Next generation physiologically based kinetic (NG-PBK) models in support of regulatory decision making.

The fields of toxicology and chemical risk assessment seek to reduce, and eventually replace, the use of animals for the prediction of toxicity in humans. In this context, physiologically based kinetic (PBK) modelling based on in vitro and in silico kinetic data has the potential to a play significant role in reducing animal testing, by providing a methodology capable of incorporating in vitro human data to facilitate the development of in vitro to in vivo extrapolation of hazard information. In the present article, we discuss the challenges in: 1) applying PBK modelling to support regulatory decision making under the toxicology and risk-assessment paradigm shift towards animal replacement; 2) constructing PBK models without in vivo animal kinetic data, while relying solely on in vitro or in silico methods for model parameterization; and 3) assessing the validity and credibility of PBK models built largely using non-animal data. The strengths, uncertainties, and limitations of PBK models developed using in vitro or in silico data are discussed in an effort to establish a higher degree of confidence in the application of such models in a regulatory context. The article summarises the outcome of an expert workshop hosted by the European Commission Joint Research Centre (EC-JRC) - European Union Reference Laboratory for Alternatives to Animal Testing (EURL ECVAM), on "Physiologically-Based Kinetic modelling in risk assessment - reaching a whole new level in regulatory decision-making" held in Ispra, Italy, in November 2016, along with results from an international survey conducted in 2017 and recently reported activities occurring within the PBK modelling field. The discussions presented herein highlight the potential applications of next generation (NG)-PBK modelling, based on new data streams.

Evidence for non-linear metabolism at low benzene exposures? A reanalysis of data.

The presence of a high-affinity metabolic pathway for low level benzene exposures of less than one part per million (ppm) has been proposed although a pathway has not been identified. The variation of metabolite molar fractions with increasing air benzene concentrations was suggested as evidence of significantly more efficient benzene metabolism at concentrations <0.1 ppm The evidence for this pathway is predicated on a rich data set from a study of Chinese shoe workers exposed to a wide range of benzene concentrations (not just "low level"). In this work we undertake a further independent re-analysis of this data with a focus on the evidence for an increase in the rate of metabolism of benzene exposures of less than 1 ppm. The analysis dataset consisted of measurements of benzene and toluene from personal air samplers, and measurements of unmetabolised benzene and toluene and five metabolites (phenol hydroquinone, catechol, trans, trans-muconic acid and s-phenylmercapturic acid) from post-shift urine samples for 213 workers with an occupational exposure to benzene (and toluene) and 139 controls. Measurements from control subjects were used to estimate metabolite concentrations resulting from non-occupational sources, including environmental sources of benzene. Data from occupationally exposed subjects were used to estimate metabolite concentrations as a function of benzene exposure. Correction for background (environmental exposure) sources of metabolites was achieved through a comparison of geometric means in occupationally exposed and control populations. The molar fractions of the five metabolites as a function of benzene exposure were computed. A supra-linear relationship between metabolite concentrations and benzene exposure was observed over the range 0.1-10 ppm benzene, however over the range benzene exposures of between 0.1 and 1 ppm only a modest departure from linearity was observed. The molar fractions estimated in this work were near constant over the range 0.1-10 ppm. No evidence of high affinity metabolism at these low level exposures was observed. Our reanalysis brings in to question the appropriateness of the dataset for commenting on low dose exposures and the use of a purely statistical approach to the analysis.

Effects of psychological priming, video, and music on anaerobic exercise performance.

Peak performance videos accompanied by music can help athletes to optimize their pre-competition mindset and are often used. Priming techniques can be incorporated into such videos to influence athletes' motivational state. There has been limited empirical work investigating the combined effects of such stimuli on anaerobic performance. The present study examined the psychological and psychophysiological effects of video, music, and priming when used as a pre-performance intervention for an anaerobic endurance task. Psychological measures included the main axes of the circumplex model of affect and liking scores taken pre-task, and the Exercise-induced Feeling Inventory, which was administered post-task. Physiological measures comprised heart rate variability and heart rate recorded pre-task. Fifteen males (age = 26.3 ± 2.8 years) were exposed to four conditions prior to performing the Wingate Anaerobic Test: music-only, video and music, video with music and motivational primes, and a no-video/no-music control. Results indicate that the combined video, music, and primes condition was the most effective in terms of influencing participants' pre-task affect and subsequent anaerobic performance; this was followed by the music-only condition. The findings indicate the utility of such stimuli as a pre-performance technique to enhance athletes' or exercisers' psychological states.

