracting pulsed beams, respectively. The formulations are structured upon the recently formulated “beam-frame” theorem that establishes these phase-space beam sets as frame sets everywhere in the propagation domain and not only over the aperture plane as in previous formulations. The expansion coefficients are obtained by projecting the source distributions over the dual beam-frame sets that have essentially the same structure as the basic sets. As such, these formulations constitute local generalization to the conventional plane waves or Green's function formulations, and also reduce the overall degrees of freedom needed to describe the radiated field. As demonstrated by the numerical examples, they resolve the local features of the source distributions in space time, and hence provide a basis for a new local inverse scattering theory to be presented subsequently.

Time-dependent infarct volume affects the benefit of recanalization.

ObjectivesThe benefit threshold of infarct volume from recanalization remains unclear. We assumed that the threshold decreased over time, and then investigated the benefit curve of infarct volume during different time periods.MethodsWe reviewed prospectively collected clinical and imaging data from acute ischemic stroke patients with internal carotid artery and M1 occlusion who underwent angiography before and 24 h after reperfusion therapy. Ordinal analyses of modified Rankin Scale scores were performed and curves were fitted.ResultsOf the included 445 patients, the median age was 71 years and 157 (35.3%) were women. The mean time from onset to treatment (OTT) was 248 ± 142 min. The median baseline infarct core volume was 49 (IQR 22–85) ml. Follow-up angiography revealed recanalization in 265 (59.6%) patients. The fitting curves showed that patients with an OTT ≤3 h would benefit from recanalization no matter how large the infarct volume was, whereas patients with an OTT between 3 and 4.5 h and with an infarct volume ≥ 125 ml, and those with an OTT ≥ 4.5 h and with an infarct volume ≥ 80 ml did not benefit from recanalization.ConclusionsWe established a time-dependent benefit threshold of infarct volume from recanalization, and thus suggested to estimate infarct core volume to select patients for reperfusion therapy in those with an OTT beyond 3 h.

Quantitative Analysis of Parotid Gland Secretion Function in Sjögren's Syndrome Patients with Dynamic Magnetic Resonance Sialography.

ObjectiveTo evaluate the secretory function of parotid glands by dynamic magnetic resonance (MR) sialography and determine the clinical performance of this technique in diagnosing and evaluating Sjögren's syndrome (SS) patients.Materials and MethodsThis study enrolled 29 healthy volunteers (25 women and 4 men; mean age, 34.8 ± 6.3 years; age range, 26–47 years) and 25 primary SS (pSS) patients (23 women and 2 men; mean age, 37.7 ± 7.9 years; age range, 25–50 years) with decreased secretory function. The volume of the parotid gland ducts was precisely measured for both groups at single pre- and 6 post-gustatory-stimulated phases. Time-dependent volume change ratio curves were generated, four parameters were derived from the curves: the slope of the increase in the first post-stimulation phase (slope1st), the peak value, the time-to-peak, the total saliva secretion post-stimulation. All values were used to quantitatively evaluate the secretory function of the parotid gland. The repeated measurement analysis, Mann-Whitney U test and receiver operating characteristic curve were applied.ResultsTime-dependent volume change ratio curves demonstrated that there is a statistically significant difference between the two groups (F = 8.750; p = 0.005). A quickly increasing curve was shown in the volunteer group, whereas a slowly increasing curve was shown in the pSS patient group. The slope1st, peak value and total saliva secretion post-stimulation of the patient group were significantly lower than those of the volunteer group (p = 0.005, p = 0.003, and p = 0.002, respectively). The time-to-peak between the two groups was not significantly different (p = 0.383). The slope1st can be used as a discriminator to diagnose SS patients (p = 0.015; odds ratio = 4.234; area under the curve = 0.726).ConclusionDynamic MR sialography is proven to be an effective method in evaluating salivary gland function and has a great potential in diagnosing and evaluating pSS patients.

Influence of Microcracks on Flow and Storage Capacities of Gas Shales at Core Scale