Development of PK- and PBPK-based modeling tools for derivation of biomonitoring guidance values.

There are numerous programs ongoing to analyze environmental exposure of humans to xenobiotic chemicals via biomonitoring measurements (e.g.: EU ESBIO, COPHES; US CDC NHANES; Canadian Health Measures Survey). The goal of these projects is to determine relative trends in exposure to chemicals, across time and subpopulations. Due to the lack of data, there is often little information correlating biomarker concentrations with exposure levels and durations. As a result, it can be difficult to utilize biomonitoring data to evaluate if exposures adhere to or exceed hazard/exposure criteria such as the Derived No-Effect Level values under the EU REACH program, or Reference Dose/Concentration values of the US EPA. A tiered approach of simple, arithmetic pharmacokinetic (PK) models, as well as more standardized mean-value, physiologically-based (PBPK) models, have therefore been developed to estimate exposures from biomonitoring results. Both model types utilize a user-friendly Excel spreadsheet interface. QSPR estimations of chemical-specific parameters have been included, as well as accommodation of variations in urine production. Validation of each model's structure by simulations of published datasets and the impact of assumptions of major model parameters will be presented.

Physiologically based pharmacokinetic modelling of human exposure to 2-butoxyethanol.

A physiologically based pharmacokinetic (PBPK) model describing the disposition of 2-butoxyethanol (2-BE) was developed in order to predict the urinary concentration of its major metabolite, butoxyacetic acid (BAA) under a range of exposure scenarios. Based on Corley et al. [Corley, R.A., Bormett, G.A., Ghanayem, B.I., 1994. Physiologically based pharmacokinetics of 2-butoxyethanol and its major metabolite, 2-butoxyacetic acid, in rats and humans. Toxicol. Appl. Pharmacol. 129, 61-79], the model included such features as multiple entry routes into the body, varying workload conditions, metabolism in the liver and elimination of free BAA in urine by glomerular filtration and acid transport. A bladder compartment simulating the fluctuations in metabolite concentration in urine caused by micturition formed a novel aspect of the model. Good agreement between model predictions and existing experimental data of total BAA levels in the blood and urine over various exposure conditions were observed. The mechanistically based PBPK model allowed comparison of disparate studies and also enabled the prediction of urinary concentrations of BAA post-shift. By calculating the total amount of BAA, any inter-individual variability in conjugation is taken into account. This led us to conclude that a biological monitoring guidance value should be proposed for total rather than free BAA with a value of 250 mmol/mol of creatinine (post-shift), based on an 8h exposure to 25 ppm 2-BE at resting working conditions.

Homology modelling of human CYP2E1 based on the CYP2C5 crystal structure: investigation of enzyme-substrate and enzyme-inhibitor interactions.

The construction of a homology model of human cytochrome P450 2E1 (CYP2E1) is reported, based on the CYP2C5 crystallographic template. A relatively high degree of primary sequence homology (identity=59%), as expected for proteins of the same CYP family, ensured a straightforward generation of the 3-dimensional model due to relatively few deletions and insertions of amino acid residues with respect to the CYP2C5 crystal structure. Probing the CYP2E1 model with typical substrates of the enzyme showed a good agreement with experimental information in the form of positions of metabolism for substrates, and with site-directed mutagenesis data on certain residues. Furthermore, quantitative relationships between substrate binding affinity and various structural parameters associated with the substrate molecules facilitated the formulation of a procedure for estimating relative binding energy and, consequently, K(m) or K(D) values towards the CYP2E1 enzyme. This method has been based on a consideration of the active site interactions between substrates and key amino acid residues lining the haem pocket, together with compound lipophilicity data from partition coefficients.

The application of physiologically based pharmacokinetic modelling in the analysis of occupational exposure to perchloroethylene.