This work investigates the interplay of gas transport in the microcracks and matrix of shale using He and $$\hbox {CO}_2$$ via transient upstream pressure-pulse-decay experiments. The microcracks are natural in the sense no laboratory pretreatment created them. A novel setup of the pressure-pulse-decay experiment was used to determine the storage and flow capacities simultaneously. Experimentally, the pressure signals are used to define time-dependent pore volume partitioning between the microcracks and the matrix. A dual continuum simulator is constructed to decouple the flow and storage capacities at early time where the gas flows simultaneously in both media. A history-matching process quantifies the pore volume partitioning of the microcracks and matrix, the permeability in the microcracks, and the diffusivity of gas in the matrix. A series of experiments were conducted on four shale samples from Eagle Ford and Haynesville shale plays at constant net effective stress of 500 psi and at increasing net effective stresses of 500, 1000, and 2000 psi. Results show that the pore volume of the microcracks in sample 180Ha (Eagle Ford) was from 8.0 to 24.1% of the total pore volume in all experiments at constant and variable net effective stresses. The Haynesville samples (TWG 1-3 & 3-3) had between 68.9 and 87.0% of the total porosity in the microcracks while the remaining percentage represents the pore volume in the matrix. These proportions are related to the magnitude of permeability and void volumes in the system. The values of Klinkenberg-corrected permeability estimated using He vary between 396 and 1.6 $$\upmu \mathrm{D}$$ . The role of adsorption was also investigated using $$\hbox {CO}_2$$ in the same experimental apparatus and conditions. It was found that the influence of pore volume partitioning is suppressed by the large adsorption capacity in the shale samples. It was found that adsorption occurs in the microcracks in permeabilities less than 50 $$\upmu \mathrm{D}$$ along with adsorption in the matrix. For permeabilities greater than that value, adsorption had no importance on microcracks. Adsorption in both media required different adsorption–pressure functions. The $$\hbox {CO}_2$$ permeability was found to be smaller than the He permeability by 2–3 factors. $$\hbox {CO}_2$$ injection into the core caused a reduction in permeability that was observed with He injection after $$\hbox {CO}_2$$ flooding. The He permeability dropped by a factor of 2–3. This permeability hysteresis is attributed to the reduction in pore size accessed by the gas. This observation provides another reason why $$\hbox {CO}_2$$ storage in depleted shale gas reservoirs is practical. Reduction in gas permeability may be essential to keep $$\hbox {CO}_2$$ underground for as long as possible.

Ignition of hydrogen-air mixtures under volumetric expansion

A better understanding of chemical kinetics under volumetric expansion is important for a number of situations relevant to industrial safety including detonation diffraction and direct initiation, reflected shock-ignition at obstacles, ignition behind a decaying shock, among others. The ignition of stoichiometric hydrogen-air mixtures was studied using 0D numerical simulations with time-dependent specific volume variations. The competition between chemical energy release and expansion-induced cooling was characterized for different cooling rates and mathematical forms describing the shock decay rate. The critical conditions for reaction quenching were systematically determined, and the thermo-chemistry dynamics were analyzed near the critical conditions.

Radiation-induced hippocampal atrophy in patients with nasopharyngeal carcinoma early after radiotherapy: a longitudinal MR-based hippocampal subfield analysis.

Increasing evidence indicates that radiation-induced injury to the hippocampus may play a critical role in neurocognitive dysfunction in patients with nasopharyngeal carcinoma (NPC). However, few studies have assessed RT-induced hippocampal structural alterations in these patients early after radiotherapy (RT). In this study, 58 NPC patients were longitudinally followed up prior to treatment initiation as well as 3 and 6months after RT, respectively. Twenty comparable normal controls were recruited and followed up in parallel. A novel magnetic resonance imaging (MRI)-based automated method was used to label hippocampal subfields. The linear mixed model was employed to evaluate longitudinal changes in the volumes of the whole hippocampus and seven hippocampal subfields. Time-dependent volume reduction was observed in the bilateral hippocampus, as well as in the bilateral granule cell layer (GCL), bilateral cornu ammonis 1 (CA1), bilateral molecular layer (ML), and bilateral subiculum (SUB) in NPC patients, but not in controls. Moreover, volume deficits in the bilateral hippocampus, bilateral GCL, and right ML showed dose-dependent patterns, and high volume losses in the bilateral hippocampus, bilateral GCL, left SUB, and right ML were associated with a rapid decline in cognitive function. Our findings demonstrated that the hippocampal subfields were selectively injured by irradiation-related early neurotoxic effects, which might account for cognitive impairment in NPC patients at an early stage after RT. Further, structural MRI could serve as a potential noninvasive imaging biomarker for the early detection of radiation effects on the hippocampus in NPC patients after RT.

An approximate analytical approach to estimate the diffusivity of toxic chemicals in polymer barrier materials from the time evolution of sessile drop profiles

Any practical technique to determine the diffusivity of chemical warfare agents in protective barrier materials should require handling only miniscule volume of the chemicals and also require data analysis methods that are not computationally burdensome. Such an approach is described here based on imaging the time evolution of ~ 1 μL sessile drop profiles on the barrier surface and using an approximate analytical approach to analyze the time-dependent drop volume and contact angle data for extracting the diffusivity. The approximate analytical approach is validated by comparison against results from computationally intensive finite element simulations available in the literature. The domain of reliable use of the sessile drop technique in terms of the relative importance of the simultaneous evaporation and absorption is assessed using measurements on three challenging toxic chemicals in air and in butyl rubber. The ability of this approach to provide reasonable diffusivity estimates even if the substrate undergoes swelling is explored by studying water sessile drops on a Nafion membrane. The results allow one to conclude that the sessile drop technique coupled to the approximate analytical approach can be reliably used for rapid screening of new barrier materials for protection against chemical warfare agents if the screening is implemented using simulants with low vapor pressures.