A physiologically based pharmacokinetic model for perchloroethylene was parameterized, calibrated and validated using anatomic, physiologic, biochemical and physicochemical data obtained from the literature. The model was used to analyse human exposure data obtained under controlled conditions and from dry cleaning establishments in the Padua area of northern Italy. Whilst the model satisfactorily simulated the urinary excretion of trichloroacetic acid, following experimental inhalation exposure to 10, 20 and 40 ppm perchloroethylene under controlled conditions the opposite was true for the occupational exposure data. However, further model refinement to incorporate inter-individual variability of anatomical, physiological and biochemical parameters which have an impact on model output, would further improve the predictive capabilities of the model. The possibility of perchloroethylene and trichloroethylene co-exposure in the occupational setting was indicated by the model.

The effects of alcohol and diallyl sulphide on CYP2E1 activity in humans: a phenotyping study using chlorzoxazone.

The effects of acute administration of dietary levels of ethanol and the garlic oil extract, diallyl sulphide (DAS), on cytochrome P450 2E1 (CYP2E1) activity in volunteers were studied using the selective probe substrate, chlorzoxazone (CZX). The ratio of the CZX metabolite 6- hydroxychlorzoxazone (6-OHCZX) to CZX was taken to indicate CYP2E1 activity. The mean differences between the baseline and DAS-treated (0.2 mg/kg) CYP2E1 activities were significantly different (two-tailed p value = 0.0242, n = 8). Likewise, the mean differences between the baseline and ethanol-treated (0.8 g/kg) CYP2E1 activities were also significantly different (two-tailed p value = 0.0005, n = 7). The reduction in in vivo CYP2E1 activity by DAS is consistent with reported inhibition observed in vitro. The marked reduction in CYP2E1 activity following acute ingestion of ethanol is consistent with a competitive inhibition mechanism of CZX metabolism. The inhibitory effect of DAS maybe additive with daily consumption of Allium vegetables in particular. This may explain the lower 6-OHCZX/CZX metabolic ratios measured in various European and Mexican cohorts and is consistent with the lower incidence of stomach, liver and colon cancers observed in southern Europeans.

The utility of PBPK in the safety assessment of chloroform and carbon tetrachloride.

Occupational exposure limits (OELs) for individual substances are established on the basis of the available toxicological information at the time of their promulgation, expert interpretation of these data in light of industrial use, and the framework in which they sit. In the United Kingdom, the establishment of specific OELs includes the application of uncertainty factors to a defined starting point, usually the NOAEL from a suitable animal study. The magnitude of the uncertainty factors is generally determined through expert judgment including a knowledge of workplace conditions and management of exposure. PBPK modeling may help in this process by informing on issues relating to extrapolation between and within species. This study was therefore designed to consider how PBPK modeling could contribute to the establishment of OELs. PBPK models were developed for chloroform (mouse and human) and carbon tetrachloride (rat and human). These substances were chosen for examination because of the extent of their toxicological databases and availability of existing PBPK models. The models were exercised to predict the rate (chloroform) or extent (carbon tetrachloride) of metabolism of these substances, in both rodents and humans. Monte Carlo analysis was used to investigate the influence of variability within the human and animal model populations. The ratio of the rates/extent of metabolism predicted for humans compared to animals was compared to the uncertainty factors involved in setting the OES. Predictions obtained from the PBPK models indicated that average rat and mouse metabolism of carbon tetrachloride and chloroform, respectively, are much greater than that of the average human. Application of Monte Carlo analysis indicated that even those people who have the fastest rates or most extensive amounts of metabolism in the population are unlikely to generate the levels of metabolite of these substances necessary to produce overt toxicity in rodents. This study highlights the value that the use of PBPK modeling may add to help inform and improve toxicological aspects of a regulatory process.

Estimation of the dermal absorption of m-xylene vapor in humans using breath sampling and physiologically based pharmacokinetic analysis.

A physiologically-based pharmacokinetic model, containing a skin compartment, was derived and used to simulate experimentally determined exposure to m-xylene, using human volunteers exposed under controlled conditions. Biological monitoring was conducted by sampling, in exhaled alveolar air and blood, m-xylene and urinary methyl hippuric acid concentrations. The dermal absorption of m-xylene vapor was successfully and conveniently studied using a breath sampling technique, and the contribution to m-xylene body burden from the dermal route of exposure was estimated to be 1.8%. The model was used to investigate the protection afforded by an air-fed, half-face mask. By iteratively changing the dermal exposure concentration, it was possible to predict the ambient concentration that was required to deliver the observed urinary excretion of methylhippuric acid, during and following inhalation exposure to 50 ppm m-xylene vapor. This latter extrapolation demonstrates how physiologically-based pharmacokinetic modeling can be applied in a practical and occupationally relevant way, and permitted a further step not possible with biological monitoring alone. The ability of the model to extrapolate an ambient exposure concentration was dependent upon human metabolism data, thereby demonstrating the mechanistic toxicological basis of model output. The methyl hydroxylation of m-xylene is catalyzed by the hepatic mixed function oxidase enzyme, cytochrome P450 2E1 and is active in the occupationally relevant, (<100 ppm) exposure range of m-xylene. The use of a scaled-up in vitro maximum rate of metabolism (Vmaxc) in the model also demonstrates the increasingly valuable potential utility of biokinetic data determined using alternative, non-animal methods in human chemical-risk assessment.

An AI approach to dynamic visual field testing.

Visual field test results are crucial to the accuracy and efficiency of diagnosing blinding diseases such as glaucoma. Herein, a method of integrating self-organizing neural networks and empirical heuristics is used to perform visual field tests via a dynamic test strategy, which can lead to a reduction in the number of trials in a perimetric test. Experiments performed using clinical test records show that we are able to reduce by 20% to 30% the number of trials per test without much adverse effect on the accuracy of the tests.

Physiologically based pharmacokinetic analysis of the concentration-dependent metabolism of halothane.

1. Previous studies with the halothane analogue and chlorofluorocarbon replacement 2,2-dichloro-1,1,1-trifluoroethane (HCFC-123) have shown that there are concentration-dependent, sex-specific differences in the rate of uptake during inhalation exposure in rat. Since it is well established that there are sex-specific differences in the control of enzyme activity in drug metabolism, male and female rats were exposed by inhalation to halothane concentrations ranging from 500 to 4000 ppm. 2. A physiologically based pharmacokinetic model describing the concentration-dependent reduction in uptake and metabolism of halothane in male and female rats was developed. The in vivo metabolic rate constants obtained were: for male rats, Km = 0.4 mg litre-1 (2.03 mumol litre-1) and Vmaxc = 9.2 mg kg1 h-1 (46.6 mumol kg1 h-1); for female rats, Km = 0.4 mg litre-1 (2.03 mumol litre-1) and Vmaxc = 10.2 mg kg-1 h-1 (51.7 mumol kg-1 h-1). 3. An equation describing the concentration-dependent decrease of hepatic metabolism of halothane successfully simulated the gas-uptake data. Simulation of cumulative urinary excretion of the major metabolite, trifluoroacetic acid, required introduction of a proportionality constant to limit the extent of reduction of halothane metabolism to 20% of the amount of enzyme activity. Good simulation of urinary excretion data was achieved, which was interpreted to indicate that, when only 20% of the enzyme is inactivated, the rate of enzyme resynthesis was adequate to replenish enzyme activity within 24 h. 4. A rapidly reversible, non-biological inactivation mechanism called "physical toxicity' is discussed as a possible explanation of concentration-dependent gas uptake.

Gas-uptake pharmacokinetics and metabolism of 2-chloro-1,1,1,2-tetrafluoroethane (HCFC-124) in the rat, mouse, and hamster.

Gas-uptake pharmacokinetics and metabolism of the chlorofluorocarbon replacement 2-chloro-1,1,1,2-tetrafluoroethane (HCFC-124) were investigated in rats, mice, and hamsters. Species differences in the rate of uptake of HCFC-124 and urinary excretion of trifluoroacetic acid were observed. In rats and mice, the uptake of HCFC-124 was described by both saturable and first-order components, whereas in the hamster only first-order uptake was observed. The in vivo metabolic rate constants obtained from computer simulation of the gas-uptake data were: for rats-KM = 1.2 mg liter-1 (8.79 mmol liter-1, Vmaxc = 0.35 +/- 0.01 mg kg-1 hr-1 (2.56 +/- 0.01 mmol kg-1 hr-1), and kfc = 1.25 +/- 0.01 hr-1 kg231; for mice-KM = 1.2 mg liter-1 (8.79 mmol liter-1), Vmaxc = 1.78 +/- 0.01 mg kg-1 hr-1 (13.0 +/- 0.007 mmol kg-1 hr-1), and kfc = 4.08 +/- 0.01 hr-1 kg-1; and for hamsters-kfc = 1.47 +/- 0.02 hr-1 kg-1. The production and excretion of trifluoroacetic acid, the major urinary metabolite of HCFC-124, were also simulated in rats and mice, but not in hamsters, by the physiologically based pharmacokinetic model when the in vivo metabolic rate constants obtained in the gas-uptake simulation studies were used. The blood:air partition coefficient of HCFC-124 in the hamster was lower than in the rat or mouse. A low blood:air partition coefficient may limit the pulmonary uptake of volatile chemicals.(ABSTRACT TRUNCATED AT 250 WORDS)

Gas-uptake pharmacokinetics of 2,2-dichloro-1,1,1-trifluoroethane (HCFC-123).