Microstructural evolution and compositional homogenization of a low Re-bearing Ni-based single crystal superalloy during through progression of heat treatment

Microstructural and chemical heterogeneities may exacerbate the high-temperature mechanical properties of single crystal turbine blades. The evolution of microstructure and compositional homogeneity of an as-cast low Re-containing multicomponent DD6 alloy during through progression of heat treatment was detailedly studied. Using optical microscope, field emission gun scanning electron microscope, electron probe microanalysis, differential scanning calorimetry, and complementary thermodynamic calculations, we investigated the time-dependent size and volume fraction variations of γ/γ′ eutectic during specific solution heat treatment steps, the evolution of γ′ precipitates, carbides, and compositional homogeneity, and the thermo-physical changes of as-cast DD6 alloy during through progression of heat treatment. Based on our experimental and theoretical results, the mechanism underlying the microstructure formation and chemical segregation of as-cast DD6 alloy was clarified. The mechanism of microstructural evolution and compositional homogenization, and the diffusion mechanism of elements that occurs between dendritic and interdendritic regions of as-cast DD6 alloy during through progression of heat treatment were elucidated. The correlations between microstructural evolution, compositional homogenization, and thermo-physical changes of as-cast DD6 alloy during through progression of heat treatment were established.

A time-dependent method for the measurement of mass flow rate of gases in microchannels

Accurate measurement methods are required in the analysis of thermodynamic non-equilibrium effects associated with rarefied gas flows. For example, for the specific case of accommodation coefficients measurements, the quantity of interest is often the mass flow rate along the channel. For this purpose, the following paper presents a new time-dependent version of the well-known constant volume method for the accurate measurement of mass flow rates of gases in microchannels. The technique proposed here is an improvement in respect to the classic technique since it can be used for transient experiments. Moreover, it can be applied to configurations with arbitrary upstream and downstream reservoirs dimensions. Particularly, the proposed method relies on the assumption that the flow conductance varies linearly during the experiments and thus the pressure variations in the reservoirs can be fitted by a well-defined exponential function. Then, the instantaneous mass flow rate through the channel can be determined directly from the pressure fitting coefficients. A great advantage of the time-dependent constant volume method is that the measurements can be obtained from a single experiment for a wide range of rarefaction conditions, since the mean pressure between the reservoirs is allowed to change with time. Moreover, this technique represents a convenient manner to provide raw data of pressure variation with time in the reservoirs and transient mass flow rate in the channel by simply providing the fitting coefficients of the theoretically derived functions. A clear geometric criterion is also presented to determine when such mass flow rate measurements can be considered as quasi-steady, corresponding closely to results obtained under steady conditions, when ideally the channel connects two infinite reservoirs at different pressures. Results for flows of nitrogen through a stainless steel microtube (L=92.22±0.01 mm and D=435.5±3.5 µm) were obtained from 118 independent experiments, provided in this work, using two different experimental setups and three different configurations of the system volumes. As expected, no deviation was observed between all experimental campaigns in terms of the reduced mass flow rate. In addition, all results indicate that nitrogen can be considered completely accommodated at the surface of the microtube used, with α=0.986±0.019 when the diffuse-specular gas-surface interaction model is adopted and αt=0.991±0.020 (αn=1) when the Cercignani-Lampis model is adopted.

Flow Characteristics around Circular Cylinders with a Normal Slit

Alterations of boundary layer separation along the upper-rear surface of a baseline and slit cylinder and the formation of a vortex in the near-wake are investigated by particle image velocimetry (PIV) at Reynolds number 1000. The slit ratio (S/D) is 0.3. The phase-lock flow structures are referred to the time-dependent volume flux at the slit exit and are achieved by the modified phase-averaged technique. The alterations and the evolution of boundary-layer flow along the upper-rear surface are demonstrated by the phase-lock flow structures. It is found that the alternate blowing and suction at the slit exit serves as a perturbation to the boundary layer near the shoulder of the slit cylinder leading to a significant delay of flow separation and the flow reattachment of boundary-layer flow along the upper-rear surface of the cylinder. After perturbation, the vortex street behind a slit cylinder is more organized and stronger than that behind a baseline cylinder at Reynolds number 1000.

Bayesian approach for quantifying silica fume and water reducer impact on creep of concrete

The behaviour of concrete structures is affected by the time-dependent volume changes of concrete caused by sustained loads and moisture loss. Creep deformations have an important effect on long-term integrity of reinforced concrete structures. Incorrect or inaccurate prediction of these volume changes leads to undesirable and harmful consequences for the structures. Knowing that the Eurocode 2 does not take admixtures and additives into consideration in predicting creep deformations, the aim of this study is to apply correction coefficients to the Eurocode 2 creep formulas to take into account the effect of additives and admixtures, especially silica fume and water reducer. A large database from international laboratories and research centres is used in this study. The first step consists of comparing experimental measurements of creep deformations with creep values predicted by the Eurocode 2 formulas. The second step lies in introducing correction coefficients to the creep formulas of Eurocode 2, taking into consideration the percentage of admixtures and additives. These new correction coefficients are introduced in the Eurocode 2 formulas using an approximate Bayesian computation method based on a rejection algorithm.