The in vivo metabolic rate constants for the metabolism of the chlorofluorocarbon replacement 2,2-dichloro-1,1,1-trifluoroethane (HCFC-123) were determined for both male and female rats with a physiologically based pharmacokinetic model. Uptake studies with 500-5,000 ppm HCFC-123 indicated that a single saturable component was involved in both sexes, and no significant differences were observed in in vivo metabolic rate constants between male and female rats. The in vivo metabolic rate constants obtained from computer simulation studies were: for male rats--KM = 1.2 mg liter-1 (7.85 mumol liter-1) and Vmaxc = 7.20 +/- 0.28 mg kg-1 hr-1 (47.1 +/- 1.83 mumol kg-1 hr-1); for female rats--KM = 1.2 mg liter-1 (7.85 mumol liter-1) and Vmaxc = 7.97 +/- 0.30 mg kg-1 hr-1 (52.1 +/- 1.96 mumol kg-1 hr-1). The physiologically based pharmacokinetic model failed to simulate the reduction in HCFC-123 uptake in female rats at 2,000-5,000 ppm. The production and excretion of trifluoroacetic acid, the major urinary metabolite of HCFC-123, was also predicted by the physiologically based pharmacokinetic model with in vivo metabolic rate constants obtained in the gas-uptake simulation studies. Diallyl sulfide, a selective, mechanism-based inhibitor of cytochrome P450 2E1, inhibited the metabolism of HCFC-123, as indicated by a decreased uptake of HCFC-123 and by a lowered urinary excretion of trifluoroacetic acid in diallyl sulfide-treated rats.

Moving through a mirage.

We consider image motion in the visual field of an observer moving through a rigid environment under mirage conditions, in which light rays follow curved trajectories as a result of a nonuniform atmospheric refractive index. We show that, under these conditions, the translation motion field does not conform to the simple pattern observed in the absence of refraction; instead it depends on the transfer characteristic of the mirage. An interesting finding is that if the mirage has a transfer characteristic that folds, as occurs for example in the desert mirage, the motion field will, in general, be infinite at the fold point.

Gas-uptake pharmacokinetics and biotransformation of 1,1-dichloro-1-fluoroethane (HCFC-141b).

A physiologically based pharmacokinetic model was used to determine the in vivo metabolic constants of the candidate chlorofluorocarbon replacement 1,1-dichloro-1-fluoroethane (HCFC-141b). Rats were exposed by inhalation to HCFC-141b concentrations ranging from 1,000 to 10,000 ppm. Uptake studies of HCFC-141b in the rat indicated the involvement of saturable and first-order components. The in vivo metabolic constants for HCFC-141b were: KM = 7.0 mg liter-1 (59.9 mumol liter-1), Vmax = 0.2 mg kg-1 hr-1 (1.71 mumol kg-1 hr-1), and k = 0.5 hr-1. In rats exposed to HCFC-141b, 2,2-dichloro-2-fluoroethanol was excreted in the urine as its glucuronide conjugate, and the rate of 2,2-dichloro-2-fluoroethanol excretion increased linearly with increasing HCFC-141b exposure concentrations. Diallyl sulfide, a selective, mechanism-based inhibitor of cytochrome P-450 2E1, inhibited the metabolism of HCFC-141b, as indicated by a decreased uptake of HCFC-141b and by a lowered urinary excretion of 2,2-dichloro-2-fluoroethanol in diallyl-sulfide-treated rats. In vitro biotransformation studies with microsomes from rats treated with pyridine, an inducer of cytochrome P-450 2E1, confirmed that cytochrome P-450 2E1 is involved in the metabolism of HCFC-141b. The in vitro metabolic rate constants for the biotransformation of HCFC-141b to 2,2-dichloro-2-fluoroethanol were: KM = 0.39 +/- 0.11 mM and Vmax = 2.08 +/- 0.23 nmol mg protein-1 hr-1.(ABSTRACT TRUNCATED AT 250 WORDS)

Adenosine 5'-(2-fluorodiphosphate) is a selective agonist at P2-purinoceptors mediating relaxation of smooth muscle.

The pharmacological effects of ATP and of an adenine nucleotide analogue, adenosine 5'-(2-fluorodiphosphate) (ADP-beta-F) on various guinea-pig isolated smooth muscle preparations were investigated. ATP relaxed the taenia coli and the aorta, and contracted the urinary bladder and vas deferens. ADP-beta-F mimicked the effects of ATP on the taenia coli and aorta, but did not contract the bladder or vas deferens. The lack of potency of ADP-beta-F on the bladder was not due to its rapid degradation by ectonucleotidases, as it was degraded more slowly than ATP by both the bladder and the taenia coli. These results are consistent with the suggestion that the P2-purinoceptors mediating contraction of smooth muscle (P2X) are different from those mediating inhibitory effects (P2Y), and suggest that ADP-beta-F is a specific agonist at the P2Y-purinoceptor.

Circadian variation in the uptake of tryptophan by cortical synaptosomes of the rat brain.

The kinetics of the high affinity uptake system for L-tryptophan (L-Try) have been measured over 24 hr in cortical synaptosome preparations of rat brain. Both the Km and Vmax of the uptake process showed a statistically significant 24 hr variation. The highest Km value, 6.71 X 10(-5) M, was measured at the beginning of the light phase and the lowest value, 4.23 X 10(-5) M, 6 hr into the dark phase. Vmax was highest at the end of the dark phase (10.43 nmol/mg/5 min) and lowest (4.80 nmol/mg/5 min) 3 hr into the dark phase. In contrast, there was no variation over 24 hr in the Vmax/Km ratio. These results suggest that the high affinity uptake process serves to ensure a constant rate of L-tryptophan entry into the neuron in the face of circadian or ultradian variations in extracellular concentration of tryptophan.

Pharmacological effects of isopolar phosphonate analogues of ATP on P2-purinoceptors in guinea-pig taenia coli and urinary bladder.

Isopolar methylene phosphonate analogues of adenosine triphosphate (ATP) were synthesized and tested on the guinea-pig isolated taenia coli (where ATP causes relaxation) and urinary bladder (where ATP causes contraction), to see if restoration of the electronegativity of the methylene linkage would enhance pharmacological potency. The compounds used were the dichloromethylene and difluoromethylene analogues of adenosine 5'-(beta,gamma-methylene)triphosphonate (AMP-PCP), L-adenosine 5'-(beta,gamma-methylene)triphosphonate (L-AMP-PCP) and 2-methylthioadenosine 5'-(beta,gamma-methylene)-triphosphonate (2-methylthio-AMP-PCP). The order of potency of the analogues depended on the tissue, and was independent of the nature of the purine or ribose moieties. None of the analogues was degraded by ectonucleotidases on either tissue. In the taenia coli the order of potency for relaxation was difluoromethylene greater than or equal to dichloromethylene greater than methylene, and this reflected the order of electronegativity of the analogues. The isopolar analogues of L-AMP-PCP were inactive in the taenia coli. In the bladder the order of potency for contraction was difluoromethylene greater than or equal to methylene greater than dichloromethylene, suggesting that electronegativity is of lesser importance here. The isopolar analogues of L-AMP-PCP were active in this tissue. The differences between the two tissues in the order of potency for these non-degradable analogues supports suggestions that P2-purinoceptors in the taenia coli (P2Y) are different from those in the bladder (P2X). The isopolar analogues of L-AMP-PCP, like L-AMP-PCP itself, were selective agonists at the P2X-purinoceptor.

The nearest neighbor and the bayes error rates.

The (k, l) nearest neighbor method of pattern classification is compared to the Bayes method. If the two acceptance rates are equal then the asymptotic error rates satisfy the inequalities Ek,l + 1 ¿ E*(¿) ¿ Ek,l dE*(¿), where d is a function of k, l, and the number of pattern classes, and ¿ is the reject threshold for the Bayes method. An explicit expression for d is given which is optimal in the sense that for some probability distributions Ek,l and dE* (¿) are equal.